To Robert Moody Energy-momentum tensor for the toroidal Lie algebras. Yuly Billig * School of Mathematics & Statistics 2 0 Carleton University 0 1125 Colonel By Drive 2 Ottawa, Ontario, K1S 5B6, Canada n e-mail: [email protected] a J 1 3 ] T Abstract. We construct vertex operator representations for the full (N +1)-toroidal Lie R algebra g. We associate with g a toroidal vertex operator algebra, which is a tensor product . h of an affine VOA, a sub-VOA of a hyperbolic lattice VOA, affine sl VOA and a twisted N t a Heisenberg-Virasoro VOA. The modules for the toroidal VOA are also modules for the toroidal m Lie algebra g. We also construct irreducible modules for an important subalgebra g of the div [ toroidalLiealgebrathatcorresponds tothedivergencefreevectorfields. Thissubalgebra carries 1 a non-degenerate invariant bilinear form. The VOA that controls the representation theory of v g is a tensor product of an affine VOA V (c) at level c, a sub-VOA of a hyperbolic lattice 3 div g˙ 1 VOA, affine sl VOA and a Virasoro VOA at level c with the following condition on the 3 N L 1 central charges: 2(N +1)+rank Vg˙(c)+cLb= 26. 0 2 0 b / h 0. Introduction. t a m Toroidal Lie algebras are very natural multi-variable generalizations of affine Kac-Moody : v algebras. The theory of affine Lie algebras is rich and beautiful, and has many important i X applications in physics. By large, applications of toroidal Lie algebras in physics are still r to be discovered. We should mention however the papers [IKUX], [IKU], where the toroidal a symmetry is discussed in the context of a 4-dimensional conformal field theory. We hope that the development of the representation theory of toroidal Lie algebras willhelp to find the proper place for these algebras in physical theories. The construction of a toroidal Lie algebra is totally parallel to the well-known construction of an (untwisted) affine Kac-Moody algebra [K1]. One starts with a finite-dimensional simple Lie algebra g˙ and considers maps from an N +1-dimensional torus into g˙. We may identify the algebra of functions on a torus with the Laurent polynomial algebra R = C[t±,t±,...,t±], by 0 1 N taking the Fourier basis, setting tk = eixk. The Lie algebra of the g˙-valued maps from a torus will then become C[t±,t±,...,t±]⊗g˙. When N = 0, this yields the loop algebra. 0 1 N * Research supported by the Natural Sciences and Engineering Research Council of Canada. 1 Just as in affine case, one builds the universal central extension (R ⊗ g˙) ⊕ K of R ⊗ g˙. However when N ≥ 1, the center K is infinite-dimensional. The infinite-dimensional center makes this Lie algebra highly degenerate. One can show, for example, that in an irreducible bounded weight module, most of the center should act trivially. To eliminate this degeneracy, we add the Lie algebra of the vector fields on a torus, D = Der(R) to (R⊗g˙)⊕K. The resulting algebra, g = (R⊗g˙)⊕K⊕D is called the toroidal Lie algebra (see Section 1 for details). The action of D on K is non-trivial, making the center of the toroidal Lie algebra g finite-dimensional. This enlarged algebra will have a much better representation theory. A major obstruction to building the representation theory of toroidal Lie algebras is that these algebras, being ZN+1 graded, do not possess a triangular decomposition whenever N ≥ 1. For this reason, the standard construction of the highest weight modules fails to work. The modules for the toroidal Lie algebras built by forcing the highest weight condition are not attractive [BC]. The true representation theory for toroidal Lie algebras was originated by Moody, Rao and Yokonuma in [MRY] and [EM], where they constructed a vertex operator representation in a homogeneous realization. The principal realization