The Atiyah-Bott formula and connectivity in chiral Koszul duality
aa r X i v : . [ m a t h . AG ] O c t THE ATIYAH-BOTT FORMULA AND CONNECTIVITY IN CHIRAL KOSZUL DUALITY
Q.P. H`ÔA
BSTRACT . We prove a family of results regarding connectivity in the theory of chiral Koszul duality. This providesnew examples of Koszul duality being an equivalence, even when the base category is not pro-nilpotent in the senseof [ FG11 ] . Based on ideas sketched in [ Gai11 ] , we show that these results also offer a simpler alternative to one of thetwo main steps in the proof of the Atiyah-Bott formula given in [ GL14 ] and [ Gai15 ] . C ONTENTS
1. Introduction 21.1. History 21.2. Prerequisites and guides to the literature 21.3. A sketch of Gaitsgory and Lurie’s method 21.4. What does this paper do? 31.5. An outline of our results 41.6. Relation to the Atiyah-Bott formula 51.7. Acknowledgments 62. Preliminaries 62.1. Notation and conventions 62.2. Prestacks 72.3. Sheaves on prestacks 72.4. The Ran space / prestack 112.5. Koszul duality 133. Turning Koszul duality into an equivalence 153.1. The case of Lie- and ComCoAlg-algebras inside Vect 153.2. Higher enveloping algebras 183.3. The case of Lie ⋆ - and ComCoAlg ⋆ -algebras on Ran X ! ( X ) and ComAlg ! ( X ) C ∗ c ( Ran X , − ) and coChev 305.2. Verdier duality 335.3. Chev, coChev, and D Ran X Date : October 4, 2016.2010
Mathematics Subject Classification.
Primary 81R99. Secondary 18G55.
Key words and phrases.
Chiral algebras, chiral homology, factorization algebras, Koszul duality, Ran space. BG and the sheaf B A NTRODUCTION
History.
Let X be a smooth and complete curve, and G a simply-connected semi-simple algebraic groupover an algebraically closed field k . Then we know that C ∗ ( BG , Λ) ≃ Sym V for some finite dimensional vector space V , where Λ is Q ℓ when k = F p ( ℓ = p ), and Λ is any field of characteristic0 when k has characteristic 0.Let Bun G denote the moduli stack of principal G -bundles over X . In the differential geometric setting, i.e. when k = C , the cohomology ring of Bun G was computed by Atiyah and Bott in [ AB83 ] . Theorem 1.1.1 (Atiyah-Bott) . We have the following equivalenceC ∗ ( Bun G , Λ) =
Sym Λ ( C ∗ ( X , V ⊗ ω X )) , where ω X is the dualizing sheaf of X . In the recent work [ GL14 ] , Gaitsgory and Lurie gave a purely algebro-geometric proof of the theorem abovein the framework of étale cohomology (see also [ Gai15 ] for an alternative perspective). In the case where X and G come from objects over k = F q , the isomorphism in Theorem 1.1.1 was proved to be compatible with theFrobenius actions on both sides. The Grothendieck-Lefschetz trace formula for Bun G then gives an expression forthe number of k -points on Bun G and hence, confirms the conjecture of Weil that the Tamagawa number of G is 1.Following ideas suggested in [ Gai11 ] , this paper aims to provide an alternative (and simpler) proof of one ofthe two main steps in the original proofs, as given in [ GL14 ] and [ Gai15 ] . This is possible due to a family of newresults regarding connectivity in the theory of chiral Koszul duality proved in this paper.1.2. Prerequisites and guides to the literature.
For the reader’s convenience, we include a quick review of thenecessary background as well as pointers to the existing literature in §2. The readers who are unfamiliar withthe language used in the introduction are encouraged to take a quick look at §2 before returning to the currentsection.1.3.
A sketch of Gaitsgory and Lurie’s method.
We will now provide a sketch of the method employed by [ GL14 ] and [ Gai15 ] . In both cases, the proofs utilize the theory of factorization algebras. Broadly speaking, there aretwo main steps: non-abelian Poincaré duality and Verdier duality on the Ran space. This corresponds to the case of constant group G × X over X . For simplicity’s sake, we will restrict ourselves to this case in the introduction. HE ATIYAH-BOTT FORMULA AND CONNECTIVITY IN CHIRAL KOSZUL DUALITY 3
Non-abelian Poincaré duality.
For the first step, one constructs a factorizable sheaf A on Ran X from f ! ω Gr Ran X where f is the natural map f : Gr Ran X → Ran X ,and Gr Ran X is the factorizable affine Grassmannian. The crucial observation is that the natural mapGr Ran X → Bun G has homologically contractible fibers, and hence, we get an equivalence(1.3.2) C ∗ c ( Bun G , ω Bun G ) ≃ C ∗ c ( Ran X , A ) .1.3.3. Verdier duality.
The right hand side of (1.3.2) is, however, not directly computable. If one thinks of factoriz-able sheaves on Ran X as E -algebras, then one reason that makes it hard to compute the factorization homologyof A is the fact that it’s not necessarily commutative (i.e. not E ∞ ). A , however, also has a commutative co-algebrastructure, via the diagonal map Gr → Gr × Gr .Thus, its Verdier dual D Ran X A naturally has the structure of a commutative algebra. In fact, it’s proved that D Ran X A is a commutative factorization algebra.1.3.4. Computing the Verdier dual.
One can prove something even better: D Ran X A is isomorphic to the commu-tative factorization algebra B coming from C ∗ ( BG ) . Indeed, a natural map from one to the other is given by acertain pairing between A and B . Since these are factorizable, showing that this map is an equivalence amountsto showing that its restriction to X is also an equivalence. This is now a purely local problem, and hence, forexample, one can reduce it to the case of P to prove it.1.3.5. Conclusion.
Recall that B ≃ C ∗ ( BG ) ≃ Sym V is a free commutative algebra, where V is some explicit chain complex that we can compute. But factorizationhomology with coefficients in a free commutative factorization algebra is easy to compute. Hence, we conclude C ∗ ( Bun G , Q ℓ ) ≃ C ∗ c ( Bun G , ω Bun G ) ∨ ≃ C ∗ c ( Ran X , A ) ∨ ≃ C ∗ c ( Ran X , D Ran X A ) (1.3.6) ≃ C ∗ c ( Ran X , B ) ≃ C ∗ c ( Ran X , Sym V ) ≃ Sym C ∗ c ( X , V ) .1.4. What does this paper do?
The main difference between [ GL14 ] and [ Gai15 ] is in the use of Verdier dualityon the Ran space. The latter greatly simplifies and clarifies the former by formally introducing the concept ofVerdier duality on a general prestack and then applying it to the case of the Ran space.Since the Ran space is a big object, its technical properties in relation to factorization homology and factor-izability are difficult to establish. More precisely, it takes a lot of work to prove the (innocent looking) equiv-alence (1.3.6) and to a somewhat lesser extent, the fact that D Ran X A is factorizable. This results in a rathercomplicated technical heart of [ Gai15 ] .In this paper, we prove a series of new results regarding connectivity in the theory of chiral Koszul duality.These are interesting in their own rights, since they give new examples of Koszul duality being an equivalence,even when the base category is not pro-nilpotent in the sense of [ FG11 ] .Based on the ideas sketched in [ Gai11 ] , the results proved in this paper also further simplify the second step ofthe proof. More precisely, these results could be used to replace all of §8, §9, and part of §12 and §15 of [ Gai15 ] . We are eliding a minor, but technical, point about unital vs. non-unital here. [ Gai15 ] doesn’t reprove non-abelian Poincaré duality. In the terminology of [ Gai15 ] , it’s not finitary. Q.P. H`Ô
An outline of our results.
We will now state the main results proved in this paper.Many results that we prove require connectivity assumptions that are somewhat cumbersome to state. Sincethese are merely technical conditions irrelevant to the discussion of the general method, we will gloss over themin this section.
Remark . Many results in this paper could be proved in a more general setting. We avoid doing so to keepthe presentation simple. We will, however, provide remarks about this throughout the text.1.5.2.
Koszul duality for
Lie and
ComCoAlg . Let ComCoAlg ⋆ ( Ran X ) and Lie ⋆ ( Ran X ) denote the categories ofcommutative co-algebra objects and Lie algebra objects in Shv ( Ran X ) with respect to the ⊗ ⋆ -monoidal structure.The theory of Koszul duality developed in [ FG11 ] gives a pair of adjoint functors (1.5.3) Chev : Lie ⋆ ( Ran X ) ⇄ ComCoAlg ⋆ ( Ran X ) : Prim [ − ] .which restricts to a pair of adjoint functorsChev : Lie ⋆ ( X ) ⇄ coFact ⋆ ( X ) : Prim [ − ] ,where coFact ⋆ ( X ) is the category of commutative factorization co-algebras on X .Even though the pair of adjoint functors above are not mutually inverses of each other in general, they arewhen we impose certain connectivity constraints on both sides. Theorem 1.5.4 (Theorem 3.3.3) . We have the following commutative diagram
Lie ⋆ ( Ran X ) ≤ c L ChevPrim [ − ] ComCoAlg ⋆ ( Ran X ) ≤ c cA Lie ⋆ ( X ) ≤ c L ?(cid:31) O O ChevPrim [ − ] coFact ⋆ ( X ) ≤ c cA ?(cid:31) O O where ≤ c L and ≤ c cA denote some connectivity constraints, and where Chev and
Prim [ − ] are the functors comingfrom Koszul duality. Koszul duality for coLie and
ComAlg . Let ComAlg ⋆ ( Ran X ) and coLie ⋆ ( Ran X ) denote the categories ofcommutative algebra objects and co-Lie algebra objects in Shv ( Ran X ) with respect to the ⊗ ⋆ -monoidal structure.As above, we have the following pair of adjoint functors coPrim [ ] : ComAlg ⋆ ( Ran X ) ⇄ coLie ⋆ ( Ran X ) : coChev .Unlike the case of Lie ⋆ and ComCoAlg ⋆ , for a co-Lie algebra g ∈ coLie ⋆ ( X ) ,coChev ( g ) ∈ ComAlg ⋆ ( Ran X ) doesn’t necessarily live inside Fact ⋆ ( X ) . However, we have the following Theorem 1.5.6 (Theorem 4.1.3) . Restricted to the full subcategory coLie ⋆ ( X ) ≥ , where we are using the perverset-structure on X , the functor coChev factors through Fact ⋆ , i.e. coLie ⋆ ( X ) ≥ & & ▼▼▼▼▼▼▼▼▼▼▼ coChev / / ComAlg ⋆ ( Ran X ) Fact ⋆ ( X ) ) (cid:9) ♥♥♥♥♥♥♥♥♥♥♥♥ Strictly speaking, we are using the category ComCoAlg ind-nilp of ind-nilpotent commutative co-algebras. However, we will see easily that,subject to an appropriate connectivity assumption of sheaves on Ran X , this category coincides with the category ComCoAlg. See also footnote 5.
HE ATIYAH-BOTT FORMULA AND CONNECTIVITY IN CHIRAL KOSZUL DUALITY 5
Interaction between coChev and factorization homology. In [ FG11 ] , it’s proved that the functor of takingfactorization homology: C ∗ c : Shv ( Ran X ) → Vectcommutes with Chev. This is because Chev is computed as a colimit, and moreover, C ∗ c has the following twouseful properties:(i) C ∗ c is symmetric monoidal with respect to the ⊗ ⋆ -monoidal structure on Shv ( Ran X ) and the usualmonoidal structure on Vect.(ii) C ∗ c is continuous.The functor coChev, however, is constructed as a limit, so we need some extra conditions to make it behavenicely with C ∗ c . Theorem 1.5.8 (Theorem 5.1.2) . Let g ∈ coLie ⋆ ( X ) ≥ c cL , where ≥ c cL denotes some co-connectivity constraint. Thenwe have a natural equivalence C ∗ c ( Ran X , coChev g ) ≃ coChev ( C ∗ c ( Ran X , g )) .1.5.9. Chev , coChev and Verdier duality. Unsurprisingly, the functors Chev and coChev mentioned above arelinked via the Verdier duality functor on Ran X . Theorem 1.5.10 (Theorem 5.3.1) . Let g ∈ Lie ⋆ ( X ) ≤− , where we are using the perverse t-structure on X . Then wehave the following natural equivalence D Ran X Chev g ≃ coChev ( D X g ) . Remark . As we shall see, the connectivity constraint Lie ⋆ ( X ) ≤− is less strict than the connectivity con-straint Lie ⋆ ( X ) ≤ c L required by Theorem 1.5.4.As a corollary of Theorem 1.5.6, we know that when g ∈ Lie ⋆ ( X ) ≤ c L , D Ran X Chev g ≃ coChev ( D X g ) is factorizable.1.6. Relation to the Atiyah-Bott formula. A mentioned above lies in the essential image of Chev, i.e. A ≃ Chev ( a ) , for some a ∈ Lie ⋆ ( X ) ≤ c L .This is a direct result of Theorem 1.5.4 and the fact that A satisfies this connectivity constraint on the ComCoAlg ⋆ side.1.6.2. As in [ Gai15 ] , we have a pairing A ⊠ B → δ ! ω Ran X ,which induces a map B → D Ran X A ,compatible with the commutative algebra structures on both sides. Thus, we get a map B → D Ran X Chev ( a ) ≃ coChev ( D X a ) ,which we want to be an equivalence. Since both sides are factorizable, it suffices to show that they are over X ,which is now a local problem, and the same proof as in [ Gai15 ] applies. Q.P. H`Ô
Conclusion.
