Representations of Twisted Toroidal Lie Algebras from Twisted Modules over Vertex Algebras
aa r X i v : . [ m a t h . QA ] S e p REPRESENTATIONS OF TWISTED TOROIDAL LIEALGEBRAS FROM TWISTED MODULES OVERVERTEX ALGEBRAS
BOJKO BAKALOV AND SAMANTHA KIRK
Abstract.
Given a simple finite-dimensional Lie algebra and anautomorphism of finite order, one defines the notion of a twistedtoroidal Lie algebra. In this paper, we construct representations oftwisted toroidal Lie algebras from twisted modules over affine andlattice vertex algebras. Introduction
A toroidal Lie algebra is the universal central extension of the ten-sor product of a simple finite-dimensional Lie algebra g and the ringof Laurent polynomials in r variables. Moody, Rao, and Yokonumawere the first to introduce toroidal Lie algebras in [41], which led toconstructions of representations of toroidal Lie algebras from vertex op-erators in works such as [9, 13, 14, 24, 25, 42, 43]. In a fashion similarto twisted affine Lie algebras [26], Fu and Jiang [21] used diagram auto-morphisms of g of finite order to construct twisted toroidal Lie algebrasand studied their integrable modules. Some representations of twistedtoroidal Lie algebras were constructed in works such as [1, 11, 23].Vertex algebras have proven to be a useful tool in constructing vertexoperator representations of infinite-dimensional Lie algebras [10, 16, 19,27, 30, 36]. Some examples include the use of lattice vertex algebras togeneralize the Frenkel–Kac realization of affine Kac–Moody algebras interms of vertex operators [10, 18] and the use of tensor products of ver-tex algebras to create representations of untwisted toroidal Lie algebrasby Berman, Billig, and Szmigielski [5]. Twisted vertex operators firstappeared in applications such as the principal realization of affine Kac–Moody algebras [28, 38] and the Frenkel–Lepowsky–Meurman con-struction of the moonshine vertex operator algebra [19, 32]. Thesetwisted vertex operators led to the notion of a twisted module over avertex algebra [3, 12, 15, 35]. Date : September 2, 2020.
In this paper, we use twisted modules over affine and lattice vertexalgebras to construct representations of twisted toroidal Lie algebras.Our paper is influenced by Berman, Billig, and Szmigielski’s work onconstructing representations of untwisted toroidal Lie algebras in [5, 6,9] and we will rely on their notation in our description of toroidal Liealgebras.The paper is organized as follows. In Section 2, we provide back-ground information on vertex algebras. In Section 3, we connect rep-resentations of affine Kac–Moody Lie algebras with twisted modulesover affine vertex algebras. In Section 4, we use the representationsof affine Kac–Moody algebras presented in Section 3 as the frameworkto build representations of twisted toroidal Lie algebras from twistedmodules over a tensor product of an affine vertex algebra and latticevertex algebras. Unless otherwise specified, all vector spaces, linearmaps and tensor products will be over the field C of complex numbers.2. Preliminaries
The purpose of this section is to review basic definitions and establishnotation for vertex algebras. For more details, we refer the reader to[3, 16, 27, 36].2.1.
Vertex algebras.
Recall that a vertex algebra [16, 27, 36] is avector space V with a distinguished vector ∈ V ( vacuum vector ),equipped with bilinear n -th products for n ∈ Z :(2.1) V ⊗ V → V, a ⊗ b a ( n ) b, subject to the following axioms. First, for every fixed a, b ∈ V , we have a ( n ) b = 0 for sufficiently large n (denoted n ≫ plays the role of a unit in the sense that a ( − = ( − a = a, a ( n ) = 0 , n ≥ . Finally, the main axiom of a vertex algebra is the
Borcherds identity (also called Jacobi identity [19]) satisfied for all a, b, c ∈ V and k, m, n ∈ Z : ∞ X j =0 (cid:18) mj (cid:19) ( a ( k + j ) b ) ( m + n − j ) c = ∞ X i =0 (cid:18) ki (cid:19) ( − i a ( m + k − i ) ( b ( n + i ) c ) − ∞ X i =0 (cid:18) ki (cid:19) ( − k + i b ( n + k − i ) ( a ( m + i ) c ) . (2.2)Note that all sums in (2.2) are in fact finite. We can view (2.1) asdefining a sequence of linear operators a ( n ) on V , for a ∈ V , n ∈ Z , EPRESENTATIONS OF TWISTED TOROIDAL LIE ALGEBRAS 3 called the modes of a . Setting k = 0 in the Borcherds identity, weobtain the commutator formula (2.3) [ a ( m ) , b ( n ) ] = ∞ X j =0 (cid:18) mj (cid:19) ( a ( j ) b ) ( m + n − j ) , which will be very useful for us.It is convenient to organize the modes into formal power series(2.4) Y ( a, z ) = X n ∈ Z a ( n ) z − n − , a ∈ V, called fields or vertex operators. The linear map Y : V → (End V )[[ z, z − ]]is known as the state-field correspondence . Observe that Y ( , z ) = I isthe identity operator.Introduce the translation operator T ∈ End( V ) defined by T a = a ( − . Then(2.5) [ T, Y ( a, z )] = Y ( T a, z ) = ∂ z Y ( a, z ) , or equivalently,(2.6) [ T, a ( n ) ] = ( T a ) ( n ) = − na ( n − . In particular, T is a derivation of all n -th products.Another important consequence of the Borcherds identity is the ( − -st product identity ( a, b ∈ V ): Y ( a ( − b, z ) = : Y ( a, z ) Y ( b, z ):= X n< a ( n ) z − n − Y ( b, z ) + X n ≥ Y ( b, z ) a ( n ) z − n − . (2.7)The double colons in (2.7) denote the so-called normally-ordered prod-uct . Combining (2.5)–(2.7), we get(2.8) Y ( a ( − − m ) b, z ) = 1 m ! :( ∂ mz Y ( a, z )) Y ( b, z ): , m ≥ . Finally, recall that the tensor product of two vertex algebras V and V is again a vertex algebra [17] with a vacuum vector ⊗ and astate-field correspondence given by ( a ∈ V , b ∈ V ):(2.9) Y ( a ⊗ b, z ) = Y ( a, z ) ⊗ Y ( b, z ) = X k,m ∈ Z a ( k ) ⊗ b ( m ) z − k − m − . In terms of modes, we have(2.10) ( a ⊗ b ) ( n ) = X k ∈ Z a ( k ) ⊗ b ( n − k − . The translation operator in V ⊗ V is T ⊗ I + I ⊗ T . BOJKO BAKALOV AND SAMANTHA KIRK
Twisted modules over vertex algebras.
