Stability of Landau-Ginzburg branes

We evaluate the ideas of $\Pi$ -stability at the Landau-Ginzburg (LG) 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 quasihomogeneous 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 maplike flow on the gauge orbits and use it to study boundary flows in several examples. Gauge transformations of nonzero 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.