Closed string deformations in open string field theory III: ${\cal N}=2$ worldsheet localization

TL;DR

This paper demonstrates that in open string field theory with ${ m N}=2$ worldsheet localization, the effective potentials from different formalisms agree under certain conditions, enabling explicit calculations of mass deformations and system condensations.

Contribution

It shows that ${ m N}=2$ worldsheet symmetry allows localization of string field theory amplitudes, simplifying the analysis of open-closed couplings and vacuum structure.

Findings

Effective potentials from different theories match under ${ m N}=2$ conditions.

Mass deformations can be explicitly computed via localization techniques.

The D3-D(-1) system condenses with a calculable binding energy.

Abstract

In this paper, which is the last of a series of three, we first verify that the two open-closed effective potentials derived in the previous paper from the WZW theory in the large Hilbert space and the $A_\infty$ theory in the small Hilbert space have the same vacuum structure. In particular, we show that mass-term deformations given by the effective (two open)-(one closed) couplings are the same, provided the effective tadpole is vanishing to first order in the closed string deformation. We show that this condition is always realized when the worldsheet BCFT enjoys a global ${\cal N}=2$ superconformal symmetry and the deforming closed string belongs to the chiral ring in both the holomorphic and anti-holomorphic sector. In this case it is possible to explicitly evaluate the mass deformation by localizing the SFT Feynman diagrams to the boundary of world-sheet moduli space, reducing the amplitude to a simple open string two-point function. As a non-trivial check of our construction we couple a constant Kalb-Ramond closed string state to the OSFT on the $\text{D}3$--$\text{D}(-1)$ system and we show that half of the bosonic blowing-up moduli become tachyonic, making the system condense to a bound state whose binding energy we compute exactly to second order in the closed string deformation, finding agreement with the literature.