Gribov horizon, contact terms and \v{C}ech- De Rham cohomology in 2D topological gravity

TL;DR

This paper investigates the global properties of observables in 2D topological gravity, revealing the role of the Gribov horizon and contact terms, and introduces a cohomological approach to define global forms for correlators.

Contribution

It introduces a cech-De Rham cohomology framework to associate global forms to correlators in 2D topological gravity, addressing issues caused by the Gribov horizon.

Findings

Derived explicit cech-De Rham cocycles for correlators.

Resolved the problem of global definition of observables despite the Gribov horizon.

Provided a first-principles approach to contact terms in string theory.

Abstract

We point out that averages of equivariant observables of 2D topological gravity are not globally defined forms on moduli space, when one uses the functional measure corresponding to the formulation of the theory as a 2D superconformal model. This is shown to be a consequence of the existence of the Gribov horizon {\it and} of the dependence of the observables on derivatives of the super-ghost field. By requiring the absence of global BRS anomalies, it is nevertheless possible to associate global forms to correlators of observables by resorting to the \v{C}ech-De Rham notion of form cohomology. To this end, we derive and solve the ``descent'' of local Ward identities which characterize the functional measure. We obtain in this way an explicit expression for the \v{C}ech-De Rham cocycles corresponding to arbitrary correlators of observables. This provides the way to compute and understand contact terms in string theory from first principles.