The N = 1 Triplet Vertex Operator Superalgebras: Twisted Sector

We classify irreducible $\sigma$-twisted modules for the N=1 super triplet vertex operator superalgebra $\mathcal{SW}(m)$ introduced recently [Adamovic D., Milas A., Comm. Math. Phys., to appear, arXiv:0712.0379]. Irreducible graded dimensions of $\sigma$-twisted modules are also determined. These results, combined with our previous work in the untwisted case, show that the $SL(2,\mathbb{Z})$-closure of the space spanned by irreducible characters, irreducible supercharacters and $\sigma$-twisted irreducible characters is $(9m+3)$-dimensional. We present strong evidence that this is also the (full) space of generalized characters for $\mathcal{SW}(m)$. We are also able to relate irreducible $\mathcal{SW}(m)$ characters to characters for the triplet vertex algebra $\mathcal{W}(2m+1)$, studied in [Adamovic D., Milas A., Adv. Math. 217 (2008), 2664-2699, arXiv:0707.1857].


Introduction
Many constructions and results in vertex algebra theory are easily extendable to the setup of vertex superalgebras by simply adding adjective "super". Still, there are results that deviate from this "super-principle" and new ideas are needed compared to the non-super case. For example, modular invariance for vertex operator superalgebras requires inclusion of supercharacters of (untwisted) modules, and more importantly, the characters of σ-twisted modules [13], where σ is the canonical parity automorphism. Since the construction and classification of σ-twisted modules is more or less independent of the untwisted construction, many aspects of the theory need to be reworked for the twisted modules (e.g., twisted Zhu's algebra [29]). In fact, even for the free fermion vertex operator superalgebra, construction of σ-twisted modules is far from being trivial (see [13] for details).
Present work is a natural continuation of our very recent paper [5], where we introduced a new family of C 2 -cofinite N = 1 vertex operator superalgebra that we call the supertriplet family SW(m), m ≥ 1. In this installment we focus on σ-twisted SW(m)-modules and their irreducible characters. The σ-twisted SW(m)-modules are usually called modules in the Ramond Within the same setup, consider the subspace O(V ) ⊂ V , spanned by elements of the form where u ∈ V is homogeneous. It can be easily shown that Then, the vector space A σ (V ) = V /O(V ) is equipped with an associative algebra structure via (see [29]). An important difference between the untwisted associative algebra A(V ) and A σ (V ) is that the latter is Z 2 -graded, so . It is also not clear that A σ (V ) = 0, while A(V ) is always nontrivial. We shall often use [a] ∈ A σ (V ) for the image of a ∈ V under the natural map V −→ A σ (V ).
The result we need is [29]: There is a one-to-one correspondence between irreducible Z ≥0 -gradable σ-twisted V -modules and irreducible A σ (V )-modules.
In the theorem there is no reference to graded A σ (V )-modules. For practical purposes we shall need a slightly different version of the above theorem, because some modules are more natural if considered as Z 2 -graded modules (shorthand, graded modules).

Theorem 2.2.
There is a one-to-one correspondence between graded irreducible Z ≥0 -gradable σ-twisted V -module and graded irreducible A σ (V )-module.
Proof . The proof mimics the non-graded case, so we shall omit details. As in the non-graded case, by applying Zhu's theory, from an irreducible graded A σ (V )-module U we construct a σtwisted graded V -module L(U ) (but of course in the process of getting L(U ) will be moding out by the maximal Our goals are to describe the structure of A σ (SM (1)), where SM (1) is the super singlet vertex algebra [5], and to discuss A σ (SW(m)) (in fact, we have a very precise conjecture about the structure of A σ (SW(m))). Let us recall (see [5] for details) that both SM (1) and SW(m) are N = 1 superconformal vertex operator superalgebras, with the superconformal vector τ . In other words, if we let Y (τ, x) = n∈Z+1/2 G(n)x −n−3/2 , then G(n) and L(m) close the N = 1 Neveu-Schwarz superalgebra.
It is not hard to see that the following hold: where in all formulas v is a vector in N = 1 vertex operator superalgebra V .
To illustrate how to use twisted Zhu's algebra, let us classify Z-graded σ-twisted modules for the (neutral) free fermion vertex operator superalgebra F , used in [5]. The next result is known so we will not provide its proof here.
, an odd generator in the associative algebra.
This result in particular implies there are precisely two irreducible σ-twisted F -modules: M ± . These two modules can be constructed explicitly. As vector spaces where Λ * is the exterior algebra, which is also a Z ≥0 -graded module for the Clifford algebra K spanned by φ(n), n ∈ Z with anti-bracket relations {φ(m), φ(n)} = δ m+n,0 . The only difference between M + and M − is in the action of φ(0) on the one-dimensional top subspace M ± (0). More precisely we have φ(0)| M ± (0) = ± 1 √ 2 . However, notice that M ± are not Z 2 -graded thus it is more natural to examine graded A σ (F )-modules. There is a unique such module (up to parity switch), spanned by 1 R and φ(0)1 R . Thus Next we describe the twisted vertex operators Details are spelled out in [13], here we only give the explicit formula. Define first acting on M and use (fermionic!) normal ordering • and extend Y by linearity on all of F (see [13] and [14] for details, especially about normal ordering). Then, we let Let us fix the Virasoro generator Then e ∆x ω s = ω s + 1 16 x −2 1. The following lemma will be important in the rest of the paper.
where C m,n are as above. Then Proof . The lemma follows directly from the identity which can be easily checked by expanding both sides as power series in x 1 and x 2 .

