Automorphism group and twisted modules of the twisted Heisenberg-Virasoro vertex operator algebra
aa r X i v : . [ m a t h . QA ] A ug Automorphism group and twisted modules of thetwisted Heisenberg-Virasoro vertex operatoralgebra
Hongyan Guo School of Mathematics and Statistics, Central China Normal University,Wuhan 430079, China
Abstract
We first determine the automorphism group of the twisted Heisenberg-Virasoro vertex operator algebra V L ( ℓ , t >
1, weintroduce a new Lie algebra L t , and show that σ t -twisted V L ( ℓ , ℓ = 0)-modules are in one-to-one correspondence with restricted L t -modules of level ℓ , where σ t is an order t automorphism of V L ( ℓ , σ t -twisted V L ( ℓ , ℓ = 0)-modules. Let L be the twisted Heisenberg-Virasoro algebra. It is the universal central exten-sion of the Lie algebra of differential operators on a circle of order at most one (cf.[1]): { f ( t ) ddt + g ( t ) | f ( t ) , g ( t ) ∈ C [ t, t − ] } . L contains an infinite-dimensional Heisenberg algebra and the Virasoro algebra assubalgebras (cf. [1], [4]). The induced module V L ( ℓ ,
0) = U ( L ) ⊗ U ( L ( ≤ ) C ℓ is a vertex operator algebra of central charge ℓ with conformal vector ω = L − (cf. [12]). V L ( ℓ ,
0) is a nonrational vertex operator algebra and is not C -cofinite.The structure theory and representation theory of the twisted Heisenberg-Virasorovertex operator algebra V L ( ℓ ,
0) are closely related to the three scalars ℓ , ℓ , ℓ (cf. [1], [2], [3], [4], [5], [9], [12], etc.).Determining the automorphism group Aut( V ) of a vertex operator algebra V isan important subject in vertex operator algebra theory. It is related to the orbifoldtheory which studies the fixed point subalgebras of vertex operator algebras andtheir modules under certain finite subgroups of the full automorphism groups. Theorbifold conjecture says that under some conditions on V , every simple V G -moduleis contained in some g -twisted V -module, where G is a finite subgroup of Aut( V ), g ∈ G , V G is the fixed point subalgebra of V under the group G .The automorphism group of the twisted Heisenberg-Virasoro algebra L has beenstudied in [15]. In this paper, we study the automorphism group of the twistedHeisenberg-Virasoro vertex operator algebra V L ( ℓ , V L ( ℓ , ω = L − and I − . By definition, any homomorphism of a vertex Partially supported by NSFC (No.11901224) and NSF of Hubei Province (No.2019CFB160)
V, Y, , ω ) takes ω to ω . Therefore, it suffices to determine theaction on I − . It turns out that automorphisms of the twisted Heisenberg-Virasorovertex operator algebra V L ( ℓ ,
0) depend on the numbers ℓ and ℓ , and there existsautomorphism of order other than 2, which makes the study of twisted modules for V L ( ℓ ,
0) interesting.In [12], all irreducible modules for the vertex operator algebra V L ( ℓ ,
0) areclassified: that is, every irreducible module for V L ( ℓ ,
0) is isomorphic to some L L ( ℓ , h , h ), h , h ∈ C . Here, for any integer t >
1, we classify σ t -twistedirreducible modules for V L ( ℓ , σ t is an order t automorphism of V L ( ℓ , L t . It is a Lie algebra with basis { L n , I n + t , k , k | n ∈ Z } , and Lie brackets[ L m , L n ] = ( m − n ) L m + n + δ m + n, m − m k , [ L m , I n + t ] = − ( n + 1 t ) I m + n + t , [ I m + t , I n + t ] = ( m + 1 t ) δ m + n + t , δ t, k , [ L , k i ] = 0 , i = 1 , . Note that when t = 2, { I n + t | n ∈ Z } is an abelian Lie algebra. Then we con-struct irreducible L t -modules L L t ( k , k , h ) as quotient of the induced modules M L t ( k , k , h ), where k , k , h ∈ C . We show that σ t -twisted V L ( ℓ , ℓ = 0) are in one-to-one correspondence with restricted L t -modules of level ℓ .Using this result, we get a complete list of irreducible σ t -twisted V L ( ℓ , ℓ = 0, ℓ , ℓ ∈ C . Let V L ( ℓ , σ t be the fixed point subalgebra of V L ( ℓ ,
0) under the automorphism σ t . V L ( ℓ , σ t is a vertex operator subalgebraof V L ( ℓ , V L ( ℓ , σ t . We remark at the end of the paper that exceptfor the case of order 2, the complete list of irreducible modules for V L ( ℓ , σ t needsto be further investigated.This paper is organized as follows. In Section 2, we review the notions andsome results of vertex operator algebras, automorphisms and twisted modules forvertex operator algebras. In Section 3, we study the automorphism group of thetwisted Heisenberg-Virasoro vertex operator algebra V L ( ℓ , V L ( ℓ , ℓ = 0) under an order t automorphism σ t for any integer t >
1. Then we give a complete list of irreducible σ t -twisted V L ( ℓ , ℓ = 0)-modules. For later use, we recall the following result (cf. Proposition 2.3.7 of [14]).2 emma 2.1. ( x − x ) m (cid:16) ∂∂x (cid:17) n x − δ (cid:16) x x (cid:17) = 0 (2.1) for m > n , m, n ∈ N , where δ (cid:16) x x (cid:17) = X n ∈ Z x n x − n . For the definition of vertex (operator) algebra and its modules, we follow [14].