Let V be a vertexalgebra and σ be an automorphism of V , i.e., a linear operator such that σ ( ) = and σ ( a ( n ) b ) = ( σa ) ( n ) ( σb ) for all a, b ∈ V , n ∈ Z . Supposethat σ N = I for some positive integer N . Then σ is diagonalizable on V .A σ -twisted V -module [12, 15] is a vector space M endowed with alinear map Y M : V → (End M )[[ z /N , z − /N ]],(2.11) Y M ( a, z ) = X m ∈ N Z a M ( m ) z − m − , a ∈ V, subject to the following axioms. First, for every a ∈ V , v ∈ M , wehave a M ( m ) v = 0 for m ≫
0. Next, Y M ( , z ) = I and(2.12) Y M ( σa, z ) = Y M ( a, e π i z ) , where the meaning of the right-hand side is that we replace z − m − with e − π i( m +1) z − m − in each summand of (2.11). Explicitly, (2.12) meansthat if a is an eigenvector of σ , then in (2.11) we only have termswith m ∈ N Z such that σa = e − π i m a . Finally, we have the twisted Borcherds identity for any a, b ∈ V , c ∈ M , k ∈ Z , m, n ∈ N Z : ∞ X j =0 (cid:18) mj (cid:19) ( a ( k + j ) b ) M ( m + n − j ) c = ∞ X i =0 (cid:18) ki (cid:19) ( − i a M ( m + k − i ) ( b M ( n + i ) c ) − ∞ X i =0 (cid:18) ki (cid:19) ( − k + i b M ( n + k − i ) ( a M ( m + i ) c ) , (2.13)provided that σa = e − π i m a .In particular, we have the commutator formula for a, b ∈ V and m, n ∈ N Z such that σa = e − π i m a :(2.14) [ a M ( m ) , b M ( n ) ] = ∞ X j =0 (cid:18) mj (cid:19) ( a ( j ) b ) M ( m + n − j ) . The translation covariance properties (2.5) and (2.6) remain valid fortwisted modules. However, formula (2.8) does not hold for twistedmodules; it is replaced by [30, Eq. (14.16)].Let V and V be vertex algebras with finite-order automorphisms σ and σ , respectively. Then σ ⊗ σ is an automorphism of the tensorproduct vertex algebra V ⊗ V . Given σ i -twisted V i -modules M i ( i =1 , M ⊗ M is a σ ⊗ σ -twisted module of V ⊗ V with(2.15) Y M ⊗ M ( a ⊗ b, z ) = Y M ( a, z ) ⊗ Y M ( b, z ) , a ∈ V , b ∈ V , EPRESENTATIONS OF TWISTED TOROIDAL LIE ALGEBRAS 5 see [2, 17].2.3.
Lattice vertex algebras.
Let Q be an integral lattice of rank ℓ , i.e., a free abelian group of rank ℓ with a symmetric bilinear form( ·|· ) : Q × Q → Z . We will assume that Q is even, which means that | α | = ( α | α ) ∈ Z for all α ∈ Q . Let h = C ⊗ Z Q and extend the form( ·|· ) to h using bilinearity.The Heisenberg Lie algebra ˆ h is defined asˆ h = ( h ⊗ C [ t, t − ]) ⊕ C K with the Lie brackets ( h, h ′ ∈ h , m, n ∈ Z ):(2.16) [ h ⊗ t m , h ′ ⊗ t n ] = mδ m, − n ( h | h ′ ) K, [ˆ h , K ] = 0 . We will use the notation h ( m ) = h ⊗ t m . The Lie algebra ˆ h has a uniquehighest-weight representation on the so-called bosonic Fock space B = Ind ˆ hh [ t ] ⊕ C K C ≃ S ( t − h [ t − ]) , where K acts as I and h [ t ] acts trivially on C . The Fock space B hasthe structure of a vertex algebra with a vacuum vector the highest-weight vector and the state-field correspondence Y defined as follows.For h ∈ h , we identify h with h ( − ∈ B and let(2.17) Y ( h, z ) = X m ∈ Z h ( m ) z − m − , h ∈ h be the free boson fields. All other fields in B are obtained from themby applying repeatedly formula (2.8); see [16, 27, 36].There exists a bimultiplicative function ε : Q × Q → {± } such that(2.18) ε ( α, α ) = ( − | α | ( | α | +1) / , α ∈ Q. By bimultiplicativity, ε satisfies ε ( α, β ) ε ( β, α ) = ( − ( α | β ) , α, β ∈ Q. We can use ε to define the twisted group algebra C ε [ Q ] with basis { e α } α ∈ Q and multiplication e α e β = ε ( α, β ) e α + β . The representation of ˆ h can be extended to the space V Q = B ⊗ C ε [ Q ]by the action h ( m ) ( u ⊗ e β ) = ( h ( m ) u + δ m, ( h | β ) u ) ⊗ e β BOJKO BAKALOV AND SAMANTHA KIRK for h ∈ h , m ∈ Z , u ∈ B and β ∈ Q . In particular, note that e β is ahighest-weight vector for the Heisenberg Lie algebra:(2.19) h ( m ) e β = δ m, ( h | β ) e β , m ≥ , h ∈ h , β ∈ Q. We can also represent the algebra C ε [ Q ] on V Q by e α ( u ⊗ e β ) = ε ( α, β )( u ⊗ e α + β )for u ∈ B and α, β ∈ Q .For simplicity of notation, we will write e α for ⊗ e α ∈ V Q and h for h ( − ⊗ e ∈ V Q , where α ∈ Q and h ∈ h . The space V Q has thestructure of a vertex algebra called the lattice vertex algebra , with avacuum vector ⊗ e and a state-field correspondence generated bythe free boson fields (2.17) and the so-called vertex operators(2.20) Y ( e α , z ) = e α z α (0) exp ∞ X n =1 α ( − n ) z n n ! exp ∞ X n =1 α ( n ) z − n − n ! . In this formula, z α (0) acts on V Q by z α (0) ( u ⊗ e β ) = z ( α | β ) ( u ⊗ e β ) , u ∈ B, α, β ∈ Q. For future use, we also recall the action of the translation operator:(2.21)
