Werner's Measure on Self-Avoiding Loops and Representations of the Virasoro Algebra

Abstract

Werner has proven the existence and essential uniqueness of a conformally invariant family of locally-finite measures on self-avoiding loops on Riemann surfaces. The measures can be thought of as self-avoiding loop analogues of Schramm-Loewner evolution with parameter κ=8/3. This family is determined by a single measure on (normalized) holomorphic univalent functions on the unit disk. We will devise an algorithm for calculating moments of their Taylor coefficients. And in special cases, we can present closed-form solutions. Essentially, our algorithm arises as a consequence of non-degeneracy for a newly-realized family of highest-weight representations of the Virasoro algebra (we provide an explicit isomorphism between these representations and those constructed by Kirillov and Yuriev). Moreover, our algorithm leads to an alternate proof of essential uniqueness of Werner's family, as first seen in the author's joint work with Douglas Pickrell. Kontsevich and Suhov have conjectured the existence and essential uniqueness of a one-parameter deformation of Werner's family to a family of measures having values in powers of determinant line bundles (the deformation parameter is given by the real parameter κ satisfying 0 ≤ κ ≤ 4). Benoist and Dubédat recently proved the existence part of this conjecture for κ=2. We will provide an outline of how the argument for the Werner case can be adapted to prove the uniqueness part of Kontsevich and Suhov's conjecture.