was later given in [B1]. Both realizations were unified and substantially generalized in [L] and [BB]. As the first application of this representation theory, one may use the vertex operator realizations to construct hierarchies of soliton equations as it was done in [B2], [ISW] and [IT]. The modules for toroidal Lie algebras introduced in these papers have weight decompo- sitions with finite-dimensional weight spaces and are bounded, but do not possess a unique highest weight. To explain this, we consider a Z grading of g by degree with respect to t , 0 which is declared to be a special variable. The subalgebra of elements of degree 0 in this Z grading, is very close to an N-toroidal Lie algebra. For this subalgebra we may consider an irreducible module T, which is a “toroidal” module for (R ⊗g˙)⊕K (a multi-variable analog 0 0 of a loop module), and is a tensor module for D . We let the elements of positive degree act on 0 T trivially, and then induce T to the module over the whole toroidal Lie algebra. This induced module has a unique irreducible quotient that may be alternatively studied via the explicit ver- tex operator constructions. This approach to the representation theory of toroidal Lie algebras was laid out in [BB]. As we see, instead of a one-dimensional highest weight space, the whole of T will be the “top” of the resulting bounded module. The space T is infinite-dimensional, but has a ZN grading with finite-dimensional subspaces. One serious problem with this representation theory that has not been previously resolved is that the vertex operator realizations were constructed not for the full toroidal algebra, but only for its subalgebra g∗ = (R ⊗ g˙) ⊕ K ⊕ D∗, where the derivations in t are missing from 0 the derivation part D. Plausible candidates for representing the missing part yielded extremely messy cocycles with values in a certain complicated completion of K [MRY], [L]. This was leaving the whole picture incomplete and unsatisfactory from the physics per- spective, because the missing derivations are responsible for the energy-momentum tensor for these modules. The main goal of the present paper is to resolve this problem and construct a class of representations for the full toroidal Lie algebras. Often the representation theory of Lie algebras is used for the construction of the vertex 2 operator algebras. In our case it is the opposite – the representation theory is developed using the machinery of the vertex operator algebras. This has been done in [BBS] for the subalgebra g∗ of the toroidal Lie algebra g. The VOA that controls the representation theory of g∗ is a tensor product of three fairly well-known VOAs – the affine g˙ VOA V , the affine gl VOA g˙ N V and a sub-VOA V+ of a hyperbolic lattice VOA. gl Hyp N After a careful analysis using the methods of [BB], it became clebar that the irreducible modules for g∗ do not admit the action of the full toroidal Lie algebra g, and thus it is necessary b to enlarge the representation space in order to get a module for g. A natural guess is that this enlarged space should be again a VOA or a VOA module. It turns out that the missing ingredient is a VOA that corresponds to the twisted Heisenberg-Virasoro algebra HVir, and the full toroidal VOA is a tensor product of four VOAs: V = V ⊗V+ ⊗V ⊗V tor g˙ Hyp slN HVir with certain conditions on the central charges of these VOAs. b b The twisted Heisenberg-Virasoro Lie algebra has a Virasoro subalgebra and a Heisenberg subalgebra, but the natural action of the Virasoro on the Heisenberg subalgebra is twisted with a cocycle (see Section 2.4 for the precise definition). The representation theory of HVir has been studied by Arbarello et al. in [ACKP]. However one special case, namely when the central