T e α = α ( − e α , α ∈ Q. Suppose σ ∈ Aut( Q ) where σ N = I and extend σ to h by linearity.Then σ lifts to an automorphism of the Heisenberg Lie algebra ˆ h by σ ( h ( m ) ) = ( σh ) ( m ) , and to an automorphism of the vertex algebra B sothat σ = . Since the cocycles ε ( α, β ) and ε ( σα, σβ ) are equivalent,there is a map η : Q → {± } such that η ( α ) η ( β ) ε ( α, β ) = η ( α + β ) ε ( σα, σβ ) , α, β ∈ Q. The map η can be chosen so that η ( α ) = 1 if σα = α (see [2]). We canlift σ to an automorphism of V Q by σ ( e α ) = η ( α ) e σα . Notice that theorder of the lift σ ∈ Aut( V Q ) is N or 2 N . The irreducible σ -twisted V Q -modules were classified in [3] (see also [12, 35]).3. Vertex operator representations of affine Kac–Moodyalgebras
In this section, we construct representations of affine Kac–Moodyalgebras from twisted modules over affine vertex algebras.
EPRESENTATIONS OF TWISTED TOROIDAL LIE ALGEBRAS 7
Affine Kac–Moody algebras.
Let g be a simple finite-dimensionalLie algebra of type X ℓ where X = A, B, C, . . . , G . Recall that the affineKac-Moody algebra of type X (1) ℓ is defined as [26]:ˆ L ( g ) = ( g ⊗ C [ t, t − ]) ⊕ C K ⊕ C d, with the Lie brackets[ a ⊗ t m , b ⊗ t n ] = [ a, b ] ⊗ t m + n + mδ m, − n ( a | b ) K, [ K, ˆ L ( g )] = 0 , [ d, a ⊗ t m ] = ma ⊗ t m , (3.1)for a, b ∈ g and m, n ∈ Z . Here ( ·|· ) is a nondegenerate symmetricinvariant bilinear form on g , such as a scalar multiple of the Killingform. We will denote by ˆ g or ˆ L ′ ( g ) the subalgebra of ˆ L ( g ) given byˆ g = ˆ L ′ ( g ) = ( g ⊗ C [ t, t − ]) ⊕ C K. Let σ be an automorphism of the Lie algebra g of finite order N . Wecan define a subalgebra ˆ L ( g , σ ) of ˆ L ( g ) by(3.2) ˆ L ( g , σ ) = M j ∈ Z L ( g , σ ) j ⊕ C K ⊕ C d, where L ( g , σ ) j = g j ⊗ C t j , j ∈ Z , g j = { a ∈ g | σa = e π i j/N a } . (3.3)When g is simply laced (of type X = A, D, E ) and σ is a diagramautomorphism of order N = 2 or 3, then the Lie algebra ˆ L ( g , σ ) isknown as the twisted affine Kac–Moody algebra of type X ( N ) ℓ . We willdenote by ˆ L ′ ( g , σ ) the subalgebra of ˆ L ( g , σ ) given byˆ L ′ ( g , σ ) = M j ∈ Z L ( g , σ ) j ⊕ C K. If σ is an arbitrary automorphism of g (simply or non-simply laced) offinite order N , then there exists an associated diagram automorphism µ (where the order of µ is 1 , , or 3, depending on g ) and an innerautomorphism ϕ such that σ = µϕ . Then the Lie algebra ˆ L ( g , σ ) isisomorphic to ˆ L ( g , µ ); see [26, Proposition 8.1, Theorem 8.5]. BOJKO BAKALOV AND SAMANTHA KIRK
Affine vertex algebras.
Suppose g is a simple finite-dimensionalLie algebra, equipped with a nondegenerate symmetric invariant bilin-ear form ( ·|· ). Consider the Lie algebra ˆ g = ˆ L ′ ( g ) with the bracketsgiven by the first two equations in (3.1). For a fixed k ∈ C (called the level ), consider the (generalized) Verma module for ˆ g : V k ( g ) = Ind ˆ gg [ t ] ⊕ C K C , where g [ t ] acts as 0 on C and K acts as multiplication by k . Themodule V k ( g ) has the structure of a vertex algebra [20], called the universal affine vertex algebra at level k . The ˆ g -module V k ( g ) has aunique irreducible quotient V k ( g ), which is also a vertex algebra [20],known as the simple affine vertex algebra at level k .Let us review the vertex algebra structure of V = V k ( g ); the sameapplies to V = V k ( g ) as well. The vacuum vector is the highest-weight vector of the ˆ g -module V . We shall not review the full state-field correspondence Y but only the generating fields. For a ∈ g and n ∈ Z , let a ( n ) act as a ⊗ t n on V . We embed g in V so that we identify a ∈ g with a ( − ∈ V . Then we have the fields Y ( a, z ) = X n ∈ Z a ( n ) z − n − , a ∈ g , known as currents . All other fields in V are obtained from them byapplying repeatedly formula (2.8); see [16, 27, 36].For a, b ∈ g ⊂ V , their modes satisfy the commutation relations ofthe Lie algebra ˆ g :(3.4) [ a ( m ) , b ( n ) ] = [ a, b ] ( m + n ) + mδ m, − n ( a | b ) k. By the commutator formula (2.3), this is equivalent to the j -th products(3.5) a (0) b = [ a, b ] , a (1) b = ( a | b ) k , a ( j ) b = 0 ( j ≥ . One can show that a V k ( g )-module is the same as a ˆ g -module M withthe property that a ( n ) v = 0 for a ∈ g , v ∈ M and n ≫ V k ( g )-modules (cf. [30, 31]).3.3. Twisted modules over V k ( g ) and twisted affine Lie alge-bras. As in Section 3.1, let g be a simple finite-dimensional Lie algebraand σ ∈ Aut( g ) such that σ N = I . We can extend σ uniquely to anautomorphism of the universal affine vertex algebra V k ( g ) by(3.6) σ ( ) = , σ ( a ( m ) ) = ( σa ) ( m ) , a ∈ g , m ∈ Z . EPRESENTATIONS OF TWISTED TOROIDAL LIE ALGEBRAS 9
Proposition 3.1.
