Vertex algebras, chiral algebras, and factorisation algebras

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1 Vertex algebras, chiral algebras, and factorisation algebras Emily Cliff University of Illinois at Urbana Champaign 18 September, 2017

2 Section 1 Vertex algebras, motivation, and road-plan

3 Definition A vertex algebra V = (V, 0, T, Y (, z)) consists of the following data:

4 Definition A vertex algebra V = (V, 0, T, Y (, z)) consists of the following data: The space of states: a graded complex vector space V = i 0 V i.

5 Definition A vertex algebra V = (V, 0, T, Y (, z)) consists of the following data: The space of states: a graded complex vector space V = i 0 V i. The vacuum vector: 0 V 0.

6 Definition A vertex algebra V = (V, 0, T, Y (, z)) consists of the following data: The space of states: a graded complex vector space V = i 0 V i. The vacuum vector: 0 V 0. The translation operator: T : V V a map of degree 1.

7 Definition A vertex algebra V = (V, 0, T, Y (, z)) consists of the following data: The space of states: a graded complex vector space V = i 0 V i. The vacuum vector: 0 V 0. The translation operator: T : V V a map of degree 1. The vertex operators: Y (, z) : V End V [[z, z 1 ]] a linear map such that if we have A V i and write Y (A, z) = n Z A (n) z n 1, then the ( n 1)th coefficient A (n) End V is of degree n + i 1.

8 These data are subject to the following conditions:

9 These data are subject to the following conditions: The vacuum axiom: Y ( 0, z) = id V. Furthermore, for any A V, A (n) 0 = 0 for n 0, and A ( 1) 0 = A.

10 These data are subject to the following conditions: The vacuum axiom: Y ( 0, z) = id V. Furthermore, for any A V, A (n) 0 = 0 for n 0, and A ( 1) 0 = A. The translation axiom: [T, Y (A, z)] = z Y (A, z) A V, T 0 = 0.

11 These data are subject to the following conditions: The vacuum axiom: Y ( 0, z) = id V. Furthermore, for any A V, A (n) 0 = 0 for n 0, and A ( 1) 0 = A. The translation axiom: [T, Y (A, z)] = z Y (A, z) A V, T 0 = 0. The locality axiom: For any A, B V there exists N N such that (z w) N [Y (A, z), Y (B, w)] = 0 End V [[z ±1, w ±1 ]].

12 What?? Vertex algebras were discovered independently by mathematicians and physicists:

13 What?? Vertex algebras were discovered independently by mathematicians and physicists: In physics: Suppose you have a two-dimensional conformal field theory. We are interested in local operators living at different points in a Riemann surface (these are called vertex operators, and are the elements of our algebra). What happens to these operators when two particles collide? operator product expansion

14 In mathematics: Inspired by work of I. Frenkel, R. Borcherds noticed that for any lattice, one can construct a space V acted on by operators corresponding to lattice vectors. In fact, there are operators ( vertex operators ) for each element of V.

15 In mathematics: Inspired by work of I. Frenkel, R. Borcherds noticed that for any lattice, one can construct a space V acted on by operators corresponding to lattice vectors. In fact, there are operators ( vertex operators ) for each element of V. This is a lattice vertex algebra. The simplest example, corresponding to the lattice Z, is called the Heisenberg vertex algebra (or the rank 1 free boson).

16 In mathematics: Inspired by work of I. Frenkel, R. Borcherds noticed that for any lattice, one can construct a space V acted on by operators corresponding to lattice vectors. In fact, there are operators ( vertex operators ) for each element of V. This is a lattice vertex algebra. The simplest example, corresponding to the lattice Z, is called the Heisenberg vertex algebra (or the rank 1 free boson). Borcherds formalized the properties satisfied by these operators to come up with the definition of a vertex algebra.

17 In mathematics: Inspired by work of I. Frenkel, R. Borcherds noticed that for any lattice, one can construct a space V acted on by operators corresponding to lattice vectors. In fact, there are operators ( vertex operators ) for each element of V. This is a lattice vertex algebra. The simplest example, corresponding to the lattice Z, is called the Heisenberg vertex algebra (or the rank 1 free boson). Borcherds formalized the properties satisfied by these operators to come up with the definition of a vertex algebra. Think of a vertex algebra as an algebra with meromorphic multiplication parametrized by the complex plane: V V V ((z)).

18 Why?

19 Why? Many interesting applications of vertex algebras arise from studying their representation theory.

20 Why? Many interesting applications of vertex algebras arise from studying their representation theory. A module/representation for a vertex algebra is a graded vector space M equipped with a map V End M[[z, z 1 ]] subject to a bunch of axioms.

