Factorisation algebras associated to Hilbert schemes of points Emily Cliff University of Oxford 14 December, 2015
Motivation ∙ Learn about factorisation: Provide and study examples of factorisation spaces and algebras of arbitrary dimensions. ∙ Learn about Hilbert schemes: Factorisation structures formalise the intuition that a space is built out of local bits in a specific way. Factorisation structures are expected to arise, based on the work of Grojnowski and Nakajima.
Outline 1 Main constructions : ℋ ilb Ran X and ℋ Ran X 2 Chiral algebras 3 Results on ℋ Ran X
Section 1 Main constructions : ℋ ilb Ran X and ℋ Ran X
Notation ∙ Fix k an algebraically closed field of characteristic 0. ∙ Let X be a smooth variety over k of dimension d . ∙ We work in the category of prestacks: . = Fun(Sch op , ∞ -Grpd) . PreStk Sch (Yoneda embedding)
The Hilbert scheme of points Fix n ≥ 0. The Hilbert scheme of n points in X is (the scheme representing) the functor X : Sch op → Set ⊂ ∞ -Grpd Hilb n S ↦→ Hilb n X ( S ) , where {︃ 𝜊 ⊂ S × X , a closed subscheme, flat over S }︃ Hilb n X ( S ) . . = . with zero-dimensional fibres of length n
The Hilbert scheme of points Example: k -points {︃ 𝜊 ⊂ X closed zero-dimensional }︃ Hilb n X (Spec k ) = . subscheme of length n For example, for X = A 2 = Spec k [ x , y ], n = 2, some k -points are 𝜊 1 = Spec k [ x , y ] / ( x , y 2 ) 𝜊 2 = Spec k [ x , y ] / ( x 2 , y ) 𝜊 3 = Spec k [ x , y ] / ( x , y ( y − 1)) . . = ⨆︁ n ≥ 0 Hilb n Notation: let Hilb X . X .
The Ran space The Ran space is a different way of parametrising sets of points in X : Ran X ( S ) . . = { A ⊂ Hom( S , X ) , a finite, non-empty set } . Let A = { x 1 , . . . , x d | x i : S → X } be an S -point of Ran X . For each x i , let Γ x i = { ( s , x i ( s )) ∈ S × X } be its graph, and define d ⋃︂ Γ A . Γ x i ⊂ S × X , . = i =1 a closed subscheme with the reduced scheme structure.
The Ran space The Ran space is not representable by a scheme, but it is a pseudo-indscheme: I ∈ fSet op X I . Ran X = colim Here the colimit is taken in PreStk, over the closed diagonal embeddings ∆( 𝛽 ) : X J ˓ → X I induced by surjections of finite sets 𝛽 : I ։ J .
Main definition: ℋ ilb Ran X Define the prestack ilb Ran X : Sch op → Set ⊂ ∞ -Grpd ℋ S ↦→ ℋ ilb Ran X ( S ) by setting ℋ ilb Ran X ( S ) to be the set { ( A , 𝜊 ) ∈ (Ran X × Hilb X )( S ) | Supp( 𝜊 ) ⊂ Γ A ⊂ S × X } . Note: This is a set-theoretic condition. Notation: We have natural projection maps f : ℋ ilb Ran X → Ran X , 𝜍 : ℋ ilb Ran X → Hilb X .
ℋ ilb Ran X as a pseudo-indscheme For a finite set I , we define ilb X I : Sch op → Grpd ℋ ilb X I ( S ) ⊂ ( X I × Hilb X )( S ) to be by setting ℋ {︁ }︁ (( x i ) i ∈ I , 𝜊 ) | ( { x i } i ∈ i , 𝜊 ) ∈ ℋ ilb Ran X ( S ) . For 𝛽 : I ։ J , we have natural maps ℋ ilb X J → ℋ ilb X I , defined by (( x j ) j ∈ J , 𝜊 ) ↦→ (∆( 𝛽 )( x j ) , 𝜊 ). Then ℋ I ∈ fSet op ℋ ilb Ran X = colim ilb X I .
