As promised in an earlier post, I want to elaborate on how Andersen's program relates to some big conjectures and proves some big theorems. I am basically reporting on a talk he gave at MPIM in November 2014.
In my earlier post, I mentioned the following:
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?
In this post, we'll touch on the statement of number 4, and why the answer to 1 is "no," and the answer to 2 is "yes." At the end we will also talk about asymptotics in Teichmüller space.
We recall that we are dealing with a TQFT $Z_N^{(k)}$ which can be constructed in 4 separate ways, and the gist of Andersen's program is to use the geometry inherent in the quantization approach to attack the difficult conjectures.
$Z_N^{(k)}(\Sigma)$ is a vector space associated to the surface $\Sigma$ and there is a central extension of the mapping class group which induces an action $Z_N^{(k)}:\widetilde{\Gamma}_{\Sigma}\to \mathrm{End}(Z_N^{(k)}(\Sigma))$.
By TQFT axioms if $Y$ is a $3$-manifold with boundary $\Sigma$, $Z_N^{(k)}(Y)$ is an element of $Z_N^{(k)}(\Sigma)$, and furthermore if $X$ is a closed manifold which is the union of two handlebodies $H_g$ glued by a mapping class $\phi$ then
$$Z_N^{(k)}(X)=\langle Z_N^{(k)}(H_g), Z_N^{(k)}(\phi) Z_N^{(k)}(H_g)\rangle.$$
The question is now what happens asymptotically as $k$ approaches $\infty$.
Asymptotic Expansion Conjecture: Let $X$ be a closed $3$-manifold. Then $Z_N^{(k)}(X)\in\mathbb C$. The conjecture is that $\newcommand{\Znk}{Z_N^{(k)}} \Znk(X)$ has an asymptotic expansion (in the Poincare sense) in $e^{2\pi i(k+N)\alpha_j} k^d$ for $d\in\mathbb Q$ and $\alpha_1,\ldots, \alpha_{m(x)}\in\mathbb R$, (i.e. finitely many phases). More precisely
$$
\Znk(X)\sim \sum_{i=1}^{m(x)}e^{2\pi i(k+N)\alpha_i} b_i (k+N)^{d_i}(1+\sum_{\ell=1}^\infty a_{i,\ell}(k+N)^\ell)
$$
Here the $\sim$ means that if we truncate the infinite sum at $\ell=L$, the difference between the left hand side and right hand side is $O(k^{d-L-1})$. The numbers $\alpha_i$ are conjectured to be Chern Simons values of flat $SU(N)$ connections on $X$. There are finitely many because the moduli space of such connections $M_{SU(N)}(X)$ is compact and the CS invariant is constant on connected components. The numbers $b_i$ are conjectured to be related to Reidemeister torsion. The numbers $d_i$ are conjectured to be virtual dimensions of moduli spaces, and the coefficients $a_{i,\ell}$ appear to be finite type invariants of some kind.
In the remainder of the post, we will talk about two things.
1) Asymptotics in $k$, which is related to Toeplitz operators.
2) Asymptotics in Teichmüller space, which is related to Lagrangian fibrations.
The first area implies "asymptotic faithfulness."
Theorem: (Andersen, Freedman-Walker-Wang (SU(2)))
$$
\cap_{i=1}^\infty \ker \mathbb P \Znk =\begin{cases}
\{1\} & (g,n)\neq (2,2)\\
\{1,H\}& (g,n)=(2,2)
\end{cases}
$$
where $H$ is the hyperelliptic involution, and $\mathbb P$ is the projectivization.
The second area leads to a geometric construction $\Znk(H_g)\in \Znk(\Sigma)$ and a construction of a Hermitian structure in $(H^{(k)}\to\mathcal T,\nabla^H).$
Both types of asymptotics together imply that the mapping class group $\Gamma_g$ ($g\geq 2$) does not have Kazhdan's property T. See this thread.
Asymptotics in k: Recall we have a bundle $H^{(k)}\to\mathcal T$ over Teichmüller space with fibers $H^{(k)}_{\sigma}$ given by holomorphic sections $H^0(M'_\sigma,L^k)$. Let $f\in C^\infty_c(M')$. This induces a multiplication operator on sections
$$M^{(k)}_f\colon C^\infty(M',L^k)\to C^\infty(M',L^k)$$ defined by pointwise multiplication $M_f^{(k)}(s)=fs$. We also have an orthogonal projection
$$\pi_\sigma^{(k)}\colon C^\infty(M',L^k)\to H^0(M'_\sigma,L^k)$$ with respect to the hermitian inner product $(s_1,s_2)=\int_{M'}\langle s_1,s_2\rangle \frac{\omega^n}{n!}$.
