Two-dimensional gravitational anomalies, Schwinger terms and dispersion relations

TL;DR

This paper demonstrates that two-dimensional gravitational anomalies and Schwinger terms can be fully determined using dispersion relations, linking infrared features to anomaly origins without relying on renormalization or ultraviolet regularization.

Contribution

It introduces a dispersive approach to compute gravitational anomalies and Schwinger terms, establishing their independence from regularization schemes and showing their infrared origin.

Findings

Anomalies are determined by superconvergence sum rules.

Imaginary parts of formfactors approach delta functions at zero momentum.

Dispersive method aligns with dimensional regularization results.

Abstract

We are dealing with two-dimensional gravitational anomalies, specifically with the Einstein anomaly and the Weyl anomaly, and we show that they are fully determined by dispersion relations independent of any renormalization procedure (or ultraviolet regularization). The origin of the anomalies is the existence of a superconvergence sum rule for the imaginary part of the relevant formfactor. In the zero mass limit the imaginary part of the formfactor approaches a $\delta$-function singularity at zero momentum squared, exhibiting in this way the infrared feature of the gravitational anomalies. We find an equivalence between the dispersive approach and the dimensional regularization procedure. The Schwinger terms appearing in the equal time commutators of the energy momentum tensors can be calculated by the same dispersive method. Although all computations are performed in two dimensions the method is expected to work in higher dimensions too.