Modular invariance in math and physics

introduction

path integral and moduli space of Riemann surfaces
\(Z=\sum_{g=0}^{\infty} g_{s}^{-\chi(\Sigma_{g})}Z_{g}=\sum_{g=0}^{\infty} g_{s}^{2g-2}Z_{g}=\frac{1}{g_{s}^2}Z_{0}+g_{s}^{0}Z_{1}+g_{s}^2Z_{2}+\cdtos\)
\(Z_{1}\) 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