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Notes on invariant measures for loop groups

Doug Pickrell

TL;DR

This work surveys a program to construct and characterize bi-invariant measures on hyperfunction completions of complex loop groups, centered on a distinguished measure $\boldsymbol{\mu_0}$ and its level-$l$ deformations $\boldsymbol{\mu}^{|\mathcal{L}|^{2l}}$. It develops two complementary routes to these measures: (i) weak limits of Wiener-type measures in Riemann–Hilbert coordinates yielding existence and invariance, and (ii) explicit root-subgroup factorization yielding product-coordinate expressions and diagonal distributions akin to affine Harish-Chandra data. The notes outline conjectures about pushforwards to moduli spaces (Goldman/Goldman-type volumes), diagonal/branching structures, and a holomorphic Peter–Weyl framework that would support sewing for WZW models and a potential bridge to YM$_3$ via parabolic reductions and holonomy coordinates. If realized, this program could provide a rigorous analytic backbone for chiral sigma models, WZW theories, and related gauge theories, offering a measure-theoretic construction of their vacua and asymptotic behavior under renormalization-group flows. The overall aim is to connect infinite-dimensional invariant measures, root-subgroup coordinates, and moduli-space pushforwards to obtain a coherent quantum-field-theoretic picture across two and three dimensions, including sewing rules and potential new insights into YM$_3$ and gravity-coupled systems.

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 $YM_3$), 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 $YM_3$.

Notes on invariant measures for loop groups

TL;DR

This work surveys a program to construct and characterize bi-invariant measures on hyperfunction completions of complex loop groups, centered on a distinguished measure and its level- deformations . It develops two complementary routes to these measures: (i) weak limits of Wiener-type measures in Riemann–Hilbert coordinates yielding existence and invariance, and (ii) explicit root-subgroup factorization yielding product-coordinate expressions and diagonal distributions akin to affine Harish-Chandra data. The notes outline conjectures about pushforwards to moduli spaces (Goldman/Goldman-type volumes), diagonal/branching structures, and a holomorphic Peter–Weyl framework that would support sewing for WZW models and a potential bridge to YM via parabolic reductions and holonomy coordinates. If realized, this program could provide a rigorous analytic backbone for chiral sigma models, WZW theories, and related gauge theories, offering a measure-theoretic construction of their vacua and asymptotic behavior under renormalization-group flows. The overall aim is to connect infinite-dimensional invariant measures, root-subgroup coordinates, and moduli-space pushforwards to obtain a coherent quantum-field-theoretic picture across two and three dimensions, including sewing rules and potential new insights into YM and gravity-coupled systems.

Abstract

Let denote a simply connected compact Lie group and let , the complexification. It is known that there exists an bi-invariant probability measure on a natural hyperfunction completion of the complex loop group . There are various generalizations, involving positive line bundle valued measures on the hyperfunction completion, replacing with a symmetric space, replacing (the configuration space of the principal chiral model) with (the homotopy equivalent space of ) gauge equivalence classes of -connections on the 2-sphere (the configuration space of ), 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 .
Paper Structure (55 sections, 24 theorems, 340 equations)

This paper contains 55 sections, 24 theorems, 340 equations.

Table of Contents

  1. Introduction
  2. Outline of the Notes
  3. Notation
  4. Loop Groups and Completions
  5. Appendix: The Abelian Case
  6. Supplementary Notation
  7. Finite Type Structure
  8. Affine Algebra Structure
  9. Loop Groups and Extensions
  10. Existence
  11. A More Quantitative Approach
  12. Characterizing Measures and Measure Classes
  13. Computing Invariant Measures I
  14. Pushforwards to Moduli Spaces
  15. Reconstructing $\mu_0$
  16. ...and 40 more sections

Key Result

Theorem 1

Suppose that $K$ is a simply connected compact Lie group, and let $G$ denote the complexification. There exists a Borel probability measure $\mu_0$ on the formal completion, $\mathbf LG$, of the complex loop group $C^{0}(S^1,G)$ which is bi-invariant with respect to $L_{fin}K$, the group of loops in

Theorems & Definitions (73)

  • Theorem 1
  • Remark
  • Remark
  • Remark
  • Theorem 2.1
  • Lemma 1
  • Corollary 1
  • Theorem 2.2
  • Theorem 2.3
  • proof
  • ...and 63 more