charge of the Heisenberg subalgebra is zero, was not fully investigated in that paper. It happens that this is precisely the case we need for the toroidal VOA. The structure of the irreducible modules for HVir with the trivial action of the center of the Heisenberg subalgebra has been determined in [B3]. Using these ingredients we can easily write down the characters of the toroidal VOA and of its modules. This leads to the following open problem: while the explicit expressions for the characters of irreducible modules are known, there is no Weyl-type character formula for the toroidal Lie algebras. Obtaining such a formula may lead to interesting number-theoretic identities. Toroidal Lie algebras are related to the class of extended affine Lie algebras. These Lie algebras have been extensively studied during the last decade (see [AABGP] and references therein). The main features of an extended affine Lie algebra is that it is graded by a finite root system and possesses a non-degenerate symmetric invariant bilinear form. The full toroidal Lie algebra g does not possess a non-degenerate invariant form, but its subalgebra g = (R⊗g˙)⊕K⊕D div div does. Here D is the subalgebra of the divergence-free vector fields. Using the theory for the div full toroidal algebra g, we are able to construct irreducible representations for its important subalgebra g as well. div The vertex operator algebra that controls the representation theory of g is a tensor div product of an affine VOA V at level c, a sub-VOA of a hyperbolic lattice VOA V+ , and a g˙ Hyp Virasoro VOA at level c . The condition on the central charges that we get here is L b cdimg˙ 2(N +1)+ +c = 26, L c+h∨ 3 which has a striking resemblance to the formula for the critical dimension in the bosonic string theory. Another interesting fact is that when N = 12, we get an exceptional module for the Lie algebra D ⊕K. Only for this value of N we can represent D ⊕K just on a hyperbolic lattice div div sub-VOA V+ , and the structure of the module becomes exceptionally simple. The character Hyp of this module is given by the −24-th power of the Dedekind η-function and has nice modular properties. WeshouldmentionthattheclassofthemodulesforthefulltoroidalLiealgebraconstructed in this paper is not exhaustive, but could be described as a toroidal counterpart of the level 1 representations for affine Lie algebras, so more research remains to be done. Thestructureofthepaperisthefollowing. InSection1wegivethedefinitionofthetoroidal Lie algebras. In Section 2 we recall the definition and the properties of VOAs and construct the vertex operator algebras corresponding to the twisted Heisenberg-Virasoro algebra using the technique of the vertex Lie algebras. In Section 3 we describe the tensor factors of the toroidal VOAs – the affine VOA V , a sub-VOA of a hyperbolic lattice VOA V+ , and the twisted g˙ Hyp gl -Virasoro VOA V . We conclude Section 3 with the definition of the toroidal VOA. In N gl −Vir Section 4 we state andNprovbe our main result – every module for the toroidal VOA is a module fbor the toroidal Lie algebra. In the last Section we describe the structure of the irreducible b modules for the full toroidal Lie algebra, as well as for its subalgebra g . We conclude the div paper with the construction of an exceptional module for D ⊕K which is possible only when div N = 12. Acknowledgements: I am grateful to Stephen Berman for the stimulating discussions and encouragement. 