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A Physical Origin for Singular Support Conditions in Geometric Langlands Theory

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Abstract

We explain how the nilpotent singular support condition introduced into the geometric Langlands conjecture by Arinkin and Gaitsgory arises naturally from the point of view of 4-dimensional \(\mathcal {N}=4\) supersymmetric gauge theory. We define what it means in topological quantum field theory to restrict a category of boundary conditions to the full subcategory of objects compatible with a fixed choice of vacuum, both in functorial field theory and in the language of factorization algebras. For B-twisted \(\mathcal {N}=4\) gauge theory with gauge group G, the moduli space of vacua is equivalent to \(\mathfrak {h}^*/W\), and the nilpotent singular support condition arises by restricting to the vacuum \(0 \in \mathfrak {h}^*/W\). We then investigate the categories obtained by restricting to points in larger strata, and conjecture that these categories are equivalent to the geometric Langlands categories with gauge symmetry broken to a Levi subgroup, and furthermore that by assembling such for the groups \(\mathrm {GL}_n\) with \({n\ge 1}\) one finds a hidden factorization algebra structure for the geometric Langlands theory.

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Notes

  1. The monoidal structure we’ll use depends on a choice of plane \(\mathbb {R}^2 \subseteq \mathbb {R}^n\). We’ll explain this in Section 2.2.

  2. This is called the phase space because it is the space of quantum states on a codimension 1 manifold \(Y\times [0,1]\); that it has the structure of an algebra is an additional property arising from considering the same boundary condition at both ends.

  3. The same argument works if observables map to something nilpotent; these two notions lead to scheme-theoretic support and set-theoretic support, respectively. For their comparison, look at Section 2.3.2.

  4. At least for nice enough (“QCA”) stacks, a class of stacks including \(\mathrm {Flat}_G(C)\). See [DG15, Theorem 3.3.5].

  5. For instance, Abelian Chern–Simons theory should have \((\Omega ^\bullet (U), d_{\mathrm {dR}})\) on each open set U as the space of fields, but \(\Omega ^1(U)\) is not linear dual to \(\Omega ^2(U)\).

  6. From a physical point of view we can think of the differential graded structure here as arising from the BRST formalism: the space of states is graded by ghost number and the BRST operator acts as a differential.

  7. For the purpose of this paper, the essential examples end up being versions of the B-model. The category of boundary conditions of B-model with target X is usually modelled by \({{\,\mathrm{Coh}\,}}(X)\); without loss or gain of information we can alternatively consider the category \({{\,\mathrm{IndCoh}\,}}(X)\). One may just use this as the category of boundary conditions, without relying on the framework of categorified geometric quantization.

  8. The natural functor is defined only when \(\mathcal {X}\) is eventually coconnective, that is, its structure sheaf should have cohomology in bounded degrees in the derived direction.

  9. Arinkin and Gaitsgory write this as \(\mathcal {A}^{\text {int-op}}\), where “int” stands for “internal”, they write \(\mathcal {A}^{\text {ext-op}}\) for the opposite using the other (“external”) \(\mathbb {E}_1\)-structure.

  10. There’s also a smallness condition: they should be locally finitely presented.

  11. A map of derived stacks is schematic if base changing by a derived scheme produces a derived scheme.

  12. We don’t have a class of theories which have two such descriptions in mind. While it’s possible to construct factorization algebras of quantum observables explicitly for, e.g, topologically twisted supersymmetric theories, there isn’t a known procedure for building corresponding functorial field theories assigning a number to a compact n-manifold. It would be interesting to check this expectation for the example of Chern–Simons theory.

  13. We fixed such an identification when we specified the AKSZ shifted symplectic structure above: the AKSZ shifted symplectic structure required choosing a 2-shifted symplectic structure on BG: this structure is equivalent to the choice of a G-invariant isomorphism \(\mathfrak {g}\cong \mathfrak {g}^*\).

  14. One might try to define it as an affine stack in the sense of [BZN12, Toë06], but we expect this will not be a productive way to think about it. For instance there are no non-zero maps from \(\mathbb {C}[u,u^{-1}]\) to a connective cdga, so if one tries to define \({{\,\mathrm{Spec}\,}}\mathbb {C}[u,u^{-1}]\) as a functor in the naïve way, the result is the empty stack.

