Yang-Mills measure on the two-dimensional torus as a random distribution
The author aims at giving a rigorous meaning to the (Euclidean) Yang-Mills measure on the two-dimensional Torus \(\mathbb{T}^2\). Formally, this measure is given by \[ d\mu(A) = Z^{-1} e^{-S_{\mathrm{YM}(A)}}\,dA\,, \] where \(dA\) denotes a formal Lebesgue measure on the affine space \(\mathcal{A}\) of connections on a principle \(G\)-bundle \(P\) over \(\mathbb{T}^2\), where \(G\) is a compact, connected Lie group with Lie algebra \(\mathfrak{g}\). In this article, it is assumed that \(P\) is trivial, so that \(\mathcal{A}\) can be identified with the space \(\Omega^1(\mathbb{T}^2,\mathfrak{g})\) of \(\mathfrak{g}\)-valued \(1\)-forms on \(\mathbb{T}^2\). The constant \(Z\) is chosen such that \(\mu\) becomes a probability measure. Finally, \[ S_{\mathrm{YM}}(A) := \int_{\mathbb{T}^2} |F_A(x)|^2\,dx \] is the Yang-Mills action where \(F_A\) is the curvature two-form of \(A\). One possibility to understand \(\mu\) is to investigate the distributions of certain gauge invariant observables, such as Wilson loops, which are defined via holonomies. The main goal is to use a characterization of the Yang-Mills measure by \textit{T. Lévy} [Yang-Mills measures on compact surfaces. Providence, RI: American Mathematical Society (AMS) (2003; Zbl 1036.58009)] to realize \(\mu\) as a random distribution with certain regularity properties for small scales. For this purpose, a new space of distributional \(1\)-forms \(\Omega_\alpha^1\) on \(\mathbb{T}^2\) is introduced for which ``holonomies along axis paths are well-defined and induce Hölder continuous functions on line segments''. Basic properties of \(\Omega_\alpha^1\) are discussed in Section 3. The main result of this article (Theorem 1.1) is the existence of a ``\(\Omega_\alpha^1\)-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 Lévy''. In particular, it is shown that \(A\) has regularity properties (which are summarized in Lemma 3.8 and Theorem 3.26) on small scales which are expected from perturbation theoretic arguments. One of the main ingredients in the proof of Theorem 1.1 is the construction of a gauge on lattice approximations of \(\mathbb{T}^2\) through iterations of the Landau (Coulomb) gauge and of a (non-unique) random variable \(A\in\Omega^{1,(N_1)}(\mathbb{T}^2,\mathfrak{g})\) (the space of discrete \(\mathfrak{g}\)-valued \(1\)-forms on a lattice approximation) ``for which a class of Wilson loops is canonically defined and has the same joint distributions as under the Yang-Mills measure'' (Theorem 4.5). In Theorem 5.3 (which is proven using a random walk argument), the author establishes probabilistic bounds on certain discretizations of the Yang-Mills measure, thereby showing that these discretizations satisfy the conditions of Theorem 4.5. The discrete approximations of \(\mu\) cover, e.g., the Villain (heat kernel) and Wilson actions.
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