2024/01/21 by Teo, Lee-Peng
#Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.2401.12260
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 Δn be the n-Laplacian and let Nn be the Gram matrix of a basis of holomorphic n-differentials on X. The local index theorem says that on the Teichmüller space T(X), the second variation of logdetΔn-log det Nn can be written as a sum of three symplectic forms ωWP, ωTZcusp and ωTZell. These are the symplectic forms for the three Kähler 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 logdetΔn and log det Nn. 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 logdetΔn and log det Nn. 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.