Vertex Operator Algebras

Transcription

1 1. Axioms for vertex algebras. - Definition of a vertex k-algebra - Category of vertex k-algebras - Modal endomorphisms - Translation covariance - Locality Vertex Operator Algebras Geoffrey Mason 2. Existence theorems - Field-theoretic characterizations - Heisenberg algebra, or free-field theory - Virasoro algebra - Vertex operator algebras - Heisenberg VOA - Virasoro VOA - VOAs associated to affine Lie algebras 3. Characters and representations - 1-point functions - 1-point functions for the Heisenberg VOA - The partition function of a Virasoro VOA - Modules over a VOA - Finiteness theorems - Modular-invariance theorems Further browsing and reading Solutions to Selected Exercises 1 Axioms for vertex algebras Much of the basic theory of vertex operator algebras can be carried out over an arbitrary base ring k. In the first part of these talks I will work at that level for two reasons: (i) it is likely to gain prominence in future and (ii) because it costs nothing to do so. Later I switch to the case k = C, which is the default option. Fix a base ring k, a commutative, associative ring with 1. Let V be a k-module. Elements of V are often called states. We fix a distinguished element 1 V called the vacuum state, and we assume that 1.1 = 1. 1

2 1.1 First definition of a vertex k-algebra V is a vertex k-algebra if it is equipped with a countable infinity of k-bilinear products, indexed by n Z, satisfying u, v, w V : V V V, u v u(n)v, (a) (Field axiom) n 0 (u, v) 0 such that u(n)v = 0 for n n 0. (b) (Creation axiom) u( 1)1 = u, u(n)1 = 0 for n 0. (c) (Jacobi identity) r, s, t Z, ( ) r (u(t + i)v)(r + s i)w = i ( t {u(r ( 1) i) i + t i)v(s + i)w ( 1) t v(s + t i)u(r + i)w }. (a) each sum in (c) is finite, rendering (c) meaningful. Important special cases of the Jacobi identity, obtained by setting t = 0 and r = 0 respectively, are (commutator formula) u(r)v(s)w v(s)u(r)w = ( ) r (u(i)v)(r + s i)w, i (associator formula) (u(t)v)(s)w = ( t {u(t ( 1) i) i i)v(s + i)w ( 1) t v(s + t i)u(i)w }. Two obvious questions. 1. Where on earth does the Jacobi identity come from? 2. How can we construct interesting examples of vertex algebras? We will develop some alternate ways to view the axioms that will help answer these questions. Exercise 1. If (V, 1) is a vertex algebra then V = 0 if, and only if, 1 = Category of vertex k-algebras Objects are vertex algebras (V, 1). Morphisms are vacuum-preserving k-linear α maps (V, 1) (V, 1 ) satisfying α(u(n)v) = α(u)(n)α(v). 2

3 This gives us a category k-v ert of vertex k-algebras. Exercise 2. Suppose that A is a commutative, associative k-algebra with identity element 1. Prove that (A, 1) is a vertex k-algebra with u( 1)v = uv (product in k) and u(n)v = 0 (n 1). Exercise 3. Show that the category of commutative, associative, unital k- algebras is a full subcategory of k-v ert. 1.3 Modal endomorphisms For each integer n and state u V, we obtain a k-linear endomorphism of V u(n) : V V, v u(n)v. u(n) is the n th mode of u. The commutator formula shows that the set of all modes of all states closes on a Lie algebra of operators on V (with respect to the usual bracket [a, b] := ab ba of operators). The vertex operator corresponding to u V is the formal generating function of the modes of u defined by Y (u, z) := n Z u(n)z n 1 End k (V )[[z, z 1 ]]. A field on V is an element a(z) := n a nz n 1 End k (V )[[z, z 1 ]] such that for any v V we have a n (v) = 0 for n 0. The set of fields on V is denoted by F(V ). It is a k-submodule of End k (V )[[z, z 1 ]]. We write a(z)v := n a n (v)z n 1 (v V ). The field axiom for a VA V says that each vertex operator Y (u, z) F(V ). Because the products u(n)v are k-bilinear, we can promote Y to a k-linear map called the state-field correspondence Y : V F(V ), u Y (u, z). Exercise 4. Prove that the state-field correspondence is injective. 1.4 Translation covariance A particularly useful k-endomorphism of V is the following: D : V V, u u( 2)1. 3

