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Discrete Torsion

Eric R. Sharpe

TL;DR

This work provides a geometric and cohomological foundation for discrete torsion by treating it as a choice of orbifold group action on the $B$-field. It derives the classification by $H^2(\Gamma,U(1))$, derives the corresponding twisted-sector phases in string partition functions, and recovers M. Douglas's projective-brane actions within this framework, while also revealing additional shift-orbifold degrees of freedom in general. The analysis extends to D-branes, Vafa–Witten's CY moduli results, and local singularities, showing that discrete torsion is a natural, purely mathematical construct with direct physical consequences. The results hold for general (not necessarily freely acting) and nonabelian groups, and illuminate how topological data and flat/B-field twists shape stringy amplitudes and brane dynamics, with connections to reflexive sheaves on singular spaces and local orbifold phenomena.

Abstract

In this article we explain discrete torsion. Put simply, discrete torsion is the choice of orbifold group action on the B field. We derive the classification H^2(G, U(1)), we derive the twisted sector phases appearing in string loop partition functions, we derive M. Douglas's description of discrete torsion for D-branes in terms of a projective representation of the orbifold group, and we outline how the results of Vafa-Witten fit into this framework. In addition, we observe that additional degrees of freedom (known as shift orbifolds) appear in describing orbifold group actions on B fields, in addition to those classified by H^2(G, U(1)), and explain how these new degrees of freedom appear in terms of twisted sector contributions to partition functions and in terms of orbifold group actions on D-brane worldvolumes. This paper represents a technically simplified version of prior papers by the author on discrete torsion. We repeat here technically simplified versions of results from those papers, and have included some new material.

Discrete Torsion

TL;DR

This work provides a geometric and cohomological foundation for discrete torsion by treating it as a choice of orbifold group action on the -field. It derives the classification by , derives the corresponding twisted-sector phases in string partition functions, and recovers M. Douglas's projective-brane actions within this framework, while also revealing additional shift-orbifold degrees of freedom in general. The analysis extends to D-branes, Vafa–Witten's CY moduli results, and local singularities, showing that discrete torsion is a natural, purely mathematical construct with direct physical consequences. The results hold for general (not necessarily freely acting) and nonabelian groups, and illuminate how topological data and flat/B-field twists shape stringy amplitudes and brane dynamics, with connections to reflexive sheaves on singular spaces and local orbifold phenomena.

Abstract

In this article we explain discrete torsion. Put simply, discrete torsion is the choice of orbifold group action on the B field. We derive the classification H^2(G, U(1)), we derive the twisted sector phases appearing in string loop partition functions, we derive M. Douglas's description of discrete torsion for D-branes in terms of a projective representation of the orbifold group, and we outline how the results of Vafa-Witten fit into this framework. In addition, we observe that additional degrees of freedom (known as shift orbifolds) appear in describing orbifold group actions on B fields, in addition to those classified by H^2(G, U(1)), and explain how these new degrees of freedom appear in terms of twisted sector contributions to partition functions and in terms of orbifold group actions on D-brane worldvolumes. This paper represents a technically simplified version of prior papers by the author on discrete torsion. We repeat here technically simplified versions of results from those papers, and have included some new material.

Paper Structure

This paper contains 34 sections, 99 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Review of orbifold $U(1)$ Wilson lines
  3. Orbifold group actions on vector fields
  4. Differences between orbifold group actions
  5. General analysis
  6. The set of orbifold group actions is a set
  7. Derivation of $H^2(\Gamma, U(1))$
  8. Orbifold group actions on $B$ fields
  9. Differences between orbifold group actions
  10. $H^2(\Gamma, U(1))$
  11. Detailed classification of orbifold group actions
  12. Commentary
  13. Derivation of twisted sector phases
  14. Analogue for orbifold $U(1)$ Wilson lines
  15. One loop
  16. ...and 19 more sections

Figures (4)

  • Figure 1: Lift of closed loop to covering space.
  • Figure 2: A twisted sector contribution to the one-loop partition function.
  • Figure 3: Degeneration of a genus two Riemann surface.
  • Figure 4: A twisted sector contribution to the two-loop partition function.