Definition 2.2. A vertex algebra (V,Y, ) consists of a vector space V , a linear map(the vertex operator map) Y ( · , x ) : V −→ ( EndV )[[ x, x − ]] , v Y ( v, x ) = X n ∈ Z v n x − n − , and a vacuum vector such that the following conditions hold for u, v ∈ V :(1) truncation condition : u n v = 0 for n sufficiently large;(2) vacuum property : Y ( , x ) = 1;(3) creation property : Y ( v, x ) ∈ V [[ x ]] and lim x → Y ( v, x ) = v ;(4) Jacobi identity : x − δ (cid:18) x − x x (cid:19) Y ( u, x ) Y ( v, x ) − x − δ (cid:18) x − x − x (cid:19) Y ( v, x ) Y ( u, x )= x − δ (cid:18) x − x x (cid:19) Y ( Y ( u, x ) v, x ) . Let D be the endomorphism of the vertex algebra V defined by D ( v ) = v − for v ∈ V . Then we have Y ( Dv, x ) = ddx Y ( v, x ). Definition 2.3. A vertex operator algebra is a Z -graded vector space (graded byweights) V = a n ∈ Z V ( n ) for v ∈ V ( n ) , n = wt v, equipped with a vertex algebra structure ( V, Y, ) and a distinguished homogeneousvector ω (the conformal vector ) of weight 2 ( ω ∈ V (2) ) such that(1) two grading restrictions: dim V ( n ) < ∞ for n ∈ Z ,V ( n ) = 0 for n sufficiently negative;32) Virasoro algebra relations: [ L ( m ) , L ( n )] = ( m − n ) L ( m + n ) + 112 ( m − m ) δ m + n, c V for m, n ∈ Z , where Y ( ω, x ) = X n ∈ Z L ( n ) x − n − (= X n ∈ Z ω n x − n − );and c V ∈ C ( central charge or rank of V );(3) L (0) -grading: L (0) v = nv = ( wt v ) v for n ∈ Z and v ∈ V ( n ) ;(4) L ( − -derivative property: Y ( L ( − v, x ) = ddx Y ( v, x ).It can be proved that ∈ V (0) and D = L ( − Definition 2.4. An automorphism of a vertex operator algebra ( V, Y, ω, ) is alinear isomorphism σ of V such that σ ( ) = , σ ( ω ) = ω and σ ( u n v ) = σ ( u ) n σ ( v )for any u, v ∈ V , n ∈ Z .The group of all automorphisms of a vertex operator algebra V is denoted byAut( V ). Any automorphism of a vertex operator algebra V is grading-preserving ,i.e. it preserves each homogeneous subspace V ( n ) of V , n ∈ Z . Let σ be an order t automorphism of a vertex operator algebra V , t is a positive integer. Then σ actssemisimply on V . Therefore V = V ⊕ V ⊕ · · · ⊕ V t − where V k is the eigenspace of V for σ with eigenvalues η k , where η = exp( π √− t ), k = 0 , . . . , t −
1. It is easy to see that the fixed points set V := V σ = { v ∈ V | σ ( v ) = v } is a vertex operator subalgebra of V Now we recall some notions regarding to twisted modules from [13].
Definition 2.5.
Let ( V, , Y ) be a vertex algebra with an automorphism σ of order t . A σ -twisted V -module is a triple ( W, d, Y W ) consisting of a vector space W , anendomorphism d of W and a linear map Y W ( · , z ) : V −→ (End W )[[ z t , z − t ]]satisfying the following conditions:(TW1) For any v ∈ V, w ∈ W , v n w = 0 for n ∈ t Z sufficiently large;4TW2) Y W ( , z ) = Id W ;(TW3) [ d, Y W ( v, z )] = Y W ( D ( v ) , z ) = ddz Y W ( v, z ) for any v ∈ V ;(TW4) For any u, v ∈ V , the following σ -twisted Jacobi identity holds: z − δ (cid:18) z − z z (cid:19) Y W ( u, z ) Y W ( v, z ) − z − δ (cid:18) z − z − z (cid:19) Y W ( v, z ) Y W ( u, z )= z − t t − X k =0 δ (cid:18) z − z z (cid:19) t ! Y W ( Y ( σ k u, z ) v, z ) . If V is a vertex operator algebra, a σ -twisted V -module for V as a vertex algebrais called a σ -twisted weak module for V as a vertex operator algebra. Definition 2.6.
