Phases of Supersymmetric D-branes on Kaehler Manifolds and the McKay correspondence

TL;DR

This paper explores the topological zero modes of type II strings on Kähler manifolds with boundaries, connecting D-brane states, K-theory, and the McKay correspondence through a novel basis construction.

Contribution

It introduces finite bosonic and fermionic bases for the topological sector of D-branes, linking boundary descriptions to K-theory and generalizing the McKay correspondence.

Findings

Constructed bases localize on submanifolds in X

Interpolates between boundary ring of chiral fields and exceptional sheaves

Provides a method to analyze D-branes on Calabi-Yau manifolds

Abstract

We study the topological zero mode sector of type II strings on a K\"ahler manifold $X$ in the presence of boundaries. We construct two finite bases, in a sense bosonic and fermionic, that generate the topological sector of the Hilbert space with boundaries. The fermionic basis localizes on compact submanifolds in $X$. A variation of the FI terms interpolates between the description of these ground states in terms of the ring of chiral fields at the boundary at small volume and helices of exceptional sheaves at large volume, respectively. The identification of the bosonic/fermionic basis with the dual bases for the non-compact/compact K-theory group on $X$ gives a natural explanation of the McKay correspondence in terms of a linear sigma model and suggests a simple generalization of McKay to singular resolutions. The construction provides also a very effective way to describe D-brane states on generic, compact Calabi--Yau manifolds and allows to recover detailed information on the moduli space, such as monodromies and analytic continuation matrices, from the group theoretical data of a simple orbifold.