1. Toroidal Lie algebras. Toroidal Lie algebras are the natural multi-variable generalizations of affine Lie algebras. In this review of the toroidal Lie algebras we follow the work [BB]. Let g˙ be a simple finite- dimensional Lie algebra over C with a non-degenerate invariant bilinear form (·|·) and let N ≥ 1 be an integer. We consider the Lie algebra R ⊗ g˙ of maps of an N + 1 dimensional torus into g˙, where R = C[t±,t±,...,t±] is the algebra of functions on a torus (in the Fourier 0 1 N basis). The universal central extension of this Lie algebra may be described by means of the following construction which is due to Kassel [Kas]. Let Ω be the space of 1-forms on a R N torus: Ω = ⊕ Rdt . We will choose the forms {k = t−1dt |p = 0,...,N} as a basis R p p p p p=0 of this free R module. There is a natural map d from the space of functions R into Ω : R N N d(f) = ∂f dt = t ∂f k . The center K for the universal central extension (R⊗g˙)⊕K ∂t p p∂t p p p p=0 p=0 of R⊗g˙Pis realized aPs K = Ω /d(R), R and the Lie bracket is given by the formula [f (t)g ,f (t)g ] = f (t)f (t)[g ,g ]+(g |g )f d(f ). 1 1 2 2 1 2 1 2 1 2 2 1 4 Here and in the rest of the paper we will denote elements of K by the same symbols as elements of Ω , keeping in mind the canonical projection Ω → Ω /d(R). R R R Just as in affine case, we add to (R⊗g˙)⊕K the algebra D of outer derivations N D = ⊕ Rd , p p=0 where d = t ∂ . We will denote the multi-indices by bold letters r = (r ,r ,...,r ), etc., p p∂t 0 1 N p and by tr the corresponding monomials tr0tr1 ...trN. 0 1 N The natural action of D on R⊗g˙ [trd ,tmg] = m tr+mg (1.1) a a uniquely extends to the action on the universal central extension (R⊗g˙)⊕K by N [trd ,tmk ] = m tr+mk +δ r tr+mk . (1.2) a b a b ab p p p=0 X This corresponds to the Lie derivative action of the vector fields on 1-forms. ItturnsoutthatthereisstillanextradegreeoffreedomindefiningtheLiealgebrastructure on (R⊗g˙)⊕K⊕D. The Lie bracket on D may be twisted with a K-valued 2-cocycle: [trd ,tmd ] = m tr+md −r tr+md +τ(trd ,tmd ). (1.3) a b a b b a a b The space of these cocycles H2(D,K) is two-dimensional and is spanned by the following cocy- cles τ and τ : 1 2 N τ (trd ,tmd ) = m r m tr+mk , 1 a b a b p p p=0 X N τ (trd ,tmd ) = r m m tr+mk . 2 a b a b p p p=0 X We will write τ = µτ +ντ . The resulting algebra (or rather a family of algebras) is called the 1 2 toroidal Lie algebra g = g(µ,ν) = (R⊗g˙)⊕K⊕D. Notethat afteradding thealgebra of derivationsD, thecenter ofthe toroidalLiegbecomes finite-dimensional with the basis {k ,k ,...,k }. This can be seen from the action (1.2) of D 0 1 N on K, which is non-trivial. In this paper we will consider only the multiples of the first cocycle τ , and we will be 1 assuming ν = 0 for most of our results here. The toroidal Lie algebra g(µ,ν) = (R⊗g˙)⊕K ⊕D has an important subalgebra g (µ) div that has divergence free vector fields as the derivation part: g (µ) = (R⊗g˙)⊕K⊕D , div div 5 where N N ∂f p D = f (t)d t = 0 . div p p p ∂t (p=0 (cid:12) p=0 p ) X (cid:12) X (cid:12) N (cid:12) ∂f The expression i t p becomes the divergence of a vector field in the angular coordinates p∂t p p=0 (x ,...,x ) on aPtorus, where t = eixj. 0 N j Note that the cocycle τ trivializes on D , so we only get the restriction of µτ . 2 div 1 The importance of this subalgebra is explained by the fact that unlike the full toroidal Lie algebra, g is an extended affine Lie algebra [BGK], i.e., g has a non-degenerate symmetric div div invariant bilinear form. The restrictions of this form to both R ⊗ g˙ and to D ⊕ K are div non-degenerate: r m (t g1|t g2) = δr,−m(g1|g2), g1,g2 ∈ g˙, while the vector fields pair with the 1-forms: N r m apt dp|t kq = δr,−maq. (1.4) p=0 (cid:0)X (cid:1) N m m One can see that the above formula is ill-defined for the full D, since d(t ) = m t k , q q q=0 being zero in K, must be in the kernel of the form. For the subalgebra D this is pPrecisely the div case since N N N a trd | r t−rk = a r = 0. p p q q q q p=0 q=0 q=0 (cid:0)X X (cid:1) X All other values of the bilinear form