3d Chern-Simons Theory from M5-branes

TL;DR

The paper provides a direct localization derivation showing that 5d ${\cal N}=2$ SYM on $S^2\times M_3$ reduces to 3d ${G_{\mathbb{C}}}$ Chern-Simons theory on $M_3$, with the imaginary level $t$ determined by $i t = \frac{8\pi^2 r}{g^2}$. This establishes a 5d lift of the 3d/3d correspondence for general gauge groups and stems from compactifying the 6d $(2,0)$ theory, with a detailed localization analysis that yields a trivial one-loop determinant. The work also discusses holomorphic factorization, links to the 6d origin, and the potential generalization to ADE and other gauge groups, along with insights from S-duality in the strong coupling regime. Overall, the results provide a concrete, mechanism-based bridge between higher-dimensional gauge theories and complex Chern-Simons theory on 3-manifolds, with implications for broader dualities and topological field theories.

Abstract

We study 5d N=2 maximally supersymmetric Yang-Mills theory with a gauge group G on S^2 x M_3, where M_3 is a 3-manifold. By explicit localization computation we show that the path-integral of the 5d N=2 theory reduces to that of the 3d G_C Chern-Simons theory on M_3, where G_C is the complexification of G. This gives a direct derivation of the appearance of the Chern-Simons theory from the compactification of the 6d (2,0) theory, confirms the predictions from the 3d/3d correspondence for G=SU(N), and suggests the generalization of the correspondence to more general gauge groups.