Jorgen Andersen has an ambitious program for solving several open problems in topology:
1) Does the mapping class group have Kazhdan's property T?
2) Are certain quantum representations of the mapping class group asymptotically faithful?
3) Is the Volume Conjecture true?
4) Is the Asymptotic Expansion Conjecture true?
5) Do Vassiliev invariants detect knottedness?
Problems 1,2 and 4 have been claimed by Andersen, and the details seem to be complete. Number 5 has been claimed in the past by Andersen, but I do not know if all details have appeared.
Andersen gave a series of talks at MPIM in Fall 2014 on this program, which I will try to summarize here, despite it being outside my area of expertise.
Andersen mentioned that when the Jones polynomial was introduced by Vaughan Jones back in 1984 or so, Atiyah issued the challenge of finding an intrinsic three dimensional interpretation of it. Such an interpretation was found by Witten in the form of a quantum field theory, called quantum Chern Simons theory, but it was not mathematically rigorous. This set the stage for further work trying to rigorize Witten's insight.
Andersen went on to list four methods of doing this, which are now known to all give the same answer. Apparently all of these areas make sense for an arbitrary "gauge group," but the following is all for $SU(N)$.
1. Witten-Reshetikhin-Turaev TQFT:
$Z^{(k)}_N, k\geq 0, N\geq 2$ is a topological field theory called the WRT TQFT, constructed from the representation theory of the quantum group $U_q(\mathfrak{sl}_2(\mathbb C))$ at the roots of unity $q=e^{2\pi i/(k+ N)}$. Andersen mentioned that this is essentially a modular tensor category.
2. Skein interpretation:
Blanchett, Habbeger-Masbaum-Vogel built $Z_2^{(k)}$ just using skein theory (the Kauffman bracket.) Later Blanchett did this for all $N$. This construction has the advantage of being elementary and accessible to low dimensional topologists.
3. Conformal Field Theory: Andersen and Ueno built a modular functor $\overline{V}^{(k)}_{\mathfrak g}$ which leads to the TQFT $Z_N^{(k)}$.
4. Geometric quantization. The idea here is to apply geometric quantization to moduli space of flat $SU(N)$ connections. This is known to give the 2 dimensional part of $Z_N^{(k)}$. (Laszlo)
So in summary, constructions 1 and 2 are equivalent almost by definition, the equivalence of 2 and 3 is due to Andersen-Ueno, and the equivalence of 3 and 4 is due to Laszlo.
For the rest of the post, we will sketch the definition of $Z_N^{(k)}(\Sigma)$ for a surface $\Sigma$ using the geometric quantization approach. (In a follow-up post, the relations to the open problems above will be discussed.) The executive summary is that $Z_N^{(k)}(\Sigma)$ is the space of covariantly constant sections of a tensor product of line bundles $H^{(k)}\otimes \mathbb L_k$ on Teichmüller space, with respect to a flat connection $\nabla^H\otimes\nabla^{\mathbb L_k}$. The bundle $H^{(k)}$ is the Hitchin bundle with Hitchin connection $\nabla^H$, which is projectively flat. (The holonomy around small circles is multiplication by a scalar.) The new bundle and connection are the contribution of Andersen and Ueno and have the property that they make the tensor product actually flat.
Moduli space of flat $SU(N)$ connections. Let $\Sigma$ be a closed oriented surface, and let $M$ denote the moduli space of flat $SU(N)$-connections on $\Sigma$. It has the following nice description $$M=Hom(\pi_1(\Sigma),SU(N))/SU(N)$$ where we divide by conjugation, although we will work in terms of connections.
There is a stratification of $M$ which we look at part of: $M=M'\cup M''$, where $M'$ is the moduli space of irreducible representations in the strong sense that the center of the image $Z(\rho(\pi_1))$ is equal to the whole center $Z({SU(N)})$.
It turns out that $M'$ is a smooth symplectic manifold of dimension $(2g-2) \mathrm{dim}(SU(N))$ (if $g\geq 2$). The symplectic form is the "Goldman symplectic form," which we now construct.
Let $A$ be a flat $SU(N)$ connection on $\Sigma$ such that $[A]\in M'$. There is a differential
$$d_A\colon \Omega^i(\Sigma,\mathfrak g)\to \Omega^{i+1}(\Sigma,\mathfrak g),$$
where $\mathfrak{g}$ is the Lie algebra of $SU(N)$ and $\Omega^{i}(\Sigma, \mathfrak g)$ consists of $\mathfrak g$-valued $i$-forms. The fact that the curvature $F_A=0$ implies $d^2_A=0$.
Fact: $T_{[A]} M'=H^1(\Sigma,d_A)$. Let $\phi_1,\phi_2\in \Omega^1(\Sigma,\mathfrak g)$ such that $d_A\phi_i=0$, and fix an $SU(N)$-invariant form $\langle,\rangle\colon\mathfrak g\times\mathfrak g\to \mathbb R$, which is a multiple of the Killing form. Now the Goldman symplectic form is defined as
$$\omega([\phi_1,\phi_2]):=\int_\Sigma \langle\phi_1\wedge\phi_2\rangle.$$
Theorem (Goldman): $\omega\in \Omega^2(M')$ satisfies $d\omega=0$ and $\omega^{\wedge m}\neq 0$. (Here $\dim M'=2m$.) In other words, it is a symplectic form.
