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

TL;DR

This work presents a comprehensive localization framework for open-closed $B$-twisted Landau-Ginzburg models with Calabi–Yau targets. It establishes a one-parameter family of bulk localization pictures labeled by $\lambda$ and, in the boundary sector, a two-parameter family labeled by $\lambda$ and $\mu$, connected by a semigroup of homotopy flows that preserve BRST cohomology. A geometric model for bulk and boundary observables is developed, culminating in residue formulas for sphere and disk correlators in the large-area and large-boundary-length limits, respectively; the boundary residue formula extends Kap2’s result to the full boundary-coupled setting. The results unify open- and closed-string localization, relate the B-model and Jacobi-ring realizations of bulk observables, and provide explicit boundary-bulk maps and traces, enabling practical computation of correlators in open-closed LG theories and shedding light on tachyon-condensation pictures in D-brane settings.

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 $λ$ 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 $λ$ and $μ$ which measure the area and boundary length of worldsheets with disk topology. We show that setting $μ=0$ and varying $λ$ 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.