Highest weight representations of the Ramond algebra
The N = 1 Ramond algebra R is the infinite-dimensional Lie superalgebra The representation theory of the N = 1 Ramond algebra has been intensively studied first in [22] and other papers (cf. [25,12], etc.).
Assume that (c, h) ∈ C 2 such that 24h = c. Let L R (c, h) ± denote the irreducible highest weight R-module generated by the highest weight vector v ± c,h such that These modules can be considered as irreducible σ-twisted modules for the Neveu-Schwarz vertex operator superalgebra (cf. [24]). Since L R (c, h) ± are not Z 2 -graded (notice that v ± c,h are eigenvectors for G(0)), it is more useful to consider graded modules. It is not hard to show that the direct sum Since L R (c, h) does not contain non-trivial Z 2 -graded submodules, we shall say that this module is Z 2 -graded irreducible.
Remark 3.1. Details about construction of L R (c, h) and its relation to irreducible non-graded modules can be found in [12] and [19]. In Section 5 we shall present free fields realization of modules L R (c, h) and L R (c, h) ± .

Intertwining operators among twisted modules
If V is a vertex operator superalgebra and W 1 , W 2 and W 3 are three V -modules then we consider the space of intertwining operators It is perhaps less standard to study intertwining operators between twisted V -modules, so we recall the definition here (see [29, p. 120]).
Definition 3.1. Let W 1 , W 2 and W 3 be σ i -twisted V -modules, respectively, where σ i is a finite order automorphism of order ν i , with common period T . An intertwining operator of type and Jacobi identity holds and ω T is a primitive T -th root of unity.
We shall also need the following result on the fusion rules. The proof is completely analogous to that of Proposition 4.1 of [5] (see also [18]).
Similarly, for every i = 0, . . . , m − 1 and n ≥ 2 we have: the space We have analogous result for fusion rules of type Proof . The proof goes along the lines in [5] (cf. [26]), with some minor modifications due to twisting. In fact, in order to avoid the twisted version of Frenkel-Zhu's formula we can proceed in a more straightforward fashion. Because all modules in question are irreducible and because all intertwining operators we are interested in are of the form , the left-hand side in the Jacobi identity looks like the ordinary Jacobi identity for intertwining operators. Thus we can use commutator formula and the null vector conditions which hold in L(c 2m+1,1 , h 1,3 ) (here v 1,3 is the highest weight vector), to study the matrix coefficient where w and w ′ are highest weight vectors in appropriate modules. This leads to differential equations for f (x), which can be solved. The general solution is a linear combination of power functions x s , where s is a rational number. The rest follows by interpreting s in terms of conformal weights for the three modules involved in Y.
4 σ-twisted modules for the super singlet algebra SM (1) In this section σ 2 will denote the parity automorphism of F . Recall also the Heisenberg vertex operator algebra M (1). Then, as in [5] we equip the space M (1) ⊗ F with a vertex superalgebra structure such that the total central charge is 3 2 (1 − 8m 2 2m+1 ). We define the following superconformal and conformal vectors: Recall also the singlet superalgebra SM (1) obtained as the kernel of the screening operator We will also use ω and H, where the latter is proportional to G − 1 2 H (for details see [5]). Consider the automoprhism σ = 1 ⊗ σ 2 of M (1) ⊗ F , acting nontrivially on the second tensor factor. This automorphism plainly preserves SM (1), thus we can study σ-twisted SM (1)modules. It is clear that [4] and λ ∈ C.
We would like to classify irreducible SM (1)-modules by virtue of Zhu's algebra. As usual, we denote by [a] ∈ A σ (SM (1)) the image of a ∈ SM (1).

