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</div> <div class="control"> <button class="button is-small is-link">Go</button> </div> </div> </form> </div> </div> <ol class="breathe-horizontal" start="1"> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2411.14417">arXiv:2411.14417</a> <span> [<a href="https://arxiv.org/pdf/2411.14417">pdf</a>, <a href="https://arxiv.org/ps/2411.14417">ps</a>, <a href="https://arxiv.org/format/2411.14417">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> </div> <p class="title is-5 mathjax"> Construction of Lie algebra weight system kernel via Vogel algebra </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Khudoteplov%2C+D">Dmitry Khudoteplov</a>, <a href="/search/hep-th?searchtype=author&query=Lanina%2C+E">Elena Lanina</a>, <a href="/search/hep-th?searchtype=author&query=Sleptsov%2C+A">Alexey Sleptsov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2411.14417v1-abstract-short" style="display: inline;"> We develop a method of constructing a kernel of Lie algebra weight system. A main tool we use in the analysis is Vogel's $螞$ algebra and the surrounding framework. As an example of a developed technique we explicitly provide all Jacobi diagrams lying in the kernel of $\mathfrak{sl}_N$ weight system at low orders. We also discuss consequences of the presence of the kernel in Lie algebra weight syst… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.14417v1-abstract-full').style.display = 'inline'; document.getElementById('2411.14417v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2411.14417v1-abstract-full" style="display: none;"> We develop a method of constructing a kernel of Lie algebra weight system. A main tool we use in the analysis is Vogel's $螞$ algebra and the surrounding framework. As an example of a developed technique we explicitly provide all Jacobi diagrams lying in the kernel of $\mathfrak{sl}_N$ weight system at low orders. We also discuss consequences of the presence of the kernel in Lie algebra weight systems for detection of correlators in the 3D Chern-Simons topological field theory and for distinguishing of knots by the corresponding quantum knot invariants. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2411.14417v1-abstract-full').style.display = 'none'; document.getElementById('2411.14417v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">18 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2410.13676">arXiv:2410.13676</a> <span> [<a href="https://arxiv.org/pdf/2410.13676">pdf</a>, <a href="https://arxiv.org/ps/2410.13676">ps</a>, <a href="https://arxiv.org/format/2410.13676">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> </div> </div> <p class="title is-5 mathjax"> Direct proof of one-hook scaling property for Alexander polynomial from Reshetikhin-Turaev formalism </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">Andrey Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Popolitov%2C+A">Aleksandr Popolitov</a>, <a href="/search/hep-th?searchtype=author&query=Sleptsov%2C+A">Alexei Sleptsov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2410.13676v2-abstract-short" style="display: inline;"> We prove that normalized colored Alexander polynomial (the $A \rightarrow 1$ limit of colored HOMFLY-PT polynomial) evaluated for one-hook (L-shape) representation R possesses scaling property: it is equal to the fundamental Alexander polynomial with the substitution $q \rightarrow q^{|R|}$. The proof is simple and direct use of Reshetikhin-Turaev formalism to get all required R-matrices. </span> <span class="abstract-full has-text-grey-dark mathjax" id="2410.13676v2-abstract-full" style="display: none;"> We prove that normalized colored Alexander polynomial (the $A \rightarrow 1$ limit of colored HOMFLY-PT polynomial) evaluated for one-hook (L-shape) representation R possesses scaling property: it is equal to the fundamental Alexander polynomial with the substitution $q \rightarrow q^{|R|}$. The proof is simple and direct use of Reshetikhin-Turaev formalism to get all required R-matrices. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2410.13676v2-abstract-full').style.display = 'none'; document.getElementById('2410.13676v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 November, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 17 October, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">20 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2402.02553">arXiv:2402.02553</a> <span> [<a href="https://arxiv.org/pdf/2402.02553">pdf</a>, <a href="https://arxiv.org/format/2402.02553">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Group Theory">math.GR</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> </div> <p class="title is-5 mathjax"> Closed 4-braids and the Jones unknot conjecture </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Korzun%2C+D">Dmitriy Korzun</a>, <a href="/search/hep-th?searchtype=author&query=Lanina%2C+E">Elena Lanina</a>, <a href="/search/hep-th?searchtype=author&query=Sleptsov%2C+A">Alexey Sleptsov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2402.02553v2-abstract-short" style="display: inline;"> The Jones problem is a question whether there is a non-trivial knot with the trivial Jones polynomial in one variable $q$. The answer to this fundamental question is still unknown despite numerous attempts to explore it. In braid presentation the case of 4-strand braids is already open. S. Bigelow showed in 2000 that if the Burau representation for four-strand braids is unfaithful, then there is a… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.02553v2-abstract-full').style.display = 'inline'; document.getElementById('2402.02553v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2402.02553v2-abstract-full" style="display: none;"> The Jones problem is a question whether there is a non-trivial knot with the trivial Jones polynomial in one variable $q$. The answer to this fundamental question is still unknown despite numerous attempts to explore it. In braid presentation the case of 4-strand braids is already open. S. Bigelow showed in 2000 that if the Burau representation for four-strand braids is unfaithful, then there is an infinite number of non-trivial knots with the trivial two-variable HOMFLY-PT polynomial and hence, with the trivial Jones polynomial, since it is obtained from the HOMFLY-PT polynomial by the specialisation of one of the variables $A=q^2$. In this paper, we study four-strand braids and ask whether there are non-trivial knots with the trivial Jones polynomial but a non-trivial HOMFLY-PT polynomial. We have discovered that there is a whole 1-parameter family, parameterised by the writhe number, of 2-variable polynomials that can be HOMFLY-PT polynomials of some knots. We explore various properties of the obtained hypothetical HOMFLY-PT polynomials and suggest several checks to test these formulas. A generalisation is also proposed for the case of a large number of strands. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.02553v2-abstract-full').style.display = 'none'; document.getElementById('2402.02553v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 18 April, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 4 February, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">23 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2210.07874">arXiv:2210.07874</a> <span> [<a href="https://arxiv.org/pdf/2210.07874">pdf</a>, <a href="https://arxiv.org/ps/2210.07874">ps</a>, <a href="https://arxiv.org/format/2210.07874">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.physletb.2023.138138">10.1016/j.physletb.2023.138138 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Tug-the-hook symmetry for quantum 6j-symbols </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Lanina%2C+E">E. Lanina</a>, <a href="/search/hep-th?searchtype=author&query=Sleptsov%2C+A">A. Sleptsov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2210.07874v2-abstract-short" style="display: inline;"> We introduce a novel symmetry for quantum 6j-symbols, which we call the tug-the-hook symmetry. Unlike other known symmetries, it is applicable for any representations, including ones with multiplicities. We provide several evidences in favour of the tug-the-hook symmetry. First, this symmetry follows from the eigenvalue conjecture. Second, it is shown by several new examples of explicit coincidenc… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2210.07874v2-abstract-full').style.display = 'inline'; document.getElementById('2210.07874v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2210.07874v2-abstract-full" style="display: none;"> We introduce a novel symmetry for quantum 6j-symbols, which we call the tug-the-hook symmetry. Unlike other known symmetries, it is applicable for any representations, including ones with multiplicities. We provide several evidences in favour of the tug-the-hook symmetry. First, this symmetry follows from the eigenvalue conjecture. Second, it is shown by several new examples of explicit coincidence of 6j-symbols with multiplicities. Third, the tug-the-hook symmetry for Wilson loops for knots in the 3d Chern-Simons theory implies the tug-the-hook symmetry for quantum 6j-symbols. An important implication of the analysis is the generalization of the tug-the-hook symmetry for the Chern-Simons Wilson loops to the case of links. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2210.07874v2-abstract-full').style.display = 'none'; document.getElementById('2210.07874v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 September, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 14 October, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">12 pages, published version</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Physics Letters B 845 (2023) 138138 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2111.11751">arXiv:2111.11751</a> <span> [<a href="https://arxiv.org/pdf/2111.11751">pdf</a>, <a href="https://arxiv.org/format/2111.11751">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.nuclphysb.2021.115644">10.1016/j.nuclphysb.2021.115644 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Implications for colored HOMFLY polynomials from explicit formulas for group-theoretical structure </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Lanina%2C+E">E. Lanina</a>, <a href="/search/hep-th?searchtype=author&query=Sleptsov%2C+A">A. Sleptsov</a>, <a href="/search/hep-th?searchtype=author&query=Tselousov%2C+N">N. Tselousov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2111.11751v1-abstract-short" style="display: inline;"> We have recently proposed arXiv:2105.11565 a powerful method for computing group factors of the perturbative series expansion of the Wilson loop in the Chern-Simons theory with $SU(N)$ gauge group. In this paper, we apply the developed method to obtain and study various properties, including nonperturbative ones, of such vacuum expectation values. First, we discuss the computation of Vassiliev i… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2111.11751v1-abstract-full').style.display = 'inline'; document.getElementById('2111.11751v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2111.11751v1-abstract-full" style="display: none;"> We have recently proposed arXiv:2105.11565 a powerful method for computing group factors of the perturbative series expansion of the Wilson loop in the Chern-Simons theory with $SU(N)$ gauge group. In this paper, we apply the developed method to obtain and study various properties, including nonperturbative ones, of such vacuum expectation values. First, we discuss the computation of Vassiliev invariants. Second, we discuss the Vogel theorem of not distinguishing chord diagrams by weight systems coming from semisimple Lie (super)algebras. Third, we provide a method for constructing linear recursive relations for the colored Jones polynomials considering a special case of torus knots $T[2,2k+1]$. Fourth, we give a generalization of the one-hook scaling property for the colored Alexander polynomials. And finally, for the group factors we provide a combinatorial description, which has a clear dependence on the rank $N$ and the representation $R$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2111.11751v1-abstract-full').style.display = 'none'; document.getElementById('2111.11751v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 23 November, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">24 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Nuclear Physics B 974 (2022) 115644 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2105.11565">arXiv:2105.11565</a> <span> [<a href="https://arxiv.org/pdf/2105.11565">pdf</a>, <a href="https://arxiv.org/ps/2105.11565">ps</a>, <a href="https://arxiv.org/format/2105.11565">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.physletb.2021.136727">10.1016/j.physletb.2021.136727 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Chern-Simons perturbative series revisited </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Lanina%2C+E">E. Lanina</a>, <a href="/search/hep-th?searchtype=author&query=Sleptsov%2C+A">A. Sleptsov</a>, <a href="/search/hep-th?searchtype=author&query=Tselousov%2C+N">N. Tselousov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2105.11565v2-abstract-short" style="display: inline;"> A group-theoretical structure in a perturbative expansion of the Wilson loops in the 3d Chern-Simons theory with $SU(N)$ gauge group is studied in symmetric approach. A special basis in the center of the universal enveloping algebra $ZU(\mathfrak{sl}_N)$ is introduced. This basis allows one to present group factors in an arbitrary irreducible finite-dimensional representation. Developed methods ha… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2105.11565v2-abstract-full').style.display = 'inline'; document.getElementById('2105.11565v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2105.11565v2-abstract-full" style="display: none;"> A group-theoretical structure in a perturbative expansion of the Wilson loops in the 3d Chern-Simons theory with $SU(N)$ gauge group is studied in symmetric approach. A special basis in the center of the universal enveloping algebra $ZU(\mathfrak{sl}_N)$ is introduced. This basis allows one to present group factors in an arbitrary irreducible finite-dimensional representation. Developed methods have wide applications, the most straightforward and evident ones are mentioned. Namely, Vassiliev invariants of higher orders are computed, a conjecture about existence of new symmetries of the colored HOMFLY polynomials is stated, and the recently discovered tug-the-hook symmetry of the colored HOMFLY polynomial is proved. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2105.11565v2-abstract-full').style.display = 'none'; document.getElementById('2105.11565v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 26 October, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 24 May, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">10 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Physics Letters B 823 (2021) 136727 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2101.10188">arXiv:2101.10188</a> <span> [<a href="https://arxiv.org/pdf/2101.10188">pdf</a>, <a href="https://arxiv.org/ps/2101.10188">ps</a>, <a href="https://arxiv.org/format/2101.10188">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1140/epjc/s10052-021-09899-8">10.1140/epjc/s10052-021-09899-8 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Combinatorics of KP hierarchy structural constants </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Andreev%2C+A">A. Andreev</a>, <a href="/search/hep-th?searchtype=author&query=Popolitov%2C+A">A. Popolitov</a>, <a href="/search/hep-th?searchtype=author&query=Sleptsov%2C+A">A. Sleptsov</a>, <a href="/search/hep-th?searchtype=author&query=Zhabin%2C+A">A. Zhabin</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2101.10188v1-abstract-short" style="display: inline;"> Following Natanzon-Zabrodin, we explore the Kadomtsev-Petviashvili hierarchy as an infinite system of mutually consistent relations on the second derivatives of the free energy with some universal coefficients. From this point of view, various combinatorial properties of these coefficients naturally highlight certain non-trivial properties of the KP hierarchy. Furthermore, this approach allows us… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2101.10188v1-abstract-full').style.display = 'inline'; document.getElementById('2101.10188v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2101.10188v1-abstract-full" style="display: none;"> Following Natanzon-Zabrodin, we explore the Kadomtsev-Petviashvili hierarchy as an infinite system of mutually consistent relations on the second derivatives of the free energy with some universal coefficients. From this point of view, various combinatorial properties of these coefficients naturally highlight certain non-trivial properties of the KP hierarchy. Furthermore, this approach allows us to suggest several interesting directions of the KP deformation via a deformation of these coefficients. We also construct an eigenvalue matrix model, whose correlators fully describe the universal KP coefficients, which allows us to further study their properties and generalizations. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2101.10188v1-abstract-full').style.display = 'none'; document.getElementById('2101.10188v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 25 January, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">17 pages, no figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> MIPT/TH-19/20; ITEP/TH-34/20; IITP/TH-21/20 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Eur. Phys. J. C 81, 1136 (2021) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2010.11021">arXiv:2010.11021</a> <span> [<a href="https://arxiv.org/pdf/2010.11021">pdf</a>, <a href="https://arxiv.org/ps/2010.11021">ps</a>, <a href="https://arxiv.org/format/2010.11021">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.4310/ATMP.2022.v26.n4.a1">10.4310/ATMP.2022.v26.n4.a1 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Topological Recursion for the extended Ooguri-Vafa partition function of colored HOMFLY-PT polynomials of torus knots </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Dunin-Barkowski%2C+P">Petr Dunin-Barkowski</a>, <a href="/search/hep-th?searchtype=author&query=Kazarian%2C+M">Maxim Kazarian</a>, <a href="/search/hep-th?searchtype=author&query=Popolitov%2C+A">Aleksandr Popolitov</a>, <a href="/search/hep-th?searchtype=author&query=Shadrin%2C+S">Sergey Shadrin</a>, <a href="/search/hep-th?searchtype=author&query=Sleptsov%2C+A">Alexey Sleptsov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2010.11021v1-abstract-short" style="display: inline;"> We prove that topological recursion applied to the spectral curve of colored HOMFLY-PT polynomials of torus knots reproduces the n-point functions of a particular partition function called the extended Ooguri-Vafa partition function. This generalizes and refines the results of Brini-Eynard-Marino and Borot-Eynard-Orantin. We also discuss how the statement of spectral curve topological recursion… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2010.11021v1-abstract-full').style.display = 'inline'; document.getElementById('2010.11021v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2010.11021v1-abstract-full" style="display: none;"> We prove that topological recursion applied to the spectral curve of colored HOMFLY-PT polynomials of torus knots reproduces the n-point functions of a particular partition function called the extended Ooguri-Vafa partition function. This generalizes and refines the results of Brini-Eynard-Marino and Borot-Eynard-Orantin. We also discuss how the statement of spectral curve topological recursion in this case fits into the program of Alexandrov-Chapuy-Eynard-Harnad of establishing the topological recursion for general weighted double Hurwitz numbers partition functions (a.k.a. KP tau-functions of hypergeometric type). <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2010.11021v1-abstract-full').style.display = 'none'; document.getElementById('2010.11021v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 October, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Adv. Theor. Math. Phys. 26 (2023), no. 4, 793-833 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2008.06416">arXiv:2008.06416</a> <span> [<a href="https://arxiv.org/pdf/2008.06416">pdf</a>, <a href="https://arxiv.org/ps/2008.06416">ps</a>, <a href="https://arxiv.org/format/2008.06416">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/JHEP12(2020)038">10.1007/JHEP12(2020)038 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Genus expansion of matrix models and $\hbar$ expansion of KP hierarchy </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Andreev%2C+A">A. Andreev</a>, <a href="/search/hep-th?searchtype=author&query=Popolitov%2C+A">A. Popolitov</a>, <a href="/search/hep-th?searchtype=author&query=Sleptsov%2C+A">A. Sleptsov</a>, <a href="/search/hep-th?searchtype=author&query=Zhabin%2C+A">A. Zhabin</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2008.06416v2-abstract-short" style="display: inline;"> We study $\hbar$ expansion of the KP hierarchy following Takasaki-Takebe arXiv:hep-th/9405096 considering several examples of matrix model $蟿$-functions with natural genus expansion. Among the examples there are solutions of KP equations of special interest, such as generating function for simple Hurwitz numbers, Hermitian matrix model, Kontsevich model and Brezin-Gross-Witten model. We show that… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2008.06416v2-abstract-full').style.display = 'inline'; document.getElementById('2008.06416v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2008.06416v2-abstract-full" style="display: none;"> We study $\hbar$ expansion of the KP hierarchy following Takasaki-Takebe arXiv:hep-th/9405096 considering several examples of matrix model $蟿$-functions with natural genus expansion. Among the examples there are solutions of KP equations of special interest, such as generating function for simple Hurwitz numbers, Hermitian matrix model, Kontsevich model and Brezin-Gross-Witten model. We show that all these models with parameter $\hbar$ are $蟿$-functions of the $\hbar$-KP hierarchy and the expansion in $\hbar$ for the $\hbar$-KP coincides with the genus expansion for these models. Furthermore, we show a connection of recent papers considering the $\hbar$-formulation of the KP hierarchy arXiv:1509.04472, arXiv:1512.07172 with original Takasaki-Takebe approach. We find that in this approach the recovery of enumerative geometric meaning of $蟿$-functions is straightforward and algorithmic. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2008.06416v2-abstract-full').style.display = 'none'; document.getElementById('2008.06416v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 September, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 14 August, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">26 pages, no figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> ITEP/TH-15/20; IITP/TH-11/20; MIPT/TH-10/20 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. High Energ. Phys. 2020, 38 (2020) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2007.12532">arXiv:2007.12532</a> <span> [<a href="https://arxiv.org/pdf/2007.12532">pdf</a>, <a href="https://arxiv.org/format/2007.12532">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.geomphys.2020.103928">10.1016/j.geomphys.2020.103928 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Distinguishing Mutant Knots </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Bishler%2C+L">L. Bishler</a>, <a href="/search/hep-th?searchtype=author&query=Dhara%2C+S">Saswati Dhara</a>, <a href="/search/hep-th?searchtype=author&query=Grigoryev%2C+T">T. Grigoryev</a>, <a href="/search/hep-th?searchtype=author&query=Mironov%2C+A">A. Mironov</a>, <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">A. Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">An. Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Ramadevi%2C+P">P. Ramadevi</a>, <a href="/search/hep-th?searchtype=author&query=Singh%2C+V+K">Vivek Kumar Singh</a>, <a href="/search/hep-th?searchtype=author&query=Sleptsov%2C+A">A. Sleptsov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2007.12532v1-abstract-short" style="display: inline;"> Knot theory is actively studied both by physicists and mathematicians as it provides a connecting centerpiece for many physical and mathematical theories. One of the challenging problems in knot theory is distinguishing mutant knots. Mutant knots are not distinguished by colored HOMFLY-PT polynomials for knots colored by either symmetric and or antisymmetric representations of $SU(N)$. Some of the… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2007.12532v1-abstract-full').style.display = 'inline'; document.getElementById('2007.12532v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2007.12532v1-abstract-full" style="display: none;"> Knot theory is actively studied both by physicists and mathematicians as it provides a connecting centerpiece for many physical and mathematical theories. One of the challenging problems in knot theory is distinguishing mutant knots. Mutant knots are not distinguished by colored HOMFLY-PT polynomials for knots colored by either symmetric and or antisymmetric representations of $SU(N)$. Some of the mutant knots can be distinguished by the simplest non-symmetric representation $[2,1]$. However there is a class of mutant knots which require more complex representations like $[4,2]$. In this paper we calculate polynomials and differences for the mutant knot polynomials in representations $[3,1]$ and $[4,2]$ and study their properties. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2007.12532v1-abstract-full').style.display = 'none'; document.getElementById('2007.12532v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 24 July, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">22 pages + 3 Appendices</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> FIAN/TD-12/20; IITP/TH-09/20; ITEP/TH-12/20; MIPT/TH-09/20 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Journal of Geometry and Physics, 159 (2021) 103928 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2005.01188">arXiv:2005.01188</a> <span> [<a href="https://arxiv.org/pdf/2005.01188">pdf</a>, <a href="https://arxiv.org/ps/2005.01188">ps</a>, <a href="https://arxiv.org/format/2005.01188">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s00220-021-04073-3">10.1007/s00220-021-04073-3 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> A novel symmetry of colored HOMFLY polynomials coming from $\mathfrak{sl}(N|M)$ superalgebras </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Mishnyakov%2C+V">V. Mishnyakov</a>, <a href="/search/hep-th?searchtype=author&query=Sleptsov%2C+A">A. Sleptsov</a>, <a href="/search/hep-th?searchtype=author&query=Tselousov%2C+N">N. Tselousov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2005.01188v1-abstract-short" style="display: inline;"> We present a novel symmetry of the colored HOMFLY polynomial. It relates pairs of polynomials colored by different representations at specific values of $N$ and generalizes the previously known "tug-the-hook" symmetry of the colored Alexander polynomial. As we show, the symmetry has a superalgebra origin, which we discuss qualitatively. Our main focus are the constraints that such a property impos… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2005.01188v1-abstract-full').style.display = 'inline'; document.getElementById('2005.01188v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2005.01188v1-abstract-full" style="display: none;"> We present a novel symmetry of the colored HOMFLY polynomial. It relates pairs of polynomials colored by different representations at specific values of $N$ and generalizes the previously known "tug-the-hook" symmetry of the colored Alexander polynomial. As we show, the symmetry has a superalgebra origin, which we discuss qualitatively. Our main focus are the constraints that such a property imposes on the general group-theoretical structure, namely the $\mathfrak{sl}(N)$ weight system, arising in the perturbative expansion of the invariant. Finally, we demonstrate its tight relation to the eigenvalue conjecture. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2005.01188v1-abstract-full').style.display = 'none'; document.getElementById('2005.01188v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 3 May, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2020. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2004.06598">arXiv:2004.06598</a> <span> [<a href="https://arxiv.org/pdf/2004.06598">pdf</a>, <a href="https://arxiv.org/ps/2004.06598">ps</a>, <a href="https://arxiv.org/format/2004.06598">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1134/S0021364020090015">10.1134/S0021364020090015 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Difference of mutant knot invariants and their differential expansion </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Bishler%2C+L">L. Bishler</a>, <a href="/search/hep-th?searchtype=author&query=Dhara%2C+S">Saswati Dhara</a>, <a href="/search/hep-th?searchtype=author&query=Grigoryev%2C+T">T. Grigoryev</a>, <a href="/search/hep-th?searchtype=author&query=Mironov%2C+A">A. Mironov</a>, <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">A. Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">An. Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Ramadevi%2C+P">P. Ramadevi</a>, <a href="/search/hep-th?searchtype=author&query=Singh%2C+V+K">Vivek Kumar Singh</a>, <a href="/search/hep-th?searchtype=author&query=Sleptsov%2C+A">A. Sleptsov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2004.06598v2-abstract-short" style="display: inline;"> We evaluate the differences of HOMFLY-PT invariants for pairs of mutant knots colored with representations of $SL(N)$, which are large enough to distinguish between them. These mutant pairs include the pretzel mutants, which require at least the representation, labeled by the Young diagram $[4,2]$. We discuss the differential expansion for the differences, it is non-trivial in the case of mutants,… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2004.06598v2-abstract-full').style.display = 'inline'; document.getElementById('2004.06598v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2004.06598v2-abstract-full" style="display: none;"> We evaluate the differences of HOMFLY-PT invariants for pairs of mutant knots colored with representations of $SL(N)$, which are large enough to distinguish between them. These mutant pairs include the pretzel mutants, which require at least the representation, labeled by the Young diagram $[4,2]$. We discuss the differential expansion for the differences, it is non-trivial in the case of mutants, which have the non-zero defect. The most effective technical tool, in this case, turns out to be the standard Reshetikhin-Turaev approach. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2004.06598v2-abstract-full').style.display = 'none'; document.getElementById('2004.06598v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 18 April, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 14 April, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2020. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">6 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> FIAN/TD-10/20; IITP/TH-06/20; ITEP/TH-07/20; MIPT/TH-06/20 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Pis'ma v ZhETF, 111, N9 (2020) 591; JETP Letters, 111(9) (2020) 494-499 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2001.10596">arXiv:2001.10596</a> <span> [<a href="https://arxiv.org/pdf/2001.10596">pdf</a>, <a href="https://arxiv.org/ps/2001.10596">ps</a>, <a href="https://arxiv.org/format/2001.10596">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s00023-020-00980-8">10.1007/s00023-020-00980-8 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> A new symmetry of the colored Alexander polynomial </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Mishnyakov%2C+V">V. Mishnyakov</a>, <a href="/search/hep-th?searchtype=author&query=Sleptsov%2C+A">A. Sleptsov</a>, <a href="/search/hep-th?searchtype=author&query=Tselousov%2C+N">N. Tselousov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2001.10596v4-abstract-short" style="display: inline;"> We present a new conjectural symmetry of the colored Alexander polynomial, that is the specialization of the quantum $\mathfrak{sl}_N$ invariant widely known as the colored HOMFLY-PT polynomial. We provide arguments in support of the existence of the symmetry by studying the loop expansion and the character expansion of the colored HOMFLY-PT polynomial. We study the constraints this symmetry impos… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2001.10596v4-abstract-full').style.display = 'inline'; document.getElementById('2001.10596v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2001.10596v4-abstract-full" style="display: none;"> We present a new conjectural symmetry of the colored Alexander polynomial, that is the specialization of the quantum $\mathfrak{sl}_N$ invariant widely known as the colored HOMFLY-PT polynomial. We provide arguments in support of the existence of the symmetry by studying the loop expansion and the character expansion of the colored HOMFLY-PT polynomial. We study the constraints this symmetry imposes on the group theoretic structure of the loop expansion and provide solutions to those constraints. The symmetry is a powerful tool for research on polynomial knot invariants and in the end we suggest several possible applications of the symmetry. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2001.10596v4-abstract-full').style.display = 'none'; document.getElementById('2001.10596v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 16 July, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 28 January, 2020; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2020. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1912.13325">arXiv:1912.13325</a> <span> [<a href="https://arxiv.org/pdf/1912.13325">pdf</a>, <a href="https://arxiv.org/ps/1912.13325">ps</a>, <a href="https://arxiv.org/format/1912.13325">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.nuclphysb.2020.115164">10.1016/j.nuclphysb.2020.115164 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Multiplicity-free $U_q(sl_N)$ 6-j symbols: relations, asymptotics, symmetries </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Alekseev%2C+V">Victor Alekseev</a>, <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">Andrey Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Sleptsov%2C+A">Alexey Sleptsov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1912.13325v4-abstract-short" style="display: inline;"> A closed form expression for multiplicity-free quantum 6-j symbols (MFS) was proposed in arXiv:1302.5143 for symmetric representations of $U_q(sl_N)$, which are the simplest class of multiplicity-free representations. In this paper we rewrite this expression in terms of q-hypergeometric series ${}_4桅_3$. We claim that it is possible to express any MFS through the 6-j symbol for $U_q(sl_2)$ with a… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1912.13325v4-abstract-full').style.display = 'inline'; document.getElementById('1912.13325v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1912.13325v4-abstract-full" style="display: none;"> A closed form expression for multiplicity-free quantum 6-j symbols (MFS) was proposed in arXiv:1302.5143 for symmetric representations of $U_q(sl_N)$, which are the simplest class of multiplicity-free representations. In this paper we rewrite this expression in terms of q-hypergeometric series ${}_4桅_3$. We claim that it is possible to express any MFS through the 6-j symbol for $U_q(sl_2)$ with a certain factor. It gives us a universal tool for the extension of various properties of the quantum 6-j symbols for $U_q(sl_2)$ to the MFS. We demonstrate this idea by deriving the asymptotics of the MFS in terms of associated tetrahedron for classical algebra $U(sl_N)$. Next we study MFS symmetries using known hypergeometric identities such as argument permutations and Sears' transformation. We describe symmetry groups of MFS. As a result we get new symmetries, which are a generalization of the tetrahedral symmetries and the Regge symmetries for N = 2. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1912.13325v4-abstract-full').style.display = 'none'; document.getElementById('1912.13325v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 19 August, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 31 December, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2019. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">25 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1909.07601">arXiv:1909.07601</a> <span> [<a href="https://arxiv.org/pdf/1909.07601">pdf</a>, <a href="https://arxiv.org/ps/1909.07601">ps</a>, <a href="https://arxiv.org/format/1909.07601">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s11005-021-01386-1">10.1007/s11005-021-01386-1 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Interplay between symmetries of quantum 6-j symbols and the eigenvalue hypothesis </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Alekseev%2C+V">Victor Alekseev</a>, <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">Andrey Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Sleptsov%2C+A">Alexey Sleptsov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1909.07601v7-abstract-short" style="display: inline;"> The eigenvalue hypothesis claims that any quantum Racah matrix for finite-dimensional representations of $U_q(sl_N)$ is uniquely determined by eigenvalues of the corresponding quantum $\cal{R}$-matrices. If this hypothesis turns out to be true, then it will significantly simplify the computation of Racah matrices. Also due to this hypothesis various interesting properties of colored HOMFLY-PT poly… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1909.07601v7-abstract-full').style.display = 'inline'; document.getElementById('1909.07601v7-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1909.07601v7-abstract-full" style="display: none;"> The eigenvalue hypothesis claims that any quantum Racah matrix for finite-dimensional representations of $U_q(sl_N)$ is uniquely determined by eigenvalues of the corresponding quantum $\cal{R}$-matrices. If this hypothesis turns out to be true, then it will significantly simplify the computation of Racah matrices. Also due to this hypothesis various interesting properties of colored HOMFLY-PT polynomials will be proved. In addition, it allows one to discover new symmetries of the quantum 6-j symbols, about which almost nothing is known for $N>2$, with the exception of the tetrahedral symmetries, complex conjugation and transformation $q \longleftrightarrow q^{-1}$. In this paper we prove the eigenvalue hypothesis in $U_q(sl_2)$ case and show that it is equivalent to 6-j symbol symmetries (the Regge symmetry and two argument permutations). Then we apply the eigenvalue hypothesis to inclusive Racah matrices with 3 symmetric incoming representations of $U_q(sl_N)$ and an arbitrary outcoming one. It gives us 8 new additional symmetries that are not tetrahedral ones. Finally, we apply the eigenvalue hypothesis to exclusive Racah matrices with symmetric representations and obtain 4 tetrahedral symmetries. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1909.07601v7-abstract-full').style.display = 'none'; document.getElementById('1909.07601v7-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 17 March, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 17 September, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2019. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">22 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Lett Math Phys 111, 50 (2021) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1906.05813">arXiv:1906.05813</a> <span> [<a href="https://arxiv.org/pdf/1906.05813">pdf</a>, <a href="https://arxiv.org/ps/1906.05813">ps</a>, <a href="https://arxiv.org/format/1906.05813">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Exactly Solvable and Integrable Systems">nlin.SI</span> </div> </div> <p class="title is-5 mathjax"> Perturbative analysis of the colored Alexander polynomial and KP soliton $蟿$-functions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Mishnyakov%2C+V">V. Mishnyakov</a>, <a href="/search/hep-th?searchtype=author&query=Sleptsov%2C+A">A. Sleptsov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1906.05813v2-abstract-short" style="display: inline;"> In this paper we study the group theoretic structures of colored HOMFLY polynomials in a specific limit. The group structures arise in the perturbative expansion of $SU(N)$ Chern-Simons Wilson loops, while the limit is $N \rightarrow 0$. The result of the paper is twofold. First, we explain the emergence of Kadomsev-Petviashvily (KP) $蟿$-functions. This result is an extension of what we did in arX… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1906.05813v2-abstract-full').style.display = 'inline'; document.getElementById('1906.05813v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1906.05813v2-abstract-full" style="display: none;"> In this paper we study the group theoretic structures of colored HOMFLY polynomials in a specific limit. The group structures arise in the perturbative expansion of $SU(N)$ Chern-Simons Wilson loops, while the limit is $N \rightarrow 0$. The result of the paper is twofold. First, we explain the emergence of Kadomsev-Petviashvily (KP) $蟿$-functions. This result is an extension of what we did in arXiv:1805.02761, where a symbolic correspondence between KP equations and group factors was established. In this paper we prove that integrability of the colored Alexander polynomial is due to it's relation to soliton $蟿$-functions. Mainly, the colored Alexander polynomial is embedded in the action of the KP generating function on the soliton $蟿$-function. Secondly, we use this correspondence to provide a rather simple combinatoric description of the group factors in term of Young diagrams, which is otherwise described in terms of chord diagrams, where no simple description is known. This is a first step providing an explicit description of the group theoretic data of Wilson loops, which would effectively reduce them to a purely topological quantity, mainly to a collection of Vassiliev invariants. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1906.05813v2-abstract-full').style.display = 'none'; document.getElementById('1906.05813v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 4 November, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 13 June, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2019. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">23 pages</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1905.01876">arXiv:1905.01876</a> <span> [<a href="https://arxiv.org/pdf/1905.01876">pdf</a>, <a href="https://arxiv.org/format/1905.01876">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1134/S0021364018220058">10.1134/S0021364018220058 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> New symmetries for the $U_q(sl_N)$ 6-j symbols from the Eigenvalue conjecture </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">Andrey Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Sleptsov%2C+A">Alexey Sleptsov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1905.01876v1-abstract-short" style="display: inline;"> In the present paper we discuss the eigenvalue conjecture, suggested in 2012, in the particular case of $U_q(sl_2)$. The eigenvalue conjecture provides a certain symmetry for Racah coefficients and we prove that \textbf{the eigenvalue conjecture is provided by the Regge symmetry} for $U_q(sl_2)$, when three representations coincide. This in perspective provides us a kind of generalization of the R… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1905.01876v1-abstract-full').style.display = 'inline'; document.getElementById('1905.01876v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1905.01876v1-abstract-full" style="display: none;"> In the present paper we discuss the eigenvalue conjecture, suggested in 2012, in the particular case of $U_q(sl_2)$. The eigenvalue conjecture provides a certain symmetry for Racah coefficients and we prove that \textbf{the eigenvalue conjecture is provided by the Regge symmetry} for $U_q(sl_2)$, when three representations coincide. This in perspective provides us a kind of generalization of the Regge symmetry to arbitrary $U_q(sl_N)$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1905.01876v1-abstract-full').style.display = 'none'; document.getElementById('1905.01876v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 May, 2019; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2019. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">9 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> IITP/TH-16/18 ITEP/TH-27/18 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Jetp Lett. (2018) 108: 697 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1805.03916">arXiv:1805.03916</a> <span> [<a href="https://arxiv.org/pdf/1805.03916">pdf</a>, <a href="https://arxiv.org/format/1805.03916">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s00023-019-00841-z">10.1007/s00023-019-00841-z <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Multi-Colored Links From 3-strand Braids Carrying Arbitrary Symmetric Representations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Dhara%2C+S">Saswati Dhara</a>, <a href="/search/hep-th?searchtype=author&query=Mironov%2C+A">A. Mironov</a>, <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">A. Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">An. Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Ramadevi%2C+P">P. Ramadevi</a>, <a href="/search/hep-th?searchtype=author&query=Singh%2C+V+K">Vivek Kumar Singh</a>, <a href="/search/hep-th?searchtype=author&query=Sleptsov%2C+A">A. Sleptsov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1805.03916v1-abstract-short" style="display: inline;"> Obtaining colored HOMFLY-PT polynomials for knots from 3-strand braid carrying arbitrary $SU(N)$ representation is still tedious. For a class of rank $r$ symmetric representations, $[r]$-colored HOMFLY-PT $H_{[r]}$ evaluation becomes simpler. Recently it was shown that $H_{[r]}$, for such knots from 3-strand braid, can be constructed using the quantum Racah coefficients (6j-symbols) of… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1805.03916v1-abstract-full').style.display = 'inline'; document.getElementById('1805.03916v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1805.03916v1-abstract-full" style="display: none;"> Obtaining colored HOMFLY-PT polynomials for knots from 3-strand braid carrying arbitrary $SU(N)$ representation is still tedious. For a class of rank $r$ symmetric representations, $[r]$-colored HOMFLY-PT $H_{[r]}$ evaluation becomes simpler. Recently it was shown that $H_{[r]}$, for such knots from 3-strand braid, can be constructed using the quantum Racah coefficients (6j-symbols) of $U_q(sl_2)$. In this paper, we generalise it to links whose components carry different symmetric representations. We illustrate the technique by evaluating multi-colored link polynomials $H_{[r_1],[r_2]}$ for the two-component link L7a3 whose components carry $[r_1]$ and $[r_2]$ colors. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1805.03916v1-abstract-full').style.display = 'none'; document.getElementById('1805.03916v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 10 May, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">11 pages, 1 figure</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> FIAN/TD-05/18; IITP/TH-04/18; ITEP/TH-06/18 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Ann. Henri Poincar茅 20(12) (2019) 4033-4054 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1805.02761">arXiv:1805.02761</a> <span> [<a href="https://arxiv.org/pdf/1805.02761">pdf</a>, <a href="https://arxiv.org/ps/1805.02761">ps</a>, <a href="https://arxiv.org/format/1805.02761">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Exactly Solvable and Integrable Systems">nlin.SI</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.physletb.2018.06.069">10.1016/j.physletb.2018.06.069 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Colored Alexander polynomials and KP hierarchy </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Mironov%2C+A">A. Mironov</a>, <a href="/search/hep-th?searchtype=author&query=Mironov%2C+S">S. Mironov</a>, <a href="/search/hep-th?searchtype=author&query=Mishnyakov%2C+V">V. Mishnyakov</a>, <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">A. Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Sleptsov%2C+A">A. Sleptsov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1805.02761v1-abstract-short" style="display: inline;"> We discuss the relation between knot polynomials and the KP hierarchy. Mainly, we study the scaling 1-hook property of the coloured Alexander polynomial: $\mathcal{A}^\mathcal{K}_R(q)=\mathcal{A}^\mathcal{K}_{[1]}(q^{\vert R\vert})$ for all 1-hook Young diagrams $R$. Via the Kontsevich construction, it is reformulated as a system of linear equations. It appears that the solutions of this system in… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1805.02761v1-abstract-full').style.display = 'inline'; document.getElementById('1805.02761v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1805.02761v1-abstract-full" style="display: none;"> We discuss the relation between knot polynomials and the KP hierarchy. Mainly, we study the scaling 1-hook property of the coloured Alexander polynomial: $\mathcal{A}^\mathcal{K}_R(q)=\mathcal{A}^\mathcal{K}_{[1]}(q^{\vert R\vert})$ for all 1-hook Young diagrams $R$. Via the Kontsevich construction, it is reformulated as a system of linear equations. It appears that the solutions of this system induce the KP equations in the Hirota form. The Alexander polynomial is a specialization of the HOMFLY polynomial, and it is a kind of a dual to the double scaling limit, which gives the special polynomial, in the sense that, while the special polynomials provide solutions to the KP hierarchy, the Alexander polynomials provide the equations of this hierarchy. This gives a new connection with integrable properties of knot polynomials and puts an interesting question about the way the KP hierarchy is encoded in the full HOMFLY polynomial. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1805.02761v1-abstract-full').style.display = 'none'; document.getElementById('1805.02761v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 May, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">10 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> FIAN/TD-07/18; IITP/TH-09/18; ITEP/TH-11/18 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Phys.Lett. B783 (2018) 268-273 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1801.09363">arXiv:1801.09363</a> <span> [<a href="https://arxiv.org/pdf/1801.09363">pdf</a>, <a href="https://arxiv.org/format/1801.09363">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.geomphys.2018.05.020">10.1016/j.geomphys.2018.05.020 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Quantum Racah matrices up to level 3 and multicolored link invariants </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Bai%2C+C">C. Bai</a>, <a href="/search/hep-th?searchtype=author&query=Jiang%2C+J">J. Jiang</a>, <a href="/search/hep-th?searchtype=author&query=Liang%2C+J">J. Liang</a>, <a href="/search/hep-th?searchtype=author&query=Mironov%2C+A">A. Mironov</a>, <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">A. Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">An. Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Sleptsov%2C+A">A. Sleptsov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1801.09363v1-abstract-short" style="display: inline;"> This paper is a next step in the project of systematic description of colored knot and link invariants started in previous papers. In this paper, we managed to explicitly find the inclusive Racah matrices, i.e. the whole set of mixing matrices in channels $R_1\otimes R_2\otimes R_3\longrightarrow Q$ with all possible $Q$, for $|R|\leq 3$. The calculation is made possible by use of the highest weig… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1801.09363v1-abstract-full').style.display = 'inline'; document.getElementById('1801.09363v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1801.09363v1-abstract-full" style="display: none;"> This paper is a next step in the project of systematic description of colored knot and link invariants started in previous papers. In this paper, we managed to explicitly find the inclusive Racah matrices, i.e. the whole set of mixing matrices in channels $R_1\otimes R_2\otimes R_3\longrightarrow Q$ with all possible $Q$, for $|R|\leq 3$. The calculation is made possible by use of the highest weight method. The result allows one to evaluate and investigate colored polynomials for arbitrary 3-strand knots and links and to check the corresponding eigenvalue conjecture. Explicit answers for Racah matrices and colored polynomials for 3-strand knots up to 10 crossings are available at http://knotebook.org. Using the obtained inclusive Racah matrices, we also calculated the exclusive Racah matrices with the help of trick earlier suggested in the case of knots. This method is proved to be effective and gives the exclusive Racah matrices earlier obtained by another method. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1801.09363v1-abstract-full').style.display = 'none'; document.getElementById('1801.09363v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 29 January, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">29 pages, 3 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> FIAN/TD-31/17 IITP/TH-02/18 ITEP/TH-02/18 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Journal of Geometry and Physics, 132 (2018) 155-180 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1712.08614">arXiv:1712.08614</a> <span> [<a href="https://arxiv.org/pdf/1712.08614">pdf</a>, <a href="https://arxiv.org/ps/1712.08614">ps</a>, <a href="https://arxiv.org/format/1712.08614">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.4310/CNTP.2019.v13.n4.a3">10.4310/CNTP.2019.v13.n4.a3 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Combinatorial structure of colored HOMFLY-PT polynomials for torus knots </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Dunin-Barkowski%2C+P">Petr Dunin-Barkowski</a>, <a href="/search/hep-th?searchtype=author&query=Popolitov%2C+A">Aleksandr Popolitov</a>, <a href="/search/hep-th?searchtype=author&query=Shadrin%2C+S">Sergey Shadrin</a>, <a href="/search/hep-th?searchtype=author&query=Sleptsov%2C+A">Alexey Sleptsov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1712.08614v3-abstract-short" style="display: inline;"> We rewrite the (extended) Ooguri-Vafa partition function for colored HOMFLY-PT polynomials for torus knots in terms of the free-fermion (semi-infinite wedge) formalism, making it very similar to the generating function for double Hurwitz numbers. This allows us to conjecture the combinatorial meaning of full expansion of the correlation differentials obtained via the topological recursion on the B… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1712.08614v3-abstract-full').style.display = 'inline'; document.getElementById('1712.08614v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1712.08614v3-abstract-full" style="display: none;"> We rewrite the (extended) Ooguri-Vafa partition function for colored HOMFLY-PT polynomials for torus knots in terms of the free-fermion (semi-infinite wedge) formalism, making it very similar to the generating function for double Hurwitz numbers. This allows us to conjecture the combinatorial meaning of full expansion of the correlation differentials obtained via the topological recursion on the Brini-Eynard-Mari帽o spectral curve for the colored HOMFLY-PT polynomials of torus knots. This correspondence suggests a structural combinatorial result for the extended Ooguri-Vafa partition function. Namely, its coefficients should have a quasi-polynomial behavior, where non-polynomial factors are given by the Jacobi polynomials. We prove this quasi-polynomiality in a purely combinatorial way. In addition to that, we show that the (0,1)- and (0,2)-functions on the corresponding spectral curve are in agreement with the extension of the colored HOMFLY-PT polynomials data. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1712.08614v3-abstract-full').style.display = 'none'; document.getElementById('1712.08614v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 March, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 22 December, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">40 pages; section 10 addressing the quantum curve was added, as well as some remarks regarding Meixner polynomials thanks to T.Koornwinder</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Commun. Number Theory Phys. 13 (2019), no. 4, 763-826 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1712.07034">arXiv:1712.07034</a> <span> [<a href="https://arxiv.org/pdf/1712.07034">pdf</a>, <a href="https://arxiv.org/format/1712.07034">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1142/S0217751X18501051">10.1142/S0217751X18501051 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> On the block structure of the quantum R-matrix in the three-strand braids </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Bishler%2C+L">L. Bishler</a>, <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">An. Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Sleptsov%2C+A">A. Sleptsov</a>, <a href="/search/hep-th?searchtype=author&query=Shakirov%2C+S">Sh. Shakirov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1712.07034v1-abstract-short" style="display: inline;"> Quantum $\mathcal{R}$-matrices are the building blocks for the colored HOMFLY polynomials. In the case of three-strand braids with an identical finite-dimensional irreducible representation $T$ of $SU_q(N)$ associated with each strand one needs two matrices: $\mathcal{R}_1$ and $\mathcal{R}_2$. They are related by the Racah matrices… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1712.07034v1-abstract-full').style.display = 'inline'; document.getElementById('1712.07034v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1712.07034v1-abstract-full" style="display: none;"> Quantum $\mathcal{R}$-matrices are the building blocks for the colored HOMFLY polynomials. In the case of three-strand braids with an identical finite-dimensional irreducible representation $T$ of $SU_q(N)$ associated with each strand one needs two matrices: $\mathcal{R}_1$ and $\mathcal{R}_2$. They are related by the Racah matrices $\mathcal{R}_2 = \mathcal{U} \mathcal{R}_1 \mathcal{U}^{\dagger}$. Since we can always choose the basis so that $\mathcal{R}_1$ is diagonal, the problem is reduced to evaluation of $\mathcal{R}_2$-matrices. This paper is one more step on the road to simplification of such calculations. We found out and proved for some cases that $\mathcal{R}_2$-matrices could be transformed into a block-diagonal ones. The essential condition is that there is a pair of accidentally coinciding eigenvalues among eigenvalues of $\mathcal{R}_1$-matrix. The angle of the rotation in the sectors corresponding to accidentally coinciding eigenvalues from the basis defined by the Racah matrix to the basis in which $\mathcal{R}_2$ is block-diagonal is $\pm \frac蟺{4}$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1712.07034v1-abstract-full').style.display = 'none'; document.getElementById('1712.07034v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 19 December, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">21 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> ITEP/TH-38/17 IITP/TH-23/17 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> International Journal of Modern Physics AVol. 33, No. 17, 1850105 (2018) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1712.03155">arXiv:1712.03155</a> <span> [<a href="https://arxiv.org/pdf/1712.03155">pdf</a>, <a href="https://arxiv.org/ps/1712.03155">ps</a>, <a href="https://arxiv.org/format/1712.03155">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1142/S0129055X18400056">10.1142/S0129055X18400056 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Orthogonal Polynomials in Mathematical Physics </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Chan%2C+C">Chuan-Tsung Chan</a>, <a href="/search/hep-th?searchtype=author&query=Mironov%2C+A">A. Mironov</a>, <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">A. Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Sleptsov%2C+A">A. Sleptsov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1712.03155v2-abstract-short" style="display: inline;"> This is a review of ($q$-)hypergeometric orthogonal polynomials and their relation to representation theory of quantum groups, to matrix models, to integrable theory, and to knot theory. We discuss both continuous and discrete orthogonal polynomials and consider their various generalizations. The review also includes the orthogonal polynomials into a generic framework of ($q$-)hypergeometric funct… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1712.03155v2-abstract-full').style.display = 'inline'; document.getElementById('1712.03155v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1712.03155v2-abstract-full" style="display: none;"> This is a review of ($q$-)hypergeometric orthogonal polynomials and their relation to representation theory of quantum groups, to matrix models, to integrable theory, and to knot theory. We discuss both continuous and discrete orthogonal polynomials and consider their various generalizations. The review also includes the orthogonal polynomials into a generic framework of ($q$-)hypergeometric functions and their integral representations. In particular, this gives rise to relations with conformal blocks of the Virasoro algebra. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1712.03155v2-abstract-full').style.display = 'none'; document.getElementById('1712.03155v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 18 December, 2017; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 8 December, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">46 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> FIAN/TD-25/17; IITP/TH-21/17; ITEP/TH-35/17 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Ludwig Faddeev Memorial Volume (2018): pp. 119-182; see: Reviews in Mathematical Physics, Vol. 30, No. 6 (2018) 1840005 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1711.10952">arXiv:1711.10952</a> <span> [<a href="https://arxiv.org/pdf/1711.10952">pdf</a>, <a href="https://arxiv.org/format/1711.10952">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1103/PhysRevD.97.126015">10.1103/PhysRevD.97.126015 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Eigenvalue hypothesis for multi-strand braids </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Dhara%2C+S">Saswati Dhara</a>, <a href="/search/hep-th?searchtype=author&query=Mironov%2C+A">A. Mironov</a>, <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">A. Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">An. Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Ramadevi%2C+P">P. Ramadevi</a>, <a href="/search/hep-th?searchtype=author&query=Singh%2C+V+K">Vivek Kumar Singh</a>, <a href="/search/hep-th?searchtype=author&query=Sleptsov%2C+A">A. Sleptsov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1711.10952v3-abstract-short" style="display: inline;"> Computing polynomial form of the colored HOMFLY-PT for non-arborescent knots obtained from three or more strand braids is still an open problem. One of the efficient methods suggested for the three-strand braids relies on the eigenvalue hypothesis which uses the Yang-Baxter equation to express the answer through the eigenvalues of the ${\cal R}$-matrix. In this paper, we generalize the hypothesis… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1711.10952v3-abstract-full').style.display = 'inline'; document.getElementById('1711.10952v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1711.10952v3-abstract-full" style="display: none;"> Computing polynomial form of the colored HOMFLY-PT for non-arborescent knots obtained from three or more strand braids is still an open problem. One of the efficient methods suggested for the three-strand braids relies on the eigenvalue hypothesis which uses the Yang-Baxter equation to express the answer through the eigenvalues of the ${\cal R}$-matrix. In this paper, we generalize the hypothesis to higher number of strands in the braid where commuting relations of non-neighbouring $\mathcal{R}$ matrices are also incorporated. By solving these equations, we determine the explicit form for $\mathcal{R}$-matrices and the inclusive Racah matrices in terms of braiding eigenvalues (for matrices of size up to 6 by 6). For comparison, we briefly discuss the highest weight method for four-strand braids carrying fundamental and symmetric rank two $SU_q(N)$ representation. Specifically, we present all the inclusive Racah matrices for representation $[2]$ and compare with the matrices obtained from eigenvalue hypothesis. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1711.10952v3-abstract-full').style.display = 'none'; document.getElementById('1711.10952v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 25 January, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 29 November, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">23 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> FIAN/TD-24/17 IITP/TH-18/17 ITEP/TH-31/17 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Phys. Rev. D 97, 126015 (2018) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1709.09228">arXiv:1709.09228</a> <span> [<a href="https://arxiv.org/pdf/1709.09228">pdf</a>, <a href="https://arxiv.org/ps/1709.09228">ps</a>, <a href="https://arxiv.org/format/1709.09228">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.physletb.2018.01.026">10.1016/j.physletb.2018.01.026 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Differential expansion for link polynomials </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Bai%2C+C">C. Bai</a>, <a href="/search/hep-th?searchtype=author&query=Jiang%2C+J">J. Jiang</a>, <a href="/search/hep-th?searchtype=author&query=Liang%2C+J">J. Liang</a>, <a href="/search/hep-th?searchtype=author&query=Mironov%2C+A">A. Mironov</a>, <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">A. Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">An. Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Sleptsov%2C+A">A. Sleptsov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1709.09228v1-abstract-short" style="display: inline;"> The differential expansion is one of the key structures reflecting group theory properties of colored knot polynomials, which also becomes an important tool for evaluation of non-trivial Racah matrices. This makes highly desirable its extension from knots to links, which, however, requires knowledge of the $6j$-symbols, at least, for the simplest triples of non-coincident representations. Based on… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1709.09228v1-abstract-full').style.display = 'inline'; document.getElementById('1709.09228v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1709.09228v1-abstract-full" style="display: none;"> The differential expansion is one of the key structures reflecting group theory properties of colored knot polynomials, which also becomes an important tool for evaluation of non-trivial Racah matrices. This makes highly desirable its extension from knots to links, which, however, requires knowledge of the $6j$-symbols, at least, for the simplest triples of non-coincident representations. Based on the recent achievements in this direction, we conjecture a shape of the differential expansion for symmetrically-colored links and provide a set of examples. Within this study, we use a special framing that is an unusual extension of the topological framing from knots to links. In the particular cases of Whitehead and Borromean rings links, the differential expansions are different from the previously discovered. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1709.09228v1-abstract-full').style.display = 'none'; document.getElementById('1709.09228v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 26 September, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">11 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> FIAN/TD-22/17; IITP/TH-26/17; ITEP/TH-16/17 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Phys.Lett. B778 (2018) 197-206 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1709.02290">arXiv:1709.02290</a> <span> [<a href="https://arxiv.org/pdf/1709.02290">pdf</a>, <a href="https://arxiv.org/format/1709.02290">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1134/S0021364017220040">10.1134/S0021364017220040 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> On $6j$-symbols for symmetric representations of $U_q(\mathfrak{su}_N)$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Mironov%2C+A">A. Mironov</a>, <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">A. Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Sleptsov%2C+A">A. Sleptsov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1709.02290v2-abstract-short" style="display: inline;"> Explicit expressions are found for the $6j$ symbols in symmetric representations of quantum $\mathfrak{su}_N$ through appropriate hypergeometric Askey-Wilson (q-Racah) polynomials. This generalizes the well-known classical formulas for $U_q(\mathfrak{su}_2)$ and provides a link to conformal theories and matrix models. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1709.02290v2-abstract-full" style="display: none;"> Explicit expressions are found for the $6j$ symbols in symmetric representations of quantum $\mathfrak{su}_N$ through appropriate hypergeometric Askey-Wilson (q-Racah) polynomials. This generalizes the well-known classical formulas for $U_q(\mathfrak{su}_2)$ and provides a link to conformal theories and matrix models. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1709.02290v2-abstract-full').style.display = 'none'; document.getElementById('1709.02290v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 18 October, 2017; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 7 September, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">9 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> FIAN/TD-18/17; IITP/TH-14/17; ITEP/TH-23/17 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Pis'ma v ZhETF, 106 (2017) 607 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1706.00761">arXiv:1706.00761</a> <span> [<a href="https://arxiv.org/pdf/1706.00761">pdf</a>, <a href="https://arxiv.org/format/1706.00761">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.nuclphysb.2017.08.016">10.1016/j.nuclphysb.2017.08.016 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Gaussian distribution of LMOV numbers </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Mironov%2C+A">A. Mironov</a>, <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">A. Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">An. Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Sleptsov%2C+A">A. Sleptsov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1706.00761v1-abstract-short" style="display: inline;"> Recent advances in knot polynomial calculus allowed us to obtain a huge variety of LMOV integers counting degeneracy of the BPS spectrum of topological theories on the resolved conifold and appearing in the genus expansion of the plethystic logarithm of the Ooguri-Vafa partition functions. Already the very first look at this data reveals that the LMOV numbers are randomly distributed in genus (!)… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1706.00761v1-abstract-full').style.display = 'inline'; document.getElementById('1706.00761v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1706.00761v1-abstract-full" style="display: none;"> Recent advances in knot polynomial calculus allowed us to obtain a huge variety of LMOV integers counting degeneracy of the BPS spectrum of topological theories on the resolved conifold and appearing in the genus expansion of the plethystic logarithm of the Ooguri-Vafa partition functions. Already the very first look at this data reveals that the LMOV numbers are randomly distributed in genus (!) and are very well parameterized by just three parameters depending on the representation, an integer and the knot. We present an accurate formulation and evidence in support of this new puzzling observation about the old puzzling quantities. It probably implies that the BPS states, counted by the LMOV numbers can actually be composites made from some still more elementary objects. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1706.00761v1-abstract-full').style.display = 'none'; document.getElementById('1706.00761v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 2 June, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">23 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> FIAN/TD-11/17; IITP/TH-09/17; ITEP/TH-15/17 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Nuclear Physics, B924 (2017) 1-32 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1702.06316">arXiv:1702.06316</a> <span> [<a href="https://arxiv.org/pdf/1702.06316">pdf</a>, <a href="https://arxiv.org/format/1702.06316">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/JHEP08(2017)139">10.1007/JHEP08(2017)139 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Checks of integrality properties in topological strings </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Mironov%2C+A">A. Mironov</a>, <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">A. Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">An. Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Ramadevi%2C+P">P. Ramadevi</a>, <a href="/search/hep-th?searchtype=author&query=Singh%2C+V+K">Vivek Kumar Singh</a>, <a href="/search/hep-th?searchtype=author&query=Sleptsov%2C+A">A. Sleptsov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1702.06316v3-abstract-short" style="display: inline;"> Tests of the integrality properties of a scalar operator in topological strings on a resolved conifold background or orientifold of conifold backgrounds have been performed for arborescent knots and some non-arborescent knots. The recent results on polynomials for those knots colored by SU(N) and SO(N) adjoint representations are useful to verify Marino's integrality conjecture up to two boxes in… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1702.06316v3-abstract-full').style.display = 'inline'; document.getElementById('1702.06316v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1702.06316v3-abstract-full" style="display: none;"> Tests of the integrality properties of a scalar operator in topological strings on a resolved conifold background or orientifold of conifold backgrounds have been performed for arborescent knots and some non-arborescent knots. The recent results on polynomials for those knots colored by SU(N) and SO(N) adjoint representations are useful to verify Marino's integrality conjecture up to two boxes in the Young diagram. In this paper, we review the salient aspects of the integrality properties and tabulate explicitly for an arborescent knot and a link. In our knotebook website, we have put these results for over 100 prime knots available in Rolfsen table and some links. The first application of the obtained results, an observation of the Gaussian distribution of the LMOV invariants is also reported. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1702.06316v3-abstract-full').style.display = 'none'; document.getElementById('1702.06316v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 24 January, 2018; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 21 February, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">28 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> FIAN/TD-19/16; IITP/TH-14/16; ITEP/TH-20/16 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> JHEP 08 (2017) 139 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1611.03797">arXiv:1611.03797</a> <span> [<a href="https://arxiv.org/pdf/1611.03797">pdf</a>, <a href="https://arxiv.org/format/1611.03797">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> </div> <p class="title is-5 mathjax"> Quantum Racah matrices and 3-strand braids in representation [3,3] </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Shakirov%2C+S">Sh. Shakirov</a>, <a href="/search/hep-th?searchtype=author&query=Sleptsov%2C+A">A. Sleptsov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1611.03797v3-abstract-short" style="display: inline;"> This paper is a next step in the project of systematic description of colored knot polynomials started in arXiv:1506.00339. In this paper, we managed to explicitly find the $\textit{inclusive}$ Racah matrices, i.e. the whole set of mixing matrices in channels $R^{\otimes 3}\longrightarrow Q$ with all possible $Q$, for $R=[3,3]$. The case $R=[3,3]$ is a multiplicity free case as well as $R=[2,2]$ o… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1611.03797v3-abstract-full').style.display = 'inline'; document.getElementById('1611.03797v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1611.03797v3-abstract-full" style="display: none;"> This paper is a next step in the project of systematic description of colored knot polynomials started in arXiv:1506.00339. In this paper, we managed to explicitly find the $\textit{inclusive}$ Racah matrices, i.e. the whole set of mixing matrices in channels $R^{\otimes 3}\longrightarrow Q$ with all possible $Q$, for $R=[3,3]$. The case $R=[3,3]$ is a multiplicity free case as well as $R=[2,2]$ obtained in arXiv:1605.03098. The calculation is made possible by the use of highest weight method with the help of Gelfand-Tseitlin tables. The result allows one to evaluate and investigate $[3,3]$-colored polynomials for arbitrary 3-strand knots, and this confirms many previous conjectures on various factorizations, universality, and differential expansions. With the help of a method developed in arXiv:1605.04881 we manage to calculate {\it exclusive} Racah matrices $S$ and $\bar S$ in $R=[3,3]$. Our results confirm a calculation of these matrices in arXiv:1606.06015, which was based on the conjecture of explicit form of differential expansion for twist knots. Explicit answers for Racah matrices and $[3,3]$-colored polynomials for 3-strand knots up to 10 crossings are available at http://knotebook.org. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1611.03797v3-abstract-full').style.display = 'none'; document.getElementById('1611.03797v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 May, 2021; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 11 November, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2016. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">16 pages, 2 figures</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1605.04881">arXiv:1605.04881</a> <span> [<a href="https://arxiv.org/pdf/1605.04881">pdf</a>, <a href="https://arxiv.org/format/1605.04881">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.physletb.2016.06.041">10.1016/j.physletb.2016.06.041 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Racah matrices and hidden integrability in evolution of knots </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Mironov%2C+A">A. Mironov</a>, <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">A. Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">An. Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Sleptsov%2C+A">A. Sleptsov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1605.04881v1-abstract-short" style="display: inline;"> We construct a general procedure to extract the exclusive Racah matrices S and \bar S from the inclusive 3-strand mixing matrices by the evolution method and apply it to the first simple representations R =[1], [2], [3] and [2,2]. The matrices S and \bar S relate respectively the maps (R\otimes R)\otimes \bar R\longrightarrow R with R\otimes (R \otimes \bar R) \longrightarrow R and (R\otimes \bar… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1605.04881v1-abstract-full').style.display = 'inline'; document.getElementById('1605.04881v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1605.04881v1-abstract-full" style="display: none;"> We construct a general procedure to extract the exclusive Racah matrices S and \bar S from the inclusive 3-strand mixing matrices by the evolution method and apply it to the first simple representations R =[1], [2], [3] and [2,2]. The matrices S and \bar S relate respectively the maps (R\otimes R)\otimes \bar R\longrightarrow R with R\otimes (R \otimes \bar R) \longrightarrow R and (R\otimes \bar R) \otimes R \longrightarrow R with R\otimes (\bar R \otimes R) \longrightarrow R. They are building blocks for the colored HOMFLY polynomials of arbitrary arborescent (double fat) knots. Remarkably, the calculation realizes an unexpected integrability property underlying the evolution matrices. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1605.04881v1-abstract-full').style.display = 'none'; document.getElementById('1605.04881v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 16 May, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2016. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">16 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> FIAN/TD-12/16; IITP/TH-08/16; ITEP/TH-10/16 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Physics Letters B760 (2016) 45-58 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1605.03098">arXiv:1605.03098</a> <span> [<a href="https://arxiv.org/pdf/1605.03098">pdf</a>, <a href="https://arxiv.org/ps/1605.03098">ps</a>, <a href="https://arxiv.org/format/1605.03098">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1134/S0021364016130038">10.1134/S0021364016130038 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Quantum Racah matrices and 3-strand braids in irreps R with |R|=4 </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Mironov%2C+A">A. Mironov</a>, <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">A. Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">An. Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Sleptsov%2C+A">A. Sleptsov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1605.03098v1-abstract-short" style="display: inline;"> We describe the inclusive Racah matrices for the first non-(anti)symmetric rectangular representation R=[2,2] for quantum groups U_q(sl_N). Most of them have sizes 2, 3, and 4 and are fully described by the eigenvalue hypothesis. Of two 6x6 matrices, one is also described in this way, but the other one corresponds to the case of degenerate eigenvalues and is evaluated by the highest weight method.… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1605.03098v1-abstract-full').style.display = 'inline'; document.getElementById('1605.03098v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1605.03098v1-abstract-full" style="display: none;"> We describe the inclusive Racah matrices for the first non-(anti)symmetric rectangular representation R=[2,2] for quantum groups U_q(sl_N). Most of them have sizes 2, 3, and 4 and are fully described by the eigenvalue hypothesis. Of two 6x6 matrices, one is also described in this way, but the other one corresponds to the case of degenerate eigenvalues and is evaluated by the highest weight method. Together with the much harder calculation for R=[3,1] in arXiv:1605.02313 and with the new method to extract exclusive matrices S and \bar S from the inclusive ones, this completes the story of Racah matrices for |R|\leq 4 and allows one to calculate and investigate the corresponding colored HOMFLY polynomials for arbitrary 3-strand and arborescent knots. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1605.03098v1-abstract-full').style.display = 'none'; document.getElementById('1605.03098v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 10 May, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2016. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">7 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> FIAN/TD-11/16; IITP/TH-07/16; ITEP/TH-09/16 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> JETP Lett. 104 (2016) 56-61, Pisma Zh.Eksp.Teor.Fiz. 104 (2016) 52-57 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1605.02313">arXiv:1605.02313</a> <span> [<a href="https://arxiv.org/pdf/1605.02313">pdf</a>, <a href="https://arxiv.org/ps/1605.02313">ps</a>, <a href="https://arxiv.org/format/1605.02313">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/JHEP09(2016)134">10.1007/JHEP09(2016)134 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> HOMFLY polynomials in representation [3,1] for 3-strand braids </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Mironov%2C+A">A. Mironov</a>, <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">A. Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">An. Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Sleptsov%2C+A">A. Sleptsov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1605.02313v1-abstract-short" style="display: inline;"> This paper is a new step in the project of systematic description of colored knot polynomials started in arXiv:1506.00339. In this paper, we managed to explicitly find the inclusive Racah matrix, i.e. the whole set of mixing matrices in channels R^3->Q with all possible Q, for R=[3,1]. The calculation is made possible by the use of a newly-developed efficient highest-weight method, still it remain… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1605.02313v1-abstract-full').style.display = 'inline'; document.getElementById('1605.02313v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1605.02313v1-abstract-full" style="display: none;"> This paper is a new step in the project of systematic description of colored knot polynomials started in arXiv:1506.00339. In this paper, we managed to explicitly find the inclusive Racah matrix, i.e. the whole set of mixing matrices in channels R^3->Q with all possible Q, for R=[3,1]. The calculation is made possible by the use of a newly-developed efficient highest-weight method, still it remains tedious. The result allows one to evaluate and investigate [3,1]-colored polynomials for arbitrary 3-strand knots, and this confirms many previous conjectures on various factorizations, universality, and differential expansions. We consider in some detail the next-to-twist-knots three-strand family (n,-1|1,-1) and deduce its colored HOMFLY. Also confirmed and clarified is the eigenvalue hypothesis for the Racah matrices, which promises to provide a shortcut to generic formulas for arbitrary representations. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1605.02313v1-abstract-full').style.display = 'none'; document.getElementById('1605.02313v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 8 May, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2016. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">21 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> FIAN/TD-08/16; IITP/TH-05/16; ITEP/TH-04/16 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Journal of High Energy Physics, 2016 (2016) 134 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1601.04199">arXiv:1601.04199</a> <span> [<a href="https://arxiv.org/pdf/1601.04199">pdf</a>, <a href="https://arxiv.org/format/1601.04199">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Representation Theory">math.RT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1088/1751-8121/aa5574">10.1088/1751-8121/aa5574 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Tabulating knot polynomials for arborescent knots </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Mironov%2C+A">A. Mironov</a>, <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">A. Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">An. Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Ramadevi%2C+P">P. Ramadevi</a>, <a href="/search/hep-th?searchtype=author&query=Singh%2C+V+K">Vivek Kumar Singh</a>, <a href="/search/hep-th?searchtype=author&query=Sleptsov%2C+A">A. Sleptsov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1601.04199v2-abstract-short" style="display: inline;"> Arborescent knots are the ones which can be represented in terms of double fat graphs or equivalently as tree Feynman diagrams. This is the class of knots for which the present knowledge is enough for lifting topological description to the level of effective analytical formulas. The paper describes the origin and structure of the new tables of colored knot polynomials, which will be posted at the… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1601.04199v2-abstract-full').style.display = 'inline'; document.getElementById('1601.04199v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1601.04199v2-abstract-full" style="display: none;"> Arborescent knots are the ones which can be represented in terms of double fat graphs or equivalently as tree Feynman diagrams. This is the class of knots for which the present knowledge is enough for lifting topological description to the level of effective analytical formulas. The paper describes the origin and structure of the new tables of colored knot polynomials, which will be posted at the dedicated site. Even if formal expressions are known in terms of modular transformation matrices, the computation in finite time requires additional ideas. We use the "family" approach, and apply it to arborescent knots in the Rolfsen table by developing a Feynman diagram technique associated with an auxiliary matrix model field theory. Gauge invariance in this theory helps to provide meaning to Racah matrices in the case of non-trivial multiplicities and explains the need for peculiar sign prescriptions in the calculation of [21]-colored HOMFLY polynomials. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1601.04199v2-abstract-full').style.display = 'none'; document.getElementById('1601.04199v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 1 April, 2016; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 16 January, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2016. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">19 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> FIAN/TD-01/16; IITP/TH-01/16; ITEP/TH-02/16 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. Phys. A: Math. Theor. 50 (2017) 085201 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1512.07192">arXiv:1512.07192</a> <span> [<a href="https://arxiv.org/pdf/1512.07192">pdf</a>, <a href="https://arxiv.org/format/1512.07192">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1142/S0217751X16501566">10.1142/S0217751X16501566 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Vassiliev invariants for pretzel knots </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Sleptsov%2C+A">A. Sleptsov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1512.07192v1-abstract-short" style="display: inline;"> We compute Vassiliev invariants up to order six for arbitrary pretzel knots, which depend on $g+1$ parameters $n_1,\ldots,n_{g+1}$. These invariants are symmetric polynomials in $n_1,\ldots,n_{g+1}$ whose degree coincide with their order. We also discuss their topological and integer-valued properties. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1512.07192v1-abstract-full" style="display: none;"> We compute Vassiliev invariants up to order six for arbitrary pretzel knots, which depend on $g+1$ parameters $n_1,\ldots,n_{g+1}$. These invariants are symmetric polynomials in $n_1,\ldots,n_{g+1}$ whose degree coincide with their order. We also discuss their topological and integer-valued properties. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1512.07192v1-abstract-full').style.display = 'none'; document.getElementById('1512.07192v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 22 December, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2015. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">14 pages, 3 figures. arXiv admin note: text overlap with arXiv:1112.5406</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> ITEP/TH-36/15, IITP/TH-21/15 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1508.02870">arXiv:1508.02870</a> <span> [<a href="https://arxiv.org/pdf/1508.02870">pdf</a>, <a href="https://arxiv.org/ps/1508.02870">ps</a>, <a href="https://arxiv.org/format/1508.02870">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1142/S0217751X15501699">10.1142/S0217751X15501699 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Colored knot polynomials. HOMFLY in representation [2,1] </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Mironov%2C+A">A. Mironov</a>, <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">A. Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">An. Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Sleptsov%2C+A">A. Sleptsov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1508.02870v1-abstract-short" style="display: inline;"> This paper starts a systematic description of colored knot polynomials, beginning from the first non-(anti)symmetric representation R=[2,1]. The project involves several steps: (i) parametrization of big families of knots a la arXiv:1506.00339, (ii) evaluating Racah/mixing matrices for various numbers of strands in various representations a la arXiv:1112.2654, (iii) tabulating and collecting the r… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1508.02870v1-abstract-full').style.display = 'inline'; document.getElementById('1508.02870v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1508.02870v1-abstract-full" style="display: none;"> This paper starts a systematic description of colored knot polynomials, beginning from the first non-(anti)symmetric representation R=[2,1]. The project involves several steps: (i) parametrization of big families of knots a la arXiv:1506.00339, (ii) evaluating Racah/mixing matrices for various numbers of strands in various representations a la arXiv:1112.2654, (iii) tabulating and collecting the results at www.knotebook.org. In this paper we discuss only representation R=[2,1] and construct all necessary ingredients that allow one to evaluate knot/links represented by three strand closed parallel braids with inserted double-fat fingers. In particular, it is used to evaluate knots from a 7-parametric family: this family contains over 80% of knots with up to 10 intersections, but does not include mutants. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1508.02870v1-abstract-full').style.display = 'none'; document.getElementById('1508.02870v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 12 August, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2015. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">25 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> FIAN/TD-5/15; IITP/TH-10/15; ITEP/TH-17/15 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Int. J. Mod. Phys. A 30 (2015) 1550169 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1412.8432">arXiv:1412.8432</a> <span> [<a href="https://arxiv.org/pdf/1412.8432">pdf</a>, <a href="https://arxiv.org/format/1412.8432">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/JHEP07(2015)069">10.1007/JHEP07(2015)069 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Colored HOMFLY polynomials for the pretzel knots and links </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Mironov%2C+A">A. Mironov</a>, <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">A. Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Sleptsov%2C+A">A. Sleptsov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1412.8432v2-abstract-short" style="display: inline;"> With the help of the evolution method we calculate all HOMFLY polynomials in all symmetric representations [r] for a huge family of (generalized) pretzel links, which are made from g+1 two strand braids, parallel or antiparallel, and depend on g+1 integer numbers. We demonstrate that they possess a pronounced new structure: are decomposed into a sum of a product of g+1 elementary polynomials, whic… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1412.8432v2-abstract-full').style.display = 'inline'; document.getElementById('1412.8432v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1412.8432v2-abstract-full" style="display: none;"> With the help of the evolution method we calculate all HOMFLY polynomials in all symmetric representations [r] for a huge family of (generalized) pretzel links, which are made from g+1 two strand braids, parallel or antiparallel, and depend on g+1 integer numbers. We demonstrate that they possess a pronounced new structure: are decomposed into a sum of a product of g+1 elementary polynomials, which are obtained from the evolution eigenvalues by rotation with the help of rescaled SU_q(N) Racah matrix, for which we provide an explicit expression. The generalized pretzel family contains many mutants, undistinguishable by symmetric HOMFLY polynomials, hence, the extension of our results to non-symmetric representations R is a challenging open problem. To this end, a non-trivial generalization of the suggested formula can be conjectured for entire family with arbitrary g and R. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1412.8432v2-abstract-full').style.display = 'none'; document.getElementById('1412.8432v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 2 April, 2015; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 29 December, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2014. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">26 pages + tables of pretzel knots up to 10 crossings</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> FIAN/TD-20/14; ITEP/TH-47/14 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> JHEP 07 (2015) 069 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1412.2616">arXiv:1412.2616</a> <span> [<a href="https://arxiv.org/pdf/1412.2616">pdf</a>, <a href="https://arxiv.org/format/1412.2616">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Quantum Algebra">math.QA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.physletb.2015.02.029">10.1016/j.physletb.2015.02.029 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Colored knot polynomials for Pretzel knots and links of arbitrary genus </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Galakhov%2C+D">D. Galakhov</a>, <a href="/search/hep-th?searchtype=author&query=Melnikov%2C+D">D. Melnikov</a>, <a href="/search/hep-th?searchtype=author&query=Mironov%2C+A">A. Mironov</a>, <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">A. Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Sleptsov%2C+A">A. Sleptsov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1412.2616v1-abstract-short" style="display: inline;"> A very simple expression is conjectured for arbitrary colored Jones and HOMFLY polynomials of a rich $(g+1)$-parametric family of Pretzel knots and links. The answer for the Jones and HOMFLY polynomials is fully and explicitly expressed through the Racah matrix of U_q(SU_N), and looks related to a modular transformation of toric conformal block. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1412.2616v1-abstract-full" style="display: none;"> A very simple expression is conjectured for arbitrary colored Jones and HOMFLY polynomials of a rich $(g+1)$-parametric family of Pretzel knots and links. The answer for the Jones and HOMFLY polynomials is fully and explicitly expressed through the Racah matrix of U_q(SU_N), and looks related to a modular transformation of toric conformal block. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1412.2616v1-abstract-full').style.display = 'none'; document.getElementById('1412.2616v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 8 December, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2014. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">5 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> FIAN/TD-19/14; ITEP/TH-42/14 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Physics Letters B743 (2015) 71-74 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1310.7622">arXiv:1310.7622</a> <span> [<a href="https://arxiv.org/pdf/1310.7622">pdf</a>, <a href="https://arxiv.org/format/1310.7622">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.nuclphysb.2014.11.003">10.1016/j.nuclphysb.2014.11.003 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> On genus expansion of superpolynomials </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Mironov%2C+A">A. Mironov</a>, <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">A. Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Sleptsov%2C+A">A. Sleptsov</a>, <a href="/search/hep-th?searchtype=author&query=Smirnov%2C+A">A. Smirnov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1310.7622v2-abstract-short" style="display: inline;"> Recently it was shown that the (Ooguri-Vafa) generating function of HOMFLY polynomials is the Hurwitz partition function, i.e. that the dependence of the HOMFLY polynomials on representation R is naturally captured by symmetric group characters (cut-and-join eigenvalues). The genus expansion and expansion through Vassiliev invariants explicitly demonstrate this phenomenon. In the present letter we… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1310.7622v2-abstract-full').style.display = 'inline'; document.getElementById('1310.7622v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1310.7622v2-abstract-full" style="display: none;"> Recently it was shown that the (Ooguri-Vafa) generating function of HOMFLY polynomials is the Hurwitz partition function, i.e. that the dependence of the HOMFLY polynomials on representation R is naturally captured by symmetric group characters (cut-and-join eigenvalues). The genus expansion and expansion through Vassiliev invariants explicitly demonstrate this phenomenon. In the present letter we claim that the superpolynomials are not functions of such a type: symmetric group characters do not provide an adequate linear basis for their expansions. Deformation to superpolynomials is, however, straightforward in the multiplicative basis: the Casimir operators are 尾-deformed to Hamiltonians of the Calogero-Moser-Sutherland system. Applying this trick to the genus and Vassiliev expansions, we observe that the deformation is fully straightforward only for the thin knots. Beyond the family of thin knots additional algebraically independent terms appear in the Vassiliev and genus expansions. This can suggest that the superpolynomials do in fact contain more information about knots than the colored HOMFLY and Kauffman polynomials. However, even for the thin knots the beta-deformation is non-innocent: already in the simplest examples it seems inconsistent with the postivity of colored superpolynomials in non-(anti)symmetric representations, which also happens in I.Cherednik's (DAHA-based) approach to the torus knots. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1310.7622v2-abstract-full').style.display = 'none'; document.getElementById('1310.7622v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 September, 2014; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 28 October, 2013; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2013. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">17 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> FIAN/TD-14/13; ITEP/TH-41/13 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Nuclear Physics, B889 (2014) 757-777 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1304.7499">arXiv:1304.7499</a> <span> [<a href="https://arxiv.org/pdf/1304.7499">pdf</a>, <a href="https://arxiv.org/format/1304.7499">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1140/epjc/s10052-013-2492-9">10.1140/epjc/s10052-013-2492-9 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> On genus expansion of knot polynomials and hidden structure of Hurwitz tau-functions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Mironov%2C+A">A. Mironov</a>, <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">A. Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Sleptsov%2C+A">A. Sleptsov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1304.7499v1-abstract-short" style="display: inline;"> In the genus expansion of the HOMFLY polynomials their representation dependence is naturally captured by symmetric group characters. This immediately implies that the Ooguri-Vafa partition function (OVPF) is a Hurwitz tau-function. In the planar limit involving factorizable special polynomials, it is actually a trivial exponential tau-function. In fact, in the double scaling Kashaev limit (the on… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1304.7499v1-abstract-full').style.display = 'inline'; document.getElementById('1304.7499v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1304.7499v1-abstract-full" style="display: none;"> In the genus expansion of the HOMFLY polynomials their representation dependence is naturally captured by symmetric group characters. This immediately implies that the Ooguri-Vafa partition function (OVPF) is a Hurwitz tau-function. In the planar limit involving factorizable special polynomials, it is actually a trivial exponential tau-function. In fact, in the double scaling Kashaev limit (the one associated with the volume conjecture) dominant in the genus expansion are terms associated with the symmetric representations and with the integrability preserving Casimir operators, though we stop one step from converting this fact into a clear statement about the OVPF behavior in the vicinity of q=1. Instead, we explain that the genus expansion provides a hierarchical decomposition of the Hurwitz tau-function, similar to the Takasaki-Takebe expansion of the KP tau-functions. This analogy can be helpful to develop a substitute for the universal Grassmannian description in the Hurwitz tau-functions. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1304.7499v1-abstract-full').style.display = 'none'; document.getElementById('1304.7499v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 April, 2013; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2013. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">8 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> FIAN/TD-07/13; ITEP/TH-11/13 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> The European Physical Journal C 73 (2013) 2492 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1303.1015">arXiv:1303.1015</a> <span> [<a href="https://arxiv.org/pdf/1303.1015">pdf</a>, <a href="https://arxiv.org/ps/1303.1015">ps</a>, <a href="https://arxiv.org/format/1303.1015">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s11232-013-0115-0">10.1007/s11232-013-0115-0 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Genus expansion of HOMFLY polynomials </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Mironov%2C+A">A. Mironov</a>, <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">A. Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Sleptsov%2C+A">A. Sleptsov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1303.1015v1-abstract-short" style="display: inline;"> In the planar limit of the 't Hooft expansion, the Wilson-loop average in 3d Chern-Simons theory (i.e. the HOMFLY polynomial) depends in a very simple way on representation (the Young diagram), so that the (knot-dependent) Ooguri-Vafa partition function becomes a trivial KP tau-function. We study higher genus corrections to this formula in the form of expansion in powers of z = q-q^{-1}. Expansion… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1303.1015v1-abstract-full').style.display = 'inline'; document.getElementById('1303.1015v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1303.1015v1-abstract-full" style="display: none;"> In the planar limit of the 't Hooft expansion, the Wilson-loop average in 3d Chern-Simons theory (i.e. the HOMFLY polynomial) depends in a very simple way on representation (the Young diagram), so that the (knot-dependent) Ooguri-Vafa partition function becomes a trivial KP tau-function. We study higher genus corrections to this formula in the form of expansion in powers of z = q-q^{-1}. Expansion coefficients are expressed through the eigenvalues of the cut-and-join operators, i.e. symmetric group characters. Moreover, the z-expansion is naturally exponentiated. Representation through cut-and-join operators makes contact with Hurwitz theory and its sophisticated integrability properties. Our formulas describe the shape of genus expansion for the HOMFLY polynomials, which for their matrix model counterparts is usually controlled by Virasoro like constraints and AMM/EO topological recursion. The genus expansion differs from the better studied weak coupling expansion at finite number of colors N, which is described in terms of the Vassiliev invariants and Kontsevich integral. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1303.1015v1-abstract-full').style.display = 'none'; document.getElementById('1303.1015v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 5 March, 2013; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 2013. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">34 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> FIAN/TD-05/13; ITEP/TH-04/13 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Theor.Math.Phys. 177 (2013) 1435-1470 (Teor.Mat.Fiz. 177 (2013) 179-221) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1208.2324">arXiv:1208.2324</a> <span> [<a href="https://arxiv.org/pdf/1208.2324">pdf</a>, <a href="https://arxiv.org/ps/1208.2324">ps</a>, <a href="https://arxiv.org/format/1208.2324">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.nuclphysb.2013.03.008">10.1016/j.nuclphysb.2013.03.008 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> NSR superstring measures in genus 5 </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Dunin-Barkowski%2C+P">Petr Dunin-Barkowski</a>, <a href="/search/hep-th?searchtype=author&query=Sleptsov%2C+A">Alexey Sleptsov</a>, <a href="/search/hep-th?searchtype=author&query=Stern%2C+A">Abel Stern</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1208.2324v5-abstract-short" style="display: inline;"> Currently there are two proposed ansatze for NSR superstring measures: the Grushevsky ansatz and the OPSMY ansatz, which for genera g<=4 are known to coincide. However, neither the Grushevsky nor the OPSMY ansatz leads to a vanishing two point function in genus four, which can be constructed from the genus five expressions for the respective ansatze. This is inconsistent with the known properties… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1208.2324v5-abstract-full').style.display = 'inline'; document.getElementById('1208.2324v5-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1208.2324v5-abstract-full" style="display: none;"> Currently there are two proposed ansatze for NSR superstring measures: the Grushevsky ansatz and the OPSMY ansatz, which for genera g<=4 are known to coincide. However, neither the Grushevsky nor the OPSMY ansatz leads to a vanishing two point function in genus four, which can be constructed from the genus five expressions for the respective ansatze. This is inconsistent with the known properties of superstring amplitudes. In the present paper we show that the Grushevsky and OPSMY ansatze do not coincide in genus five. Then, by combining these ansatze, we propose a new ansatz for genus five, which now leads to a vanishing two-point function in genus four. We also show that one cannot construct an ansatz from the currently known forms in genus 6 that satisfies all known requirements for superstring measures. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1208.2324v5-abstract-full').style.display = 'none'; document.getElementById('1208.2324v5-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 12 March, 2013; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 11 August, 2012; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2012. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">several corrections</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> ITEP-TH-40/12 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Nucl.Phys.B 872 (2013), pp. 106-126 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1201.3339">arXiv:1201.3339</a> <span> [<a href="https://arxiv.org/pdf/1201.3339">pdf</a>, <a href="https://arxiv.org/ps/1201.3339">ps</a>, <a href="https://arxiv.org/format/1201.3339">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/JHEP05(2012)070">10.1007/JHEP05(2012)070 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Interplay between MacDonald and Hall-Littlewood expansions of extended torus superpolynomials </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Mironov%2C+A">A. Mironov</a>, <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">A. Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Shakirov%2C+S">Sh. Shakirov</a>, <a href="/search/hep-th?searchtype=author&query=Sleptsov%2C+A">A. Sleptsov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1201.3339v2-abstract-short" style="display: inline;"> In arXiv:1106.4305 extended superpolynomials were introduced for the torus links T[m,mk+r], which are functions on the entire space of time variables and, at expense of reducing the topological invariance, possess additional algebraic properties, resembling those of the matrix model partition functions and the KP/Toda tau-functions. Not surprisingly, being a suitable extension it actually allows o… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1201.3339v2-abstract-full').style.display = 'inline'; document.getElementById('1201.3339v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1201.3339v2-abstract-full" style="display: none;"> In arXiv:1106.4305 extended superpolynomials were introduced for the torus links T[m,mk+r], which are functions on the entire space of time variables and, at expense of reducing the topological invariance, possess additional algebraic properties, resembling those of the matrix model partition functions and the KP/Toda tau-functions. Not surprisingly, being a suitable extension it actually allows one to calculate the superpolynomials. These functions are defined as expansions into MacDonald polynomials, and their dependence on k is entirely captured by the action of the cut-and-join operator, like in the HOMFLY case. We suggest a simple description of the coefficients in these character expansions, by expanding the initial (at k=0) conditions for the k-evolution into the new auxiliary basis, this time provided by the Hall-Littlewood polynomials, which, hence, play a role in the description of the dual m-evolution. For illustration we list manifest expressions for a few first series, mk\pm 1, mk\pm 2, mk\pm 3, mk\pm 4. Actually all formulas were explicitly tested up to m=17 strands in the braid. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1201.3339v2-abstract-full').style.display = 'none'; document.getElementById('1201.3339v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 17 February, 2012; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 16 January, 2012; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2012. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">8 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> FIAN/TD-22/11; ITEP/TH-01/12 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> JHEP 2012 (2012) 70 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1201.0025">arXiv:1201.0025</a> <span> [<a href="https://arxiv.org/pdf/1201.0025">pdf</a>, <a href="https://arxiv.org/format/1201.0025">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1088/1751-8113/45/38/385204">10.1088/1751-8113/45/38/385204 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Explicit computation of Drinfeld associator in the case of the fundamental representation of gl(N) </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Dunin-Barkowski%2C+P">Petr Dunin-Barkowski</a>, <a href="/search/hep-th?searchtype=author&query=Sleptsov%2C+A">Alexey Sleptsov</a>, <a href="/search/hep-th?searchtype=author&query=Smirnov%2C+A">Andrey Smirnov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1201.0025v3-abstract-short" style="display: inline;"> We solve the regularized Knizhnik-Zamolodchikov equation and find an explicit expression for the Drinfeld associator. We restrict to the case of the fundamental representation of $gl(N)$. Several tests of the results are presented. It can be explicitly seen that components of this solution for the associator coincide with certain components of WZW conformal block for primary fields. We introduce t… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1201.0025v3-abstract-full').style.display = 'inline'; document.getElementById('1201.0025v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1201.0025v3-abstract-full" style="display: none;"> We solve the regularized Knizhnik-Zamolodchikov equation and find an explicit expression for the Drinfeld associator. We restrict to the case of the fundamental representation of $gl(N)$. Several tests of the results are presented. It can be explicitly seen that components of this solution for the associator coincide with certain components of WZW conformal block for primary fields. We introduce the symmetrized version of the Drinfeld associator by dropping the odd terms. The symmetrized associator gives the same knot invariants, but has a simpler structure and is fully characterized by one symmetric function which we call the Drinfeld prepotential. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1201.0025v3-abstract-full').style.display = 'none'; document.getElementById('1201.0025v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 1 September, 2012; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 29 December, 2011; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2012. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">14 pages, 2 figures; several flaws indicated by referees corrected</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> ITEP/TH-64/11 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> J. Phys. A: Math. Theor. 45 385204 (2012) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1112.5406">arXiv:1112.5406</a> <span> [<a href="https://arxiv.org/pdf/1112.5406">pdf</a>, <a href="https://arxiv.org/format/1112.5406">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Combinatorics">math.CO</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1142/S0217751X13300251">10.1142/S0217751X13300251 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Kontsevich integral for knots and Vassiliev invariants </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Dunin-Barkowski%2C+P">Petr Dunin-Barkowski</a>, <a href="/search/hep-th?searchtype=author&query=Sleptsov%2C+A">Alexey Sleptsov</a>, <a href="/search/hep-th?searchtype=author&query=Smirnov%2C+A">Andrey Smirnov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1112.5406v3-abstract-short" style="display: inline;"> We review quantum field theory approach to the knot theory. Using holomorphic gauge we obtain the Kontsevich integral. It is explained how to calculate Vassiliev invariants and coefficients in Kontsevich integral in a combinatorial way which can be programmed on a computer. We discuss experimental results and temporal gauge considerations which lead to representation of Vassiliev invariants in ter… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1112.5406v3-abstract-full').style.display = 'inline'; document.getElementById('1112.5406v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1112.5406v3-abstract-full" style="display: none;"> We review quantum field theory approach to the knot theory. Using holomorphic gauge we obtain the Kontsevich integral. It is explained how to calculate Vassiliev invariants and coefficients in Kontsevich integral in a combinatorial way which can be programmed on a computer. We discuss experimental results and temporal gauge considerations which lead to representation of Vassiliev invariants in terms of arrow diagrams. Explicit examples and computational results are presented. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1112.5406v3-abstract-full').style.display = 'none'; document.getElementById('1112.5406v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 21 January, 2012; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 22 December, 2011; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2011. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">25 pages, 17 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> ITEP/TH-63/11 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Int. J. Mod. Phys. A 28, 1330025 (2013) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1106.4305">arXiv:1106.4305</a> <span> [<a href="https://arxiv.org/pdf/1106.4305">pdf</a>, <a href="https://arxiv.org/ps/1106.4305">ps</a>, <a href="https://arxiv.org/format/1106.4305">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Geometric Topology">math.GT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/JHEP03(2013)021">10.1007/JHEP03(2013)021 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Superpolynomials for toric knots from evolution induced by cut-and-join operators </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Dunin-Barkowski%2C+P">P. Dunin-Barkowski</a>, <a href="/search/hep-th?searchtype=author&query=Mironov%2C+A">A. Mironov</a>, <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">A. Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Sleptsov%2C+A">A. Sleptsov</a>, <a href="/search/hep-th?searchtype=author&query=Smirnov%2C+A">A. Smirnov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1106.4305v4-abstract-short" style="display: inline;"> The colored HOMFLY polynomials, which describe Wilson loop averages in Chern-Simons theory, possess an especially simple representation for torus knots, which begins from quantum R-matrix and ends up with a trivially-looking split W representation familiar from character calculus applications to matrix models and Hurwitz theory. Substitution of MacDonald polynomials for characters in these formula… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1106.4305v4-abstract-full').style.display = 'inline'; document.getElementById('1106.4305v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1106.4305v4-abstract-full" style="display: none;"> The colored HOMFLY polynomials, which describe Wilson loop averages in Chern-Simons theory, possess an especially simple representation for torus knots, which begins from quantum R-matrix and ends up with a trivially-looking split W representation familiar from character calculus applications to matrix models and Hurwitz theory. Substitution of MacDonald polynomials for characters in these formulas provides a very simple description of "superpolynomials", much simpler than the recently studied alternative which deforms relation to the WZNW theory and explicitly involves the Littlewood-Richardson coefficients. A lot of explicit expressions are presented for different representations (Young diagrams), many of them new. In particular, we provide the superpolynomial P_[1]^[m,km\pm 1] for arbitrary m and k. The procedure is not restricted to the fundamental (all antisymmetric) representations and the torus knots, still in these cases some subtleties persist. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1106.4305v4-abstract-full').style.display = 'none'; document.getElementById('1106.4305v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 2 December, 2016; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 21 June, 2011; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2011. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">23 pages + Tables (51 pages)</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> FIAN/TD-10/11; ITEP/TH-21/11 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> JHEP 03 (2013) 021 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1006.4322">arXiv:1006.4322</a> <span> [<a href="https://arxiv.org/pdf/1006.4322">pdf</a>, <a href="https://arxiv.org/format/1006.4322">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1016/j.difgeo.2015.01.007">10.1016/j.difgeo.2015.01.007 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> On the Homology of Certain Smooth Covers of Moduli Spaces of Algebraic Curves </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Dunin-Barkowski%2C+P">Petr Dunin-Barkowski</a>, <a href="/search/hep-th?searchtype=author&query=Popolitov%2C+A">Alexander Popolitov</a>, <a href="/search/hep-th?searchtype=author&query=Shabat%2C+G">George Shabat</a>, <a href="/search/hep-th?searchtype=author&query=Sleptsov%2C+A">Alexei Sleptsov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1006.4322v3-abstract-short" style="display: inline;"> We suggest a general method of computation of the homology of certain smooth covers $\hat{\mathcal{M}}_{g,1}(\mathbb{C})$ of moduli spaces $\mathcal{M}_{g,1}\br{\mathbb{C}}$ of pointed curves of genus $g$. Namely, we consider moduli spaces of algebraic curves with level $m$ structures. The method is based on the lifting of the Strebel-Penner stratification $\mathcal{M}_{g,1}\br{\mathbb{C}}$. We ap… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1006.4322v3-abstract-full').style.display = 'inline'; document.getElementById('1006.4322v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1006.4322v3-abstract-full" style="display: none;"> We suggest a general method of computation of the homology of certain smooth covers $\hat{\mathcal{M}}_{g,1}(\mathbb{C})$ of moduli spaces $\mathcal{M}_{g,1}\br{\mathbb{C}}$ of pointed curves of genus $g$. Namely, we consider moduli spaces of algebraic curves with level $m$ structures. The method is based on the lifting of the Strebel-Penner stratification $\mathcal{M}_{g,1}\br{\mathbb{C}}$. We apply this method for $g\leq 2$ and obtain Betti numbers; these results are consistent with Penner and Harer-Zagier results on Euler characteristics. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1006.4322v3-abstract-full').style.display = 'none'; document.getElementById('1006.4322v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 27 December, 2011; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 22 June, 2010; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2010. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">15 pages, 12 figures; proof added, corrections</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> ITEP/TH-97/09 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Differential Geometry and its Applications, Volume 40 (2015), Pages 86-102 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/0908.2113">arXiv:0908.2113</a> <span> [<a href="https://arxiv.org/pdf/0908.2113">pdf</a>, <a href="https://arxiv.org/ps/0908.2113">ps</a>, <a href="https://arxiv.org/format/0908.2113">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Algebraic Geometry">math.AG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Number Theory">math.NT</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1088/1126-6708/2009/10/072">10.1088/1126-6708/2009/10/072 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Lattice Theta Constants vs Riemann Theta Constants and NSR Superstring Measures </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Dunin-Barkowski%2C+P">P. Dunin-Barkowski</a>, <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">A. Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Sleptsov%2C+A">A. Sleptsov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="0908.2113v3-abstract-short" style="display: inline;"> We discuss relations between two different representations of hypothetical holomorphic NSR measures, based on two different ways of constructing the semi-modular forms of weight 8. One of these ways is to build forms from the ordinary Riemann theta constants and another -- from the lattice theta constants. We discuss unexpectedly elegant relations between lattice theta constants, corresponding t… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0908.2113v3-abstract-full').style.display = 'inline'; document.getElementById('0908.2113v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="0908.2113v3-abstract-full" style="display: none;"> We discuss relations between two different representations of hypothetical holomorphic NSR measures, based on two different ways of constructing the semi-modular forms of weight 8. One of these ways is to build forms from the ordinary Riemann theta constants and another -- from the lattice theta constants. We discuss unexpectedly elegant relations between lattice theta constants, corresponding to 16-dimensional self-dual lattices, and Riemann theta constants and present explicit formulae expressing the former ones through the latter. Starting from genus 5 the modular-form approach to construction of NSR measures runs into serious problems and there is a risk that it fails completely already at genus 6. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0908.2113v3-abstract-full').style.display = 'none'; document.getElementById('0908.2113v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 October, 2009; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 14 August, 2009; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2009. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">16 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Report number:</span> ITEP/TH-35/09 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> JHEP 0910:072,2009 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/0801.4293">arXiv:0801.4293</a> <span> [<a href="https://arxiv.org/pdf/0801.4293">pdf</a>, <a href="https://arxiv.org/ps/0801.4293">ps</a>, <a href="https://arxiv.org/format/0801.4293">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="High Energy Physics - Theory">hep-th</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s11232-009-0005-7">10.1007/s11232-009-0005-7 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Geometric Hamiltonian Formalism for Reparametrization Invariant Theories with Higher Derivatives </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Dunin-Barkowski%2C+P">Petr Dunin-Barkowski</a>, <a href="/search/hep-th?searchtype=author&query=Sleptsov%2C+A">Alexei Sleptsov</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="0801.4293v2-abstract-short" style="display: inline;"> Reparametrization invariant Lagrangian theories with higher derivatives are considered. We investigate the geometric structures behind these theories and construct the Hamiltonian formalism in a geometric way. The Legendre transformation which corresponds to the transition from the Lagrangian formalism to the Hamiltonian formalism is non-trivial in this case. The resulting phase bundle, i.e. the… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0801.4293v2-abstract-full').style.display = 'inline'; document.getElementById('0801.4293v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="0801.4293v2-abstract-full" style="display: none;"> Reparametrization invariant Lagrangian theories with higher derivatives are considered. We investigate the geometric structures behind these theories and construct the Hamiltonian formalism in a geometric way. The Legendre transformation which corresponds to the transition from the Lagrangian formalism to the Hamiltonian formalism is non-trivial in this case. The resulting phase bundle, i.e. the image of the Legendre transformation, is a submanifold of some cotangent bundle. We show that in our construction it is always odd-dimensional. Therefore the canonical symplectic two-form from the ambient cotangent bundle generates on the phase bundle a field of the null-directions of its restriction. It is shown that the integral lines of this field project directly to the extremals of the action on the configuration manifold. Therefore this naturally arising field is what is called the Hamilton field. We also express the corresponding Hamilton equations through the generilized Nambu bracket. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0801.4293v2-abstract-full').style.display = 'none'; document.getElementById('0801.4293v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 19 February, 2008; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 28 January, 2008; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2008. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">19 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Theor.Math.Phys.158:61-81, 2009 </p> </li> </ol> <div class="is-hidden-tablet">  <span class="help" style="display: inline-block;"><a href="https://github.com/arXiv/arxiv-search/releases">Search v0.5.6 released 2020-02-24</a>  </span> </div> </div> </main> <footer> <div class="columns is-desktop" role="navigation" aria-label="Secondary">  <div class="column"> <div class="columns"> <div class="column"> <ul class="nav-spaced"> <li><a href="https://info.arxiv.org/about">About</a></li> <li><a href="https://info.arxiv.org/help">Help</a></li> </ul> </div> <div class="column"> <ul class="nav-spaced"> <li> <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon filter-black" role="presentation"><title>contact arXiv</title><desc>Click here to contact arXiv</desc><path d="M502.3 190.8c3.9-3.1 9.7-.2 9.7 4.7V400c0 26.5-21.5 48-48 48H48c-26.5 0-48-21.5-48-48V195.6c0-5 5.7-7.8 9.7-4.7 22.4 17.4 52.1 39.5 154.1 113.6 21.1 15.4 56.7 47.8 92.2 47.6 35.7.3 72-32.8 92.3-47.6 102-74.1 131.6-96.3 154-113.7zM256 320c23.2.4 56.6-29.2 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