Let V ∈ Vect such that Sym ( V ⊗ ω X ) ≃ B where Sym is taken inside Shv ( Ran X ) using the ⊗ ⋆ -monoidal structure. Then, we haveSym C ∗ c ( X , V ⊗ ω X ) ≃ C ∗ c ( Ran X , Sym ( V ⊗ ω X )) ≃ C ∗ c ( Ran X , B ) ≃ C ∗ c ( Ran X , coChev Ran X D X a ) ≃ coChev ( C ∗ c ( X , D X a )) ≃ coChev ( C ∗ c ( X , a ) ∨ ) ≃ Chev ( C ∗ c ( X , a )) ∨ ≃ C ∗ c ( Ran X , Chev a ) ∨ ≃ C ∗ c ( Ran X , A ) ∨ ≃ C ∗ c ( Bun G , ω Bun G ) ∨ ≃ C ∗ ( Bun G , Q ℓ ) . Remark . It is interesting to note that many technical results about Verdier duality are proved only for thecase of curves in [ Gai15 ] , while results stated here about Koszul duality are for arbitrary dimension (even thoughin the end, they serve a similar purpose regarding the Atiyah-Bott formula). This is in part because [ Gai15 ] workswith more general sheaves on the Ran space, whereas we mostly concern ourselves with sheaves of special shapes,i.e. they are all of the form Chev g or coChev g .1.7. Acknowledgments.
The author would like to express his gratitude to D. Gaitsgory, without whose tirelessguidance and encouragement in pursuing this problem, this work would not have been possible.The author is grateful to his advisor B.C. Ngô for many years of patient guidance and support.2. P
RELIMINARIES
In this section, we will set up the language and conventions used throughout the paper. Since the materialcovered here are used in various places, the readers should feel free to skip it and backtrack when necessary.The mathematical content in this section has already been treated elsewhere. Hence, results are stated withoutany proof, and we will do our best to provide the necessary references. It is important to note that it is not ouraim to be exhaustive. Rather, we try to familiarize the readers with the various concepts and results used in thetext, as well as to give pointers to the necessary references for the background materials.2.1.
Notation and conventions.
Category theory.
We will use DGCat to denote the ( ∞ , 1 ) -category of stable infinity categories, DGCat pres to denote the full subcategory of DGCat consisting of presentable categories, and DGCat pres,cont the (non-full)subcategory of DGCat pres where we restrict to continuous functors, i.e. those commuting with colimits. Spc willbe used to denote the category of spaces, or more precisely, ∞ -groupoids.The main references for this subject are [ Lur15 ] and [ Lur14 ] . For a slightly different point of view, seealso [ GR ] .2.1.2. Algebraic geometry.
Throughout this paper, k will be an algebraically closed ground field. We will denoteby Sch the ∞ -category obtained from the ordinary category of separated schemes of finite type over k . All ourschemes will be objects of Sch. In most cases, we will use the calligraphic font to denote prestacks, for eg. X , Y etc., and the usual font to denote schemes, for eg. X , Y etc.2.1.3. t-structures. Let C be a stable infinity category, equipped with a t -structure. Then we have the followingdiagram of adjoint functors C ≤ i ≤ / / C tr ≤ o o tr ≥ / / C ≥ i ≥ o o HE ATIYAH-BOTT FORMULA AND CONNECTIVITY IN CHIRAL KOSZUL DUALITY 7
We use τ ≤ and τ ≥ to denote τ ≤ = i ≤ ◦ tr ≤ : C → C and τ ≥ = i ≥ ◦ tr ≥ : C → C respectively.Shifts of these functors, for e.g. τ ≥ n and τ ≤ n , are defined in the obvious ways.2.2. Prestacks.
The theory of sheaves on prestacks has been developed in [ GL14 ] and [ Gai15 ] . In this subsectionand the next, we will give a brief review of this theory, including the definition of the category of sheaves as well asvarious pull and push functors. We will state them as facts, without any proof, which (unless otherwise specified),could all be found in [ Gai15 ] .2.2.1. A prestack is a contravariant functor from Sch to Spc, i.e. a prestack Y is a functor Y : Sch op → Spc.Let PreStk be the ∞ -category of prestacks. Then by Yoneda’s lemma, we have a fully-faithful embeddingSch , → PreStk.2.2.2.
Properties of prestacks.
Due to categorical reasons, any prestack Y can be written as a colimit of schemes Y ≃ colim i ∈ I Y i .2.2.3. A prestack is said to be is a pseudo-scheme if it could be written as a colimit of schemes, where allmorphisms are proper.2.2.4. A prestack is pseudo-proper if it could be written as a colimit of proper schemes. It is straightforward tosee that pseudo-proper prestacks are pseudo-schemes.2.2.5. A prestack is said to be finitary if it could be expressed as a finite colimit of schemes.2.2.6. We also have relative versions of the definitions above in an obvious manner. Namely, we can speak ofa morphism f : Y → S , where Y is a prestack and S is a scheme, being pseudo-schematic (resp. pseudo-proper,finitary).2.2.7. More generally, a morphism f : Y → Y is said to be pseudo-schematic (resp. pseudo-proper, finitary) if for any scheme S , equipped with a morphism S → Y , the morphism f S in the following pull-back diagram S × Y Y f S (cid:15) (cid:15) / / Y (cid:15) (cid:15) S / / Y is pseudo-schematic (resp. pseudo-proper, finitary).2.3. Sheaves on prestacks.
As we mentioned above, proofs of all the results in mentioned in this section, unlessotherwise specified, could be found in [ Gai15 ] . Q.P. H`Ô
Sheaves on schemes.
We will adopt the same conventions as in [ Gai15 ] , except that for simplicity, we willrestrict ourselves to the “constructible setting.” Namely, for a scheme S ,(i) when the ground field is C , and Λ is an arbitrary field of characteristic 0, we take Shv ( S ) to be the ind-completion of the category of constructible sheaves on S with Λ -coefficients.(ii) for any ground field k in general, and Λ = Q ℓ , Q ℓ with ℓ = char k , we take Shv ( S ) to be the ind-completionof the category of constructible ℓ -adic sheaves on S with Λ -coefficients. See also [ GL14, §4 ] , [ LZ12 ] ,and [ LZ14 ] .The theory of sheaves on schemes is equipped with the various pairs of adjoint functors f ! ⊣ f ! and f ∗ ⊣ f ∗ for any morphism f : S → S between schemes. Moreover, we also have box-product ⊠ and hence, also ⊗ and ! ⊗ .2.3.2. Throughout the text, we will use the perverse t -structure on Shv ( S ) , when S is a scheme.2.3.3. We will also use Vect to denote the category of sheaves on a point, i.e. Vect denotes the (infinity derived)category of chain complexes in vector spaces over Λ .2.3.4. Sheaves on prestacks.
For a prestack Y , the category Shv ( Y ) is defined byShv ( Y ) = lim S ∈ ( Sch op / Y ) Shv ( S ) ,where the transition functor we use is the shriek-pullback.Thus, an object F ∈ Shv ( Y ) is the same as the following data(i) A sheaf F S , y ∈ Shv ( S ) for each S ∈ Sch and y : S → Y (i.e. y ∈ Y ( S ) ).(ii) An equivalence of sheaves F S ′ , f ( y ) → f ! F S , y for each morphism of schemes f : S ′ → S .Moreover, we require that this assignment satisfies a homotopy-coherent system of compatibilities.2.3.5. More formally, one can define Shv ( Y ) as the right Kan extension ofShv : Sch op → DGCat pres,cont along the Yoneda embedding Sch op , → PreStk op .Thus, by formal reasons, the functor Shv : PreStk op → DGCat pres,cont preserves limits. In other words, we have Shv ( colim i Y i ) ≃ lim i Shv ( Y i ) .In particular, if a prestack Y ≃ colim i ∈ I Y i is a colimit of schemes, then Shv ( Y ) ≃ lim i ∈ I Shv ( Y i ) .2.3.6. Now, if we replace all the transition functors by their left adjoints, namely the !-pushforward, then wehave a diagram I op → DGCat pres,cont ,and we have a natural equivalence Shv ( Y ) ≃ colim i ∈ I op Shv ( Y i ) where the colimit is taken inside DGCat pres,cont . HE ATIYAH-BOTT FORMULA AND CONNECTIVITY IN CHIRAL KOSZUL DUALITY 9 Y = colim i Y i be a prestack, and denote ins i : Y i → Y the canonical map. Then, for any sheaf F ∈ Shv ( Y ) , we have the following natural equivalence(2.3.8) F ≃ colim i ins i ! ins ! i F f ! ⊣ f ! . Let f : Y → Y be a morphism between prestacks. Then by restriction, we get a functor f ! : Shv ( Y ) → Shv ( Y ) ,which commutes with both limits and colimits. In particular, f ! admits a left adjoint f ! . The functor f ! is generally not computable. However, there are a couple of cases where it is.2.3.10. The first instance is when the target of f is a scheme f : Y → S ,and suppose that Y ≃ colim i Y i .Then, by (2.3.8), we have f ! F ≃ colim f ! ins i ! ins ! i F ≃ colim f i ! ins ! i F .where f i : Y i → Y → S is just a morphism between schemes.2.3.11. The second case is where f is pseudo-proper, then f ! satisfies the base change theorem with respect tothe ( − ) ! -pullback. Namely, for any pull-back diagram of prestacks Y ′ f (cid:15) (cid:15) g / / Y f (cid:15) (cid:15) Y ′ g / / Y and any sheaf F ∈ Shv ( Y ) , we have a natural equivalence g ! f ! F ≃ f ! g ! F .Thus, in particular, if we have a pull-back diagram S × Y Y f S (cid:15) (cid:15) i S / / Y f (cid:15) (cid:15) S i S / / Y where S is a scheme, then i ! S f ! F ≃ f S ! i ! S F and as discussed above, f S ! could be computed as an explicit colimit. It also admits a right adjoint. However, we do not make use of it in this paper. F ∈ Shv ( Y ) . Then we denote by C ∗ c ( Y , F ) = s ! F ,where s : Y → Spec k is the structural map of Y to a point.2.3.13. In case where F ≃ ω Y is the dualizing sheaf on Y (characterized by the property that its ( − ) ! -pullbackto any scheme is the dualizing sheaf on that scheme), then we write C ∗ ( Y ) = C ∗ c ( Y , ω Y ) ,and C red ∗ ( Y ) = Fib ( C ∗ ( Y ) → Λ) .2.3.14. f ∗ ⊣ f ∗ . When f : Y → Y is a schematic morphism between prestacks, one can also define a pair of adjoint functors (see [ Gai15 ] wherethe functor f ∗ is defined, and [ Ho15 ] where the adjunction is constructed) f ∗ : Shv ( Y ) ⇄ Shv ( Y ) : f ∗ .2.3.15. The behavior of f ∗ is easy to describe, due to the fact that f ∗ satisfies the base change theorem withrespect to the ( − ) ! -pullback functor. Namely, suppose F ∈ Shv ( Y ) and we have a pullback square where S (andhence, S ) is a scheme S g / / f S (cid:15) (cid:15) Y f (cid:15) (cid:15) S g / / Y Then, the pullback could be described in classical terms, since g ! f ∗ F ≃ f S ∗ g ! F ,where f S is just a morphism between schemes.2.3.16. The functor f ∗ is slightly more complicated to describe. However, when f : Y → Y is étale, which is the case where we need, we have a natural equivalence (see [ Ho15, Prop. 2.7.3 ] )(2.3.17) f ! ≃ f ∗ .2.3.18. We will also need the following fact in the definition of commutative factorizable co-algebras: let U f / / Z g / / X be morphisms between prestacks, where g is finitary pseudo-proper, f and h = g ◦ f are schematic. Then we havea natural equivalence (see [ Ho15, Prop. 2.10.4 ] )(2.3.19) g ! ◦ f ∗ ≃ ( g ◦ f ) ∗ ≃ h ∗ . HE ATIYAH-BOTT FORMULA AND CONNECTIVITY IN CHIRAL KOSZUL DUALITY 11
Monoidal structure.
The theory of sheaves on prestacks discussed so far naturally inherits the box-tensorstructure from the theory of sheaves on schemes. Namely, let F i ∈ Shv ( Y i ) where Y i ’s are prestacks, for i =
1, 2.Then, for any pair of schemes S , S equipped with maps f i : S i → Y i ,we have ( f × f ) ! ( F ⊠ F ) ≃ f !1 F ⊠ f !2 F .Pulling back along the diagonal δ : Y → Y × Y for any prestack Y , we get the ! ⊗ -symmetric monoidal structure on Y in the usual way. More explicitly, for F , F ∈ Shv ( Y ) , we define F ⊗ F = δ ! ( F ⊠ F ) .2.4. The
Ran space / prestack. The Ran space (or more precisely, prestack) of a scheme plays a central role inthis paper. The Ran space, along with various objects on it, was first studied in the seminal book [ BD04 ] in thecase of curves, and was generalized to higher dimensions in [ FG11 ] . In what follows, we will quickly review themain definitions and results. For proofs, unless otherwise specified, we refer the reader to [ Gai15 ] and [ FG11 ] .The topologically inclined reader could also find an intuitive introduction in [ Ho15, §1 ] .2.4.1. For a scheme X ∈ Sch, we will use Ran X to denote the following prestack: for each scheme S ∈ Sch, ( Ran X )( S ) = { non-empty finite subsets of X ( S ) } Alternatively, one has Ran X ≃ colim I ∈ fSet surj,op X I where fSet surj denotes the category of non-empty finite sets, where morphisms are surjections.Using the fact that X is separated, one sees easily that Ran X is a pseudo-scheme. Moreover, when X is proper,Ran X is pseudo-proper.2.4.2. The ⊗ ⋆ monoidal structure. There is a special monoidal structure on Ran X which we will use throughoutthe text: the ⊗ ⋆ -monoidal structure.Consider the following map union : Ran X × Ran X → Ran X given by the union of non-empty finite subsets of X . One can check that union is finitary pseudo-proper. Giventwo sheaves F , G ∈ Shv ( Ran X ) , we define F ⊗ ⋆ G = union ! ( F ⊠ G ) .This defines the ⊗ ⋆ -monoidal structure on Shv ( Ran X ) .2.4.3. Since union is pseudo-proper, F ⊗ ⋆ G has an easy presentation. Namely, for F , F , . . . , F k ∈ Shv ( Ran X ) ,and any non-empty finite set I , we have the following(2.4.4) ( F ⊗ ⋆ F ⊗ ⋆ · · · ⊗ ⋆ F k ) | ◦ X I ≃ M I = S ki = I i ∆ ! ⊔ ki = I i ։ ∪ ki = I i ( F ⊠ · · · ⊠ F k ) | ( Q ◦ X Ii ) disj where ( Q ◦ X I i ) disj denotes the open subscheme of Q i X I i where no two “coordinates” are equal, and where ∆ ⊔ ki = I i ։ ∪ ki = I i : X I , → Y i X I i is the map induced by the surjection k G i = I i ։ k [ i = I i ≃ I . Factorizable sheaves.
