3d TQFT from 6d SCFT

TL;DR

The paper derives a direct 3d TQFT $\mathbb{T}[X]$ from the 6d $(2,0)$ SCFT on $S^1\times S^2\times M$ by twisting along $M$ and reducing to 5d MSYM on $S^2\times M$, with the infinite-volume limit yielding a complex Chern-Simons–type theory on $M$. The F-term is taken to be the Chern-Simons functional $W=\frac{1}{2}\int_M CS(\mathcal{A})$, supplemented by corrections to preserve half of the $\mathcal{N}=(2,2)$ supersymmetry on $S^2$, and localization is used to relate the theory to a 3d path integral over magnetic flux sectors. In the infinite-volume limit, the partition function becomes a sum over $B$ of complex Chern-Simons path integrals for $(G_B)_{\mathbb{C}}$ with level $k=0$ and coupling $s=\frac{8\pi^2 r}{e^2}$, with a modified measure from one-loop determinants and a semistable integration domain. The work thus provides a concrete six-dimensional origin for the 3d/3d picture with complex Chern-Simons, clarifies measure and semistability issues, and lays groundwork for explicit calculations on general three-manifolds $M$.

Abstract

We study the six-dimensional (2,0) superconformal field theory on S^1 x S^2 x M via compactification to five dimensions, where M is a three-manifold. Twisted along M, the five-dimensional theory has a half of N = (2,2) supersymmetry on S^2, the other half being broken by a superpotential. We show that in the limit where M is infinitely large, the twisted theory reduces to a three-dimensional topological quantum field theory which is closely related to Chern-Simons theory for the complexified gauge group.