On the Semi-Relative Condition for Closed (TOPOLOGICAL) Strings

TL;DR

This paper offers a Lagrangian interpretation of the semi-relative condition in closed string theory, linking it to cohomology classes of BRS operators and deriving covariant expressions for topological gravity observables.

Contribution

It introduces a covariant framework for understanding the semi-relative condition and derives new expressions for topological gravity observables in string theory.

Findings

Semi-relative condition is equivalent to cohomology classes of BRS operators.

Explicitly relates trivial states to BRS variations of non-globally defined operators.

Proves a formula connecting gravitational descendants to moduli space integrals.

Abstract

We provide a simple lagrangian interpretation of the meaning of the $b_0^-$ semi-relative condition in closed string theory. Namely, we show how the semi-relative condition is equivalent to the requirement that physical operators be cohomology classes of the BRS operators acting on the space of local fields {\it covariant} under world-sheet reparametrizations. States trivial in the absolute BRS cohomology but not in the semi-relative one are explicitly seen to correspond to BRS variations of operators which are not globally defined world-sheet tensors. We derive the covariant expressions for the observables of topological gravity. We use them to prove a formula that equates the expectation value of the gravitational descendant of ghost number 4 to the integral over the moduli space of the Weil-Peterson K\"ahler form.