Using the ⊗ ⋆ -monoidal structure on Shv ( Ran X ) , one can talk about various types ofalgebras / coalgebras in Shv ( Ran X ) . The ones that are of importance to us in this papers areComAlg ⋆ ( Ran X ) , Lie ⋆ ( Ran X ) , ComCoAlg ⋆ ( Ran X ) , and coLie ⋆ ( Ran X ) .As the name suggests, these are used, respectively, to denote the categories of commutative algebras, Lie algebras,commutative co-algebras and co-Lie algebras in Shv ( Ran X ) with respect to the ⊗ ⋆ -monoidal structure definedabove.2.4.6. We use Lie ⋆ ( X ) and coLie ⋆ ( X ) to denote the full subcategories of Lie ⋆ ( Ran X ) and coLie ⋆ ( Ran X ) respec-tively, consisting of objects whose supports are inside the diagonalins X : X , → Ran X of Ran X .2.4.7. Let j : ( Ran X ) n disj → ( Ran X ) n where ( Ran X ) n disj is the open sub-prestack of ( Ran X ) n defined by the following condition: for each scheme S , ( Ran X ) n ( S ) consists of n non-empty subsets of X ( S ) , whose graphs are pair-wise disjoint.2.4.8. Let A ∈ ComCoAlg ⋆ ( Ran X ) .Then, by definition, we have the following map (which is the co-multiplication of the commutative co-algebrastructure) A → A ⊗ ⋆ A ⊗ ⋆ · · · ⊗ ⋆ A ≃ union ! ( A ⊠ · · · ⊠ A ) .Using the the unit map of the adjunction j ∗ ⊣ j ∗ , we get the following mapunion ! ( A ⊠ · · · ⊠ A ) → union ! j ∗ j ∗ ( A ⊠ · · · ⊠ A ) ≃ ( union ◦ j ) ∗ j ! ( A ⊠ · · · ⊠ A ) ,where for the equivalence, we made use of (2.3.17) and (2.3.19).Altogether, we get a map A → ( union ◦ j ) ∗ j ! ( A ⊠ · · · ⊠ A ) and hence, by adjunction and (2.3.17), we get a map(2.4.9) j ! union ! A → j ! ( A ⊠ · · · ⊠ A ) . Definition 2.4.10. A is a commutative factorization algebra if the map (2.4.9) is an equivalence for all n ’s.We use coFact ⋆ ( X ) to denote the full subcategory of ComAlg ⋆ ( Ran X ) consisting of commutative factorizationco-algebras.2.4.11. Let B ∈ ComAlg ⋆ ( Ran X ) .Then, by definition, we have the following map (which is the multiplication of the commutative algebra structure)union ! ( B ⊠ B ⊠ · · · ⊠ B ) ≃ B ⊗ ⋆ B ⊗ ⋆ · · · ⊗ ⋆ B → B .This induces the following map of sheaves B ⊠ · · · ⊠ B → union ! B on ( Ran X ) n , and hence, a map of sheaves(2.4.12) j ! ( B ⊠ · · · ⊠ B ) → j ! union ! B .on ( Ran X ) n disj . Definition 2.4.13. B is a commutative factorization algebra if the map (2.4.12) is an equivalence for all n ’s. Weuse Fact ⋆ ( X ) to denote the full subcategory of ComAlg ⋆ ( Ran X ) consisting of commutative factorization algebras. HE ATIYAH-BOTT FORMULA AND CONNECTIVITY IN CHIRAL KOSZUL DUALITY 13
Koszul duality.
In this subsection, we will quickly review various concepts and results in the theory of Koszulduality that are relevant to us. This theory, initially developed in [ Qui69 ] , illuminates the duality between com-mutative co-algebras and Lie algebras. It was further developed and generalized in the operadic setting in [ GK94 ] .In the chiral / factorizable setting, the paper [ FG11 ] provides us with necessary technical tools and language tocarry out many topological arguments in the context of algebraic geometry. The results and definitions we reviewbelow could be found in [ FG11 ] and [ GR ] .2.5.1. Symmetric sequences.
Let Vect Σ denote the category of symmetric sequences. Namely, its objects are collec-tions O = { O ( n ) , n ≥ } ,where each O ( n ) is an object of Vect, acted on by the symmetric group Σ n .The infinity category Vect Σ is equipped with a natural monoidal structure which makes the functorVect Σ → Fun ( Vect, Vect ) given by the following formula O ⋆ V = M n ( O ( n ) ⊗ V ⊗ n ) Σ n symmetric monoidal.2.5.2. Operads and co-operads.
By an operad (resp. co-operad), we will mean an augmented associative algebra(resp. co-algebra) object in Vect Σ , with respect to the monoidal structure described above. We use Op (resp. coOp)to denote the categories of operads (resp. co-operads).In general, the Bar and coBar construction gives us the following pair of adjoint functorsBar : Op ⇄ coOp : coBar .For an operad O (resp. co-operad P ), we also use O ∨ (resp. P ∨ ) to denote Bar ( O ) (resp. coBar ( P ) ). Remark . In what follows, we will adopt the following convention: all our operads / co-operads will havethe property that the augmentation map is an equivalence, when restricted to O ( ) (resp. P ( ) ). And under thisrestriction, one can show that the following unit map is an equivalence O → coBar ◦ Bar ( O ) or in a slightly different notation O → ( O ∨ ) ∨ when O ∈ Op satisfying the assumption above.2.5.4.
Algebras and co-algebras.
Let C be a stable presentable symmetric monoidal ∞ -category compatibly ten-sored over Vect. Then, an operad O (resp. co-operad P ) naturally defines a monad (resp. comonad) on C .Thus, for an operad O (resp. co-operad P ), one can talk about the category of algebras O -alg ( C ) (resp. co-algebras P -coalg ( C ) ) in C with respect to the operad O (resp. co-operad P ).As usual (as for any augmented monad), one has the following pairs of adjoint functorsFree O : C ⇄ O -alg ( C ) : oblv O and Bar O : O -coalg ( C ) ⇄ C : triv O for an operad O , and similarly, the following pairs of adjoint functorsoblv P : P -coalg ( C ) ⇄ C : coFree P and cotriv P : C ⇄ P -coalg ( C ) : coBar P for a co-operad P .2.5.5. Koszul duality.
The functors mentioned above could be lifted to get a pair of adjoint functors(2.5.6) Bar enh : O -alg ( C ) ⇄ P -coalg ( C ) : coBar enh where oblv P ◦ Bar enh O ≃ Bar O and oblv O ◦ coBar enh P ≃ coBar P . Turning Koszul duality into an equivalence.
In general, the pair of adjoint functors at (2.5.6) is not anequivalence. One of the main achievements of [ FG11 ] is to formulate a precise condition on the base category C ,namely the pro-nilpotent condition, which turns (2.5.6) into an equivalence.One of the main technical points of our paper is to prove another case where Koszul duality is still an equiva-lence, even when the categories involved are not pro-nilpotent.The two main instances of Koszul duality that are important in this paper are the duality between Lie-algebrasand ComCoAlg-algebras, and coLie-algebras and ComAlg-algebras.2.5.8. The case of
Lie and
ComCoAlg . We have the following equivalence of co-operads (see [ FG11 ] ):Lie ∨ ≃ ComCoAlg [ ] ,where ComCoAlg [ ]( n ) ≃ k [ n − ] is equipped with the sign action of the symmetric group Σ n .2.5.9. Equivalently, the functor [ ] : C → C gives rise to an equivalence of categories [ ] : ComCoAlg [ ]( C ) ≃ ComCoAlg ( C ) .2.5.10. This gives us the following diagramLie ( C ) Chev & & ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ [ ] (cid:15) (cid:15) Bar
Lie / / ComCoAlg [ ]( C ) coBar ComCoAlg [ ] o o [ ] (cid:15) (cid:15) Lie [ − ]( C ) [ − ] O O Bar
Lie [ − ] / / ComCoAlg ( C ) coBar ComCoAlg o o [ − ] O O Prim [ − ] f f ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ We usually use Chev to denote [ ] ◦ Bar
Lie ≃ ◦
Bar
Lie [ − ] ◦ [ ] and Prim [ − ] to denote(2.5.11) coBar ComCoAlg [ ] ◦ [ − ] ≃ [ − ] ◦ coBar ComCoAlg .2.5.12.
The case of coLie and
ComAlg . Dually, we have the following equivalence of co-operadsComAlg ∨ ≃ coLie [ ] ,and similar to the above, the functor [ ] : C → C gives rise to an equivalence of categories [ ] : coLie [ ]( C ) ≃ coLie ( C ) . The interested reader could read more about this in [ FG11 ] , since we do not need this fact in the current work. HE ATIYAH-BOTT FORMULA AND CONNECTIVITY IN CHIRAL KOSZUL DUALITY 15 ( C ) coPrim [ ] & & ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ [ ] (cid:15) (cid:15) Bar
ComAlg / / coLie [ ]( C ) coBar coLie [ ] o o [ ] (cid:15) (cid:15) ComAlg [ − ]( C ) [ − ] O O Bar
ComAlg [ − ] / / coLie ( C ) coBar coLie o o [ − ] O O coChev f f ▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲▲ As above, we usually use coChev to denote [ − ] ◦ coBar coLie ≃ coBar coLie [ ] ◦ [ − ] and coPrim [ ] to denote [ ] ◦ Bar
ComAlg ≃ Bar
ComAlg [ − ] ◦ [ ] .3. T URNING K OSZUL DUALITY INTO AN EQUIVALENCE
The goal of this section is to prove Theorem 1.5.4. We will start by examining the special case where X isjust a point, i.e. Shv ( Ran X ) ≃ Shv ( X ) ≃ Vect, and prove that Koszul duality provides a natural equivalence ofcategories Chev : Lie ( Vect ≤− ) ≃ ComCoAlg ( Vect ≤− ) : Prim [ − ] .Even though this case is not strictly needed in the proof of the general case, it is interesting in its own right,as it allows us to predict the correct connectivity condition needed in the general case, whose precise statementand proof are presented in the final subsection. We recommend the reader to first read the case of Vect, since itshares the same strategy as the main proof without the additional numerical complexity.3.1. The case of
Lie - and
ComCoAlg -algebras inside
Vect . We will now prove the following
Theorem 3.1.1.
Chev and
Prim [ − ] give rise to a pair of mutually inverse functors Chev : Lie ( Vect ≤− ) ⇄ ComCoAlg ( Vect ≤− ) : Prim [ − ] Remark . Since Chev is defined as a colimit, it is easy to see that Chev | Lie ( Vect ≤− ) lands in the correct subcat-egory cut out by the connectivity assumption Vect ≤− . However, a priori, the same is not obvious for Prim [ − ] ,being defined as a limit. It is, however, clear from the proof below that this in fact holds. Remark . Unless otherwise specified, our functors will be automatically restricted to the subcategories withthe appropriate connectivity conditions. For example, we will write Chev instead of Chev | Lie ( Vect ≤− ) in most cases. Remark . Note that Theorem 3.1.1 can be proved more generally for a presentable symmetric monoidalstable infinity category with a t -structure satisfying some mild properties. The pair of operad and co-operad Lieand ComCoAlg could also be made more general. The curious readers could take a look at the remarks at the endof this subsection.3.1.5. We follow a similar strategy as in [ FG11 ] . Namely, to prove that Chev and Prim [ − ] are mutually inversefunctors, it suffices to show that the left adjoint functor, Chev, is fully-faithful, and the right adjoint functor,Prim [ − ] is conservative.We start with the following Lemma 3.1.6.
The functor
Prim [ − ] | ComCoAlg ( Vect ≤− ) satisfies the following conditions(i) Prim [ − ] commutes with sifted colimits.(ii) The natural map Free
Lie → Prim [ − ] ◦ triv ComCoAlg is an equivalence.
As in [ FG11, §4.1.8 ] , this immediately implies the following corollary. For the sake of completeness, we includethe proof here. Corollary 3.1.7.
Chev | Lie ( Vect ≤− ) is fully faithful.Proof. It suffices to show that the unit map id → Prim [ − ] ◦ Chevis an equivalence. Since Prim [ − ] commutes with sifted colimits by part (i) of Lemma 3.1.6, it suffices to showthat the following is an equivalence Free Lie → Prim [ − ] ◦ Chev ◦ Free
Lie ,since any Lie-algebra could be written as a sifted colimit of the free ones. However, we know that (even withoutthe connectivity condition) Chev ◦ Free
Lie ≃ triv Lie and hence, it suffices to show that Free
Lie → Prim [ − ] ◦ Chev .But now, we are done due to part (ii) of Lemma 3.1.6.3.1.8. Before proving Lemma 3.1.6, we start with a couple of preliminary observations. In essence, the lemma isa statement about commuting limits and colimits. In a stable infinity category, if, for instance, the limit is a finiteone, then one can always do that. In our situation, coBar is causing troubles because it is defined as an infinitelimit.The main idea of the proof is that when c ∈ ComCoAlg ( Vect ≤− ) ,then even though coBar ComCoAlg ( c ) is computed as an infinite limit, each of its cohomological degree will be controlled by finitely many of terms inthe limit.3.1.9. For brevity’s sake, we will use P to denote the co-operad ComCoAlg. Recall that in general, for any c ∈ ComCoAlg ( Vect ) ,we have coBar P ( c ) = Tot ( coBar • P ( c )) where coBar • P ( c ) is a co-simplicial object.Let coBar n P ( c ) = Tot ( coBar • P ( c ) | ∆ ≤ n ) be the limit over the restriction of the co-simplicial object to ∆ ≤ n . Then we have the following tower c ≃ coBar P ( c ) ← coBar P ( c ) ← · · · ← coBar n P ( c ) ← · · · and coBar P ( c ) ≃ lim n coBar n P ( c ) . Lemma 3.1.10.
