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
\[ U(w;q)=\sum_{n=0}^{\infty}(wq;q)_{n}(w^{-1}q;q)_{n}q^{n+1} \]
\[R(w;q)=\sum_{n=0}^\infty \frac{q^{n^2}}{(wq;q)_n(w^{-1}q;q)_n}\]
\[C(w;q)=\frac{(q)_{\infty}}{(wq;q)_{\infty}(w^{-1}q;q)_{\infty}}\]
- 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
computational resource
expositions
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.