# Implications of gauge symmetry breaking on the spectral side of geometric Langlands?

Let $G$ be a complex reductive algebraic group and $X$ be a smooth compact complex curve. It's easy to see that the space of vacua in B-twisted $N=4$ SUSY Yang--Mills theory is $\mathfrak{h}^*[2]/W$ (where $\mathfrak{h}$ is the Cartan subalgebra of $\mathfrak{g}$ and $W$ is the Weyl group).

There is a theorem due to Elliott and Yoo stating that the full category of boundary conditions compatible with the vacuum $0 \in \mathfrak{h}^*[2]/W$ is equivalent to $IndCoh_{\mathcal{N}_G}(LocSys_G)$, the category of ind-coherent sheaves on the moduli space of principal $G$-bundles equipped with flat connection on $X$ with singular support contained in the global nilpotent cone. Note that the former is the spectral side of geometric Langlands correspondence due to Arinkin--Gaitsgory.

This theorem gives a physical interpretation to $IndCoh_{\mathcal{N}_G}(LocSys_G)$ (and makes it clear that it's the right category to consider in geometric Langlands correspondence, as opposed to the category of quasi-coherent sheaves $QCoh(LocSys_G)$). Elliott and Yoo formulated following conjecture expressing the effect of gauge symmetry breaking on the categories under consideration: the full subcategory of objects in $IndCoh(LocSys_G)$ compatible with a vacuum $u\in \mathfrak{h}^*/W$ is equivalent to $IndCoh_{\mathcal{N}_L}(LocSys_L)$ where $L$ is the stabilizer of $u$.

My question is: what are the mathematical implications of this conjecture in the context of geometric Langlands program?

## 1 Answer

We discussed some conjectural implications in Section 4.2 of the paper. I wouldn't say that the category of sheaves with nilpotent singular support was necessarily the "right" category to consider from a gauge theoretic point of view. Instead I'd say that if one considers the equivalence for the whole category of coherent objects (or its completion) the categories on both sides of the equivalence form a flat family over the moduli space of vacua with the equivalence taking place compatibly with this family, which is to say equivariantly for the action of the algebra of local operators on the two sides. Of course there's a missing ingredient here: a concrete understanding of the action of the algebra of local operators on the category of ind-coherent / renormalized D-modules on $\mathrm{Bun}_G$.

One possible implication we proposed was that, at least for $G = \mathrm{GL}_n$, these equivalences should "assemble" in the limit $n \to \infty$ to define an equivalence of factorization categories over $\mathbb{C}$. Physically we can think of this as the family of twisted theories on arbitrary collections of parallel D3 branes in type IIB factorizing nicely whenever two coincident branes are moved apart. The existence of this factorization category (and the prediction that duality preserves the factorization structures) is, however, a purely mathematical expectation. It's distinct from the factorization structure on the curve that usually appears in the geometric Langlands literature.

• Isn't the action of local operators on the automorphic side given by the identification of local operators with H^*(BG), and the linearity of the category of D-modules on Bun_G given by a choice of point in the Riemann surface (giving a map from Bun_G to BG)? Jul 2, 2018 at 22:12
• In that context I'd like to mention the work of Beraldo on "H-modules", which enables one to explain the automorphic versions of various singular support conditions on the spectral side -- eg what it means to be a quasicoherent sheaf on Loc rather than Ind-coherent -- more intrinsically, ie without choosing a point in the Riemann surface Jul 2, 2018 at 22:14
• Hi David. To clarify (and I'm not claiming this is necessarily hard, just that I haven't done the calculation yet) that is what the action looks like, but I haven't worked out an explicit enough description to characterize which coherent D-modules are supported at non-zero points in $\mathfrak{h}^*/W$ (when this makes sense: say after adjoining an invertible degree 2 parameter). Arinkin and Gaitsgory explain the support condition at 0 in their Remark 12.8.8 but we still need to work out the analogue for other vacua. Jul 3, 2018 at 20:00
• is this factorization structure somehow related/inspired by Langlands functoriality in the version described here: webusers.imj-prg.fr/~michael.harris/Takagi.pdf#page=11 ?
– user74900
May 15, 2019 at 17:54