"Quantum modular forms"의 두 판 사이의 차이

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1번째 줄: 1번째 줄:
 
==Kontsevich's strange function==
 
==Kontsevich's strange function==
 
* definition
 
* definition
−
$$
+
:<math>
 
F(q)=\sum_{n=0}^{\infty}(q)_n
 
F(q)=\sum_{n=0}^{\infty}(q)_n
−
$$
+
</math>
 
* originated from quantum invariants of trefoil knot
 
* originated from quantum invariants of trefoil knot
−
* if $F(x)=F(e^{2\pi i x})$, then
+
* if <math>F(x)=F(e^{2\pi i x})</math>, then
−
$$
+
:<math>
 
\zeta_{24k}^{-1}F(\frac{-1}{k})\sim \sqrt{-i}k^{3/2}\zeta_{24}^kF(k)+g(k)
 
\zeta_{24k}^{-1}F(\frac{-1}{k})\sim \sqrt{-i}k^{3/2}\zeta_{24}^kF(k)+g(k)
−
$$
+
</math>
 
* theorem (Zagier)
 
* theorem (Zagier)
 
Let  
 
Let  
−
$$
+
:<math>
 
\phi(x)=e^{\pi i x /12}F(e^{2\pi i x})
 
\phi(x)=e^{\pi i x /12}F(e^{2\pi i x})
−
$$
+
</math>
−
$\phi : \mathbb{Q} \to \mathbb{C}$ satisfies
+
<math>\phi : \mathbb{Q} \to \mathbb{C}</math> satisfies
−
$$
+
:<math>
 
\phi(-x)+(-ix)^{-3/2}\phi(1/x)=g(x)
 
\phi(-x)+(-ix)^{-3/2}\phi(1/x)=g(x)
−
$$
+
</math>
−
where $g:\mathbb{R}\to \mathbb{C}$ is a $C^{\infty}$ function
+
where <math>g:\mathbb{R}\to \mathbb{C}</math> is a <math>C^{\infty}</math> function
 
* Strange identity
 
* Strange identity
−
$$
+
:<math>
 
F(q^{-1})=-\frac{1}{2}\sum_{n=1}^{\infty}n \left(\frac{12}{n}\right)q^{-\frac{n^2-1}{24}}
 
F(q^{-1})=-\frac{1}{2}\sum_{n=1}^{\infty}n \left(\frac{12}{n}\right)q^{-\frac{n^2-1}{24}}
−
$$
+
</math>
−
with $q=e^{2\pi i x}$
+
with <math>q=e^{2\pi i x}</math>
−
* related to the partial theta function $\tilde{\eta}(q)$
+
* related to the partial theta function <math>\tilde{\eta}(q)</math>
  
  
30번째 줄: 30번째 줄:
 
==generating function of unimodal sequences==
 
==generating function of unimodal sequences==
 
* generating function of unimodal sequences
 
* generating function of unimodal sequences
−
$$
+
:<math>
 
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}
−
$$
+
</math>
 
* [[Dyson's rank generating function]]
 
* [[Dyson's rank generating function]]
−
$$R(w;q)=\sum_{n=0}^\infty \frac{q^{n^2}}{(wq;q)_n(w^{-1}q;q)_n}$$
+
:<math>R(w;q)=\sum_{n=0}^\infty \frac{q^{n^2}}{(wq;q)_n(w^{-1}q;q)_n}</math>
 
* [[Andrews-Garvan crank modular form]]  
 
* [[Andrews-Garvan crank modular form]]  
−
$$C(w;q)=\frac{(q)_{\infty}}{(wq;q)_{\infty}(w^{-1}q;q)_{\infty}}$$
+
:<math>C(w;q)=\frac{(q)_{\infty}}{(wq;q)_{\infty}(w^{-1}q;q)_{\infty}}</math>
−
* 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
+
* limit formula <math>\zeta_b=e^{2\pi i/b}</math>, <math>1\le a <b</math>, for every root of unity <math>\zeta</math>, there exists an integer <math>c</math> such that
−
$$
+
:<math>
 
\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)
 
