Modified pure spinors and mirror symmetry

TL;DR

This work extends mirror symmetry to brane configurations on SU(3)-structure manifolds by introducing gauge-field–modified pure spinors that act as moment maps for A- and B-model symmetries. The authors show that these modified spinors are exchanged under fiberwise T-duality on T^3-fibered spaces, using a Fourier--Mukai framework and a Clifford(6,6) formalism to connect open- and closed-string sectors. A central advance is the incorporation of non-Lagrangian (coisotropic) A-branes via a gauge-field–dependent deformation of the moment map, yielding a mirror counterpart to deformed Hermitian Yang–Mills stability on B-branes. They provide explicit T-duality formulas with Q = $e^{F}$ and discuss open-string, non-Abelian generalizations, linking brane stability conditions to the generalized geometry picture and K-theory charges.

Abstract

It has been argued recently that mirror symmetry exchanges two pure spinors characterizing a generic manifold with SU(3)-structure. We show how pure spinors are modified in the presence of topological D-branes, so that they are still exchanged by mirror symmetry. This exchange emerges from the fact that the modified pure spinors come out as moment maps for the symmetries of A and B-models. The modification by the gauge field is argued to ensure the inclusion into the mirror exchange of the A-model non-Lagrangian branes endowed with a non-flat connection. Treating the connection as a distribution on an ambient six-manifold, assumed to be T^3-fibered, the proposed mirror formula is established by fiberwise T-duality.