A Non-local "Boundary'' Term for Two-Point Amplitudes in String Field Theory

TL;DR

This paper introduces a new boundary term in bosonic string field theory to restore cyclicity and correctly reproduce two-point amplitudes, verified explicitly for open and closed strings.

Contribution

It proposes a novel boundary term involving a commutator with the BRST operator, ensuring a well-defined variational principle and correct two-point amplitudes.

Findings

Reproduces correct tree-level two-point amplitudes for open and closed strings.

Restores cyclicity broken by a stringy step-function operator.

Ensures a well-defined variational principle for the action.

Abstract

We propose a new ``boundary'' term in free bosonic string field theory. Here, the term ``boundary'' refers to a contribution introduced to restore the cyclicity broken by the insertion of a stringy step-function operator, although it is not strictly localized as it involves an operator that selects positive-energy modes. We construct this term as a bilinear form involving the commutator of the BRST operator with the step-function operator. The resulting action has a well-defined variational principle. When evaluated on on-shell string fields, the proposed term reproduces the correct tree-level two-point amplitude for both open and closed strings. We verify this explicitly in the tachyon and massless sectors.