The Theory of C2-Cofinite Vertex Operator Algebras

The theory of C2-coﬁnite vertex operator algebras

Thomas Creutzig and Terry Gannon

Abstract Despite several efforts, the theory of rational VOAs is still much more complete than that of C2-cofinite ones. We address this by focusing on the analogue of the categorical Verlinde formula and its relation to the modular S-matrix. We explain that the role of the Hopf link is played by the open Hopf link, which defines a representation of the fusion ring on each End(W ). We extract numbers, called logarithmic Hopf link invariants, from those operators. From this a Verlinde formula can be obtained that depends on the endomorphism ring structure. We apply our ideas directly to the triplet algebras Wp. Using the restricted quantum group of sl2, we compute all open Hopf link operators and the logarithmic Hopf link invariants and compare them successfully with the modular S- matrix of Wp. In particular, S-matrix coefficients involving pseudo-trace functions relate to logarithmic Hopf link invariants with nilpotent endomorphism insertion. We show how the S-matrix specifies the fusion ring (not merely the Grothendieck ring) uniquely.

1 The representation theory of nice VOAs

For reasons of simplicity, we will restrict attention in this paper to vertex operator algebras `∞ (VOAs) V of CFT-type — this means V has L0-grading V = n=0 Vn with V0 = C1 and dim Vn < ∞. We also require V, when regarded as a V-module in the usual way, to be simple and isomorphic to its contragredient V∨. By Modgr(V) we mean the category of grading-restricted weak V-modules M, by which we mean L0 decomposes M into a direct sum of ﬁnite-dimensional generalized eigenspaces with eigenvalues bounded from below. When L0 is in fact diagonalizable over M, M is called ordinary. This terminology is explained in more detail in Section 2 below. For convenience we will refer to any grading-restricted weak V-module, simply as a V-module.

1.1 The rational modular story Let V be a strongly-rational vertex operator algebra — i.e. regular, in addition to the aforementioned restrictions. The theory of strongly-rational VOAs is well-understood. Such a VOA enjoys three fundamental properties: (Cat) its category of modules Modgr(V) forms a modular tensor category [T, H2]; (Mod) its 1-point toroidal functions are modular [Z]; and (Ver) Verlinde’s formula holds [V, MS, H1].

1 For strongly-rational V, all M ∈ Modgr(V) are ordinary. A strongly-rational V has finitely many simple modules in Modgr(V), up to isomorphism. Fix representatives M0,M1,...,Mn of these classes, with M0 = V. Every (ordinary) V-module will be isomor- k phic to a unique direct sum of these Mj. The fusion coefficients Nij ∈ Z≥0 are defined as the dimension of the space of intertwiners of type Mk . Tensor products of modules Mi Mj can be defined (see e.g. [HL] and references therein), realizing those fusion coefficients:

n ∼ M k Mi ⊗ Mj = Nij Mk . k=0

This deﬁnes a ring structure (the tensor ring or fusion ring) on the formal Z-span of the representatives Mj, with unit M0. For each i, deﬁne the fusion matrix Ni by (Ni)jk = k Nij . They yield a representation of the tensor ring:

X k Ni Nj = Nij Nk . k

k We also have Nij = dim(HomV (Mi ⊗ Mj,Mk)). Given a V-module M, the 1-point function FM on the torus is

c L0− FM (τ, v) := trM o(v) q 24 , (1) for any τ in the upper half-plane H and v ∈ V, where o(v) ∈ End(M) is the zero-mode. The character of a module is ch[M](τ) = FM (τ, 1). Zhu proved [Z] that for v ∈ V[k] (using the L[0]-grading introduced in [Z]), these 1-point functions form the components of a vector-valued modular form of weight k, holomorphic in H and meromorphic at the cusps, for the modular group SL(2,Z). Hence the characters form a vector-valued modular form of weight 0 for SL(2, Z). (The reason for preferring the more general 1-point functions over the more familiar characters is that the latter are usually not linearly independent.) In particular, the transformation τ 7→ −1/τ is called the modular S-transformation and deﬁnes a matrix Sχ via n k X χ FMi (−1/τ, v) = τ Sij FMj (τ, v) . j=0

The eigenvalues of L0 on a simple V-module Mj lie in hj + Z≥0 where hj is called the conformal weight of Mj. [AnM] for instance show that these hj, together with the central charge c, are rational, using the modularity of the characters. k In 1988, Verlinde [V] proposed that these so-called fusion coeﬃcients Nij are related to the matrix Sχ by what is now called Verlinde’s formula:

n χ χ χ −1 X Si`Sj` S N k = `k . (2) ij Sχ `=0 0` This is one of the most exciting outcomes of the mathematics of rational conformal ﬁeld χ χ −1 theory (CFT). As S is a unitary matrix, (S )`k here can be replaced with the complex χ ∗ conjugate Sk` . The Verlinde formula thus has three aspects:

