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Wiener distribution on holonomy groups

Yuguang Zhang

TL;DR

The paper addresses how holonomy—an invariant of connections—behaves under convergence of the underlying geometric data. It constructs a measure $\mu_x(\nabla)$ on the holonomy group $O(r)$ as the push-forward of the Wiener measure on loop space through the stochastic holonomy map, and establishes a convergence theorem: if a sequence of connections $\nabla^k$ converges to $\nabla$ (with compatible metric convergence), then $\mu_x(\nabla^k) \rightharpoonup \mu_x(\nabla)$ in distribution. It also introduces finite-dimensional approximations $\mu_x^m(\nabla)$ that converge to $\mu_x(\nabla)$ and proves an approximation theorem, linking the limit measure to the closure of the holonomy group. The results apply to settings where holonomy changes in the limit, including flat $U(1)$-connections on Lagrangian submanifolds and consequences for Bohr-Sommerfeld conditions, providing a probabilistic framework to study holonomy convergence and its geometric implications.

Abstract

This paper proves a convergence theorem for the push-forward Wiener measures on holonomy groups via stochastic parallel transports along convergent metric connections.

Wiener distribution on holonomy groups

TL;DR

The paper addresses how holonomy—an invariant of connections—behaves under convergence of the underlying geometric data. It constructs a measure on the holonomy group as the push-forward of the Wiener measure on loop space through the stochastic holonomy map, and establishes a convergence theorem: if a sequence of connections converges to (with compatible metric convergence), then in distribution. It also introduces finite-dimensional approximations that converge to and proves an approximation theorem, linking the limit measure to the closure of the holonomy group. The results apply to settings where holonomy changes in the limit, including flat -connections on Lagrangian submanifolds and consequences for Bohr-Sommerfeld conditions, providing a probabilistic framework to study holonomy convergence and its geometric implications.

Abstract

This paper proves a convergence theorem for the push-forward Wiener measures on holonomy groups via stochastic parallel transports along convergent metric connections.

Paper Structure

This paper contains 6 sections, 9 theorems, 63 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Wiener measure
  4. Stochastic parallel transports
  5. Finite dimensional approximation
  6. Proofs

Key Result

Theorem 1.1

Let $(M,g)$ be a smooth connected compact Riemannian $n$-dimensional manifold, and $x\in M$. Consider a real vector bundle $\pi: V \rightarrow M$ of rank $r$ on $M$, which is equipped with a metric connection $\nabla$ on $V$ that is compatible with a bundle metric $h$. Suppose that $\nabla^k$ is a f and $h^k|_x \rightarrow h|_x$ in $V_x^* \otimes V_x^*$, when $k\rightarrow \infty$. Then $\mu_x(\n

Theorems & Definitions (19)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Lemma 2.2
  • Definition 2.3: Wiener distribution
  • Proposition 2.4
  • proof : Sketch of Proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 9 more