Hypotrochoids in conformal restriction systems and Virasoro descendants

Abstract

A conformal restriction system is a commutative, associative, unital algebra equipped with a representation of the groupoid of univalent conformal maps on connected open sets of the Riemann sphere, and a family of linear functionals on subalgebras, satisfying a set of properties including conformal invariance and a type of restriction. This embodies some expected properties of expectation values in conformal loop ensembles CLE. In the context of conformal restriction systems, we study certain algebra elements associated with hypotrochoid simple curves (including the ellipse). These have the CLE interpretation of being "renormalized random variables" that are nonzero only if there is at least one loop of hypotrochoid shape. Each curve has a center w, a scale ε and a rotation angle θ, and we analyze the renormalized random variable as a function of u=εe^{iθ} and w. We find that it has an expansion in positive powers of u and u*, and that the coefficients of pure u (u*) powers are holomorphic in w (w*). We identify these coefficients (the "hypotrochoid fields") with certain Virasoro descendants of the identity field in conformal field theory, thereby showing that they form part of a vertex operator algebraic structure. This largely generalizes works by the author (in CLE), and the author with his collaborators V. Riva and J. Cardy (in SLE 8/3 and other restriction measures), where the case of the ellipse, at the order u^2, led to the stress-energy tensor of CFT. The derivation uses in an essential way the Virasoro vertex operator algebra structure of conformal derivatives established recently by the author. The results suggest in particular the exact evaluation of CLE expectations of products of hypotrochoid fields as well as non-trivial relations amongst them through the vertex operator algebra, and further shed light onto the relationship between CLE and CFT.