Let c ∈ ComCoAlg ( Vect ≤− ) . Then, for all n ≥ , the following natural map tr ≥− n + + n + coBar n P ( c ) → tr ≥− n + + n + coBar n − P ( c ) . is an equivalence. This fact applies to the category of algebras over any operad in general.
HE ATIYAH-BOTT FORMULA AND CONNECTIVITY IN CHIRAL KOSZUL DUALITY 17
Proof.
Let F n ( c ) denote the difference between coBar n P ( c ) and coBar n − P ( c ) , F n ( c ) = Fib ( coBar n P ( c ) → coBar n − P ( c )) .Then for c ∈ ComCoAlg ( Vect ≤− ) ,we see that F n ( c ) ∈ Vect ≤− · n + n ≃ Vect ≤− n + + n .Indeed, this is because of the fact that c ∈ Vect ≤− and in the direct sumcoBar • P ( c )([ n ]) = M m ≥ P ⋆ n ( m ) ⊗ S m c ⊗ m , m = n is the first summand where we have non-degenerate “(co-)cells.”As a consequence, tr ≥− n + + n + coBar n P ( c ) → tr ≥− n + + n + coBar n − P ( c ) is an equivalence and we are done.Using the fact that infinite products preserve Vect ≤ , the lemma above directly implies the following Corollary 3.1.11.
Let c ∈ ComCoAlg ( Vect ≤− ) . Then, for any n, the following natural map tr ≥− n coBar P ( c ) → tr ≥− n coBar m P ( c ) is an equivalence for all m ≫ , where the bound depends only on n.Proof of Lemma 3.1.6. The proof is now simple. In fact, we will only prove part (i), as the other one is almost iden-tical. Note that due to (2.5.11), what we prove about coBar P implies the corresponding statement of Prim [ − ] ,up to a shift.It suffices to show that for all n , we havetr ≥− n coBar P ( colim α c α ) ≃ tr ≥− n colim α coBar P ( c α ) where α runs over some sifted diagram. But now, from Corollary 3.1.11, for all m ≫
0, we havetr ≥− n coBar P ( colim α c α ) ≃ tr ≥− n coBar m P ( colim α c α ) ≃ tr ≥− n colim α coBar m P ( c α ) ≃ colim α tr ≥− n coBar m P ( c α ) ≃ colim α tr ≥− n coBar P ( c α ) ≃ tr ≥− n colim α coBar P ( c α ) . Remark . The cohomological estimate done above implies thatcoBar
ComCoAlg ( c ) ∈ Lie [ − ]( Vect ≤− ) ,or equivalently Prim [ − ]( c ) ∈ Lie ( Vect ≤− ) ,when c ∈ ComCoAlg ( Vect ≤− ) .Indeed, from Corollary 3.1.11, we know that for some m ≫ ≥− coBar P ( c ) ≃ tr ≥− coBar m P ( c ) ,and moreover, a downward induction using Lemma 3.1.10 shows thattr ≥− coBar m P ( c ) ≃ tr ≥− coBar P ( c ) ≃ tr ≥− c ≃ Lemma 3.1.14.
The functor
Prim [ − ] : ComCoAlg ( Vect ≤− ) → Lie ( Vect ≤− ) is conservative.Proof. It suffices to show that coBar P : ComCoAlg ( Vect ≤− ) → Lie [ − ]( Vect ≤− ) is conservative, and we will prove that by contradiction. Namely, let f : c → c be a morphism in ComCoAlg ( Vect ≤− ) such that f is not an equivalence. Suppose thatcoBar P ( f ) : coBar P ( c ) → coBar P ( c ) is an equivalence, we will derive a contradiction.Let k be the smallest number such thattr ≥− k ( f ) : tr ≥− k c → tr ≥− k c is not an equivalence. Now, by Corollary 3.1.11, we know that there is some m ≫ ≥− k coBar P ( c i ) ≃ tr ≥− k coBar m P ( c i ) for i ∈ {
1, 2 } . Thus, we know that tr ≥− k coBar m P ( c ) → tr ≥− k coBar m P ( c ) is an equivalence.By an estimate similar to that of Lemma 3.1.10, we see thattr ≥− k F n ( c ) ≃ tr ≥− k F n ( c ) for all n ≥
1. Indeed, the difference between F n ( c ) and F n ( c ) lies in cohomological degrees ≤ − ( n − ) − k + n = − n + − k + n + < − k , ∀ n ≥ n = m , using the diagram F n ( c ) (cid:15) (cid:15) / / coBar n P ( c ) (cid:15) (cid:15) / / coBar n − P ( c ) (cid:15) (cid:15) F n ( c ) / / coBar n P ( c ) / / coBar n − P ( c ) implies that τ ≥− k c ≃ τ ≥− k c ,which contradicts our original assumption. Hence, we are done. Remark . Note that the proof we gave above could be carried out in a more general setting. Namely, theonly properties of Vect that we used are(i) The symmetric monoidal structure is right exact (namely, it preserved Vect ≤ ).(ii) The t -structure on Vect is left separated.(iii) Infinite products preserve Vect ≤ . Remark . We can also replace the operad Lie by any operad O such that(i) O is classical, i.e. it lies in the heart of the t -structure of Vect.(ii) O ∨ [ − ] is also classical.(iii) O ( ) ≃ Λ (as we already assume throughout this paper).3.2. Higher enveloping algebras.
This subsection serves as the topological analogue of the results proved inthe next one. The main reference of this part is [ GR ] . HE ATIYAH-BOTT FORMULA AND CONNECTIVITY IN CHIRAL KOSZUL DUALITY 19 g ∈ Lie ( Vect ) .Then one can form its E n -universal enveloping algebra U E n ( g ) ∈ E n ( Vect ) by applying the following sequence of functorsLie ( Vect ) Ω × n ≃ [ − n ] / / E n ( Lie ( Vect )) E n ( Chev ) / / E n ( ComCoAlg ( Vect )) oblv ComCoAlg / / E n ( Vect ) where E n ( Lie ( Vect )) and E n ( ComCoAlg ( Vect )) are categories of E n -algebras with respect to the Cartesian monoidalstructure on Lie ( Vect ) and ComCoAlg ( Vect ) respectively (note that the latter on is just the given by ⊗ in Vect).3.2.2. It is proved in [ GR ] that [ − n ] induces an equivalence [ − n ] : Lie ( Vect ) ≃ E n ( Lie ( Vect )) : [ n ] .Moreover, we know from Theorem 3.1.1 that E n ( Chev ) : E n ( Lie ( Vect ≤− )) → E n ( ComCoAlg ( Vect ≤− )) .Thus, we get the following equivalence of categories(3.2.3) Lie ( Vect ≤− n − ) ≃ E n ( ComCoAlg ( Vect ≤− )) .3.2.4. The equivalence (3.2.3) is precisely what we are looking for in the context of factorization algebras onthe Ran space in the following subsection. One part of the work is to find connectivity assumptions on Shv ( Ran X ) which mirror those in Vect ≤− n − and Vect ≤− respectively.3.3. The case of
Lie ⋆ - and ComCoAlg ⋆ -algebras on Ran X . We come to the precise formulation and the proof ofTheorem 1.5.4.
Definition 3.3.1.
Let Shv ( Ran X ) ≤ c cA and Shv ( Ran X ) ≤ c L denote the full subcategory of Shv ( Ran X ) consisting ofsheaves F such that for all non-empty finite sets I , F | ◦ X I ∈ Shv ( ◦ X I ) ≤ ( − − d ) | I |− ,and respectively, F | ◦ X I ∈ Shv ( ◦ X I ) ≤ ( − − d ) | I | .Here, we use the perverse t -structure, and X is a scheme of pure dimension d . Notation 3.3.2.
We will use Lie ⋆ ( Ran X ) ≤ c L and ComCoAlg ⋆ ( Ran X ) ≤ c cA to denote Lie ⋆ ( Shv ( Ran X ) ≤ c L ) and ComCoAlg ⋆ ( Ran X ) ≤ c cA respectively.With these connectivity assumptions in mind, we will prove the following Theorem 3.3.3.
We have the following commutative diagram (3.3.4) Lie ⋆ ( Ran X ) ≤ c L ChevPrim [ − ] ComCoAlg ⋆ ( Ran X ) ≤ c cA Lie ⋆ ( X ) ≤ c L ?(cid:31) O O ChevPrim [ − ] coFact ⋆ ( X ) ≤ c cA ?(cid:31) O O where ≤ c L and ≤ c cA denote the connectivity constraints given in Definition 3.3.1, and where Chev and
Prim [ − ] are the functors coming from Koszul duality. Remark . As in the case of Vect, our functors will be automatically restricted to the subcategories withthe appropriate connectivity conditions, unless otherwise specified. For example, we will write Chev instead ofChev | Lie ⋆ ( Ran X ) ≤ cL in most cases. Remark . As in the case of Vect, the pair of operad and co-operad Lie and ComCoAlg could be replaced byan operad O and its Koszul dual O ∨ such that (i) O is classical, i.e. it lies in the heart of the t -structure of Vect.(ii) O ∨ [ − ] is also classical.(iii) O ( ) ≃ Λ (as we already assume throughout this paper).We start with a preliminary lemma, which ensures that the categoriesLie ⋆ ( Ran X ) ≤ c L and ComCoAlg ⋆ ( Ran X ) ≤ c cA are actually well-defined. Lemma 3.3.7.
The subcategories
Shv ( Ran X ) ≤ c L and Shv ( Ran X ) ≤ c cA are preserved under the ⊗ ⋆ -monoidal structureon Shv ( Ran X ) .Proof of Lemma 3.3.7. Recall from (2.4.4) that if F , . . . , F k ∈ Shv ( Ran X ) ,then from the definition of ⊗ ⋆ , we have(3.3.8) ( F ⊗ ⋆ · · · ⊗ ⋆ F k ) | ◦ X I ≃ M I = ∪ ki = I i ∆ ! ⊔ ki = I i ։ ∪ ki = I i ( F ⊠ · · · ⊠ F k ) | ( Q ki = ◦ X Ii ) disj .Now, suppose that F , . . . , F k ∈ Shv ( Ran X ) ≤ c L ,then we see that each summand in (3.3.8) lies in perverse cohomological degrees ≤ ( − − d ) k X i = | I i | + d k X i = | I i | − | I | ! ≤ − k X i = | I i | − d | I |≤ ( − − d ) | I | .Here, the first inequality is due to the fact that the map ◦ X I → k Y i = ( ◦ X I i ) disj is a regular embedding, and that the (perverse) cohomological amplitude of the !-pullback along a regular em-bedding is equal to the codimension. Thus, this implies that F ⊗ ⋆ · · · ⊗ ⋆ F k ∈ Shv ( Ran X ) ≤ c L .Similarly, suppose that F , . . . , F k ∈ Shv ( Ran X ) ≤ c cA , Note that for a general operad O , only the first row of (3.3.4) makes sense. HE ATIYAH-BOTT FORMULA AND CONNECTIVITY IN CHIRAL KOSZUL DUALITY 21 then each summand in (3.3.8) lies in perverse cohomological degrees ≤ ( − − d ) k X i = | I i | − k + d k X i = | I i | − | I | ! (3.3.9) ≤ − k X i = | I i | − k − d | I |≤ ( − − d ) | I | − F ⊗ ⋆ · · · ⊗ ⋆ F k ∈ Shv ( Ran X ) ≤ c cA ,which concludes the proof.3.3.10. Back to Theorem 3.3.3. First, we will prove the equivalence on the top row of (3.3.4). And then, we willshow that it induces an equivalence between the corresponding sub-categories on the bottom row.As in the case of Vect, to prove that Chev and Prim [ − ] are mutually inverse functors, it suffices to show thatChev is fully-faithful, and Prim [ − ] is conservative. The following lemma will help us achieve this goal. Lemma 3.3.11.
The functor
Prim [ − ] | ComCoAlg ⋆ ( Ran X ) ≤ ccA satisfies the following conditions (see Remark 3.3.5)(i) Prim [ − ] commutes with sifted colimits.(ii) The natural map Free
Lie → Prim [ − ] ◦ triv ComCoAlg is an equivalence.
As in Corollary 3.1.7, this immediately implies the following
Corollary 3.3.12.
Chev | Lie ⋆ ( Ran X ) ≤ cL is fully faithful. ◦ X I for each non-empty finite set I .3.3.14. In general, for any A ∈ ComCoAlg ⋆ ( Ran X ) ≤ c cA ,we have coBar ComCoAlg ( A ) = Tot ( coBar • ComCoAlg ( A )) ,where coBar • ComCoAlg ( A ) is a co-simplicial object.Let coBar n ComCoAlg ( A ) = Tot ( coBar • ComCoAlg ( A ) | ∆ ≤ n ) .Then, we have the following tower A ≃ coBar ( A ) ← coBar ( A ) ← · · · and coBar ComCoAlg ( A ) ≃ lim n coBar n ComCoAlg ( A ) . F n ( A ) = Fib ( coBar n ComCoAlg ( A ) → coBar n − ( A )) ,and let I be a non-empty finite set. Using the same argument as in the case of Vect in combination with thecohomological estimate (3.3.9), we see that F n ( A ) | ◦ X I lives in cohomological degrees ≤ ( − − d ) n X i = | I i | − n + d n X i = | I i | − | I | ! + n = − n X i = | I i | − n − d | I | + n ≤ − n + − d | I | + n which goes to −∞ when n → ∞ .This gives us the following analog of Lemma 3.1.10. Lemma 3.3.16.