21 Example (The moonshine module)

22 Example (The moonshine module) The vertex algebra associated to the Leech lattice has a particular representation V, whose automorphism group is the Monster group M. [FLM 1988]

23 Example (The moonshine module) The vertex algebra associated to the Leech lattice has a particular representation V, whose automorphism group is the Monster group M. [FLM 1988] This approach allowed Borcherds to prove Conway and Norton s Moonshine Conjecture in 1992.

24 Example (Modular tensor categories)

25 Example (Modular tensor categories) If V is a sufficiently nice vertex algebra ( rational ) its representation category is a modular tensor category [Huang 2005]:

26 Example (Modular tensor categories) If V is a sufficiently nice vertex algebra ( rational ) its representation category is a modular tensor category [Huang 2005]: Finitely semi-simple Braided monoidal, with duals ( fusion product ) The braiding satisfies a non-degeneracy condition...

27 Example (Modular tensor categories) If V is a sufficiently nice vertex algebra ( rational ) its representation category is a modular tensor category [Huang 2005]: Finitely semi-simple Braided monoidal, with duals ( fusion product ) The braiding satisfies a non-degeneracy condition... A modular tensor category is equivalent to a modular functor.

28 Example (Modular tensor categories) If V is a sufficiently nice vertex algebra ( rational ) its representation category is a modular tensor category [Huang 2005]: Finitely semi-simple Braided monoidal, with duals ( fusion product ) The braiding satisfies a non-degeneracy condition... A modular tensor category is equivalent to a modular functor. We can build modular functors out of vertex algebras using spaces of conformal blocks.

29 Example (Modular tensor categories) If V is a sufficiently nice vertex algebra ( rational ) its representation category is a modular tensor category [Huang 2005]: Finitely semi-simple Braided monoidal, with duals ( fusion product ) The braiding satisfies a non-degeneracy condition... A modular tensor category is equivalent to a modular functor. We can build modular functors out of vertex algebras using spaces of conformal blocks. Important example: representations of affine Lie algebras and quantum groups [KL 1993, F 1996]

30 Plan

31 So far we have seen that: Plan

32 Plan So far we have seen that: Vertex algebras are interesting and important, and turn up in different areas of maths and physics having a vaguely geometric flavour.

33 Plan So far we have seen that: Vertex algebras are interesting and important, and turn up in different areas of maths and physics having a vaguely geometric flavour. The definition is not transparently geometric or tractable.

34 Plan So far we have seen that: Vertex algebras are interesting and important, and turn up in different areas of maths and physics having a vaguely geometric flavour. The definition is not transparently geometric or tractable. Luckily for us, Beilinson and Drinfeld reformulated the definition in more geometric language, to give us the notions of a factorisation algebra and a chiral algebra.

35 Plan

36 Plan Step 1 Cover some basics on the geometry of prestacks.

37 Plan Step 1 Cover some basics on the geometry of prestacks. Step 2 Learn the definitions and examples of factorisation algebras, and also factorisation spaces.

38 Plan Step 1 Cover some basics on the geometry of prestacks. Step 2 Learn the definitions and examples of factorisation algebras, and also factorisation spaces. Step 3 Learn about chiral algebras, and how they are related to vertex algebras and to factorisation algebras.

39 Plan Step 1 Cover some basics on the geometry of prestacks. Step 2 Learn the definitions and examples of factorisation algebras, and also factorisation spaces. Step 3 Learn about chiral algebras, and how they are related to vertex algebras and to factorisation algebras. Step 4 Learn about some properties of factorisation algebras.

40 Section 2 Preliminaries on the geometry of prestacks

41 Conventions We ll work over C. By Sch, we mean the category of schemes of finite type over C.

42 Key perspective: Grothendieck s functor of points

43 Key perspective: Grothendieck s functor of points Given a scheme X, we obtain a functor X : Sch op Set S Hom(S, X ).

44 Key perspective: Grothendieck s functor of points Given a scheme X, we obtain a functor X : Sch op Set S Hom(S, X ). We can recover everything we want to know about the scheme X by studying the properties of this functor.

