  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.