Let A ∈ ComCoAlg ⋆ ( Ran X ) ≤ c cA . Then, for any n and I, the following natural map tr ≥− n + − d | I | + n + ( coBar n ComCoAlg ( A ) | ◦ X I ) → tr ≥− n + − d | I | + n + ( coBar n − ( A ) | ◦ X I ) is an equivalence. This implies the following result, which is parallel to Corollary 3.1.11.
Corollary 3.3.17.
Let A ∈ ComCoAlg ⋆ ( Ran X ) ≤ c cA . Then, for any n and I, the following natural map tr ≥− n ( coBar ComCoAlg ( A ) | ◦ X I ) → tr ≥− n ( coBar m ComCoAlg ( A ) | ◦ X I ) is an equivalence, when m ≫ depending only on n and I. Remark . Note that when X is a point, namely when d = dim X =
0, the cohomological estimates inLemma 3.3.16 recover those of Lemma 3.1.10.To finish with the top equivalence in (3.3.4), we need the following
Lemma 3.3.20.
The functor
Prim [ − ] : ComCoAlg ⋆ ( Ran X ) ≤ c cA → Lie ⋆ ( Ran X ) ≤ c L is conservative.Proof. It suffices to show thatcoBar
ComCoAlg : ComCoAlg ⋆ ( Ran X ) ≤ c cA → Lie ⋆ [ − ]( Ran X ) ≤ c cA is conservative, and we will do so by contradiction. Namely, let f : A → A be a morphism in ComCoAlg ⋆ ( Ran X ) ≤ c cA that is not an equivalence. Suppose thatcoBar ComCoAlg ( f ) : coBar ComCoAlg ( A ) → coBar ComCoAlg ( A ) is an equivalence, we will derive a contradiction.Let I be the smallest set such that the map f | ◦ X I : A | ◦ X I → A | ◦ X I HE ATIYAH-BOTT FORMULA AND CONNECTIVITY IN CHIRAL KOSZUL DUALITY 23 is not an equivalence. Let k ≥ ≥ ( − − d ) | I |− − k ( A | ◦ X I ) → tr ≥ ( − − d ) | I |− − k ( A | ◦ X I ) is not an equivalence.By Corollary 3.3.17, we know that there exists some m ≫ ≥ ( − − d ) | I |− − k ( coBar ComCoAlg ( A i ) | ◦ X I ) ≃ tr ≥ ( − − d ) | I |− − k ( coBar m ComCoAlg ( A i ) | ◦ X I ) for i ∈ {
1, 2 } . Thus, we get the following equivalencetr ≥ ( − − d ) | I |− − k ( coBar m ComCoAlg ( A ) | ◦ X I ) ≃ tr ≥ ( − − d ) | I |− − k ( coBar m ComCoAlg ( A ) | ◦ X I ) .But observe that if we let F n ( A i ) = Fib ( coBar n ComCoAlg ( A i ) → coBar n − ( A i )) then the difference between F n ( A ) | ◦ X I and F n ( A ) | ◦ X I lies in cohomological degrees ≤ ( − − d ) | I | − − k + ( − − d ) n − X i = | I i | − ( n − ) + n + d | I | + n − X i = | I i | − | I | ! ≤ ( − − d ) | I | − − k − n − X i = | I i | − n + + n < ( − − d ) | I | − − k .This implies that for n ≥
1, tr ≥ ( − − d ) | I |− − k ( F n ( A ) | ◦ X I ) ≃ tr ≥ ( − − d ) | I |− − k ( F n ( A ) | ◦ X I ) .Thus, as in the case of Vect, a downward induction implies thattr ≥ ( − − d ) | I |− − k ( A | ◦ X I ) ≃ tr ≥ ( − − d ) | I |− − k ( A | ◦ X I ) ,which contradicts our original assumption, and we are done.3.3.21. Corollary 3.3.12 and Lemma 3.3.20 together prove the equivalence on the top row of diagram (3.3.4).For the equivalence in the bottom row, it suffices to show that for g ∈ Lie ⋆ ( Ran X ) ≤ c L ,Chev ( g ) is factorizable if and only if g ∈ Lie ⋆ ( X ) ≤ c L .3.3.22. For the “if” direction, recall that as a consequence of [ FG11, Thm. 6.4.2 and 5.2.1 ] , we know that thefunctor Chev : Lie ⋆ ( X ) → ComCoAlg ⋆ ( Ran X ) lands inside the full-subcategory coFact ⋆ ( X ) of factorizable co-algebras. We thus get a functorChev : Lie ⋆ ( X ) ≤ c L → coFact ⋆ ( X ) ≤ c cA ,which settles the “if” direction.3.3.23. For the “only if” direction, let g ∈ Lie ⋆ ( Ran X ) ≤ c L whose support does not lie in X . We will show that Chev g is not factorizable.Using the ass-gr ◦ addFil trick (see §A), it suffices to prove for the case where g is a trivial (i.e. abelian) Liealgebra. In that case, we know that Chev g = Sym > ( g [ ]) ,where Sym is taken using the ⊗ ⋆ -monoidal structure.Let I be the smallest set, with | I | >
1, such that g | ◦ X I
0. Now, it’s easy to see that Sym > ( g [ ]) fails thefactorizability condition at ◦ X I , which concludes the “only if” direction.
4. F
ACTORIZABILITY OF coChevIn this section, we will prove Theorem 1.5.6, which asserts that when g ∈ coLie ⋆ ( X ) satisfies a certain co-connectivity constraint, the commutative algebracoChev ( g ) ∈ ComAlg ⋆ ( Ran X ) is factorizable.Note that an analog of this result, where coChev is replaced by Chev, has been proved in [ FG11 ] (and in fact,we used this result in the previous section). The main difficulties of the coChev case stem from the fact that,unlike Chev, coChev is defined as a limit, and most of the functors that we want it to interact with don’t generallycommute with limits.As above, our main strategy is to introduce a certain co-connectivity condition to ensure that when one takesthe limit of a diagram involving objects satisfying it, the answer, in some sense, converges instead of running offto infinity, so we still have a good control over it.We start with the precise statement of the theorem. Then, after a quick digression on the various notionsrelated to the convergence of a limit, we will present the main strategy. Finally, the proof itself will be given.4.1. The statement.
We start with the co-connectivity conditions.
Definition 4.1.1.
Let Shv ( Ran X ) ≥ n denote the full subcategory of Shv ( Ran X ) consisting of sheaves F such thatfor all non-empty finite sets I , F | ◦ X I ∈ Shv ( ◦ X I ) ≥ n ,As before, we use the perverse t -structure. Notation 4.1.2.
We will use coLie ⋆ ( Ran X ) ≥ n and ComAlg ⋆ ( Ran X ) ≥ n to denote coLie ⋆ ( Shv ( Ran X ) ≥ n ) and ComAlg ⋆ ( Shv ( Ran X ) ≥ n ) respectively.We will prove the following Theorem 4.1.3.
Restricted to the full subcategory coLie ⋆ ( X ) ≥ , the functor coChev factors through Fact ⋆ , i.e. coLie ⋆ ( X ) ≥ & & ▼▼▼▼▼▼▼▼▼▼▼ coChev / / ComAlg ⋆ ( Ran X ) Fact ⋆ ( X ) ) (cid:9) ♥♥♥♥♥♥♥♥♥♥♥♥ In other words, coChev g is factorizable when g ∈ coLie ⋆ ( X ) ≥ . Stabilizing co-filtrations and decaying sequences (a digression).
In this subsection, we describe a condi-tion on co-filtered and graded objects which make them behave nicely with respect to taking limits.
Definition 4.2.1.
Let C be a stable infinity category equipped with a t -structure. Then, a co-filtered object c ∈ C coFil > (see §B) is said to stabilize if for all n , tr ≤ n c m → tr ≤ n c m + is an equivalence for all m ≫ c ∈ C gr > is said to be decaying if for all n iftr ≤ n c m ≃ m ≫ HE ATIYAH-BOTT FORMULA AND CONNECTIVITY IN CHIRAL KOSZUL DUALITY 25
Notation 4.2.2.
We use C coFil > ,stab and C gr > ,decay to denote the subcategories of C coFil > and C gr > consisting ofstabilizing and decaying objects respectively.We have the following lemmas, whose proofs are straightforward. Lemma 4.2.3.
Let c ∈ C coFil > . Then c ∈ C coFil > ,stab if and only if ass-gr c ∈ C gr > ,decay . Lemma 4.2.4.
If c ∈ C coFil > ,stab , then for each n, the natural map τ ≤ n oblv coFil c → τ ≤ n c m is an equivalence when m ≫ .Proof. By throwing away finitely many terms at the beginning, without loss of generality, we can assume that thenatural maps τ ≤ n + c i → τ ≤ n + c j , ∀ i ≥ j > τ ≤ n lim i c i → τ ≤ n c .Equivalently, it suffices to show that Fib ( lim i c i → c ) ∈ C ≥ n + .However, Fib ( lim i c i → c ) ≃ lim i ( Fib ( c i → c )) ∈ C ≥ n + because Fib ( c i → c ) ∈ C ≥ n + , ∀ i .Hence, we are done, since i ≥ n + : C ≥ n + → C commutes with limits (see §2.1.3). Lemma 4.2.5.
The natural transformation M → Y between functors C gr > ,decay → C is an equivalence.Proof. Note that Y i c i ≃ lim k M i ≤ k c i .Moreover, since the sequence we are taking the limit over stabilizes, the result follows as a direct corollary ofLemma 4.2.4. Definition 4.2.7.
A co-filtered sheaf F ∈ Shv ( Ran X ) coFil > is said to stabilize if for any non-empty finite set I , F | ◦ X I ∈ Shv ( ◦ X I ) coFil > ,stab .Similarly, a graded sheaf F ∈ Shv ( Ran X ) gr > is said to be decaying if for any non-empty finite set I , F | ◦ X I ∈ Shv ( ◦ X I ) gr > ,decay . Notation 4.2.8.
We use Shv ( Ran X ) coFil > ,stab and Shv ( Ran X ) gr > ,decay to denote the sub-categories of Shv ( Ran X ) coFil > and Shv ( Ran X ) gr > consisting of stabilizing and decaying objects, respectively.It’s straightforward to see that the following analogs of the lemmas above still hold in this setting. Lemma 4.2.9.
Let F ∈ Shv ( Ran X ) coFil > . Then F ∈ Shv ( Ran X ) coFil > ,stab if and only if ass-gr F ∈ Shv ( Ran X ) gr > ,decay . Lemma 4.2.10. If F ∈ Shv ( Ran X ) coFil > ,stab , then for each I and n, the natural map τ ≤ n oblv coFil F | ◦ X I → τ ≤ n F m | ◦ X I is an equivalence when m ≫ . Lemma 4.2.11.
The natural transformation M → Y between functors Shv ( Ran X ) gr > ,decay → Shv ( Ran X ) is an equivalence. Strategy.
To prove that Chev g is factorizable when g ∈ Lie ⋆ ( X ) , [ FG11 ] uses the addFil trick (see §A) toreduce to the case where g is a trivial. In that case,Chev g ≃ Sym > g ,and the result can be seen directly. In the case of coChev, while the core strategy remains the same, it is morecomplicated to carry out since many commutative diagrams needed for the addFil trick to work (see (A.3.3))don’t commute in general in this new setting. The co-connectivity constraints are what needed to make thesediagrams commute and hence, to allow us to reduce to the trivial case. Note that oblv coFil commutes with restricting to ◦ X I for any non-empty, finite set I . Thus, the LHS is free of ambiguity. HE ATIYAH-BOTT FORMULA AND CONNECTIVITY IN CHIRAL KOSZUL DUALITY 27 ⋆ ( X ) ≥ (cid:15) (cid:15) coChev / / ComAlg ⋆ ( Ran X ) ≥ coLie ⋆ ( X ) ≥ > ,stab coChev coFil / / ass-gr (cid:15) (cid:15) ComAlg ⋆ ( Ran X ) ≥ > ,stabass-gr (cid:15) (cid:15) oblv coFil O O coLie ⋆ ( X ) ≥ > ,decay coChev gr / / Q (cid:15) (cid:15) ComAlg ⋆ ( Ran X ) ≥ > ,decay Q (cid:15) (cid:15) coLie ⋆ ( X ) ≥ / / ComAlg ⋆ ( Ran X ) ≥ Suppose also that oblv coFil preserves factorizability, and that ass-gr and Q are conservative with respect to fac-torizability. Then by the same reasoning as in the addFil trick, to prove that coChev g is factorizable, it suffices toassume that g has a trivial coLie-structure. In that case,coChev g ≃ Sym > ( g [ − ]) ,and as in the Chev case, we are done.4.3.3. The rest of this section will be devoted to the execution of the strategy outlined above.4.4. Well-definedness of functors.
Before proving that the diagram commutes, we need to first make sense ofit. A priori, the functors written in the diagram are not necessarily well-defined. For instance, we haven’t shownthat all the four instances of coChev land in the correct target categories. Moreover, we also don’t know thatoblv coFil , ass-gr, and Q preserve the algebra / co-algebra structures.We start with the following straight-forward observation which settles the latter question. Lemma 4.4.1.
