Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 1/4 Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules Christopher Sadowski Rutgers University Department of Mathematics New Brunswick, NJ http://dimax.rutgers.edu/ sadowski sadowski@dimax.rutgers.edu

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 2/4 Outline Classical Algebras: associative algebras and Lie algebras Nonclassical Algebras: vertex algebras Affine Lie algebras and modules Changing the vertex operator algebra module actions with (H, x) Results

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 3/4 Classical Algebras Associative Algebras and Lie Algebras

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 4/4 Vector Space (over a field F): A vector space, V, over a field F is a set equipped with two operations: vector addition and scalar multiplication such that... Associative (u + v) + w = u + (v + w) for all u, v, w V. Identity There exists 0 V such that v + 0 = v = 0 + v for all v V. Inverses For each v V there exists v V such that v + ( v) = 0 = ( v) + v. Commutative u + v = v + u for all u, v V. Distributive c(u + v) = cu + cv for all u, v V and c F. Distributive (a + b)v = av + bv for all v V and a, b F. Associative (ab)v = a(bv) for all v V and a, b F. Identity 1v = v for all v V. Examples: R 3 over R, C 3 over C, or R[x] over R.

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 5/4 Algebras (over F): An Algebra, A, is a vector space (over some field F) equipped with a multiplication map: m : A A A such that m is bilinear. Instead of writing m(u,v), we usually use juxtaposition to denote multiplication: m(u,v) = uv bilinear means: for all u,v,w A and c F... Linear on the Left (u + v)w = uw + vw and (cu)v = c(uv) Linear on the Right u(v +w) = uv +uw and u(cv) = c(uv)

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 6/4 Examples: Real Matrix Algebras Let A = R n n be all n n matrices with real entries. 1 1 2 0 = 3 1 0 1 1 1 1 1 Polynomial Algebras Let A = R[x, y] be all polynomials in two indeterminants (x and y) with real coefficients. (3x 2 + xy + 2)(2y x) = 6x 2 y 3x 3 + 2xy 2 x 2 y + 4y 2 2x Cross Product Algebra u v. Let A = R 3 here we multiply vectors by taking their cross product: i j k 1,2,3 1,0, 1 = det 1 2 3 = 2,4, 2 1 0 1

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 7/4 Special Properties: Let A be an algebra (over some field F). Associative A is Associative if u(vw) = (uv)w for all u, v, w A. Unital A is Unital or an algebra with identity if there exists some 1 A such that 1v = v1 = v for all v A. Commutative A is Commutative if uv = vu for all u, v A. Examples: A = R[x, y] is a commutative associative unital algebra. Its identity is the polynomial 1. A = R n n is an associative unital algebra, but it is not commutative (unless n = 1). Its identity is the identity matrix I n. A = R 3 equipped with the cross product is a non-associative non-commutative algebra and has no identity. So what kind of algebra is this?

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 8/4 Lie Algebras Let L be an algebra (over some field F). Instead of using juxtaposition, let s denote multiplication with a bracket: m(u, v) = [u, v]. L is called a Lie Algebra if the following axioms hold: Skew-Commutative [v, v] = 0 for all v L. Jacobi Identity [[u, v], w] + [[v, w], u] + [[w, u], v] = 0 for all u, v, w L. Examples: A = R 3 equipped with the cross product is a Lie algebra. Remember that v v = 0 (the cross product of parallel vectors is zero). A tedious calculation shows that the Jacobi identity holds as well. If we give the matrix algebra R n n a different multiplication, called the commutator bracket, defined by [A, B] = AB BA, it becomes a Lie algebra. To remind ourselves that we are using a different multiplication we call this algebra gl(n, R).

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 9/4 Lie Algebras What do the axioms really say? The first axiom, [v,v] = 0, implies the following: 0 = [u+v, u+v] = [u, u+v]+[v, u+v] = [u, u]+[u, v]+[v, u]+[v, v] = [u, v]+[v, u] Therefore, [u, v] = [v, u] (almost commutative!) The Jacobi identity says much more. Using the above property we can re-write the Jocobi identity as follows: [[u,v],w] + [[v,w],u] + [[w,u],v] = 0

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 10/4 Lie Algebras What do the axioms really say? The first axiom, [v,v] = 0, implies the following: 0 = [u+v, u+v] = [u, u+v]+[v, u+v] = [u, u]+[u, v]+[v, u]+[v, v] = [u, v]+[v, u] Therefore, [u, v] = [v, u] (almost commutative!) The Jacobi identity says much more. Using the above property we can re-write the Jocobi identity as follows: [[u,v],w] + [[w,u],v] = [[v,w],u]

