Vertex algebras, chiral algebras, and factorisation algebras


 Ann Bryant
 3 years ago
 Views:
Transcription
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 roadplan
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 twodimensional 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 semisimple Braided monoidal, with duals ( fusion product ) The braiding satisfies a nondegeneracy 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 semisimple Braided monoidal, with duals ( fusion product ) The braiding satisfies a nondegeneracy 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 semisimple Braided monoidal, with duals ( fusion product ) The braiding satisfies a nondegeneracy 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 semisimple Braided monoidal, with duals ( fusion product ) The braiding satisfies a nondegeneracy 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 sheaflike properties (with respect to your favourite topology, e.g. fppf, étale).
53 Stacks A stack is a prestack that has nice sheaflike properties (with respect to your favourite topology, e.g. fppf, étale). Example (The stack of Gbundles on a curve)
54 Stacks A stack is a prestack that has nice sheaflike properties (with respect to your favourite topology, e.g. fppf, étale). Example (The stack of Gbundles 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 sheaflike properties (with respect to your favourite topology, e.g. fppf, étale). Example (The stack of Gbundles on a curve) Let G be a reductive group and X a smooth projective curve. Then Bun G is the stack of principal Gbundles on X.
56 Stacks A stack is a prestack that has nice sheaflike properties (with respect to your favourite topology, e.g. fppf, étale). Example (The stack of Gbundles on a curve) Let G be a reductive group and X a smooth projective curve. Then Bun G is the stack of principal Gbundles on X. As a prestack it is the functor
57 Stacks A stack is a prestack that has nice sheaflike properties (with respect to your favourite topology, e.g. fppf, étale). Example (The stack of Gbundles on a curve) Let G be a reductive group and X a smooth projective curve. Then Bun G is the stack of principal Gbundles on X. As a prestack it is the functor Bun G : Sch op Grpd S {P S X P a principal Gbundle}.
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 Gbundle, 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 Gbundle, 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 Gbundle, 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 Gbundle, 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 nonempty finite sets and surjections.
68 Example: the Ran space Let X be a separated scheme. Let fset be the category of nonempty 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 nonempty 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 nonempty 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 nonempty 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 nonempty 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 nonempty 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 nonempty 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 nonempty sets of maps S X }. In particular, when S = pt, Ran X (pt) = {finite nonempty sets of points in X }.
80 (Pseudo)Indschemes
81 (Pseudo)Indschemes Definition: A pseudoindscheme is a prestack which can be expressed as a colimit (in PreStk) of schemes over closed embeddings.
82 (Pseudo)Indschemes Definition: A pseudoindscheme 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 pseudoindscheme 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 pseudoindscheme 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 Dmodules on schemes
86 Dmodules on schemes Pretend you know what a Dmodule on a scheme X is.
87 Dmodules on schemes Pretend you know what a Dmodule on a scheme X is. Key properties of D(X ), the dg category of Dmodules Objects are complexes of sheaves with some extra structure, like a flat connection on a vector bundle.
88 Dmodules on schemes Pretend you know what a Dmodule on a scheme X is. Key properties of D(X ), the dg category of Dmodules Objects are complexes of sheaves with some extra structure, like a flat connection on a vector bundle. We have an exterior tensor product:
89 Dmodules on schemes Pretend you know what a Dmodule on a scheme X is. Key properties of D(X ), the dg category of Dmodules 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 Dmodules on schemes Pretend you know what a Dmodule on a scheme X is. Key properties of D(X ), the dg category of Dmodules 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 Dmodules on schemes Pretend you know what a Dmodule on a scheme X is. Key properties of D(X ), the dg category of Dmodules 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 Dmodules 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 Dmodules 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 Dmodules 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 Dmodules 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 Dmodules 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 Dmodules on prestacks
98 Dmodules on prestacks If Y : Sch op Grpd is any prestack, we define the dg category D(Y) by right Kan extension.
99 Dmodules 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 Dmodules 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 Dmodules 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 Dmodules 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 Dmodules 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 Dmodules on a prestack is the same as the category of Dmodules on the stackification. x f α y
104 Dmodules on pseudoinschemes
105 Dmodules on pseudoinschemes Luckily, when Y colim I S Z(I ) is a colimit of schemes:
106 Dmodules on pseudoinschemes 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 Dmodules on pseudoinschemes 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 Dmodules on pseudoinschemes 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 Dmodules on pseudoindschemes So we can consider colim I S D(Z(I )) Δ(J)! Δ(I )! D(Z(J)) D(Z(I )) Z(α)!