For any σ -twisted V k ( g ) -module M , the Lie algebraspanned by the modes of the fields Y M ( a, z ) for a ∈ g form a repre-sentation of the twisted affine Kac–Moody algebra ˆ L ′ ( g , σ − ) on M oflevel k . Proof.
We will define a Lie algebra homomorphism from ˆ L ′ ( g , σ − ) tothe modes of the fields Y M ( a, z ). By linearity, since σ is diagonalizable,we can assume that a ∈ g is an eigenvector of σ . Suppose that σa = e − π i j/N a for some j ∈ Z . Then by (3.3), we have a ⊗ t j ∈ ˆ L ( g , σ − ) j .On the other hand, the property (2.12) of twisted modules implies thatin the expansion (2.11) of Y M ( a, z ), we only have modes a M ( m ) such that m ∈ jN + Z . We define a linear map from ˆ L ′ ( g , σ − ) to the span of thesemodes, by sending a ⊗ t j to a M ( j/N ) . We also send K to kI as a linearoperator on M .To check that this is a Lie algebra homomorphism, we compute thecommutator of modes, which is given by the σ -twisted commutatorformula (2.14). Using (3.5), we obtain[ a M ( m ) , b M ( n ) ] = [ a, b ] M ( m + n ) + mδ m, − n ( a | b ) k, for a, b ∈ g and m, n ∈ N Z such that σa = e − π i m a and σb = e − π i n b .As this coincides with the Lie bracket (3.1) in ˆ L ′ ( g , σ − ) with K = kI ,the claim follows. Remark 3.2.
When σ is an inner automorphism of g , then ˆ L ′ ( g , σ − ) ≃ ˆ L ′ ( g ) = ˆ g is an untwisted affine Lie algebra.Suppose now g is a simple Lie algebra of type X ℓ where X = A, D, E (simply laced). Let ∆ be its root system and Q = Z ∆ its root lattice.Let σ be an automorphism of the lattice Q of finite order N . Recallfrom Section 2.3 that we can use Q to construct the lattice vertexalgebra V Q , and we can lift σ to an automorphism of the lattice vertexalgebra V Q . Corollary 3.3 (cf. [29, 30]) . For any σ -twisted V Q -module M , themodes of the free bosons Y M ( h, z ) for h ∈ h and the modes of thevertex operators Y M ( e α , z ) for α ∈ ∆ span a representation of thetwisted affine Kac–Moody algebra ˆ L ′ ( g , σ − ) on M . Proof.
By the Frenkel–Kac construction [18], the lattice vertex algebra V Q is isomorphic to the simple affine vertex algebra V ( g ) (see [27,Theorem 5.6 (c)]). The map σ induces an automorphism of g andhence of V ( g ). Recall that V ( g ) is a quotient of the universal affinevertex algebra V ( g ). Thus any σ -twisted V ( g )-module M is also a σ -twisted module for V ( g ). The claim then follows from Proposition3.1. Remark 3.4. If g is a non-simply laced Lie algebra, then g can beembedded in a simply laced Lie algebra g as the set of fixed pointsunder a diagram automorphism µ of g . We can use Corollary 3.3 toconstruct representations of ˆ L ′ ( g , σ − ) and then restrict them to thesubalgebra ˆ L ′ ( g ); see [8, 22]. Level-one representations of affine Kac–Moody algebras associated to non-simply laced Lie algebras have beenexplored in papers such as [34, 39, 40, 44].4. Vertex operator representations of toroidal Liealgebras
In this section, we construct representations of toroidal Lie algebrasfrom twisted modules over a tensor product of an affine vertex algebraand a certain lattice vertex algebra.4.1.
Untwisted toroidal Lie algebras.