are trivial: (R⊗g˙|D ⊕K) = 0, (D |D ) = 0, (K|K) = 0. div div div It is easy to verify that the resulting symmetric bilinear form is invariant and non-degenerate. The study representation theory of toroidal Lie algebras has begun in [MRY] and [EM], with further developments in [B1], [L], [BB],[BBS].In all of these papers there was one common difficulty that has not been resolved – the representations constructed there were not for the full toroidal algebra g, but only for a subalgebra N g∗ = (R⊗g˙)⊕K⊕ ⊕ Rd , p p=1 (cid:18) (cid:19) where the piece Rd that corresponds to the toroidal energy-momentum tensor was missing. 0 This left the theory in a somewhat incomplete form, and the goal of the present paper is to construct a class of representations for the full toroidal Lie algebra. 6 2. Vertex operator algebra associated with the twisted Heisenberg- Virasoro Lie algebra. 2.1. Definitions and properties of a VOA. Let us recall the basic notions of the theory of the vertex operator algebras. Here we are following [K2] and [Li]. Definition. A vertex algebra is a vector space V with a distinguished vector 1 (vacuum vector) in V, an operator D (infinitesimal translation) on the space V, and a linear map Y (state-field correspondence) Y(·,z) : V → (EndV)[[z,z−1]], a 7→ Y(a,z) = a z−n−1 (where a ∈ EndV), (n) (n) Z nX∈ such that the following axioms hold: (V1) For any a,b ∈ V, a b = 0 for n sufficiently large; (n) (V2) [D,Y(a,z)] = Y(D(a),z) = d Y(a,z) for any a ∈ V; dz (V3) Y(1,z) = Id ; V (V4) Y(a,z)1 ∈ (EndV)[[z]] and Y(a,z)1| = a for any a ∈ V (self-replication); z=0 (V5) For any a,b ∈ V, the fields Y(a,z) and Y(b,z) are mutually local, that is, (z −w)n[Y(a,z),Y(b,w)] = 0, for n sufficiently large. A vertex algebra V is called a vertex operator algebra (VOA) if, in addition, V contains a vector ω (Virasoro element) such that (V6) The components L(n) = ω of the field (n+1) Y(ω,z) = ω z−n−1 = L(n)z−n−2 (n) Z Z nX∈ nX∈ satisfy the Virasoro algebra relations: n3 −n [L(n),L(m)] = (n−m)L(n+m)+δ (rank V)Id, where rank V ∈ C; (2.1) n,−m 12 (V7) D = L(−1); (V8) V is graded by the eigenvalues of L(0): V = ⊕ V with L(0) = nId. Z n Vn n∈ This completes the definition of a VOA. (cid:12) (cid:12) As a consequence of the axioms of the vertex algebra we have the following important commutator formula: n 1 ∂ z [Y(a,z ),Y(b,z )] = Y(a b,z ) z−1 δ 2 . (2.2) 1 2 n! (n) 2 1 ∂z z n≥0 (cid:20) (cid:18) 2(cid:19) (cid:18) 1(cid:19)(cid:21) X 7 As usual, the delta function is δ(z) = zn. Z nX∈ By (V1), the sum in the right hand side of the commutator formula is actually finite. Allthevertexoperatoralgebrasthatappearinthispaperhavethegradingsbynon-negative ∞ integers: V = ⊕ V . In this case the sum in the right hand side of the commutator formula n n=0 (2.2) runs from n = 0 to n = deg (a)+deg (b)−1, because deg (a b) = deg (a)+deg (b)−n−1, (2.3) (n) and the elements of negative degree vanish. The commutator formula (2.2) may be written as the commutator relations between the components of the vertex operators: n [a ,b ] = (a b) . (2.4) (n) (m) j (j) (n+m−j) j≥0(cid:18) (cid:19) X Equivalently, n a b = b a + (a b) , (2.5) (n) (m) (m) (n) j (j) (n+m−j) j≥0(cid:18) (cid:19) X and also m a b = b a − (b a) , (2.6) (n) (m) (m) (n) j (j) (n+m−j) j≥0(cid:18) (cid:19) X Another consequence of the axioms of a vertex algebra is the Borcherds’ identity: m (a b) c j (k+j) (m+n−j) j≥0(cid:18) (cid:19) X k k = (−1)k+j+1 b a c+ (−1)j a b c, k,m,n ∈ Z. j (n+k−j) (m+j) j (m+k−j) (n+j) j≥0 (cid:18) (cid:19) j≥0 (cid:18) (cid:19) X X (2.7) We will particularly need its special case when m = 0 and k = −1: (a b) c = b a c+ a b c, k ∈ Z. (2.8) (−1) (n) (n−j−1) (j) (−1−j) (n+j) j≥0 j≥0 X X Let us list some other consequences of the axioms of a vertex algebra