D-Branes And Mirror Symmetry

TL;DR

This paper explores the relationship between D-branes, mirror symmetry, and various supersymmetric models, revealing new mechanisms for brane creation, connections to soliton numbers, and geometric realizations of algebraic structures.

Contribution

It introduces novel boundary conditions and interactions for D-branes in supersymmetric models, linking soliton numbers to R-charges and providing geometric interpretations of algebraic structures.

Findings

Identified D-branes preserving half of the supercharges in multiple models

Linked soliton numbers to R-charges at UV fixed points

Provided geometric realizations of Verlinde algebra and modular matrices

Abstract

We study (2,2) supersymmetric field theories on two-dimensional worldsheet with boundaries. We determine D-branes (boundary conditions and boundary interactions) that preserve half of the bulk supercharges in non-linear sigma models, gauged linear sigma models, and Landau-Ginzburg models. We identify a mechanism for brane creation in LG theories and provide a new derivation of a link between soliton numbers of the massive theories and R-charges of vacua at the UV fixed point. Moreover we identify Lagrangian submanifolds that arise as the mirror of certain D-branes wrapped around holomorphic cycles of K\"ahler manifolds. In the case of Fano varieties this leads to the explanation of Helix structure of the collection of exceptional bundles and soliton numbers, through Picard-Lefshetz theory applied to the mirror LG theory. Furthermore using the LG realization of minimal models we find a purely geometric realization of Verlinde Algebra for SU(2) level k as intersection numbers of D-branes. This also leads to a direct computation of modular transformation matrix and provides a geometric interpretation for its role in diagonalizing the Fusion algebra.