Vertex algebras, chiral algebras, and factorisation algebras

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

Section 1 Vertex algebras, motivation, and road-plan

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

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.

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.

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.

These data are subject to the following conditions:

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.

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 ]].

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

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

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.

Why?

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

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.

Example (The moonshine module)

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]

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.

Example (Modular tensor categories)

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

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.

Plan

So far we have seen that: Plan

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.

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.

Plan

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

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.

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.

Section 2 Preliminaries on the geometry of prestacks

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

Key perspective: Grothendieck s functor of points

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

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.

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

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

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

Stacks

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

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)

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.

Example (continued)

Example (continued) Functoriality:

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

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:

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

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.

Example: the Ran space

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

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

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.

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).

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

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

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 )

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

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 }.

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,

(Pseudo-)Indschemes

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

(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.

(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.

D-modules on schemes

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

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:

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! ).

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!.

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

D-modules on prestacks

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

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

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

D-modules on pseudo-inschemes

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

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

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 ):

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)!

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 ))

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.

Example (D-modules on Ran X )

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

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

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 ).

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.

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 ).

Section 3 Factorisation spaces and factorisation algebras

Motivation

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.

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,

Definition: A factorisation space over X

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

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

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.

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

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 ) }.

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.

Definition: A factorisation algebra over X

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

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.

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 ).

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.

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 ).

Example 1: the Hilbert scheme of points

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

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)).

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 }.

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

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

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.

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

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

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

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.

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.

Example 2: the Beilinson Drinfeld Grassmannian (1991)

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

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),

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

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.

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.

Application 1: Geometric Langlands (Gaitsgory 2012)

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 ).

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.

Application 2: Representation theory of affine Lie algebras

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.

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.