Prequantum line bundle: Consider a complex line bundle $L\to M'$, and a connection $\nabla$ in $L$ over $M'$. Let $(,)$ be a hermitian structure in $L$, such that $d(S_1,S_2)=(\nabla s_1,s_2)+(s_2\nabla s_2)$ and $F_\nabla=-i\omega$. This is called a prequantum line bundle, and exists iff $[\omega]\in im(H^2((M',\mathbb Z))\to H^2(M',\mathbb R))$. The fact is that one can choose our invariant form $\langle,\rangle$ such that this is satisfied. If $\pi_1(M')$ is simply connected, this line bundle is unique, and indeed $M'$ is simply connected in our case.
Now define $\mathcal H^{(k)}=C^{\infty}(M',L^k)$ for $k\in \mathbb Z_+$, where $L^k$ is the $k$th tensor power of the line bundle $L$. We have
$$P^{(k)}\colon C^\infty(M')\to Op(\mathcal H^{(k)})$$ defined by $P^{(k)}_f=\frac{1}{k}\nabla_{X_f}+if,$ where $i_{X_f}\omega=df$ defines $X_f$. We have
$$[P^{(k)}_f,P^{(k)}_g]=P_{\{f,g\}},$$ where $\{f,g\}=\omega(X_f,X_g)$ is the Poisson bracket.
Connection with Chern-Simons: Let $\mathcal L$ be the bundle $(\mathcal A_{\Sigma}\times\mathbb C )\to \mathcal A_{\Sigma}$ where $\mathcal A_\Sigma$ is the space of all flat $SU(N)$ connections on $\Sigma$. It is acted on by the gauge group $\mathcal G_\Sigma$. Choose a path in $\mathcal A_\Sigma$ from $A$ to $g^*A$ for $g\in\mathcal G$. We can think of $A_t$ as a connection on $\Sigma\times I$. Now consider multiplication by $e^{2\pi iCS(A_t)}$ as a map $\mathbb C\to \mathbb C$, and let $L=\mathcal L/\mathcal G_{\Sigma}|_{M'}$. This gives an explicit construction of $L$. This is independent of the path chosen since $CS(A)\in\mathbb Z$ for closed three manifolds.
Polarization: Choose a complex structure on $M'$. That is, choose an $I\colon TM'\to TM'$ where $I^2=-id$, $\omega(IX,IY)=\omega(X,Y),$ for $X,Y\in TM'$ and $I$ is integrable. (Locally there exist complex coordinates where $I$ acts as $i$.)
$\ker(I-i)=E(I,i)\subset TM'\otimes\mathbb C$ is a lagrangian subbundle. (Here the notation $E(A,\lambda)$ refers to the $\lambda$-eigenspace of the operator $A$.)
Let $\mathcal T$ be Teichmüller space and $*_\sigma\colon\Omega^1(\Sigma, \mathfrak g)\to \Omega^1(\Sigma, \mathfrak g)$ the associated Hodge star operator, where $*_\sigma^2=-id$. Hodge theory tells us that
$$H^1(\Sigma,d_A)=\ker \Delta_A=\ker d_A\cap\ker d_A^*$$
where $\Delta_A=d_Ad_A^*+d_A^*d_A$ and $d_A^*=-*_\sigma d_A*_\sigma$. Now define $I_\sigma=-*_\sigma$.
Define $H^{(k)}=\{s\in C^\infty(M',L^k):\nabla_zs=0 \,\forall z\in C^\infty(M',T'_\sigma)\},$ where $(T'_\sigma)_A=E(I_\sigma,i)_A\subset H^1(\Sigma,d_A)\otimes\mathbb C$. $\nabla^{0,1}_\sigma=\frac{1}{2}(Id+|\frac{i}{2}I_\sigma)$, $\nabla=\overline{\partial}$ operator in $L^k$.
The $k$th tensor power $L_\sigma^k=L^k$ is a holomorphic line bundle over $(M',I_\sigma)$. Then $H^{(k)}_\sigma=H^0(M'_\sigma, L^k)$, the space of holomorphic sections.
In summary, we have a bundle $\mathcal T\times\mathcal H^{(k)}\to \mathcal T$ and $H^{(k)}\subset \mathcal T\times\mathcal H^{(k)}$.
Hitchin Connection: $\mathbb\nabla^H$ is a connection in the bundle $H^{(k)}$ over Teichmüller space, which is projectively flat. (I.e. transporting around small circles results in a scalar times the identity.) The curvature satisfies $F_{\nabla^H}=-\omega_{\mathcal T}^{(k)}\otimes id$ where $\omega_{\mathcal T}^{(k)}$ is a $2$-form on Teichmüller space.
This allows one to define the projective version $\mathbb P Z^{(k)}_N(\Sigma)$ as the covariant constant sections of $\mathbb PH^{(k)}$ with respect to the Hitchin connection.
Theorem (Andersen-Ueno): There exists a complex line bundle $\mathbb L_k$ and a $\tilde{\Gamma}_g$-equivariant connection $\nabla^{\mathbb L_k}$ on Teichmuller space such that $F_{\nabla^{\mathbb L_k}}=-\omega_{\mathcal T}^{(k)}$. Here $\tilde{\Gamma}_g$ is a central extension of the mapping class group $\Gamma_g$.
The value of the TQFT on the surface $\Sigma$, $Z_N^{(k)}(\Sigma)$ can now be defined as the covariantly constant sections of $\nabla^H\otimes\nabla^{\mathbb L_k}$ on $H^{(k)}\otimes \mathbb L_k$.
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