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 <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$H=s-{n\over 2}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mi>H</mi> <mo>=</mo> <mi>s</mi> <mo>−</mo> <mrow> <mfrac> <mi>n</mi> <mn>2</mn> </mfrac> </mrow> </math></EquationSource> </InlineEquation>.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<Stack> s k </Stack>(M)) is given by <InlineEquation ID="IEq2"> <EquationSource Format="TEX">${( - \Delta )^{ - {s \over 2}}}{W_k}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mi mathvariant="normal">Δ</mi> <msup> <mo stretchy="false">)</mo> <mrow> <mo>−</mo> <mrow> <mfrac> <mi>s</mi> <mn>2</mn> </mfrac> </mrow> </mrow> </msup> </mrow> <mrow> <msub> <mi>W</mi> <mi>k</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, where Wk is a k-form-valued white noise. We write <Equation ID="Equ1"> <EquationSource Format="TEX">$\text{FGF}_{s}^{k}(M)_{d=0}\qquad \text{and} \qquad \text{FGF}_{s}^{k}(M)_{d=0}$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <msubsup> <mtext>FGF</mtext> <mrow> <mi>s</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>M</mi> <msub> <mo stretchy="false">)</mo> <mrow> <mi>d</mi> <mo>=</mo> <mn>0</mn> </mrow> </msub> <mspace width="2em" /> <mtext>and</mtext> <mspace width="2em" /> <msubsup> <mtext>FGF</mtext> <mrow> <mi>s</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi>M</mi> <msub> <mo stretchy="false">)</mo> <mrow> <mi>d</mi> <mo>∗</mo> <mo>=</mo> <mn>0</mn> </mrow> </msub> </math></EquationSource> </Equation> for the L2 orthogonal projections of FGF<Stack> s k </Stack>(M) onto the space of k-forms on which d (resp. d) vanishes. We explain how FGF<Stack> s k </Stack>(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 FGF<Stack> 1 1 </Stack>(M) and its gauge-fixed projection FGF<Stack> 1 1 </Stack>(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 FGF<Stack> 1 1 </Stack>(M)d=0, which can be interpreted as a random connection whose curvature is (a Lie-algebra valued analog of) FGF<Stack> 2 0 </Stack>(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 FGF<Stack> 1 1 </Stack>(M)d=0 into big-surface integrals of FGF<Stack> 2 0 </Stack>(M)d=0 or FGF<Stack> 2 0 </Stack>(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 FGF<Stack> 1 1 </Stack>(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.

Fractional Gaussian Forms and Gauge Theory: An Overview

期刊论文
发表日期：

2025年11月20日

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Sky Cao,
Scott Sheffield

摘要

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 $H=s-{n\over 2}$ $H = s - \frac{n}{2}$ .

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 ${( - \Delta )^{ - {s \over 2}}}{W_k}$ $(- Δ)^{- \frac{s}{2}} W_{k}$ , where W_k is a k-form-valued white noise. We write $\text{FGF}_{s}^{k}(M)_{d=0}\qquad \text{and} \qquad \text{FGF}_{s}^{k}(M)_{d*=0}$ ${FGF}_{s}^{k} (M)_{d = 0} and {FGF}_{s}^{k} (M)_{d * = 0}$ 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 FGF ₁ ¹ (M) and its gauge-fixed projection FGF ₁ ¹ (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 FGF ₁ ¹ (M)_d*=0, which can be interpreted as a random connection whose curvature is (a Lie-algebra valued analog of) FGF ₂ ⁰ (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 FGF ₁ ¹ (M)_d*=0 into big-surface integrals of FGF ₂ ⁰ (M)_d=0 or FGF ₂ ⁰ (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 FGF ₁ ¹ (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.