Recently Tim Cochran passed away unexpectedly at the age of 59. Everyone who knew him was shocked and saddened at his untimely death, and my deepest condolences extend to all who were affected by this. Coincidentally, I was thinking about a cool paper he wrote in 1984 where he introduced a method he would later explore in much greater depth in an AMS Memoir book, the method of "derived links."
Suppose you have a link of two components with linking number $0$. Each component spans a Seifert surface, and the two surfaces intersect generically along a $1$-manifold. It is not hard to show that you can arrange for this $1$-manifold to be connected: so it is a circle. Either of the two surfaces determines an integer framing of this circle. This framing is called the Sato-Levine invariant and is independent of the choice of Seifert surfaces. We will denote this invariant by $\beta^1$. It is a fun exercise to compute this for the Whitehead link. (The answer is $\pm1$ depending on conventions.)
Cochran's idea was to iterate this construction to define a sequence of higher order invariants $\beta^i$ defined as follows. Form a new link (a derived link) from the original link $L$ by tossing out the second component and replacing it with the intersection curve of the two surfaces. We will call this link $D(L)$. Now we define $\beta^2(L)=\beta^1(D(L))$, and it turns out this is well-defined. In general $D^k(L)$ is defined by keeping the first component fixed and replacing the second component by intersection of the seifert surfaces of $D^{k-1}(L)$. Then $\beta^k(L)=\beta^1(D^{k-1}(L))$. The amazing thing is that this is a well-defined integer.
Cochran proves that $\beta^i(L)$ are integer lifts of Milnor's $\mu$ invariants $\mu_{1122...22}$, where there are $2i$ "2"s. This is interesting because normally $\mu$ invariants are only well-defined modulo the gcd of lower order invariants, and it also gives a cool topological interpretation of them. Cochran would later go on to show that one could define all $\mu$ invariants in this way using derived links, but one cannot in general remove the indeterminacy as in this case.
Cochran also proves that these $\beta^i(L)$ are essentially equivalent to a polynomial invariant called the $\eta$ function, due to Kojima. Briefly, let $L=(M,K)$ be a two component link as above, and consider the infinite cyclic cover of the complement, $Y$, and let $\Delta(t)$ be the symmetrized Alexander polynomial of the second component $K$. Let $z$ be a lift of $M$ to $Y$, $z_0$ a nearby lift of an untwisted parallel of $K$ and $t_*$ a generator of the covering transformation group. Then $A(t_*)$ kills $z\in H_1(Y)$ and $A(t_*)(z)=\partial d$ for some 2-chain $d$ in $Y$. Then Kojima's $\eta$ function is defined as $$\eta(L)=\sum_{n=-\infty}^\infty\frac{1}{A(t)}(z_0\cdot t_*^nd)t^n.$$ Cochran proves that under the change of variables $x=(1-t)(1-t^{-1})$, $$\eta(L)=\sum_{i=1}^\infty \beta^i(L)x^i.$$ This brings me to why I was interested in this. My collaborators Rob Schneiderman and Peter Teichner were studying the Kirk invariant of immersed $2$-spheres in the $4$-sphere, which turns out to be related to the Kojima invariant in the following way. Take a link of $2$ components with linking number $0$ and both components unknotted. Think of this link as lying on a copy of $S^3$ in $S^4$. Cap off the first component into the upper hemisphere of $S^4$ with a standard disk, and homotope the second component to a trivial knot in the complement of the first component, also rising up into the upper hemisphere. Do the same thing into the lower hemisphere but with components reversed. This gives two disjoint self-intersecting spheres in the $4$-ball. It turns out that the Kirk invariant of this link of $2$-spheres is the $\eta$ invariant of the classical link that we started with! This was noted by Kirk in his original paper.
