Boundary terms in string field theory

TL;DR

This paper introduces boundary terms into string field theory to ensure a well-defined variational principle, revealing how boundary contributions affect stress-energy conservation and the structure of the theory.

Contribution

It explicitly formulates boundary terms in free closed string field theory and relates them to the low-energy effective action with Gibbons-Hawking-York boundary terms.

Findings

Boundary terms restore the variational principle in string field theory.

Failure of stress-energy conservation is linked to boundary contributions.

The boundary-augmented action matches low-energy effective actions with boundary terms.

Abstract

We supplement the string field theory action with boundary terms to make its variational principle well-posed. Central to our considerations is the violation of the stress-energy tensor conservation in non-compact CFTs due to the boundary terms. This manifests as the failure of the cyclicity of the BRST operator, which encodes the target space integration by parts identities at the level of the worldsheet. Using this failure, we argue that the free closed string field theory action admits a well-posed variational principle upon including an additional boundary contribution. We explicitly work out the resulting action up to the massless level and show that it is related to the expansion of the low-energy effective string action endowed with the Gibbons-Hawking-York term on a flat background. We also discuss the structure of the boundary terms in the interacting theory.