타원 초기하 적분 (elliptic hypergeometric integrals)
개요
- 정리 (스피리도노프 Spiridonov).
복소수 \(t_1, \dots ,t_6,p,q\)가 \(|t_1|, \dots , |t_6|,|p|,|q| <1\)이고, \(\prod_{i=1}^6 t_i=pq\)을 만족한다고 하자. 다음이 성립한다. \begin{equation} \label{betaint} \frac{(p;p)_\infty (q;q)_\infty}{2} \int_{\mathbb{T}} \frac{\prod_{i=1}^6 \Gamma(t_i z ;p,q)\Gamma(t_i z^{-1} ;p,q)}{\Gamma(z^{2};p,q) \Gamma(z^{-2};p,q)} \frac{dz}{2 \pi i z} = \prod_{1 \leq i < j \leq 6} \Gamma(t_i t_j;p,q), \end{equation} 여기서 \(\mathbb{T}\)는 단위원 (양의 방향)이고 \(\Gamma\)는 타원 감마 함수 \[ \Gamma (z;p,q) = \prod_{m=0}^\infty \prod_{n=0}^\infty \frac{1-p^{m+1}q^{n+1}/z}{1-p^m q^n z} \]
- \(p \rightarrow 0\)일 때, 나스랄라-라만 적분을 얻는다
\begin{equation} \frac{(q;q)_\infty}{2} \int_{\mathbb{T}}\frac{(z \prod_{i=1}^5 t_i;q)_\infty (z^{-1} \prod_{i=1}^5 t_i;q)_\infty (z^2;q)_\infty (z^{-2};q)_\infty}{\prod_{i=1}^5 (t_i z)_\infty (t_i z^{-1})_\infty} \frac{dz}{2\pi i z} \ = \ \frac{\prod_{j=1}^5 (\frac{t_1 t_2 t_3 t_4 t_5}{t_j};q)_\infty}{\prod_{1 \leq i < j \leq 5} (t_i t_j;q)_\infty} \end{equation}
매스매티카 파일 및 계산 리소스
리뷰, 에세이, 강의노트
- Gahramanov, Ilmar. “Mathematical Structures behind Supersymmetric Dualities.” arXiv:1505.05656 [hep-Th, Physics:math-Ph], May 21, 2015. http://arxiv.org/abs/1505.05656.
- van Diejen, J. F., and V. P. Spiridonov. “Elliptic Beta Integrals and Mudular Hypergeometric Sums: An Overview.” Rocky Mountain Journal of Mathematics 32, no. 2 (June 2002): 639–56. doi:10.1216/rmjm/1030539690.
관련논문
- Gahramanov, Ilmar, and Grigory Vartanov. “Extended Global Symmetries for 4d N=1 SQCD Theories.” Journal of Physics A: Mathematical and Theoretical 46, no. 28 (July 19, 2013): 285403. doi:10.1088/1751-8113/46/28/285403.
- Spiridonov, V. P. “On the Elliptic Beta Function.” Rossi\uı Skaya Akademiya Nauk. Moskovskoe Matematicheskoe Obshchestvo. Uspekhi Matematicheskikh Nauk 56, no. 1(337) (2001): 181–82. doi:10.1070/rm2001v056n01ABEH000374.