A Proof of Local Background Independence of Classical Closed String Field Theory

TL;DR

This work proves local background independence for classical closed string field theory by constructing a symplectic diffeomorphism that maps master actions and BV structures between nearby conformal field theories. A key device is a new family of string vertices, notably a three-punctured sphere ${\cal V}'_3$, which obeys Jacobi-like identities and enables a controlled interpolation (via ${\cal B}_N$) between standard and primed vertices, thereby generating the full diffeomorphism order by order. The linear component of the field redefinition defines a theory-space connection $\Gamma_\mu$ arising from the off-shell three-string vertex, forging a deep link between Riemann surface geometry and CFT theory-space geometry. The paper also demonstrates a robust covering of moduli space using the interpolating and standard vertices, and discusses extensions to finite-distance backgrounds, open strings, and the prospect of quantum background independence, outlining multiple avenues for manifest background-independent formulations in string theory.

Abstract

We give a complete proof of local background independence of the classical master action for closed strings by constructing explicitly, for any two nearby conformal theories in a CFT theory space, a symplectic diffeomorphism between their state spaces mapping the corresponding non-polynomial string actions into each other. We uncover a new family of string vertices, the lowest of which is a three string vertex satisfying exact Jacobi identities with respect to the original closed string vertices. The homotopies between the two sets of string vertices determine the diffeomorphism establishing background independence. The linear part of the diffeomorphism is implemented by a CFT theory-space connection determined by the off-shell three closed string vertex, showing how string field theory induces a natural interplay between Riemann surface geometry and CFT theory space geometry. (Three figures are contained in a separate tar compressed uuencoded figures file. See the TeX file for instructions for printing the figures.)