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

Domenico Fiorenza, Hisham Sati, Urs Schreiber

TL;DR

This work identifies a seven-dimensional nonabelian Chern-Simons theory on twisted String-2-connections as the holographic dual to the six-dimensional $(0,2)$ M5-brane theory, with anomaly cancellation in M-theory motivating the global moduli via higher differential cohomology. The authors construct a globally defined action on the full moduli 2-stack of String-2-connections, incorporating the $I_8$-polynomial and Pontryagin-class refinements, and show that locally the theory is governed by a 7d CS term with fields valued in affine Kac-Moody extensions of loop algebras, while the global structure encodes nontrivial instanton sectors. A detailed framework is developed using higher stacks, $L_ ext{infty}$-algebras, and differential characteristic maps to realize higher CS functionals and their level quantization, including cup-product constructions and twisted String structures (e.g., String^{2a}, String^{2DD}). The analysis covers ADE generalizations, the role of E8 twists, and the relation to D-brane boundary phenomena, offering a comprehensive, gauge-invariant scheme for higher CS theories across dimensions with potential implications for holography, anomaly cancellation, and nonabelian 2-form dynamics on M-branes.

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 deduce from anomaly cancellation a natural candidate for the holographic dual of this nonabelian 2-form field, under AdS7/CFT6-duality. We find this way a 7-dimensional nonabelian Chern-Simons theory of String 2-connection fields, which, in a certain higher gauge, are given locally by non-abelian 2-forms with values in an affine Kac-Moody 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.

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

TL;DR

This work identifies a seven-dimensional nonabelian Chern-Simons theory on twisted String-2-connections as the holographic dual to the six-dimensional M5-brane theory, with anomaly cancellation in M-theory motivating the global moduli via higher differential cohomology. The authors construct a globally defined action on the full moduli 2-stack of String-2-connections, incorporating the -polynomial and Pontryagin-class refinements, and show that locally the theory is governed by a 7d CS term with fields valued in affine Kac-Moody extensions of loop algebras, while the global structure encodes nontrivial instanton sectors. A detailed framework is developed using higher stacks, -algebras, and differential characteristic maps to realize higher CS functionals and their level quantization, including cup-product constructions and twisted String structures (e.g., String^{2a}, String^{2DD}). The analysis covers ADE generalizations, the role of E8 twists, and the relation to D-brane boundary phenomena, offering a comprehensive, gauge-invariant scheme for higher CS theories across dimensions with potential implications for holography, anomaly cancellation, and nonabelian 2-form dynamics on M-branes.

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 deduce from anomaly cancellation a natural candidate for the holographic dual of this nonabelian 2-form field, under AdS7/CFT6-duality. We find this way a 7-dimensional nonabelian Chern-Simons theory of String 2-connection fields, which, in a certain higher gauge, are given locally by non-abelian 2-forms with values in an affine Kac-Moody 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.

Paper Structure

This paper contains 44 sections, 1 theorem, 13 equations.

Table of Contents

  1. Introduction
  2. Evidence for and ingredients of the 7d nonabelian gerbe theory
  3. The M5-brane worldvolume theory: loop groups and the String group
  4. Nonabelian gerbes and 2-gerbes with connections.
  5. Loop group and String 2-group degrees of freedom on the M5-brane.
  6. M-branes and $L_\infty$-algebras, Lie $n$-algebras and "3-algebras"
  7. Holography and Chern-Simons theory
  8. $\mathrm{AdS}$-compactifications.
  9. The minimal supergravity in seven dimensions.
  10. The (0, 1) theory as dual to AdS${}_7$.
  11. The anomaly-corrected nonabelian 7d Chern-Simons term.
  12. The abelian theory.
  13. The nonabelian theory.
  14. Which gauge group(s)?
  15. Why loop bundles?
  16. ...and 29 more sections

Key Result

Proposition 4.5.1

(i) in terms of the strict $\mathfrak{string}$ Lie 2-algebra $(\hat{\Omega}\mathfrak{so}\to P_*\mathfrak{so})$, an object in $\mathbf{H}(\Sigma, \mathbf{B} \mathrm{String}_{\mathrm{conn}})$ is the datum of a pair of nonabelian differential forms $A \in \Omega^1(\Sigma, P_* \mathfrak{so})$, $B \in \O

Theorems & Definitions (2)

  • Remark 3.6.1
  • Proposition 4.5.1