On a coordinate independent description of string worldsheet theory

TL;DR

This paper develops a background-independent, coordinate-invariant framework for analyzing string worldsheet theory in curved spacetime, introducing DeWitt-Virasoro generators and exploring their algebra to understand conformal invariance.

Contribution

It introduces a coordinate-independent formulation of string worldsheet theory using DeWitt's quantum mechanics approach, enabling invariant analysis of Virasoro algebra in curved backgrounds.

Findings

DeWitt-Virasoro algebra reduces to Witt algebra with anomalies vanishing in Ricci-flat backgrounds

Framework allows calculation of invariant matrix elements between scalar states

Provides insights into conformal invariance in specific curved spacetimes like pp-waves

Abstract

We study worldsheet conformal invariance for bosonic string propagating in a curved background using the hamiltonian formalism. In order to formulate the problem in a background independent manner we first rewrite the worldsheet theory in a language where it describes a single particle moving in an infinite-dimensional curved spacetime. This language is developed at a formal level without regularizing the infinite-dimensional traces. Then we adopt DeWitt's (Phys.Rev.85:653-661,1952) coordinate independent formulation of quantum mechanics in the present context. Given the expressions for the classical Virasoro generators, this procedure enables us to define the coordinate invariant quantum analogues which we call DeWitt-Virasoro generators. This framework also enables us to calculate the invariant matrix elements of an arbitrary operator constructed out of the DeWitt-Virasoro generators between two arbitrary scalar states. Using these tools we further calculate the DeWitt-Virasoro algebra in spin-zero representation. The result is given by the Witt algebra with additional anomalous terms that vanish for Ricci-flat backgrounds. Further analysis need to be performed in order to precisely relate this with the beta function computation of Friedan and others. Finally, we explain how this analysis improves the understanding of showing conformal invariance for certain pp-wave that has been recently discussed using hamiltonian framework.