### Multiple M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory

#### Posted by Urs Schreiber

We are in the process of finalizing a little article:

Domenico Fiorenza, Hisham Sati, Urs Schreiber,

*Multiple M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory*

Below is the abstract, and, below the fold, the beginning of the introduction. A pdf with the current version is behind the above link.

We would be grateful for comments.

The article is written in a non-formal style with an audience of certain physicists in mind, but ample pointers are given to places where all the ingredients are spelled out in detail and precisely.

## Abstract

The worldvolume theory of coincident M5-branes is expected to contain a nonabelian 2-form/nonabelian gerbe gauge theory that is a higher analog of self-dual Yang-Mills theory. But the precise details – in particular the global moduli / instanton / magnetic charge structure – have remained elusive.

Here we argue that the *holographic dual* of this nonabelian 2-form field, under $\mathrm{AdS}_7/\mathrm{CFT}_6$ duality, can be deduced from anomaly cancellation.

We find this way a 7-dimensional nonabelian Chern-Simons theory of *twisted String 2-connection* fields, which, in a certain higher gauge, are given locally by non-abelian 2-forms with values in a Kac-Moody loop Lie algebra. We construct the corresponding action functional on the entire smooth moduli 2-stack of field configurations, thereby defining the theory globally, at all levels and with the full instanton structure, which is nontrivial due to the twists imposed by the quantum corrections. Along the way we explain some general phenomena of higher nonabelian gauge theory that we need.

## Introduction

The quantum field theory (QFT) on the worldvolume of M5-branes is known to be a 6-dimensional (2,0)-superconformal theory that contains a 2-form potential field $B_2$, whose 3-form field strength $H_3$ is self-dual. Whatever it is precisely and in generality, this QFT has been argued to be the source of deep physical and mathematical phenomena, such as Montonen-Olive S-duality, geometric Langlands duality, and Khovanov homology. Yet, and despite this interest, a complete description of the precise details of this QFT is still lacking.

In particular, as soon as one considers the worldvolume theory of *several coincident* M5-branes, the 2-form appearing locally in this 6d QFT is expected to be *nonabelian* (to take values in a nonabelian Lie algebra). But a description of this *nonabelian gerbe theory* has been elusive. Here we present a consistent formulation of nonabelian 2-form fields and propose dynamics for them under holography.

On the other hand, for a single M5-brane the Lagrangian of the theory has been formulated. Furthermore, in this abelian case there is a *holographic dual* description of the 6d theory by 7-dimensional abelian Chern-Simons theory, as part of $\mathrm{AdS}_7/\mathrm{CFT}_6$-duality.

We give here an argument, following Witten96, Witten98 but taking the quantum anomaly cancellation of the M5-brane in 11-dimensional supergravity into account, that in the general case the $\mathrm{AdS}_7$/$\mathrm{CFT}_6$-duality involves a 7-dimensional nonabelian Chern-Simons action that is evaluated on higher nonabelian gauge fields which we identify as _twisted 2-connections over the String-2-group.

Then we give a precise description of a certain canonically existing 7-dimensional nonabelian gerbe-theory on boundary values of quantum-corrected supergravity field configurations in terms of nonabelian differential cohomology. We show that this has the properties expected from the quantum anomaly structure of 11-dimensional supergravity. In particular, we discuss that there is a higher gauge in which these field configurations locally involve non-abelian 2-forms with values in the Kac-Moody central extension of the loop Lie algebra of the special orthogonal Lie algebra $\mathfrak{so}$ and of the exceptional Lie algebra of E8. We also describe the global structure of the smooth moduli 2-stack of field configurations, which is more subtle.

## Re: Multiple M5-branes, String 2-connections, and 7d nonabelian Chern-Simons theory

Does any extra light get shone on geometric Langlands duality and Khovanov homology from your paper?

Typos:

p. 48 different but but equivalent

p. 50 The degeewise ordinary