Definition (Toeplitz Operator): $T^{(k)}_{f,\sigma}=\pi^{(k)}_{\sigma}\circ M_f^{(k)}\colon L^2(M',L^k)\to H^0(M'_\sigma,L^k)$. By restriction we can also think of $T_{f,\sigma}^{(k)}\in \mathrm{End}(H^0(M'_\sigma,L^k)).$
Facts about Toeplitz Operators:
Three authors (whose names I was not able to get down) have proven some very strong properties about these operators. The first fact is that the limit of the operator norms is just the sup norm. $\displaystyle\lim_{k\to\infty} ||T^{(k)}_{f,\sigma}||=\sup_{x\in M'}|f(x)|.$ The operator norm is just the usual spectral radius since $H^0(M'_\sigma, L^k)$ is finite dimensional. We also have the following, related to deformation quantization: $T^{(k)}_{f,\sigma}\cdot T^{(k)}_{g,\sigma}\sim T^{(k)}_{f*g,\sigma}$. Where there exist unique $c_\ell\colon C^\infty_c(M')\times C^\infty_c(M')\to C^\infty_c(M')$ with $f*_{\sigma}g\sum_{\ell=0}^\infty c_{\ell,\sigma}(f,g)h^\ell$ a star-product. That is
$$\forall L\geq 0 ||T^{(k)}_{f,\sigma}T^{(k)}_{g,\sigma}-T^{(k)}_{\sum_{\ell=0}^L c_\ell(f,g)(\frac{1}{k})^\ell}||=O(k^{-L-1}).$$ Here $c_0(f,g)=fg$ and $c_1(f,g)-c_1(g,f)=\{f,g\}$ the Poisson bracket.
Applying to geometric quantization: Recall that $\nabla^H$ is the Hitchin connection in $H^{(k)}\to \mathcal T$ and let $\nabla^{H,e}$ be the induced connection in $\mathrm{End}(H^{(k)})\to \mathcal T$. Because the Hitchin connection is projectively flat, this induced connection is actually flat. Note that the Toeplitz operators give a family of sections of the endomorphism bundle.
Theorem (Andersen): $||\nabla^{H,e}T^{(k)}_f||=O(k^{-1}).$
Theorem (Andersen): There is a unique formal Hitchin connection $D^H$ on the bundle $\mathcal T\times C^\infty_c(M')[[h]]\to\mathcal T$ such that $\nabla^{H,e}T^{(k)}_f\sim T^{(k)}_{D^H(f)}$ for $f\colon \mathcal T\to C^{\infty}_c(M')$.
From this we can deduce asymptotic faithfulness. See Andersen's Annals paper.
Asymptotics in Teichmüller space: Pick an element $\sigma_0\in\mathcal T$, and choose an element $P$ of the Thurston compactification $\overline{\mathcal T }$ corresponding to a pants decomposition. (Aside: Let $\mathcal S$ be the set of isotopy classes of simple closed curves in the surface. There is a map $\mathcal T\to\mathbb P(\mathbb R^s_{\geq 0}))$ given by taking the lengths of the curves with respect to the metric structure. This is actually an embedding and the closure of the image is the Thurston compactification. ) Define a path $\sigma_t$ from $\sigma_0$ out to the point $P$ on the boundary by inserting flat cylinders of length $t$ at the curves in the pants decomposition. This gives a unique conformal structure, giving a well-defined element of Teichmüller space.
Question: What happens to the parallel transport operator $P_{\nabla^H,\sigma_0,\sigma_t}\colon H^{(k)}_{\sigma_0}\to H^{(k)}_{\sigma_t}$ as $t\to\infty$?
Andersen was running out of time, and at this point restricted to the $N=2$ case. Recalling that $M$ is the moduli space of flat $SU(N)$ connections, let $H_P\colon M\to [-2,2]^{3g-3}$ denote the holonomy defined by $H_P([A])=(\mathrm{hol}_A(\gamma_i))_{1\leq i\leq 3g-3}.$ (The curves $\gamma_i$ are the curves in the pants decomposition.) Let $B^{(k)}\subset H_P(M)\subset[-2,2]^{3g-3}$ be defined by
$$B^{(k)}=\{b\in H_P(M)\,|\,(L^k,\nabla)|_{H_P^{-1}(B)}\text{ is trivial})\}.$$ These are called Bohr-Somerfeld fibers. Now for $b\in B^{(k)}$ we have $s_b\in H^0(H_P^{-1}(b),(L^k,\nabla)|_{H_P^{-1}(B)})$, and define
$H_P=\oplus_{b\in B^{(k)}} H^0(H^{-1}_P(b),(L^k,\nabla)|_{H_P^{-1}(B)})$.
Theorem(Andersen): $P_{\nabla^H,\sigma_0,\sigma_t}\colon H^{(k)}_{\sigma+_0}\to H^{(k)}_{\sigma_t}$ has a limiting isomorphism $P_{\nabla^H,\sigma_0,P}\colon H^{(k)}_{\sigma_0}\to H^{(k)}_P$.
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