Gerbes, M5-Brane Anomalies and E_8 Gauge Theory

TL;DR

The paper develops a geometric framework of gerbes to capture higher-form gauge data in string/M-theory, starting from abelian gerbes for the NS-NS 3-form $H$ and extending to abelian 2-gerbes for the M-theory 4-form $G$. It then introduces twisted nonabelian gerbes, specifically twisted $\tilde{\Omega} E_8$-gerbes, as the appropriate structure describing global anomalies for stacks of M5-branes and their cancellation via anomaly inflow from 11-dimensional supergravity. The authors show how these twisted nonabelian gerbes encode the nonabelian 2-form gauge data on multiple M5-branes and relate them to twisted bundles arising on D-branes after dimensional reduction to Type IIA. The construction leverages Deligne cohomology, holonomy, and Chern-Simons 2-gerbes to formulate consistent actions and anomaly-cancellation conditions, unifying brane physics with topological higher-gauge structures. Overall, the work provides a rigorous topological toolkit for understanding M5-brane global anomalies and their relation to $E_8$-based descriptions of the $C$-field and to D-branes in lower dimensions.

Abstract

Abelian gerbes and twisted bundles describe the topology of the NS-NS 3-form gauge field strength H. We review how they have been usefully applied to study and resolve global anomalies in open string theory. Abelian 2-gerbes and twisted nonabelian gerbes describe the topology of the 4-form field strength G of M-theory. We show that twisted nonabelian gerbes are relevant in the study and resolution of global anomalies of multiple coinciding M5-branes. Global anomalies for one M5-brane have been studied by Witten and by Diaconescu, Freed and Moore. The structure and the differential geometry of twisted nonabelian gerbes (i.e. modules for 2-gerbes) is defined and studied. The nonabelian 2-form gauge potential living on multiple coinciding M5-branes arises as curving (curvature) of twisted nonabelian gerbes. The nonabelian group is in general $\tildeΩE_8$, the central extension of the E_8 loop group. The twist is in general necessary to cancel global anomalies due to the nontriviality of the 11-dimensional 4-form G field strength and due to the possible torsion present in the cycles the M5-branes wrap. Our description of M5-branes global anomalies leads to the D4-branes one upon compactification of M-theory to Type IIA theory.