D-branes, open string vertex operators, and Ext groups

TL;DR

This paper explicitly links Ext groups to massless open string modes on D-branes in Calabi-Yau manifolds, providing concrete physical tests of derived category descriptions of D-branes.

Contribution

It explicitly computes boundary vertex operators in open string B models and clarifies their correspondence with Ext groups, confirming theoretical proposals.

Findings

Vertex operators correspond to Ext group elements

Spectral sequences are realized in BRST cohomology

Results support derived category descriptions of D-branes

Abstract

In this paper we explicitly work out the precise relationship between Ext groups and massless modes of D-branes wrapped on complex submanifolds of Calabi-Yau manifolds. Specifically, we explicitly compute the boundary vertex operators for massless Ramond sector states, in open string B models describing Calabi-Yau manifolds at large radius, directly in BCFT using standard methods. Naively these vertex operators are in one-to-one correspondence with certain sheaf cohomology groups (as is typical for such vertex operator calculations), which are related to the desired Ext groups via spectral sequences. However, a subtlety in the physics of the open string B model has the effect of physically realizing those spectral sequences in BRST cohomology, so that the vertex operators are actually in one-to-one correspondence with Ext group elements. This gives an extremely concrete physical test of recent proposals regarding the relationship between derived categories and D-branes. We check these results extensively in numerous examples, and comment on several related issues.