Quantum modular forms - 편집 역사

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@@ 1번째 줄: / 1번째 줄: @@
 ==Kontsevich's strange function==
 * definition
-$$
+:<math>
 F(q)=\sum_{n=0}^{\infty}(q)_n
-$$
+</math>
 * 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
-$$
+:<math>
 \zeta_{24k}^{-1}F(\frac{-1}{k})\sim \sqrt{-i}k^{3/2}\zeta_{24}^kF(k)+g(k)
-$$
+</math>
 * theorem (Zagier)
 Let
-$$
+:<math>
 \phi(x)=e^{\pi i x /12}F(e^{2\pi i x})
-$$
+</math>
-$\phi : \mathbb{Q} \to \mathbb{C}$ satisfies
+<math>\phi : \mathbb{Q} \to \mathbb{C}</math> satisfies
-$$
+:<math>
 \phi(-x)+(-ix)^{-3/2}\phi(1/x)=g(x)
-$$
+</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
-$$
+:<math>
 F(q^{-1})=-\frac{1}{2}\sum_{n=1}^{\infty}n \left(\frac{12}{n}\right)q^{-\frac{n^2-1}{24}}
-$$
+</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
-$$
+:<math>
 U(w;q)=\sum_{n=0}^{\infty}(wq;q)_{n}(w^{-1}q;q)_{n}q^{n+1}
-$$
+</math>
 * [[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]]
-$$C(w;q)=\frac{(q)_{\infty}}{(wq;q)_{\infty}(w^{-1}q;q)_{\infty}}$$
+:<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
-$$
+:<math>
 \lim_{q\to \zeta} R(\zeta_{b}^{a};q)-\zeta_{b^2}^{c} C(\zeta_{b}^{a};q)=-(1-\zeta_{b}^{a})(1-\zeta_{b}^{-a})U(\zeta_{b}^{a};\zeta)
-$$
+</math>
 ===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
-$$
+:<math>
 U(-1;\zeta)=\sum_{n=0}^{k-1} (1+\zeta)^2(1+\zeta^2)^2\cdots (1+\zeta^n)^2\zeta^{n+1}
-$$
+</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
-$$
+:<math>
 \lim_{q\to \zeta} f(q)-(-1)^k b(q)=-4\sum_{n=0}^{k-1} (1+\zeta)^2(1+\zeta^2)^2\cdots (1+\zeta^n)^2\zeta^{n+1}
-$$
+</math>
 ===Kontsevich's strange function===
 * Bryson-Ono-Pitman-Rhoades
-$$U(q)=F(q^{-1})$$
+:<math>U(q)=F(q^{-1})</math>
 ===non-holomorphic modular form===
 * thm (Andrews-Rhoades-Zwegers)
-$$
+:<math>
 q^{-1/24}U(q)+\int +\int
-$$
+</math>
 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
-$$
+:<math>
 \frac{1}{2x+1}f(\frac{x}{2x+1})=e^{\pi i/12}f(x)+h(x)
-$$
+</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==
-* Kathrin Bringmann, Jeremy Lovejoy, Larry Rolen, On some special families of $q$-hypergeometric Maass forms, http://arxiv.org/abs/1603.01783v1
+* 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.

← 이전 판		2020년 11월 14일 (토) 02:25 판
106번째 줄:		106번째 줄:
	[[분류:Mock modular forms]]		[[분류:Mock modular forms]]
	[[분류:TQFT]]		[[분류:TQFT]]
		+	[[분류:migrate]]

@@ 95번째 줄: / 95번째 줄: @@
 ==articles==
 * 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.

← 이전 판		2015년 11월 19일 (목) 09:08 판
104번째 줄:		104번째 줄:

	[[분류:Mock modular forms]]		[[분류:Mock modular forms]]
		+	[[분류:TQFT]]

@@ 95번째 줄: / 95번째 줄: @@
 ==articles==
 * 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.

← 이전 판		2013년 12월 20일 (금) 18:09 판
88번째 줄:		88번째 줄:
	==computational resource==		==computational resource==
	* https://docs.google.com/file/d/0B8XXo8Tve1cxWmFPWkZTMVdBeDA/edit		* https://docs.google.com/file/d/0B8XXo8Tve1cxWmFPWkZTMVdBeDA/edit
		+
		+
		+	==expositions==
		+	* Folsom, https://docs.google.com/file/d/0B8XXo8Tve1cxRGhqVHR2YWV2OEk/edit

← 이전 판		2013년 8월 6일 (화) 20:02 판
76번째 줄:		76번째 줄:

	==WRT invariant of the Poincare sphere==		==WRT invariant of the Poincare sphere==
−	* ~~[[Chern-Simons gauge theory and Witten's invariant]]~~	+	* see [[WRT (Witten-Reshetikhin-Turaev) 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~~

@@ 80번째 줄: / 80번째 줄: @@
 * $W: \{\text{root of unity}\} \to \mathbb{C}$ defined by
 $$
-W(q)=\frac{1}{2G}\sum_{\beta \pmod 60K} \frac{(1-\zeta^{24\beta})(1-\zeta^{40\beta})}{1-\zeta^{60\beta}}\zeta^{-(\beta+1)^2}
+W(q)=\frac{1}{2G}\sum_{\beta \pmod {60K}} \frac{(1-\zeta^{24\beta})(1-\zeta^{40\beta})}{1-\zeta^{60\beta}}\zeta^{-(\beta+1)^2}
 $$
 where $\zeta$ satisfies $\zeta^{120}=q$ and $G=\sum_{\beta}\zeta^{-\beta^2}$

@@ 77번째 줄: / 77번째 줄: @@
 ==WRT invariant of the Poincare sphere==
 * [[Chern-Simons gauge theory and Witten's invariant]]
 * $W: \{\text{root of unity}\} \to \mathbb{C}$ defined by
 $$