2 ⊗ (V1) There is a matrix S simultaneously diagonalizing all fusion matrices Ni. Each ⊗ −1 ⊗ diagonal entry ρl(Mi) := (S NiS )ll deﬁnes a one-dimensional representation

X k ρl(Mi)ρl(Mj) = Nij ρl(Mk) k

of the tensor ring. All of these ρl are distinct. (V2) The Hopf link invariants

m M ∈ Sij := Mi j C

for any 0 ≤ i, j ≤ n give one-dimensional representations of the fusion ring:

m m n m S Sj` X S i` = N k k` . S m S m ij S m 0` 0` k=0 0`

m m Moreover, each representation ρl appearing in (V1) equals one of these Mi 7→ Si`/S0`. (V3) The deepest fact: these three S-matrices are essentially the same. More precisely, χ χ m ⊗ χ ⊗ m m Sij/S00 = Sij, and S = S works in (V1). For this choice of S , ρl(Mi) = Sil /S0l. Much of (V1)-(V3) is automatic in any modular tensor category. In particular, Theo- rem 4.5.2 in [T] says that in any such category,

m m m ∗ X Si`Sj`Sk` N k = D−2 , ij S m ` 0`

2 P m 2 χ −1 m where D = i S0i . So the remaining content of (2) is that S = D S . Categorically, ∼ the space of 1-point functions FM is identiﬁed with the space Hom(1, H) where H = ∨ ∨ ⊕iMi ⊗ Mi (throughout this paper, M denotes the contragredient or dual of M). This object H is naturally a Hopf algebra and a Frobenius algebra. Using this structure, Section 6 of [Ly1] shows End(H) carries a projective action of SL(2, Z); it is the coend of [Ly1], and [Sh] suggests to interpret it (or its dual) as the adjoint algebra of the category. The SL(2, Z)-action on the FM correspond to the subrepresentation on Hom(1, H), which is generated by S-matrix D−1S m and T -matrix coming from the ribbon twist. The other subrepresentations Hom(Mj, H) correspond to the projective SL(2, Z)-representations on the space of 1-point functions with insertions v ∈ Mj, where the vertex operator YM (v, z) M0 implicit in (1) is replaced with an intertwining operator of type ∨ . As in [FS], the Mi Mi character FM of (1) can be interpreted categorically as the partial trace of the ‘adjoint’ action H ⊗ M → M. It turns out to be far easier to construct modular tensor categories, than to construct VOAs or CFTs. This is related to the fact that the subfactor picture is essentially equivalent to the categorical one, at least if one assumes unitarity. This can be used to probe just how complete the lists of known (strongly-rational) VOAs and CFTs are. More precisely, given a fusion category (essentially a modular tensor category without the braiding), taking its Drinfeld double or centre construction yields a modular tensor category. Fusion

3 categories can be constructed and classiﬁed relatively easily, and their doubles worked out, at least when their fusion rules are relatively simple. Examples of this strategy are provided in e.g. [EG1]. This body of work suggests that the zoo of known modular tensor categories is quite incomplete, and hence that the zoo of known strongly-rational VOAs and CFTs is likewise incomplete.

1.2 The finite logarithmic story An important challenge is to extend the aforementioned results beyond the semi-simplicity of the associated tensor categories. A VOA or CFT is called logarithmic if at least one of its modules is indecomposable but reducible. The name refers to logarithmic singularities appearing in their correlation functions and operator product algebras of intertwining operators. In this paper we are interested in logarithmic VOAs with only finitely many simple modules. Of these, the best studied class is the family of Wp-triplet algebras parameterized by p ∈ Z>1 and this will be our main example too. See e.g. [CR4, FS] for introductions on logarithmic VOAs. By a strongly-finite VOA, we mean a simple C2-cofinite VOA of CFT-type with V isomorphic to V∨. These have finitely many simple modules, though they may have uncountably many indecomposable ones (see Corollary 5.2 below). Any strongly-rational VOA is strongly-finite. Gaberdiel–Goddard have an interesting conjecture that a strongly- finite VOA is strongly-rational iff Zhu’s algebra A0(V) is semi-simple. This subsection reviews what is known about them. See Section 2.2 below for more details. The key hypothesis here is C2-cofiniteness; many of the following aspects will persist even when simplicity or CFT-type is lost. 2 The Wp models are strongly-finite. They have central charge c = 1 − 6(p − 1) /p, and are generated by the conformal vector and 3 other states. The symplectic fermions form a logarithmic vertex operator superalgebra with c = −2d for any d ∈ Z>0 (the number of + pairs of fermions); their even part SFd is a strongly-finite VOA [Ab]. The Wp,p0 -models are C2-cofinite but not simple, so are not strongly-finite. (Cat)0 A tensor product theory for strongly-finite VOAs has been developed by Huang, Lepowsky and Zhang (see e.g. [HLZ] and references therein), see also Miyamoto [Miy2]. The corresponding tensor category Modgr(V) is braided. Rigidity of this category is proven so far only for the Wp models [TW] and symplectic fermions + SFd [DR]. The notion of ordinary V-module was generalized to logarithmic modules (equivalent to what we call grading-restricted weak modules) in [Mil]; see Section 2.2. The space W of intertwiners of type UV , for any V-modules U, V, W , is isomorphic to HomV (U ⊗ V,W ), where the notion of intertwiner must be generalized appropriately [Mil]. In this W nonsemi-simple setting though, we will in general only have the inequality NU,V ≤ dim(HomV (U ⊗ V,W )). [FS] suggest that the generalization of modular tensor category appropriate to strongly-finite VOAs is what we will call a log-modular tensor category and define in Section 1.4, though the notion was first introduced in [KL]. The category of course can no longer be semi-simple, but the subtlety in its definition is the precise form nondegeneracy takes. Miyamoto [Miy3] proposed a weakened (and more complicated) notion of rigidity (‘semi-rigidity’) for nonsemi-simple VOA contexts, but the Wp-models

