Chern-Simons theory from M5-branes and calibrated M2-branes

Abstract

We study a sector of the 5d maximally supersymmetric Yang-Mills theory on $S^5$ consisting of $1/8$-BPS Wilson loop operators contained within a great $S^3$ inside $S^5$. We conjecture that these observables are described by a 3d Chern Simons theory on $S^3$, analytically continued to a pure imaginary Chern-Simons level. Therefore, the expectation values of these 5d Wilson loops compute knot invariants. We verify this conjecture in the weakly-coupled regime from explicit Feynman diagram computations. At strong coupling, these Wilson loop operators lift to $1/8$-BPS surface operators in the 6d $(2,0)$ theory on $S^1\times S^5$. Using AdS/CFT, we show that these surface operators are dual to M2-branes subject to certain calibration conditions required in order to preserve supersymmetry. We compute the renormalized action of a large class of calibrated M2-branes and obtain a perfect match with the field theory prediction. Finally, we present a derivation of the 3d Chern-Simons theory from 5d super-Yang-Mills theory using supersymmetric localization, modulo a subtle issue that we discuss.

64 pages + appendices, 4 figures

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• AdS${}_7$ Black-Hole Entropy and 5D $\mathcal{N}=2$ Yang-Mills
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• Surface operators in the 6d $\mathcal{N}=(2,0)$ theory
• Topological Correlators and Surface Defects from Equivariant Cohomology
• Phases of five-dimensional supersymmetric gauge theories
• Observations on BPS observables in 6d
• Ironing out the crease
• Five-dimensional cohomological localization and squashed $q$-deformations of two-dimensional Yang-Mills theory
• Anyons on M5-Probes of Seifert 3-Orbifolds via Flux Quantization
• BPS surface operators and calibrations
• Surface Operators in Superspace
• Generating functions for Higgs/Coulomb branch operators from 1d-3d cohomological equivalence
• On the D5-brane description of $\frac{1}{4}$-BPS Wilson loops in $\mathcal{N}=4$ super Yang-Mills theory