Modified pure spinors and mirror symmetry

TL;DR

This paper explores how pure spinors are modified in the presence of topological D-branes to preserve mirror symmetry, using T-duality on T^3-fibered manifolds to establish the modified mirror relations.

Contribution

It introduces a modification of pure spinors accounting for gauge fields on D-branes, ensuring mirror symmetry exchanges remain valid in more general settings.

Findings

Modified pure spinors act as moment maps for A and B-model symmetries.

Mirror symmetry exchanges these modified pure spinors even with D-brane gauge fields.

The mirror formula is derived via fiberwise T-duality on T^3-fibered manifolds.

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.