Then the twisted generators act as
Proof . Recall from [5] the formulas As in [5] we have Then we get The last formula is proven similarly by using e ∆x operator.
Here are the main result of this section.
In particular, Zhu's algebra is commutative.
Proof . First notice (cf. [5]) that We also have the relation because H = v 1,3 is a highest weight vector. By using (2.1) we obtain On the other hand, by using skew-symmetry we also have Combined, we obtain Notice that relation (4.1) can be written as Let C[a, b] denote the Z 2 -graded complex commutative associative algebra generated by odd vectors a, b.
Theorem 4.2. The associative algebra A σ (SM (1)) is isomorphic to the Z 2 -graded commutative associative algebra Proof . The proof is similar to that of Theorem 6.1 of [2]. First we notice that we have a surjective homomorphism It is easy to see that Ker Φ is a Z 2 -graded ideal. We shall now prove that Ker Φ = H(a, b) . Evidently the generating element H(a, b) is even, so Theorem 4.1 gives H(a, b) ⊂ Ker Φ. Assume now that K(a, b) ∈ Ker Φ. By using division algorithm we get . We also notice that R(a, b) ∈ Ker Φ. Since Ker Φ is Z 2 -graded ideal, we can assume that R(a, b) is homogeneous. If R(a, b) is an even element we have that A has odd degree and B has even degree, and therefore The case when R(a, b) is odd element again leads to formula (4.2). As in [2] we now shall evaluate R(a, b) on A σ (SM (1))-modules and get This implies that deg(B) − deg(A) = 2m. Contradiction. Therefore R(a, b) = 0 and K(a, b) ∈ H(a, b) . The proof follows.
Remark 4.1. By using the same arguments as in [2] and [5], we can conclude that every irreducible Z ≥0 -gradable σ-twisted SM (1)-module is isomorphic to an irreducible subquotient of M (1, λ) ⊗ M ± . By using the structure of twisted Zhu's algebra A σ (SM (1)) and the methods developed in [3], we can also construct logarithmic σ-twisted SM (1)-modules. 5 The N = 1 Ramond module structure of twisted V L ⊗ F -modules In this section we shall assume that the reader is familiar with basic results on twisted representations of lattice vertex superalgebras. Details can be found in [9,10,16] and [29]. We shall use the same notation as in [5]. Let L = Zα be a rank one lattice with nondegenerate form given by α, α = 2m+1, where m ∈ Z >0 . Let V L be the corresponding vertex superalgebra.
Remark 5.1. It is important to notice that σ 1 -twisted V L -module (V γ R i +L , Y γ R i +L ) can be constructed from untwisted module (V γ i +L , Y γ i +L ) as follows (cf. [29]): Let F be the fermionic vertex operator superalgebra with central charge 1/2 and σ 2 its parity map. Let M be the σ 2 twisted F -module (cf. [14]).
Then σ = σ 1 ⊗ σ 2 is the parity automorphism of order two of the vertex superalgebra V L ⊗ F and From the Jacobi identity for σ-twisted modules it follows that the coefficients of define a representation of the N = 1 Ramond algebra.
Proposition 5.1. Assume that n ∈ Z. Then e γ R i −nα ⊗ 1 R is a singular vector for the N = 1 Ramond algebra R and
Lemma 5.1. The (screening) operator commutes with the action of R. (for details and some applications see [7]).
We shall first present results on the structure of σ-twisted V L ⊗F -modules, viewed as modules for the N = 1 Ramond algebra. Each V L+γ R i ⊗ M is a direct sum of super Feigin-Fuchs modules via Since operators Q j , j ∈ Z >0 , commute with the action of the Ramond algebra, they are actually (Lie superalgebra) intertwiners between super Feigin-Fuchs modules inside V L+γ R i ⊗ M .
To simplify the notation, we shall identify e β with e β ⊗ 1 R for every β ∈ L + γ R i .
Since wt e γ R i +(j−n)α > wt e γ R i −nα if j > 2n, we conclude that One can similarly see that for m ≤ i ≤ 2m: Q j e γ R i −nα = 0 for j > 2n + 1.

