As pointed out in the comments, this is Theorem 6.3.1.2 of Hilburn-Raskin. (It certainly was known much earlier, but I'm not sure what to give as a reference.)

Their proof is stated quite elegantly, in a way that makes the canonicity of the equivalence clear. Let me give a less canonical explanation, which you may or may not find more digestible. (But note that if you want to show good properties of this equivalence it really does help to have a canonical description.)

The point is that $L\mathbb{G}_m$ and $\operatorname{LocSys}_{\mathbb{G}_m}(\mathbb{D})$ are relatively simple and can be described explicitly.

Let's start with $L\mathbb{G}_m$. Actually, let's start with its reduced (ind-)scheme $(L\mathbb{G}_m)_{\operatorname{red}}.$ By Kashiwara's lemma, this has the same category of D-modules as does $L\mathbb{G}_m.$ I claim there is an isomorphism

$$(L\mathbb{G}_m)_{\operatorname{red}}\cong\mathbb{Z}\times\mathbb{G}_m\times\mathbb{A}^{\infty}_{\operatorname{pro}}$$

where $\mathbb{A}^{\infty}_{\operatorname{pro}}$ is $\operatorname{Spec}$ of a polynomial ring in infinitely many generators. The -pro subscript reflects that this is a scheme which is an inverse limit of finite-dimensional affine spaces; later we will see its cousin $\mathbb{A}^{\infty}_{\operatorname{ind}}$ which is the ind-scheme that is a colimit of finite-dimensional affine spaces.

Let me justify this isomorphism on $\mathbb{C}$-points and leave verifying it at the level of functors of points as an exercise. The $\mathbb{C}$-points of $L\mathbb{G}_m$ parametrize invertible elements of $\mathbb{C}((t))$. There are of the form

$$ct^d(1+a_1t+a_2t^2+\cdots)$$

with $d\in\mathbb{Z}$, $c$ an invertible complex number, and $a_i$ complex numbers. The $d$ corresponds to the $\mathbb{Z}$-factor, the $c$ gives the $\mathbb{G}_m$, and the $a_i$ give the $\mathbb{A}^{\infty}_{\operatorname{pro}}.$

Now we turn to $\operatorname{LocSys}_{\mathbb{G}_m}(\mathbb{D}).$ I claim there is an isomorphism

$$\operatorname{LocSys}_{\mathbb{G}_m}(\mathbb{D})\cong B\mathbb{G}_m\times(\mathbb{A}^1/\mathbb{Z})\times(\mathbb{A}^{\infty}_{\operatorname{ind}})_{\operatorname{dR}}.$$

Here $B\mathbb{G}_m$ is the classifying stack of $\mathbb{G}_m$ and $\mathbb{A}^1/\mathbb{Z}$ is the quotient of the affine line by the
translation $\mathbb{Z}$-action. The last term is a little more subtle. As mentioned before, there is an ind-scheme $\mathbb{A}^{\infty}_{\operatorname{ind}}$ defined as a colimit of finite dimensional affine spaces. I take its de Rham stack, which has functor of points

$$\operatorname{Hom}(X,(\mathbb{A}^{\infty}_{\operatorname{ind}})_{\operatorname{dR}})\cong\operatorname{Hom}(X_{\operatorname{red}},\mathbb{A}^{\infty}_{\operatorname{ind}}).$$

The main property I will use of de Rham stacks is that $\operatorname{QCoh}((\mathbb{A}^{\infty}_{\operatorname{ind}})_{\operatorname{dR}})\cong\operatorname{D-mod}(\mathbb{A}^{\infty}_{\operatorname{ind}})$.

Once again, let me check this expression for $\operatorname{LocSys}_{\mathbb{G}_m}(\mathbb{D})$ at the level of $\mathbb{C}$-points. Of course, you should be fundamentally unsatisfied with this, given that the de Rham stack is invisible if you only look at $\mathbb{C}$-points, but I leave the functor of points check as an exercise. The moduli space of local systems is the quotient

$$(dt+\mathbb{C}((t)))/\mathbb{C}((t))^{\times}$$

where $\mathbb{C}((t))^{\times}$ acts by gauge transformations. Because we are in an abelian setting, the gauge action is super simple; the action of $f$ is just translation by $\frac{df}{f}.$ Again, write

$$f=ct^d(1+a_1t+a_2t^2+\cdots).$$

Then we have

$$\frac{df}{f}=\frac{d}{t}+\operatorname{dlog}(1+a_1t+a_2t^2+\cdots).$$

The $c$ does not appear in the final expression, indicating that the $\mathbb{G}_m$ corresponding to $c$ acts trivial, which contributes a $B\mathbb{G}_m$. The $\operatorname{dlog}(1+a_1t+a_2t^2+\cdots)$ can be any formal power series, so all in all the $\mathbb{C}$-points of our quotient are elements of

$$B\mathbb{G}_m\times\mathbb{C}((t))/(\mathbb{Z}\frac{1}{t}+\mathbb{C}[[t]]).$$

The coefficient of $\frac{1}{t}$ gives a $\mathbb{A}^1/\mathbb{Z}$, and the coefficients of $\frac{1}{t^n}$, $n\geq 2$, give a $(\mathbb{A}^{\infty}_{\operatorname{ind}})_{\operatorname{dR}}.$ To see that we get $\mathbb{A}^{\infty}_{\operatorname{ind}}$ and not $\mathbb{A}^{\infty}_{\operatorname{pro}},$ note that only finitely many coordinates are allowed to be nonzero. (If you want to see where the de Rham stack comes from, you'll have to do this calculation at the level of functors of points.) So that gives our expression for $\operatorname{LocSys}_{\mathbb{G}_m}(\mathbb{D}).$

Once we have these formulas, we can see that geometric local class field theory is the product of three simpler equivalences:

$$\operatorname{D-mod}(\mathbb{Z})\cong\operatorname{QCoh}(B\mathbb{G}_m)$$

$$\operatorname{D-mod}(\mathbb{G}_m)\cong\operatorname{QCoh}(\mathbb{A}^1/\mathbb{Z})$$

$$\operatorname{D-mod}(\mathbb{A}^{\infty}_{\operatorname{pro}})\cong\operatorname{QCoh}((\mathbb{A}^{\infty}_{\operatorname{ind}})_{\operatorname{dR}})\cong\operatorname{D-mod}(\mathbb{A}^{\infty}_{\operatorname{ind}}).$$

The first equivalence follows because $\operatorname{QCoh}(B\mathbb{G}_m)$ is the representation category of $\mathbb{G}_m$, which can be described as the category of $\mathbb{Z}$-graded vector spaces.

The second equivalence is the Fourier transform for D-modules on $\mathbb{G}_m$ (it sends the D-module $Dt^{\lambda}$ on $\mathbb{G}_m$ to $\lambda$.)

The third equivalence is a limit of finite-dimensional additive Fourier transforms of D-modules.

Combined they give the desired equivalence.

invertible elementsin $A((t))$? Or are you considering the loop space of the addtive group $\mathbb{G}_a$? $\endgroup$1more comment