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Analogues of Discrete Torsion for the M-Theory Three-Form

Eric R. Sharpe

TL;DR

This work develops a mathematical framework to describe analogues of discrete torsion for the M-theory three-form C by treating C as a 2-gerbe with connection and analyzing orbifold group actions via Čech data. It shows that, in a natural simplifying regime, differences between lifts of the orbifold group to the C-field are classified by the group cohomology $H^3(\Gamma,U(1))$, with 3-cocycles realized as higher-order maps between gerbes. The paper also computes membrane twisted-sector phases on $T^3$ and demonstrates their invariance under $SL(3,Z)$, providing a membrane analogue of modular invariance. It discusses limitations due to neglecting flux quantization and E8 Chern–Simons structure and notes potential local orbifold degrees of freedom, pointing to further avenues for fully physical incorporating of these degrees of freedom.

Abstract

In this article we shall outline a derivation of the analogue of discrete torsion for the M-theory three-form potential. We find that some of the differences between orbifold group actions on the C field are classified by H^3(G, U(1)). We also compute the phases that the low-energy effective action of a membrane on T^3 would see in the analogue of a twisted sector, and note that they are invariant under the obvious SL(3,Z) action.

Analogues of Discrete Torsion for the M-Theory Three-Form

TL;DR

This work develops a mathematical framework to describe analogues of discrete torsion for the M-theory three-form C by treating C as a 2-gerbe with connection and analyzing orbifold group actions via Čech data. It shows that, in a natural simplifying regime, differences between lifts of the orbifold group to the C-field are classified by the group cohomology , with 3-cocycles realized as higher-order maps between gerbes. The paper also computes membrane twisted-sector phases on and demonstrates their invariance under , providing a membrane analogue of modular invariance. It discusses limitations due to neglecting flux quantization and E8 Chern–Simons structure and notes potential local orbifold degrees of freedom, pointing to further avenues for fully physical incorporating of these degrees of freedom.

Abstract

In this article we shall outline a derivation of the analogue of discrete torsion for the M-theory three-form potential. We find that some of the differences between orbifold group actions on the C field are classified by H^3(G, U(1)). We also compute the phases that the low-energy effective action of a membrane on T^3 would see in the analogue of a twisted sector, and note that they are invariant under the obvious SL(3,Z) action.

Paper Structure

This paper contains 18 sections, 87 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Orbifold group actions on 2-gerbes
  3. The set of orbifold group actions
  4. Differences between orbifold group actions
  5. Residual gauge invariances
  6. First level of residual gauge invariances
  7. Second level of residual gauge invariances
  8. $H^3(\Gamma, U(1))$
  9. Detailed classification of orbifold group actions
  10. Commentary
  11. Membrane twisted sector phases
  12. Twisted sector phases on $T^3$
  13. "Modular invariance" on $T^3$
  14. Review of modular invariance on $T^2$
  15. Calculation on $T^3$
  16. ...and 3 more sections

Figures (4)

  • Figure 1: Three-torus seen as open box on covering space.
  • Figure 2: Four of the twelve edges, all descending to same line.
  • Figure 3: Four of the twelve edges, all descending to same line.
  • Figure 4: Four of the twelve edges, all descending to same line.