For V a vertex operator algebra, a σ -twisted V -module W is a σ -twisted weak module for V as a vertex algebra and W = ` h ∈ C W ( h ) such that(TW5) L (0) w = hw for h ∈ C , w ∈ W ( h ) ;(TW6) For any fixed h, W ( h + n ) = 0 for n ∈ t Z sufficiently small;(TW7) dim W ( h ) < ∞ for any h ∈ C . Remark 2.7.
Let W be a σ -twisted V -module. Then[ L (0) , Y W ( v, z )] = zY W ( L ( − v, z ) + Y W ( L (0) v, z ) (2.2)for v ∈ V . Hence, if v ∈ V is homogeneous, v n W ( h ) ⊆ W ( h + wt v − n − for n ∈ t Z , h ∈ C . It follows that a σ -twisted V -module W decomposes into twisted submodules cor-responding to the congruence classes mod t Z : For h ∈ C / t Z , let W [ h ] = a α + t Z = h W ( α ) . (2.3)Then W = a h ∈ C / t Z W [ h ] . (2.4)In particular, if W is irreducible, then W = W [ h ] (2.5)for some h . 5 emark 2.8. Let W be a σ -twisted V -module. For u ∈ V k , v ∈ V , there is thefollowing twisted iterate formula (cf. (2.32) of [13]) Y W ( Y ( u, z ) v, z ) = Res z (cid:18) z − z z (cid:19) kt · X (2.6)where X = z − δ (cid:18) z − z z (cid:19) Y W ( u, z ) Y W ( v, z ) − z − δ (cid:18) z − z − z (cid:19) Y W ( v, z ) Y W ( u, z ) . Then it can be easily deduced that for any n ∈ Z , Y W ( u n v, z ) = Res z ∞ X j =0 ( − j (cid:18) kt j (cid:19) z kt − j z − kt · Y j (2.7)where Y j = (cid:16) ( z − z ) n + j Y W ( u, z ) Y W ( v, z ) − ( − z + z ) n + j Y W ( v, z ) Y W ( u, z ) (cid:17) , (cid:18) kt j (cid:19) = kt ( kt − · · · ( kt − j + 1) j ! . Note that when k = 0, kt = 0, we have j = 0, andthen (2.7) is the usual formula for (untwisted) V -modules (cf. (3.8.16) of [14]). Definition 2.9. A homomorphism between two σ -twisted weak V -modules M and W is a linear map f : M −→ W such that for any v ∈ V , f Y M ( v, z ) = Y W ( v, z ) f. (2.8)If V is a vertex operator algebra, then a V -module homomorphism f is compatiblewith the gradings: f ( M ( h ) ) ⊆ W ( h ) for h ∈ C . (2.9)Using the formula (2.7), similarly as Proposition 4.5.1 of [14], there is the fol-lowing result. Proposition 2.10.
Let W and W be σ -twisted V -modules and let ψ ∈ Hom C ( W , W ) .Suppose that Y ( a, z ) ψ = ψY ( a, z ) , for a ∈ S, where S is a given generating set of V . Then ψ is a σ -twisted V -module homomor-phism. In the following, we review from Section 3 of [13] the local systems of twistedvertex operators. 6 efinition 2.11.
Let W be a vector space, let t be a fixed positive integer. A Z t -twisted weak vertex operator on W is a formal series a ( z ) = P n ∈ t Z a n z − n − ∈ (End W )[[ z t , z − t ]] such that for any w ∈ W , a n w = 0 for n ∈ t Z sufficiently large. Definition 2.12.
Two Z t -twisted weak vertex operators a ( z ) and b ( z ) are said tobe mutually local if there is a positive integer n such that( z − z ) n a ( z ) b ( z ) = ( z − z ) n b ( z ) a ( z ) . A Z t -twisted weak vertex operator is called a Z t -twisted vertex operator if it is localwith itself.Denote by F ( W, t ) the space of all Z t -twisted weak vertex operators on W . Let σ be the endomorphism of (End W )[[ z t , z − t ]] defined by: σf ( z t ) = f ( η − z t ).Denote by F ( W, t ) k = { f ( z ) ∈ F ( W, t ) | σf ( z ) = η k f ( z ) } for 0 ≤ k ≤ t −
1. Forany mutually local Z t -twisted vertex operators a ( z ) , b ( z ) on W , define a ( z ) n b ( z ) asfollows (cf. Definition 3.7 of [13]). Definition 2.13.
Let W be a vector space and let a ( z ) and b ( z ) be mutually local Z t -twisted vertex operators on W such that a ( z ) ∈ F ( W, t ) k . Then for any integer n we define a ( z ) n b ( z ) as an element of F ( W, t ) as follows: a ( z ) n b ( z ) = Res z Res z (cid:18) z − z z (cid:19) kt z n · X (2.10)where X = z − δ (cid:18) z − zz (cid:19) a ( z ) b ( z ) − z − δ (cid:18) z − z − z (cid:19) b ( z ) a ( z ) . For any set S of mutually local Z t -twisted vertex operators on W , by Zorn’sLemma, there exists a local system A of Z t -twisted vertex operators on W (cf.Section 3 of [13]). Denote by h S i the vertex algebra generated by S inside A via theoperations a ( z ) n b ( z ), n ∈ Z . Furthermore, there is the following result (cf. Corollary3.15 of [13]). Proposition 2.14.