110 Dmodules on pseudoindschemes 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 Dmodules on pseudoindschemes 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 Dmodules on pseudoindschemes 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 Dmodules on pseudoindschemes 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 (Dmodules on Ran X )
115 Example (Dmodules on Ran X ) For each I fset we must have F X I D(X I ). ( Δ(I )! F )
116 Example (Dmodules 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 (Dmodules 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 (Dmodules 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 pseudoindscheme. The Ran space of X is a pseudoindscheme parametrising finite nonempty subsets of X. So we can describe its category of Dmodules: the objects are compatible families of Dmodules 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 Dmodules 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 Dmodules 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 Dmodules 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 infinitedimensional 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 Dmodule A I D(X I ).
138 Definition: A factorisation algebra over X 1 For every I fset a Dmodule 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 Dmodule 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 Dmodule 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 Dmodule 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 comultiplication map.
141 Definition: A factorisation algebra over X 1 For every I fset a Dmodule 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 Dmodule 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 comultiplication 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 0dimensional subschemes of X of finite length.
144 Example 1: the Hilbert scheme of points The Hilbert scheme of points of X parametrizes 0dimensional 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 settheoretically 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 settheoretically 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 Gbundle 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 fullyfaithful 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 fullyfaithful 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 Gbundles.
Introduction to Chiral Algebras
Introduction to Chiral Algebras Nick Rozenblyum Our goal will be to prove the fact that the algebra End(V ac) is commutative. The proof itself will be very easy  a version of the Eckmann Hilton argument
More informationHomology and Cohomology of Stacks (Lecture 7)
Homology and Cohomology of Stacks (Lecture 7) February 19, 2014 In this course, we will need to discuss the ladic homology and cohomology of algebrogeometric objects of a more general nature than algebraic
More informationNOTES ON GEOMETRIC LANGLANDS: STACKS
NOTES ON GEOMETRIC LANGLANDS: STACKS DENNIS GAITSGORY This paper isn t even a paper. I will try to collect some basic definitions and facts about stacks in the DG setting that will be used in other installments
More informationA padic GEOMETRIC LANGLANDS CORRESPONDENCE FOR GL 1
A padic GEOMETRIC LANGLANDS CORRESPONDENCE FOR GL 1 ALEXANDER G.M. PAULIN Abstract. The (de Rham) geometric Langlands correspondence for GL n asserts that to an irreducible rank n integrable connection
More informationPART II.2. THE!PULLBACK AND BASE CHANGE
PART II.2. THE!PULLBACK AND BASE CHANGE Contents Introduction 1 1. Factorizations of morphisms of DG schemes 2 1.1. Colimits of closed embeddings 2 1.2. The closure 4 1.3. Transitivity of closure 5 2.
More informationIndCoh Seminar: Indcoherent sheaves I
IndCoh Seminar: Indcoherent sheaves I Justin Campbell March 11, 2016 1 Finiteness conditions 1.1 Fix a cocomplete category C (as usual category means category ). This section contains a discussion of
More informationPART IV.2. FORMAL MODULI
PART IV.2. FORMAL MODULI Contents Introduction 1 1. Formal moduli problems 2 1.1. Formal moduli problems over a prestack 2 1.2. Situation over an affine scheme 2 1.3. Formal moduli problems under a prestack
More informationLOCAL VS GLOBAL DEFINITION OF THE FUSION TENSOR PRODUCT
LOCAL VS GLOBAL DEFINITION OF THE FUSION TENSOR PRODUCT DENNIS GAITSGORY 1. Statement of the problem Throughout the talk, by a chiral module we shall understand a chiral Dmodule, unless explicitly stated
More informationarxiv: v6 [math.ag] 29 Nov 2013
DG INDSCHEMES arxiv:1108.1738v6 [math.ag] 29 Nov 2013 DENNIS GAITSGORY AND NICK ROZENBLYUM To Igor Frenkel on the occasion of his 60th birthday Abstract. We develop the notion of indscheme in the context
More informationThree Descriptions of the Cohomology of Bun G (X) (Lecture 4)
