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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/2407.04810">arXiv:2407.04810</a> <span> [<a href="https://arxiv.org/pdf/2407.04810">pdf</a>, <a href="https://arxiv.org/ps/2407.04810">ps</a>, <a href="https://arxiv.org/format/2407.04810">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> <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/10.21468/SciPostPhys.17.4.119">10.21468/SciPostPhys.17.4.119 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Supersymmetric polynomials and algebro-combinatorial duality </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Galakhov%2C+D">Dmitry Galakhov</a>, <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">Alexei Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Tselousov%2C+N">Nikita 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="2407.04810v2-abstract-short" style="display: inline;"> In this note we develop a systematic combinatorial definition for constructed earlier supersymmetric polynomial families. These polynomial families generalize canonical Schur, Jack and Macdonald families so that the new polynomials depend on odd Grassmann variables as well. Members of these families are labeled by respective modifications of Young diagrams. We show that the super-Macdonald polynom… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.04810v2-abstract-full').style.display = 'inline'; document.getElementById('2407.04810v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.04810v2-abstract-full" style="display: none;"> In this note we develop a systematic combinatorial definition for constructed earlier supersymmetric polynomial families. These polynomial families generalize canonical Schur, Jack and Macdonald families so that the new polynomials depend on odd Grassmann variables as well. Members of these families are labeled by respective modifications of Young diagrams. We show that the super-Macdonald polynomials form a representation of a super-algebra analog $\mathsf{T}(\widehat{\mathfrak{gl}}_{1|1})$ of Ding-Ioahara-Miki (quantum toroidal) algebra, emerging as a BPS algebra of D-branes on a conifold. A supersymmetric modification for Young tableaux and Kostka numbers are also discussed. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.04810v2-abstract-full').style.display = 'none'; document.getElementById('2407.04810v2-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 September, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 5 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 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">26 pages, 2 figures, v2: typos corrected, references added</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> SciPost Phys. 17, 119 (2024) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2407.03301">arXiv:2407.03301</a> <span> [<a href="https://arxiv.org/pdf/2407.03301">pdf</a>, <a href="https://arxiv.org/ps/2407.03301">ps</a>, <a href="https://arxiv.org/format/2407.03301">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.2024.138911">10.1016/j.physletb.2024.138911 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Macdonald polynomials for super-partitions </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Galakhov%2C+D">Dmitry Galakhov</a>, <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">Alexei Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Tselousov%2C+N">Nikita 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="2407.03301v1-abstract-short" style="display: inline;"> We introduce generalization of famous Macdonald polynomials for the case of super-Young diagrams that contain half-boxes on the equal footing with full boxes. These super-Macdonald polynomials are polynomials of extended set of variables: usual $p_k$ variables are accompanied by anti-commuting Grassmann variables $胃_k$. Starting from recently defined super-Schur polynomials and exploiting orthogon… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.03301v1-abstract-full').style.display = 'inline'; document.getElementById('2407.03301v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2407.03301v1-abstract-full" style="display: none;"> We introduce generalization of famous Macdonald polynomials for the case of super-Young diagrams that contain half-boxes on the equal footing with full boxes. These super-Macdonald polynomials are polynomials of extended set of variables: usual $p_k$ variables are accompanied by anti-commuting Grassmann variables $胃_k$. Starting from recently defined super-Schur polynomials and exploiting orthogonality relations with triangular decompositions we are able to fully determine super-Macdonald polynomials. These new polynomials have similar properties to canonical Macdonald polynomials -- they respect two different orderings in the set of (super)-Young diagrams simultaneously. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2407.03301v1-abstract-full').style.display = 'none'; document.getElementById('2407.03301v1-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 July, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2406.20074">arXiv:2406.20074</a> <span> [<a href="https://arxiv.org/pdf/2406.20074">pdf</a>, <a href="https://arxiv.org/ps/2406.20074">ps</a>, <a href="https://arxiv.org/format/2406.20074">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/10.1007/JHEP08(2024)209">10.1007/JHEP08(2024)209 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Algorithms for representations of quiver Yangian algebras </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Galakhov%2C+D">Dmitry Galakhov</a>, <a href="/search/hep-th?searchtype=author&query=Gavshin%2C+A">Alexei Gavshin</a>, <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">Alexei Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Tselousov%2C+N">Nikita 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="2406.20074v1-abstract-short" style="display: inline;"> In this note, we aim to review algorithms for constructing crystal representations of quiver Yangians in detail. Quiver Yangians are believed to describe an action of the BPS algebra on BPS states in systems of D-branes wrapping toric Calabi-Yau three-folds. Crystal modules of these algebras originate from molten crystal models for Donaldson-Thomas invariants of respective three-folds. Despite the… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.20074v1-abstract-full').style.display = 'inline'; document.getElementById('2406.20074v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2406.20074v1-abstract-full" style="display: none;"> In this note, we aim to review algorithms for constructing crystal representations of quiver Yangians in detail. Quiver Yangians are believed to describe an action of the BPS algebra on BPS states in systems of D-branes wrapping toric Calabi-Yau three-folds. Crystal modules of these algebras originate from molten crystal models for Donaldson-Thomas invariants of respective three-folds. Despite the fact that this subject was originally at the crossroads of algebraic geometry with effective supersymmetric field theories, equivariant toric action simplifies applied calculations drastically. So the sole pre-requisite for this algorithm's implementation is linear algebra. It can be easily taught to a machine with the help of any symbolic calculation system. Moreover, these algorithms may be generalized to toroidal and elliptic algebras and exploited in various numerical experiments with those algebras. We illustrate the work of the algorithms in applications to simple cases of $\mathsf{Y}(\mathfrak{sl}_2)$, $\mathsf{Y}(\widehat{\mathfrak{gl}}_{1})$ and $\mathsf{Y}(\widehat{\mathfrak{gl}}_{1|1})$. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2406.20074v1-abstract-full').style.display = 'none'; document.getElementById('2406.20074v1-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 June, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 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">41 pages, 9 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> JHEP 08 (2024) 209 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2403.14600">arXiv:2403.14600</a> <span> [<a href="https://arxiv.org/pdf/2403.14600">pdf</a>, <a href="https://arxiv.org/ps/2403.14600">ps</a>, <a href="https://arxiv.org/format/2403.14600">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="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/JHEP05(2024)118">10.1007/JHEP05(2024)118 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Wall-Crossing Effects on Quiver BPS Algebras </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Galakhov%2C+D">Dmitry Galakhov</a>, <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">Alexei Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Tselousov%2C+N">Nikita 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="2403.14600v2-abstract-short" style="display: inline;"> BPS states in supersymmetric theories can admit additional algebro-geometric structures in their spectra, described as quiver Yangian algebras. Equivariant fixed points on the quiver variety are interpreted as vectors populating a representation module, and matrix elements for the generators are then defined as Duistermaat-Heckman integrals in the vicinity of these points. The well-known wall-cros… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.14600v2-abstract-full').style.display = 'inline'; document.getElementById('2403.14600v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2403.14600v2-abstract-full" style="display: none;"> BPS states in supersymmetric theories can admit additional algebro-geometric structures in their spectra, described as quiver Yangian algebras. Equivariant fixed points on the quiver variety are interpreted as vectors populating a representation module, and matrix elements for the generators are then defined as Duistermaat-Heckman integrals in the vicinity of these points. The well-known wall-crossing phenomena are that the fixed point spectrum establishes a dependence on the stability (Fayet-Illiopolous) parameters $味$, jumping abruptly across the walls of marginal stability, which divide the $味$-space into a collection of stability chambers -- ``phases'' of the theory. The standard construction of the quiver Yangian algebra relies heavily on the molten crystal model, valid in a sole cyclic chamber where all the $味$-parameters have the same sign. We propose to lift this restriction and investigate the effects of the wall-crossing phenomena on the quiver Yangian algebra and its representations -- starting with the example of affine super-Yangian $\mathsf{Y}(\widehat{\mathfrak{gl}}_{1|1})$. In addition to the molten crystal construction more general atomic structures appear, in other non-cyclic phases (chambers of the $味$-space). We call them glasses and also divide in a few different classes. For some of the new phases we manage to associate an algebraic structure again as a representation of the same affine Yangian $\mathsf{Y}(\widehat{\mathfrak{gl}}_{1|1})$. This observation supports an earlier conjecture that the BPS algebraic structures can be considered as new wall-crossing invariants. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2403.14600v2-abstract-full').style.display = 'none'; document.getElementById('2403.14600v2-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> 15 April, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 21 March, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> March 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">36 pages, 7 figures, minor corrections, references added</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> JHEP05(2024)118 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2402.05920">arXiv:2402.05920</a> <span> [<a href="https://arxiv.org/pdf/2402.05920">pdf</a>, <a href="https://arxiv.org/ps/2402.05920">ps</a>, <a href="https://arxiv.org/format/2402.05920">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/10.1140/epjc/s10052-024-12952-x">10.1140/epjc/s10052-024-12952-x <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Simple Representations of BPS Algebras: the case of $Y(\widehat{\mathfrak{gl}}_2)$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Galakhov%2C+D">Dmitry Galakhov</a>, <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">Alexei Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Tselousov%2C+N">Nikita 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="2402.05920v2-abstract-short" style="display: inline;"> BPS algebras are the symmetries of a wide class of brane-inspired models. They are closely related to Yangians -- the peculiar and somewhat sophisticated limit of DIM algebras. Still they possess some simple and explicit representations. We explain here that for $Y(\widehat{\mathfrak{gl}}_r)$ these representations are related to Uglov polynomials, whose families are also labeled by natural $r$. Th… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.05920v2-abstract-full').style.display = 'inline'; document.getElementById('2402.05920v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2402.05920v2-abstract-full" style="display: none;"> BPS algebras are the symmetries of a wide class of brane-inspired models. They are closely related to Yangians -- the peculiar and somewhat sophisticated limit of DIM algebras. Still they possess some simple and explicit representations. We explain here that for $Y(\widehat{\mathfrak{gl}}_r)$ these representations are related to Uglov polynomials, whose families are also labeled by natural $r$. They arise in the limit $\hbar\longrightarrow 0$ from Macdonald polynomials, and generalize the well-known Jack polynomials ($尾$-deformation of Schur functions), associated with $r=1$. For $r=2$ they approximate Macdonald polynomials with the accuracy $O(\hbar^2)$, so that they are eigenfunctions of {\it two} immediately available commuting operators, arising from the $\hbar$-expansion of the first Macdonald Hamiltonian. These operators have a clear structure, which is easily generalizable, -- what provides a technically simple way to build an explicit representation of Yangian $Y(\widehat{\mathfrak{gl}}_2)$, where $U^{(2)}$ are associated with the states $|位\rangle$, parametrized by chess-colored Young diagrams. An interesting feature of this representation is that the odd time-variables $p_{2n+1}$ can be expressed through mutually commuting operators from Yangian, however even time-variables $p_{2n}$ are inexpressible. Implications to higher $r$ become now straightforward, yet we describe them only in a sketchy way. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2402.05920v2-abstract-full').style.display = 'none'; document.getElementById('2402.05920v2-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 February, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 8 February, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2024. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2311.00760">arXiv:2311.00760</a> <span> [<a href="https://arxiv.org/pdf/2311.00760">pdf</a>, <a href="https://arxiv.org/ps/2311.00760">ps</a>, <a href="https://arxiv.org/format/2311.00760">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="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/10.1103/PhysRevD.109.066001">10.1103/PhysRevD.109.066001 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Towards the theory of Yangians </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Galakhov%2C+D">Dmitry Galakhov</a>, <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">Alexei Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Tselousov%2C+N">Nikita 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="2311.00760v1-abstract-short" style="display: inline;"> We review the main ideas underlying the emerging theory of Yangians -- the new type of