Factorisation Consider ( ℋ ilb Ran X ) disj = { ( A = A 1 ⊔ A 2 , 𝜊 ) ∈ ℋ ilb Ran X } . . = 𝜊 ∩ ̂︁ Suppose that in fact Γ A 1 ∩ Γ A 2 = ∅ , so that if we set 𝜊 i . Γ A i , we see that 1 𝜊 = 𝜊 1 ⊔ 𝜊 2 2 ( A i , 𝜊 i ) ∈ ℋ ilb Ran X for i = 1 , 2. Proposition ( ℋ ilb Ran X ) disj ≃ ( ℋ ilb Ran X × ℋ ilb Ran X ) disj .
Factorisation In particular, when A = { x 1 } ⊔ { x 2 } , we can express this formally as follows: j . = X 2 ∖ ∆( X ) ∙ Set U . → X 2 . − − − − ˓ ∙ Then the proposition specialises to the statement that there exists a canonical isomorphism c : ℋ ilb X 2 × X 2 U − ∼ → ( ℋ ilb X × ℋ ilb X ) × X × X U . We have similar isomorphisms c ( 𝛽 ) associated to any surjection of finite sets I ։ J . These are called factorisation isomorphisms.
Factorisation Theorem f : ℋ ilb Ran X → Ran X defines a factorisation space on X. If X is proper, f is an ind-proper morphism.
Linearisation of ℋ ilb Ran X Set-up: Let 𝜇 I ∈ ( ℋ ilb X I ) be a family of (complexes of) -modules compatible with the factorisation structure. {︁ }︁ . = ( f I ) ! 𝜇 I ∈ ( X I ) X I . Then the family defines a factorisation algebra on X . More precisely: For every 𝛽 : I = ⨆︁ j ∈ J I j ։ J , we have isomorphisms 1 v ( 𝛽 ) : ∆( 𝛽 ) ! X I − ∼ → X J ⇒ { X I } give an object “colim X I ” of (Ran X ), which we’ll denote by f ! 𝜇 . → j ( 𝛽 ) ∗ (︁ )︁ 2 c ( 𝛽 ) : j ( 𝛽 ) ∗ ( X I ) ∼ − ⊠ j ∈ J X Ij
Linearisation of ℋ ilb Ran X Definition Set ℋ X I . . = ( f I ) ! 𝜕 H ilb XI . This gives a factorisation algebra ℋ Ran X = f ! 𝜕 H ilb Ran X . Goal for the rest of the talk: study this factorisation algebra.
Section 2 Chiral algebras
Chiral algebras A chiral algebra on X is a -module X on X equipped with a Lie bracket 𝜈 A : j ∗ j ∗ ( X ⊠ X ) → ∆ ! X ∈ ( X × X ) .
Factorisation algebras and chiral algebras Theorem (Beilinson–Drinfeld, Francis–Gaitsgory) We have an equivalence of categories {︃ factorisation algebras }︃ {︃ chiral algebras }︃ − ∼ → . on X on X
Idea of the proof Let { X I } be a factorisation algebra. j ∗ j ∗ ( X ⊠ X ) ∼ j ∗ j ∗ ( X 2 ) ∆ ! ∆ ! X 2 X 2 ∼ ∆ ! X
Idea of the proof Let { X I } be a factorisation algebra. j ∗ j ∗ ( X ⊠ X ) ∼ j ∗ j ∗ ( X 2 ) ∆ ! ∆ ! X 2 X 2 ∼ ∆ ! X This defines 𝜈 A : j ∗ j ∗ ( X ⊠ X ) → ∆ ! X . To check the Jacobi identity, we use the factorisation isomorphisms for I = { 1 , 2 , 3 } .
