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

Abstract

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 × 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.