Conformally Exact Metric and Dilaton in String Theory on Curved Spacetime

TL;DR

This paper develops a Hamiltonian method to compute the exact metric and dilaton fields in string theory on curved spacetimes, valid to all orders in the inverse level expansion, for various superstring models based on coset spaces.

Contribution

It introduces a general Hamiltonian approach for calculating conformally exact metric and dilaton in string models on cosets, extending previous semi-classical and perturbative results to all orders.

Findings

Exact expressions for metric and dilaton in specific coset models.

Relations between superstring and bosonic string metrics and dilatons.

Consistency with 1-loop perturbation theory in the semiclassical limit.

Abstract

Using a Hamiltonian approach to gauged WZW models, we present a general method for computing the conformally exact metric and dilaton, to all orders in the $1/k$ expansion, for any bosonic, heterotic, or type-II superstring model based on a coset $G/H$. We prove the following relations: (i) For type-II superstrings the conformally exact metric and dilaton are identical to those of the non-supersymmetric {\it semi-classical} bosonic model except for an overall renormalization of the metric obtained by $k\to k- g$. (ii) The exact expressions for the heterotic superstring are derived from their exact bosonic string counterparts by shifting the central extension $k\to 2k-h$ (but an overall factor $(k-g)$ remains unshifted). (iii) The combination $e^\Phi\sqrt{-G}$ is independent of $k$ and therefore can be computed in lowest order perturbation theory as required by the correct formulation of a conformally invariant path integral measure. The general formalism is applied to the coset models $SO(d-1,2)_{-k}/SO(d-1,1)_{-k}$ that are relevant for string theory on curved spacetime. Explicit expressions for the conformally exact metric and dilaton for the cases $d=2,3,4$ are given. In the semiclassical limit $(k\to \infty)$ our results agree with those obtained with the Lagrangian method up to 1-loop in perturbation theory.