Aside: chiral algebras and vertex algebras Let ( V , Y ( · , z ) , | 0 ⟩ ) be a quasi-conformal vertex algebra, and let C be a smooth curve. We can use this data to construct a chiral algebra ( 𝒲 C , 𝜈 ) on C . This procedure works for any smooth curve C , and gives a compatible family of chiral algebras. Together, all of these chiral algebras form a universal chiral algebra of dimension 1.
Lie ⋆ algebras A Lie ⋆ algebra on X is a -module ℒ on X equipped with a Lie bracket ℒ ⊠ ℒ → ∆ ! ℒ . Example: we have a canonical embedding X ⊠ X → j ∗ j ∗ ( X ⊠ X ) . So every chiral algebra X is a Lie ⋆ algebra.
Universal chiral enveloping algebras The resulting forgetful functor F : { chiral algebras } → { Lie ⋆ algebras } has a left adjoint U ch : { Lie ⋆ algebras } → { chiral algebras } . U ch ( ℒ ) is the universal chiral envelope of ℒ . 1 U ch ( ℒ ) has a natural filtration, and there is a version of the PBW theorem. 2 U ch ( ℒ ) has a structure of chiral Hopf algebra.
Commutative chiral algebras A chiral algebra X is commutative if the underlying Lie ⋆ bracket is zero. Translation into factorisation language: j ∗ j ∗ ( X ⊠ X ) ∼ j ∗ j ∗ ( X 2 ) ∆ ! ∆ ! X 2 X 2 ∼ ∆ ! X
Commutative chiral algebras A chiral algebra X is commutative if the underlying Lie ⋆ bracket is zero. Translation into factorisation language: j ∗ j ∗ ( X ⊠ X ) X ⊠ X ∼ j ∗ j ∗ ( X 2 ) ∆ ! ∆ ! X 2 X 2 ∼ 0 ∆ ! X
Commutative chiral algebras A chiral algebra X is commutative if the underlying Lie ⋆ bracket is zero. Translation into factorisation language: j ∗ j ∗ ( X ⊠ X ) X ⊠ X ∼ j ∗ j ∗ ( X 2 ) ∆ ! ∆ ! X 2 X 2 ∼ ∆ ! X
Commutative factorisation algebras A factorisation algebra { X I } is commutative if every factorisation isomorphism c ( 𝛽 ) − 1 : j ∗ (︁ )︁ → j ∗ X I ∼ ⊠ j ∈ J X Ij − extends to a map of -modules on all of X I : ⊠ j ∈ J X Ij → X I . Proposition (Beilinson–Drinfeld) We have equivalences of categories ⎧ ⎫ ⎧ ⎫ {︃ commutative }︃ commuative commutative ⎨ ⎬ ⎨ ⎬ ⎭ ≃ ⎭ ≃ factorisation chiral . X -algebras ⎩ ⎩ algebras algebras
Section 3 Results on ℋ Ran X
Chiral homology Let p Ran X : Ran X → pt. The chiral homology of a factorisation algebra Ran X is defined by ∫︂ Ran X . . = p RanX , ! Ran X . It is a derived formulation of the space of conformal blocks of a vertex algebra V : ∫︂ H 0 ( 𝒲 Ran X ) = space of conformal blocks of V .
The chiral homology of ℋ Ran X ∫︂ ℋ Ran X . Goal: compute . = p Ran X , ! f ! 𝜕 H ilb Ran X . ℋ ilb Ran X ρ f ∫︂ ⇒ ℋ Ran X ≃ p Hilb X , ! 𝜍 ! 𝜕 H ilb Ran X Hilb X Ran X ≃ p Hilb X , ! 𝜍 ! 𝜍 ! 𝜕 Hilb X . p Hilb X p Ran X pt
The chiral homology of ℋ Ran X Theorem 𝜍 ! : (Hilb X ) → (Hilb Ran X ) is fully faithful, and hence 𝜍 ! ∘ 𝜍 ! → id D (Hilb X ) is an equivalence. Corollary ∫︂ . = H • ℋ Ran X ≃ p Hilb X , ! 𝜕 Hilb X . dR (Hilb X ) .
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