Chern-Simons reciprocity for boundary-parabolic representations

From papers

Let MM be an oriented compact 33-manifold with a torus boundary, and let RSL2(C)para(M)\mathrm{R}^{\rm para}_{\mathrm{SL}_2(\mathbb{C})}(M) be the set of conjugacy classes of boundary-parabolic representations π1(M)SL2(C)\pi_1(M)\to\mathrm{SL}_2(\mathbb{C}). Assume this set is finite, and let CS(ρ)C/Z\mathrm{CS}(\rho)\in\mathbb{C}/\mathbb{Z} denote the Chern-Simons invariant. Boundary-parabolic Chern-Simons reciprocity conjecture. If RSL2(C)para(M)<|\mathrm{R}^{\rm para}_{\mathrm{SL}_2(\mathbb{C})}(M)|<\infty, then

24ρRSL2(C)para(M)CS(ρ)=0.24\sum_{\rho\in\mathrm{R}^{\rm para}_{\mathrm{SL}_2(\mathbb{C})}(M)}\mathrm{CS}(\rho)=0.

The statement is posed as an analogue of the closed-manifold reciprocity problem for manifolds with torus boundary; the supplied passage gives no proof or resolution.

Progress summary

Nothing recorded yet. Refresh searches the literature and the public web for attempts on this problem, and writes the first summary here.

Sources & referencesView supporting material

Primary source

Takefumi Nosaka, “Reciprocity of the Chern-Simons invariants of 3-manifolds”, arXiv:2212.07569 (2022).

Solutions 0

No solutions have been posted yet.