For any n, the functors oblv coFil : Shv ( Ran X ) ≥ n ,coFil > ,stab → Shv ( Ran X ) ≥ n ass-gr : Shv ( Ran X ) ≥ n ,coFil > → Shv ( Ran X ) ≥ n ,gr > Y ≃ M : Shv ( Ran X ) ≥ n ,gr > ,decay → Shv ( Ran X ) ≥ n are symmetric monoidal with respect to the ⊗ ⋆ -monoidal structure on Ran
X . C ≥ n is preserved under limits for any stable infinitycategory C with a t -structure (since i ≥ n commutes with limits, see §2.1.3).4.4.4. By the same token, we know that the essential images of coChev coFil and coChev gr satisfy the co-connectivityassumption (i.e. live in (perverse) cohomological degree ≥ Corollary 4.4.5.
We have the following commutative diagram coLie ⋆ ( X ) ≥ > ,stab coChev coFil / / ass-gr (cid:15) (cid:15) ComAlg ⋆ ( Ran X ) ≥ > ass-gr (cid:15) (cid:15) coLie ⋆ ( X ) ≥ > ,decay coChev gr / / ComAlg ⋆ ( Ran X ) ≥ > Now, by Lemma 4.2.9, to show that coChev coFil and coChev gr satisfy the stab and decay conditions respectively,it suffices to show that coChev gr satisfies the decay condition. However, this is also a direct consequence of thefact that the shriek-pullback functor is left exact and C > n is preserved under limits (for any stable infinity category C with a t -structure), and we are done.4.5. Commutative diagrams.
We will now proceed to prove that the diagram (4.3.2) commutes. First note thatwe have just settled the commutativity of the middle diagram of (4.3.2) at the end of the previous subsection.4.5.1. The commutativity of the bottom diagram of (4.3.2) is clear if we know that Q is symmetric monoidal.However, by Lemma 4.2.11, we have Y ≃ M and we know that L is symmetric monoidal.4.5.2. Finally, to show that the top diagram of (4.3.2) commutes, it suffices to show that the following diagramcommutes(4.5.3) coLie ⋆ ( X ) ≥ / / ComAlg ⋆ ( Ran X ) ≥ coLie ⋆ ( X ) ≥ > ,staboblv coFil O O coChev coFil / / ComAlg ⋆ ( Ran X ) ≥ > ,staboblv coFil O O since the composition coLie ⋆ ( X ) ≥ −→ coLie ⋆ ( X ) ≥ > ,stab oblv coFil −→ coLie ⋆ ( X ) ≥ is the identity functor (see also §A.3.1). However, this is clear since the functoroblv coFil : Shv ( Ran X ) ≥ n ,coFil > ,stab → Shv ( Ran X ) ≥ n commutes with limit for any n , and moreover it is symmetric monoidal with respect to the ⊗ ⋆ -monoidal structureon Shv ( Ran X ) by Lemma 4.4.1.4.6. Relation to factorizability.
It is easy to see thatass-gr : ComAlg ⋆ ( Ran X ) ≥ > ,stab → ComAlg ⋆ ( Ran X ) ≥ > ,decay reflects factorizability. Moreover, as we’ve discussed above, we have the equivalence Y ≃ M as functors ComAlg ⋆ ( Ran X ) ≥ > ,decay → ComAlg ⋆ ( Ran X ) ≥ .But now it’s clear that Q reflects factorizability, since L does.Finally, since oblv coFil : ComAlg ⋆ ( Ran X ) ≥ > ,stab → ComAlg ⋆ ( Ran X ) ≥ is compatible with ⊠ (for the same reason that it is compatible with ⊗ ⋆ ) , and moreover ( − ) ! commutes withlimits (being a right adjoint), we see easily that oblv coFil preserves factorizability. Thus, we conclude the proof ofTheorem 4.1.3. HE ATIYAH-BOTT FORMULA AND CONNECTIVITY IN CHIRAL KOSZUL DUALITY 29
Relation to coLie ! ( X ) and ComAlg ! ( X ) . In this subsection, we will discuss the various links between ob-jects defined on X such as coLie ! ( X ) and ComAlg ! ( X ) and objects defined on Ran X such as coLie ⋆ ( Ran X ) ,ComAlg ⋆ ( Ran X ) and Fact ⋆ ( X ) . This subsection is not used anywhere in the paper. We include it here for thesake of completeness.4.7.1. Recall that on a scheme X , there are two symmetric monoidal structures, ⊗ and ! ⊗ . Thus, we could talkabout various algebra / co-algebra objects defined on itLie ∗ ( X ) , coLie ! ( X ) , ComAlg ! ( X ) ,where Lie ∗ ( X ) (not to be confused with Lie ⋆ ( X ) ) is the category of Lie-algebra objects in Shv ( X ) with respect tothe ⊗ -monoidal structure, and coLie ! ( X ) (resp. ComAlg ! ( X ) ) is the category of coLie-algebra (resp. commutativealgebra) objects in Shv ( X ) with respect to the ! ⊗ -monoidal structure.4.7.2. The following observations are straightforward, and are both based on the fact that the functorsins ∗ X : Shv ( Ran X ) ⊗ ⋆ → Shv ( X ) ⊗ and ins ! X : Shv ( Ran X ) ⊗ ⋆ → Shv ( X ) ! ⊗ are symmetric monoidal, where ins X : X → Ran X is the diagonal embedding. Lemma 4.7.3.
We have a pair of adjoint functors ins ∗ X : Lie ⋆ ( Ran X ) ⇄ Lie ∗ ( X ) : ins X ∗ which induces an equivalence of categories Lie ⋆ ( X ) ≃ Lie ∗ ( X ) . Lemma 4.7.4.
We have a pair of adjoint functors ins X ! : coLie ! ( X ) ⇄ coLie ⋆ ( Ran X ) : ins ! X which induces an equivalence of categories coLie ! ( X ) ≃ coLie ⋆ ( X ) .4.7.5. We also have the following functorins ! X : ComAlg ⋆ ( Ran X ) → ComAlg ! ( X ) which commutes with limits. Thus, we get a pair of adjoint functors(4.7.6) ins X ? : ComAlg ! ( X ) ⇄ ComAlg ⋆ ( Ran X ) : ins ! X .We have the following result from [ GL14, Thm. 5.6.4 ] . Theorem 4.7.7.
The pair of adjoint functors in (4.7.6) induces an equivalence of categories
ComAlg ! ( X ) ≃ Fact ⋆ ( X ) .4.7.8. The first link between coLie ! ( X ) , coLie ⋆ ( X ) , ComAlg ! ( X ) , ComAlg ⋆ ( Ran X ) and Fact ⋆ ( X ) is given by thefollowing diagram(4.7.9) coLie ! ( X ) coChev (cid:15) (cid:15) coLie ⋆ ( X ) ≃ ins ! X o o coChev (cid:15) (cid:15) ComAlg ! ( X ) ComAlg ⋆ ( Ran X ) ins ! X o o whose commutativity is straightforward due to the fact that ins ! X commutes with limits and that it’s monoidal. Proposition 4.7.11.
We have the following commutative diagram coLie ! ( X ) ≥ (cid:15) (cid:15) ≃ ins X ! / / coLie ⋆ ( X ) ≥ (cid:15) (cid:15) ComAlg ! ( X ) ins X ? / / Fact ⋆ ( X ) Proof.
For any g ∈ coLie ! ( X ) , we have a natural mapins X ? ◦ coChev → coChev ◦ ins X ! .of objects in ComAlg ⋆ ( Ran X ) . Now, we know from Theorem 4.7.7 that the LHS is factorizable. Moreover, when g ∈ coLie ! ( X ) ≥ , we know from Theorem 4.1.3 that the RHS is also factorizable. Thus, to show that the mapabove is an equivalence when g ∈ coLie ! ( X ) ≥ , it suffices to show that they are equivalence on the diagonal.However, that is clear from (4.7.9) and we are done.5. I NTERACTIONS BETWEEN VARIOUS FUNCTORS ON THE R AN SPACE
In this section, we tie together the links between the various functors on the Ran spaces: Chev, coChev, C ∗ c ( Ran X , − ) ,and D Ran , the functor of Verdier duality on the Ran space.5.1. C ∗ c ( Ran X , − ) and coChev . In this subsection, we will prove Theorem 1.5.8, which gives us a criterion forthe commutativity of the functor coChev and the functor C ∗ c ( Ran X , − ) . Note that it has been proved in [ FG11 ] that Chev always commutes with C ∗ c ( Ran X , − ) . The main reason is that C ∗ c ( Ran X , − ) is continuous and monoidalwith respect to the ⊗ ⋆ -monoidal structure on Shv ( Ran X ) and the usual monoidal structure on Vect. As before,our main difficulty comes from the fact that coChev is defined as a limit, and for that to behave well with respectto C ∗ c ( Ran X , − ) , we need to impose a certain co-connectivity assumption.5.1.1. Throughout this subsection, X will be assumed to be a proper scheme of pure dimension d . Theorem 5.1.2.
For any g ∈ coLie ⋆ ( X ) ≥ + d , the natural mapC ∗ c ( Ran X , coChev g ) → coChev ( C ∗ c ( X , g )) is an equivalence. P ) as a sequential limit, and then establish a certain stability condition onthe sequence we take the limit over. The main point is to show that for any n , tr ≥− n of our limit is just tr ≥− n of theterms when we go sufficiently far in the sequence. And at a finite step, commuting with a colimit is automatic.5.1.4. Our current situation is the dual of that. Namely, we will express C ∗ c ( Ran X , − ) as the colimit of a sequence satisfying a certain stability condition, which allows us, after truncating on the rightvia tr ≤ n for each n , to commute it with the limit defining coChev. Since Supp g ⊂ X ⊂ Ran X , C ∗ c ( Ran X , g ) ≃ C ∗ c ( X , g ) . HE ATIYAH-BOTT FORMULA AND CONNECTIVITY IN CHIRAL KOSZUL DUALITY 31 K → C could be written as a sequential limit (resp. colimit) if we have a functor K → Z . We can then use left (resp.right) Kan extension to produce a new diagram Z → C ,and the original limit (resp. colimit) could be written as a sequential limit (resp. colimit) of this new diagram.For example, in the case of limit over a co-simplicial object, the functor to Z is simply ∆ → Z ≥ [ n ] n .And in the case of the Ran space, the functor is fSet surj → Z > I
7→ | I | .5.1.6. Truncated Ran space.
Now we can apply the remark above to the case of the Ran space. For any scheme X and any positive integer n , we define Ran ≤ n X ≃ colim I ∈ fSet surj | I |≤ n X I .Then Ran X ≃ colim Ran ≤ n X ≃ colim ( X → Ran ≤ X → Ran ≤ X → · · · ) ,and hence, for any F ∈ Shv ( Ran X ) , C ∗ c ( Ran X , F ) ≃ colim n C ∗ c ( Ran ≤ n , F | Ran ≤ n X ) .The following observation, which gives the link among the cohomology groups C ∗ c ( Ran ≤ n X , F | Ran ≤ n X ) for various n ’s, comes from [ Gai15, Cor. 9.1.4 ] . Lemma 5.1.7.
We have the following natural equivalenceC ∗ ( ◦ X I , F | ◦ X I ) Σ I ≃ coFib ( C ∗ c ( Ran ≤| I |− X , F | Ran ≤| I |− X ) → C ∗ c ( Ran ≤| I | X , F | Ran ≤| I | X )) .5.1.8. When g ∈ coLie ⋆ ( X ) ≥ + d ,using the addCoFil trick (4.3.2), we can also express coChev g as a sequential limitcoChev g ≃ oblv coFil coChev coFil addCoFil g ≃ lim i ( coChev coFil addCoFil g ) i .Where ( coChev coFil addCoFil g ) i is the i -th step in the co-filtration.5.1.9. For brevity’s sake, we will denotecoChev i g = ( coChev coFil addCoFil g ) i and so we have coChev g ≃ lim i coChev i g . i vary nicely withrespect to i . Namely, for any non-negative integer i ,Supp coChev i g ⊂ Ran ≤ i X and for all non-empty finite set I such that | I | ≤ i , ( coChev i g ) | ◦ X I lives in perverse cohomological degrees ≥ i ( d + ) +
1. This gives us the following observations.
Lemma 5.1.11.
For any g ∈ coLie ⋆ ( X ) ≥ + d and any non-empty finite set I, ( coChev g ) | ◦ X I lives in cohomological degrees ≥ ( + d ) | I | + . Corollary 5.1.12.
For any g ∈ coLie ⋆ ( X ) ≥ + d and any non-empty finite set I,C ∗ ( ◦ X I , ( coChev g ) | ◦ X I ) Σ I lives in cohomological degrees ≥ | I | + . Lemma 5.1.13.
For any g ∈ coLie ⋆ ( X ) ≥ + d , any positive integer i, and any non-empty finite set I,C ∗ ( ◦ X I , ( coChev i g ) | ◦ X I ) Σ I lives in cohomological degrees ≥ max ( | I | + i + ) . With these observations, we are ready for the proof of Theorem 5.1.2.
Proof of Theorem 5.1.2.
