A-branes, foliations and localization

TL;DR

This paper introduces a new way to count stable A-branes using topological string theory, relating it to spectral networks, Euler characteristics, and BPS invariants, and demonstrating their agreement through equivariant localization.

Contribution

It provides a natural definition of stable A-brane counts via the Witten index and relates these to spectral networks, Euler characteristics, and Donaldson-Thomas invariants.

Findings

Agreement between Witten index counts and spectral network BPS invariants

Reduction of Euler characteristic computation via equivariant localization

Correspondence with Donaldson-Thomas invariants through mirror symmetry

Abstract

This paper studies a notion of enumerative invariants for stable $A$-branes, and discusses its relation to invariants defined by spectral and exponential networks. A natural definition of stable $A$-branes and their counts is provided by the string theoretic origin of the topological $A$-model. This is the Witten index of the supersymmetric quantum mechanics of a single $D3$ brane supported on a special Lagrangian in a Calabi-Yau threefold. Geometrically, this is closely related to the Euler characteristic of the $A$-brane moduli space. Using the natural torus action on this moduli space, we reduce the computation of its Euler characteristic to a count of fixed points via equivariant localization. Studying the $A$-branes that correspond to fixed points, we make contact with definitions of spectral and exponential networks. We find agreement between the counts defined via the Witten index, and the BPS invariants defined by networks. By extension, our definition also matches with Donaldson-Thomas invariants of $B$-branes related by homological mirror symmetry.