45 Key perspective: Grothendieck s functor of points Given a scheme X, we obtain a functor X : Sch op Set S Hom(S, X ). We can recover everything we want to know about the scheme X by studying the properties of this functor. So instead of studying schemes, we look at all functors Sch op Y Set Grpd -Grpd Y Y

46 Key perspective: Grothendieck s functor of points Given a scheme X, we obtain a functor X : Sch op Set S Hom(S, X ). We can recover everything we want to know about the scheme X by studying the properties of this functor. So instead of studying schemes, we look at all functors Sch op Y Set Grpd -Grpd Y Y

47 Key perspective: Grothendieck s functor of points Given a scheme X, we obtain a functor X : Sch op Set S Hom(S, X ). We can recover everything we want to know about the scheme X by studying the properties of this functor. So instead of studying schemes, we look at all functors Sch op Y Set Grpd -Grpd Y Y

48 Definition: PreStk = Fun(Sch op, -Grpd). Prestacks

49 Prestacks Definition: PreStk = Fun(Sch op, -Grpd). Example (Schemes)

50 Prestacks Definition: PreStk = Fun(Sch op, -Grpd). Example (Schemes) We have the Yoneda embedding Sch PreStk X X.

51 Stacks

52 Stacks A stack is a prestack that has nice sheaf-like properties (with respect to your favourite topology, e.g. fppf, étale).

53 Stacks A stack is a prestack that has nice sheaf-like properties (with respect to your favourite topology, e.g. fppf, étale). Example (The stack of G-bundles on a curve)

54 Stacks A stack is a prestack that has nice sheaf-like properties (with respect to your favourite topology, e.g. fppf, étale). Example (The stack of G-bundles on a curve) Let G be a reductive group and X a smooth projective curve.

55 Stacks A stack is a prestack that has nice sheaf-like properties (with respect to your favourite topology, e.g. fppf, étale). Example (The stack of G-bundles on a curve) Let G be a reductive group and X a smooth projective curve. Then Bun G is the stack of principal G-bundles on X.

56 Stacks A stack is a prestack that has nice sheaf-like properties (with respect to your favourite topology, e.g. fppf, étale). Example (The stack of G-bundles on a curve) Let G be a reductive group and X a smooth projective curve. Then Bun G is the stack of principal G-bundles on X. As a prestack it is the functor

57 Stacks A stack is a prestack that has nice sheaf-like properties (with respect to your favourite topology, e.g. fppf, étale). Example (The stack of G-bundles on a curve) Let G be a reductive group and X a smooth projective curve. Then Bun G is the stack of principal G-bundles on X. As a prestack it is the functor Bun G : Sch op Grpd S {P S X P a principal G-bundle}.

58 Example (continued)

59 Example (continued) Functoriality:

60 Example (continued) Functoriality: Given f : S T, we need to define f * : Bun G (T ) Bun G (S).

61 Example (continued) Functoriality: Given f : S T, we need to define f * : Bun G (T ) Bun G (S). If P T X is a principal G-bundle, then f * (P).= (f id X ) * P:

62 Example (continued) Functoriality: Given f : S T, we need to define f * : Bun G (T ) Bun G (S). If P T X is a principal G-bundle, then f * (P).= (f id X ) * P: f * (P) P S X f id X. T X

63 Example (continued) Functoriality: Given f : S T, we need to define f * : Bun G (T ) Bun G (S). If P T X is a principal G-bundle, then f * (P).= (f id X ) * P: f * (P) P S X f id X. T X Actually we don t care that much about stacks.

64 Example (continued) Functoriality: Given f : S T, we need to define f * : Bun G (T ) Bun G (S). If P T X is a principal G-bundle, then f * (P).= (f id X ) * P: f * (P) P S X f id X. T X Actually we don t care that much about stacks. What we care about (i.e. what we can understand) are indschemes.

65 Example: the Ran space

66 Example: the Ran space Let X be a separated scheme.

67 Example: the Ran space Let X be a separated scheme. Let fset be the category of non-empty finite sets and surjections.

68 Example: the Ran space Let X be a separated scheme. Let fset be the category of non-empty finite sets and surjections. Given I fset, we can form X I = X X X.

69 Example: the Ran space Let X be a separated scheme. Let fset be the category of non-empty finite sets and surjections. Given I fset, we can form X I = X X X. Given α : I J we obtain Δ(α) : X J X I (x j ) (y i ) where y i = x α(i).