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 11/4 Lie Algebras What do the axioms really say? The first axiom, [v,v] = 0, implies the following: 0 = [u+v, u+v] = [u, u+v]+[v, u+v] = [u, u]+[u, v]+[v, u]+[v, v] = [u, v]+[v, u] Therefore, [u, v] = [v, u] (almost commutative!) The Jacobi identity says much more. Using the above property we can re-write the Jocobi identity as follows: [[u,v],w] + [[w,u],v] = [u, [v,w]]

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 12/4 Lie Algebras What do the axioms really say? The first axiom, [v,v] = 0, implies the following: 0 = [u+v, u+v] = [u, u+v]+[v, u+v] = [u, u]+[u, v]+[v, u]+[v, v] = [u, v]+[v, u] Therefore, [u, v] = [v, u] (almost commutative!) The Jacobi identity says much more. Using the above property we can re-write the Jocobi identity as follows: [[u,v],w] + [[w,u],v] = [u, [v,w]]

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 13/4 Lie Algebras What do the axioms really say? The first axiom, [v,v] = 0, implies the following: 0 = [u+v, u+v] = [u, u+v]+[v, u+v] = [u, u]+[u, v]+[v, u]+[v, v] = [u, v]+[v, u] Therefore, [u, v] = [v, u] (almost commutative!) The Jacobi identity says much more. Using the above property we can re-write the Jocobi identity as follows: [[u,v],w] [v, [w,u]] = [u, [v,w]]

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 14/4 Lie Algebras What do the axioms really say? The first axiom, [v,v] = 0, implies the following: 0 = [u+v, u+v] = [u, u+v]+[v, u+v] = [u, u]+[u, v]+[v, u]+[v, v] = [u, v]+[v, u] Therefore, [u, v] = [v, u] (almost commutative!) The Jacobi identity says much more. Using the above property we can re-write the Jocobi identity as follows: [[u,v],w] + [v, [w,u]] = [u, [v,w]]

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 15/4 Lie Algebras What do the axioms really say? The first axiom, [v,v] = 0, implies the following: 0 = [u+v, u+v] = [u, u+v]+[v, u+v] = [u, u]+[u, v]+[v, u]+[v, v] = [u, v]+[v, u] Therefore, [u, v] = [v, u] (almost commutative!) The Jacobi identity says much more. Using the above property we can re-write the Jocobi identity as follows: [[u,v],w] + [v, [u,w]] = [u, [v,w]]

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 16/4 Lie Algebras What do the axioms really say? The Jacobi identity says much more. Using the above property we can re-write the Jacobi identity as follows: [u, [v, w]] = [[u, v], w] + [v, [u, w]] d dt [f(t)g(t)] = d dt [f(t)] g(t) + f(t) d dt [g(t)] Let A be an algebra and : A A. If (uv) = (u)v + u (v) for all u, v A, then we say that is a derivation of A. The Jacobi identity simply says, The multiplication operators of L are derivations.

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 17/4 Non-Classical Algebras: Vertex (Operator) Algebras

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 18/4 Origins of Vertex Operator Algebras: 1970s Vertex operators appear in the study of String Theory. 1980s Mathematicians use vertex operators to study certain representations of affine Lie algebras. 1984 I. Frenkel, J. Lepowsky, and A. Meurman construct V. 1986 R. Borcherds introduces a set of axioms for a notion which he calls a vertex algebra. 1992 R. Borcherds proves the Conway-Norton conjectures.

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 19/4 The Definition of a Vertex Algebra Notation: V [x] polynomials in x with coefficients in V. V [x, x 1 ] Laurent polynomials in x with coefficients in V. V [[x]] power series in x with coefficients in V. V ((x)) lower truncated Laurent series with coefficients in V. V [[x, x 1 ]] Laurent series in x with coefficients in V. WARNING: C[[x, x 1 ]] is not an algebra! Sometimes mutiplication isn t well defined. Example: δ(x) = n Z xn is the formal delta function. Notice that (δ(x)) 2 is undefined.

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 20/4 The Definition of a Vertex Algebra Algebras (over C): Let V be a vector space over C. Vertex Algebras: Let V be a vector space over C.

