Fractional Gaussian forms and gauge theory: an overview
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2024
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| author | Cao, Sky Sheffield, Scott |
| author_facet | Cao, Sky Sheffield, Scott |
| contents | 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 $(-Δ)^{-\frac{s}{2}} W_k,$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_19321 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Fractional Gaussian forms and gauge theory: an overview Cao, Sky Sheffield, Scott Probability Mathematical Physics 60G60, 81T13 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 $(-Δ)^{-\frac{s}{2}} W_k,$ 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. |
| title | Fractional Gaussian forms and gauge theory: an overview |
| topic | Probability Mathematical Physics 60G60, 81T13 |
| url | https://arxiv.org/abs/2406.19321 |