Two-loop Loewner potentials

TL;DR

This paper generalizes the two-loop Loewner potential to pairs of non-intersecting Jordan curves, explores its properties, and connects it with conformal field theory and the geometry of the determinant line bundle.

Contribution

It introduces four equivalent definitions of the two-loop Loewner potential and analyzes its finiteness, minimizers, and divergence issues, linking it to CFT and geometric structures.

Findings

Potential is finite iff both loops are Weil-Petersson quasicircles.

Minimizers of the potential are pairs of circles.

Potential diverges as circles move apart or merge.

Abstract

We study a generalization of the Schramm-Loewner evolution loop measure to pairs of non-intersecting Jordan curves on the Riemann sphere. We also introduce four equivalent definitions for a two-loop Loewner potential: respectively expressing it in terms of normalized Brownian loop measure, zeta-regularized determinants of the Laplacian, an integral formula generalizing universal Liouville action, and Loewner-Kufarev energy of a foliation. Moreover, we prove that the potential is finite if and only if both loops are Weil-Petersson quasicircles, that it is an Onsager-Machlup functional for the two-loop SLE, and a variational formula involving Schwarzian derivatives. Addressing the question of minimization of the two-loop Loewner potential, we find that any such minimizers must be pairs of circles. However, the potential is not bounded, diverging to negative infinity as the circles move away from each other and to positive infinity as the circles merge, thus preventing a definition of two-loop Loewner energy for the prospective large deviations theory for the two-loop SLE. To remedy the divergence, we study a way of generalizing the two-loop Loewner potential by taking into account how conformal field theory (CFT) partition functions depend on the modulus of the annulus between the loops. This generalization is motivated by the correspondence between SLE and CFT, and it also emerges from the geometry of the real determinant line bundle as introduced by Kontsevich and Suhov.