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

## TL;DR

This paper explores the connection between 5d supersymmetric Yang-Mills theory, 3d Chern-Simons theory, and M2-branes, demonstrating how Wilson loops relate to knot invariants and how surface operators correspond to calibrated M2-branes via AdS/CFT.

## Contribution

It proposes a novel link between 5d SYM Wilson loops and 3d Chern-Simons theory, and matches M2-brane calculations with field theory predictions using supersymmetric localization.

## Key findings

- Wilson loop expectation values compute knot invariants.
- Calibrated M2-branes match field theory predictions.
- Derived 3d Chern-Simons from 5d super-Yang-Mills.

## 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.