Vertex Algebras Associated to Toroidal Algebras

Transcription

1 Vertex Algebras Associated to Toroidal Algebras Jackson Walters June 26, 2017

2 Outline Factorization algebras

3 Outline Factorization algebras Vertex Algebras Associated to Factorization Algebras

4 Outline Factorization algebras Vertex Algebras Associated to Factorization Algebras Toroidal Algebras Goal: Reconstruct Toroidal Vertex Algebras

5 Factorization algebras Factorization algebras are a formalism for describing the algebra of observables in classical/quantum field theories.

6 Factorization algebras Factorization algebras are a formalism for describing the algebra of observables in classical/quantum field theories. A factorization algebra F on a manifold M with values in a category C assigns for each: open U F(U) C inclusion U V, F(U) F(V ) disjoint U1,..., U k V, F(U 1 )... F(U k ) F(V )

7 Factorization algebras Factorization algebras are a formalism for describing the algebra of observables in classical/quantum field theories. A factorization algebra F on a manifold M with values in a category C assigns for each: open U F(U) C inclusion U V, F(U) F(V ) disjoint U1,..., U k V, F(U 1 )... F(U k ) F(V ) F satisfies natural coherence and gluing conditions.

8 Recovering a vertex algebra Suppose F is a factorization algebra on C.

9 Recovering a vertex algebra Suppose F is a factorization algebra on C. If F is translation invariant and carries an S 1 action (in a precise sense, see [Costello-Gwilliam, 2016]), one can recover a vertex algebra.

10 Kac-Moody algebras For g a finite dim. simple Lie algebra, can form affine Kac-Moody algebra ĝ.

11 Kac-Moody algebras For g a finite dim. simple Lie algebra, can form affine Kac-Moody algebra ĝ. ĝ = Lg CK is a central extension of the loop algebra Lg = g C [t ± ].

12 Kac-Moody algebras For g a finite dim. simple Lie algebra, can form affine Kac-Moody algebra ĝ. ĝ = Lg CK is a central extension of the loop algebra Lg = g C [t ± ]. [, K] = 0, i.e. K is central.

13 Kac-Moody algebras For g a finite dim. simple Lie algebra, can form affine Kac-Moody algebra ĝ. ĝ = Lg CK is a central extension of the loop algebra Lg = g C [t ± ]. [, K] = 0, i.e. K is central. [A f (t), B g(t)] = [A, B] f (t)g(t) Res t=0 fdg(a, B)K where (, ) = 1 2h (, ) K.

14 Vertex algebras associated to Kac-Moody algebras There are vertex algebras naturally associated to representations of Kac-Moody algeabras.

15 Vertex algebras associated to Kac-Moody algebras There are vertex algebras naturally associated to representations of Kac-Moody algeabras. The simplest VA is the vacuum module V k (g) = Indĝ Lg + CK C k = U(ĝ) U(Lg+ CK) C k of level k where C k is 1-dim. rep n of Lg + CK ĝ.

16 Vertex algebras associated to Kac-Moody algebras There are vertex algebras naturally associated to representations of Kac-Moody algeabras. The simplest VA is the vacuum module V k (g) = Indĝ Lg + CK C k = U(ĝ) U(Lg+ CK) C k of level k where C k is 1-dim. rep n of Lg + CK ĝ. Lg + = g C [t] acts by zero and K acts by k.

17 Recovering V k (g) OTOH, we can naturally associate a factorization algebra to g via the factorization envelope construction [CG, 2016].

18 Recovering V k (g) OTOH, we can naturally associate a factorization algebra to g via the factorization envelope construction [CG, 2016]. Let κ : g 2 C be a g-invariant pairing.

19 Recovering V k (g) OTOH, we can naturally associate a factorization algebra to g via the factorization envelope construction [CG, 2016]. Let κ : g 2 C be a g-invariant pairing. g κ : U (Ωc 0, (U) g, ) CK is a cosheaf of dg-lie algebras.

20 Recovering V k (g) OTOH, we can naturally associate a factorization algebra to g via the factorization envelope construction [CG, 2016]. Let κ : g 2 C be a g-invariant pairing. g κ : U (Ωc 0, (U) g, ) CK is a cosheaf of dg-lie algebras. Obtain factorization algebra by taking Chevalley-Eilenberg cochains, F κ : U C (g κ (U)) = Sym(Ω 0, c (U) g [1], + d CE ).

