Corner contributions to Neumann jump determinants: three model calculations and a BFK conjecture

Abstract

We formulate a local corner-factor conjecture for the determinant of the Neumann jump operator on a piecewise real-analytic cutting curve. For the mirror double of a simply connected geodesic polygon with interior angles $\pi\alpha_1,\ldots,\pi\alpha_N$, the conjectural determinant is \[ \Det_{\angle}'\cN = \frac{\length(\partial P)}2 \prod_{j=1}^N\alpha_j^{-1/2}. \] Here $\Det_{\angle}'$ denotes an intrinsic determinant, still to be constructed, that is required to satisfy a Burghelea--Friedlander--Kappeler gluing formula; only its quotient by $\length(\partial P)$ is purely angle-dependent. Three model calculations support the conjecture: a flat polygon and its double give $\frac12\prod_j\alpha_j^{-1/2}$; a spherical spindle split into congruent lunes gives $1/(2\alpha)$ for two angles $\pi\alpha$; and every spherical Coxeter triangle with angles $(\pi/p,\pi/q,\pi/r)$ gives $\frac12\sqrt{pqr}$, including $\sqrt2$ for the octant. The conjecture also reduces the Dirichlet determinant of a constant-curvature polygon to that of its closed double. Although the singular anomaly formula applies to the double in general, an explicit evaluation in nonzero curvature requires solving its uniformization problem. We discuss connections with the Neumann jump determinants in the work of Wiegmann--Zabrodin and Wang and with the recent Grunsky-operator approach to Coulomb gases on domains with corners.