Quantum modular forms

Kontsevich's strange function

• definition

\[ F(q)=\sum_{n=0}^{\infty}(q)_n \]

• originated from quantum invariants of trefoil knot
• if \(F(x)=F(e^{2\pi i x})\), then

\[ \zeta_{24k}^{-1}F(\frac{-1}{k})\sim \sqrt{-i}k^{3/2}\zeta_{24}^kF(k)+g(k) \]

• theorem (Zagier)

Let \[ \phi(x)=e^{\pi i x /12}F(e^{2\pi i x}) \] \(\phi : \mathbb{Q} \to \mathbb{C}\) satisfies \[ \phi(-x)+(-ix)^{-3/2}\phi(1/x)=g(x) \] where \(g:\mathbb{R}\to \mathbb{C}\) is a \(C^{\infty}\) function

• Strange identity

\[ F(q^{-1})=-\frac{1}{2}\sum_{n=1}^{\infty}n \left(\frac{12}{n}\right)q^{-\frac{n^2-1}{24}} \] with \(q=e^{2\pi i x}\)

• related to the partial theta function \(\tilde{\eta}(q)\)

generating function of unimodal sequences

• generating function of unimodal sequences

\[ U(w;q)=\sum_{n=0}^{\infty}(wq;q)_{n}(w^{-1}q;q)_{n}q^{n+1} \]

• Dyson's rank generating function

\[R(w;q)=\sum_{n=0}^\infty \frac{q^{n^2}}{(wq;q)_n(w^{-1}q;q)_n}\]

• Andrews-Garvan crank modular form

\[C(w;q)=\frac{(q)_{\infty}}{(wq;q)_{\infty}(w^{-1}q;q)_{\infty}}\]

• limit formula \(\zeta_b=e^{2\pi i/b}\), \(1\le a <b\), for every root of unity \(\zeta\), there exists an integer \(c\) such that

\[ \lim_{q\to \zeta} R(\zeta_{b}^{a};q)-\zeta_{b^2}^{c} C(\zeta_{b}^{a};q)=-(1-\zeta_{b}^{a})(1-\zeta_{b}^{-a})U(\zeta_{b}^{a};\zeta) \]

special case

• If \(b=2\) and \(a=1\), then \(\zeta_{b}^{a}=-1\)
• \(U(-1;\zeta)\) becomes a finite sum if \(\zeta\) is a root of unity

\[ U(-1;\zeta)=\sum_{n=0}^{k-1} (1+\zeta)^2(1+\zeta^2)^2\cdots (1+\zeta^n)^2\zeta^{n+1} \]

• \(R(-1;q)=f(q)\) and \(C(-1;q)=b(q)\) in 3rd order mock theta functions
• Thus if \(\zeta\) be even \(2k\) order root of unity

\[ \lim_{q\to \zeta} f(q)-(-1)^k b(q)=-4\sum_{n=0}^{k-1} (1+\zeta)^2(1+\zeta^2)^2\cdots (1+\zeta^n)^2\zeta^{n+1} \]

Kontsevich's strange function

• Bryson-Ono-Pitman-Rhoades

\[U(q)=F(q^{-1})\]

non-holomorphic modular form

• thm (Andrews-Rhoades-Zwegers)

\[ q^{-1/24}U(q)+\int +\int \] is a non-holomorphic modular form of weight 3/2

\(\sigma\) and \(\sigma^{*}\)

• \(\sigma(q)=2\sum_{n=0}^{\infty}(-1)^n (q;q)_n\)
• (Cohen) \(\sigma(q)=-\sigma^{*}(q^{-1})\) for every root of unity
• let \(f(x)=q^{1/24}\sigma(q)\) where \(q=e^{2\pi i x}\)
• (Lewis-Zagier) \(f : \mathbb{Q} \to \mathbb{C}\) satisfies

\[ \frac{1}{2x+1}f(\frac{x}{2x+1})=e^{\pi i/12}f(x)+h(x) \] where \(h\) is \(C^{\infty}\) on \(\mathbb{R}\) and real analytic except at \(x=-1/2\)

WRT invariant of the Poincare sphere

• see WRT (Witten-Reshetikhin-Turaev) invariant

related items

• quantum dilogarithm
• Chern-Simons gauge theory and Witten's invariant

computational resource

• https://docs.google.com/file/d/0B8XXo8Tve1cxWmFPWkZTMVdBeDA/edit

expositions

• Folsom, https://docs.google.com/file/d/0B8XXo8Tve1cxRGhqVHR2YWV2OEk/edit

articles

• Kathrin Bringmann, Jeremy Lovejoy, Larry Rolen, On some special families of \(q\)-hypergeometric Maass forms, http://arxiv.org/abs/1603.01783v1
• Dimofte, Tudor, and Stavros Garoufalidis. “Quantum Modularity and Complex Chern-Simons Theory.” arXiv:1511.05628 [hep-Th], November 17, 2015. http://arxiv.org/abs/1511.05628.
• Bringmann, Kathrin, and Larry Rolen. “Half-Integral Weight Eichler Integrals and Quantum Modular Forms.” arXiv:1409.3781 [math], September 12, 2014. http://arxiv.org/abs/1409.3781.
• Rolen, Larry, and Robert P. Schneider. 2013. “A ‘Strange’ Vector-Valued Quantum Modular Form.” arXiv:1304.1210 (April 3). http://arxiv.org/abs/1304.1210.
• Bryson, Jennifer, Ken Ono, Sarah Pitman, and Robert C. Rhoades. 2012. “Unimodal Sequences and Quantum and Mock Modular Forms.” Proceedings of the National Academy of Sciences 109 (40) (October 2): 16063–16067. doi:10.1073/pnas.1211964109.
• Zagier, Don. 2010. “Quantum Modular Forms.” In Quanta of Maths, 11:659–675. Clay Math. Proc. Providence, RI: Amer. Math. Soc. http://www.ams.org/mathscinet-getitem?mr=2757599.
  • http://people.mpim-bonn.mpg.de/zagier/files/qmf/fulltext.pdf