Let W be any vector space. Let S be a set of mutually local Z t -twisted vertex operators on W . Then h S i is a vertex algebra with an automorphism σ of order t such that W is a σ -twisted h S i -module in the sense Y W ( a ( z ) , z ) = a ( z ) . In this section, we first recall the definition of the twisted Heisenberg-Virasoro al-gebra L and the construction of the twisted Heisenberg-Virasoro vertex operatoralgebra V L ( ℓ , V L ( ℓ , L (see [1] or [4]).7 efinition 3.1. The twisted Heisenberg-Virasoro algebra L is a Lie algebra withbasis { L n , I n , c , c , c | n ∈ Z } , and the following Lie brackets:[ L m , L n ] = ( m − n ) L m + n + δ m + n, m − m c , (3.1)[ L m , I n ] = − nI m + n − δ m + n, ( m + m ) c , (3.2)[ I m , I n ] = mδ m + n, c , [ L , c i ] = 0 , i = 1 , , . (3.3)Clearly, Span { L n , c | n ∈ Z } is a Virasoro algebra, Span { I n , c | n ∈ Z \{ }} isan infinite-dimensional Heisenberg algebra, we denote them by V ir , H respectively.Let L ( z ) = X n ∈ Z L n z − n − , I ( z ) = X n ∈ Z I n z − n − , then the defining relations of L become to be[ L ( z ) , L ( z )]= X m,n ∈ Z ( m − n ) L m + n z − m − z − n − + X m ∈ Z m − m c z − m − z m − = ddz (cid:16) L ( z ) (cid:17) z − δ (cid:18) z z (cid:19) + 2 L ( z ) ∂∂z z − δ (cid:18) z z (cid:19) + c (cid:18) ∂∂z (cid:19) z − δ (cid:18) z z (cid:19) , (3.4)[ L ( z ) , I ( z )]= − X m,n ∈ Z nI m + n z − m − z − n − − X m ∈ Z ( m + m ) c z − m − z m − = ddz (cid:16) I ( z ) (cid:17) z − δ (cid:18) z z (cid:19) + I ( z ) ∂∂z z − δ (cid:18) z z (cid:19) − (cid:18) ∂∂z (cid:19) z − δ (cid:18) z z (cid:19) c , (3.5)[ I ( z ) , I ( z )] = X m ∈ Z mc z − m − z m − = ∂∂z z − δ (cid:18) z z (cid:19) c . (3.6)We recall the construction of the twisted Heisenberg-Virasoro vertex operatoralgebra V L ( ℓ ,
0) from [12]. Let L ( ≤ = a n ≤ C L − n ⊕ a n ≤ C I − n ⊕ X i =1 C c i , ( ≥ = a n ≥ C L − n ⊕ a n ≥ C I − n . They are graded subalgebras of L and L = L ( ≤ ⊕ L ( ≥ . Let ℓ i , i = 1 , , , be any complex numbers. Consider C as an L ( ≤ -modulewith c i acting as the scalar ℓ i , i = 1 , , , and with ` n ≤ C L − n ⊕ ` n ≤ C I − n actingtrivially. Denote this L ( ≤ -module by C ℓ . Form the induced module V L ( ℓ ,
0) = U ( L ) ⊗ U ( L ( ≤ ) C ℓ , (3.7)where U ( · ) denotes the universal enveloping algebra of a Lie algebra. Set =1 ⊗ ∈ V L ( ℓ , V L ( ℓ ,
0) is a vertex operator algebra with vacuum vector and conformal vector ω = L − . And { ω = L − , I := I − } is a generating subsetof V L ( ℓ , V L ( ℓ , V L ( ℓ ,
0) = a n ≥ V L ( ℓ , ( n ) , where V L ( ℓ , (0) = C and V L ( ℓ , ( n ) , n ≥
1, has a basis consisting of thevectors I − k · · · I − k s L − m · · · L − m r for r, s ≥ m ≥ · · · ≥ m r ≥ k ≥ · · · ≥ k s ≥ r P i =1 m i + s P j =1 k j = n. Now we give our first main result. The automorphism group of V L ( ℓ ,
0) isdetermined in the following theorem.
Theorem 3.2. (1) If ℓ = 0 , then Aut ( V L ( ℓ , { id } . (2) If ℓ = 0 and ℓ = 0 , then Aut ( V L ( ℓ , ∼ = Z .(3) If both ℓ and ℓ are 0, then Aut ( V L ( ℓ , ∼ = C × = C \{ } . Proof.