Three Descriptions of the Cohomology of Bun G (X) (Lecture 4) February 5, 2014 Let k be an algebraically closed field, let X be a algebraic curve over k (always assumed to be smooth and complete), and
More informationh M (T ). The natural isomorphism η : M h M determines an element U = η 1
MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY 7 2.3. Fine moduli spaces. The ideal situation is when there is a scheme that represents our given moduli functor. Definition 2.15. Let M : Sch Set be a moduli
More informationGalois to Automorphic in Geometric Langlands
Galois to Automorphic in Geometric Langlands Notes by Tony Feng for a talk by TsaoHsien Chen April 5, 2016 1 The classical case, G = GL n 1.1 Setup Let X/F q be a proper, smooth, geometrically irreducible
More informationALGEBRAIC GROUPS JEROEN SIJSLING
ALGEBRAIC GROUPS JEROEN SIJSLING The goal of these notes is to introduce and motivate some notions from the theory of group schemes. For the sake of simplicity, we restrict to algebraic groups (as defined
More information1 Notations and Statement of the Main Results
An introduction to algebraic fundamental groups 1 Notations and Statement of the Main Results Throughout the talk, all schemes are locally Noetherian. All maps are of locally finite type. There two main
More informationThe Affine Grassmannian
1 The Affine Grassmannian Chris Elliott March 7, 2013 1 Introduction The affine Grassmannian is an important object that comes up when one studies moduli spaces of the form Bun G (X), where X is an algebraic
More informationNOTES ON FLAT MORPHISMS AND THE FPQC TOPOLOGY
NOTES ON FLAT MORPHISMS AND THE FPQC TOPOLOGY RUNE HAUGSENG The aim of these notes is to define flat and faithfully flat morphisms and review some of their important properties, and to define the fpqc
More informationGeometry of Conformal Field Theory
Geometry of Conformal Field Theory Yoshitake HASHIMOTO (Tokyo City University) 2010/07/10 (Sat.) AKB Differential Geometry Seminar Based on a joint work with A. Tsuchiya (IPMU) Contents 0. Introduction
More informationCHAPTER I.2. BASICS OF DERIVED ALGEBRAIC GEOMETRY
CHAPTER I.2. BASICS OF DERIVED ALGEBRAIC GEOMETRY Contents Introduction 2 0.1. Why prestacks? 2 0.2. What do we say about prestacks? 3 0.3. What else is done in this Chapter? 5 1. Prestacks 6 1.1. The
More informationOn the geometric Langlands duality
On the geometric Langlands duality Peter Fiebig Emmy Noether Zentrum Universität Erlangen Nürnberg Schwerpunkttagung Bad Honnef April 2010 Outline This lecture will give an overview on the following topics:
More informationWhat are stacks and why should you care?
What are stacks and why should you care? Milan Lopuhaä October 12, 2017 Todays goal is twofold: I want to tell you why you would want to study stacks in the first place, and I want to define what a stack
More informationINTRODUCTION TO PART IV: FORMAL GEOMTETRY
INTRODUCTION TO PART IV: FORMAL GEOMTETRY 1. What is formal geometry? By formal geometry we mean the study of the category, whose objects are PreStk laftdef, and whose morphisms are nilisomorphisms of
More informationSEMINAR NOTES: QUANTIZATION OF HITCHIN S INTEGRABLE SYSTEM AND HECKE EIGENSHEAVES (SEPT. 8, 2009)
SEMINAR NOTES: QUANTIZATION OF HITCHIN S INTEGRABLE SYSTEM AND HECKE EIGENSHEAVES (SEPT. 8, 2009) DENNIS GAITSGORY 1. Hecke eigensheaves The general topic of this seminar can be broadly defined as Geometric
More informationCONFORMAL FIELD THEORIES
CONFORMAL FIELD THEORIES Definition 0.1 (Segal, see for example [Hen]). A full conformal field theory is a symmetric monoidal functor { } 1 dimensional compact oriented smooth manifolds {Hilbert spaces}.
More informationProof of Langlands for GL(2), II
Proof of Langlands for GL(), II Notes by Tony Feng for a talk by Jochen Heinloth April 8, 016 1 Overview Let X/F q be a smooth, projective, geometrically connected curve. The aim is to show that if E is
More informationConstruction of M B, M Dol, M DR
Construction of M B, M Dol, M DR Hendrik Orem Talbot Workshop, Spring 2011 Contents 1 Some Moduli Space Theory 1 1.1 Moduli of Sheaves: Semistability and Boundedness.............. 1 1.2 Geometric Invariant
More informationCONTRACTIBILITY OF THE SPACE OF RATIONAL MAPS
CONTRACTIBILITY OF THE SPACE OF RATIONAL MAPS DENNIS GAITSGORY For Sasha Beilinson Abstract. We define an algebrogeometric model for the space of rational maps from a smooth curve to an algebraic group
More informationInfinite root stacks of logarithmic schemes
Infinite root stacks of logarithmic schemes Angelo Vistoli Scuola Normale Superiore, Pisa Joint work with Mattia Talpo, Max Planck Institute Brown University, May 2, 2014 1 Let X be a smooth projective
More informationALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 9: SCHEMES AND THEIR MODULES.
