# Localization of effective actions in open superstring field theory: small Hilbert space

## TL;DR

This paper proves localization of effective couplings for open superstring massless modes in the small Hilbert space, showing they are determined by boundary contributions and exploring marginal deformations in D3/D(-1) systems.

## Contribution

It provides a proof of localization in the small Hilbert space and connects algebraic couplings to boundary moduli space contributions, advancing understanding of superstring effective actions.

## Key findings

- Effective couplings are localized at boundary of moduli space.
- Quartic potentials exhibit flat directions indicating marginal deformations.
- Evidence for exactly marginal deformations in D3/D(-1) system.

## Abstract

We consider the algebraic effective couplings for open superstring massless modes in the framework of the $A_\infty$ theory in the small Hilbert space. Focusing on quartic algebraic couplings, we reduce the effective action of the $A_\infty$ theory to the Berkovits one where we have already shown that such couplings are fully computed from contributions at the boundary of moduli space, when the massless fields under consideration are appropriately charged under an ${\cal N}\!=\!2$ $R$-symmetry. Here we offer a proof of localization which is in the small Hilbert space. We also discuss the flat directions of the obtained quartic potentials and give evidence for the existence of exactly marginal deformations in the $D3/D(-1)$ system in the framework of string field theory.