A-branes, foliations and localization

Sibasish Banerjee, Pietro Longhi, Mauricio Romo

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.