4 We claim that D satisfies the following identity (as operators on V ): [D, u(t)] = tu(t 1). To see this, apply the associator formula with w = 1, s = 2 to obtain Du(t)v = (u(t)v)( 2)1 = ( t {u(t ( 1) i) i i)v( 2 + i)1 ( 1) t v( 2 + t i)u(i)1 }. Then use of the creation axiom yields Du(t)v = 1 ( ) t ( 1) i u(t i)v( 2 + i)1 = u(t)dv tu(t 1)v, i i=0 and this is what the desired identity says when applied to a state v V. In terms of vertex operators, this identity is called translation covariance (with respect to D). Explicitly, it says (with obvious notation) [D, Y (u, z)] = z Y (u, z) (formal derivative). Exercise 5. Prove that D1 = 0. Exercise 6. Prove that Y (1, z) = Id V. Exercise 7. Show that (k, 1) is an initial object in the category k-v ert. 1.5 Locality Let t 0 be large enough so that u(t + i)v = 0 for i 0. The Jacobi identity then reads ( t {u(r ( 1) i) i + t i)v(s + i)w ( 1) t v(s + t i)u(r + i)w } = 0. Now notice that (z 1 z 2 ) t Y (u, z 1 )Y (v, z 2 ) = ( ) t ( 1) i z1 t i z2 i u(m)v(n)z1 m 1 z2 n 1 i m,n = { ( ) } t ( 1) i u(r + t i)v(s + i) z1 r 1 z2 s 1. i r,s Similarly, (z 1 z 2 ) t Y (v, z 2 )Y (u, z 1 ) = ( 1) t r,s { ( ) } t ( 1) i v(s + t i)u(r + i) z2 s 1 z1 r 1. i 4

5 Thus the first displayed identity is equivalent to the more provocative formulation (z 1 z 2 ) t [Y (u, z 1 ), Y (v, z 2 )] = 0 (t n 0 (u, v)). This remarkable identity is called locality. Locality is the single most fundamental fact about vertex operators. Two fields a(z), b(z) F(V ) are mutually local if, for some t 0, we have (z 1 z 2 ) t [a(z 1 ), b(z 2 )] = 0. We write this as a(z) t b(z), or simply a(z) b(z). Thus for a VA we have Y (u, z) Y (v, z) (u, v V ). Examples. Y (u, z) 0 Y (v, z) [u(r), v(s)] = 0 Y (u, z) 1 Y (v, z) [u(r + 1), v(s)] [u(r), v(s + 1)] = 0 Y (u, z) 2 Y (v, z) [u(r + 2), v(s)] 2[u(r + 1), v(s + 1)] + [u(r), v(s + 2)] = 0 2 Existence theorems From now on k = C, though this is hardly necessary. A vertex algebra is a vertex C-algebra Field-theoretic characterizations Theorem. Let (V, 1, Y, D) consist of a k-module V, an element 1 V, a k-linear map Y : V F(V ), u Y (u, z) = n u(n)z n 1, and a k-endomorphism D End k (V ). Then V is a vertex algebra if u, v V, (a) Y (1, z) = Id V (b) Y (u, z)1 = u + n 2 u(n)1z n 1 (c) D1 = 0, [D, Y (u, z)] = z Y (u, z) (d) Y (u, z) Y (v, z) We have already shown that the first definition of vertex algebra (a) -(d). The converse amounts to showing that the full Jacobi identity can be established on the basis of locality. Theorem. Let (V, 1, D) consist of a k-module V, an element 1 V, and a k-endomorphism D End k (V ). Suppose given a subset U V and a map U F(V ), u Y (u, z) = n u(n)z n 1 such that (b) -(d) in the previous Theorem are satisfied u, v U. Suppose further that V = span u 1 (n 1 )...u n (n k )1 u i U, n i Z. 5

