Moduli stack of G-local systems

Definition

Let $X$ be a smooth projective curve over a characteristic-zero field and let $G$ be a reductive algebraic group. The de Rham moduli stack of $G$ -local systems is the derived mapping stack

\operatorname{LocSys}^{\mathrm{dR}}_G(X) =\operatorname{Map}(X_{\mathrm{dR}},BG).

It parametrizes principal $G$ -bundles with flat connection in families.

When $X$ is a compact Riemann surface, the Betti moduli stack is the derived character stack

\operatorname{LocSys}^{\mathrm{B}}_G(X) =\operatorname{Map}(X_{\mathrm{Betti}},BG),

whose classical points are representations of $\pi_1(X)$ in $G$ modulo conjugation. Riemann–Hilbert comparison relates the analytifications of the de Rham and Betti stacks; it does not identify their algebraic structures.

Why the derived stack matters

Automorphisms and deformation-obstruction groups are part of the moduli problem. The derived enhancement records them. For a smooth projective curve, $\operatorname{LocSys}^{\mathrm{dR}}_G(X)$ is quasi-smooth, so coherent sheaves on it have a singular support.

Spectral role

Geometric Langlands uses $\operatorname{LocSys}_{\widehat G}(X)$ , where $\widehat G$ is the dual group. Its relevant sheaf category is generally $\operatorname{IndCoh}_{\mathcal N}$ , not merely quasi-coherent sheaves.

References

Dima Arinkin and Dennis Gaitsgory, “Singular support of coherent sheaves, and the geometric Langlands conjecture,” Selecta Mathematica 21 (2015), 1–199. arXiv.