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Holonomy on D-Branes

Alan L. Carey, Stuart Johnson, Michael K. Murray

TL;DR

This work treats the $B$-field on D-branes as a Deligne cohomology (differential character) class to build anomaly-free worldsheet actions. It develops a two-pronged strategy: for torsion $B$-fields on the D-brane $Q$, bundle gerbes and bundle gerbe modules provide the necessary corrective holonomies to neutralize the global anomaly, while in the non-torsion case, infinite-dimensional bundle gerbe modules and trace-class holonomies extend Kapustin's framework to yield a well-defined action. Transgression to loop space supplies line bundles whose sections (e.g., Pfaffians, traces of holonomy) combine with worldsheet data to produce an action independent of auxiliary choices. The analysis unifies Deligne cohomology, bundle gerbes, and $C^*$-algebras, clarifying the dependence on gauge choices and establishing a geometric pathway to twisted $K$-theory and anomaly cancellation in D-brane settings.

Abstract

This paper shows how to construct anomaly free world sheet actions in string theory with $D$-branes. Our method is to use Deligne cohomology and bundle gerbe theory to define geometric objects which are naturally associated to $D$-branes and connections on them. The holonomy of these connections can be used to cancel global anomalies in the world sheet action.

Holonomy on D-Branes

TL;DR

This work treats the -field on D-branes as a Deligne cohomology (differential character) class to build anomaly-free worldsheet actions. It develops a two-pronged strategy: for torsion -fields on the D-brane , bundle gerbes and bundle gerbe modules provide the necessary corrective holonomies to neutralize the global anomaly, while in the non-torsion case, infinite-dimensional bundle gerbe modules and trace-class holonomies extend Kapustin's framework to yield a well-defined action. Transgression to loop space supplies line bundles whose sections (e.g., Pfaffians, traces of holonomy) combine with worldsheet data to produce an action independent of auxiliary choices. The analysis unifies Deligne cohomology, bundle gerbes, and -algebras, clarifying the dependence on gauge choices and establishing a geometric pathway to twisted -theory and anomaly cancellation in D-brane settings.

Abstract

This paper shows how to construct anomaly free world sheet actions in string theory with -branes. Our method is to use Deligne cohomology and bundle gerbe theory to define geometric objects which are naturally associated to -branes and connections on them. The holonomy of these connections can be used to cancel global anomalies in the world sheet action.

Paper Structure

This paper contains 23 sections, 9 theorems, 109 equations.

Table of Contents

  1. Introduction
  2. Action building
  3. Deligne cohomology
  4. Local description
  5. Holonomy of a Deligne class
  6. Local formulae
  7. Differential characters
  8. Deligne class of a torsion cohomology class
  9. Line bundles on loop space.
  10. Sections of the line bundle on loop space.
  11. A geometric interpretation
  12. Bundle gerbes
  13. Torsion bundle gerbes
  14. Lifting bundle gerbes
  15. Stable isomorphism of bundle gerbes
  16. ...and 8 more sections

Key Result

Proposition 3.1

In the situation above

Theorems & Definitions (19)

  • Example 3.1
  • Example 3.2
  • Example 3.3
  • Example 3.4
  • Example 3.5
  • Example 3.6
  • Proposition 3.1: Holonomy gluing property
  • proof
  • Theorem 4.1: Mur
  • Proposition 4.2: Mur
  • ...and 9 more