Dilaton shift under duality and torsion of elliptic complex

TL;DR

This paper demonstrates that the ratio of determinants in 2D sigma model dualities can be exactly computed as a torsion of an elliptic complex, reproducing the dilaton shift and providing explicit determinant calculations.

Contribution

It establishes a novel connection between determinant ratios in dual sigma models and elliptic complex torsion, enabling exact computations of the dilaton shift.

Findings

Determinant ratio expressed as elliptic complex torsion

Exact computation of determinants in duality transformations

Reproduction of the dilaton shift under duality

Abstract

We observe that the ratio of determinants of $2d$ Laplacians which appear in the duality transformation relating two sigma models with abelian isometries can be represented as a torsion of an elliptic (DeRham) complex. As a result, this ratio can be computed exactly and is given by the exponential of a local functional of $2d$ metric and target space metric. In this way the well known dilaton shift under duality is reproduced. We also present the exact computation of the determinant which appears in the duality transformation in the path integral.