The following lemma is useful for constructing singular vectors in V L+γ
Proof . The proof uses the results on fusion rules from Proposition 3.1 and is completely analogous to that of Lemma 6.1 in [5].
As in the Virasoro algebra case the N = 1 Feigin-Fuchs modules are classified according to their embedding structure. For the purposes of our paper we shall focus only on modules of certain types (Type 4 and 5 in [20]). These modules are either semisimple (Type 5) or they become semisimple after quotienting with the maximal semisimple submodule (Type 4).
The following result follows directly from Lemma 5.2 and the structure theory of super Feigin-Fuchs modules, after some minor adjustments of parameters (cf. Type 4 embedding structure in [20]).
(iii) For the quotient module we have

Moreover, we have
These vectors satisfy the following relations:

The submodule generated by singular vectors Sing i is isomorphic to
(ii) For the quotient module we have Theorem 5.3.

⊗M is completely reducible and generated by the family of singular vectors
and it is isomorphic to ⊗ M ± generated by the singular vectors Then Remark 5.3. In this section we actually constructed explicitly all the non-trivial intertwining operators from Proposition 3.1.

The σ-twisted SW(m)-modules
Since SW(m) ⊂ V L ⊗ F is σ-invariant, then every σ-twisted V L ⊗ F -module is also a σ-twisted module for the vertex operator superalgebra SW(m). In this section we shall consider σ-twisted V L ⊗ F -modules from Section 5 as σ-twisted SW(m)-modules. In what follows we shall classify all the irreducible σ-twisted SW(m)-modules by using Zhu's algebra A σ (SW(m)). Following [4] and [5], we first notice the following important fact: where t = λ, α , and Proof . First we notice that The proof of Proposition 8.3 from [5] gives that so it remains to examine o(w 2 ). Recall Lemma 2.1, so that Now, observe that the expression in (6.2) is precisely −o(w 1 )·v ± λ , while (6.3) and (6.4) are equal. Consequently, Now, we expand the generalized rational function in the last formula and obtain The sum in the last formula can be evaluated as in [4] and [5]. We have where A m is above. The proof follows.
A direct consequence of Proposition 6.1 and relation (6.1) is the following important result: Theorem 6.1. In Zhu's algebra A σ (SW(m)) we have the following relation In parallel with [5] we conjecture that f R m (x) is in fact the minimal polynomial of [ω] in A σ (SW(m)).
We have the following irreducibility result. The proof is similar to that of Theorem 3.7 in [4].

Moreover, both [ω] and [τ ] are units in
, for X ∈ {E, F, H}, , where q is a certain polynomial.
Equipped with all these results we are now ready to classify irreducible σ-twisted SW(m)modules.
(i) The set provides, up to isomorphism, all irreducible σ-twisted SW(m)-modules.
(ii) The set So the vertex operator algebra SW(m) contains only finitely many irreducible modules. But one can easily see that modules V L+γ R i ⊗ M and V L+γ R m+i ⊗ M (0 ≤ i ≤ m − 1) constructed in Theorems 5.1 and 5.2 are not completely reducible. Thus we have: Corollary 6.2. The vertex operator superalgebra SW(m) is not σ-rational, i.e., the category of σ-twisted SW(m)-modules is not semisimple. Remark 6.2. In our forthcoming paper [7] we shall prove that SW(m) also contains logarithmic σ-twisted representations.

Modular properties of characters of σ-twisted SW(m)-modules
We first introduce some basic modular forms needed for description of irreducible twisted SW(m) characters. The Dedekind η-function is usually defined as the infinite product an automorphic form of weight 1 2 . As usual in all these formulas q = e 2πiτ , τ ∈ H. We also introduce Let us recall Jacobi Θ-function where j, k ∈ 1 2 Z. If j ∈ Z + 1 2 or k ∈ N + 1 2 it will be useful to use the formula Observe that Θ j+2k,k (τ, z) = Θ j,k (τ, z).
(i) For 0 ≤ i ≤ m − 1, we have: (ii) For 0 ≤ i ≤ m, we have: Now, we recall also formulas for irreducible SW(m) characters and supercharacters obtained in [5].
But this is well below the conjectural dimension, because 10m + 8-dimensional part cannot control possible logarithmic modules. Thus, as in the case of ordinary SW(m)-modules, we expect to have 2m non-isomorphic (non-graded) logarithmic modules with two-dimensional top component. This then leads to the following conjecture: Conjecture 8.1. For every m ∈ N, dim(A σ (SW(m))) = 12m + 8.
If we assume the existence of m logarithmic modules so that Conjecture 8.1 holds true, then the following fact is expected.
Thus, generalized SW(m) characters, supercharacters and σ-twisted characters together should give rise to a 9m + 3-dimensional modular invariant space.