U-Duality and the Compactified Gauss-Bonnet Term

TL;DR

This paper thoroughly analyzes the toroidal compactification of the Gauss-Bonnet term in string theory, focusing on its symmetry properties under U-duality and the structure of dilaton exponents in the reduced action.

Contribution

It extends previous work by explicitly computing the compactified Gauss-Bonnet term, including dilaton contributions, and clarifies the representation theory aspects related to U-duality symmetry.

Findings

Dilaton exponents form weights of sl(n+1,R)

Overall exponential dilaton factor should be excluded from the representation

Dilaton weights remain on the positive side of the root lattice

Abstract

We present the complete toroidal compactification of the Gauss-Bonnet Lagrangian from D dimensions to (D-n) dimensions. Our goal is to investigate the resulting action from the point of view of the "U-duality" symmetry SL(n+1,R) which is present in the tree-level Lagrangian when D-n=3. The analysis builds upon and extends the investigation of the paper [arXiv:0706.1183], by computing in detail the full structure of the compactified Gauss-Bonnet term, including the contribution from the dilaton exponents. We analyze these exponents using the representation theory of the Lie algebra sl(n+1,R) and determine which representation seems to be the relevant one for quadratic curvature corrections. By interpreting the result of the compactification as a leading term in a large volume expansion of an SL(n+1,Z)-invariant action, we conclude that the overall exponential dilaton factor should not be included in the representation structure. As a consequence, all dilaton exponents correspond to weights of sl(n+1,R), which, nevertheless, remain on the positive side of the root lattice.