6 Then there is a unique extension of Y to a state-field correspondence Y : V F(V ) such that (V, Y, 1, D) is a vertex algebra. In this situation, we say that U generates V, written V = U. Exercise 8. Suppose that a(z) = a(n)z n 1 F(V ) satisfies (b), (c). Then Then a(z)1 = 0 a( 1)1 = 0. Exercise 9. Let a(z) be as in Exercise 8 with a( 1)1 = 0. Suppose also that b(z) = b(n)z n 1 F(V ), a(z) b(z) and b(z)1 = n 1 b(n)1z n 1. Then a(z)b( 1)1 = Heisenberg algebra, or free field theory The set-up is as follows: V = k[x 1, x 2,...], 1 = 1 Y (x 1, z) = n 1(nx n z n 1 + xn z n 1 ) D = n 1(n + 1)x n+1 xn Clearly Y (x 1, z) F(V ) and Y (x 1, z)1 = x 1 + O(z), so we have a field that satisfies (b). It is also translation covariant (cf. Exercise 10). These and other calculations are facilitated by setting so that a( n) = nx n, a(n) = xn (n 1), a(0) = 0, a(z) := Y (x 1, z) = n a(n)z n 1, D = n 0 a( 1 n)a(n) We have [ xn, x n ] = [x n, xn ] = Id V, and all other brackets are 0. Thus The calculation [a(m), a(n)] = mδ m+n,0 Id V. [a(r + 2), a(s)] 2[a(r + 1), a(s + 1)] + [a(r), a(s + 2)] = 0 readily follows, showing that a(z) 2 a(z) (cf. Subsection 1.5). Now the last Theorem applies, and we conclude Theorem. (V, 1, Y, D) is a vertex algebra (the rank 1 Heisenberg algebra). 6

7 The states a( n 1 )...a( n k )1 (1 n 1... n k ) are a basis of V and Y (a( n 1 )...a( n k )1, z) =: 1 1 (n 1 1)! n 1 1 z a(z)... (n k 1)! n k 1 z a(z) :, where the normal ordering product : a 1 (z)...a k (z) : for fields is defined as : a(z)b(z) := a 2 (z) a n z n 1 + a n z n 1 b(z) n 0 n<0 : a(z)b(z)c(z) : = : a(z)(: b(z)c(z) :) : Exercise 10. Prove that [D, Y (x 1, z)] = z Y (x 1, z). Exercise 11. Prove that z : a(z)b(z) :=: ( z a(z))b(z) : + : a(z) z b(z) : 2.2 Virasoro algebra This is the Lie algebra V ir := n Z kl n kk ( ) m + 1 [L m, L n ] = (m n)l m+n + 1/2 K, [L m, K] = 0. 3 Exercise 12. Let M be a V ir-module such that K acts on M as a scalar c. Set L(z) := n Z L nz n 2, and assume that L(z) F(M) (identifying L n with its representation on M). Prove: (a) [L 1, L(z)] = z L(z), (b) L(z) 4 L(z). 2.3 Vertex operator algebras A vertex operator algebra (VOA) is a vertex algebra (V, 1, Y, D) with a distinguished state ω V (the conformal vector, or Virasoro vector) such that Y (ω, z) = n L(n)z n 2 satisfies: (i) [L(m), L(n)] = (m n)l(m + n) + 1/2 ( ) m+1 cidv for some c k 3 (ii) D = L( 1) (iii) L(0) is semisimple with an integral spectrum that is bounded below with finite multiplicities. (iii) means that V has a conformal Z-grading by L(0)-eigenvalues: V = n n0 V n V n = {v V L(0)v = nv}, dim V n <, V n = 0 (n < n 0 ) (ii) means that [L( 1), Y (v, z)] = z Y (v, z) (v V ). (i) means that the modes of Y (ω, z) close on (a representation of) the Virasoro 7

8 algebra with the central element K acting as a scalar c, called the central charge of V. The VOA is denoted (V, 1, Y, ω). Implicit in all of this is that the field Y (ω, z) is mutually local with all fields Y (v, z) (v V ), and in particular with itself, as well as being translation covariant with respect to L( 1). The point of Exercise 12 is that these assumptions are consistent. The VOA axioms have several elementary, but important, consequences involving the new ingredients introduced by the Virasoro field. Here are a few of them: 1 V 0 ω = L( 2)1 V 2 v V k v(n) : V m V m+k n 1 A morphism of VOAs is a morphism of VAs that preserves conformal vectors. For each c C we have a category V OA c whose objects are VOAs of central charge c. It is a subcategory of k-vert, but not a full subcategory. Exercise 13. Prove the last displayed statements. 2.4 Heisenberg VOA Theorem. The Heisenberg VA (V, 1, Y, D) (cf. Subsection 2.1) is a VOA of central charge c = 1 if we take ω = 1/2x 2 1. It has conformal grading V = n 0 V n = k1 kx 1 k x 2 1, x 2..., V n = {weighted homogeneous polynomials of degree n; deg x i = i} Refer back to Subsection 2.1 for notation. We are taking ω := 1/2x 2 1 = 1/2a( 1) 2 1, and have to show this state is a Virasoro vector with the relevant integrality properties. L(r)z r 2 := 1/2Y (a( 1) 2 1, z) = 1/2 : a(z) 2 : r = 1/2 { n 1 a(n)z n 1 m L(r) = a(m)z m 1 + m a(m)z m 1 n 0 { 1/2 n Z a(r n)a(n), r 0 n 1 a( n)a(n), r = 0 a(n)z n 1 } In particular L( 1) = n 0 a( 1 n)a(n) = D. That the operators L(r) close on the Virasoro algebra with central charge c = 1 is Exercise 13 below. 8

