Two-Loop Loewner Potentials

Abstract We study a generalization of the Schramm–Loewner evolution (SLE) loop measure to pairs of non-intersecting Jordan curves on the Riemann sphere. We introduce four equivalent definitions of two-loop Loewner potential for smooth pairs: 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 an Onsager–Machlup functional for the two-loop SLE, and a variational formula involving Schwarzian derivatives. The first, third and fourth definitions are finite if and only if both loops are Weil–Petersson quasicircles. 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 principle 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.