Modular invariance in math and physics
imported>Pythagoras0님의 2013년 7월 14일 (일) 11:13 판
introduction
- Is it useful?
- Why is it important?
- Kac http://www.ams.org/publications/online-books/hmbrowder-hmbrowder-kac.pdf
- Modular invariance in lattice statistical mechanics http://aflb.ensmp.fr/AFLB-26j/aflb26jp287.pdf
path integral in string theory
- path integral and moduli space of Riemann surfaces \[Z=\sum_{g=0}^{\infty} g_{s}^{-\chi(\Sigma_{g})}Z_{g}=\sum_{g=0}^{\infty} g_{s}^{2g-2}Z_{g}=\frac{1}{g_{s}^2}Z_{0}+g_{s}^{0}Z_{1}+g_{s}^2Z_{2}+\cdots\]
- \(Z_{1}\) is an integral over \(M_1 = \mathbb{H}/SL(2,\mathbb{Z})\) i.e. the fundamental domain.
- string theory (symmetries, modular group) has a natural covariant UV cutoff!
circle method
- modular invariant partition functions
- Kac-Peterson modular S-matrix
- Mock theta and physics
- Blackhole theory
- Hardy-Ramanujan tauberian theorem
expositions
- Nikolov, Nikolay M., and Ivan T. Todorov. 2004. “Lectures on Elliptic Functions and Modular Forms in Conformal Field Theory”. ArXiv e-print math-ph/0412039. http://arxiv.org/abs/math-ph/0412039.