Colliding Wave Solutions, Duality, and Diagonal Embedding of General Relativity in Two-Dimensional Heterotic String Theory

TL;DR

This paper explores the embedding of Einstein theory into two-dimensional heterotic string theory, revealing duality symmetries and generating new colliding string wave solutions through the inverse scattering method.

Contribution

It introduces a diagonal embedding of Einstein-Maxwell theory into heterotic string theory and demonstrates how duality transformations relate to string symmetries and solution generation.

Findings

Embedded backgrounds satisfy the Ernst equation.

SL(2,Z) acts as modular and S-duality transformations.

New colliding string wave solutions are constructed.

Abstract

The non-linear sigma model of the dimensionally reduced Einstein (-Maxwell) theory is diagonally embedded into that of the two-dimensional heterotic string theory. Consequently, the embedded string backgrounds satisfy the (electro-magnetic) Ernst equation. In the pure Einstein theory, the Matzner-Misner SL(2,{\bf R}) transformation can be viewed as a change of conformal structure of the compactified flat two-torus, and in particular its integral subgroup SL(2,{\bf Z}) acts as the modular transformation. The Ehlers SL(2,{\bf R}) and SL(2,{\bf Z}) similarly act on another torus whose conformal structure is induced through the Kramer-Neugebauer involution. Either of the Matzner-Misner and the Ehlers SL(2,{\bf Z}) can be embedded to a special T-duality, and if the former is chosen, then the Ehlers SL(2,{\bf Z}) is shown to act as the S-duality on the four-dimensional sector. As an application we obtain some new colliding string wave solutions by using this embedding as well as the inverse scattering method.