70 Example: the Ran space Let X be a separated scheme. Let fset be the category of non-empty finite sets and surjections. Given I fset, we can form X I = X X X. Given α : I J we obtain Δ(α) : X J X I (x j ) (y i ) where y i = x α(i). We consider colim I fset op X I Ran X. Δ(J) Δ(I ) X J Δ(α) X I

71 Example: the Ran space Let X be a separated scheme. Let fset be the category of non-empty finite sets and surjections. Given I fset, we can form X I = X X X. Given α : I J we obtain Δ(α) : X J X I (x j ) (y i ) where y i = x α(i). We consider colim I fset op X I Ran X. Δ(J) Δ(I ) X J Δ(α) X I

72 Example: the Ran space Let X be a separated scheme. Let fset be the category of non-empty finite sets and surjections. Given I fset, we can form X I = X X X. Given α : I J we obtain Δ(α) : X J X I (x j ) (y i ) where y i = x α(i). We consider colim I fset op X I Ran X. Δ(J) Δ(I ) X J Δ(α) X I

73 Remark: The colimit is taken in the category Fun(Sch op, -Grpd).

74 Remark: The colimit is taken in the category Fun(Sch op, -Grpd). RanX (S) = colim fset op X I (S)

75 Remark: The colimit is taken in the category Fun(Sch op, -Grpd). RanX (S) = colim fset op X I (S) = colim fset op Hom(S, X I )

76 Remark: The colimit is taken in the category Fun(Sch op, -Grpd). RanX (S) = colim fset op X I (S) = colim Hom(S, X I ) fset op = colim Hom(S, X )I op fset

77 Remark: The colimit is taken in the category Fun(Sch op, -Grpd). RanX (S) = colim fset op X I (S) = colim Hom(S, X I ) fset op = colim Hom(S, X )I op fset = {finite non-empty sets of maps S X }.

78 Remark: The colimit is taken in the category Fun(Sch op, -Grpd). RanX (S) = colim fset op X I (S) = colim Hom(S, X I ) fset op = colim Hom(S, X )I op fset = {finite non-empty sets of maps S X }. In particular, when S = pt,

79 Remark: The colimit is taken in the category Fun(Sch op, -Grpd). RanX (S) = colim fset op X I (S) = colim Hom(S, X I ) fset op = colim Hom(S, X )I op fset = {finite non-empty sets of maps S X }. In particular, when S = pt, Ran X (pt) = {finite non-empty sets of points in X }.

80 (Pseudo-)Indschemes

81 (Pseudo-)Indschemes Definition: A pseudo-indscheme is a prestack which can be expressed as a colimit (in PreStk) of schemes over closed embeddings.

82 (Pseudo-)Indschemes Definition: A pseudo-indscheme is a prestack which can be expressed as a colimit (in PreStk) of schemes over closed embeddings. It is an indscheme if it is a filtered colimit.

83 (Pseudo-)Indschemes Definition: A pseudo-indscheme is a prestack which can be expressed as a colimit (in PreStk) of schemes over closed embeddings. It is an indscheme if it is a filtered colimit. We like pseudo indschemes because it is easier for us to do geometry with them than with arbitrary prestacks.

84 (Pseudo-)Indschemes Definition: A pseudo-indscheme is a prestack which can be expressed as a colimit (in PreStk) of schemes over closed embeddings. It is an indscheme if it is a filtered colimit. We like pseudo indschemes because it is easier for us to do geometry with them than with arbitrary prestacks. Study their (derived/dg) categories of sheaves; Study their cohomology, other invariants.

85 D-modules on schemes

86 D-modules on schemes Pretend you know what a D-module on a scheme X is.

87 D-modules on schemes Pretend you know what a D-module on a scheme X is. Key properties of D(X ), the dg category of D-modules Objects are complexes of sheaves with some extra structure, like a flat connection on a vector bundle.

88 D-modules on schemes Pretend you know what a D-module on a scheme X is. Key properties of D(X ), the dg category of D-modules Objects are complexes of sheaves with some extra structure, like a flat connection on a vector bundle. We have an exterior tensor product:

89 D-modules on schemes Pretend you know what a D-module on a scheme X is. Key properties of D(X ), the dg category of D-modules Objects are complexes of sheaves with some extra structure, like a flat connection on a vector bundle. We have an exterior tensor product: (F D(X ), G D(Y )) F G D(X Y ).

90 D-modules on schemes Pretend you know what a D-module on a scheme X is. Key properties of D(X ), the dg category of D-modules Objects are complexes of sheaves with some extra structure, like a flat connection on a vector bundle. We have an exterior tensor product: (F D(X ), G D(Y )) F G D(X Y ). Given f : X Y we have f! : D(Y ) D(X )

91 D-modules on schemes Pretend you know what a D-module on a scheme X is. Key properties of D(X ), the dg category of D-modules Objects are complexes of sheaves with some extra structure, like a flat connection on a vector bundle. We have an exterior tensor product: (F D(X ), G D(Y )) F G D(X Y ). Given f : X Y we have such that (f g)! g! f! etc. f! : D(Y ) D(X )

92 Key properties of D(X ), the dg category of D-modules If f is an open embedding, then f! has a right adjoint f * (and we sometimes write f * = f! ).

