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Lectures on D-branes and Sheaves

E. Sharpe

TL;DR

This work surveys how D-branes in the open string B model on Calabi–Yau manifolds can be modeled by coherent sheaves, with open-string spectra counted by Ext groups and boundary OPE structures linked to Yoneda products. It develops multiple routes to realize Ext computations on the worldsheet, including vertex operators, tachyon condensation and spectral sequences, and extends the framework to flat B-field twists, orbifolds, and Higgs vevs. The notes further connect these physical constructions to derived categories, generalized complexes, and stability notions (notably pi-stability), and illustrate practical methods for calculating spectra in challenging geometries. The treatment highlights the deep interplay between physics and algebraic geometry, offering a toolkit for leveraging mathematical structures to count states and understand D-brane dynamics in string compactifications.

Abstract

These notes are a writeup of lectures given at the twelfth Oporto meeting on ``Geometry, Topology, and Physics,'' and at the Adelaide workshop ``Strings and Mathematics 2003,'' primarily geared towards a physics audience. We review current work relating boundary states in the open string B model on Calabi-Yau manifolds to sheaves. Such relationships provide us with a mechanism for counting open string states in situations where the physical spectrum calculation is nearly intractable -- after translating to mathematics, such calculations become easy. We describe several different approaches to these models, and also describe how these models are changed by varying physical circumstances -- flat B field backgrounds, orbifolds, and nonzero Higgs vevs. We also discuss mathematical interpretations of operator products, and how such mathematical interpretations can be checked physically. One of the motivations for this work is to understand the precise physical relationship between boundary states in the open string B model and derived categories in mathematics, and we outline what is currently known of the relationship.

Lectures on D-branes and Sheaves

TL;DR

This work surveys how D-branes in the open string B model on Calabi–Yau manifolds can be modeled by coherent sheaves, with open-string spectra counted by Ext groups and boundary OPE structures linked to Yoneda products. It develops multiple routes to realize Ext computations on the worldsheet, including vertex operators, tachyon condensation and spectral sequences, and extends the framework to flat B-field twists, orbifolds, and Higgs vevs. The notes further connect these physical constructions to derived categories, generalized complexes, and stability notions (notably pi-stability), and illustrate practical methods for calculating spectra in challenging geometries. The treatment highlights the deep interplay between physics and algebraic geometry, offering a toolkit for leveraging mathematical structures to count states and understand D-brane dynamics in string compactifications.

Abstract

These notes are a writeup of lectures given at the twelfth Oporto meeting on ``Geometry, Topology, and Physics,'' and at the Adelaide workshop ``Strings and Mathematics 2003,'' primarily geared towards a physics audience. We review current work relating boundary states in the open string B model on Calabi-Yau manifolds to sheaves. Such relationships provide us with a mechanism for counting open string states in situations where the physical spectrum calculation is nearly intractable -- after translating to mathematics, such calculations become easy. We describe several different approaches to these models, and also describe how these models are changed by varying physical circumstances -- flat B field backgrounds, orbifolds, and nonzero Higgs vevs. We also discuss mathematical interpretations of operator products, and how such mathematical interpretations can be checked physically. One of the motivations for this work is to understand the precise physical relationship between boundary states in the open string B model and derived categories in mathematics, and we outline what is currently known of the relationship.

Paper Structure

This paper contains 54 sections, 212 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Overview of mathematics of sheaves and Ext groups
  3. Complexes and exact sequences
  4. Sheaves
  5. Sheaf cohomology
  6. Ext groups
  7. Overview of the B model topological field theory
  8. Basics
  9. Two anomalies
  10. Open string analogue of the Calabi-Yau condition
  11. The Freed-Witten anomaly
  12. Closed string B model spectra
  13. Vertex operators for Ext groups
  14. Basic analysis
  15. A spectral sequence and its physical realization
  16. ...and 39 more sections

Figures (1)

  • Figure 1: Example of generalized complex. Each arrow is labelled by the degree of the corresponding vertex operator.