For each i , we know that coChev i g is computed as a finite limit. Thus, we have thefollowing natural equivalence C ∗ c ( Ran X , coChev i g ) ≃ coChev i ( C ∗ c ( Ran X , g )) .Taking the limit over i on both sides, we observe that it suffices to prove thatlim i C ∗ c ( Ran X , coChev i g ) ≃ C ∗ c ( Ran X , lim i coChev i g ) .For that, it suffices to show that for each m , we have an equivalencetr ≤ m lim i C ∗ c ( Ran X , coChev i g ) ≃ tr ≤ m C ∗ c ( Ran X , lim i coChev i g ) .But now, for some M ≫
0, depending only on m , we havetr ≤ m C ∗ c ( Ran X , lim i coChev i g ) ≃ tr ≤ m colim n C ∗ c ( Ran ≤ n X , lim i coChev i g | Ran ≤ n X ) ≃ tr ≤ m C ∗ c ( Ran ≤ M X , lim i coChev i g | Ran ≤ M X ) (5.1.14) ≃ tr ≤ m lim i C ∗ c ( Ran ≤ M X , coChev i g | Ran ≤ M X ) (5.1.15) ≃ lim i tr ≤ m C ∗ c ( Ran ≤ M X , coChev i g | Ran ≤ M X ) ≃ lim i tr ≤ m C ∗ c ( Ran X , coChev i g ) (5.1.16) ≃ tr ≤ m lim i C ∗ c ( Ran X , coChev i g ) .Here, we used Lemma 5.1.7 in both (5.1.14) and (5.1.16). Moreover, (5.1.14) and (5.1.16) use Corollary 5.1.12and Lemma 5.1.13 respectively. Finally, (5.1.15) is due to the fact that C ∗ c ( Ran ≤ M X , − ) commutes with limits. Indeed, this functor is computed as a finite colimit of functors of the form C ∗ c ( X I , − ) . HE ATIYAH-BOTT FORMULA AND CONNECTIVITY IN CHIRAL KOSZUL DUALITY 33
But these functors commute with limits since X I are all complete due to our assumption on X , i.e. C ∗ c ( X I , − ) ≃ C ∗ ( X I , − ) .5.2. Verdier duality.
Before studying the link between Chev and coChev, we start with a quick recollection ofVerdier duality on prestacks along with various useful properties. The main reference is [ Gai15 ] . However, sincewe only use basic properties of D Ran , we’ll provide the complete proof in most cases.5.2.1. Let Y be a prestack such that the diagonal mapdiag Y : Y → Y × Y is pseudo-proper. Given F , G ∈ Shv ( Y ) , by a pairing between them, we shall mean a map F ⊠ G → diag Y ! ω Y .We define the Verdier dual, D Y G , of G to represent the functor F Hom ( F ⊠ G , diag Y ! ω Y ) .Namely, we have the following natural equivalenceHom ( F , D Y G ) ≃ Hom ( F ⊠ G , diag Y ! ω Y ) .The following lemma is immediate from the definition. Lemma 5.2.2.
Let F ∈ Shv ( Y ) , such that F ≃ colim i ∈ I F i . Then D Y F ≃ lim i ∈ I op D Y F i .5.2.3. We will now study the link between Verdier duality and ⊠ . Proposition 5.2.4.
Let Y and Y be finitary pseudo-schemes, and F i ∈ Shv ( Y i ) . Then, we have a natural equivalence D Y F ⊠ D Y F ≃ D Y × Y ( F ⊠ F ) . Proof.
First, note that the result holds when both Y and Y are schemes.For the general case of finitary pseudo-schemes, we write Y ≃ colim i Y i and Y ≃ colim j Y j .Then, F ≃ colim i ins i ! ins !1 i F and F ≃ colim j ins j ! ins !2 j F .Thus, D Y × Y ( F ⊠ F ) ≃ D Y × Y colim i , j ( ins i × ins j ) ! ( ins i × ins j ) ! ( F ⊠ F ) ≃ lim i , j ( ins i × ins j ) ! D Y i × Y j ( ins !1 i F ⊠ ins !2 j F ) (5.2.5) ≃ lim i , j ( ins i × ins j ) ! ( D Y i ins !1 i F ⊠ D Y j ins !2 j F ) (5.2.6) ≃ ( lim i ins i ! D Y i ins !1 i F ) ⊠ ( lim j ins j ! D Y j ins !1 j F ) (5.2.7) ≃ ( D Y colim i ins i ! ins !1 i F ) ⊠ ( D Y ins j ! ins !2 j F ) (5.2.8) ≃ D Y F ⊠ D Y F .Here, – (5.2.6) is due to the fact that the statement we are trying to prove holds for the case of schemes.– (5.2.7) is due to the fact that the limits we are taking are all finite (due to the finitary assumption).– (5.2.5) and (5.2.8) are both due to Lemma 5.2.2 and Proposition 5.2.9 below. Proposition 5.2.9.
Let f : Y → Y be a finitary pseudo-proper map between pseudo-schemes, each having a finitarydiagonal. Then, the natural transformation f ! ◦ D Y → D Y ◦ f ! is an equivalence.Proof. See [ Gai15, Cor. 7.5.6 ] . Remark . One direct corollary of this proposition is the fact that for any sheaf F ∈ Shv ( X ) , we have thefollowing natural equivalence δ X ! D X F ≃ D Ran X δ X ! F . Corollary 5.2.11.
Let F , F , · · · , F k ∈ Shv ( Ran X ) with finite supports, i.e. there exists an n such that all the F i ’scome from Shv ( Ran ≤ n X ) . Then, we have the following natural equivalenceD Ran X ( F ⊗ ⋆ F ⊗ · · · ⊗ ⋆ F k ) ≃ ( D Ran X F ) ⊗ ⋆ ( D Ran X F ) ⊗ ⋆ · · · ⊗ ⋆ ( D Ran X F k ) .5.3. Chev , coChev , and D Ran X . We will now turn to Theorem 1.5.10, which provides the link between the twofunctors Chev and coChev via the functor of taking Verdier duality on the Ran space.
Theorem 5.3.1.
Let g ∈ Lie ⋆ ( X ) ≤− . Then we have a natural equivalence coChev ( D X g ) ≃ D Ran X Chev ( g ) , of objects in ComAlg ⋆ ( Ran X ) , where D Ran X is the functor of taking Verdier duality on Ran
X .
Note that this is the only place we use Verdier duality on the Ran space. However, we essentially use it in arather minimal way: not much besides the definition itself.
Proof.
We will employ ideas originated from the addFil and addCoFil tricks (see also §A). First, observe that forany g ∈ Lie ⋆ ( X ) , we have a canonical equivalenceaddCoFil D Ran X g ≃ D Ran X addFil g .We denote Chev i g and coChev i D Ran X g to be the i -th piece in the filtration / co-filtration ofChev ( addFil g ) and coChev ( addCoFil D Ran X g ) respectively.From §A and the top part of the commutative diagram (4.3.2), we have the following natural equivalencesChev g ≃ colim i Chev i g ,coChev ( D Ran X g ) ≃ lim i coChev i ( D Ran X g ) .At the same time, by Lemma 5.2.2, we know that D Ran X colim i Chev i g ≃ lim i D Ran X Chev i g .Thus, it suffices to show that D Ran X Chev i g ≃ coChev i D Ran X g .Now, it’s an immediate consequence of Corollary 5.2.11. Corollary 5.3.2.
Let g ∈ Lie ⋆ ( X ) ≤− . Then D Ran X Chev ( g ) is a factorizable commutative algebra on Ran
X .
HE ATIYAH-BOTT FORMULA AND CONNECTIVITY IN CHIRAL KOSZUL DUALITY 35
Proof.
This is a direct consequence of Theorem 5.3.1 and Theorem 1.5.6.5.4. coChev and open embeddings.
We end the section with the following easy observation.
Proposition 5.4.1.
Let j : X ′ → Xbe an open embedding of schemes, which induces an open embedding of prestacksj
Ran : Ran X ′ → Ran X . Then for any g ′ ∈ coLie ⋆ ( X ′ ) , we have the following natural equivalence ( j Ran ) ∗ coChev ( g ′ ) ≃ coChev ( j ∗ g ′ ) . Proof (Sketch).
The result is a direct consequence of the fact that f ∗ , being a right adjoint, commutes with limitsfor any schematic morphism f between prestacks. Moreover, if f i : X ′ i → X i are open embeddings of schemes,and F i ∈ Shv ( X ′ i ) for i =
1, 2, then we have a natural equivalence ( f × f ) ∗ ( F ⊠ F ) ≃ f ∗ F ⊠ f ∗ F .6. A N APPLICATION TO THE A TIYAH -B OTT FORMULA
We will now give an application of the results proved so far to the Atiyah-Bott formula. As mentioned in theintroduction, these results allow us to simplify the second of the two main steps in the original proofs givenin [ GL14 ] and [ Gai15 ] . In what follows, §6.1–§6.4 are intended to orient the readers with the existing resultsproved in [ GL14 ] and [ Gai15 ] , whereas the purpose of the last part, §6.5, is to explain how the results we’veproved so far fit in with the rest.6.1. The statement.
From now on, X is a smooth and complete curve over an algebraically closed field k , and G a smooth, fiber-wise connected group-scheme over X , whose generic fiber is semi-simple simply connected. Dueto [ GL14, Lem. 7.1.1 and Prop A.3.11 ] , we can (and from now on we will) assume that G is semi-simple simplyconnected over an open dense subset j : X ′ , → X ,and moreover, the fibers of G over any point in X − X ′ are homologically trivial.We will also use j Ran : Ran X ′ → Ran X to denote the corresponding open embedding on the Ran space and Γ j Ran : Ran X ′ → Ran X ′ × Ran X to denote its graph.6.1.1. Let G be the split form of G . Then it is well-known that(6.1.2) C ∗ ( BG ) ≃ Sym M is a free commutative algebra, for some M ∈ Vect. In the case of ℓ -adic sheaves in positive characteristic setting,this equivalence is compatible with the geometric Frobenius action, where M ≃ M e Λ[ − e ]( − e ) ,and e ’s are the exponents of G .The assignment G M is functorial with respect to automorphisms of G , and hence, for a general G (subject to the assumptions mentioned above), we get a local system M ∈ Shv ( X ′ ) , Namely, all the results stated in these subsections could be found in [ GL14 ] or [ Gai15 ] . The readers should be warned that we providea mere overview of the development given in these two papers, with many technical points elided. whose !-fiber at each geometric point x ∈ X is equivalent to M .Below is the statement of the Atiyah-Bott formula. Theorem 6.1.3.
Let G , X as above. Then(a) We have an equivalence C ∗ ( Bun G ) ≃ Sym ( C ∗ ( X ′ , M )) . (b) When k = F q , and X and G are defined over F q , the above equivalence can be chosen to be compatible with theFrobenius actions. BG and the sheaf B . B that we will describe now encodes the reduced cohomology BG , the classifying stack of G . For each I ∈ Ran X ( S ) , let D I ⊂ S × X be the corresponding Cartier divisor. Let BG I denote the Artin stackclassifying G -bundles over D I and f I : BG I → S the forgetful map. Then, we define e B S , I = D S ( Fib ( f I ! f ! I Λ S → Λ S )) ,where D S is the functor of taking Verdier duality on S . These sheaves, assembled together, give rise to a sheaf(see also [ GL14, Prop. 5.4.3 ] ) e B ∈ Shv ( Ran X ) .6.2.2. Note that for any finite set of points { x , . . . , x n } ∈ ( Ran X )( k ) , the !-fiber of e B at this point is(6.2.3) coFib Λ → n O i = C ∗ ( BG x i ) ! .6.2.4. Using a variant of the diagonal map BG → BG × BG ,we can equip e B with the structure of an object inComAlg ⋆ ( Ran X ) .However, we see easily from (6.2.3) that e B is not factorizable. The functor TakeOut developed in [ Gai15 ] allowsus to remove all the extra components in it and construct out of it a new object B ∈ Fact ⋆ ( X ) with the correct!-fibers at a point { x , . . . , x n } ∈ ( Ran X )( k ) n O i = C ∗ red ( BG x i ) .Moreover, B has the same cohomology along Ran X as the original sheaf e B (see also [ Gai15, Cor. 5.3.5 ] ) C ∗ c ( Ran X , B ) ≃ C ∗ c ( Ran X , e B ) .6.2.5. B and Bun G . For every S ∈ Sch and I ∈ ( Ran X )( S ) , we have a map of prestacks over S by restricting thebundle to the divisor D I (6.2.6) S × Bun G → BG I .This induces a map e B S , I → ω S ⊗ C ∗ red ( Bun G ) and hence, also a map e B → ω Ran X ⊗ C ∗ red ( Bun G ) .Applying the functor C ∗ c ( Ran X , − ) and using the fact that Ran X is homologically contractible, we get a map(6.2.7) C ∗ c ( Ran X , B ) ≃ C ∗ c ( Ran X , e B ) → C ∗ red ( Bun G ) . HE ATIYAH-BOTT FORMULA AND CONNECTIVITY IN CHIRAL KOSZUL DUALITY 37 G , i.e. homologically contractible fibers outside of X ’, onegets an equivalence(6.2.9) B ≃ ( j Ran X ) ∗ B ′ ≃ Sym > ( j ∗ M ) where B ′ is the restriction of B to Ran X ′ and, the symmetric algebra is taken inside Shv ( Ran X ) using the ⊗ ⋆ -monoidal structure.6.2.10. Using the equivalence (6.2.9) and the fact that C ∗ c ( Ran X , − ) commutes with Sym, we get an explicitpresentation of the LHS of (6.2.7)(6.2.11) C ∗ c ( Ran X , B ) ≃ Sym > C ∗ c ( X , j ∗ M ) ≃ Sym > C ∗ ( X ′ , M ) ,which appears in the statement of the Atiyah-Bott formula as stated in Theorem 6.1.3.6.2.12. Now, we are done if we could show that the map in (6.2.7) is an equivalence.6.3. Affine Grassmannian and the sheaf A . Unfortunately, one does not know how to directly prove that (6.2.7)is an equivalence. Instead, [ GL14 ] proceeds with an equivalence of a dual nature, which we will now brieflyrecall.6.3.1. The main player in this step is the affine Grassmannian, or more precisely, a factorizable version of theaffine Grassmannian. Let G , X as above. The factorizable affine Grassmannian of G , denoted by Gr Ran X ′ , is theprestack such that for each scheme S , Gr Ran X ′ ( S ) = { ( P , I , α ) } ,where(i) P is a G -bundle over S × X (ii) I is a non-empty finite subset of X ′ ( S ) (iii) α is a trivialization of P on the complement of the graph of I .6.3.2. From the definition, we have the following natural morphism g : Gr Ran X ′ → Ran X ′ ,where we forget everything, except for the set I , and similarly another natural morphism u : Gr Ran X ′ → Bun G ,we we only remember the bundle P .6.3.3. The map g allows us to define e A ′ ≃ Fib ( g ! ( ω Gr Ran X ′ ) → ω Ran X ′ ) ∈ Shv ( Ran X ′ ) ,and the map u induces a map at the cohomology level, namely(6.3.4) C red ∗ ( Gr Ran X ′ ) → C red ∗ ( Bun G ) .Together, we get the following map(6.3.5) C ∗ c ( Ran X ′ , e A ′ ) → C red ∗ ( Bun G ) .6.3.6. Note that since Gr Ran X ′ → Ran X ′ is pseudo-proper, e A ′ is easy to describe. Namely for any finite set of points { x , x , . . . , x n } ⊂ X ( k ) , the !-fiber of e A ′ at this point is(6.3.7) Fib n O i = C ∗ ( Gr G xi ) → Λ ! . Note that this is a special case of the fact that C ∗ c ( Ran X , − ) commutes with Chev. And in fact, both are due to the same reasons: that C ∗ c ( Ran X , − ) is continuous and that it’s symmetric monoidal. A and Bun G . The equivalence of a dual nature that we alluded to earlier is given by the following impor-tant result (see [ GL14, Thm. 3.2.13 ] ). Theorem 6.3.9.
