Conformally exact metric and dilation in string theory on curved spacetime

Using a Hamiltonian approach to gauged Wess-Zumino-Witten models, we present a general method for computing the conformally exact metric and dilaton, to all orders in the $\frac{1}{k}$ expansion, for any bosonic, heterotic, or type-II superstring model based on a coset $\frac{G}{H}$. We prove the following relations: (i) For type-II superstrings the conformally exact metric and dilaton are identical to those of the nonsupersymmetric semiclassical 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. The general formalism is applied to the coset models $\frac{{\mathrm{SO}\mathrm{}(d-1,2\mathrm{})\mathrm{}}_{-k}}{{\mathrm{SO}\mathrm{}(d-1,1\mathrm{})\mathrm{}}_{-k}}$ that are relevant for string theory on curved spacetime. Explicit expressions for the conformally exact metric and dilaton for the cases $d=2,3,4\mathrm{}$ are given. In the semiclassical limit ($k\to \infty$) our results agree with those obtained with the Lagrangian method up to one loop in perturbation theory.