Stability of Landau-Ginzburg branes

TL;DR

This paper investigates the stability of Landau-Ginzburg B-branes using matrix factorizations, proposing a new notion of R-stability linked to unitarity and analyzing boundary flows and gauge transformations.

Contribution

It introduces R-stability for matrix factorizations, relates it to moduli space problems, and studies boundary flows and gauge transformations in the Landau-Ginzburg B-brane category.

Findings

R-stability is proposed as a unitarity condition at the IR fixed point.

A flow on gauge orbits is defined to study boundary flows.

Gauge transformations of non-zero degree influence brane-antibrane dynamics.

Abstract

We evaluate the ideas of Pi-stability at the Landau-Ginzburg point in moduli space of compact Calabi-Yau manifolds, using matrix factorizations to B-model the topological D-brane category. The standard requirement of unitarity at the IR fixed point is argued to lead to a notion of "R-stability" for matrix factorizations of quasi-homogeneous LG potentials. The D0-brane on the quintic at the Landau-Ginzburg point is not obviously unstable. Aiming to relate R-stability to a moduli space problem, we then study the action of the gauge group of similarity transformations on matrix factorizations. We define a naive moment map-like flow on the gauge orbits and use it to study boundary flows in several examples. Gauge transformations of non-zero degree play an interesting role for brane-antibrane annihilation. We also give a careful exposition of the grading of the Landau-Ginzburg category of B-branes, and prove an index theorem for matrix factorizations.