Arc space & formal loops in motivic integration

Question

One of the most essential ingredients in the theory of motivic integration are the space of arcs of a given $k$-variety $X$. This is a scheme, whose $k$-rational points are the $k[[t]]$-valued points of $X$.

In german the space of arcs is also called the "Raum der formalen Schleifen": literally that translates as "space of formal loops".

My question is if there any geometrical reason & motivation to call these objects "arcs" or "formal loops"? Does there exist any analogy to the topological loops and loop spaces which motivates the choice of the name "arcs" or "formal loops" for these objects occuring in motivic integration?

if $char \ k =0$ we can also associate to $X$ the topological loop space by taking the space $\Omega X(\mathbb{C}):=Hom(S^1, X(\mathbb{C}))$ where the set of $\mathbb{C}$-valued points $X(\mathbb{C})$ can be endowed with analytic topology if $X$ nice enough. $\Omega X(\mathbb{C})$ is the topological loop space of $X$. Is there any meaningful relation between this topological loop space and the space of arcs as defined above justifying the choice of similar names?

Answer 1

I believe Nash originally chose the term "arc" to mean a short path, rather than a circle. The ring of functions on the completion of an algebraic curve over $k$ at a smooth $k$-point is isomorphic to $k[[t]]$, so maps from its spectrum to a variety $X$ can be viewed as infinitesimal pieces of a smooth curve in $X$. I do not know why it was translated to "Schleife" in German. To me, it seems to be a potentially misleading choice.

The term "formal loop" in English is typically reserved for a map from the spectrum of the formal Laurent series field $k((t))$ (see, e.g., Kapranov, Vasserot "Vertex algebras and the formal loop space"). While the spectrum of $k((t))$ is a point, there are good reasons to think of it as a small circle. First, if we view the spectrum of $k[[t]]$ as an infinitesimal disc or a small piece of a Riemann surface (roughly, the place where formal power series converge), then inverting $t$ removes the central point, so we may imagine the remaining space as an infinitesimal annulus or circle. Second, $\operatorname{Spec} k((t))$ looks like a point with "Zariski glasses", but with "étale glasses", we can see finer structure coming from finite extensions. In particular, if $k$ is algebraically closed of characteristic zero, then any finite extension of $k((t))$ is isomorphic to $k((t^{1/n}))$ for some $n \geq 1$, so we can identify the étale fundamental group with that of a punctured complex disc or a circle.

One memorable difference between formal arcs and formal loops is that for $X$ a variety of positive dimension, the space of formal loops in $X$ is not representable by a scheme because of its large size.