9 Finally, it is clear that V is the direct sum of the (weighted) homogeneous polynomials of degree n 0. Moreover, L(0) = n 1 nx n xn is the Euler operator, acting on x e x em m as multiplication by ie i, which is the degree of the monomial. The Theorem follows. ( Exercise 14. Prove that [L(m), L(n)] = (m n)l(m + n) + 1 m+1 ) IdV Virasoro VOA In constructing the Heisenberg VOA we followed the following blueprint: (a) construct the Heisenberg Lie algebra H = n Z ka(n) kc (b) construct a representation of H on the k-module V (c) make V into a k-vertex algebra by finding generating mutually local fields (d) look for a conformal vector so that V is a VOA This can be repeated with propitious choices of Lie algebra and representation to yield some basic examples of VOAs. The first example uses the Virasoro algebra V ir in place of H. Vir has a decomposition into Lie subalgebras V ir = V ir V ir 0 V ir + V ir ± = n>0 kl ±n, V ir 0 = kl 0 kc The relevant representation here is an induced Vir-module constructed as follows. Make the 1-dimensional k-module k1 into a V ir + V ir 0 -module by setting L n.1 = 0 (n 0), c.1 = c1 (any c k). Then M(c) = Ind V ir V ir + V ir 0k1 is a Vir-module. By the Poincaré-Birkhoff-Witt theorem it has a natural basis {L n1...l nk 1 n 1... n k 1}. Let L(z) = n L nz n 2. For n n 1 we have L n.l n1...l nk 1 = ([L n, L n1 ] + L n1 L n )L n2...l nk 1 = ((n + n 1 )L n n1 + L n1 L n )L n2...l nk 1, and an easy induction yields L n.l n1...l nk 1 = 0 for n > n n k. Therefore L(z) F(M(c)). L(z) is not creative as it stands since L(z)1 = L 1 1z To cure this malaise, form the V ir quotient module V (c) = M(c)/V ir.l 1 1, 9

10 which has natural basis 1 {L n1...l nk 1 n 1... n k 2}. Let ω = L 2 1 and Y (ω, z) = n L(n)z n 2 where L(n) is the induced action of L n on V (c). Then L(z) F(M(c)) Y (ω, z) F(V (c)), and V (c) has been constructed to guarantee creativity: Y (ω, z)1 = L( 2)1 + O(z) = ω + O(z). Finally, locality Y (ω, z) 4 Y (ω, z) and translation covariance follow from Exercise 12. The existence theorems now show that (V (c), 1, Y, L( 1)) is a VA. Finally, the Virasoro relations easily show that V (c) n is spanned states L n1...l nk 1 with n n k = n. Hence, Theorem. (V (c), 1, Y, ω) is a VOA of central charge c generated by ω. This shows the existence of VOAs of arbitrary central charge. Exercise 15. Show that the Virasoro VOA V (c) is an intitial object in the category V OA c of VOAs of central charge c. 2.6 VOAs associated to affine Lie algebra We follow the blueprint layed out in Subsection 2.5, now taking the underlying Lie algebra to be an affine Lie algebra. Begin with a finite-dimensional Lie algebra L equipped with a nondegenerate symmetric k-bilinear form, on L. The associated affine Lie algebra is with brackets (a, b L) ˆL = L C[t, t 1 ] kc [a t m, b t n ] = [a, b] t m+n + m a, b δ m+n,0 c, [ˆL, c] = 0 Eg., the Heisenberg Lie algebra is the case L = k, a, b = ab. We shall here take L to be a simple Lie algebra and, to be the Killing form κ. The triangular decomposition is ˆL = L tk[t] }{{}} L t {{ 0 kc } L t 1 k[t 1 ] }{{} ˆL + ˆL 0 ˆL There is a derivation on ˆL defined by d(a t n ) := na t n 1, d(c) = 0. The representation is an induced module as before; for l k, let k1 be the 1-dimensional space annihilated by ˆL + L t 0 and with c.1 = l1. The ˆL-module is V (l) = IndˆḼ L + ˆL 0 k1. Unlike the Virasoro case, there is no need to take a quotient of V (l) - it is already the underlying space of a VA that we call the affine algebra VA of level 1 States in V (c) are really cosets of V ir.l 1 1; we routinely abuse notation in this regard. 10