21 Recovering V k (g) OTOH, we can naturally associate a factorization algebra to g via the factorization envelope construction [CG, 2016]. Let κ : g 2 C be a g-invariant pairing. g κ : U (Ωc 0, (U) g, ) CK is a cosheaf of dg-lie algebras. Obtain factorization algebra by taking Chevalley-Eilenberg cochains, F κ : U C (g κ (U)) = Sym(Ω 0, c (U) g [1], + d CE ). The vertex algebra recovered from F κ is isomorphic to the vertex algebra V κ (g).

22 Toroidal algebras Toroidal algebras are a certain N-variable generalization of Kac-Moody algebras.

23 Toroidal algebras Toroidal algebras are a certain N-variable generalization of Kac-Moody algebras. MLg = g C [ t ±1 1,..., t± N ] is a multi-loop algebra.

24 Toroidal algebras Toroidal algebras are a certain N-variable generalization of Kac-Moody algebras. MLg = g C [ t 1 ±1,..., ] t± N is a multi-loop algebra. A central extension is given by g = MLg Ω 1 R /dr where R = C [ t 1 ±,..., ] t± N.

25 Toroidal algebras Toroidal algebras are a certain N-variable generalization of Kac-Moody algebras. MLg = g C [ t 1 ±1,..., ] t± N is a multi-loop algebra. A central extension is given by g = MLg Ω 1 R /dr where R = C [ t 1 ±,..., ] t± N. For N = 1, this is just the affine Kac-Moody algebra. For N > 1, the central term is infinite dimensional.

26 Toroidal algebras Toroidal algebras are a certain N-variable generalization of Kac-Moody algebras. MLg = g C [ t 1 ±1,..., ] t± N is a multi-loop algebra. A central extension is given by g = MLg Ω 1 R /dr where R = C [ t 1 ±,..., ] t± N. For N = 1, this is just the affine Kac-Moody algebra. For N > 1, the central term is infinite dimensional. There have been vertex algebras associated to representations of this Lie algebra [Berman-Billig-Szmigielski, 2013].

27 Goal: Recover toroidal vertex algebras via factorization algebras Let Y = C X with X = (C ) N be the trivial torus fibration over C with natural projection π : Y C.

28 Goal: Recover toroidal vertex algebras via factorization algebras Let Y = C X with X = (C ) N be the trivial torus fibration over C with natural projection π : Y C. For U C open, π 1 (U) = U X open in Y.

29 Goal: Recover toroidal vertex algebras via factorization algebras Let Y = C X with X = (C ) N be the trivial torus fibration over C with natural projection π : Y C. For U C open, π 1 (U) = U X open in Y. Ω(U) = Ω 0, (U X, t0 + X ) = smooth forms of type (0, ) which are zero outside K X where K U.

30 Goal: Recover toroidal vertex algebras via factorization algebras Let Y = C X with X = (C ) N be the trivial torus fibration over C with natural projection π : Y C. For U C open, π 1 (U) = U X open in Y. Ω(U) = Ω 0, (U X, t0 + X ) = smooth forms of type (0, ) which are zero outside K X where K U.

31 Goal: Recover toroidal vertex algebras via factorization algebras Let Y = C X with X = (C ) N be the trivial torus fibration over C with natural projection π : Y C. For U C open, π 1 (U) = U X open in Y. Ω(U) = Ω 0, (U X, t0 + X ) = smooth forms of type (0, ) which are zero outside K X where K U. F(U) = C (Ω(U)) defines a factorization algebra on C.

32 Goal: Recover toroidal vertex algebras via factorization algebras Easy to show F is translation invariant carries and carries appropriate S 1 action, so can recover a vertex algebra.

33 Goal: Recover toroidal vertex algebras via factorization algebras Easy to show F is translation invariant carries and carries appropriate S 1 action, so can recover a vertex algebra. Idea: Relate this vertex algebra to vertex algebras associated to toroidal algebras as found in literature.

34 References I K. Costello, O. Gwilliam. Factorization Algebras in Quantum Field Theory S. Berman, Y. Billig, J. Szmigielski. Vertex operator algebras and the representation theory of toroidal algebras