\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)
−
$$
+
</math>
 
===special case===
 
===special case===
−
* If $b=2$ and  $a=1$, then $\zeta_{b}^{a}=-1$
+
* If <math>b=2</math> and  <math>a=1</math>, then <math>\zeta_{b}^{a}=-1</math>
−
* $U(-1;\zeta)$ becomes a finite sum if $\zeta$ is a root of unity
+
* <math>U(-1;\zeta)</math> becomes a finite sum if <math>\zeta</math> is a root of unity
−
$$
+
:<math>
 
U(-1;\zeta)=\sum_{n=0}^{k-1} (1+\zeta)^2(1+\zeta^2)^2\cdots (1+\zeta^n)^2\zeta^{n+1}
 
U(-1;\zeta)=\sum_{n=0}^{k-1} (1+\zeta)^2(1+\zeta^2)^2\cdots (1+\zeta^n)^2\zeta^{n+1}
−
$$
+
</math>
−
* $R(-1;q)=f(q)$ and $C(-1;q)=b(q)$ in [[3rd order mock theta functions]]
+
* <math>R(-1;q)=f(q)</math> and <math>C(-1;q)=b(q)</math> in [[3rd order mock theta functions]]
−
* Thus if $\zeta$ be even $2k$ order root of unity
+
* Thus if <math>\zeta</math> be even <math>2k</math> order root of unity
−
$$
+
:<math>
 
\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}
−
$$
+
</math>
 
===Kontsevich's strange function===
 
===Kontsevich's strange function===
 
* Bryson-Ono-Pitman-Rhoades
 
* Bryson-Ono-Pitman-Rhoades
−
$$U(q)=F(q^{-1})$$
+
:<math>U(q)=F(q^{-1})</math>
 
===non-holomorphic modular form===
 
===non-holomorphic modular form===
 
* thm (Andrews-Rhoades-Zwegers)
 
* thm (Andrews-Rhoades-Zwegers)
−
$$
+
:<math>
 
q^{-1/24}U(q)+\int +\int
 
q^{-1/24}U(q)+\int +\int
−
$$
+
</math>
 
is a non-holomorphic modular form of weight 3/2
 
is a non-holomorphic modular form of weight 3/2
  
  
  
−
==$\sigma$ and $\sigma^{*}$==
+
==<math>\sigma</math> and <math>\sigma^{*}</math>==
−
* $\sigma(q)=2\sum_{n=0}^{\infty}(-1)^n (q;q)_n$
+
* <math>\sigma(q)=2\sum_{n=0}^{\infty}(-1)^n (q;q)_n</math>
−
* (Cohen) $\sigma(q)=-\sigma^{*}(q^{-1})$ for every root of unity
+
* (Cohen) <math>\sigma(q)=-\sigma^{*}(q^{-1})</math> for every root of unity
−
* let $f(x)=q^{1/24}\sigma(q)$ where $q=e^{2\pi i x}$
+
* let <math>f(x)=q^{1/24}\sigma(q)</math> where <math>q=e^{2\pi i x}</math>
−
* (Lewis-Zagier) $f : \mathbb{Q} \to \mathbb{C}$ satisfies
+
* (Lewis-Zagier) <math>f : \mathbb{Q} \to \mathbb{C}</math> satisfies
−
$$
+
:<math>
 
\frac{1}{2x+1}f(\frac{x}{2x+1})=e^{\pi i/12}f(x)+h(x)
 
\frac{1}{2x+1}f(\frac{x}{2x+1})=e^{\pi i/12}f(x)+h(x)
−
$$
+
</math>
−
where $h$ is $C^{\infty}$ on $\mathbb{R}$ and real analytic except at $x=-1/2$
+
where <math>h</math> is <math>C^{\infty}</math> on <math>\mathbb{R}</math> and real analytic except at <math>x=-1/2</math>
  
  
95번째 줄: 95번째 줄:
  
 
==articles==
 
==articles==
 +
* Kathrin Bringmann, Jeremy Lovejoy, Larry Rolen, On some special families of <math>q</math>-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.
 
* 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.
 
* 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.
103번째 줄: 105번째 줄:
  
 
[[분류:Mock modular forms]]
 
[[분류:Mock modular forms]]
 +
[[분류:TQFT]]
 +
[[분류:migrate]]

2020년 11월 16일 (월) 04:34 기준 최신판

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

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

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

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


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