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 21/4 The Definition of a Vertex Algebra Algebras (over C): Equip V with a bilinear map : V V V (a multiplication map). (u,v) u v Vertex Algebras: Equip V with infinitely many bilinear maps n : V V V (where n Z). (u,v) u n v

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 22/4 The Definition of a Vertex Algebra Algebras (over C): Equip V with a bilinear map : V V V (a multiplication map). (u,v) u v Vertex Algebras: Equip V with a bilinear map Y (,x) : V V V [[x,x 1 ]] (V [[x,x 1 ]] are Laurent series with coefficients in V ). (u,v) Y (u,x)v = u n v x n 1 n Z

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 23/4 The Definition of a Vertex Algebra Algebras (over C): Equip V with a bilinear map : V V V (a multiplication map). (u,v) u v Vertex Algebras: Equip V with a bilinear map Y (,x) : V V V ((x)) (V ((x)) are lower truncated Laurent series with coefficients in V ). (u,v) Y (u,x)v = u n v x n 1 n Z where u n v = 0 for n 0.

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 24/4 The Definition of a Vertex Algebra Algebras (over C): To be a Lie algebra V s multiplication must be skew-symmetric and satisfy the Jacobi identity. (uv)w + (vw)u + (wu)v = 0 Vertex Algebras: V s vertex operators must satisfy the Jacobi identity. ( ) ( ) x 1 0 δ x1 x 2 Y (u, x 1 )Y (v, x 2 )w x 1 0 x δ x2 x 1 Y (v, x 2 )Y (u, x 1 )w 0 x 0 ( ) = x 1 2 δ x1 x 0 Y (Y (u, x 0 )v, x 2 )w x 2

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 25/4 The Definition of a Vertex Algebra Algebras (over C): To be a Lie algebra V s multiplication must be skew-symmetric and satisfy the Jacobi identity. XXXXXXXXXXXXXX (uv)w + (vw)u + (wu)v = 0 u(vw) = (uv)w + v(uw) (the product rule) Vertex Algebras: V s vertex operators must satisfy the Jacobi identity. x 1 0 δ ( x1 x 2 x 0 ) Y (u, x 1 )(Y (v, x 2 )w) = ( ) ( x 1 2 δ x1 x 0 Y (Y (u, x 0 )v, x 2 )w+x 1 0 x δ x2 x 1 2 x 0 ) Y (v, x 2 )(Y (u, x 1 )w)

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 26/4 The Definition of a Vertex Algebra Algebras (over C): To be a unital algebra V must have an identity vector 1. 1u = u1 = u Vertex Algebras: A vertex algebra has a vaccuum vector 1. Y (1,x)u = u and Y (u,x)1 = e xd u In particular, Y (u, 0)1 = u.

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 27/4 The Definition of a Vertex Algebra Algebras (over C): V is a commutative algebra if... uv = vu Vertex Algebras: Vertex operators satisfy a property called locality. (x 1 x 2 ) N Y (u,x 1 )Y (v,x 2 ) = (x 1 x 2 ) N Y (v,x 2 )Y (u,x 1 ) for some N 0

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 28/4 The Definition of a Vertex Algebra Algebras (over C): V is an associative algebra if... (uv)w = u(vw) Vertex Algebras: Vertex operators satisfy a property called weak associativity. (x 1 x 2 ) N Y (Y (u,x 1 )v,x 2 )w = (x 1 +x 2 ) N Y (u,x 1 +x 2 )(Y (v,x 2 )w) for some N 0

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 29/4 Trivial Example: Holomorphic Vertex Algebras Let A be a commutative associative unital algebra (whose identity is 1) and let D : A A be a derivation of A. Define Y (u,x) = e xd u. That is: Y (u,x)v = n=0 u n 1 vx n where u n 1 v = 1 n! Dn (u)v Then, A equipped with this vertex operator map, Y (,x), becomes a vertex algebra with vacuum vector 1. Note: If D = 0, we simply have Y (u,x)v = uv. That is: All commutative associative unital algebras are vertex algebras.

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 30/4 Cartan Matrices Define a matrix C = (a ij ) 1 i,j l.

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 30/4 Cartan Matrices Define a matrix C = (a ij ) 1 i,j l. We call C a Cartan matrix iff:

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 30/4 Cartan Matrices Define a matrix C = (a ij ) 1 i,j l. We call C a Cartan matrix iff: For all i, a ii = 2

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 30/4 Cartan Matrices Define a matrix C = (a ij ) 1 i,j l. We call C a Cartan matrix iff: For all i, a ii = 2 For all i j, a ij 0 For all i,j, a ij = 0 iff a ji = 0 C is a positive definite matrix

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 31/4 Notation: Simple Lie algebras have a nice decomposition. We pick a Cartan subalgebra h g.

