Fix a "nice" curve $X$ (smooth, projective, proper, geometrically connected, what-have-you) and an algebraic torus $G$, both over a field of characteristic $0$ (possibly algebraically closed?).


From what I have been able to gather from several lecture videos by Gaitsgory (such as [Gaitsgory - Singular support of coherent sheaves](https://youtu.be/S7f_SSZIXQQ)), the categorical-geometric Langlands Correspondence $G$ is supposed to be an equivalence of categories:
$$\mathsf{DMod}\left(\mathsf{Bun}\_G(X)\right) \cong \mathsf{QCoh}(\mathsf{LocSys}^{{}^LG}(X))$$
wherein $\mathsf{DMod}\left(\mathsf{Bun}\_G(X)\right)$ is the ($\infty$-)category of D-modules on the moduli stack $\mathsf{Bun}\_G(X)$ of $G$-bundles on $X$, and $\mathsf{QCoh}(\mathsf{LocSys}^{{}^LG})$ is the ($\infty$-)category of quasi-coherent modules on the stack of ${}^L G$-equivariant local systems on $X$, where ${}^L G$ is the Langlands dual of $G$.


My questions are then:


1. Is this the correct statement? Specifically, I would like to understand if we need to replace $\mathsf{QCoh}$ with $\mathsf{IndCoh}$ (see part II of [A study in derived algebraic geometry, Volume I: Correspondences and duality](http://people.math.harvard.edu/%7Egaitsgde/GL/Vol1.pdf) by Gaitsgory and Rozenblyum for an introduction to ind-coherent sheaves), and why (or why not)?
2. What is a standard reference that I can consult? Most references that I have been able to find (by Gaitsgory, Laumon, etc.) are about the Langlands Correspondence for $\operatorname{GL}\_2$, $\operatorname{GL}\_n$, or more general kinds of reductive groups, and I don't think I have what it takes yet to extract the special case of tori from these papers.



Categorical-geometric Langlands for tori
In case $G=T$ is a torus and $G^\vee=T^\vee$ is the dual torus, the geometric Langlands conjecture — or “categorical geometric class field theory” (for a smooth projective curve $C$ over $\mathbb C$) becomes a theorem, and has the form you stated: a derived equivalence
$$\mathsf{DMod}(\mathsf{Bun}\_T)\simeq \mathsf{QCoh}(\mathsf{LocSys}\_{T^\vee}).$$
It is due (in a form at least very close to this) to Laumon ("[Transformation de Fourier Generalisée](https://arxiv.org/abs/alg-geom/9603004)") and Rothstein ("[Sheaves with Connection on Abelian Varieties](https://doi.org/10.1215/S0012-7094-96-08418-5)"), who independently extended the Fourier–Mukai transform from coherent sheaves on an abelian variety to D-modules.


Indeed $\mathsf{Bun}\_T$ is the product of a lattice, an abelian variety and $\mathrm{pt}/T$. $\mathsf{LocSys}$ is the product of the rigidified scheme of flat connections, a factor $\mathrm{pt}/T^\vee$ from automorphisms and a constant derived factor $\mathfrak t^\*[-1]$. The lattice matches with the stacky factor $\mathrm{pt}/T^\vee$, while the $\mathrm{pt}/T$-factor matches with the constant derived structure of $\mathsf{LocSys}$. The "interesting" part is the abelian variety and the dual rigidified space of flat connections, which is the Laumon–Rothstein theorem.


The reason why you see $\mathsf{QCoh}$ rather than $\mathsf{IndCoh}$ is the manner in which you treat the stacky factor on $\mathsf{Bun}$, ie by what you mean by D-modules on a stack — you can change ("renormalize") the definition to get all of $\mathsf{IndCoh}$. (You can see it from Koszul duality, lining up usual $D$-modules on $\mathrm{pt}/T$ with quasicoherent rather than ind-coherent modules over the Koszul dual exterior algebra.) For general $G$ you can match up D-modules with ind-coherent sheaves with nilpotent singular support (which in the abelian case recovers $\mathsf{QCoh}$), as developed in the (first) paper [Singular support of coherent sheaves and the geometric Langlands conjecture](https://doi.org/10.1007/s00029-014-0167-5) of Arinkin–Gaitsgory. Or you can "renormalize" D-modules to get something that conjecturally recovers all of $\mathsf{IndCoh}$.


