Teichmuller TQFT vs Chern-Simons Theory

TL;DR

This work develops a concrete, physically grounded definition of Teichmüller TQFT as an analytically continued Chern-Simons theory with a special integration cycle governed by Kapustin–Witten flow and Nahm pole boundaries. It demonstrates that Teichmüller TQFT is dual to complex SL(2,C) Chern-Simons theory at integer level k=1 and derives two variants CS-I and CS-II via six-dimensional and holographic reductions, clarifying their parameter relations. The paper also constructs a brane-based quantization of Teichmüller space, connects Teichmüller TQFT to Liouville conformal blocks, and analyzes hyperbolic 3-manifold asymptotics through Andersen–Kashaev-type conjectures, Lefschetz thimbles, and perturbative checks. Together these results illuminate the role of Teichmüller TQFT within the 3d–3d correspondence, modular functor structures, and holographic descriptions of Virasoro blocks, while outlining key open questions about integration cycles, dualities, and generalizations to other gauge groups.

Abstract

Teichmüller TQFT is a unitary 3d topological theory whose Hilbert spaces are spanned by Liouville conformal blocks. It is related but not identical to PSL(2,R) Chern-Simons theory. To physicists, it is known in particular in the context of 3d-3d correspondence and also in the holographic description of Virasoro conformal blocks. We propose that this theory can be defined by an analytically-continued Chern-Simons path-integral with an unusual integration cycle. On hyperbolic three-manifolds, this cycle is singled out by the requirement of invertible vielbein. Mathematically, our proposal translates a known conjecture by Andersen and Kashaev into a conjecture about the Kapustin-Witten equations. We further explain that Teichmüller TQFT is dual to complex SL(2,C) Chern-Simons theory at integer level k=1, clarifying some puzzles previously encountered in the 3d-3d correspondence literature. We also present a new simple derivation of complex Chern-Simons theories from the 6d (2,0) theory on a lens space with a transversely-holomorphic foliation.

Figures (13)

• Figure 1: The worldsheet of a string, whose space of states is the quantization of the Teichmüller space. The brane $\mathcal{B}_c$ on the right is coisotropic, and the brane $\mathcal{B}_{\mathcal{T}}$ on the left is Lagrangian, supported on the Hitchin section.
• Figure 2: Teichmüller TQFT on a three-manifold $W$ can be engineered by the four-dimensional $\mathcal{N}=4$ super Yang-Mills theory on $W\times\mathcal{I}$ with the D5-type and the NS5-type boundary conditions at the two ends of the interval.
• Figure 3: The functional $h$ grows in the upward vertical direction. The integration cycles $\mathcal{S}_{y_0}$ and $\mathcal{S}_{y_1}$, $y_0<y_1$, are obtained by flowing downward from the Nahm pole. If $y_1$ is large, the flows (shown in blue) that end up at finite points on $\mathcal{S}_{y_1}$ tend to spend most of their time near critical points $\mathbb{a}_1$, $\mathbb{a}_2$. This defines the decomposition of $\mathcal{S}_{y_1}$ into the Lefschetz thimbles $\mathcal{C}_{{\mathbb a}_1}$ and $\mathcal{C}_{{\mathbb a}_2}$.
• Figure 4: We can stretch and cut the interval $\mathcal{I}$ to factorize the partition function on $W\times\mathcal{I}$ into products of partition functions in two half-spaces $W\times{\mathbb R}^+$. The partition functions in the left and the right half-space, respectively, are $Z^{\rm D5}_{\bf a}(\widetilde{q})$ and $Z^{\rm NS5,top}_{\bf a}(b^2)$.
• Figure 5: Profile of the function $h(f,g)$. The upper and the lower critical points $(1,0)$ and $(-1,0)$ correspond to the geometric and the conjugate geometric flat connections, respectively. The model solution descends from the hill on the left and ends at $\mathcal{A}_{\overline{\rm geom}}$.