Complex Chern-Simons from M5-branes on the Squashed Three-Sphere

Clay Cordova, Daniel L. Jafferis

TL;DR

This work shows that dimensionally reducing the 6d $(2,0)$ M5-brane theory on a squashed $S^3_$ yields complex Chern-Simons theory with gauge group $rak{g}_{}$ in three dimensions. The squashing parameter $$ sets the imaginary CS level to $=$ with the real part fixed at $k=1$, and the fermionic sector naturally acts as Faddeev-Popov ghosts for the emergent noncompact gauge symmetry. A $Q$-exact deformation localizes the path integral, and integrating out heavy modes produces the FP determinant, revealing a nonperturbative contour that defines the complex CS path integral. The result implies a precise equality between the squashed-sphere partition function of the 3d $$-theory $T_{rak{g}}(M_3)$ and the CS partition function on $M_3$ with gauge group $rak{g}_{}$, with broad implications for the 3d-3d correspondence and related dualities.

Abstract

We derive an equivalence between the (2,0) superconformal M5-brane field theory dimensionally reduced on a squashed three-sphere, and Chern-Simons theory with complex gauge group. In the reduction, the massless fermions obtain an action which is second order in derivatives and are reinterpreted as ghosts for gauge fixing the emergent non-compact gauge symmetry. A squashing parameter in the geometry controls the imaginary part of the complex Chern-Simons level.

Complex Chern-Simons from M5-branes on the Squashed Three-Sphere

TL;DR

This work shows that dimensionally reducing the 6d

M5-brane theory on a squashed

yields complex Chern-Simons theory with gauge group

in three dimensions. The squashing parameter

sets the imaginary CS level to

with the real part fixed at

, and the fermionic sector naturally acts as Faddeev-Popov ghosts for the emergent noncompact gauge symmetry. A

-exact deformation localizes the path integral, and integrating out heavy modes produces the FP determinant, revealing a nonperturbative contour that defines the complex CS path integral. The result implies a precise equality between the squashed-sphere partition function of the 3d

-theory

and the CS partition function on

with gauge group

, with broad implications for the 3d-3d correspondence and related dualities.

Abstract

We derive an equivalence between the (2,0) superconformal M5-brane field theory dimensionally reduced on a squashed three-sphere, and Chern-Simons theory with complex gauge group. In the reduction, the massless fermions obtain an action which is second order in derivatives and are reinterpreted as ghosts for gauge fixing the emergent non-compact gauge symmetry. A squashing parameter in the geometry controls the imaginary part of the complex Chern-Simons level.