  15. If G is semisimple we can define \(\widetilde{v}\) as the pullback of v under \(\mathfrak {g}^*/G \rightarrow \mathfrak {h}^*/W\), but this map doesn’t quite make sense if G is Abelian.

  16. We should remark that the question of why the relationship of boundary conditions on either side of these parabolic domain walls should be preserved by S-duality is beyond the scope of this paper.

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Acknowledgements

This project began with a suggestion of David Nadler and David Ben-Zvi, and would not have been possible without David Ben-Zvi generously sharing his ideas about the nature of vacua in the context of functorial quantum field theories. We would like to thank Kevin Costello for many helpful conversations about all aspects of this work. We would also like to thank Dima Arinkin, Davide Gaiotto, Dennis Gaitsgory, Saul Glasman, Owen Gwilliam, Pavel Safronov, Claudia Scheimbauer, and Brian Williams for useful conversations and comments, and the anonymous referees for detailed comments and criticism. This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Industry Canada and by the Province of Ontario through the Ministry of Economic Development & Innovation. CE acknowledges the support of IHÉS. The research of CE on this project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (QUASIFT Grant Agreement 677368). PY acknowledges the support of IHÉS during his visit in 2017 and funding from ERC QUASIFT Grant 677368.

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  1. Institut des Hautes Études Scientifiques, 35 Route de Chartres, Bures-sur-Yvette, 91440, France

    Chris Elliott

  2. Department of Mathematics, Yale University, 10 Hillhouse Ave., New Haven, CT, 06511, USA

    Philsang Yoo

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Appendices

Appendices

Singular Support for Global Complete Intersections

In this “Appendix” we’ll describe the proof of Proposition 2.49 from Section 2.2.2. The argument we’ll give is only a slight modification of the arguments in [AG15, Section 9.2]. We give a proof here, following Arinkin and Gaitsgory closely, in order to justify the modification where the auxiliary degree 2 parameter is included.

Recall the setup: \(\mathcal {Z}\) was a global complete intersection stack, which meant that it could be written as a fiber product smooth stacks

where \(\mathcal {X} \rightarrow \mathcal {V}\) is a section of a smooth schematic map \(\mathcal {V}\rightarrow \mathcal {X}\). One can associate to such a global complete intersection a pair of groupoids which we’ll use heavily: namely \(\mathcal {G}_{\mathcal {Z}/\mathcal {U}} := \mathcal {Z} \times _{\mathcal {U}} \mathcal {Z}\) as a derived group scheme over \(\mathcal {Z}\) and \(\mathcal {G}_{\mathcal {X}/\mathcal {V}} := \mathcal {X} \times _{\mathcal {V}} \mathcal {X}\), a derived group scheme over \(\mathcal {X}\).

There is a canonical morphism \(f :\mathrm {Sing}(\mathcal {Z}) \rightarrow \mathrm {Sing}(\mathcal {G}_{\mathcal {X}/\mathcal {V}} \times _{\mathcal {X}} \mathcal {U}) \cong V \times _{\mathcal {X}} \mathcal {U}\), obtained by first embedding \(\mathrm {Sing}(\mathcal {Z})\) into the total space \(V \times _{\mathcal {X}} \mathcal {Z}\) of the relative tangent complex \(\mathbb {T}_{\mathcal {Z} /\mathcal {X}}\), where V is the total space of the restriction of \(\mathbb {T}_{\mathcal {Z} /\mathcal {X}}\) to \(\mathcal {X}\), and then composing with the defining map \(\mathcal {Z} \rightarrow \mathcal {U}\). This definition of V is equivalent to the definition of V given in Section 2.2.2 by [AG15, 9.1.4].

