Modular invariance in math and physics

introduction

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!
- compare with http://www.physics.thetangentbundle.net/wiki/Quantum_field_theory/Schwinger_proper_time_formalism

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.
Lepowsky, J. “Affine Lie Algebras and Combinatorial Identities.” In Lie Algebras and Related Topics, edited by David Winter, 130–56. Lecture Notes in Mathematics 933. Springer Berlin Heidelberg, 1982. http://link.springer.com/chapter/10.1007/BFb0093358.
Lepowsky, J. "Lie algebras and combinatorics." Proc. Internat. Congr. Math.(Helsinki, 1978)(to appear) (1978). http://www.mathunion.org/ICM/ICM1978.2/Main/icm1978.2.0579.0584.ocr.pdf