Fractional Gaussian forms and gauge theory: an overview

TL;DR

This paper provides an overview of fractional Gaussian fields and their differential form analogs, exploring their mathematical properties, transformations, and connections to gauge theories, while highlighting open problems and conjectures in the field.

Contribution

It introduces fractional Gaussian $k$-forms, analyzes their transformations, and discusses their relevance to gauge theories and open problems in scaling limits.

Findings

Defined fractional Gaussian $k$-forms and their projections.

Analyzed transformation properties under differential operators.

Connected $1$-form FGF to gauge theories and outlined open problems.

Abstract

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 $\textit{differential form}$ analogs of these objects are equally natural: for example, instead of considering an instance $h(x)$ of the GFF on $\mathbb R^2$, one might write $h_1(x)dx_1 + h_2(x) dx_2$ where $h_1$ and $h_2$ are independent GFF instances. In general, given $k \in \{0,1,\ldots,n\}$, an instance of the $\textit{fractional Gaussian $k$-form}$ with parameter $s \in \mathbb R$ (abbreviated $\mathrm{FGF}_s^k(M)$) is given by $(-\Delta)^{-\frac{s}{2}} W_k,$ where $W_k$ is a $k$-form-valued white noise. We write $$\textrm{FGF}_s^k(M)_{d=0} \quad \textrm{and} \quad \textrm{FGF}_s^k(M)_{d^*=0}$$ for the $L^2$ orthogonal projections of $\textrm{FGF}_s^k(M)$ onto the space of $k$-forms on which $d$ (resp.\ $d^*$) vanishes. We explain how $\mathrm{FGF}_s^k(M)$ 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 $\textrm{FGF}_1^1(M)$ and its $\textit{gauge-fixed}$ projection $\textrm{FGF}_1^1(M)_{d^*=0}$ 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.