Semiclassical Analysis of the 3d/3d Relation

TL;DR

This work provides quantitative evidence for a 3d/3d correspondence by showing that the semiclassical limit of the 3d $\mathcal{N}=2$ partition function on a duality wall matches the hyperbolic volume and Chern-Simons invariant of a mapping torus built from a once-punctured torus. The trace of a mapping class group element in quantum Teichmüller theory, analyzed via the quantum dilogarithm formalism, yields a geometric potential $V_{\rm geom}$ that reproduces the hyperbolic data at the saddle point, while $h$-dependent deformations encode the longitude and lead to the A-polynomial; subleading Gaussian fluctuations reproduce Reidemeister torsion. The results are obtained without relying on unproven dualities, linking exact 3d localization to classical hyperbolic geometry through explicit saddle-point analysis for a class of mapping tori. The study opens avenues for extending the approach to other Riemann surfaces and exploring perturbative corrections, strengthening the landscape of 3d/3d correspondences and their geometric content.

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.

Figures (4)

• Figure 1: The boundary torus for canonical triangulation of the figure eight knot complement. The gray region represents the fundamental region of the torus, and each of the two strips (consisting of four triangles) represents the boundary of a tetrahedron. The modulus of the tetrahedra are parametrized by two complex numbers $x$ and $y.$ The horizontal (vertical) dotted arrow represents the longitude (meridian) of the torus.
• Figure 2: All the four vertices of an ideal tetrahedron gather around the puncture of the torus. When we cut the tetrahedron around a puncture, the boundary is a union of four triangles, colored gray. See Gueritaud.
• Figure 3: The figure for the boundary torus, for an example $\varphi=L^2 R$. The fundamental region of the torus is shown in gray. We express $\varphi$ as a product of $L$ and $R$, and we layer the four triangles of Figure \ref{['aroundvertex']} in different ways depending $L$ or $R$. The angles in the figure are not depicted correctly, and angles with the same symbol should really the same. A more complicated example can be found in Figure 4 of Gueritaud.
• Figure 4: An ideal tetrahedron in $\mathbb{H}^3$ has all the four vertices on the boundary of $\mathbb{H}^3$, which we can take to be $\{0,1,z,\infty\} \in \mathbb{C}\cup \{\infty\}$.