Dilation shift under duality and torsion of elliptic complex

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 local functional of 2d metric and target space metric. In this way the well-known dilation shift under duality is reproduced. We also present the exact computation of the determinant which appears in the duality transformation in the path integral.