11 l. To justify this one has to check some details, none of them difficult. V (l) is spanned by states {a 1 ( n 1 )...a k ( n k )1 a i L, n 1... n k 1} where for a L, a(n) represents the action of a t n on V (l). Identify a with a( 1)1 and set Y (a, z) = n a(n)z n 1. Using the relations in ˆL, we find that Y (a, z) 2 Y (b, z) (a, b L) and each Y (a, z) is creative and translation covariant with respect to D, the endomorphism of V (l) naturally induced by d. Now the existence theorems apply. There is a natural grading on V (l) by nonnegative integers, V (l) n being spanned by those states a 1 ( n 1 )...a k ( n k )1 with n n k = n. We would like a Virasoro vector whose conformal grading gives exactly these homogeneous spaces. This is possible in most cases (Sugawara construction). Omitting details, we have Theorem. For any l k, (V (l), 1, Y, D) is a VA. It becomes a VOA if we choose ω = 1 2(l + h ) u i ( 1)u i where {u i } is any o.n. basis of a Cartan subalgebra of L, h is the dual Coxeter number of the root system attached to L, and l+h 0. The L(0)-eigenspaces are the spaces V (l) n, and the central charge is dim L/(l + h ). Taking a finite-dimensional abelian Lie algebra H, equipped with nondegenerate symmetric bilinear form,, in place of the simple Lie algebra L in the construction of V (l), we obtain in a completely similar way a VA of level l which, if l 0, is a VOA of central charge dim H. This is the Heisenberg VOA (or free field theory) of central charge dim H. Exercise 16. Give the details of the proof that V (l) is a VA. Exercise 17. Give the detailed construction of Heisenberg VOA of central charge dim H outlined in the previous paragraph. 3 Characters and representations In this final Section we discuss modules over a VOA and their characters (often called correlation functions). There will be less detail compared to the previous two Sections. i 11

12 3.1 1-point functions Let (V, 1, Y, ω) be a VOA with conformal grading V = n n0 V n and central charge c. Because each V n is finite-dimensional we may consider its formal generating function, or partition function, n dim V nq n (q a formal variable). It is fruitful to include an extra factor q c/24 to obtain Z V (q) = q c/24 n n 0 dim V n q n, also called the partition function of V. This can be generalized. Recall from Subsection 2.3 that v V k v(n) : V m V m+k n 1. Taking n = k 1 yields an operator o(v) := v(k 1) called the zero mode 2 of v with the property that o(v) acts on each homogeneous space V m. The linear extension of v o(v) to V yields zero modes for all states in V with the same property. We again take traces to obtain a formal generating function Z V (v, q) = q c/24 n n 0 tr V o(v)q n. Because Y (1, z) = Id V, the zero mode o(1) of the vacuum is also Id V Z V (1, q) reduces to the partition function Z V (q). In this way, we have constructed a map which is a sort of character, often called the 1-point correlation function, Z V : V q c/24 {formal Laurent series in q} v Z V (v, q). A main goal is to understand the properties of this function. They depend in subtle ways on the nature of V itself point functions for the Heisenberg VOA Here we take V to be the Heisenberg VOA of central charge 1. First we look at the partition function. V n consists of homogeneous polynomials of degree n with deg x i = i (cf. Subsection 2.4) so has a basis consisting of monomials v λ := x m x m k k where λ is the unrestricted partition of n with each part 3 i occurring with multiplicity m i. If p(n) is the number of unrestricted partitions of n then Z V (q) = q 1/24 n 0 p(n)q n. and 2 This is a misnomer, since according to our earlier nomenclature v(0) is the zero mode. 3 One usually requires that the parts of a partition occur with positive multiplicity, so this language is slightly nonstandard. 12