Peter and Rob were at that point unaware of the connection to Cochran's work, and had independently defined the $\eta$ invariant in this simple case. They conjectured that it was related to well-known invariants like $\mu$ invariants. After discussing this with them, I realized that the only possible $\mu$-invariants that this could be related to were the $\mu_{1122...22}$ invariants, and that a basis for $2$-component links of the above form modulo equivalence of their $\eta$ invariants was given by clasper surgeries on the unlink by trees $t_k$ defined as follows. Take a line segment with the two endpoints labeled by "1" and attach $k$ hairs in a row labeled by "2." I was puzzled by the fact that after calculating $\mu_{112...2}$ on my tree basis, I was getting
$$
\begin{array}{c|cccccc}
&t_1&t_2&t_3&t_4&t_5&t_6\\
\mu_{1122} & 1& 2& 0& 0& 0& 0&\\
\mu_{112222} &*& *& 1 &2& 0& 0&\\
\mu_{11222222} &* &* &*& * & 1 &2\\
\end{array}
$$
The *'s indicate places where $\mu$ invariants are not defined due to their indeterminacy. In my head, I was thinking of this table (incorrectly) like this:
$$
\begin{array}{c|cccccc}
&t_1&t_2&t_3&t_4&t_5&t_6\\
\mu_{1122} & 1& 2& 0& 0& 0& 0&\\
\mu_{112222} &0& 0& 1 &2& 0& 0&\\
\mu_{11222222} &0 &0 &0& 0 & 1 &2\\
\end{array}
$$
This indicated that the $\mu$ invariants were not a dual basis, since this matrix was not diagonalizable and in fact half of the columns were lacking pivots. So there were some other mysterious invariants floating around, but what the heck could they be? In a sense this is indeed true, there are other mysterious invariants floating around! Namely, Cochran's $\beta^i$ invariants, which are integer lifts of the $\mu$ invariants. In fact, one can fill in the chart above with the $\beta^i$'s as follows:
$$
\begin{array}{c|cccccc}
&t_1&t_2&t_3&t_4&t_5&t_6\\
\beta^1 & 1& 2& 0& 0& 0& 0&\\
\beta^2 &0& 1& 1 &2& 0& 0&\\
\beta^3 &0 &0 &-3& -4 & 1 &2\\
\end{array}
$$
which is perfectly consistent with the $\beta^i$'s forming a complete invariant system. Cochran had already figured out the problem we were studying 30 years ago.
Tuesday, December 23, 2014
Friday, December 5, 2014
More on Quantum Chern-Simons
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 $P_{\nabla^H,\sigma_0,P}\colon H^{(k)}_{\sigma_0}\to H^{(k)}_P$.
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 $P_{\nabla^H,\sigma_0,P}\colon H^{(k)}_{\sigma_0}\to H^{(k)}_P$.
Thursday, December 4, 2014
A one variable application of the multivariable chain rule
In the last post the theme was using the one variable chain rule in place of the multivariable version. I wanted to relay a pretty cool trick that goes in the other direction. Suppose you want to differentiate $x^x$. You remember that $\frac{d}{dx} x^k=kx^{k-1}$ and $\frac{d}{dx}b^x=\ln b b^x$. Now think of $x^x=u^v$ where $u=v=x$. The chain rule now says
\begin{align*}
\frac{d}{dx} u^v&=(\ln u) u^v u_x+v u^{v-1}v_x\\
&=(\ln x) x^x+ x\, x^{x-1}\\
&=(\ln x+1)x^x
\end{align*}
And we get the right answer as if by magic. In general, you can take a derivative by isolating every occurrence of the variable $x$, then differentiating while thinking of all other occurrences as constant, and then adding up the results.
\begin{align*}
\frac{d}{dx} u^v&=(\ln u) u^v u_x+v u^{v-1}v_x\\
&=(\ln x) x^x+ x\, x^{x-1}\\
&=(\ln x+1)x^x
\end{align*}
And we get the right answer as if by magic. In general, you can take a derivative by isolating every occurrence of the variable $x$, then differentiating while thinking of all other occurrences as constant, and then adding up the results.