ALGEBRAIC GEOMETRY COURSE NOTES, LECTURE 9: SCHEMES AND THEIR MODULES. ANDREW SALCH 1. Affine schemes. About notation: I am in the habit of writing f (U) instead of f 1 (U) for the preimage of a subset
More informationWhat is an indcoherent sheaf?
What is an indcoherent sheaf? Harrison Chen March 8, 2018 0.1 Introduction All algebras in this note will be considered over a field k of characteristic zero (an assumption made in [Ga:IC]), so that we
More informationALGEBRAIC GEOMETRY: GLOSSARY AND EXAMPLES
ALGEBRAIC GEOMETRY: GLOSSARY AND EXAMPLES HONGHAO GAO FEBRUARY 7, 2014 Quasicoherent and coherent sheaves Let X Spec k be a scheme. A presheaf over X is a contravariant functor from the category of open
More informationDuality, Residues, Fundamental class
Duality, Residues, Fundamental class Joseph Lipman Purdue University Department of Mathematics lipman@math.purdue.edu May 22, 2011 Joseph Lipman (Purdue University) Duality, Residues, Fundamental class
More informationPART II.1. INDCOHERENT SHEAVES ON SCHEMES
PART II.1. INDCOHERENT SHEAVES ON SCHEMES Contents Introduction 1 1. Indcoherent sheaves on a scheme 2 1.1. Definition of the category 2 1.2. tstructure 3 2. The direct image functor 4 2.1. Direct image
More informationElementary (haha) Aspects of Topos Theory
Elementary (haha) Aspects of Topos Theory Matt Booth June 3, 2016 Contents 1 Sheaves on topological spaces 1 1.1 Presheaves on spaces......................... 1 1.2 Digression on pointless topology..................
More informationMODULI TOPOLOGY. 1. Grothendieck Topology
MODULI TOPOLOG Abstract. Notes from a seminar based on the section 3 of the paper: Picard groups of moduli problems (by Mumford). 1. Grothendieck Topology We can define a topology on any set S provided
More informationConformal blocks for a chiral algebra as quasicoherent sheaf on Bun G.
Conformal blocks for a chiral algebra as quasicoherent sheaf on Bun G. Giorgia Fortuna May 04, 2010 1 Conformal blocks for a chiral algebra. Recall that in Andrei s talk [4], we studied what it means
More informationQUANTIZATION VIA DIFFERENTIAL OPERATORS ON STACKS
QUANTIZATION VIA DIFFERENTIAL OPERATORS ON STACKS SAM RASKIN 1. Differential operators on stacks 1.1. We will define a Dmodule of differential operators on a smooth stack and construct a symbol map when
More informationAlgebraic Geometry
MIT OpenCourseWare http://ocw.mit.edu 18.726 Algebraic Geometry Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 18.726: Algebraic Geometry
More informationINTRODUCTION TO PART V: CATEGORIES OF CORRESPONDENCES
INTRODUCTION TO PART V: CATEGORIES OF CORRESPONDENCES 1. Why correspondences? This part introduces one of the two main innovations in this book the (, 2)category of correspondences as a way to encode
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 24
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 24 RAVI VAKIL CONTENTS 1. Vector bundles and locally free sheaves 1 2. Toward quasicoherent sheaves: the distinguished affine base 5 Quasicoherent and coherent sheaves
More informationConnecting Coinvariants
Connecting Coinvariants In his talk, Sasha taught us how to define the spaces of coinvariants: V 1,..., V n = V 1... V n g S out (0.1) for any V 1,..., V n KL κg and any finite set S P 1. In her talk,
More informationDERIVED CATEGORIES OF STACKS. Contents 1. Introduction 1 2. Conventions, notation, and abuse of language The lisseétale and the flatfppf sites
DERIVED CATEGORIES OF STACKS Contents 1. Introduction 1 2. Conventions, notation, and abuse of language 1 3. The lisseétale and the flatfppf sites 1 4. Derived categories of quasicoherent modules 5
