Topological Correlators in Landau-Ginzburg Models with Boundaries

TL;DR

This work extends Vafa's open-string topological formula to Landau-Ginzburg models with boundaries by identifying B-branes with CDG-modules over the CDG algebra $(O,0,W)$. The physical construction via brane–anti-brane tachyons enforces holomorphic boundary data and yields a boundary differential $D$ with $D^2=W$, so open-string states are described by the cohomology of $D$ and the boundary OPE by endomorphism composition. The authors derive a closed-form disk correlator as a generalized residue and generalize to arbitrary genus $g$ and $h$ boundaries, expressing correlators via Hessians and supertraces of boundary data, thereby providing an open-closed topological TFT framework for LG models. They also discuss localization to the critical locus (Orlov) and outline extensions to LG orbifolds, offering a pathway toward exact open- and closed-sector superpotentials and links to Homological Mirror Symmetry in simple LG settings.

Abstract

We compute topological correlators in Landau-Ginzburg models on a Riemann surface with arbitrary number of handles and boundaries. The boundaries may correspond to arbitrary topological D-branes of type B. We also allow arbitrary operator insertions on the boundary and in the bulk. The answer is given by an explicit formula which can be regarded as an open-string generalization of C. Vafa's formula for closed-string topological correlators. We discuss how to extend our results to the case of Landau-Ginzburg orbifolds.