Through the rest of the pa-per, we fix a positive integer r and a simple finite-dimensional Liealgebra g equipped with a nondegenerate symmetric invariant bilinearform ( ·|· ). We will use variables t , t , . . . , t r and multi-index notation t = ( t , . . . , t r ) , m = ( m , . . . , m r ) ∈ Z r , t m = t m · · · t m r r . Consider the loop algebra L r +1 ( g ) in r + 1 variables L r +1 ( g ) = g ⊗ O , O = C [ t ± , t ± , . . . , t ± r ] , with the Lie bracket given by ( a, b ∈ g , m , n ∈ Z , m , n ∈ Z r ):[ a ⊗ t m t m , b ⊗ t n t n ] = [ a, b ] ⊗ t m + n t m + n . Next, we create a central extension ˆ L ′ r +1 ( g ) of L r +1 ( g ):ˆ L ′ r +1 ( g ) = L r +1 ( g ) ⊕ K , where(4.1) K = (cid:16) r M i =0 C K i ⊗ O (cid:17). span C n r X i =0 m i K i ⊗ t m t m (cid:12)(cid:12)(cid:12) m i ∈ Z o . The Lie brackets in ˆ L ′ r +1 ( g ) are given by:[ a ⊗ t m t m , b ⊗ t n t n ] = [ a, b ] ⊗ t m + n t m + n (4.2) + ( a | b ) r X i =0 m i K i ⊗ t m + n t m + n , [ K , ˆ L ′ r +1 ( g )] = 0 . (4.3) EPRESENTATIONS OF TWISTED TOROIDAL LIE ALGEBRAS 11
By Kassel’s Theorem, the central extension ˆ L ′ r +1 ( g ) of L r +1 ( g ) is uni-versal [33] (setting K i = t − i dt i allows us to identify K with O /d O ).The Lie algebra ˆ L ′ r +1 ( g ) is known as the (untwisted) toroidal Lie alge-bra [41].It will be convenient to slightly modify the definition of ˆ L ′ r +1 ( g ) asfollows. For a given complex number k (called the level ), we replacethe bracket (4.2) with:[ a ⊗ t m t m , b ⊗ t n t n ] = [ a, b ] ⊗ t m + n t m + n + k ( a | b ) r X i =0 m i K i ⊗ t m + n t m + n . (4.4)The resulting Lie algebra L r +1 ( g ) ⊕ K with brackets (4.3), (4.4) will bedenoted as ˆ L ′ r +1 ,k ( g ) and called the (untwisted) toroidal Lie algebra oflevel k . Notice that, for k = 0, formulas (4.2) and (4.4) are equivalentafter rescaling the bilinear form ( ·|· ) or rescaling the central elements K , . . . , K r .Now we will add derivations to our toroidal Lie algebra, as in [9].We let D be the Lie algebra of derivations of O given by D = n r X i =0 d i ⊗ f i (cid:12)(cid:12)(cid:12) f i ∈ O o , d i = t i ∂∂t i , and D + be the subalgebra of D given by D + = n r X i =1 d i ⊗ f i (cid:12)(cid:12)(cid:12) f i ∈ O o . Then the elements of D + extend uniquely to derivations of the Liealgebra ˆ L ′ r +1 ( g ) by ( a ∈ g , 1 ≤ i, j ≤ r ):( d i ⊗ t m t m )( a ⊗ t n t n ) = n i a ⊗ t m + n t m + n , ( d i ⊗ t m t m )( K j ⊗ t n t n ) = n i K j ⊗ t m + n t m + n + δ i,j r X l =0 m l K l ⊗ t m + n t m + n . The Lie algebra we will consider in this paper, which we will refer toagain as a toroidal Lie algebra of level k , isˆ L r +1 ,k ( g ) = ˆ L ′ r +1 ,k ( g ) ⊕ D + with the Lie brackets given by (4.3), (4.4) and ( a ∈ g , m l , n l ∈ Z ,1 ≤ i, j ≤ r ):[ d i ⊗ t m t m , a ⊗ t n t n ] = n i a ⊗ t m + n t m + n , (4.5) [ d i ⊗ t m t m , K j ⊗ t n t n ] = n i K j ⊗ t m + n t m + n (4.6) + δ i,j r X l =0 m l K l ⊗ t m + n t m + n , [ d i ⊗ t m t m , d j ⊗ t n t n ] = n i d j ⊗ t m + n t m + n (4.7) − m j d i ⊗ t m + n t m + n − n i m j r X l =0 m l K l ⊗ t m + n t m + n . The last term in (4.7) corresponds to a K -valued 2-cocycle on D + . Formore information on derivations of toroidal Lie algebras and 2-cocycles,see [6]. Notice that, for k = 0, if we rescale the generators K , . . . , K r in order to replace (4.4) with (4.2), the 2-cocycle gets rescaled by 1 /k .4.2. Twisted toroidal Lie algebras.
As before, fix a level k ∈ C ,and let σ be an automorphism of order N of a simple finite-dimensionalLie algebra g . As in (3.3), we denote by g j ( j ∈ Z ) the eigenspace of σ with eigenvalue e π i j/N . Note that the σ -invariance of ( ·|· ) implies(4.8) ( a | b ) = 0 , a ∈ g m , b ∈ g n , m + n N. Consider the subalgebra L r +1 ( g , σ ) of the loop algebra L r +1 ( g ) givenby L r +1 ( g , σ ) = M m ∈ Z L r +1 ( g , σ ) m , where L r +1 ( g , σ ) m = g m ⊗ span C (cid:8) t m t m (cid:12)(cid:12) m ∈ Z , m ∈ Z r (cid:9) . Let K ′ = span C (cid:8) K i ⊗ t Nm t m (cid:12)(cid:12) m ∈ Z , m ∈ Z r , i = 0 , . . . , r (cid:9) span C (cid:8)(cid:0) N m K + P ri =1 m i K i (cid:1) ⊗ t Nm t m (cid:12)(cid:12) m i ∈ Z (cid:9) . We can identify K ′ as the subspace of K given by the image ofspan C (cid:8) K i ⊗ t Nm t m (cid:12)(cid:12) m ∈ Z , m ∈ Z r , i = 0 , . . . , r (cid:9) under the quotient map L ri =0 C K i ⊗ O → K (cf. (4.1)). Then thecentral extensionˆ L ′ r +1 ,k ( g , σ ) = L r +1 ( g , σ ) ⊕ K ′ ⊂ ˆ L ′ r +1 ,k ( g )is a subalgebra of ˆ L ′ r +1 ,k ( g ), thanks to (4.4), (4.8). When σ is a diagramautomorphism, ˆ L ′ r +1 , ( g , σ ) is known as the twisted toroidal Lie algebra [21]. We will continue to use that terminology for an arbitrary finite-order automorphism σ of g . EPRESENTATIONS OF TWISTED TOROIDAL LIE ALGEBRAS 13