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 31/4 Notation: Simple Lie algebras have a nice decomposition. We pick a Cartan subalgebra h g. The dimension of h is l. We call this the rank of g. We can pick a basis Π = {α 1,...,α l } for the dual space h* We call these elements fundamental roots.

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 32/4 Notation: We fix the basis {λ 1,...,λ l } which is dual to Π (i.e. λ i (H j ) = δ i,j ).

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 32/4 Notation: We fix the basis {λ 1,...,λ l } which is dual to Π (i.e. λ i (H j ) = δ i,j ). These λ i s are called fundamental weights. Define the fundamental coweights {H (1),...,H (l) } h to be the basis dual to {α 1,...,α l }.

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 33/4 The Affinization of a Lie Algebra Define ĝ as follows: where c is central and ĝ := g C[t,t 1 ] Cc [a t m,b t n ] = [a,b] t m+n + m a,b δ m+n,0 c for every a,b g and m,n Z. ĝ is the (untwisted) affine Lie algebra associated with g.

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 34/4 From g to ĝ Given λ h and k C, we can define a linear functional (k,λ) ĥ = (h Cc) by the following: For all h h, let (k,λ)(h) = λ(h). On Cc, let (k,λ)(c) = k. Recall that λ i (1 i l) are the fundamental weights of g. For convenience, let λ 0 = 0. Then, define kλ i = (k,λ i ) for 0 i l.

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 35/4 Standard Modules Irreducible Representations for g are determined by their highest weights ( eigenvalue for a special highest weight vector"). If our module is to be finite dimensional, then the highest weight λ = l i=1 m iλ i where m i Z 0. Likewise, the nice (what are called standard or highest weight integrable) irreducible modules for ĝ are determined by their highest weights Λ = l i=0 m iλ i where m i Z 0

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 36/4 The Simple VOA L(kΛ 0 ) Theorem: Let k Z >0. Then, L(kΛ 0 ) = L(k, 0) has the structure of a simple vertex operator algebra. Also, its modules (VOA modules) are exactly the standard modules for ĝ of level k (i.e. c v = kv).

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 37/4 (H, x) and Li s Theorem Let H = l i=1 m ih (i) where m i Z.

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 37/4 (H, x) and Li s Theorem Let H = l i=1 m ih (i) where m i Z. ( ) Define (H,x) = x H(0) H(k) exp k ( x) k, k=1 where H(k) denotes the action of H t k. Theorem [Li]: For any irreducible L(kΛ 0 )-module W, W (H) = (W,Y W ( (H,x),x) is also an irreducible L(kΛ 0 )-module.

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 38/4 A Property of (H, x) It s not hard to see that (H + H,x) = (H,x) (H,x) and (0,x) = Id

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 38/4 A Property of (H, x) It s not hard to see that (H + H,x) = (H,x) (H,x) and (0,x) = Id So, if we know what each H (j) does, we know everything!

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 39/4 Method While the plays nicely with the VOA structure, it does some strange things to the ĝ-module structure.

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 39/4 Method While the plays nicely with the VOA structure, it does some strange things to the ĝ-module structure. To determine which module we ended up with, we searched for the "new" highest weight vectors.

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 40/4 We found the following For A 1, the case of sl 2 (C), we have L (H(1)) (k,n) is isomorphic to L(k,k n).

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 40/4 We found the following For A 1, the case of sl 2 (C), we have L (H(1)) (k,n) is isomorphic to L(k,k n). For A 2, the case of sl 3 (C), we have L (H(1)) (k,aλ 1 + bλ 2 ) is isomorphic to L(k, (k a b)λ 1 + aλ 2 ) and L (H(2)) (k,aλ 1 + bλ 2 ) is isomorphic to L(k,bλ 1 + (k a b)λ 2 ).

Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 41/4 We found the following Fortunately, the cases of the simple Lie algebras G 2, F 4 and E 8 turned out to be trivial, as the coroot and coweight lattices were equal, so, by Li s Theorem, takes each module back to itself. For each simple Lie algebra, we obtained nice Weyl group elements that determine the new highest weight vector our gives us. Unfortunaley, we had to do it case by case for each type, and have yet to find a pattern that will work for any simple Lie algebra.

The End Vertex Operator Algebra Structure of Standard Affine Lie Algebra Modules p. 42/4