Let ϕ : V L ( ℓ , −→ V L ( ℓ , V L ( ℓ , ϕ ( ) = and ϕ ( ω ) = ω. Since V L ( ℓ ,
0) is generated by ω and I = I − , it suffices to determine ϕ ( I ). ϕ is grading-preserving, so ϕ ( I ) = aI for some a ∈ C × .Then, on the one hand, we have ϕ ( L I ) = aL I = a [ L , I − ] = − aℓ , on the other hand, we have ϕ ( L I ) = ϕ ([ L , I − ] ) = − ℓ . ℓ = 0, we get that a = 1, i.e. when ℓ = 0, Aut( V L ( ℓ , { id } only consists of the identity map.Suppose now ℓ = 0. Let’s consider ϕ ( I I ). On the one hand, ϕ ( I I ) = ϕ ( I ) ϕ ( I ) = a I I − = a [ I , I − ] = a ℓ , on the other hand, ϕ ( I I ) = ϕ ([ I , I − ] ) = ℓ . So if ℓ = 0, we get a = 1, i.e. a = 1 or a = −
1, then Aut( V L ( ℓ , ∼ = Z .Now let ℓ = 0 and ℓ = 0, then a can be any nonzero complex number, soAut( V L ( ℓ , ∼ = C × . σ t -twisted V L ( ℓ , -modules In this section, we always require ℓ = 0. We study twisted modules for the vertexoperator algebra V L ( ℓ , t >
1, we introduce aninfinite-dimensional Lie algebra L t . We show that there is a one-to-one correspon-dence between restricted L t -modules of level ℓ and σ t -twisted V L ( ℓ , σ t is an order t automorphism of V L ( ℓ , σ t -twisted V L ( ℓ , ℓ = 0 and ℓ = 0, then t can only be the integer 2 (Theorem 3.2).As we need, we introduce the following Lie algebra. Definition 4.1.
Let L t be a Lie algebra with basis { L n , I n + t , k , k | n ∈ Z } , andthe Lie brackets are given by:[ L m , L n ] = ( m − n ) L m + n + δ m + n, m − m k , (4.1)[ L m , I n + t ] = − ( n + 1 t ) I m + n + t , (4.2)[ I m + t , I n + t ] = ( m + 1 t ) δ m + n + t , δ t, k , [ L , k i ] = 0 , i = 1 , . (4.3)Note that if t = 2, then [ I m + t , I n + t ] = 0 for any m, n ∈ Z .Form the generating function as L ( z ) = X n ∈ Z L n z − n − , I σ t ( z ) = X n ∈ Z I n + t z − n − t − . L ( z ) , L ( z )]= ddz ( L ( z )) z − δ (cid:18) z z (cid:19) + 2 L ( z ) ∂∂z z − δ (cid:18) z z (cid:19) + 112 (cid:18) ∂∂z (cid:19) z − δ (cid:18) z z (cid:19) k , (4.4)[ L ( z ) , I σ t ( z )]= − X m,n ∈ Z ( n + 1 t ) I m + n + t z − m − z − n − t − = ddz (cid:16) I σ t ( z ) (cid:17) z − δ (cid:18) z z (cid:19) + I σ t ( z ) ∂∂z z − δ (cid:18) z z (cid:19) , (4.5)[ I σ t ( z ) , I σ t ( z )] = X m ∈ Z ( m + 1 t ) z − m − t − z m + t − δ t, k = ∂∂z z − δ (cid:18) z z (cid:19) (cid:18) z z (cid:19) t ! δ t, k . (4.6)Now we construct irreducible L t -modules (cf. [12], [14], etc.). Let( L t ) ≥ = ( a m ≥ C L m ) ⊕ ( a n ≥ C I n + t ) ⊕ C k ⊕ C k . It is a subalgebra of L t . Let C be an ( L t ) ≥ -module, where L m , I n + t act trivially for all m ≥ n ≥ L , k , k act as scalar multiplications by h, k , k respectively. Denote this( L t ) ≥ -module by C k ,h . Form the induced module M L t ( k , k , h ) = U ( L t ) ⊗ U (( L t ) ≥ ) C k ,h . Set k ,h = 1 ∈ C k ,h ⊂ M L t ( k , k , h ) . Then M L t ( k , k , h ) is C -graded by L -eigenvalues: M L t ( k , k , h ) = a n ≥ M L t ( k , k , h ) n + h , where M L t ( k , k , h ) ( h ) = C k ,h and M L t ( k , k , h ) ( n + h ) is the L -eigenspace of eigen-value n + h for n > M L [ − ( k , k , h ) ( n + h ) has a basis consisting of I − k + t · · · I − k s + t L − m · · · L − m r k ,h for r, s ≥ m ≥ · · · ≥ m r ≥ k ≥ · · · ≥ k s ≥ r P i =1 m i + s P j =1 ( k j − t ) = n , n >
0. 11 emark 4.2.