4 obey the usual rigidity and the nonsemi-simple finite tensor categories [EO] of Etingof– Ostrik have the usual rigidity built in. We expect strongly-finite VOAs to obey rigidity, though of course it may be easier to prove semi-rigidity. Fix representatives M0,...,Mn of isomorphism classes of simple V-modules as before. Let Pi be their projective covers (defined in Section 2.2). Likewise fix representatives of gr isomorphism classes Mλ of indecomposable V-modules. Any V-module M ∈ Mod (V) is isomorphic to a direct sum of finitely many Mλ, in a unique way. The tensor product defines a ring structure Fusfull(V), called the (full) tensor ring, on the formal span over gr Z of the Mλ. This ring is large – Mod (V) can be expected to have continuous families of indecomposables. There are two smaller versions of Fusfull(V). The smallest is the Grothendieck ring FusGr(V), by which is meant the quotient of the tensor ring Fusfull(V) by the formal differences M − M 0 − M 00 for each short exact sequence 0 → M 0 → M → M 00 → 0 in Modgr(V). We will write [M] for the image of a V-module M in the Grothendieck ring, and +, ⊗g for the sum, product in the Grothendieck ring. The Grothendieck ring has gr P basis [M0],..., [Mn] over Z. Explicitly, for any module M ∈ Mod (V), [M] = i mi[Mi] where mi ∈ Z≥0 are the Jordan–Hölder multiplicities of M. Finally, we have the subring simp full Fus (V) of Fus (V) generated by the simple V-modules Mi and their projective covers Pi. In the special case where V is strongly-rational, the Grothendieck ring, full tensor ring and simple-projective subring all coincide. We prefer to avoid calling any of these the fusion ring, as the term ‘fusion ring’ is used in different senses in the literature. The 1-point functions (1) can still be defined, but as we explain in Section 2.2 will P P satisfy FM = i miFMi for any logarithmic module M, where [M] = i mi[Mi]. The

problem is that the C-span of characters ch[Mi](τ) = FMi (τ, 1) is no longer SL(2, Z)- aτ+b invariant, although we still have that each ch[Mi] cτ+d lies in the C[τ]-span of the characters. Many authors interpret this as saying that e.g. Sχ is now τ-dependent. We prefer to say that the ordinary trace functions (1) must be augmented by pseudo-trace functions associated to reducible V-modules. The main result here is due to Miyamoto [Miy1]:

0 (Mod) The C-span of ordinary trace and pseudo-trace functions (see Section 2.2 for more details) is SL(2, Z)-invariant. Corollary 5.10 of [Miy1] uses this to argue that conformal weights and central charges in strongly-finite VOAs must also be rational. The dimension of the resulting SL(2, Z)- representation is dimAm/[Am,Am] − dimAm−1/[Am−1,Am−1], where Ak = Ak(V) is the kth Zhu algebra (a finite-dimensional associative algebra), and m is sufficiently large. It is possible to state explicitly how large m must be, but this isn’t terribly useful as the algebras Ak(V) are very hard to identify in practise. There have been several proposals for a strongly-finite Verlinde formula beyond rationality, mainly in the physics literature, see [R, GabR, FHST]. All these proposals are guided by analogy to the rational setting and they do not connect to the tensor category point of view (recall that equation (2) is a theorem in any modular tensor category, when χ −1 m S there is replaced with D S ). Define the tensor resp. Grothendieck matrices Nλ ν k resp. [N ]i, with entries (Nλ)µν = Nλµ and ([N ]i)jk = [N ]ij defined by the structure

5 constants

M ν X k Mλ ⊗ Mµ = Nλµ ·Mν , [Mi] ⊗g [Mj] = [N ]ij [Mk] . ν j

As before, they deﬁne representations of the simple-projective tensor subring respectively Grothendieck ring.