Wednesday, December 3, 2014
Chain rules
Suppose we want to differentiate the expression $f(x,y)=x^2y^3$ with respect to $t$, where $x$ and $y$ are both functions of $t$. The multivariable calculus method is to use the multivariable chain rule:
$$\frac{df}{dt}=\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt},$$
which would leave us with $2xy^3 x'+3x^2y^2y'$. However, one could do this completely with single variable calculus:
\begin{align*}
\frac{d}{dt}(x^2y^3)&=(\frac{d}{dt} x^2)y^3+x^2(\frac{d}{dt}y^3)\\
&=(2x \frac{dx}{dt}) y^3+x^2(3y^2\frac{dy}{dt})
\end{align*}
which gives the same answer. So the question arises as to whether one can derive the multivariable chain rule this way. At least if you have a function which is a composition of standard functions like addition, multiplication, exponentiation, trig functions and so forth, one can prove the multivariable chain rule by noticing that one can use the product rule, chain rule and so forth, to push your derivative through the function until it hits pure functions of $x$ or $y$. When you take the derivative at this final stage, you will end up multiplying by $\frac{dx}{dt}$ or $\frac{dy}{dt}$ accordingly. Collecting the coefficients of $\frac{dx}{dt}$ together will collect all those instances where you hit a pure function of $x$, which will form the function $\frac{\partial f}{\partial x}$, and similarly for the $y$ part.
One could make this argument more rigorous by arguing recursively. Show it is true for pure functions of $x$ and $y$ and then show that if it works on a given function, then it will work if you apply another one variable function to it, or if you multiply two such functions together, etc. For example, suppose that we know the multivariable chain rule for $f(x,y)$ and $g(x,y)$, then we can show it works for $f(x,y)g(x,y)$ by differentiating in the one variable sense:
\begin{align*}
\frac{d}{dt}(f(x,y)g(x,y))&=f_t g+f g_t\\
&=(f_x x_t+f_y y_t)g+f(g_xx_t+g_yy_t)\\
&=(f_x g+f g_x)x_t+(f_yg+fg_y)y_t\\
&=(fg)_xx_t+(fg)_yy_t
\end{align*}
I don't see a way to prove the multivariable chain rule as a consequence of the one variable case for general functions as opposed to some recursively constructed class. I would be interested if anyone has any ideas.
$$\frac{df}{dt}=\frac{\partial f}{\partial x}\frac{dx}{dt}+\frac{\partial f}{\partial y}\frac{dy}{dt},$$
which would leave us with $2xy^3 x'+3x^2y^2y'$. However, one could do this completely with single variable calculus:
\begin{align*}
\frac{d}{dt}(x^2y^3)&=(\frac{d}{dt} x^2)y^3+x^2(\frac{d}{dt}y^3)\\
&=(2x \frac{dx}{dt}) y^3+x^2(3y^2\frac{dy}{dt})
\end{align*}
which gives the same answer. So the question arises as to whether one can derive the multivariable chain rule this way. At least if you have a function which is a composition of standard functions like addition, multiplication, exponentiation, trig functions and so forth, one can prove the multivariable chain rule by noticing that one can use the product rule, chain rule and so forth, to push your derivative through the function until it hits pure functions of $x$ or $y$. When you take the derivative at this final stage, you will end up multiplying by $\frac{dx}{dt}$ or $\frac{dy}{dt}$ accordingly. Collecting the coefficients of $\frac{dx}{dt}$ together will collect all those instances where you hit a pure function of $x$, which will form the function $\frac{\partial f}{\partial x}$, and similarly for the $y$ part.
One could make this argument more rigorous by arguing recursively. Show it is true for pure functions of $x$ and $y$ and then show that if it works on a given function, then it will work if you apply another one variable function to it, or if you multiply two such functions together, etc. For example, suppose that we know the multivariable chain rule for $f(x,y)$ and $g(x,y)$, then we can show it works for $f(x,y)g(x,y)$ by differentiating in the one variable sense:
\begin{align*}
\frac{d}{dt}(f(x,y)g(x,y))&=f_t g+f g_t\\
&=(f_x x_t+f_y y_t)g+f(g_xx_t+g_yy_t)\\
&=(f_x g+f g_x)x_t+(f_yg+fg_y)y_t\\
&=(fg)_xx_t+(fg)_yy_t
\end{align*}
I don't see a way to prove the multivariable chain rule as a consequence of the one variable case for general functions as opposed to some recursively constructed class. I would be interested if anyone has any ideas.
Monday, December 1, 2014
Andersen's Approach to Quantum Chern-Simons Theory
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 $H^{(k)}\otimes \mathbb L_k$.
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 $H^{(k)}\otimes \mathbb L_k$.
Subscribe to:
Posts (Atom)