As a module for L t , M L t ( k , k , h ) is generated by k ,h with therelations L k ,h = h k ,h , k i = k i , i = 1 , , and L m k ,h = 0 , I n + t k ,h = 0 for m ≥ , n ≥ .M L t ( k , k , h ) is universal in the sense that for any L t -module W of level k equipped with a vector v such that L v = hv, L m v = 0 , I n + t v = 0 for m ≥ , n ≥ L t -module map M L t ( k , k , h ) −→ W sending k ,h to v .In general, M L t ( k , k , h ) as an L t -module may be reducible. Since C k ,h gen-erate M L t ( k , k , h ) as L t -module, for any proper submodule U of M L t ( k , k , h ), U ( h ) = U T M L t ( k , k , h ) ( h ) = 0. Hence there exists a maximal proper L t -submodule T L t ( k , k , h ). Set L L t ( k , k , h ) = M L t ( k , k , h ) /T L t ( k , k , h ) . Then L L t ( k , k , h ) is an irreducible L t -module. Definition 4.3. An L t -module W is said to be restricted if for any w ∈ W, n ∈ Z , L n w = 0 and I n + t w = 0 for n sufficiently large. We say an L t -module W is of level k if the central element k i acts as scalar k i for i = 1 , . It is easy to see that M L t ( k , k , h ), L L t ( k , k , h ) are restricted L t -modules oflevel k , for any h ∈ C . Now we are going to relate L t -modules with twisted V L ( ℓ , Theorem 4.4. If W is a restricted L t -module of level ℓ , then W is a σ t -twisted V L ( ℓ , -module for V L ( ℓ , as a vertex algebra with Y σ t ( L − , z ) = L ( z ) = X n ∈ Z L n z − n − ,Y σ t ( I − , z ) = I σ t ( z ) = X n ∈ Z I n + t z − n − t − . Proof.
Let U W = { L ( z ) , I σ t ( z ) , W } , where W is the identity operator on W .Clearly, L ( z ) , I σ t ( z ) are Z t -twisted weak vertex operators on W . From (4.4), (4.5),(4.6), using (2.1), we see that L ( z ) , I σ t ( z ) are mutually local Z t -twisted vertex oper-ators on W . Hence, by Proposition 2.14, h U W i is a vertex algebra with W a faithful σ -twisted module, where σ is an order t automorphism of the vertex algebra h U W i .To say that W is a σ t -twisted module for V L ( ℓ , V L ( ℓ , h U W i .By Lemma 2.11 of [13], Y ( L ( z ) , z ) and Y ( I σ t ( z ) , z ) satisfy the twisted Heisenberg-Virasoro relations (3.4), (3.5), (3.6). Then h U W i is an L -module with L n , I n actingas L ( z ) n +1 , I σ t ( z ) n for n ∈ Z , c i acting as ℓ i with ℓ = 0, i = 1 , , .
12y the universal property of V L ( ℓ ,
0) (c.f. Remark 2.7 of [12]), there exists aunique L -module homomorphism ψ : V L ( ℓ , −→ h U W i ; W . Then ψ ( ω n v ) = L ( z ) n ψ ( v ) = ψ ( ω ) n ψ ( v ) ,ψ ( I n v ) = I σ t ( z ) n ψ ( v ) = ψ ( I ) n ψ ( v ) . for all v ∈ V L ( ℓ , n ∈ Z . Hence ψ is a vertex algebra homomorphism. Therefore, W is a weak σ t -twisted V L ( ℓ , Y σ t ( L − , z ) = L ( z ), Y σ t ( I − , z ) = I σ t ( z ) . Conversely, we have
Theorem 4.5. If W is a σ t -twisted V L ( ℓ , ( ℓ = 0 )-module, then W is a re-stricted L t -module of level ℓ with L ( z ) = Y W ( L − , z ) , I σ t ( z ) = Y W ( I − , z ) . Proof.
Let W be a σ t -twisted V L ( ℓ , ℓ = 0)-module. Recall the following for-mula (c.f. (2.40) of [13])[ Y W ( a, z ) , Y W ( b, z )] = ∞ X j =0 j ! (cid:18) ∂∂z (cid:19) j z − δ (cid:18) z z (cid:19) (cid:18) z z (cid:19) kt ! Y W ( a j b, z ) , where k is determined by a . In our case, when a = L − , k = 0, when a = I − , k = 1 . For a = b = L − , we have ((2.21) of [12])( L − ) j L − = ( j + 1) L j − + δ j − , ( j − − ( j − c , so [ Y W ( L − , z ) , Y W ( L − , z )]= Y W ( L − , z ) z − δ (cid:18) z z (cid:19) + 2 Y W ( L − , z ) (cid:18) ∂∂z (cid:19) z − δ (cid:18) z z (cid:19) + 112 (cid:18) ∂∂z (cid:19) z − δ (cid:18) z z (cid:19) ℓ . (4.7)For a = L − , b = I − , we have ((2.22) of [12] with ℓ = 0)( L − ) j I − = I j − ,
13o [ Y W ( L − , z ) , Y W ( I − , z )]= Y W ( I − , z ) z − δ (cid:18) z z (cid:19) + Y W ( I − , z ) (cid:18) ∂∂z (cid:19) z − δ (cid:18) z z (cid:19) , (4.8)For a = b = I − , we have ((2.23) of [12])( I − ) j I − = jδ j − , c , so [ Y W ( I − , z ) , Y W ( I − , z )] = (cid:18) ∂∂z (cid:19) z − δ (cid:18) z z (cid:19) (cid:18) z z (cid:19) t ℓ . (4.9)Note that for t = 2, we have to require ℓ = 0 (Theorem 3.2). Therefore,with (4.4), (4.5) and (4.6), W is a L t -module with L ( z ) = Y W ( L − , z ), I σ t ( z ) = Y W ( I − , z ), k i = ℓ i , i = 1 ,
3. Then W is restricted of level ℓ is clear.Let L (0) t = C L ⊕ C k ⊕ C k , L ( n ) t = C L − n for 0 = n ∈ Z , L ( − t + n ) t = C I − n + t , L ( k ) t = 0 for k ∈ t Z , k / ∈ Z , − t + Z . Then L = ` n ∈ Z L ( nt ) t is a 1 t Z -graded Lie algebra, and the grading is given by ad L -eigenvalues.By Theorem 4.4 and Theorem 4.5 we have the following result. Theorem 4.6.
