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

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.