Sibasish BanerjeeWeyertal 86-90, Department of Mathematics, University of Cologne, 50679, Cologne, Germany; Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, GermanyPietro LonghiInstitute for Theoretical Physics, ETH Zurich, 8093, Zurich, SwitzerlandMauricio Andrés Romo JorqueraYau Mathematical Sciences Center, Tsinghua University, Beijing, 100084, China
Symplectic GeometryAlgebraic GeometryarXiv subject: High Energy Physics - Theory (hep-th)mathscidoc:2207.34001
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.
We prove a version of equivariant split generation of Fukaya category when a symplectic manifold admits a free action of a finite group <i>G</i>. Combining this with some generalizations of Seidel's algebraic frameworks from , we obtain new cases of homological mirror symmetry for some symplectic tori with non-split symplectic forms, which we call <i>special isogenous tori</i>. This extends the work of AbouzaidSmith . We also show that derived Fukaya categories are complete invariants of special isogenous tori.