Phases Of N=2 Theories In 1+1 Dimensions With Boundary

TL;DR

This paper investigates B-type D-branes in linear sigma models with Abelian gauge groups, introducing a grade restriction rule that unifies various geometric phases and mathematical equivalences of D-brane categories.

Contribution

It introduces the grade restriction rule for classifying gauge representations on D-branes, connecting different phases and unifying previous mathematical results.

Findings

Established the grade restriction rule for D-branes.

Connected D-branes across geometric, orbifold, and Landau-Ginzburg phases.

Unified mathematical frameworks like McKay correspondence and Orlov's construction.

Abstract

We study B-type D-branes in linear sigma models with Abelian gauge groups. The most important finding is the grade restriction rule. It classifies representations of the gauge group on the Chan-Paton factor, which can be used to define a family of D-branes over a region of the K\"ahler moduli space that connects special points of different character. As an application, we find a precise, transparent relation between D-branes in various geometric phases as well as free orbifold and Landau-Ginzburg points. The result reproduces and unifies many of the earlier mathematical results on equivalences of D-brane categories, including the McKay correspondence and Orlov's construction.