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Yang-Mills fields and random holonomy along Brownian bridges

by Marc Arnaudon and Anton Thalmaier

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 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.
*Ann. Probab. ***31** (2003) 769-790

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