Proposition A.1

If \(Y \subseteq \mathrm {Sing}(\mathcal {Z})\) is a closed subset of the form \(f^{-1}(\widetilde{Y} \times _{\mathcal {X}}\mathcal {U})\), then there is a canonical equivalence

$$\begin{aligned}&\mathrm {IndCoh}_{Y[t,t^{-1}]}(\mathcal {Z})[t,t^{-1}] \cong \mathrm {IndCoh}(\mathcal {Z})[t,t^{-1}] \\&\quad \otimes _{\mathrm {HC}^\bullet (\mathcal {X} / \mathcal {V})[t,t^{-1}]^{\mathrm {op}}\text {-mod}} \mathrm {HC}^\bullet (\mathcal {X} / \mathcal {V})[t,t^{-1}]^{\mathrm {op}}\text {-mod}_{\widetilde{Y}[t,t^{-1}]} , \end{aligned}$$

where the category \(\mathrm {HC}^\bullet (\mathcal {X} / \mathcal {V})^{\mathrm {op}}\text {-mod}\) acts on \(\mathrm {IndCoh}(Z)\) by first using Koszul duality to identify it with the category \(\mathrm {IndCoh}(\mathcal {G}_{\mathcal {X}/\mathcal {V}})\), then using the monoidal functor

$$\begin{aligned}\mathrm {IndCoh}(\mathcal {G}_{\mathcal {X}/\mathcal {V}}) \otimes _{{{\,\mathrm{QC}\,}}(\mathcal {X})} {{\,\mathrm{QC}\,}}(\mathcal {U}) \rightarrow \mathrm {IndCoh}(\mathcal {G}_{\mathcal {Z}/\mathcal {U}})\end{aligned}$$

to define an action of \(\mathrm {IndCoh}(\mathcal {G}_{\mathcal {X}/\mathcal {V}})\) on \(\mathrm {IndCoh}(\mathcal {Z})\).

Proof

We’ll start by recalling the main ideas exploited by Arinkin and Gaitsgory, then introduce the parameter t. According to [AG15, Section 9.2], in this context we can understand singular support conditions for \(\mathrm {IndCoh}(\mathcal {Z})\) in terms of an action of the monoidal category \(\mathrm {IndCoh}(\mathcal {G}_{\mathcal {X}/\mathcal {V}})\). Indeed, the groupoid \(\mathcal {G}_{\mathcal {Z}/\mathcal {U}}\) acts on \(\mathcal {Z}\), and so the category \(\mathrm {IndCoh}(\mathcal {Z})\) is a module over the monoidal category \(\mathrm {IndCoh}(\mathcal {G}_{\mathcal {Z}/\mathcal {U}})\). There’s a natural monoidal pullback functor

$$\begin{aligned}\mathrm {IndCoh}(\mathcal {G}_{\mathcal {X}/\mathcal {V}}) \otimes _{{{\,\mathrm{QC}\,}}(\mathcal {X})} {{\,\mathrm{QC}\,}}(\mathcal {U}) \rightarrow \mathrm {IndCoh}(\mathcal {G}_{\mathcal {Z}/\mathcal {U}})\end{aligned}$$

which makes \(\mathrm {IndCoh}(\mathcal {Z})\) into a module over \(\mathrm {IndCoh}(\mathcal {G}_{\mathcal {X}/\mathcal {V}}) \otimes _{\mathrm {IndCoh}(\mathcal {X})} \mathrm {IndCoh}(\mathcal {U})\), or indeed over \(\mathrm {IndCoh}(\mathcal {G}_{\mathcal {X}/\mathcal {V}})\). In order to understand singular support in terms of this action, we first identify the category on the left hand side as

$$\begin{aligned} \mathrm {IndCoh}(\mathcal {G}_{\mathcal {X}/\mathcal {V}}) \otimes _{{{\,\mathrm{QC}\,}}(\mathcal {X})} {{\,\mathrm{QC}\,}}(\mathcal {U}) \cong&\, \mathrm {IndCoh}(\mathcal {G}_{\mathcal {X}/\mathcal {V}}) \\&\otimes _{{{\,\mathrm{QC}\,}}(\mathcal {G}_{\mathcal {X}/\mathcal {V}})} \left( {{\,\mathrm{QC}\,}}(\mathcal {G}_{\mathcal {X}/\mathcal {V}}) \otimes _{{{\,\mathrm{QC}\,}}(\mathcal {X})} {{\,\mathrm{QC}\,}}(\mathcal {U}) \right) \\ \cong&\, \mathrm {IndCoh}(\mathcal {G}_{\mathcal {X}/\mathcal {V}}) \otimes _{{{\,\mathrm{QC}\,}}(\mathcal {G}_{\mathcal {X}/\mathcal {V}})} {{\,\mathrm{QC}\,}}(\mathcal {G}_{\mathcal {X}/\mathcal {V}} \times _{\mathcal {X}} \mathcal {U})\\ \cong&\, \mathrm {IndCoh}(\mathcal {G}_{\mathcal {X}/\mathcal {V}} \times _{\mathcal {X}} \mathcal {U}). \end{aligned}$$