13 This can be written more suggestively in the form (Euler) Z V (q) 1 = q 1/24 n=1 (1 q n ). The significance (and justification for including the extra factor!) is that the rhs is the q-expansion for Dedekind s eta-function η, which is a modular form of weight 1/2. This in turn means the following: (i) for τ in the complex upper half-plane H and q = e 2πiτ, Z V (q) 1 defines a holomorphic function η(τ). (ii) With respect to the action of the modular group SL 2 (Z) on H by Möbius transformations ( ) a b τ = aτ + b c d cτ + d, we have η ( ) aτ + b = ɛ(a, b, c, d) τη(τ) cτ + d where ɛ(a, b, c, d) is a 24 th root of unity. The general 1-point function is a bit more complicated, and we give just enough background to state the result. There are Eisenstein series denoted by P, Q, R by Ramanujan; they are holomorphic functions in H with q-expansions P = 1 24 dq n, Q = d 3 q n, R = d 5 q n n=1 d n n=1 d n n=1 d n Q and R are modular forms of weights 4, 6 respectively; P is a quasimodular form of weight 2. These functions are algebraically independent, and generate the algebra of quasimodular forms C[P, Q, R]. Theorem. The 1-point function defines a surjection Z V : V η(τ) 1 C[P, Q, R]. (Z V (v, q) = A(τ)/η(τ) with A(τ) C[P, Q, R], and every A(τ) so arises.) Exercise 18. Let V be the Heisenberg VOA of central charge 1. Prove that Z V (ω, q) = 1 P 24η. 13

14 3.3 The partition functions of a Virasoro VOA We start by computing the partition function of the Virasoro VOA V (c) of central charge c (cf. Subsection 2.5). Recall that the induced module M(c) (loc cit) has a natural basis L n1...l nk 1 indexed by unrestricted partitions (n 1,..., n k ). Much as in the case of the Heisenberg VOA, we find that the partition function of M(c) (which is itself not a VA) is q c/24 n 1 (1 qn ) 1. To arrive at V (c) we had to quotient by the V ir-submodule spanned by those basis vectors that involve L 1. At the level of partitions, we are are then left with those whose smallest part is 2. We thus obtain Z V (c) (q) 1 = q c/24 n=2 (1 q n ). This is not modular in any sense of the word, but neither is it the end of the story. To say more we must briefly return to morphisms of a VOA, to which up to now we have merely paid lip service (cf. Subsection 1.2). The kernel N of a morphism α : V V of VOAs has the property that u(n)v, v(n)u N whenever u N, v V, and (as is to be expected) the quotient V/N naturally carries the structure of a VA, and even a VOA if ω / N (which is the case if N V ). We call N an ideal of V. Suppose that V is a VOA of CFT-type, which means that in the conformal grading there are no nonzero states of negative weight and only one (up to scalars) of weight zero (the vacuum state). Thus V = k1 V 1... Many of the most basic VOAs enjoy this property, including the VOAs that we have discussed in these Notes. If V is of CFT-type and N V is an ideal, then v N ω(1)v = L(0)v N, i.e. N is invariant under L(0). This implies that N is a graded submodule of V. Moreover, if N is a proper ideal then it cannot contain 1 by the creativity axiom, so that N n 1 V n (using the CFT property). This applies to all proper ideals, and shows that a VOA of CFT-type has a unique maximal proper ideal. Call this maximal ideal J. The quotient V/J is then the unique simple quotient VOA of V. (Simple means there are no nonzero proper ideals.) It can be difficult to ascertain the nature of J; even deciding if it is nonzero may be nontrivial. Now we return to consideration of the Virasoro VOA V (c). We have the following omnibus result. 14

15 Theorem. Let V (c) be the Virasoro VOA of central charge c and J the unique maximal proper ideal of V (c). Then J 0 if, and only if, c = 1 6(s t)2 st with coprime integers s, t satisfying 2 s < t. If this condition is satisfied, set L(c) = V (c)/j. Then Z L(c) (q) is a modular function of weight zero. In particular, ( ) aτ + b Z L(c) = Z L(c) (τ) cτ + d ( ) a b for all in some congruence subgroup of SL c d 2 (Z). (These values of c and the corresponding VOAs L(c) are called the discrete series.) Exercise 19. Prove that the Heisenberg VOA of central charge 1 is simple. 3.4 Modules over a VOA Let (V, 1, Y, D) be a VA. A V -module is a pair (M, Y M ) where M is a k-module, Y M : V F(M), u Y M (u, z) = n u M(n)z n 1, and u, v V, w M: (a) Y M (1, z) = Id M (c) (Jacobi identity) r, s, t Z, ( ) r (u(t + i)v) M (r + s i)w = i ( t {um ( 1) i) i (r + t i)v M (s + i)w ( 1) t v M (s + t i)u M (r + i)w }. The idea is to mimic as far as possible the definition of a VA, but now with the modes of fields operating on M. There are no analogs of creativity or translation covariance. One can deduce analogs of many of the properties of the fields Y (u, z) much as we did for V in Section 1. For example, we have Y M (u, z) Y M (v, z) for all u, v V. (V, Y ) is obviously an example of a V -module, called the adjoint module or representation. There is also a self-evident notion of morphism of V -modules and direct sum of V -modules. Thus V -modules are the objects of an abelian category V -mod. Now assume that V is a VOA of central charge c and M a V -module in the previous sense. It can be shown that the modes of Y M (ω, z) = n L M(n)z n 2 again close on the Virasoro algebra of central charge c. It is then natural to impose grading conditions on V -modules. It turns out that to require an 15