More informationLectures on Galois Theory. Some steps of generalizations
= Introduction Lectures on Galois Theory. Some steps of generalizations Journée Galois UNICAMP 2011bis, ter Ubatuba?=== Content: Introduction I want to present you Galois theory in the more general frame
More informationExercises of the Algebraic Geometry course held by Prof. Ugo Bruzzo. Alex Massarenti
Exercises of the Algebraic Geometry course held by Prof. Ugo Bruzzo Alex Massarenti SISSA, VIA BONOMEA 265, 34136 TRIESTE, ITALY Email address: alex.massarenti@sissa.it These notes collect a series of
More informationThe moduli stack of vector bundles on a curve
The moduli stack of vector bundles on a curve Norbert Hoffmann norbert.hoffmann@fuberlin.de Abstract This expository text tries to explain briefly and not too technically the notions of stack and algebraic
More informationDESCENT THEORY (JOE RABINOFF S EXPOSITION)
DESCENT THEORY (JOE RABINOFF S EXPOSITION) RAVI VAKIL 1. FEBRUARY 21 Background: EGA IV.2. Descent theory = notions that are local in the fpqc topology. (Remark: we aren t assuming finite presentation,
More informationVertex Algebras and Algebraic Curves
Mathematical Surveys and Monographs Volume 88 Vertex Algebras and Algebraic Curves Edward Frenkei David BenZvi American Mathematical Society Contents Preface xi Introduction 1 Chapter 1. Definition of
More informationChern classes à la Grothendieck
Chern classes à la Grothendieck Theo Raedschelders October 16, 2014 Abstract In this note we introduce Chern classes based on Grothendieck s 1958 paper [4]. His approach is completely formal and he deduces
More information0.1 Spec of a monoid
These notes were prepared to accompany the first lecture in a seminar on logarithmic geometry. As we shall see in later lectures, logarithmic geometry offers a natural approach to study semistable schemes.
More informationSUMMER SCHOOL ON DERIVED CATEGORIES NANTES, JUNE 2327, 2014
SUMMER SCHOOL ON DERIVED CATEGORIES NANTES, JUNE 2327, 2014 D. GAITSGORY 1.1. Introduction. 1. Lecture I: the basics 1.1.1. Why derived algebraic geometry? a) Fiber products. b) Deformation theory. c)
More informationPARABOLIC SHEAVES ON LOGARITHMIC SCHEMES
PARABOLIC SHEAVES ON LOGARITHMIC SCHEMES Angelo Vistoli Scuola Normale Superiore Bordeaux, June 23, 2010 Joint work with Niels Borne Université de Lille 1 Let X be an algebraic variety over C, x 0 X. What
More informationThe geometric Satake isomorphism for padic groups
The geometric Satake isomorphism for padic groups Xinwen Zhu Notes by Tony Feng 1 Geometric Satake Let me start by recalling a fundamental theorem in the Geometric Langlands Program, which is the Geometric
More information1 Categorical Background
1 Categorical Background 1.1 Categories and Functors Definition 1.1.1 A category C is given by a class of objects, often denoted by ob C, and for any two objects A, B of C a proper set of morphisms C(A,
More informationFORMAL GLUEING OF MODULE CATEGORIES
FORMAL GLUEING OF MODULE CATEGORIES BHARGAV BHATT Fix a noetherian scheme X, and a closed subscheme Z with complement U. Our goal is to explain a result of Artin that describes how coherent sheaves on
More informationBASIC MODULI THEORY YURI J. F. SULYMA
BASIC MODULI THEORY YURI J. F. SULYMA Slogan 0.1. Groupoids + Sites = Stacks 1. Groupoids Definition 1.1. Let G be a discrete group acting on a set. Let /G be the category with objects the elements of
More informationMODULI STACKS FOR LINEAR CATEGORIES
MODULI STACKS FOR LINEAR CATEGORIES THOMAS POGUNTKE Abstract. We propose a simple definition of the moduli stack of objects in a locally proper Cauchy complete linear category and study its basic properties.