By [AG15, Lemma 9.2.6] the category \(\mathrm {IndCoh}_Y(\mathcal {Z})\) is equivalent to the restricted category

$$\begin{aligned}\mathrm {IndCoh}(\mathcal {Z}) \otimes _{\mathrm {IndCoh}(\mathcal {G}_{\mathcal {X}/\mathcal {V}} \times _{\mathcal {X}} \mathcal {U})} \mathrm {IndCoh}_{f(Y)}(\mathcal {G}_{\mathcal {X}/\mathcal {V}} \times _{\mathcal {X}} \mathcal {U}).\end{aligned}$$

We’ll want a slightly modified version of this statement.

Let us bring the auxiliary parameter t into the story. We claim that there is an equivalence

$$\begin{aligned}&\mathrm {IndCoh}_{f^{-1}(Y)[t,t^{-1}]}(\mathcal {Z})[t,t^{-1}] \cong \mathrm {IndCoh}(\mathcal {Z})[t,t^{-1}] \otimes _{\mathrm {IndCoh}(\mathcal {G}_{\mathcal {X}/\mathcal {V}} \times _{\mathcal {X}} \mathcal {U})[t,t^{-1}]}\\&\quad \times \mathrm {IndCoh}_{Y[t,t^{-1}]}(\mathcal {G}_{\mathcal {X}/\mathcal {V}} \times _{\mathcal {X}} \mathcal {U})[t,t^{-1}] \end{aligned}$$

where now \(Y \subseteq \mathrm {Sing}(\mathcal {G}_{\mathcal {X}/\mathcal {V}} \times _{\mathcal {X}} \mathcal {U})\) is a Zariski-closed subset.

This is proven in exactly the same way as [AG15, Lemma 9.2.6]. As they argue, it suffices to prove the claim in the case where \(\mathcal {Z}\) is affine. This argument is not affected by the introduction of the parameter t since the descent takes place independently of this parameter (heuristically we’re performing descent for maps \(Z \times {{\,\mathrm{Spec}\,}}\mathbb {C}[t,t^{-1}] \rightarrow \mathcal {Z} \times {{\,\mathrm{Spec}\,}}\mathbb {C}[t,t^{-1}]\) which are constant in the second factor). The embedding \(f :\mathrm {Sing}(\mathcal {Z}) \rightarrow V \times _{\mathcal {X}} \mathcal {U}\) induces a map \(f^*:{{\,\mathrm{Sym}\,}}(V)[t,t^{-1}] \otimes _{\mathcal {O}(\mathcal {X})} \mathcal {O}(\mathcal {U}) \rightarrow \mathcal {O}(\mathrm {Sing}(\mathcal {Z}))[t,t^{-1}]\). We then note that the triangle

figure b

defined using this map and the action of \(\mathrm {IndCoh}(\mathcal {G}_{\mathcal {X}/\mathcal {V}} \times \mathcal {U})[t,t^{-1}]\) on \(\mathrm {IndCoh}(\mathcal {Z})[t,t^{-1}]\) commutes. This implies the claim that we want since we can equivalently take the restriction to \(f^{-1}(Y)[t,t^{-1}]\) with respect to the \(\mathcal {O}(\mathrm {Sing}(\mathcal {Z}))[t,t^{-1}]\)-action, or the restriction to its preimage under \(f^*\), i.e. the restriction to \(f(f^{-1}(Y))[t,t^{-1}] = Y[t,t^{-1}]\) with respect to the \({{\,\mathrm{Sym}\,}}(V)[t,t^{-1}] \otimes \mathcal {O}(\mathcal {U})\)-action.

