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
- Chern-Simons gauge theory and Witten's invariant
- WRT (Witten-Reshetikhin-Turaev) invariant
- $W: \{\text{root of unity}\} \to \mathbb{C}$ defined by
$$ W(q)=\frac{1}{2G}\sum_{\beta \pmod {60K}} \frac{(1-\zeta^{24\beta})(1-\zeta^{40\beta})}{1-\zeta^{60\beta}}\zeta^{-(\beta+1)^2} $$ where $\zeta$ satisfies $\zeta^{120}=q$ and $G=\sum_{\beta}\zeta^{-\beta^2}$
- $q$ is a root of unity of order $K$
theorem by Ma-Rhoades
- for every root of unity
$$ W(q)=1-Q(q)=\sum_{n=1}^{\infty}q^{n}(q^n;q)_n $$
- moreover,
$$ Q(q^{-1}) = \begin{cases} \phi_0(-q), & \text{if $q$ is a root of unity of odd order}\\ 1-F_0(q), & \text{if $q$ is a root of unity of even order} \end{cases} $$ where $\phi_0$ and $F_0$ are two of Ramanujan's fifth order mock theta function
computational resource
articles
- 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.