More informationPICARD GROUPS OF MODULI PROBLEMS II
PICARD GROUPS OF MODULI PROBLEMS II DANIEL LI 1. Recap Let s briefly recall what we did last time. I discussed the stack BG m, as classifying line bundles by analyzing the sense in which line bundles may
More informationCharacters in Categorical Representation Theory
Characters in Categorical Representation Theory David BenZvi University of Texas at Austin Symplectic Algebraic eometry and Representation Theory, CIRM, Luminy. July 2012 Overview Describe ongoing joint
More informationLECTURE 1: SOME GENERALITIES; 1 DIMENSIONAL EXAMPLES
LECTURE 1: SOME GENERALITIES; 1 DIMENSIONAL EAMPLES VIVEK SHENDE Historically, sheaves come from topology and analysis; subsequently they have played a fundamental role in algebraic geometry and certain
More informationABSTRACT DIFFERENTIAL GEOMETRY VIA SHEAF THEORY
ABSTRACT DIFFERENTIAL GEOMETRY VIA SHEAF THEORY ARDA H. DEMIRHAN Abstract. We examine the conditions for uniqueness of differentials in the abstract setting of differential geometry. Then we ll come up
More informationRepresentation theory of vertex operator algebras, conformal field theories and tensor categories. 1. Vertex operator algebras (VOAs, chiral algebras)
Representation theory of vertex operator algebras, conformal field theories and tensor categories YiZhi Huang 6/29/20107/2/2010 1. Vertex operator algebras (VOAs, chiral algebras) Symmetry algebras
More informationAn overview of Dmodules: holonomic Dmodules, bfunctions, and V filtrations
An overview of Dmodules: holonomic Dmodules, bfunctions, and V filtrations Mircea Mustaţă University of Michigan Mainz July 9, 2018 Mircea Mustaţă () An overview of Dmodules Mainz July 9, 2018 1 The
More informationLAFFORGUE BACKGROUND SEMINAR PART 3  UNIFORMIZATION OF Bun G
LAFFORGUE BACKGROUND SEMINAR PART 3  UNIFORMIZATION OF Bun G EVAN WARNER 1. More about the moduli stack of Gbundles Recall our setup from before: k is a field 1, X a projective connected smooth curve
More informationMODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY
MODULI PROBLEMS AND GEOMETRIC INVARIANT THEORY VICTORIA HOSKINS Abstract In this course, we study moduli problems in algebraic geometry and the construction of moduli spaces using geometric invariant theory.
More informationAPPENDIX TO [BFN]: MORITA EQUIVALENCE FOR CONVOLUTION CATEGORIES
APPENDIX TO [BFN]: MORITA EQUIVALENCE FOR CONVOLUTION CATEGORIES DAVID BENZVI, JOHN FRANCIS, AND DAVID NADLER Abstract. In this brief postscript to [BFN], we describe a Morita equivalence for derived,
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 24
FOUNDATIONS OF ALGEBRAIC GEOMETR CLASS 24 RAVI VAKIL CONTENTS 1. Normalization, continued 1 2. Sheaf Spec 3 3. Sheaf Proj 4 Last day: Fibers of morphisms. Properties preserved by base change: open immersions,
More information1. Algebraic vector bundles. Affine Varieties
0. Brief overview Cycles and bundles are intrinsic invariants of algebraic varieties Close connections going back to Grothendieck Work with quasiprojective varieties over a field k Affine Varieties 1.
More informationLecture 3: Flat Morphisms
Lecture 3: Flat Morphisms September 29, 2014 1 A crash course on Properties of Schemes For more details on these properties, see [Hartshorne, II, 15]. 1.1 Open and Closed Subschemes If (X, O X ) is a
More informationAFFINE PUSHFORWARD AND SMOOTH PULLBACK FOR PERVERSE SHEAVES
AFFINE PUSHFORWARD AND SMOOTH PULLBACK FOR PERVERSE SHEAVES YEHAO ZHOU Conventions In this lecture note, a variety means a separated algebraic variety over complex numbers, and sheaves are Clinear. 1.
