Effective superpotentials for B-branes in Landau-Ginzburg models

TL;DR

This paper calculates the effective superpotential for B-branes in Landau-Ginzburg models using a perturbative approach to the partition function, establishing connections to matrix factorizations and confirming conjectures in minimal models and mirror symmetry.

Contribution

It introduces a perturbative method to compute the disk partition function as the effective superpotential, linking it to matrix factorizations and validating conjectures in specific models.

Findings

Computed the disk partition function for B-models.

Proved a conjecture for A-type minimal models.

Confirmed the superpotential matches mirror symmetry predictions.

Abstract

We compute the partition function for the topological Landau-Ginzburg B-model on the disk. This is done by treating the worldsheet superpotential perturbatively. We argue that this partition function as a function of bulk and boundary perturbations may be identified with the effective D-brane superpotential in the target spacetime. We point out the relationship of this approach to matrix factorizations. Using these methods, we prove a conjecture for the effective superpotential of Herbst, Lazaroiu and Lerche for the A-type minimal models. We also consider the Landau-Ginzburg theory of the cubic torus where we show that the effective superpotential, given by the partition function, is consistent with the one obtained by summing up disk instantons in the mirror A-model. This is done by explicitly constructing the open-string mirror map.