93 Key properties of D(X ), the dg category of D-modules If f is an open embedding, then f! has a right adjoint f * (and we sometimes write f * = f! ). If f is proper, then f! has a left adjoint f!.

94 Key properties of D(X ), the dg category of D-modules If f is an open embedding, then f! has a right adjoint f * (and we sometimes write f * = f! ). If f is proper, then f! has a left adjoint f!. If we have X = Z U with i : Z X closed and j : U X open, then

95 Key properties of D(X ), the dg category of D-modules If f is an open embedding, then f! has a right adjoint f * (and we sometimes write f * = f! ). If f is proper, then f! has a left adjoint f!. If we have X = Z U with i : Z X closed and j : U X open, then j * i! = 0, and

96 Key properties of D(X ), the dg category of D-modules If f is an open embedding, then f! has a right adjoint f * (and we sometimes write f * = f! ). If f is proper, then f! has a left adjoint f!. If we have X = Z U with i : Z X closed and j : U X open, then j * i! = 0, and we have distinguished triangles for each F D(X ) i! i! F F j * j * F +1.

97 D-modules on prestacks

98 D-modules on prestacks If Y : Sch op -Grpd is any prestack, we define the dg category D(Y) by right Kan extension.

99 D-modules on prestacks If Y : Sch op -Grpd is any prestack, we define the dg category D(Y) by right Kan extension. D(Y) lim ( S Sch Aff /Y x! y! ) op D(S) S T D(S) D(T ) Y f! x f α y

100 D-modules on prestacks If Y : Sch op -Grpd is any prestack, we define the dg category D(Y) by right Kan extension. D(Y) lim ( S Sch Aff /Y x! y! ) op D(S) S T D(S) D(T ) Y f! x f α y

101 D-modules on prestacks If Y : Sch op -Grpd is any prestack, we define the dg category D(Y) by right Kan extension. D(Y) lim ( S Sch Aff /Y x! y! ) op D(S) S T D(S) D(T ) Y f! Here the commutativity of the right hand diagram is given by a natural isomorphism y! f! x!. x f α y

102 D-modules on prestacks If Y : Sch op -Grpd is any prestack, we define the dg category D(Y) by right Kan extension. D(Y) lim ( S Sch Aff /Y x! y! ) op D(S) S T D(S) D(T ) Y f! Here the commutativity of the right hand diagram is given by a natural isomorphism y! f! x!. Motivation: descent for sheaves on schemes. x f α y

103 D-modules on prestacks If Y : Sch op -Grpd is any prestack, we define the dg category D(Y) by right Kan extension. D(Y) lim ( S Sch Aff /Y x! y! ) op D(S) S T D(S) D(T ) Y f! Here the commutativity of the right hand diagram is given by a natural isomorphism y! f! x!. Motivation: descent for sheaves on schemes. Remark: The category of D-modules on a prestack is the same as the category of D-modules on the stackification. x f α y

104 D-modules on pseudo-inschemes

105 D-modules on pseudo-inschemes Luckily, when Y colim I S Z(I ) is a colimit of schemes:

106 D-modules on pseudo-inschemes Luckily, when Y colim I S Z(I ) is a colimit of schemes: it s enough to consider the limit only over the objects Z(I ) Sch /Y. D(Y) lim D(Z(I )). I Sop

107 D-modules on pseudo-inschemes Luckily, when Y colim I S Z(I ) is a colimit of schemes: it s enough to consider the limit only over the objects Z(I ) Sch /Y. D(Y) lim D(Z(I )). I Sop Even more luckily, when the diagram defining Y has only closed embeddings Z(α) : Z(J) Z(I ):

108 D-modules on pseudo-inschemes Luckily, when Y colim I S Z(I ) is a colimit of schemes: it s enough to consider the limit only over the objects Z(I ) Sch /Y. D(Y) lim D(Z(I )). I Sop Even more luckily, when the diagram defining Y has only closed embeddings Z(α) : Z(J) Z(I ): all the maps Z(α)! have left adjoints Z(α)!.

