Growth rate conjecture for Chern–Simons asymptotics

Assume the asymptotic expansion conjecture holds for a closed oriented $3$-manifold $M$. Let $\mathcal{M}(M)$ be the moduli space of flat $\operatorname{SU}(N)$-connections, and let $\mathcal{M}_j$ be the subspace on which the Chern–Simons action has value $c_j$. For a flat connection $A$, let $h_A^i=\dim H^i(M,\operatorname{Ad}_P)$, where $H^i(M,\operatorname{Ad}_P)$ is the cohomology of the flat-connection complex. Growth rate conjecture. If $c_j$ and $d_j$ are the quantities in the asymptotic expansion conjecture, then

$$d_j=\frac12\max_{[A]\in\mathcal{M}_j}(h_A^1-h_A^0),$$

where the maximum is taken over Zariski-open subsets of $\mathcal{M}_j$ on which $h_A^1-h_A^0$ is constant.

This conjecture refines the expected asymptotic expansion by relating the exponent $d_j$ of each Chern–Simons contribution to the local cohomological dimensions of flat connections. It is conditional on the asymptotic expansion conjecture, and the supplied text gives no resolution status.