hidden symmetry in string-inspired models. Their classification by quivers is a far-going generalization of simple Lie algebras classification by Dynkin diagrams. However, this is still a kind of project, while a more constructive approach goes through toric Calabi-Yau spaces, related supersymmetric systems and… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.00760v1-abstract-full').style.display = 'inline'; document.getElementById('2311.00760v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2311.00760v1-abstract-full" style="display: none;"> We review the main ideas underlying the emerging theory of Yangians -- the new type of hidden symmetry in string-inspired models. Their classification by quivers is a far-going generalization of simple Lie algebras classification by Dynkin diagrams. However, this is still a kind of project, while a more constructive approach goes through toric Calabi-Yau spaces, related supersymmetric systems and the Duistermaat-Heckmann/equivariant integrals between the fixed points in the ADHM-like moduli spaces. These fixed points are classified by crystals (Young-type diagrams) and Yangian generators describe ``instanton'' transitions between them. Detailed examples will be presented elsewhere. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2311.00760v1-abstract-full').style.display = 'none'; document.getElementById('2311.00760v1-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 November, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Phys. Rev. D 109, 066001 (2024) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2307.03150">arXiv:2307.03150</a> <span> [<a href="https://arxiv.org/pdf/2307.03150">pdf</a>, <a href="https://arxiv.org/ps/2307.03150">ps</a>, <a href="https://arxiv.org/format/2307.03150">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="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/JHEP08(2023)049">10.1007/JHEP08(2023)049 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Super-Schur Polynomials for Affine Super Yangian $\mathsf{Y}(\widehat{\mathfrak{gl}}_{1|1})$ </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Galakhov%2C+D">Dmitry Galakhov</a>, <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">Alexei Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Tselousov%2C+N">Nikita 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="2307.03150v2-abstract-short" style="display: inline;"> We explicitly construct cut-and-join operators and their eigenfunctions -- the Super-Schur functions -- for the case of the affine super-Yangian $\mathsf{Y}(\widehat{\mathfrak{gl}}_{1|1})$. This is the simplest non-trivial (semi-Fock) representation, where eigenfunctions are labeled by the superanalogue of 2d Young diagrams, and depend on the supertime variables $(p_k,胃_k)$. The action of other ge… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2307.03150v2-abstract-full').style.display = 'inline'; document.getElementById('2307.03150v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2307.03150v2-abstract-full" style="display: none;"> We explicitly construct cut-and-join operators and their eigenfunctions -- the Super-Schur functions -- for the case of the affine super-Yangian $\mathsf{Y}(\widehat{\mathfrak{gl}}_{1|1})$. This is the simplest non-trivial (semi-Fock) representation, where eigenfunctions are labeled by the superanalogue of 2d Young diagrams, and depend on the supertime variables $(p_k,胃_k)$. The action of other generators on diagrams is described by the analogue of the Pieri rule. As well we present generalizations of the hook formula for the measure on super-Young diagrams and of the Cauchy formula. Also a discussion of string theory origins for these relations is provided. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2307.03150v2-abstract-full').style.display = 'none'; document.getElementById('2307.03150v2-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, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 6 July, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> July 2023. </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">27 pages, 3 figures</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> JHEP08(2023)049 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2305.12282">arXiv:2305.12282</a> <span> [<a href="https://arxiv.org/pdf/2305.12282">pdf</a>, <a href="https://arxiv.org/ps/2305.12282">ps</a>, <a href="https://arxiv.org/format/2305.12282">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/10.1007/JHEP11(2023)165">10.1007/JHEP11(2023)165 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> 3-Schurs from explicit representation of Yangian $Y(\hat{\mathfrak{gl}}_1)$. Levels 1-5 </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">A. Morozov</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="2305.12282v1-abstract-short" style="display: inline;"> We suggest an ansatz for representation of affine Yangian $Y(\hat{ \mathfrak{gl}}_1)$ by differential operators in the triangular set of time-variables ${\bf P}_{a,i}$ with $1\leqslant i\leqslant a$, which saturates the MacMahon formula for the number of $3d$ Young diagrams/plane partitions. In this approach the 3-Schur polynomials are defined as the common eigenfunctions of an infinite set of com… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2305.12282v1-abstract-full').style.display = 'inline'; document.getElementById('2305.12282v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2305.12282v1-abstract-full" style="display: none;"> We suggest an ansatz for representation of affine Yangian $Y(\hat{ \mathfrak{gl}}_1)$ by differential operators in the triangular set of time-variables ${\bf P}_{a,i}$ with $1\leqslant i\leqslant a$, which saturates the MacMahon formula for the number of $3d$ Young diagrams/plane partitions. In this approach the 3-Schur polynomials are defined as the common eigenfunctions of an infinite set of commuting "cut-and-join" generators $蠄_n$ of the Yangian. We manage to push this tedious program through to the 3-Schur polynomials of level 5, and this provides a rather big sample set, which can be now investigated by other methods. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2305.12282v1-abstract-full').style.display = 'none'; document.getElementById('2305.12282v1-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> 20 May, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2023. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2211.14956">arXiv:2211.14956</a> <span> [<a href="https://arxiv.org/pdf/2211.14956">pdf</a>, <a href="https://arxiv.org/ps/2211.14956">ps</a>, <a href="https://arxiv.org/format/2211.14956">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/10.1016/j.physletb.2023.137887">10.1016/j.physletb.2023.137887 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Hunt for 3-Schur polynomials </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">A. Morozov</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="2211.14956v2-abstract-short" style="display: inline;"> This paper describes our attempt to understand the recent success of Na Wang in constructing the 3-Schur polynomials, associated with the plane partitions. We provide a rather detailed review and try to figure out the new insights, which allowed to overcome the problems of the previous efforts. In result we provide a very simple definition of time-variables ${\bf P}_{i\geqslant j}$ and the cut-and… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2211.14956v2-abstract-full').style.display = 'inline'; document.getElementById('2211.14956v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2211.14956v2-abstract-full" style="display: none;"> This paper describes our attempt to understand the recent success of Na Wang in constructing the 3-Schur polynomials, associated with the plane partitions. We provide a rather detailed review and try to figure out the new insights, which allowed to overcome the problems of the previous efforts. In result we provide a very simple definition of time-variables ${\bf P}_{i\geqslant j}$ and the cut-and-join operator $\hat W_2$, which generates the set of $3$-Schur functions. Some coefficients in $\hat W_2$ remain undefined and require more effort to be fixed. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2211.14956v2-abstract-full').style.display = 'none'; document.getElementById('2211.14956v2-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> 20 May, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 27 November, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2022. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2205.12238">arXiv:2205.12238</a> <span> [<a href="https://arxiv.org/pdf/2205.12238">pdf</a>, <a href="https://arxiv.org/ps/2205.12238">ps</a>, <a href="https://arxiv.org/format/2205.12238">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/10.1140/epjc/s10052-022-10851-7">10.1140/epjc/s10052-022-10851-7 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Differential Expansion for antiparallel triple pretzels: the way the factorization is deformed </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">A. Morozov</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="2205.12238v1-abstract-short" style="display: inline;"> For a peculiar family of double braid knots there is a remarkable factorization formula for the coefficients of the differential (cyclotomic) expansion (DE), which nowadays is widely used to construct the exclusive Racah matrices $S$ and $\bar S$ in arbitrary representations. The origins of the factorization remain obscure and the special role of double braids remains a mystery. In an attempt to b… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2205.12238v1-abstract-full').style.display = 'inline'; document.getElementById('2205.12238v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2205.12238v1-abstract-full" style="display: none;"> For a peculiar family of double braid knots there is a remarkable factorization formula for the coefficients of the differential (cyclotomic) expansion (DE), which nowadays is widely used to construct the exclusive Racah matrices $S$ and $\bar S$ in arbitrary representations. The origins of the factorization remain obscure and the special role of double braids remains a mystery. In an attempt to broaden the perspective, we extend the family of double braids to antiparallel triple pretzels, which are obtained by the defect-preserving deformation from the trefoil and all have defect zero. It turns out that factorization of DE coefficients is violated quite strongly, still remains described by an elegant formula, at least for all symmetric representations. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2205.12238v1-abstract-full').style.display = 'none'; document.getElementById('2205.12238v1-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 May, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2022. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2204.05977">arXiv:2204.05977</a> <span> [<a href="https://arxiv.org/pdf/2204.05977">pdf</a>, <a href="https://arxiv.org/ps/2204.05977">ps</a>, <a href="https://arxiv.org/format/2204.05977">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/10.1140/epjc/s10052-022-10705-2">10.1140/epjc/s10052-022-10705-2 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Evolution properties of the knot's defect </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">A. Morozov</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="2204.05977v1-abstract-short" style="display: inline;"> The defect of differential (cyclotomic) expansion for colored HOMFLY-PT polynomials is conjectured to be invariant under any antiparallel evolution and change linearly with the evolution in any parallel direction. In other words, each ${\cal R}$-matrix can be substituted by an entire 2-strand braid in two different ways: the defect remains intact when the braid is antiparallel and changes by half… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2204.05977v1-abstract-full').style.display = 'inline'; document.getElementById('2204.05977v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2204.05977v1-abstract-full" style="display: none;"> The defect of differential (cyclotomic) expansion for colored HOMFLY-PT polynomials is conjectured to be invariant under any antiparallel evolution and change linearly with the evolution in any parallel direction. In other words, each ${\cal R}$-matrix can be substituted by an entire 2-strand braid in two different ways: the defect remains intact when the braid is antiparallel and changes by half of the added length when the braid is parallel. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2204.05977v1-abstract-full').style.display = 'none'; document.getElementById('2204.05977v1-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 April, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2022. </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2202.11683">arXiv:2202.11683</a> <span> [<a href="https://arxiv.org/pdf/2202.11683">pdf</a>, <a href="https://arxiv.org/ps/2202.11683">ps</a>, <a href="https://arxiv.org/format/2202.11683">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="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.2022.137193">10.1016/j.physletb.2022.137193 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Polynomial representations of classical Lie algebras and flag varieties </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">A. Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Reva%2C+M">M. Reva</a>, <a href="/search/hep-th?searchtype=author&query=Tselousov%2C+N">N. Tselousov</a>, <a href="/search/hep-th?searchtype=author&query=Zenkevich%2C+Y">Y. Zenkevich</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="2202.11683v3-abstract-short" style="display: inline;"> Recently we have started a program to describe the action of Lie algebras associated with Dynkin-type diagrams on generic Verma modules in terms of polynomial vector fields. In this paper we explain that the results for the classical ABCD series of Lie algebras coincide with the more conventional approach, based on the knowledge of the entire algebra, not only the simple roots. We apply the coset… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2202.11683v3-abstract-full').style.display = 'inline'; document.getElementById('2202.11683v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2202.11683v3-abstract-full" style="display: none;"> Recently we have started a program to describe the action of Lie algebras associated with Dynkin-type diagrams on generic Verma modules in terms of polynomial vector fields. In this paper we explain that the results for the classical ABCD series of Lie algebras coincide with the more conventional approach, based on the knowledge of the entire algebra, not only the simple roots. We apply the coset description, starting with a large representation and then reducing it with the help of the algebra, commuting with the original one. The irreducible representations are then obtained by gauge fixing this residual symmetry. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2202.11683v3-abstract-full').style.display = 'none'; document.getElementById('2202.11683v3-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 May, 2022; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 23 February, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2022. </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/2106.03638">arXiv:2106.03638</a> <span> [<a href="https://arxiv.org/pdf/2106.03638">pdf</a>, <a href="https://arxiv.org/format/2106.03638">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/10.1140/epjc/s10052-021-09676-7">10.1140/epjc/s10052-021-09676-7 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Irreducible representations of simple Lie algebras by differential operators </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">A. Morozov</a>, <a href="/search/hep-th?searchtype=author&query=Reva%2C+M">M. Reva</a>, <a href="/search/hep-th?searchtype=author&query=Tselousov%2C+N">N. Tselousov</a>, <a href="/search/hep-th?searchtype=author&query=Zenkevich%2C+Y">Y. Zenkevich</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="2106.03638v1-abstract-short" style="display: inline;"> We describe a systematic method to construct arbitrary highest-weight modules, including arbitrary finite-dimensional representations, for any finite dimensional simple Lie algebra $\mathfrak{g}$. The Lie algebra generators are represented as first order differential operators in $\frac{1}{2} \left(\dim \mathfrak{g} - \text{rank} \, \mathfrak{g}\right) $ variables. All rising generators ${\bf e}$… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2106.03638v1-abstract-full').style.display = 'inline'; document.getElementById('2106.03638v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2106.03638v1-abstract-full" style="display: none;"> We describe a systematic method to construct arbitrary highest-weight modules, including arbitrary finite-dimensional representations, for any finite dimensional simple Lie algebra $\mathfrak{g}$. The Lie algebra generators are represented as first order differential operators in $\frac{1}{2} \left(\dim \mathfrak{g} - \text{rank} \, \mathfrak{g}\right) $ variables. All rising generators ${\bf e}$ are universal in the sense that they do not depend on representation, the weights enter (in a very simple way) only in the expressions for the lowering operators ${\bf f}$. We present explicit formulas of this kind for the simple root generators of all classical Lie algebras. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2106.03638v1-abstract-full').style.display = 'none'; document.getElementById('2106.03638v1-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 June, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2021. </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 (2021) 10, 898 </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/2010.02165">arXiv:2010.02165</a> <span> [<a href="https://arxiv.org/pdf/2010.02165">pdf</a>, <a href="https://arxiv.org/format/2010.02165">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/10.1140/epjc/s10052-020-08745-7">10.1140/epjc/s10052-020-08745-7 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Are Maxwell knots integrable? </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/hep-th?searchtype=author&query=Morozov%2C+A">A. Morozov</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="2010.02165v2-abstract-short" style="display: inline;"> We review properties of the null-field solutions of source-free Maxwell equations. We focus on the electric and magnetic field lines, especially on limit cycles, which actually can be knotted and/or linked at every given moment. We analyse the fact that the Poynting vector induces self-consistent time evolution of these lines and demonstrate that the Abelian link invariant is integral of motion. T… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2010.02165v2-abstract-full').style.display = 'inline'; document.getElementById('2010.02165v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2010.02165v2-abstract-full" style="display: none;"> We review properties of the null-field solutions of source-free Maxwell equations. We focus on the electric and magnetic field lines, especially on limit cycles, which actually can be knotted and/or linked at every given moment. We analyse the fact that the Poynting vector induces self-consistent time evolution of these lines and demonstrate that the Abelian link invariant is integral of motion. The same is expected to be true also for the non-Abelian invariants (like Jones and HOMFLY-PT polynomials or Vassiliev invariants), and many integrals of motion can imply that the Poynting evolution is actually integrable. We also consider particular examples of the field lines for the particular family of finite energy source-free "knot" solutions, attempting to understand when the field lines are closed -- and can be discussed in terms of knots and links. Based on computer simulations we conjecture that Ranada's solution, where every pair of lines forms a Hopf link, is rather exceptional. In general, only particular lines (a set of measure zero) are limit cycles and represent closed lines forming knots/links, while all the rest are twisting around them and remain unclosed. Still, conservation laws of Poynting evolution and associated integrable structure should persist. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2010.02165v2-abstract-full').style.display = 'none'; document.getElementById('2010.02165v2-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 December, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 5 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> The European Physical Journal C volume 80, Article number: 1182 (2020) </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/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> </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" 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