Semiclassical Analysis of the 3d/3d Relation

TL;DR

This paper provides evidence for a conjectured equivalence between 3d N=2 gauge theory partition functions and SL(2,R) Chern-Simons theory on mapping tori, connecting quantum field theory with hyperbolic geometry.

Contribution

It demonstrates the classical limit of the 3d N=2 partition function reproduces hyperbolic volume and Chern-Simons invariants, and shows subleading corrections match Reidemeister torsion.

Findings

Partition function limit reproduces hyperbolic volume.

Partition function limit reproduces Chern-Simons invariant.

Subleading correction reproduces Reidemeister torsion.

Abstract

We provide quantitative evidence for our previous conjecture which states an equivalence of the partition function of a 3d N=2 gauge theory on a duality wall and that of the SL(2,R) Chern-Simons theory on a mapping torus, for a class of examples associated with once-punctured torus. In particular, we demonstrate that a limit of the 3d N=2 partition function reproduces the hyperbolic volume and the Chern-Simons invariant of the mapping torus. This is shown by analyzing the classical limit of the trace of an element of the mapping class group in the Hilbert space of the quantum Teichmuller theory. We also show that the subleading correction to the partition function reproduces the Reidemeister torsion.