The map (6.3.4) , and hence (6.3.5) , is an equivalence. → Gr × Gr,one can equip e A ′ with the structure of an object inComCoAlg ⋆ ( Ran X ′ ) .However, note that the sheaf e A ′ is not factorizable, since its !-fiber, as described in (6.3.7), is too big, i.e. it’snot equivalent to(6.3.11) n O i = C red ∗ ( Gr G xi ) .Using a similar reasoning as in the case of e B and B , we can construct an object A ′ ∈ coFact ⋆ ( X ′ ) with the correct!-fiber as given in (6.3.11), and moreover, A ′ has the property that C ∗ c ( Ran X ′ , e A ′ ) ≃ C ∗ c ( Ran X ′ , A ′ ) .6.3.12. Altogether, we have the following Proposition 6.3.13.
We have a natural equivalenceC ∗ c ( Ran X ′ , A ′ ) ≃ C red ∗ ( Bun G ) .6.4. Pairing.
We will now describe how the equivalence given by Proposition 6.3.13 helps us show that (6.2.7)is an equivalence.6.4.1. For any schemes S , S ′ ∈ Sch and any non-empty finite subsets I ⊂ X ( S ) and I ′ ⊂ X ′ ( S ′ ) , we have a naturalmap (which is just a more elaborate variant of (6.2.6))Gr I ′ × S → Bun G × S ′ × S → S ′ × BG I ,which induces a map A ′ ⊠ B → ω Ran X ′ × Ran X ,and hence, a pairing (using TakeOut) A ′ ⊠ B → Γ j Ran ! ω Ran X ′ .6.4.2. Restricting this map to Ran X ′ × Ran X ′ gives us the following map A ′ ⊠ B ′ → ( δ Ran X ′ ) ! ω Ran X ′ ,and hence, using the definition of Verdier duality, a map(6.4.3) B ′ → D Ran X ′ A ′ between objects in ComAlg ⋆ ( Ran X ′ ) .6.4.4. It is proved, in fact twice (using different methods), in §17 and §18 of [ Gai15 ] , that the restrictionof (6.4.3) to the diagonal X ′ of Ran X ′ is an equivalence. Namely, we have(6.4.5) B ′ | X ′ ≃ ( D Ran X ′ A ′ ) | X ′ . HE ATIYAH-BOTT FORMULA AND CONNECTIVITY IN CHIRAL KOSZUL DUALITY 39
The last steps.
The results that we have just proved in this paper appear in two places in the concludingsteps, which are given by Proposition 6.5.1 and 6.5.4. Together, they imply the Atiyah-Bott formula.
Proposition 6.5.1. D Ran X ′ A ′ is factorizable, i.e.D Ran X ′ A ′ ∈ Fact ⋆ ( X ′ ) ⊂ ComAlg ⋆ ( Ran X ′ ) . Proof.
It is well-known that for a split semi-simple simply connected group G , C red ∗ ( Gr G , Λ) lives in cohomologi-cal degrees ≤ −
2. Using the fact that Gr
Ran X ′ → Ran X ′ is pseudo-proper and that A ′ is factorizable, we see that for each non-empty finite set I , A ′ | ◦ X ′ I lives in (perverse)cohomological degrees ≤ − | I | .Now, by Theorem 3.3.3, we know that there exists an object a ′ ∈ Lie ⋆ ( X ′ ) ≤ c L such that A ′ ≃ Chev ( a ′ ) .Theorem 5.3.1 then implies that D Ran X ′ Chev ( a ′ ) ≃ coChev ( D Ran X ′ a ′ ) ,which is known to be factorizable by Theorem 4.1.3 Corollary 6.5.2.
The map given in (6.4.3) is an equivalence, i.e. (6.5.3) B ′ ≃ D Ran X ′ A ′ , and hence B ≃ ( j Ran ) ∗ coChev D X ′ a ′ ≃ coChev j ∗ D X ′ a ′ . Proof.
The first statement is a direct consequence of the proposition above and the equivalence (6.4.5), where asthe second statement is the result of Proposition 5.4.1.
Proposition 6.5.4.
We have the following equivalence induced by Proposition 6.5.1C ∗ c ( Ran X , B ) ≃ C ∗ c ( Ran X ′ , A ′ ) ∨ . Proof.
We have the following equivalences C ∗ c ( Ran X , B ) ≃ C ∗ c ( Ran X , coChev j ∗ D X ′ a ′ ) (6.5.5) ≃ coChev C ∗ c ( X , j ∗ D X ′ a ′ ) (6.5.6) ≃ coChev C ∗ ( X , D X ′ a ′ ) ≃ coChev ( C ∗ c ( X , a ′ ) ∨ ) ≃ ( Chev ( C ∗ c ( X ′ , a ′ ))) ∨ (6.5.7) ≃ C ∗ c ( Ran X ′ , Chev a ′ ) ∨ ≃ C ∗ c ( Ran X ′ , A ′ ) ∨ .Here, (6.5.5), (6.5.6) and (6.5.7) are due to Corollary 6.5.2, Theorem 5.1.2 and Theorem 5.3.1 (applied to apoint) respectively.6.5.8. Finally, as a corollary, we have the Atiyah-Bott formula. Indeed, we have C red ∗ ( Bun G ) ∨ ≃ C ∗ c ( Ran X ′ , A ′ ) ∨ ≃ C ∗ c ( Ran X , B ) ≃ Sym > C ∗ ( X ′ , M ) where the first, second and third equivalences are due to Proposition 6.3.13, Proposition 6.5.4, and (6.2.11)respectively. A PPENDIX
A. T HE addFil TRICK
In this appendix, we will quickly recall, without proof, a useful construction taken from [ GR, §IV.2 ] , whichallows us to reduce many statements about P -algebras to trivial P -algebras, where P is an operad in Vect.Throughout this subsection, all categories without any further description will be assumed to be presentable,symmetric monoidal stable infinity over a field k of characteristic 0. Moreover, functors between these categoriesare assumed to be continuous.All such categories, along with continuous functors between them, form a category, which we will useDGCat SymMonpres,cont ,to denote, or for simplicity DGCat
SymMon .A.1.
Notations.
For a symmetric monoidal category C , we denote the category of filtered objects in CC Fil = Fun ( Z , C ) ,the category of functors from Z to C . Here, Z is a ordered set, viewed as a category. Similarly, we denote thecategory of graded objects C gr = Fun ( Z set , C ) ,where Z set is a the discrete category, whose underlying underlying objects are the integers. A.2.
Functors.
Now, we will recall several familiar functors between C , C Fil , and C gr .A.2.1. Let V = · · · → V n − → V n → V n + → · · · ,be an object in C Fil . Then, we define ass-gr : C Fil → C gr to be the functor of taking the associated graded objectass-gr ( V ) n = coFib ( V n − → V n ) ,and oblv Fil : C Fil → C to be the left Kan extension along Z → pt.Namely oblv Fil ( V ) = colim n ∈ Z V n .A.2.2. We also use ( gr → Fil ) : C gr → C Fil and M : C gr → C to denote the functor obtained by taking the left Kan extension along Z set → Z ,and Z set → ptrespectively.A.2.3. Note that the categories C Fil and C gr are equipped with a natural symmetric monoidal structure comingfrom C , and moreover, the functors ass-gr, oblv Fil , gr → Fil, and L are naturally symmetric monoidal. In [ GR ] , it’s called Z Spc . HE ATIYAH-BOTT FORMULA AND CONNECTIVITY IN CHIRAL KOSZUL DUALITY 41
A.2.4.
Adding a filtration.
Let addFil : C → C Fil be the functor defined as follows: for an object V in C ,addFil ( V ) n = ¨ V , when n ≥ M ◦ ass-gr ◦ addFil ≃ oblv Fil ◦ addFil ≃ id C .A.3. Interactions with algebras over an operad.
Let P be an operad in Vect. Then we have the following pairof functors addFil : P -alg ( C ) → P -alg ( C Fil > ) and oblv Fil : P -alg ( C Fil > ) → P -alg ( C ) .A.3.1. Let F : DGCat SymMon → C at ∞ be a functor, where C at ∞ is the ∞ -category of all ∞ -categories. Suppose we have a continuous natural transfor-mation Φ : P -alg ( − ) → F ( − ) ,i.e. morphisms between two objects in Fun ( DGCat
SymMon , C at ∞ ) .Then from what we’ve discussed above, we have the following commutative diagram P -alg ( C ) Φ / / F ( C ) P -alg ( C Fil ) oblv Fil O O Φ / / F ( C Fil ) oblv Fil O O which, combined with the fact that oblv Fil ◦ addFil ≃ id C ,implies that the following diagram also commutes P -alg ( C ) addFil (cid:15) (cid:15) Φ / / F ( C ) P -alg ( C Fil ) Φ / / F ( C Fil ) oblv Fil O O A.3.2. Further composing the diagram above with ass-gr and L gives us the following commutative diagram(A.3.3) P -alg ( C ) Φ / / addFil (cid:15) (cid:15) F ( C ) P -alg ( C Fil > ) ass-gr (cid:15) (cid:15) Φ Fil / / F ( C Fil > ) oblv Fil O O ass-gr (cid:15) (cid:15) P -alg ( C gr > ) L (cid:15) (cid:15) Φ gr / / F ( C gr > ) L (cid:15) (cid:15) P -alg ( C ) Φ / / F ( C ) We will refer to this as the fundamental commutative diagram of the addFil trick . A.3.4. Now, suppose there are two natural transformations Φ , Φ : P -alg ( − ) → F ( − ) equipped with a morphism between them α : Φ → Φ .Or more concretely, we have a compatible family of morphisms in F ( C )Φ ( c ) → Φ ( c ) parametrized by pairs ( C , c ) where c ∈ C and C ∈ DGCat
SymMon , and we want to prove that α is an equivalence.A.3.5. The top square of the commutative diagram above implies that it suffices to show that Φ Fil1 ◦ addFil → Φ Fil2 ◦ addFilis an equivalence. But since ass-gr and L are conservative, it suffices to show that M ◦ ass-gr ◦ Φ Fil1 ◦ addFil → M ◦ ass-gr ◦ Φ Fil2 ◦ addFilis an equivalence, which, due to the commutativity of the diagrams, is equivalent to Φ ◦ M ◦ ass-gr ◦ addFil → Φ ◦ M ◦ ass-gr ◦ addFilbeing an equivalence.A.3.6. The crucial observation of [ GR, Prop. IV.2.1.4.6 ] is the following Proposition A.3.7.
The functor M ◦ ass-gr ◦ addFil : P -alg ( C ) → F ( C ) is canonically equivalent to triv P ◦ oblv P , i.e. P -alg ( C ) oblv P −→ C triv P −→ P -alg ( C ) .A.3.8. This implies that it suffices to prove that Φ ( c ) → Φ ( c ) is an equivalence only for the case where c is a trivial algebra.A.4. A general principle.
More generally, suppose we want to prove a property of Φ( c ) for some c ∈ P -alg ( C ) .Moreover, suppose this property is preserved under under oblv Fil , and is conservative under L and ass-gr. Then,it suffices to prove the case where c has a trivial algebra structure.A PPENDIX
B. C O - FILTRATION AND addCoFilIn this appendix, we will collect various notions that are dual to the one in §A. These are used in the body ofthe paper to give a proof of the addCoFil trick in a special case.B.1.
Notations.
For a symmetric monoidal category C , we denote the category of co-filtered objects in CC coFil = Fun ( Z op , C ) .B.2. Functors.
As in the case of filtration, there are several familiar functors between C , C coFil , and C gr . HE ATIYAH-BOTT FORMULA AND CONNECTIVITY IN CHIRAL KOSZUL DUALITY 43
B.2.1. Let V = · · · → V n + → V n → V n − → · · · ,be an object in C coFil . Then we define ass-gr : C coFil → C gr to be the functor of taking the associated graded objectass-gr ( V ) n = Fib ( V n → V n − ) ,and oblv coFil : C coFil → C to be the right Kan extension along Z op → pt.Namely oblv coFil ( V ) = lim n ∈ Z op V n .B.2.2. Note that the category C coFil naturally inherits the monoidal structure coming from C . Moreover, thefunctor ass-gr is monoidal.B.2.3. We also use Y : C gr → C to denote the right Kan extension along Z set → pt.Namely Y (( V n ) n ∈ Z ) = Y n ∈ Z V n .B.2.4. Adding a co-filtration.
We will use addCoFil : C → C coFil to denote a functor defined as follows: for an object V in C ,addCoFil ( V ) n = ¨ V , when n ≥ EFERENCES [ AB83 ] M. F. Atiyah and R. Bott,
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