More informationBasic results on Grothendieck Duality
Basic results on Grothendieck Duality Joseph Lipman 1 Purdue University Department of Mathematics lipman@math.purdue.edu http://www.math.purdue.edu/ lipman November 2007 1 Supported in part by NSA Grant
More informationTHE MODULI STACK OF GBUNDLES JONATHAN WANG
THE MODULI STACK OF GBUNDLES JONATHAN WANG Contents 1. Introduction 1 1.1. Acknowledgments 2 1.2. Notation and terminology 2 2. Quotient stacks 3 2.1. Characterizing [Z/G] 4 2.2. Twisting by torsors 7
More informationThe Canonical Sheaf. Stefano Filipazzi. September 14, 2015
The Canonical Sheaf Stefano Filipazzi September 14, 015 These notes are supposed to be a handout for the student seminar in algebraic geometry at the University of Utah. In this seminar, we will go over
More informationTopics in Algebraic Geometry
Topics in Algebraic Geometry Nikitas Nikandros, 3928675, Utrecht University n.nikandros@students.uu.nl March 2, 2016 1 Introduction and motivation In this talk i will give an incomplete and at sometimes
More informationTHE KEEL MORI THEOREM VIA STACKS
THE KEEL MORI THEOREM VIA STACKS BRIAN CONRAD 1. Introduction Let X be an Artin stack (always assumed to have quasicompact and separated diagonal over Spec Z; cf. [2, 1.3]). A coarse moduli space for
More informationALGEBRAIC KTHEORY HANDOUT 5: K 0 OF SCHEMES, THE LOCALIZATION SEQUENCE FOR G 0.
ALGEBRAIC KTHEORY HANDOUT 5: K 0 OF SCHEMES, THE LOCALIZATION SEQUENCE FOR G 0. ANDREW SALCH During the last lecture, we found that it is natural (even just for doing undergraduatelevel complex analysis!)
More informationON COSTELLO S CONSTRUCTION OF THE WITTEN GENUS: L SPACES AND DGMANIFOLDS
ON COSTELLO S CONSTRUCTION OF THE WITTEN GENUS: L SPACES AND DGMANIFOLDS RYAN E GRADY 1. L SPACES An L space is a ringed space with a structure sheaf a sheaf L algebras, where an L algebra is the homotopical
More informationSUMMER COURSE IN MOTIVIC HOMOTOPY THEORY
SUMMER COURSE IN MOTIVIC HOMOTOPY THEORY MARC LEVINE Contents 0. Introduction 1 1. The category of schemes 2 1.1. The spectrum of a commutative ring 2 1.2. Ringed spaces 5 1.3. Schemes 10 1.4. Schemes
More informationDerived Morita theory and Hochschild Homology and Cohomology of DG Categories
Derived Morita theory and Hochschild Homology and Cohomology of DG Categories German Stefanich In this talk we will explore the idea that an algebra A over a field (ring, spectrum) k can be thought of
More informationMotivic integration on Artin nstacks
Motivic integration on Artin nstacks Chetan Balwe Nov 13,2009 1 / 48 Prestacks (This treatment of stacks is due to B. Toën and G. Vezzosi.) Let S be a fixed base scheme. Let (Aff /S) be the category of
More informationHochschild homology and Grothendieck Duality
Hochschild homology and Grothendieck Duality Leovigildo Alonso Tarrío Universidade de Santiago de Compostela Purdue University July, 1, 2009 Leo Alonso (USC.es) Hochschild theory and Grothendieck Duality
More informationHolomorphic symplectic fermions
Holomorphic symplectic fermions Ingo Runkel Hamburg University joint with Alexei Davydov Outline Investigate holomorphic extensions of symplectic fermions via embedding into a holomorphic VOA (existence)
More informationPART III.3. INDCOHERENT SHEAVES ON INDINFSCHEMES
PART III.3. INDCOHERENT SHEAVES ON INDINFSCHEMES Contents Introduction 1 1. Indcoherent sheaves on indschemes 2 1.1. Basic properties 2 1.2. tstructure 3 1.3. Recovering IndCoh from indproper maps
More informationTHE MODULI OF SUBALGEBRAS OF THE FULL MATRIX RING OF DEGREE 3
THE MODULI OF SUBALGEBRAS OF THE FULL MATRIX RING OF DEGREE 3 KAZUNORI NAKAMOTO AND TAKESHI TORII Abstract. There exist 26 equivalence classes of ksubalgebras of M 3 (k) for any algebraically closed field
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 37
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 37 RAVI VAKIL CONTENTS 1. Motivation and game plan 1 2. The affine case: three definitions 2 Welcome back to the third quarter! The theme for this quarter, insofar
More informationNOTES ON ATIYAH S TQFT S
NOTES ON ATIYAH S TQFT S J.P. MAY As an example of categorification, I presented Atiyah s axioms [1] for a topological quantum field theory (TQFT) to undergraduates in the University of Chicago s summer
More informationA QUICK NOTE ON ÉTALE STACKS