As in Section 4.1, we can add to ˆ L ′ r +1 ,k ( g , σ ) a subalgebra of the Liealgebra of derivations D of O . We define(4.9) ˆ L r +1 ,k ( g , σ ) = ˆ L ′ r +1 ,k ( g , σ ) ⊕ D ′ + where D ′ + = span C (cid:8) d i ⊗ t Nm t m (cid:12)(cid:12) m ∈ Z , m ∈ Z r , i = 1 , . . . , r (cid:9) . It is easy to see from (4.5)–(4.7) that ˆ L r +1 ,k ( g , σ ) is a subalgebra of thetoroidal Lie algebra ˆ L r +1 ,k ( g ). We will also call ˆ L r +1 ,k ( g , σ ) the twistedtoroidal Lie algebra of level k .4.3. Twisted modules over V k ( g ) ⊗ V J and twisted toroidal Liealgebras. We will now explore the relationship between twisted toroidalLie algebras in r + 1 variables and the tensor product of the universalaffine vertex algebra V k ( g ) with the lattice vertex algebra correspond-ing to r copies of a certain rank-2 lattice. The level k ∈ C will be fixedthrough the end of the section.As before, consider an automorphism σ of finite order N of a simplefinite-dimensional Lie algebra g , and the twisted toroidal Lie algebra(of level k ) ˆ L r +1 ,k ( g , σ ) defined by (4.9). For i = 1 , . . . , r , let J i be thelattice given by(4.10) J i = Z δ i ⊕ Z Λ i , ( δ i | Λ i ) = 1 , ( δ i | δ i ) = (Λ i | Λ i ) = 0 . We define a bimultiplicative function ε : J i × J i → {± } satisfying(2.18) by ε ( δ i , Λ i ) = − ε = 1 for all other pairs of generators.Then we can form the lattice vertex algebra V J i as in Section 2.3.Introduce the orthogonal direct sum(4.11) J = J ⊕ · · · ⊕ J r , and extend ε to J × J by ε ( δ i , Λ j ) = − i = j and ε = 1 for all otherpairs of generators. Then the lattice vertex algebra V J is isomorphicto the tensor product:(4.12) V J ≃ V J ⊗ · · · ⊗ V J r . As preparation for our main theorem, we need to calculate some n -thproducts in V J . We will use the notation pδ = r X i =1 p i δ i , p = ( p , . . . , p r ) ∈ Z r . Lemma 4.1.
The lattice vertex algebra V J has the following n -th prod-ucts for p , q ∈ Z r and i, j = 1 , . . . , r : e pδ ( − e qδ = e ( p + q ) δ , e pδ ( − e qδ = ( pδ ) ( − e ( p + q ) δ , (cid:0) u ( − e pδ (cid:1) ( n ) (cid:0) v ( − e qδ (cid:1) = 0 for n ≥ , u, v ∈ { , δ , . . . , δ r } , (cid:0) Λ i − e pδ (cid:1) (0) e qδ = q i e ( p + q ) δ , (cid:0) Λ i − e pδ (cid:1) ( n ) e qδ = 0 for n ≥ , (cid:0) Λ i − e pδ (cid:1) (0) (cid:0) δ j ( − e qδ (cid:1) = q i δ j ( − e ( p + q ) δ + δ i,j ( pδ ) ( − e ( p + q ) δ , (cid:0) Λ i − e pδ (cid:1) (1) (cid:0) δ j ( − e qδ (cid:1) = δ i,j e ( p + q ) δ , (cid:0) Λ i − e pδ (cid:1) ( n ) (cid:0) δ j ( − e qδ (cid:1) = 0 for n ≥ , (cid:0) Λ i − e pδ (cid:1) (0) (cid:0) Λ j − e qδ (cid:1) = (cid:0) − p j Λ i − + q i Λ j − − q i p j ( pδ ) ( − (cid:1) e ( p + q ) δ , (cid:0) Λ i − e pδ (cid:1) (1) (cid:0) Λ j − e qδ (cid:1) = − q i p j e ( p + q ) δ , (cid:0) Λ i − e pδ (cid:1) ( n ) (cid:0) Λ j − e qδ (cid:1) = 0 for n ≥ . Proof.
The n -th product (cid:0) Λ i − e pδ (cid:1) ( n ) (cid:0) Λ j − e qδ (cid:1) is the coefficient of z − n − in Y (cid:0) Λ i − e pδ , z (cid:1) Λ j − e qδ . Applying the ( − Y (Λ i − e pδ , z )Λ j − e qδ = : Y (Λ i , z ) Y ( e pδ , z ): Λ j − e qδ = X l< Λ i l ) z − l − ! e pδ z ( pδ ) (0) exp X m> ( pδ ) ( − m ) z m m ! ×× exp X m> ( pδ ) ( m ) z − m − m ! Λ j − e qδ + e pδ z ( pδ ) (0) exp X m> ( pδ ) ( − m ) z m m ! exp X m> ( pδ ) ( m ) z − m − m ! ×× X l ≥ Λ i l ) z − l − ! Λ j − e qδ . Next, we use the brackets (2.16) in the Heisenberg Lie algebra and itsaction (2.19) on its highest-weight vectors, to get:[( pδ ) ( m ) , Λ j − ] = δ m, p j , [Λ i l ) , Λ j − ] = 0 , and ( pδ ) ( m ) e qδ = 0 , Λ i l ) e qδ = δ l, q i e qδ , m, l ≥ . EPRESENTATIONS OF TWISTED TOROIDAL LIE ALGEBRAS 15
Moreover, since ε ( δ i , δ j ) = 1, we have e pδ e qδ = e ( p + q ) δ . From here, weobtain: Y (Λ i − e pδ , z )Λ j − e qδ = X l< Λ i l ) z − l − ! exp X m> ( pδ ) ( − m ) z m m !(cid:0) Λ j − − p j z − (cid:1) e ( p + q ) δ + exp X m> ( pδ ) ( − m ) z m m !(cid:0) q i Λ j − z − − q i p j z − (cid:1) e ( p + q ) δ . To find (cid:0) Λ i − e pδ (cid:1) ( n ) (cid:0) Λ j − e qδ (cid:1) for n ≥
0, we extract the terms withnegative powers of z from the above expression: (cid:0) − p j Λ i − z − + q i Λ j − z − − q i p j z − − q i p j ( pδ ) ( − z − (cid:1) e ( p + q ) δ This gives us (cid:0) Λ i − e pδ (cid:1) (0) (cid:0) Λ j − e qδ (cid:1) = (cid:0) − p j Λ i − + q i Λ j − − q i p j ( pδ ) ( − (cid:1) e ( p + q ) δ , (cid:0) Λ i − e pδ (cid:1) (1) (cid:0) Λ j − e qδ (cid:1) = − q i p j e ( p + q ) δ , (cid:0) Λ i − e pδ (cid:1) ( n ) (cid:0) Λ j − e qδ (cid:1) = 0 for n ≥ . The other n -th products follow from similar reasoning.Recall the universal affine vertex algebra V k ( g ) of level k , definedin Section 3.2. We will consider the tensor product of vertex algebras V k ( g ) ⊗ V J ; see (2.9). We can extend σ to an automorphism σ of V k ( g )of order N as described in (3.6). Then we extend σ to an automorphismof the vertex algebra V k ( g ) ⊗ V J , again of order N , by letting(4.13) σ ( a ⊗ b ) = σa ⊗ b. Let M be a σ -twisted V k ( g )-module, and M ′ be a V J -module (un-twisted). Then M = M ⊗ M ′ is a σ -twisted V k ( g ) ⊗ V J -module with astate-field correspondence Y M given by (2.15). Now we can formulateour main theorem, which uses twisted V k ( g ) ⊗ V J -modules to createrepresentations of the twisted toroidal Lie algebra. Theorem 4.2.
