Yang-Mills measure on the two-dimensional torus as a random distribution


Chevyrev, I

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We introduce a space of distributional one-forms $\Omega^1_\alpha$ on the
torus $\mathbf{T}^2$ for which holonomies along axis paths are well-defined and
induce H\"older continuous functions on line segments. We show that there
exists an $\Omega^1_\alpha$-valued random variable $A$ for which Wilson loop
observables of axis paths coincide in law with the corresponding observables
under the Yang-Mills measure in the sense of L\'evy (2003). It holds
furthermore that $\Omega^1_\alpha$ embeds into the H\"older-Besov space
$\mathcal{C}^{\alpha-1}$ for all $\alpha\in(0,1)$, so that $A$ has the correct
small scale regularity expected from perturbation theory. Our method is based
on a Landau-type gauge applied to lattice approximations.

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Journal Article