Let's consider $K_t(M)$, the Kauffman bracket skein module (see this and this papers) of a three-manifold $M$. When $t=-1$, $K_t(M)$ is easily seen to be isomorphic to the ring of functions on the character variety $X=\operatorname{Hom}(\pi_1(M),\operatorname{SL}(2))/\\!/\operatorname{SL}(2)$, and the isomorphism sends a loop $\gamma$ in $K_t(M)$ to the function on $X$ given by $\rho\mapsto\operatorname{tr}(\rho(\gamma))$. The proof of this amounts to checking the following three identities:

1) $\operatorname{tr}(A)=\operatorname{tr}(A^{-1})$

2) $\operatorname{tr}(AB)+\operatorname{tr}(AB^{-1})=\operatorname{tr}(A)\operatorname{tr}(B)$

3) $\operatorname{tr}(\operatorname{id})=2$

for $A,B\in\operatorname{SL}(2)$. (1) corresponds to the loops in the Kauffman bracket skein module being unoriented. (2) and (3) correspond to the skein relations. Note I'm being a bit cavalier and forgetting that actually everything is framed, but it isn't particularly important for this question.

Question: Is there a natural class of objects to allow $A$ and $B$ to range over, and a natural "function" $\operatorname{tr}$ on this class of objects, such that the following more general identities are satisfied?

1) $\operatorname{tr}(A)=\operatorname{tr}(A^{-1})$

2) $-t\operatorname{tr}(AB)-t^{-1}\operatorname{tr}(AB^{-1})=\operatorname{tr}(A)\operatorname{tr}(B)$

3) $\operatorname{tr}(\operatorname{id})=-(t+t^{-1})$

Notice that here $\operatorname{tr}$ can't be a genuine function, since (2) (along with (1)) implies that product operation on trace values is noncommutative.

Of course, the motivation of this question is to obtain a description of $K_t(M)$ just in terms of $\pi_1(M)$, and it is natural to try something involving $U_q(\mathfrak s\mathfrak l_2)$.

  • $\begingroup$ This might sound pedantic, but it isn't clear whether it should be possible to describe $K_t(M)$ in terms of $\pi_1(M)$, since switching the orientation of $M$ inverts $t$ but has no effect on $\pi_1(M)$. $\endgroup$ Aug 28 '13 at 14:43

The old answer is that the trace identity you give in 2) is not quite right, Let $R=\sum_i a_i\otimes b_i$ be the $R$ matrix for $U_q(sl_2)$ and let $t$ be the $4$th root of $q$, then

$$t tr(XY)+ t^{-1}tr(S(X)Y)=\sum_itr(a_iX)tr(b_iY),$$

where $S$ is the antipode and $tr$ is the ordinary trace in the fundamental representation of $U_q(sl_2)$. That discovery really cost me, as I spent a long time trying to find the identity using the quantum trace, and it wasn't true. The appearence of the $R$-matrix on the right hand side of the equation is because diagrammatically, that is where the crossing is.

You can find it in a paper of Bullock, Frohman and Bartoszynska in Communications in Mathematical Physics in the late 90's where we proved that the space of observables for $U_q(sl_2)$ lattice gauge field theory based on a fat graph is the Kauffman bracket skein algebra of the surface which is a regular neighborhood of your graph.

The skeins are functions on the space of connections on the lattice. We based our construction of lattice gauge field theory in the work of Alexeev, Schomerus and Grosse. Also we were inspired somewhat by the work of Buffenoir and Roche on lattice gauge field theory. In both their approaches, the algebra of observables were defined via generators and relations. They used a Wick ordering to get functions. We saw that the connections were actually a coalgebra, and the dual product on the observables satisfied the relators given by the authors in the physical literature. This allowed us to give a coordinate free construction of lattice gauge field theory based on a quantum group, that led to structural control over the algebra of observables. Falling back on an observation of Fock we were able to show that the algebra of observables quantizes the characters of the underlying surface group with respect to a Poisson structure constructed by Goldman.

To get the signs to work like you want you need to work with $-tr$. The minus sign has been explained nicely by Bonahon and Wong. It comes from the fact that you are looking at $PSL_2(\mathbb{C})$ representations and lifting them to to $SL_{2}(\mathbb{C})$ representations.

Our ultimate step in this direction was "The Yang-Mills measure in the Kauffman bracket skein module". The lattice models were equipped with a "path" integral which is a trace on the algebra of observables that is topologically invariant. Once we had identified the algebra of observables as the Kauffman bracket skein algebra, we carried the path integral over to the purely topological picture to get a trace which deforms either integration against Haar measure in the case where the surface has boundary, or integration against the symplectic measure if the surface is closed. Although the physicists treated the path integral as a formal power series, we were able to show that as long as Planck's constant does not lie on the unit circle, or if it does, it is a root of unity, that the integral actually converges.

In more modern terms, quantum Teichm\"{u}ller theory as developed by Fock, Checkov, Bonahon and Kashaev, constructs a dual lattice gauge field theory, whose representation theory has been worked out by Bonahon and his collaborators. What is nice about this is you can emulate steps of the proof of the geometrization conjecture in the quantum setting and find fixed representations. Bonahon and Wong recently proved that the space of observables contains a large subalgebra which is the Kauffman bracket skein algebra of the underlying surface.

The shortcoming of all of this is that it doesn't address characters of closed surfaces. Generally the way you deal with that is by defining a projector. Ultimately more geometric approaches via a line bundle over the character variety will probably give more satisfactory answers in that case.

The quantum hyperbolic invariants of Baseilhac and Bennedetti end up assigning quantum invariants to knots and links in manifolds with a $PSL_2(\mathbb{C})$ representation. When the underlying representation is trivial these are the invariants of Kashaev which is have been equated with evaluations of the colored Jones polynomial by Murakami and Murakami.

Some citations.

Bullock, Doug; Frohman, Charles; Kania-Bartoszyńska, Joanna Topological interpretations of lattice gauge field theory. Comm. Math. Phys. 198 (1998), no. 1, 47–81.

Bullock, Doug; Frohman, Charles; Kania-Bartoszynska, Joanna The Yang-Mills measure in the Kauffman bracket skein module. Comment. Math. Helv. 78 (2003), no. 1, 1–17.

Bonahon, Francis; Liu, Xiaobo Representations of the quantum Teichmüller space and invariants of surface diffeomorphisms. Geom. Topol. 11 (2007), 889–937.

Kashaev, Rinat M. On quantum moduli space of flat PSL2(R)-connections on a punctured surface. Handbook of Teichmüller theory. Vol. I, 761–782, IRMA Lect. Math. Theor. Phys., 11, Eur. Math. Soc., Zürich, 2007

Baseilhac, Stephane; Benedetti, Riccardo Quantum hyperbolic geometry. Algebr. Geom. Topol. 7 (2007), 845–917.

An alternate approach to quantum hyperbolic invariants:

Kashaev, R.; Reshetikhin, N. Invariants of tangles with flat connections in their complements. Graphs and patterns in mathematics and theoretical physics, 151–172, Proc. Sympos. Pure Math., 73, Amer. Math. Soc., Providence, RI, 2005.

Kauffman brackets, character varieties, and triangulations of surfaces Francis Bonahon (USC), Helen Wong (Carleton College) arXiv:1009.0084v1 [math.GT]

  • $\begingroup$ This is a great answer! Thanks a lot! $\endgroup$ Jul 2 '11 at 3:22

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