Let σ be an automorphism of order N of a simplefinite-dimensional Lie algebra g , and let M = M ⊗ M ′ be a σ -twisted V k ( g ) ⊗ V J -module. Then the Lie algebra of modes of the fields Y M ( a ⊗ e pδ , z ) , Y M ( ⊗ e pδ , z ) ,Y M ( ⊗ δ i ( − e pδ , z ) , Y M ( ⊗ Λ i − e pδ , z ) , where a ∈ g and p ∈ Z r , form a representation of the twisted toroidalLie algebra ˆ L r +1 ,k ( g , σ − ) of level k on M . Explicitly, we have a Liealgebra homomorphism ϕ : ˆ L r +1 ,k ( g , σ − ) → End M given by : a ⊗ t m t p (cid:0) a ⊗ e pδ (cid:1) M ( mN ) ,K ⊗ t Nm t p N (cid:0) ⊗ e pδ (cid:1) M ( m − ,K i ⊗ t Nm t p (cid:0) ⊗ δ i ( − e pδ (cid:1) M ( m ) ,d i ⊗ t Nm t p (cid:0) ⊗ Λ i − e pδ (cid:1) M ( m ) , (4.14) for p ∈ Z r , ≤ i ≤ r , and a ∈ g , m ∈ Z such that σa = e − π i m/N a . Proof.
Recall that L r +1 ( g , σ − ) m for m ∈ Z is spanned by all elementsof the form a ⊗ t m t p where p ∈ Z r and a ∈ g with σ − a = e π i m/N a . Thelatter condition is equivalent to σa = e − π i m/N a . Hence, ˆ L r +1 ,k ( g , σ − )is spanned by all elements in the left-hand side of (4.14), subject tothe relations: N mK ⊗ t Nm t p + r X i =1 p i K i ⊗ t Nm t p = 0 . For the map ϕ to be well defined, we need to check that N m N (cid:0) ⊗ e pδ (cid:1) M ( m − + r X i =1 p i (cid:0) ⊗ δ i ( − e pδ (cid:1) M ( m ) = 0 , or equivalently, m (cid:0) ⊗ e pδ (cid:1) M ( m − + (cid:0) ⊗ ( pδ ) ( − e pδ (cid:1) M ( m ) = 0 . This follows from the translation covariance property (2.6): − m (cid:0) ⊗ e pδ (cid:1) M ( m − = (cid:0) T ( ⊗ e pδ ) (cid:1) M ( m ) = (cid:0) ⊗ T e pδ (cid:1) M ( m ) = (cid:0) ⊗ ( pδ ) ( − e pδ (cid:1) M ( m ) , where we used (2.21) and that T acts as T ⊗ I + I ⊗ T on a tensorproduct of vertex algebras.To show that ϕ is a homomorphism, we need to check that thedefining Lie brackets (4.3)–(4.7) of the twisted toroidal Lie algebraˆ L r +1 ,k ( g , σ − ) match with the commutators of modes in the right-hand side of (4.14). The latter are determined by the commutatorformula (2.14). We can apply (2.14) because σa = e − π i m/N a im-plies σ ( a ⊗ e pδ ) = e − π i m/N ( a ⊗ e pδ ), since σ acts as the identity EPRESENTATIONS OF TWISTED TOROIDAL LIE ALGEBRAS 17 on V J . Similarly, if b ∈ g and n ∈ Z satisfy σb = e − π i n/N a , then σ ( b ⊗ e qδ ) = e − π i n/N ( b ⊗ e qδ ) for q ∈ Z r . Hence, h ( a ⊗ e pδ ) M ( mN ) , ( b ⊗ e qδ ) M ( nN ) i = ∞ X l =0 (cid:18) mN l (cid:19)(cid:0) ( a ⊗ e pδ ) ( l ) ( b ⊗ e qδ ) (cid:1) M ( mN + nN − l ) . The l -th product ( a ⊗ e pδ ) ( l ) ( b ⊗ e qδ ) is the coefficient of z − l − in theexpression Y ( a ⊗ e pδ , z )( b ⊗ e qδ ) = Y ( a, z ) b ⊗ Y ( e pδ , z ) e qδ . We are only interested in the negative powers of z . By the products(3.5) in the affine vertex algebra V k ( g ), we have Y ( a, z ) b = ( a | b ) k z − + [ a, b ] z − + · · · , where · · · denote terms with higher powers of z . The products in thelattice vertex algebra V J were computed in Lemma 4.1. In particular,we have Y ( e pδ , z ) e qδ = e ( p + q ) δ + ( pδ ) ( − e ( p + q ) δ z + · · · . Putting these together, we get Y ( a ⊗ e pδ , z )( b ⊗ e qδ ) = z − ( a | b ) k ⊗ e ( p + q ) δ + z − [ a, b ] ⊗ e ( p + q ) δ + z − ( a | b ) k ⊗ ( pδ ) ( − e ( p + q ) δ + · · · . Hence, we obtain the bracket (cid:2) ϕ (cid:0) a ⊗ t m t p (cid:1) , ϕ (cid:0) b ⊗ t n t q (cid:1)(cid:3) = h(cid:0) a ⊗ e pδ (cid:1) M ( mN ) , (cid:0) b ⊗ e qδ (cid:1) M ( nN ) i = (cid:0) [ a, b ] ⊗ e ( p + q ) δ (cid:1) M ( mN + nN ) + ( a | b ) k (cid:0) ⊗ ( pδ ) ( − e ( p + q ) δ (cid:1) M ( mN + nN ) + mN ( a | b ) k (cid:0) ⊗ e ( p + q ) δ (cid:1) M ( mN + nN − = ϕ (cid:0) [ a, b ] ⊗ t m + n t p + q (cid:1) + k ( a | b ) r X i =1 p i ϕ (cid:0) K i ⊗ t m + n t p + q (cid:1) + k ( a | b ) mϕ (cid:0) K ⊗ t m + n t p + q (cid:1) . By (4.2), the last expression is exactly ϕ ([ a ⊗ t m t p , b ⊗ t n t q ]).We compute the other brackets in a similar fashion, and we obtainfor m, n ∈ Z , a ∈ C ⊕ g , 1 ≤ i, j ≤ r , p , q ∈ Z r : h(cid:0) ⊗ u ( − e pδ (cid:1) M ( m ) , (cid:0) a ⊗ e qδ (cid:1) M ( nN ) i = h(cid:0) ⊗ u ( − e pδ (cid:1) M ( m ) , (cid:0) ⊗ δ j ( − e qδ (cid:1) M ( n ) i = 0 , u ∈ { , δ i } , h(cid:0) ⊗ Λ i − e pδ (cid:1) M ( m ) , (cid:0) a ⊗ e qδ (cid:1) M ( nN ) i = q i (cid:0) a ⊗ e ( p + q ) δ (cid:1) M ( m + nN ) , h(cid:0) ⊗ Λ i − e pδ (cid:1) M ( m ) , (cid:0) ⊗ δ j ( − e qδ (cid:1) M ( n ) i = q i (cid:0) ⊗ δ j ( − e ( p + q ) δ (cid:1) M ( m + n ) + δ i,j (cid:0) ⊗ ( pδ ) ( − e ( p + q ) δ (cid:1) M ( m + n ) + mδ i,j (cid:0) ⊗ e ( p + q ) δ (cid:1) M ( m + n − , h(cid:0) ⊗ Λ i − e pδ (cid:1) M ( m ) , (cid:0) ⊗ Λ j − e qδ (cid:1) M ( n ) i = q i (cid:0) ⊗ Λ j − e ( p + q ) δ (cid:1) M ( m + n ) − p j (cid:0) ⊗ Λ i − e ( p + q ) δ (cid:1) M ( m + n ) − q i p j (cid:0) ⊗ ( pδ ) ( − e ( p + q ) δ (cid:1) M ( m + n ) − mq i p j (cid:0) ⊗ e ( p + q ) δ (cid:1) M ( m + n − . A close inspection shows that these brackets agree with the brackets(4.3)–(4.7) in the twisted toroidal Lie algebra ˆ L r +1 ,k ( g , σ − ). This com-pletes the proof of the theorem.4.4. Twisted modules over lattice vertex algebras and twistedtoroidal Lie algebras.
Now let g be a simple Lie algebra of type X ℓ where X = A, D, E (simply laced), with a root system ∆ and a rootlattice Q = Z ∆. Let σ be an automorphism of the lattice Q of finiteorder N .Consider the orthogonal direct sum of lattices L = Q ⊕ J, where, as before, J is defined by (4.10), (4.11). We extend σ to anautomorphism of L , acting as the identity on J , and we lift it to anautomorphism of the lattice vertex algebra V L ≃ V Q ⊗ V J . Finally, let H = C ⊗ Z L = h ⊕ span { δ i , Λ i } ≤ i ≤ r , where h = C ⊗ Z Q is the Cartan subalgebra of g .With the above notation, using the Frenkel–Kac construction [18],we can reformulate Theorem 4.2 as follows. Corollary 4.3.
For any σ -twisted V L -module M , the Lie algebra ofmodes of the fields Y M ( e α + pδ , z ) , Y M ( h ( − e pδ , z ) ( α ∈ ∆ ∪ { } , h ∈ H , p ∈ Z r ) form a representation of the twisted toroidal Lie algebra ˆ L r +1 , ( g , σ − ) of level on M . Proof.
The proof is the same as the proof of Corollary 3.3.
EPRESENTATIONS OF TWISTED TOROIDAL LIE ALGEBRAS 19
In the special case when σ is a Coxeter element in the Weyl groupof g , the above corollary recovers Billig’s construction from [9].Observe that, for r = 1, the lattice Q ⊕ Z δ is the root lattice of theaffine Kac–Moody algebra ˆ g of type X (1) ℓ , while H = h ⊕ C δ ⊕ C Λ is theCartan subalgebra of ˆ g . Moreover, the set { α + pδ | α ∈ ∆ ∪{ } , p ∈ Z } is the union of { } and the root system of ˆ g . A natural question iswhether one can generalize the statement of Corollary 4.3 to the casewhen σ is an infinite-order automorphism of L , such as, for example,an element of the Weyl group of ˆ g (cf. [4]). Acknowledgments.
We are grateful to Kailash Misra and NaihuanJing for valuable discussions. The first author was supported in partby a Simons Foundation grant 584741.
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