Mathematical Aspects of D-branes

TL;DR

This paper reviews how D-branes with nonzero Higgs vevs can be described using sheaves, linking physical configurations to mathematical objects and verifying this with open string spectrum comparisons.

Contribution

It introduces a sheaf-based mathematical framework for D-branes with Higgs vevs, expanding the class of sheaves that can represent D-branes.

Findings

Open string spectra match Ext groups between sheaves.

The sheaf description extends previous understanding of D-brane configurations.

Examples demonstrate the effectiveness of the sheaf encoding.

Abstract

In these notes we review recent work on describing D-branes with nonzero Higgs vevs in terms of sheaves, which gives a physical on-shell D-brane interpretation to more sheaves than previously understood as describing D-branes. The mathematical ansatz for this encoding is checked by comparing open string spectra between D-branes with nonzero Higgs vevs to Ext groups between the corresponding sheaves. We illustrate the general methods with a few examples.