109 D-modules on pseudo-indschemes So we can consider colim I S D(Z(I )) Δ(J)! Δ(I )! D(Z(J)) D(Z(I )) Z(α)!

110 D-modules on pseudo-indschemes So we can consider colim I S D(Z(I )) Δ(J)! Δ(I )! D(Z(J)) D(Z(I )) Z(α)! Fact: D(Y) lim S op D! (Z(I )) colim S D! (Z(I )) Δ(I )! Δ(I )! D(Z(I ))

111 D-modules on pseudo-indschemes So we can consider colim I S D(Z(I )) Δ(J)! Δ(I )! D(Z(J)) D(Z(I )) Z(α)! Fact: D(Y) lim S op D! (Z(I )) colim S D! (Z(I )) Δ(I )! Δ(I )! D(Z(I ))

112 D-modules on pseudo-indschemes So we can consider colim I S D(Z(I )) Δ(J)! Δ(I )! D(Z(J)) D(Z(I )) Z(α)! Fact: D(Y) lim S op D! (Z(I )) colim S D! (Z(I )) Δ(I )! Δ(I )! D(Z(I ))

113 D-modules on pseudo-indschemes So we can consider colim I S D(Z(I )) Δ(J)! Δ(I )! D(Z(J)) D(Z(I )) Z(α)! Fact: D(Y) lim S op D! (Z(I )) colim S D! (Z(I )) Δ(I )! Δ(I )! D(Z(I )) and (Δ(I )!, Δ(I )! ) form an adjoint pair.

114 Example (D-modules on Ran X )

115 Example (D-modules on Ran X ) For each I fset we must have F X I D(X I ). ( Δ(I )! F )

116 Example (D-modules on Ran X ) For each I fset we must have F X I D(X I ). ( Δ(I )! F ) For α : I J, we have Δ(α) : X J X I

117 Example (D-modules on Ran X ) For each I fset we must have F X I D(X I ). ( Δ(I )! F ) For α : I J, we have Δ(α) : X J X I, and we require an isomorphism F(α) : F X J Δ(α)! F X I D(X J ).

118 Example (D-modules on Ran X ) For each I fset we must have F X I D(X I ). ( Δ(I )! F ) For α : I J, we have Δ(α) : X J X I, and we require an isomorphism F(α) : F X J Δ(α)! F X I D(X J ). These isomorphisms must be compatible with compositions I J K.

119 Takeaways on prestacks The definition of prestacks is easy. The definition of sheaves on prestacks is hard. But it s a lot easier if the prestack is a pseudo-indscheme. The Ran space of X is a pseudo-indscheme parametrising finite non-empty subsets of X. So we can describe its category of D-modules: the objects are compatible families of D-modules F X I D(X I ).

120 Section 3 Factorisation spaces and factorisation algebras

121 Motivation

122 Motivation Factorisation algebras are going to be special D-modules on Ran X, but before we get into the definition, let s recall what we re trying to capture.

123 Motivation Factorisation algebras are going to be special D-modules on Ran X, but before we get into the definition, let s recall what we re trying to capture. Recall that in conformal field theory, we re interested in local operators living at a collection of points (x 1,... x n ) X n,

124 Motivation Factorisation algebras are going to be special D-modules on Ran X, but before we get into the definition, let s recall what we re trying to capture. Recall that in conformal field theory, we re interested in local operators living at a collection of points (x 1,... x n ) X n, and we want to understand what happens when these points collide that is, when we approach the diagonal Δ X n.

125 Definition: A factorisation space over X

126 Definition: A factorisation space over X 1 For every I fset an indscheme Y X I X I.

127 Definition: A factorisation space over X 1 For every I fset an indscheme Y X I X I. 2 Ran s condition.

128 Definition: A factorisation space over X 1 For every I fset an indscheme Y X I X I. 2 Ran s condition. For every α : I J, we have an embedding X J X I, and we require an isomorphism ν α : Y X J Y X I X J.

129 Definition: A factorisation space over X 1 For every I fset an indscheme Y X I X I. 2 Ran s condition. For every α : I J, we have an embedding X J X I, and we require an isomorphism ν α : Y X J Y X I X J. 3 Factorisation isomorphisms.

130 Definition: A factorisation space over X 1 For every I fset an indscheme Y X I X I. 2 Ran s condition. For every α : I J, we have an embedding X J X I, and we require an isomorphism ν α : Y X J Y X I X J. 3 Factorisation isomorphisms. For example, let j : U X 2 be the complement of the diagonal. We require an isomorphism c : Y 2 U (Y 1 Y 1 ) U.

