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

Sheldon Katz, Eric Sharpe

TL;DR

This paper establishes a concrete bridge between D-brane physics and derived-category mathematics by showing that massless Ramond sector open-string states between branes wrapped on complex submanifolds are counted by Ext groups. The authors compute boundary vertex operators in the open-string B model and demonstrate how BRST cohomology realizes the spectral sequences relating sheaf cohomology to Ext groups, with the Freed–Witten anomaly playing a crucial role in matching physical states to Ext elements. They analyze parallel coincident branes, branes of differing dimensions, and general intersecting branes, uncovering subtle boundary-condition effects and a new selection rule for BPS configurations. The work highlights how Serre duality is preserved when the FW twisting is properly included and provides a framework for Ext groups of complexes, advancing the physical interpretation of derived categories in D-brane physics. Overall, the paper tests and refines the derived-category–D-brane correspondence, offering a systematic BCFT approach to counting open-string states in diverse brane configurations.

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.

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

TL;DR

This paper establishes a concrete bridge between D-brane physics and derived-category mathematics by showing that massless Ramond sector open-string states between branes wrapped on complex submanifolds are counted by Ext groups. The authors compute boundary vertex operators in the open-string B model and demonstrate how BRST cohomology realizes the spectral sequences relating sheaf cohomology to Ext groups, with the Freed–Witten anomaly playing a crucial role in matching physical states to Ext elements. They analyze parallel coincident branes, branes of differing dimensions, and general intersecting branes, uncovering subtle boundary-condition effects and a new selection rule for BPS configurations. The work highlights how Serre duality is preserved when the FW twisting is properly included and provides a framework for Ext groups of complexes, advancing the physical interpretation of derived categories in D-brane physics. Overall, the paper tests and refines the derived-category–D-brane correspondence, offering a systematic BCFT approach to counting open-string states in diverse brane configurations.

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.

Paper Structure

This paper contains 32 sections, 137 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Review of the open string B model
  3. Actions and boundary conditions
  4. Two anomalies
  5. Open string analogue of the Calabi-Yau condition
  6. The Freed-Witten anomaly
  7. Parallel coincident branes on $S \hookrightarrow X$
  8. Basic analysis of massless boundary Ramond spectra
  9. Examples
  10. First subtlety: mathematics
  11. Second subtlety: physics
  12. Nontriviality of the spectral sequence
  13. Details of the differentials
  14. Physical realization of the spectral sequence
  15. Parallel branes on submanifolds of different dimension
  16. ...and 17 more sections

Figures (1)

  • Figure 1: Open string realizing map between simple complexes.