A QUICK NOTE ON ÉTALE STACKS DAVID CARCHEDI Abstract. These notes start by closely following a talk I gave at the Higher Structures Along the Lower Rhine workshop in Bonn, in January. I then give a taste
More informationSTACKY HOMOTOPY THEORY
STACKY HOMOTOPY THEORY GABE ANGEINIKNO AND EVA BEMONT 1. A stack by any other name... argely due to the influence of Mike Hopkins and collaborators, stable homotopy theory has become closely tied to moduli
More informationSupercategories. Urs July 5, Odd flows and supercategories 4. 4 Braided monoidal supercategories 7
Supercategories Urs July 5, 2007 ontents 1 Introduction 1 2 Flows on ategories 2 3 Odd flows and supercategories 4 4 Braided monoidal supercategories 7 1 Introduction Motivated by the desire to better
More informationFOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 5
FOUNDATIONS OF ALGEBRAIC GEOMETRY CLASS 5 RAVI VAKIL CONTENTS 1. The inverse image sheaf 1 2. Recovering sheaves from a sheaf on a base 3 3. Toward schemes 5 4. The underlying set of affine schemes 6 Last
More informationNon characteristic finiteness theorems in crystalline cohomology
Non characteristic finiteness theorems in crystalline cohomology 1 Non characteristic finiteness theorems in crystalline cohomology Pierre Berthelot Université de Rennes 1 I.H.É.S., September 23, 2015
More informationVertex operator algebras as a new type of symmetry. Beijing International Center for Mathematical Research Peking Universty
Vertex operator algebras as a new type of symmetry YiZhi Huang Department of Mathematics Rutgers University Beijing International Center for Mathematical Research Peking Universty July 8, 2010 1. What
More informationAlgebraic Cobordism. 2nd GermanChinese Conference on Complex Geometry East China Normal University ShanghaiSeptember 1116, 2006.
Algebraic Cobordism 2nd GermanChinese Conference on Complex Geometry East China Normal University ShanghaiSeptember 1116, 2006 Marc Levine Outline: Describe the setting of oriented cohomology over a
More informationThe Picard Scheme and the Dual Abelian Variety
The Picard Scheme and the Dual Abelian Variety Gabriel DorfsmanHopkins May 3, 2015 Contents 1 Introduction 2 1.1 Representable Functors and their Applications to Moduli Problems............... 2 1.2 Conditions
More informationPROBLEMS FOR VIASM MINICOURSE: GEOMETRY OF MODULI SPACES LAST UPDATED: DEC 25, 2013
PROBLEMS FOR VIASM MINICOURSE: GEOMETRY OF MODULI SPACES LAST UPDATED: DEC 25, 2013 1. Problems on moduli spaces The main text for this material is Harris & Morrison Moduli of curves. (There are djvu files
More informationAn Atlas For Bun r (X)
An Atlas For Bun r (X) As told by Dennis Gaitsgory to Nir Avni October 28, 2009 1 Bun r (X) Is Not Of Finite Type The goal of this lecture is to find a smooth atlas locally of finite type for the stack
More information370 INDEX AND NOTATION
Index and Notation action of a Lie algebra on a commutative! algebra 1.4.9 action of a Lie algebra on a chiral algebra 3.3.3 action of a Lie algebroid on a chiral algebra 4.5.4, twisted 4.5.6 action of
More informationTOPICS IN ALGEBRA COURSE NOTES AUTUMN Contents. Preface Notations and Conventions
TOPICS IN ALGEBRA COURSE NOTES AUTUMN 2003 ROBERT E. KOTTWITZ WRITTEN UP BY BRIAN D. SMITHLING Preface Notations and Conventions Contents ii ii 1. Grothendieck Topologies and Sheaves 1 1.1. A Motivating
More informationDmanifolds and derived differential geometry
Dmanifolds and derived differential geometry Dominic Joyce, Oxford University September 2014 Based on survey paper: arxiv:1206.4207, 44 pages and preliminary version of book which may be downloaded from
More informationDerivations and differentials
Derivations and differentials Johan Commelin April 24, 2012 In the following text all rings are commutative with 1, unless otherwise specified. 1 Modules of derivations Let A be a ring, α : A B an A algebra,
More informationConstructible isocrystals (London 2015)
Constructible isocrystals (London 2015) Bernard Le Stum Université de Rennes 1 March 30, 2015 Contents The geometry behind Overconvergent connections Construtibility A correspondance Valuations (additive
More informationThe Hecke category (part I factorizable structure)
The Hecke category (part I factorizable structure) Ryan Reich 16 February 2010 In this lecture and the next, we will describe the Hecke category, namely, the thing which acts on Dmodules on Bun G and
More information