131 Definition: A factorisation space over X 1 For every I fset an indscheme Y X I X I. 2 Ran s condition. For every α : I J, we have an embedding X J X I, and we require an isomorphism ν α : Y X J Y X I X J. 3 Factorisation isomorphisms. For example, let j : U X 2 be the complement of the diagonal. We require an isomorphism c : Y 2 U (Y 1 Y 1 ) U. We require these isomorphisms to be compatible with composition of diagonal embeddings.

132 Definition: A factorisation space over X 1 For every I fset an indscheme Y X I X I. 2 Ran s condition. For every α : I J, we have an embedding X J X I, and we require an isomorphism ν α : Y X J Y X I X J. 3 Factorisation isomorphisms. For example, let j : U X 2 be the complement of the diagonal. We require an isomorphism c : Y 2 U (Y 1 Y 1 ) U. We require these isomorphisms to be compatible with composition of diagonal embeddings. Remark: This is an infinite-dimensional phenomenon.

133 Factorisation isomorphisms More generally, for any α : I J, we have I = j J I j.

134 Factorisation isomorphisms More generally, for any α : I J, we have I = j J I j. We can consider the open embedding j(α) : U(α) X I, where U(α) = { x I X I x i1 x i 2 unless α(i 1 ) = α(i 2 ) }.

135 Factorisation isomorphisms More generally, for any α : I J, we have I = j J I j. We can consider the open embedding j(α) : U(α) X I, where U(α) = { x I X I x i1 x i 2 unless α(i 1 ) = α(i 2 ) }. Then we require an isomorphism c α : j(α) * Y X I j(α) * j J Y X I j.

136 Definition: A factorisation algebra over X

137 Definition: A factorisation algebra over X 1 For every I fset a D-module A I D(X I ).

138 Definition: A factorisation algebra over X 1 For every I fset a D-module A I D(X I ). 2 Ran s condition. ν α : A J Δ(α)! A I.

139 Definition: A factorisation algebra over X 1 For every I fset a D-module A I D(X I ). 2 Ran s condition. ν α : A J Δ(α)! A I. 3 Factorisation isomorphisms. c α : j(α) * (A I ) j(α) * ( j J A Ij ).

140 Definition: A factorisation algebra over X 1 For every I fset a D-module A I D(X I ). 2 Ran s condition. ν α : A J Δ(α)! A I. 3 Factorisation isomorphisms. c α : j(α) * (A I ) j(α) * ( j J A Ij ). Remark The data of (1) and (2) is a D-module A on Ran X. By adjunction, c α corresponds to a map A I j(α) * j(α) * ( ) j J A Ij, which we think of as a co-multiplication map.

141 Definition: A factorisation algebra over X 1 For every I fset a D-module A I D(X I ). 2 Ran s condition. ν α : A J Δ(α)! A I. 3 Factorisation isomorphisms. c α : j(α) * (A I ) j(α) * ( j J A Ij ). Remark The data of (1) and (2) is a D-module A on Ran X. By adjunction, c α corresponds to a map A I j(α) * j(α) * ( j J A Ij ), which we think of as a co-multiplication map. The data of (3) gives us a coalgebra structure on A in D(RanX ) (equipped with the chiral monoidal structure ).

142 Example 1: the Hilbert scheme of points

143 Example 1: the Hilbert scheme of points The Hilbert scheme of points of X parametrizes 0-dimensional subschemes of X of finite length.

144 Example 1: the Hilbert scheme of points The Hilbert scheme of points of X parametrizes 0-dimensional subschemes of X of finite length. For example, for X = A 2 = Spec C[x, y], some points (of length 2) are ξ 1 = Spec C[x, y]/(x, y 2 ) ξ 2 = Spec C[x, y]/(x 2, y) ξ 3 = Spec C[x, y]/(x, y(y 1)).

145 Example 1: the Hilbert scheme of points As a prestack, Hilb X : Sch op Set { S ξ S X closed, flat over S with zerodimensional fibres of finite length }.

146 Example 1: the Hilbert scheme of points Define Hilb X I to be the space parametrising pairs (x I, ξ), where

147 Example 1: the Hilbert scheme of points Define Hilb X I to be the space parametrising pairs (x I, ξ), where x I = (x 1,..., x n ) X I

148 Example 1: the Hilbert scheme of points Define Hilb X I to be the space parametrising pairs (x I, ξ), where x I = (x 1,..., x n ) X I ξ Hilb X is supported set-theoretically on the set {x 1,... x n }.