The σ t -twisted modules for V L ( ℓ , ( ℓ = 0 ) viewed as a vertex op-erator algebra (i.e. C -graded by L -eigenvalues and with the two grading restrictions(TW6), (TW7)) are exactly those restricted modules for the Lie algebra L t of level ℓ that are C -graded by L -eigenvalues and with the two grading restrictions. Further-more, for any σ t -twisted V L ( ℓ , -module W , the σ t -twisted V L ( ℓ , -submodulesof W are exactly the submodules of W for L t , and these submodules are in particulargraded. Hence irreducible restricted L t -modules of level ℓ corresponds to irreducible σ t -twisted V L ( ℓ , h ∈ C , L L t ( ℓ , ℓ , h ) is an irreduciblerestricted L t -module of level ℓ , so it is an irreducible σ t -twisted V L ( ℓ , ℓ = 0)-module.Now we give the complete list of irreducible σ t -twisted V L ( ℓ , Theorem 4.7.
Let ℓ = 0 , ℓ , ℓ ∈ C . Then { L L t ( ℓ , ℓ , h ) | h ∈ C } is a completelist of irreducible σ t -twisted V L ( ℓ , -modules. roof. Let W = ` r ∈ C W ( r ) be an irreducible σ t -twisted V L ( ℓ , W is an irreducible restricted L t -module of level ℓ . So k i acts on W as ascalar ℓ i for i = 1 ,
3. From Remark 2.7, there exists h ∈ C such that W ( h ) = 0 and W ( h − n ) = 0 for all n ∈ t Z ≥ . Let 0 = w ∈ W ( h ) . Then L w = hw, L m w = 0 , I n + t w = 0for m ≥ , n ≥
0. In view of Remark 4.2, there is a unique L t -module homomorphism ϕ : M L t ( ℓ , ℓ , h ) −→ W such that ϕ ( k ,h ) = w . By Proposition 2.10, ϕ is a σ -twisted h U W i -module ho-momorphism (since L t generates the vertex algebra h U W i ), where σ is an order t automorphism of the vertex algebra h U W i . Recall that M L t ( ℓ , ℓ , h ) is a weak σ t -twisted V L ( ℓ , ψ in Theorem 4.4.So ϕ can be viewed as a σ t -twisted V L ( ℓ , W is ir-reducible and T L t ( ℓ , ℓ , h ) is the (unique) largest proper submodule of M L t ( ℓ , ℓ , h ),it follows that ϕ ( M L t ( ℓ , ℓ , h )) = W and Ker ϕ = T L t ( ℓ , ℓ , h ) . Thus ϕ reduces to a σ t -twisted V L ( ℓ , L L t ( ℓ , ℓ , h )to W .It is interesting and important to classify the irreducible modules for the fixedpoint subalgebra V L ( ℓ , σ t := { v ∈ V L ( ℓ , | σ t ( v ) = v } , t ∈ Z ≥ . In the caseof ℓ = 0 and ℓ = 0, we have t = 2. Then σ t is the order 2 automorphism of V L ( ℓ ,
0) which is induced from its Heisenberg vertex operator subalgebra. Denoteby σ = σ . Precisely, the automorphism σ : V L ( ℓ , −→ V L ( ℓ , I − k · · · I − k s L − m · · · L − m r ( − s I − k · · · I − k s L − m · · · L − m r , and extended linearly, where r, s ≥ m ≥ · · · ≥ m r ≥ k ≥ · · · ≥ k s ≥
1. Let V L ( ℓ , + = { v ∈ V L ( ℓ , | σ ( v ) = v } be the fixed point subalgebra under σ .For ℓ = 0, denote by c ˜ V ir = ℓ − ℓ ℓ . V H ( ℓ ,
0) be the vertex operator algebra constructed from the Heisenberg subal-gebra H which is equipped with the nonstandard conformal vector ω H = ℓ I − I − + ℓ ℓ I − (of central charge 1 − ℓ ℓ ). Let ˜ V ir be the Virasoro algebra constructedby ˜ ω = ω − ω H . Let V ˜ V ir ( c ˜ V ir ,
0) be the corresponding Virasoro vertex operatoralgebra. Recall that when ℓ = 0, we have (cf. Theorem 3.16 of [12]) V L ( ℓ , ∼ = V H ( ℓ , ⊗ V ˜ V ir ( c ˜ V ir , V H ( ℓ , σ : V H ( ℓ , −→ V H ( ℓ , I − k · · · I − k s ( − s I − k · · · I − k s and extended linearly, where s ≥ k ≥ · · · ≥ k s ≥