16 integral grading is too strong, so we define a module over a VOA V to be a V -module M in the previous sense that also carries a conformal grading M = λ C M λ such that L M (0)w = λw (w M λ ) dim M λ < M λ+n = 0 (n 0) Again, if V is a VOA then the adjoint module is a module over the VOA, and we obtain 4 the category V -mod of modules over a VOA. A V -module M is simple, or irreducible, if no proper subspace of M is invariant under all modes of all fields Y M (v, z). Now the analog of the third part of Exercise 13 holds (and is not hard to prove), viz. v V k v M (n) : M λ M λ+k n 1 This has two main consequences for us. Firstly, if M 0 is simple then the conformal grading on M takes the form M = n 0 M h+n (M h 0) for some fixed h C called the conformal grading of M. Secondly (and assuming now that M is indeed simple) the zero mode o M (v) := v M (k 1) (v V k ) leaves invariant the homogeneous subspaces M h+n and therefore defines, much as in the case M = V, a 1-point correlation function Z M : V q c/24+h {power series in q} v Z M (v, q) := q c/24+h n 0 Tr M o M (v)q n. In particular, we have the partition function of M, i.e. Z M (q) = q c/24+h n 0 dim M n+h q n. 3.5 Finiteness theorems We are interested in VOAs whose module category has only finitely many (inequivalent) simple objects. There are two major ways to guarantee this. Theorem. Let V be a VOA with C 2 (V ) := u( 2)v u, v V. Then V has only finitely many simple modules if it is C 2 -cofinite (ie. dim V/C 2 (V ) < ). 4 One should carefully distinguish between modules over a VOA and the underlying VA, but we gloss over that here. 16

17 Theorem. Suppose that 5 V -mod is a semisimple category. (Then V is called rational.) Then V has only finitely many simple modules. An important open question here asks: if V rational, is it then necessarily C 2 -cofinite? The converse is false, because there are C 2 -cofinite VOAs which have modules which are not completely reducible. Examples. 1. The Heisenberg VOA has infinitely many inequivalent simple modules. It is therefore neither C 2 -cofinite nor rational. 2. The Virasoro VOA V (c) is never rational. The simple quotient L(c) is rational (and C 2 -cofinite) just when it is in the discrete series described in Subsection The affine Lie algebra VOA V (l) of level l is rational (and C 2 -cofinite) if, and only if, l is a positive integer. 3.6 Modular-invariance theorems The importance of considering all simple modules over a rational VOA is illustrated in the following (Zhu s Theorem). Theorem. Suppose that V is both C 2 -cofinite and rational. (As explained, the first assumption ought to be redundant.) Let q = e 2πiτ, τ H (cf. Subsection 3.2). Then the linear span of the partition functions Z M (τ) (M ranging over the inequivalent simple modules for V ) is a vector-valued modular form of weight zero on SL 2 (Z) with poles at. This means the following: (a) each Z M (τ) is holomorphic in H. (b) the Z M (τ) span an SL 2 (Z)-module in the sense that ( ) aτ + b Z M = ( ) a b c MM (γ)z M (τ), γ := SL cτ + d c d 2 (Z). M can be chosen so that ρ : γ (c MM (γ)) is a representa- The constants c MM tion of SL 2 (Z). There is an extension of this to general 1-point functions Z M (v, q). It applies to any state v, but we state just a special case to avoid technicalities. Theorem. Suppose that V is both C 2 -cofinite and rational. Suppose that v V k satisfies L(1)v = L(2)v = 0. (v is called a primary state.) Then the linear span of the 1-point functions Z M (v, τ) is a vector-valued modular form of weight k on SL 2 (Z) with poles at. This means the following: 5 The actual assumptions needed for this Theorem are slightly stronger than stated. We have conceded this lacuna to avoid undue technicality in our discussion of rational VOAs 17

