Schreiber 7d Chern-Simons theory and the 5-brane

An article that we have written:

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• Domenico Fiorenza, Hisham Sati, Urs Schreiber,

  
  

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

  
  

  Advances in Theoretical and Mathematical Physics,

  Volume 18, Number 2 (2014) p. 229-321

  
  

  download:

  • arXiv:1201.5277

  • doi:10.4310/ATMP.2014.v18.n2.a1

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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 stack / global instanton / magnetic charge structure – have remained elusive. Here we argue that the holographic dual of this nonabelian 2-form field, under AdS7/CFT6 duality, can be deduced from anomaly cancellation in 11-dimensional supergravity. 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.

The companion article The moduli 3-stack of the C-field provides mathematical details on the model for the supergravity C-field used here.

Introduction

The quantum field theory (QFT) on the worldvolume of M5-branes is known to be a 6d (2,0)-supersymmetric QFT that contains a connection on a 2-bundle $B_2$, whose 3-form field strength $H_3$ is self-dual, $H_3 = \star H_3$. 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 (a gerbe is a “higher analog” of a gauge bundle. Here we present a consistent formulation of nonabelian 2-form fields and propose dynamics for them under holography.

For a single M5-brane the Lagrangian of the theory has been formulated. Furthermore, in this abelian case there is, a holograohic dual description of the 6d theory by 7-dimensional abelian Chern-Simons theory, as part of AdS7/CFT6-duality. We give here an argument, following Witten96 and Witten98 but taking the quantum anomaly cancellation of the M5-brane in 11-dimensional supergravity into account, that in the general case the AdS7/CFT6-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 $\mathfrak{e}_8$. But we also describe and explicitly construct the global structure of the moduli 2-stack of field configurations, which is more subtle.

Related projects

• differential cohomology in a cohesive topos

  • Synthetic Quantum Field Theory

    • Quantum gauge field theory in Cohesive homotopy type theory

    • Type-semantics for quantization

  • cohomology

    • Principal ∞-bundles – theory, presentations and applications

    • Fivebrane structures

  • ∞-Chern-Weil theory

    • L-∞ algebra connections

    • Cech cocycles for differential characteristic classes

    • Twisted differential structures

      • Twisted Differential String and Fivebrane Structures

      • The moduli 3-stack of the C-field

  • ∞-Chern-Simons theory

    • A higher stacky perspective on Chern-Simons theory

    • Higher Chern-Weil Derivation of AKSZ Sigma-Models

    • nonabelian 7d Chern-Simons theory and the 5-brane

    • Extended higher cup-product Chern-Simons theories

  • ∞-Wess-Zumino-Witten theory

    • The brane bouquet

    • Obstruction theory for parameterized higher WZW terms

  • Higher geometric prequantum theory

    • Classical field theory via higher differential geometry

    • Local prequantum field theory

    • L-∞ algebras of local observables from higher prequantum bundles

    • Higher extensions of mapping class groups

  • Motivic quantization of local prequantum field theory

    • Geometric quantization of symplectic and Poisson manifolds

    • Cohomological quantization of local prequantum boundary field theory

References

Relevant structures in string theory are reviewed and discussed in

• Hisham Sati, Geometric and topological structures related to M-branes , part I (arXiv:1001.5020), part II: Twisted $String$ and $String^c$ structures (arxiv/1007.5419); part III: Twisted higher structures (http://arxiv.org/abs/1008.1755)

Notes of a survey of relevant background material on infinity-Chern-Simons theory are in

• Urs Schreiber, Chern-Simons terms on higher moduli stacks, talk at Hausdorff Institute, Bonn (2011) (pdf)

For a comprehensive list of references see differential cohomology in a cohesive topos – references.