"Quantum modular forms"의 두 판 사이의 차이

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imported>Pythagoras0
1번째 줄: 1번째 줄:
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==example==
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==Kontsevich's strange function==
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* unimodular generating function
+
* definition
 +
$$
 +
F(q)=\sum_{n=0}^{\infty}(q)_n
 +
$$
 +
* 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
 +
$$
 +
\phi(x)=e^{\pi i x /12}F(e^{2\pi i x})
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$$
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$\phi : \mathbb{Q} \to \mathbb{C}$ satisfies
 +
$$
 +
\phi(-x)+(-ix)^{-3/2}\phi(1/x)=g(x)
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$$
 +
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}
 
U(w;q)=\sum_{n=0}^{\infty}(wq;q)_{n}(w^{-1}q;q)_{n}q^{n+1}
23번째 줄: 52번째 줄:
 
\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}
 
\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})$$
 +
  
 +
 +
==$\sigma$ and $\sigma^{*}$==
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* $\sigma(q)=2\sum_{n=0}^{\infty}(-1)^n (q;q)_n$
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* (Cohen) $\sigma(q)=-\sigma^{*}(q^{-1})$ for every root of unity
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* 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==
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* [[Chern-Simons gauge theory and Witten's invariant]]
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* $W: \{\text{root of unity}\} \to \mathbb{C}$ 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 $G=\sum_{\beta}\zeta^{-\beta^2}$
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* $q$ is a root of unity of order $K$
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===theorem by Ma-Rhoades===
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* for every root of unity
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$$
 +
W(q)=1-Q(q)=\sum_{n=1}^{\infty}q^{n}(q^n;q)_n
 +
$$
 +
* moreover,
 +
$$
 +
Q(q^{-1}) =
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\begin{cases}
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\phi_0(-q), & \text{if $q$ is a root of unity of odd order}\\
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1-F_0(q), & \text{if $q$ is a root of unity of even order}
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\end{cases}
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$$
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where $\phi_0$ and $F_0$ are two of Ramanujan's fifth order mock theta function
  
  
29번째 줄: 96번째 줄:
 
==related items==
 
==related items==
 
* [[quantum dilogarithm]]
 
* [[quantum dilogarithm]]
 +
* [[Chern-Simons gauge theory and Witten's invariant]]
 +
  
  

2013년 7월 27일 (토) 20:27 판

Kontsevich's strange function

  • definition

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

  • 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

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

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

Kontsevich's strange function

  • Bryson-Ono-Pitman-Rhoades

$$U(q)=F(q^{-1})$$


$\sigma$ 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

$$ W(q)=\frac{1}{2G}\sum_{\beta \pmod 60K} \frac{(1-\zeta^{24\beta})(1-\zeta^{40\beta})}{1-\zeta^{60\beta}}\zeta^{-(\beta+1)^2} $$ 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


related items


computational resource


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