On a Coordinate-Independent Description of String Worldsheet Theory

Abstract

We rewrite the bosonic worldsheet theory in curved background 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, 1952) coordinate-independent formulation of quantum mechanics in the present context. This procedure enables us to define coordinate invariant quantum analogue of classical Virasoro generators, 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 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.