Fractional Gaussian Forms and Gauge Theory: An Overview

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. They are sometimes parameterized by the Hurst parameter

H=s-\frac{n}{2}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H=s-{n\over 2}$$\end{document}

.

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. This “Gaussian-free-field-based 1-form” can be projected onto curl-free and divergence-free components, which in turn arise as gradients and dual-gradients of independent membrane models.

In general, given k ∈ {0,1,…, n}, an instance of the fractional Gaussian k-form with parameter s ∈ ℝ (abbreviated FGF_s^k(M)) is given by

(-\mathrm{\Delta }{)}^{-\frac{s}{2}}{W}_k

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${( - \Delta )^{ - {s \over 2}}}{W_k}$$\end{document}

, where W_k is a k-form-valued white noise. We write

{\text{FGF}}_s^k(M{)}_{d=0}\text{and}{\text{FGF}}_s^k(M{)}_{d\ast =0}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\text{FGF}_{s}^{k}(M)_{d=0}\qquad \text{and} \qquad \text{FGF}_{s}^{k}(M)_{d*=0}$$\end{document}

for the L² orthogonal projections of FGF_s^k(M) onto the space of k-forms on which d (resp. d*) vanishes. We explain how 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 review “massive” variants, variants on lattices, and a variant involving the Chern–Simons action. We explain the role of the higher order cohomology groups of M.

The 1-form FGF₁¹(M) and its gauge-fixed projection FGF₁¹(M)_d*=0 arise as low-temperature/small-scale limits of U(1) Yang–Mills gauge theory. When U(1) is replaced with another compact Lie group, the corresponding limit is a Lie-algebra-valued analog of FGF₁¹(M)_d*=0, which can be interpreted as a random connection whose curvature is (a Lie-algebra valued analog of) FGF₂⁰(M)_d=0.

Wilson loop observables of this connection are defined for sufficiently regular “big loops” (trajectories in the space of divergence-free 1-forms obtainable as limits of finite-length loops) in a gauge invariant way, even in the non-abelian case. We define “big surfaces” (trajectories in the space of 2-forms obtainable as limits of smooth surfaces with boundary) and note that Stokes’ theorem converts big-loop integrals of FGF₁¹(M)_d*=0 into big-surface integrals of FGF₂⁰(M)_d=0 or FGF₂⁰(M). A type of exponential correlation decay and area law applies within the slabs ℝ² × [0,1]^m but not within ℝⁿ for n > 2.

One may interpret FGF₁¹(M)_d*=0 as a random divergence-free vector field, which is conjectured to be the fine-mesh scaling limit of the n-dimensional dimer model when n > 2. (Kenyon proved this for n = 2.) 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.

Keywords

Gaussian k-forms / random distributions / Yang–Mills / 60G60 / 81T13

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