Fractional Gaussian forms and gauge theory: an overview

Fractional Gaussian fields are scalar-valued random functions or generalized functions on an n-dimensional manifold M, indexed by a parameter s. They include white noise ($s = 0$), Brownian motion ($s=1, n=1$), the 2D Gaussian free field ($s = 1, n=2$) and the membrane model ($s = 2$). These simple objects are ubiquitous in math and science, and can be used as a starting point for constructing non-Gaussian theories. The differential form analogs of these objects are equally natural: for example, instead of considering an instance $h(x)$ of the GFF on R², one might write h₁(x)dx₁ + h₂(x) dx₂ where h₁ and h₂ are independent GFF instances. In general, given k ∈ \0,1,…,n\, an instance of the fractional Gaussiank-form with parameter s ∈ R (abbreviated FGFₛᵏ(M)) is given by (-Δ)-s/2 Wₖ, where Wₖ is a k-form-valued white noise. We write FGFₛᵏ(M)d=0 and FGFₛᵏ(M)d^*=0 for the L² orthogonal projections of FGFₛᵏ(M) onto the space of k-forms on which d (resp.\ d^*) vanishes. We explain how FGFₛᵏ(M) and its projections transform under d and d^*, as well as wedge/Hodge-star operators, subspace restrictions, and axial projections. We discuss how the $1$-form FGF₁¹(M) and its gauge-fixed projection FGF₁¹(M)d^*=0 are related to gauge theories, and we formulate several conjectures and open problems about scaling limits, including possible off-critical/non-Gaussian limits, whose construction in the Yang-Mills setting is a famous open problem.