# Teichmuller TQFT The Teichmüller space $\mathcal{T}_{g,n}$ of a 2D manifold $C$ with genus $g$ and $n$ boundaries is the universal covering space of $Cs moduli space $\mathcal{M}_{g,n}$. By definition,$\mathcal{M}_{g, n}=\mathcal{T}_{g, n} / \operatorname{Map}\left(\Sigma_{g, n}\right),$where $\operatorname{Map}\left(\Sigma_{g, n}\right)$ is the mapping class group. It has a symplectic structure. This symplectic structure is quantised and identified with the Hilbert space of a quantum mechanical system. ## Relation to $\mathcal{N}=4$ SYM The Teichmüller TQFT on manifolds of the form $\mathbb{R}\times C$ can be obtained by dimensionally reducing $\mathcal{N}=4$ SYM. Consider the 4d theory $\Sigma \times C$ and compactify on $C$ leads to a sigma-model on the worldsheet $\Sigma$ with the target space being the Hitchin moduli space for $C$, where $\Sigma= \mathcal{I}\times \mathbb{R}$. To define Teichmüller theory on a general 3-manifold $W$, reduce $\mathcal{N}=4$ SYM on $\mathcal{I}\times W$. In summary, Teichmüller TQFT is the four-dimensional topologically twisted $\mathcal{N} = 4$ super Yang-Mills theory, put on an interval with particular boundary conditions. ## Relation to CS theory The phase space of $PSL(2, R)$ Chern-Simons theory is the moduli space $\mathcal{M}^\mathbb{R}$ of flat $PSL(2, R)$ connections on $C$. Flat $\mathrm{SL}(2,\mathbb{R})$ bundles are classified by their Euler number. The Teichmuller space is the component with the maximal Euler number. The CS action endows $\mathcal{T}$ with a symplectic form which is a multiple of the Weil-Petersson form $\omega_\mathrm{WP}$. Using it to quantise produces a Hilbert space $\mathcal{H}\simeq L^2(\mathbb{R}^{3g-3})$. In [[2017#Mikhaylov]], it is proposed that Teichmüller TQFT can be defined by an analytically-continued Chern-Simons path integral with an unusual integration cycle. ## Relation to 3d-3d correspondence The 6d $(2, 0)$ theory of type $A_1$ on a manifold $S_b^3\times W$ where $S_b^3$ is a squashed 3-sphere with squashing parameter $b$. Reducing on the sphere leads to a CS theory on $W$. For $W=\mathbb{R}\times C$, this gives Hilbert space associated to $C$. Alternatively, reducing on $C$ first gives a 4d $\mathcal{N}=2$ $\mathcal{S}$-class theory whose Hilbert space on $S^3_b$ gives the space of Liouville conformal blocks, via [[0649 AGT correspondence|AGT]]. ## Relation to Virasoro TQFT Both Teichmüller TQFT and [[0596 Virasoro TQFT|Virasoro TQFT]] are obtained by quantising Teichmüller space. ## Refs - quantisation - [[1990#Verlinde]] (original) - [[2005#Teschner]]: factorisation properties and similarities with modular functor - [[1997#Kashaev]] - [[1999#Chekhov, Fock]] - [[2003#Teschner (Aug)]] - [[2003#Teschner (Mar)]] - [[2010#Teschner]] - triangulation formulation - [[2011#Andersen, Kashaev]] - [[2013#Andersen, Kashaev]] - Teichmüller TQFT formulated from an unusual integration circle of analytically continued [[0089 Chern-Simons theory|CS theory]] - [[2017#Mikhaylov]] - surfaces with boundary - [[2006#Luo]]: a new parametrisation of Teichmuller space for surfaces with boundary