Infrared divergences and harmonic anomalies in the two-loop superstring effective action

TL;DR

This paper investigates two-loop corrections to certain couplings in the low-energy effective action of type II string theory on a torus, addressing infrared divergences, and confirms their properties align with U-duality predictions.

Contribution

It provides a detailed analysis of infrared divergences, introduces a method to extract finite couplings, and shows these couplings satisfy Laplace eigenvalue equations with specific anomalous terms.

Findings

Renormalized couplings are Laplace eigenmodes with source terms.

Explicit prescription for removing infrared cutoff dependence.

Agreement with U-duality predictions and properties of Kawazumi-Zhang invariant.

Abstract

We analyze the pertubative contributions to the $D^4 R^4$ and $D^6 R^4$ couplings in the low-energy effective action of type II string theory compactified on a torus $T^d$, with particular emphasis on two-loop corrections. In general, it is necessary to introduce an infrared cut-off $\Lambda$ to separate local interactions from non-local effects due to the exchange of massless states. We identify the degenerations of the genus-two Riemann surface which are responsible for power-like dependence on $\Lambda$, and give an explicit prescription for extracting the $\Lambda$-independent effective couplings. These renormalized couplings are then shown to be eigenmodes of the Laplace operator with respect to the torus moduli, up to computable anomalous source terms arising in the presence of logarithmic divergences, in precise agreement with predictions from U-duality. Our results for the two-loop $D^6 R^4$ contribution also probe essential properties of the Kawazumi-Zhang invariant