Yang--Mills fields and random holonomy along Brownian bridges

Abstract

We characterize Yang-Mills connections in vector bundles in terms of covariant derivatives of stochastic parallel transport along variations of Brownian bridges on the base manifold. In particular, we prove that a connection in a vector bundle E is Yang-Mills if and only if the covariant derivative of parallel transport along Brownian bridges (in the direction of their drift) is a local martingale, when transported back to the starting point. We present a Taylor expansion up to order 3 for stochastic parallel transport in E along small rescaled Brownian bridges and prove that the connection in E is Yang-Mills if and only if all drift terms in the expansion (up to order 3) vanish or, equivalently, if and only if the average rotation of parallel transport along small bridges and loops is of order 4. Introduction.This article is concerned with the characterization of YangMills connections in a vector bundle E over a compact Riemannian manifold M in terms of stochastic parallel transport along Brownian bridges. Recall that YangMills connections in a vector bundle E with a metric over M are the critical points of the following functional (the so-called Yang-Mills action):

A probabilistic approach to the Yang–Mills heat equation

“…Now letting σ = r 1 /2 = f (r 0 ) which gives (r 1 − σ ) 4 = f (r 0 ) 4 , we get along with Eq. (4.13) f (r 0 ) 4 sup P (f (r 0 ),t 0 ,x 0 ) e 2 4 a α 2 or sup P (f (r 0 ),t 0 ,x 0 ) e 2 4 a α 2 f (r 0 ) 4 which proves the theorem. ✷ An essential tool for the proof of Theorem 4.2 is the monotonicity formula (3.19) which involves φ (t,x) (r).…”

“…By continuity of t → φ (t,x) (r), there exists ε > 0 such that r ε 1 (a −1 YM(0)) ∧ T − ε and φ (t,x) (r) aε 0 for any t ∈ [T − ε, T [. Consequently, by Theorem 4.2, sup P (f (r),t,x) e 2 4 a α 2 f (r)4 .…”

“…(2) Yang-Mills connections have been characterized in terms of the holonomy along Brownian loops or Brownian bridges in [4,7,22]. Here we characterize solutions to the Yang-Mills heat equation by the fact that (U…”

“…where the relation between t 0 , r 0 and r 1 has to be determined. Let σ 0 ∈ [0, r 1 [ such that (r 1 − σ 0 ) 4 sup…”

Lévy Laplacians and instantons

“…This approach can be also useful in the connection with the Yang-Mills equations (see [28,30,32,35,36]) and instantons (see [29,34]). Different approaches to the Yang-Mills equations based on the parallel transport but not based on the Levy Laplacian were used in [17,15,13,14,8,9]. Particularly, instantons were studied in [13,14,8].…”