In particular, suppose \(Y \subseteq V \times _{\mathcal {X}} \mathcal {Z}\) is a subset of the form \(\widetilde{Y} \times _{\mathcal {X}} \mathcal {Z}\) for some closed subset \(\widetilde{Y} \subseteq V\). Then according to [AG15, Proposition 8.4.14] there is an equivalence

$$\begin{aligned}\mathrm {IndCoh}_Y(\mathcal {G}_{\mathcal {X}/\mathcal {V}} \times _{\mathcal {X}} \mathcal {U}) \cong \mathrm {IndCoh}_{\widetilde{Y}}(\mathcal {G}_{\mathcal {X}/\mathcal {V}}) \otimes _{{{\,\mathrm{QC}\,}}(\mathcal {X})} {{\,\mathrm{QC}\,}}(\mathcal {U}).\end{aligned}$$

We can make the same argument identically including the parameter t in order to say that there is an equivalence

$$\begin{aligned}&\mathrm {IndCoh}_{Y[t,t^{-1}]}(\mathcal {G}_{\mathcal {X}/\mathcal {V}} \times _{\mathcal {X}} \mathcal {U})[t,t^{-1}] \cong \mathrm {IndCoh}_{\widetilde{Y}[t,t^{-1}]}(\mathcal {G}_{\mathcal {X}/\mathcal {V}})[t,t^{-1}] \\&\quad \otimes _{{{\,\mathrm{QC}\,}}(\mathcal {X})[t,t^{-1}]} {{\,\mathrm{QC}\,}}(\mathcal {U})[t,t^{-1}]. \end{aligned}$$

The upshot of all this is that if \(Y \subseteq \mathrm {Sing}(\mathcal {Z})\) is a closed subset of the form \(f^{-1}(\widetilde{Y} \times _{\mathcal {X}}\mathcal {U})\) then we can understand the restricted category \(\mathrm {IndCoh}_{Y[t,t^{-1}]}(\mathcal {Z})[t,t^{-1}]\) as the restriction along \(\widetilde{Y}[t,t^{-1}]\) of \(\mathrm {IndCoh}(\mathcal {Z})[t,t^{-1}]\) with respect to the action of \(\mathrm {IndCoh}(\mathcal {G}_{\mathcal {X}/\mathcal {V}})[t,t^{-1}]\). Even better, by Koszul duality [AG15, 9.1.2] (this is just the observation that the counit of the adjunction 2.25 is an equivalence), we can identify \(\mathrm {IndCoh}(\mathcal {G}_{\mathcal {X}/\mathcal {V}})\) with the category \(\mathrm {HC}^\bullet (\mathcal {X} / \mathcal {V})^{\mathrm {op}}\text {-mod}\) and thus [AG15, Corollary 9.1.7] we are able to identify \(\mathrm {IndCoh}_{Y[t,t^{-1}]}(\mathcal {Z})[t,t^{-1}]\) as the restriction along \(\widetilde{Y}[t,t^{-1}]\) of \(\mathrm {IndCoh}(\mathcal {Z})[t,t^{-1}]\) with respect to the action of \(\mathrm {HC}^\bullet (\mathcal {X} / \mathcal {V})[t,t^{-1}]^{\mathrm {op}}\text {-mod}\). Specifically here we’re using Koszul duality before taking the tensor product with \(\mathbb {C}[t,t^{-1}]\text {-mod}\); it’s still true that this induces an equivalence of categories

$$\begin{aligned}\mathrm {HC}^\bullet (\mathcal {X} / \mathcal {V})[t,t^{-1}]^{\mathrm {op}}\text {-mod}_{Y[t,t^{-1}]} \cong \mathrm {IndCoh}_{Y[t,t^{-1}]}(\mathcal {G}_{\mathcal {X}/\mathcal {V}})[t,t^{-1}]\end{aligned}$$

with specified support conditions. It suffices to check this smoothly locally, so we can assume \(\mathcal {X}\) is affine in which case it follows from commutativity of the triangle (*). \(\square \)

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Elliott, C., Yoo, P. A Physical Origin for Singular Support Conditions in Geometric Langlands Theory. Commun. Math. Phys. 368, 985–1050 (2019). https://doi.org/10.1007/s00220-019-03438-z

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