149 Example 1: the Hilbert scheme of points Define Hilb X I to be the space parametrising pairs (x I, ξ), where x I = (x 1,..., x n ) X I ξ Hilb X is supported set-theoretically on the set {x 1,... x n }. As a prestack, Hilb X I : Sch op Set S (x I, ξ) x I : S X I ; ξ Hilb X (S); Supp(ξ) i I Γ x i.

150 Example 1: the Hilbert scheme of points Theorem (C.) This is a factorisation space over X.

151 Example 1: the Hilbert scheme of points Theorem (C.) This is a factorisation space over X. Sketch of proof:

152 Example 1: the Hilbert scheme of points Theorem (C.) This is a factorisation space over X. Sketch of proof: Indschematic.

153 Example 1: the Hilbert scheme of points Theorem (C.) This is a factorisation space over X. Sketch of proof: Indschematic. Over the diagonal: ((x 1, x 1 ), ξ) ((x 1 ), ξ) Hilb X 1.

154 Example 1: the Hilbert scheme of points Theorem (C.) This is a factorisation space over X. Sketch of proof: Indschematic. Over the diagonal: ((x 1, x 1 ), ξ) ((x 1 ), ξ) Hilb X 1. Away from the diagonal: ((x 1, x 2 ), ξ) ( ((x 1 ), ξ {x1 }), ((x 2 ), ξ {x2 }) ) Hilb X 1 Hilb X 1.

155 Example 2: the Beilinson Drinfeld Grassmannian (1991)

156 Example 2: the Beilinson Drinfeld Grassmannian (1991) Let X be a curve and let G be a reductive group.

157 Example 2: the Beilinson Drinfeld Grassmannian (1991) Let X be a curve and let G be a reductive group. Define Gr G,X I where to be the space parametrising triples (x I, P, σ),

158 Example 2: the Beilinson Drinfeld Grassmannian (1991) Let X be a curve and let G be a reductive group. Define Gr G,X I where to be the space parametrising triples (x I, P, σ), x I = (x 1,..., x n ) X n P X is a principal G-bundle over X σ : X {x 1,..., x n } P is a section/trivialisation

159 Example 2: the Beilinson Drinfeld Grassmannian (1991) As a prestack Gr G,X I : S (x I, P, σ) x I : S X I ; P S X Bun G (S); σ : S X ( i I Γ x i ) P a section.

160 Example 2: the Beilinson Drinfeld Grassmannian (1991) As a prestack Gr G,X I : S (x I, P, σ) x I : S X I ; P S X Bun G (S); σ : S X ( i I Γ x i ) P a section. Theorem (BD) This is a factorisation space.

161 Application 1: Geometric Langlands (Gaitsgory 2012)

162 Application 1: Geometric Langlands (Gaitsgory 2012) The obvious maps Gr G,X n Bun G,X give rise to a fully-faithful embedding of categories D(Bun G,X ) D(Gr G,Ran X ).

163 Application 1: Geometric Langlands (Gaitsgory 2012) The obvious maps Gr G,X n Bun G,X give rise to a fully-faithful embedding of categories D(Bun G,X ) D(Gr G,Ran X ). This is a big deal in Geometric Langlands.

164 Application 2: Representation theory of affine Lie algebras

165 Application 2: Representation theory of affine Lie algebras Linearising with respect to line bundles on Gr G,X n gives rise to factorisation algebras associated to affine Lie algebras.

166 Application 2: Representation theory of affine Lie algebras Linearising with respect to line bundles on Gr G,X n gives rise to factorisation algebras associated to affine Lie algebras. One expects to recover the representations of these affine Lie algebras (in particular, integrable representations of a fixed level) geometrically.

167 Application 2: Representation theory of affine Lie algebras Linearising with respect to line bundles on Gr G,X n gives rise to factorisation algebras associated to affine Lie algebras. One expects to recover the representations of these affine Lie algebras (in particular, integrable representations of a fixed level) geometrically. Introduce the notion of a module over a factorisation space; linearising gives rise to modules of the corresponding factorisation algebras.

168 Application 2: Representation theory of affine Lie algebras Linearising with respect to line bundles on Gr G,X n gives rise to factorisation algebras associated to affine Lie algebras. One expects to recover the representations of these affine Lie algebras (in particular, integrable representations of a fixed level) geometrically. Introduce the notion of a module over a factorisation space; linearising gives rise to modules of the corresponding factorisation algebras. Examples of modules over the Grassmannian can be constructed using moduli spaces of parabolic G-bundles.

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