that we will be using in the paper. It follows from V7 and V8 that ω a = D(a) (2.9) (0) 8 and ω a = deg (a)a for a homogeneous. (2.10) (1) The map D is a derivation of the n-th product: D(a b) = (Da) b+a Db. (2.11) (n) (n) (n) It could be easily derived from V2 that (Da) = −na (2.12) (n) (n−1) and thus 1 a = (Dk(a)) , k ≥ 0. (2.13) (−1−k) (−1) k! The last formula that we quote here is the skew-symmetry identity: 1 a b = (−1)n+j+1 Dj(b a). (2.14) (n) (n+j) j! j≥0 X 2.2. Tensor products of VOAs. The toroidal VOA that we introduce at the end of Section 3 is constructed by taking a tensor product of three VOAs. Let us review here the definition of the tensor product of two VOAs (V′,Y′,ω′,1) and (V′′,Y′′,ω′′,1) (the case of an arbitrary number of factors is a trivial generalization). The tensor product space V = V′ ⊗V′′ has the VOA structure under Y(a⊗b,z) = Y′(a,z)⊗Y′′(b,z), (2.15), ω = ω′ ⊗1+1⊗ω′′, (2.16) and 1 = 1⊗1 being the identity element. It follows from (2.16) that the rank of V (see V6) is the sum of the ranks of the tensor factors. We will be later using the following simple lemma: Lemma 2.1. Let a,c ∈ V′, b,d ∈ V′′. Then (i) (a⊗1) (1⊗b) = a⊗b. (−1) (ii) (a⊗1) (1⊗b) = 0 for n ≥ 0. (n) (iii) Suppose a c = 0 for j ≥ 0. Then (a⊗b) (c⊗d) = (a c)⊗(b d). (j) (n) (−1−j) (n+j) j≥0 Proof. Part (i) follows from V3 and V4. Part (ii) is aPconsequence of the commutativity of Y(a⊗1,z ) and Y(1⊗b,z ). Part (iii) follows from (i), (ii) and (2.8). 1 2 2.3. Vertex Lie algebras. An important source of the vertex algebras is provided by the vertex Lie algebras. In presenting this construction we will be following [DLM] (see also [P], [R], [K2], [FKRW]). 9 Let L be a Lie algebra with the basis {u(n),c(−1) u ∈ U,c ∈ C,n ∈ Z} (U, C are some index sets). Define the corresponding fields in L[[z,z−1]]: (cid:12) (cid:12) u(z) = u(n)z−n−1, c(z) = c(−1)z0, u ∈ U,c ∈ C. Z nX∈ Let F be a subspace in L[[z,z−1]] spanned by all the fields u(z),c(z) and their derivatives of all orders. Definition. A Lie algebra L with the basis as above is called a vertex Lie algebra if the following two conditions hold: (1) for all u ,u ∈ U, 1 2 n j ∂ z [u (z ),u (z )] = f (z ) z−1 δ 2 , (2.17) 1 1 2 2 j 2 1 ∂z z j=0 " (cid:18) 2(cid:19) (cid:18) 1(cid:19)# X where f (z) ∈ F and n depends on u ,u , j 1 2 (2) for all c ∈ C, the elements c(−1) are central in L. This definition is a simplified version of the one from [DLM] and is not quite as general as the original definition, but it is sufficient for our purposes. Let L+ be a subspace in L with the basis {u(n) u ∈ U,n ≥ 0} and let L− be a subspace with the basis {u(n),c(−1) u ∈ U,c ∈ C,n < 0}. Then L = L+ ⊕L− and L+,L− are in fact (cid:12) subalgebras in L. (cid:12) (cid:12) The universal envelopi(cid:12)ng vertex algebra V of a vertex Lie algebra L is defined as an L induced module V = IndL (C1) = U(L−)⊗1, L L+ where C1 is a trivial 1-dimensional L+ module. Theorem 2.2. ([DLM], Theorem 4.8) Let L be a vertex Lie algebra. Then (a) V has a structure of a vertex algebra with the vacuum vector 1, infinitesimal translation L D being a natural extension of the derivation of L given by D(u(n)) = −nu(n−1), D(c(−1)) = 0, u ∈ U, c ∈ C, and the state-field correspondence map Y defined by the formula: Y (a (−1−n )...a (−1−n )a (−1−n )1,z) 1 1 k−1 k−1 k k 1 ∂ n1 1 ∂ nk−1 1 ∂ nk =: a (z) ... : a (z) a (z) : ... : , 1 k−1 k n ! ∂z n ! ∂z n ! ∂z (cid:18) 1 (cid:18) (cid:19) (cid:19) (cid:18) k−1 (cid:18) (cid:19) (cid:19)(cid:18) k (cid:18) (cid:19) (cid:19) (2.18) where a ∈ U,n ≥ 0 or a ∈ C,n = 0. j j j j (b) Any restricted L module is a vertex algebra module for V . L (c) For an arbitrary character λ : C → C, the factor module V (λ) = U(L−)1/U(L−) (c(−1)−λ(c))1 L c∈C (cid:10) (cid:11) is a quotient vertex algebra. 10