Colliding wave solutions, duality, and diagonal embedding of general relativity in two-dimensional heterotic string theory Original Research Article

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 (electromagnetic) Ernst equation. In the pure Einstein theory, the Matzner-Misner SL(2,R) transformation can be viewed as a change of the conformal structure of the compactified flat two-torus, and in particular its integral subgroup SL(2,Z) acts as the modular transformation. The Ehlers SL(2,R) and SL(2,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,Z) can be embedded to a special T-duality, and if the former is chosen, then the Ehlers SL(2,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.