The geometric Langlands categorification conjecture for Hall algebras

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Let XX be a smooth projective complex curve and let Bun(X)dR\mathbf{Bun}(X)_\mathsf{dR} and Coh(XdR)\mathbf{Coh}(X_\mathsf{dR}) denote the de Rham moduli stacks of bundles and coherent sheaves on XX. Geometric Langlands categorification conjecture. There should exist an E1\mathbb E_1-algebra structure on the dg-category Cohb(Bun(X)dR)\mathsf{Coh}^{\mathsf b}(\mathbf{Bun}(X)_\mathsf{dR}) and an E1\mathbb E_1-monoidal equivalence

Cohb(Bun(X)dR)≃Cohb(Coh(XdR)).\mathsf{Coh}^{\mathsf b}(\mathbf{Bun}(X)_\mathsf{dR})\simeq \mathsf{Coh}^{\mathsf b}(\mathbf{Coh}(X_\mathsf{dR})).

This is proposed as a categorified, de Rham analogue of the relation between the two realizations of the quantum toroidal algebra discussed in the cited literature. The source does not provide evidence of resolution.

References

Primary source

Mauro Porta and Francesco Sala, “Two-dimensional categorified Hall algebras”, arXiv:1903.07253 (2022).

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