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href="/nlab/show/automorphic+representation">automorphic representation</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Jacobi+theta+function">Jacobi theta function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Riemann+theta+function">Riemann theta function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dedekind+eta+function">Dedekind eta function</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/Mellin+transform">Mellin transform</a></strong></p> <p><strong><a class="existingWikiWord" href="/nlab/show/zeta+function">zeta function</a></strong>, <strong><a class="existingWikiWord" href="/nlab/show/L-function">L-function</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Riemann+zeta+function">Riemann zeta function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dirichlet+L-function">Dirichlet L-function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Artin+L-function">Artin L-function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/automorphic+L-function">automorphic L-function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Ruelle+zeta+function">Ruelle zeta function</a>, <a class="existingWikiWord" href="/nlab/show/Selberg+zeta+function">Selberg zeta function</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/eta+function">eta function</a></strong></p> <p><strong>in physics</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+blocks">conformal blocks</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/zeta+function+regularization">zeta function regularization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vacuum+energy">vacuum energy</a></p> </li> </ul></div></div> <h4 id="elliptic_cohomology">Elliptic cohomology</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/elliptic+cohomology">elliptic cohomology</a>, <a class="existingWikiWord" href="/nlab/show/tmf">tmf</a>, <a class="existingWikiWord" href="/nlab/show/string+theory">string theory</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/complex+oriented+cohomology+theory">complex oriented</a><a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> of <a class="existingWikiWord" href="/nlab/show/chromatic+level">chromatic level</a> 2</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+curve">elliptic curve</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/supersingular+elliptic+curve">supersingular elliptic curve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+elliptic+curve">derived elliptic curve</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/moduli+stack+of+elliptic+curves">moduli stack of elliptic curves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/modular+form">modular form</a>, <a class="existingWikiWord" href="/nlab/show/Jacobi+form">Jacobi form</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Eisenstein+series">Eisenstein series</a>, <a class="existingWikiWord" href="/nlab/show/j-invariant">j-invariant</a>, <a class="existingWikiWord" href="/nlab/show/Weierstrass+sigma-function">Weierstrass sigma-function</a>, <a class="existingWikiWord" href="/nlab/show/Dedekind+eta+function">Dedekind eta function</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/elliptic+genus">elliptic genus</a>, <a class="existingWikiWord" href="/nlab/show/Witten+genus">Witten genus</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+modular+form">topological modular form</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+orientation+of+tmf">string orientation of tmf</a></li> </ul> </li> </ul></div></div> <h4 id="complex_geometry">Complex geometry</h4> <div class="hide"><div> <p><a class="existingWikiWord" href="/nlab/show/geometry">geometry</a>, <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a>, <a class="existingWikiWord" href="/nlab/show/complex+line">complex line</a></p> <p><strong><a class="existingWikiWord" href="/nlab/show/complex+geometry">complex geometry</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+manifold">complex manifold</a>, <a class="existingWikiWord" href="/nlab/show/complex+structure">complex structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+analytic+space">complex analytic space</a></p> </li> </ul> <h3 id="variants">Variants</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+complex+geometry">generalized complex geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+supermanifold">complex supermanifold</a></p> </li> </ul> <h3 id="structures">Structures</h3> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Dolbeault+complex">Dolbeault complex</a>, <a class="existingWikiWord" href="/nlab/show/holomorphic+de+Rham+complex">holomorphic de Rham complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge+structure">Hodge structure</a>, <a class="existingWikiWord" href="/nlab/show/Hodge+filtration">Hodge filtration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hodge-filtered+differential+cohomology">Hodge-filtered differential cohomology</a></p> </li> </ul> <h3 id="examples">Examples</h3> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">dim = 1</annotation></semantics></math>: <a class="existingWikiWord" href="/nlab/show/Riemann+surface">Riemann surface</a>, <a class="existingWikiWord" href="/nlab/show/super+Riemann+surface">super Riemann surface</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Calabi-Yau+manifold">Calabi-Yau manifold</a></p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">dim = 2</annotation></semantics></math>: <a class="existingWikiWord" href="/nlab/show/K3+surface">K3 surface</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+Calabi-Yau+manifold">generalized Calabi-Yau manifold</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#as_functional_determinant_of_laplace_operator_on_elliptic_curve'>As functional determinant of Laplace operator on elliptic curve</a></li> </ul> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A <a class="existingWikiWord" href="/nlab/show/modular+form">modular form</a>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/Jacobi+theta+function">Jacobi theta function</a> for special values of its arguments…</p> <h2 id="definition">Definition</h2> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>q</mi> <mrow><mn>1</mn><mo stretchy="false">/</mo><mn>24</mn></mrow></msup><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow> <mn>∞</mn></munderover><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><msup><mi>q</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \eta(\tau) = q^{1/24} \prod_{n=1}^\infty (1-q^n) </annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>≔</mo><msup><mi>e</mi> <mrow><mn>2</mn><mi>π</mi><mi>i</mi><mi>τ</mi></mrow></msup></mrow><annotation encoding="application/x-tex">q \coloneqq e^{2\pi i \tau}</annotation></semantics></math>.</p> <h2 id="properties">Properties</h2> <h3 id="as_functional_determinant_of_laplace_operator_on_elliptic_curve">As functional determinant of Laplace operator on elliptic curve</h3> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℂ</mi><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mi>ℤ</mi><mo>⊕</mo><mi>τ</mi><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{C}/(\mathbb{Z}\oplus \tau \mathbb{Z})</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/complex+torus">complex torus</a> (complex <a class="existingWikiWord" href="/nlab/show/elliptic+curve">elliptic curve</a>) equipped with its standard flat <a class="existingWikiWord" href="/nlab/show/Riemannian+metric">Riemannian metric</a>, then the <a class="existingWikiWord" href="/nlab/show/zeta+function+of+an+elliptic+differential+operator">zeta function</a> of the corresponding <a class="existingWikiWord" href="/nlab/show/Laplace+operator">Laplace operator</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi></mrow><annotation encoding="application/x-tex">\Delta</annotation></semantics></math> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ζ</mi> <mi>Δ</mi></msub><mo>=</mo><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>2</mn><mi>s</mi></mrow></msup><mi>E</mi><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo>≔</mo><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>2</mn><mi>s</mi></mrow></msup><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>l</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>ℤ</mi><mo>×</mo><mi>ℤ</mi><mo>−</mo><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow></munder><mfrac><mn>1</mn><mrow><msup><mrow><mo stretchy="false">|</mo><mi>k</mi><mo>+</mo><mi>τ</mi><mi>l</mi><mo stretchy="false">|</mo></mrow> <mrow><mn>2</mn><mi>s</mi></mrow></msup></mrow></mfrac><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \zeta_{\Delta} = (2\pi)^{-2 s} E(s) \coloneqq (2\pi)^{-2 s} \underset{(k,l)\in \mathbb{Z}\times\mathbb{Z}-(0,0)}{\sum} \frac{1}{{\vert k +\tau l\vert}^{2s}} \,. </annotation></semantics></math></div> <p>The corresponding <a class="existingWikiWord" href="/nlab/show/functional+determinant">functional determinant</a> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>exp</mi><mo stretchy="false">(</mo><msubsup><mi>E</mi> <mi>Δ</mi> <mstyle scriptlevel="0"><mo>′</mo></mstyle></msubsup><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>Im</mi><mi>τ</mi><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><msup><mrow><mo stretchy="false">|</mo><mi>η</mi><mo stretchy="false">(</mo><mi>τ</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo></mrow> <mn>4</mn></msup><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \exp( E^\prime_{\Delta}(0) ) = (Im \tau)^2 {\vert \eta(\tau)\vert}^4 \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/Dedekind+eta+function">Dedekind eta function</a>.</p> <p>(recalled e.g. in <a href="#Todorov03">Todorov 03, page 3</a>)</p> <p>This kind of expression appears as the <a class="existingWikiWord" href="/nlab/show/partition+function">partition function</a> of the <a class="existingWikiWord" href="/nlab/show/bosonic+string">bosonic string</a> (e.g. section 6.4.2 in these lectures: <a href="http://www.damtp.cam.ac.uk/user/tong/string/six.pdf">pdf</a>)</p> <h2 id="references">References</h2> <ul> <li> <p>Wikipedia, <em><a href="http://en.wikipedia.org/wiki/Dedekind_eta_function">Dedekind eta function</a></em></p> </li> <li id="Atiyah87"> <p><a class="existingWikiWord" href="/nlab/show/Michael+Atiyah">Michael Atiyah</a>, <em>The logarithm of the Dedekind <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math>-function</em>, Math. Ann. 278, 335-380 (1987) (<a href="http://www.maths.ed.ac.uk/~aar/papers/atiyahlg.pdf">pdf</a>)</p> </li> </ul> <p>See also</p> <ul> <li id="Todorov03"><a class="existingWikiWord" href="/nlab/show/Andrey+Todorov">Andrey Todorov</a>, <em>The analogue of the Dedekind eta function for CY threefolds</em>, 2003 <a href="http://www.ma.huji.ac.il/conf/crelle.pdf">pdf</a></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on July 18, 2015 at 08:19:51. 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