Local Index Theorem for Cofinite Hyperbolic Riemann Surfaces

Abstract

We discuss the local index theorem for cofinite Riemann surfaces in a pedagogical way, from a more computational perspective. Given a cofinite Riemann surface $X$, let $\Delta_n$ be the $n$-Laplacian and let $N_n$ be the Gram matrix of a basis of holomorphic $n$-differentials on $X$. The local index theorem says that on the Teichm\"uller space $T(X)$, the second variation of $\log\det\Delta_n-\log \det N_n$ can be written as a sum of three symplectic forms $\omega_{\text{WP}}$, $\omega_{\text{TZ}}^{\text{cusp}}$ and $\omega_{\text{TZ}}^{\text{ell}}$. These are the symplectic forms for the three K\"ahler metrics on $T(X)$ -- the Weil-Petersson metric, the parabolic Takhtajan-Zograf (TZ) metric and the elliptic Takhtajan-Zograf metric. Using Ahlfors' variational formulas and projection formulas, we derive explicitly integral formulas for the variations of $\log\det\Delta_n$ and $\log \det N_n$. The integrals are regular integrals that allow explicit computations. In the spirit of the Selberg trace formula, we identify the identity, hyperbolic, parabolic and elliptic contributions to the second variations of $\log\det\Delta_n$ and $\log \det N_n$. We showed that the Weil-Petersson term comes from the identity contribution, while the parabolic TZ metric and elliptic TZ metric terms come from parabolic and elliptic contributions respectively. The hyperbolic contributions are cancelled. As a byproduct, we obtain alternative integral formulas for the parabolic TZ metric and the elliptic TZ metric.