Quantum modular forms

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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

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

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


  • 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

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

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

Kontsevich's strange function

  • Bryson-Ono-Pitman-Rhoades


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

related items

computational resource