Interpreting Witten's Asymptotic Expansion of the WRT invariant.

Question

Witten's asymptotic expansion conjecture as described in "Problems on invariants of knots and 3-manifolds" in Geometry and Topology Monographs, Volume 4 states that

$$Z_r^{SU(2)}(M)\sim_{r\rightarrow \infty} e^{-3\pi \bf{i}(1+b^1(M))/4}\times \sum_{[A]}e^{2\pi \bf{i}CS(A)} r^{(h^1_A-h^0_A)} e^{-2\pi \bf{i}(I_A/4+h^0_A/8)}\tau_M(A)^{1/2},$$

where $b^1(M)$ is the first betti number of $M$, $CS(A)$ is the chern-simons invariant of $A$, $h^i_A$ is the rank of the $i$th cohomology of $M$ with coefficients in the $su(2)$ bundle twisted by the adjoint action of the monodromy of $A$ and $I_A$ is the spectral flow of the Laplace operator along a path connecting $A$ to the trivial flat connection. Finally, $\tau_M(A)$ is the Reidemeister torsion.

Since $h^1$ is often the dimension of the Zariski tangent space of the space of gauge equivalence classes of flat connections, if $h^1>0$ in this formula, you are admitting the possibility that the moduli space of flat connections has positive dimension. In this case the sum is not really a sum, but an integral. The torsion in this case defines a volume form on $H_0(M,adA)\oplus H^1(M;adA)\oplus H_2(M,ad A)\oplus H^3(M,adA)$ ,where I mean the complex with twisted homology as appears above. Clearly what is meant by $\tau_M(A)^{1/2}$ must be a volume form on the moduli space of gauge equivalence classes of flat connections for the integral to work. That means $\tau_M(A)^{1/2}$ is a top dimensional form on $H^1(M;adA)$.

What is the explicit formula for the volume form? In other words, how do you go about cutting down the torsion and taking its square root as as a differential form?

Answer 1

In the conjecture, which should hold in some way for any other simple, simply-connected, compact Lie group $G$, $M$ is closed and $A$ is a flat connection in a trivialized $G$-bundle over $M$. It is easier to think of Reidemeister torsion as a density on $H^0(M,adA)^* \oplus H^1(M,adA) \oplus H^2(M,adA)^* \oplus H^3(M,adA)$: It is a norm on the determinant line of the cohomology, or equivalently, an element of $|\det H^0(M,adA) \otimes \det H^1(M,adA)^* \otimes \det H^2(M,adA) \otimes \det H^3(M,adA)^*|.$ I believe, it was first mentioned by Jeffrey, how to interpret the conjecture using Reidemeister torsion as a volume form or density.

We need to fix an inner product on the Lie algebra $\frak{g}$ of $G$, e.g. the Killing form, so that we can use Poincaré duality to identify $H^0(M,ad A)$ with $H^3(M,ad A)^* $ and $H^2(M,ad A)$ with $H^1(M,ad A)^* $. We denote the identification on the level of densities by $PD$. If we also assume that $A$ is irreducible, then $H^0(M,adA) = H^3(M,adA)^* = 0$ and we can write $$ \tau_M(A)=\omega_A \otimes PD^{-1}(\omega_A) $$ for some $\omega_A \in |\det H^1(M,adA)^*|$. Then $\sqrt{\tau_M} = \omega$ is a density on the moduli space of gauge-equivalence classes irreducible flat connections, which we can integrate.

If $H^0(M,adA) \neq 0$, then an orthonormal basis of $H^0(M,adA) \subset \frak{g}$ with respect to the chosen inner product on $\frak{g}$ gives a canonical element $\eta_A \in |\det H^0(M,adA)|$. Then $$ \tau_M(A) = \eta_A \otimes \omega_A \otimes PD^{-1}(\omega_A) \otimes PD(\eta_A). $$ The density on the moduli space of flat connections is therefore $\omega$, but integration needs to be made sense of. The conjecture depends on a suitable choice of inner product on $\frak{g}$, if the cohomology is not trivial.

I should add, that the conjecture you wrote is only a conjecture for the leading order term, not for the full asymptotic expansion. In my paper with Joergen Andersen we give a more precise statement of the same conjecture and prove it for finite-order mapping tori. The version using spectral flow is pre-eminent in the literature, but it becomes more compact if you use the Rho invariant $\rho$ instead: $$Z_{k}^{G}(M)\sim \sum_{c\in C} \frac{1}{|Z(G)|}\int_{A \in c} \sqrt{\tau_{M}(A)} e^{2\pi i CS(A)k} e^{\frac{\pi i}{4} \rho_A(M)} k^{d_c},$$ where $C$ is the set of all connected components of the moduli space of gauge-equivalence classes of flat connections on $M$, and $$d_c = \frac{1}{2} \max_{A \in c} \left( \dim(H^1(M,d_A)) - \dim(H^0(M,d_A)) \right),$$ where $\max$ is the maximum attained on a Zariski open subset of $c$. The Rho invariant is related to the spectral flow and the Chern-Simons invariant via the APS index theorem, which gives $r = k + h$ in your statement (where the $r$ is missing next to the Chern-Simons invariant), where $h$ is the dual Coxeter number. Note that this formula is not really the leading order term, but only the leading order term at each flat connection. Also note that my signs are a little different because I use different conventions for the spectral flow as well the normalization of the Chern-Simons invariant.