Localization and traces in open-closed topological Landau-Ginzburg models

TL;DR

This paper explores localization techniques in open-closed B-twisted Landau-Ginzburg models with Calabi-Yau targets, revealing a family of localization pictures parameterized by worldsheet area and boundary length, and connecting them via homotopy equivalences.

Contribution

It introduces a generalized family of localization pictures in open-closed Landau-Ginzburg models, extending the residue representation to arbitrary worldsheet areas and boundary conditions.

Findings

Identifies a one-parameter family of localization pictures in closed models.

Derives the boundary residue formula from explicit boundary couplings.

Shows localization pictures are related by homotopy equivalences.

Abstract

We reconsider the issue of localization in open-closed B-twisted Landau-Ginzburg models with arbitrary Calabi-Yau target. Through careful analsysis of zero-mode reduction, we show that the closed model allows for a one-parameter family of localization pictures, which generalize the standard residue representation. The parameter $\lambda$ which indexes these pictures measures the area of worldsheets with $S^2$ topology, with the residue representation obtained in the limit of small area. In the boundary sector, we find a double family of such pictures, depending on parameters $\lambda$ and $\mu$ which measure the area and boundary length of worldsheets with disk topology. We show that setting $\mu=0$ and varying $\lambda$ interpolates between the localization picture of the B-model with a noncompact target space and a certain residue representation proposed recently. This gives a complete derivation of the boundary residue formula, starting from the explicit construction of the boundary coupling. We also show that the various localization pictures are related by a semigroup of homotopy equivalences.