1. The fixed point subalgebra V H ( ℓ , + = { v ∈ V H ( ℓ , | σ ( v ) = v } has beed extensively studied (cf. [6] etc.).Then it is immediately to see that when ℓ = 0 and ℓ = 0, we have an isomor-phism of vertex operator algebras V L ( ℓ , + ∼ = V H ( ℓ , + ⊗ V ˜ V ir ( c ˜ V ir , . Up to isomorphism, irreducible modules for the vertex operator algebra V H ( ℓ , + are (cf. [6], etc.) V H ( ℓ , ± , V H ( ℓ , σ ) ± , V H ( ℓ , λ ) ∼ = V H ( ℓ , − λ ) , ∀ = λ ∈ C . Up to isomorphism, irreducible modules for the Virasoro vertex operator algebra V ˜ V ir ( c ˜ V ir ,
0) are L ˜ V ir ( c ˜ V ir , h ) for all h ∈ C (cf. [8], [16], etc.). Therefore, in the caseof ℓ = 0 and ℓ = 0, irreducible modules of V L ( ℓ , + are one-to-one correspondto the tensor product of irreducible modules of V H ( ℓ , + and irreducible modulesof V ˜ V ir ( c ˜ V ir ,
0) (cf. Proposition 4.7.2 and Theorem 4.7.4 of [7]).For other automorphism σ t , the complete list of irreducible V L ( ℓ , σ t -modulesremains to be investigated. References [1] E. Arbarello, C. De Concini, V. G. Kac, C. Procesi, Moduli spaces of curvesand representation theory,
Comm. Math. Phys. (1988), no. 1, 1-36.[2] D. Adamovi, G. Radobolja, Free eld realization of the twisted Heisenberg Vi-rasoro algebra at level zero and its applications,
J. Pure Appl. Algebra (2015), no. 10, 43224342. 163] D. Adamovi, G. Radobolja, Self-dual and logarithmic representations of thetwisted Heisenberg-Virasoro algebra at level zero,
Commun. Contemp. Math. (2019), no. 2, 1850008, 26 pp.[4] Y. Billig, Representations of the twisted Heisenberg-Virasoro algebra at levelzero, Canad. Math. Bull. (2003), no. 4, 529-537.[5] Y. Billig, A category of modules for the full toroidal Lie algebra, Int. Math.Res. Not. (2006), Art. ID. 68395, 46pp.[6] C. Dong, K. Nagatomo, Classification of irreducible modules for the vertexoperator algebra M (1) + , J. Algebra (1999), no. 1, 384404.[7] I. B. Frenkel, Y. Z. Huang, J. Lepowsky, On axiomatic approaches to vertexoperator algebras and modules,
Mem. Amer. Math. Soc. (1993), no. 494,Viii+64 pp.[8] I. B. Frenkel, Y. C. Zhu, Vertex operator algebras associated to representationsof affine and Virasoro algebras,
Duke Math. J. (1992), no. 1, 123-168.[9] I. B. Frenkel, A. M. Zeitlin, Quantum group GL q (2) and quantum Laplaceoperator via semi-infinite cohomology, J. Noncommut. Geom. (7) (2013), no. 4,10071026.[10] H. Guo, H. S. Li, S. Tan, Q. Wang, q -Virasoro algebra and vertex algebras, J.Pure Appl. Algebra (2015), no. 4, 1258-1277.[11] H. Guo, Q. Wang, Associating vertex algebras with the unitary Lie algebra,
J.Algebra (2015), 126-146.[12] H. Guo, Q. Wang, Twisted Heisenberg-Virasoro vertex operator algebra,
Glas.Mat. Ser. III (74) (2019), no. 2, 369407.[13] H. S. Li, Local systems of twisted vertex operators, vertex operator superalge-bras and twisted modules. In: Moonshine, the Monster, and related topics, pp.203236, Contemp. Math. , Amer. Math. Soc., Providence, RI, 1996.[14] J. Lepowsky, H. S. Li, Introduction to Vertex Operator Algebras and TheirRepresentations,
Progress in Mathematics , Vol. 227, Birkh¨auser, Boston, 2004.[15] R. Shen, C. Jiang, The derivation algebra and automorphism group ofthe twisted Heisenberg-Virasoro algebra,