Notes on invariant measures for loop groups

2022/07/19 by Doug Pickrell, Pickrell, Doug
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Black Holes and Theoretical Physics #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #math-ph math.FA

paper · pdf · doi:10.48550/arxiv.2207.09913

openalex publication_date 2022/07/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let K denote a simply connected compact Lie group and let G=K^{\mathbb C}, the complexification. It is known that there exists an LK bi-invariant probability measure on a natural hyperfunction completion of the complex loop group LG. There are various generalizations, involving positive line bundle valued measures on the hyperfunction completion, replacing K with a symmetric space, replacing LK (the configuration space of the principal chiral model) with (the homotopy equivalent space of ) gauge equivalence classes of K-connections on the 2-sphere (the configuration space of YM₃), and so on. The purpose of these notes is to publicize a number of conjectures and questions concerning how these measures are characterized, how they are explicitly represented, and how they are potentially relevant to quantum sigma models and YM₃.

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