18 (a) each Z M (v, τ) is holomorphic in H. (b) the Z M (v, τ) span an SL 2 (Z)-module in the sense that ( (cτ + d) k Z M v, aτ + b ) = ( ) a b c MM (γ)z M (v, τ), γ := SL cτ + d c d 2 (Z). M The constants c MM can be chosen so that ρ v : γ (c MM (γ)) is a representation of SL 2 (Z). Because SL 2 (Z) is generated by the standard matrices ( ) ( ) S =, T = the class of VOAs V we are discussing naturally give rise to modular data in the form of S- and T -matrices that generate a representation of the modular group SL 2 (Z) acting on a space whose dimension is the # simple V -modules. Under additional assumptions (in particular V is of CFT-type, cf. Subsection 3.3) these representations ρ factor through a congruence kernel, i.e. Γ(N) ker ρ. for some N. This amounts to the assertion that each 1-point function Z M (v, τ) is a modular form of weight k and level N. Furthermore, the category V -mod is a modular tensor category. Further browsing and reading. 1. Chiral Algebras, A. Beilinson and V. Drinfeld, AMS. (Very abstract approach to VAs and their generalizations based on alg geom. Not for the casual reader. Alg geom background necessary.) 2. Vertex Algebras and Algebraic Curves, E. Frenkel and D. Ben-Zvi, AMS. (Introduction to connections between VAs and algebraic geometry. Some expertise in alg geom a plus.) 3. Vertex Algebras for Beginners, V. Kac, AMS. (Introduction with a Lietheoretic bent.) 4. Introduction to Vertex Operator Algebras and Their Representations, J. Lepowsky, H. Li, Birkhäuser. (Systematic introduction to VOAs emphasizing the formalism of the Jacobi identity.) 5. Preparatory Course, CFT2011, U. Heidelberg, G. Mason. (Downloadable conference lecture notes on connections on VOAs and the Monster, 6. Vertex operators and modular forms, in A Window into Zeta and Modular Physics, G. Mason and M. Tuite, MSRI/CUP. (Extended notes on connections between VOAs, modular forms and elliptic functions. Lectures online at MSRI website.) 7. Axioms for a Vertex Algebra and the Locality of Quantum Fields, A. Matsuo, K. Nagatomo. Math. Soc. Japan. (Concise introduction to the calculus of quantum fields. Hard to locate copies.) 18

19 4 Solutions to Selected Exercises Exercise 1. If (V, 1) is a vertex algebra then V = 0 if, and only if, 1 = 0. Exercise 2. Suppose that A is a commutative, associative k-algebra with identity element 1. Prove that (A, 1) is a k-vertex algebra with u( 1)v = uv (product in k) and u(n)v = 0 (n 1). Exercise 3. Show that the category of commutative, associative, unital k- algebras is a full subcategory of k-v ert. Exercise 4. Prove that the state-field correspondence is injective. Exercise 5. Prove that D1 = 0. Exercise 6. Prove that Y (1, z) = Id V. Exercise 7. Show that (k, 1) is an initial object in the category k-v ert. Exercise 8. Suppose that a(z) = a(n)z n 1 F(V ) satisfies (b), (c). Then Then a(z)1 = 0 a( 1)1 = 0. Exercise 9. Let a(z) be as in Exercise 8 with a( 1)1 = 0. Suppose also that b(z) = b(n)z n 1 F(V ), a(z) b(z) and b(z)1 = n 1 b(n)1z n 1. Then a(z)b( 1)1 = 0. Exercise 10. Prove that [D, Y (x 1, z)] = z Y (x 1, z). Exercise 11. Prove that z : a(z)b(z) :=: ( z a(z))b(z) : + : a(z) z b(z) : Exercise 12. Let M be a V ir-module such that K acts on M as a scalar c. Set L(z) := n Z L nz n 2, and assume that L(z) F(M). Prove the following: (a) [L 1, L(z)] = z L(z), (b) L(z) 4 L(z). Exercise 13. Prove (i) 1 V 0 ; (ii) ω = L( 2)1 V 2 ; (iii) v V k v(n) : V m V m+k n 1. 19

20 ( Exercise 14. Prove that [L(m), L(n)] = (m n)l(m + n) + 1 m+1 ) IdV. 2 3 Exercise 15. Show that the Virasoro VOA V (c) is an initial object in the category V OA c of VOAs of central charge c. Exercise 16. Give the details of the proof that V (l) is a VA. Exercise 17. Give the detailed construction of Heisenberg VOA of central charge dim H. Exercise 18. Let V be the Heisenberg VOA of central charge 1. Prove that Z V (ω, q) = 1 P 24η. Exercise 19. Prove that the Heisenberg VOA of central charge 1 is simple. 20