Large $N$ twisted partition functions in 3d-3d correspondence and Holography

TL;DR

This work analyzes the large-$N$ limit of twisted partition functions $Z_{g,p}$ for 3D $\mathcal{N}=2$ SCFTs $T_N[M]$ arising from M5-branes on closed hyperbolic 3-manifolds $M$, using the 3D-3D correspondence to relate Bethe vacua to irreducible $SL(N,\mathbb{C})$ flat connections. By exploiting perturbative invariants of complex Chern-Simons theory around two distinguished flat connections, the authors derive universal large-$N$ expressions proportional to $\mathrm{vol}(M)$ for the two Bethe vacua, and show these match the on-shell actions of Bolt-type $AdS_4$ solutions in the holographic dual. They connect these results to the squashed $S^3$ partition functions, establishing universal relations between twisted ptns and $S^3_b$ ptns in the large-$N$ limit, and highlight the role of 3D-3D mathematics (Chern-Simons perturbation theory, Ray-Singer torsion) in encoding holographic data. The findings provide a robust, geometry-driven picture of black-hole entropy counts in AdS$_4$/CFT$_3$ and suggest avenues for accessing subleading quantum corrections.

Abstract

We study the large $N$ limit of twisted partition functions on $\mathcal{M}_{g,p}$, the $S^1$ bundle of degree $p$ over a Riemann surface of genus $g$, for 3D $\mathcal{N}=2$ superconformal field theories arising as low-energy limit of wrapped $N$ M5-branes on hyperbolic 3-manifold $M$. We study contributions from two Bethe vacua which correspond to two canonical irreducible $SL(N, \mathbb{C})$ flat connections on $M$ via 3D-3D correspondence. Using mathematical results on perturbtaive Chern-Simons invariants around the flat connections, we find universal expressions for the large $N$ twisted partition functions contributed from the two Bethe vacua in term of the hyperbolic volume of $M$. The two large $N$ partition functions perfectly match the on-shell actions for two Bolt-type solutions in the holographic dual $AdS_4$ gravity respectively.