0 ⊗ ⊗g (V1) There are matrices S resp. [S ] which simultaneously put the Nλ resp. [N ]i into block diagonal form. These blocks deﬁne indecomposable representations of the tensor resp. Grothendieck rings.

0 (V1) is not deep, and follows because the matrices Nλ resp. [N ]i pairwise commute. In [R], both the tensor matrices Nλ and the Grothendieck matrices [N ]i for the Wp models have been explicitly block-diagonalized: the latter involves only blocks of size 1 and 2 [PRR], while those of the former involve also blocks of size 3 [R]. We also should block- diagonalise the full tensor ring Fusfull(V), in the sense of [CR1],[CR2]. To our knowledge this has never been explored. To our knowledge, no analogue of (V2)0 has been explored. The Hopf link invariants of (V2) are still deﬁned, but vanish on projective modules (at least if the category is rigid). If they did not vanish they would no longer yield tensor or Grothendieck ring representations. The reason for this is that dim End(Mλ) can be > 1. Nothing general about (V3)0 is known or has been explicitly conjectured. However, Lyubashenko [Ly1] explained that the modular group acts also on certain nonsemi-simple braided tensor categories. [FGST1] observed that the SL(2, Z) action on the center of the restricted quantum group Uq(sl2) of sl2 at 2p-th root of unity q, coincides with the one on the space of trace and pseudo-trace functions of the Wp-triplet algebra. The categorical SL(2, Z)-action discussed last subsection extends here as follows. Write C = Modgr(V). According to [Sh], the coend/Hopf algebra/Frobenius algebra/adjoint algebra H is simply UR(V), where U : Z(C) → C is the forgetful functor from the Drinfeld centre of C, and R is its right adjoint. The space of 1-point functions on the torus is HomV (V, H). The ordinary trace functions (1) span a subspace (of dimension equal to the number of simple V-modules).

1.3 Our observations We propose a general answer to (V2)0, using what we call logarithmic Hopf link invariants. Very much like ordinary Hopf link invariants in a (semi-simple) modular tensor category immediately imply the Verlinde formula [T], the logarithmic Hopf link invariants also define a logarithmic Verlinde formula for structure constants of the tensor or Grothendieck ring provided that a certain S-matrix is invertible. Before detailing this idea below, let us provide some historical background. One of us (TC) has several collaborations with Antun Milas, David Ridout and Simon Wood on logarithmic VOAs with infinitely many simple objects. These VOAs have the advantage over the C2-cofinite ones that the generic indecomposable module is simple and there is a natural Verlinde formula, which we have verified to a certain extent in the cases of the affine vertex algebra of sl2 at admissibible but non-integer level [CR1, CR2], the affine vertex super algebra of gl(1|1) [CR3] and its extensions [AC] as well as the

6 singlet algebra [CM1, RW1]. But there has not yet been any connection made to the categorical picture. Since the non-generic singlet algebra module characters are partial theta functions, they have no nice modular S-transformation. This forced us in [CM1] to regularize characters and then the modular transformation had a correction term. In the rational setting (at least when there is a unique Mi with lowest conformal weight), the Verlinde formula implies that the map

ch[M](τ) C → C,M 7→ lim , τ→0 ch[M0](τ) where M0 is the vacuum module, is a representation of the fusion ring. In the setting studied in [CM1], this asymptotic dimension depended on the regime of the regularization parameter. If the correction term dominated, then the asymptotic dimensions seemed to correspond to the semi-simplification of the fusion ring. This picture was further elaborated in [CMW], where the new insight is that in these non-finite logarithmic theories the asymptotic dimensions of characters conjecturally define representations of the tensor ring and they contain detailed information on that ring and its quotient structure. Higher rank analogues are currently being studied [CM2] and reviews on our ideas are given in [CR4, RW2, C]. A natural question is thus: Can we give a categorical interpretation for these representations of the fusion ring? Our answer, in the context of strongly-finite VOAs, is discussed next. Let C be a finite rigid and braided but nonsemi-simple tensor category (see Section 2.1 for the definitions). For each object W , define the map

ΦW : C → End(W ),U 7→ ΦU,W , where ΦU,W is the open Hopf link operator

ΦU,W = U (3)

As we prove in Theorem 2.2 below, naturality of the braiding implies that ΦU,W defines a representation of the tensor ring in the finite-dimensional algebra End(W ): ΦU⊗V,W = full ΦU,W ◦ ΦV,W . We would expect that the representation U 7→ ΦU,W of Fus (V) is indecomposable iff W is. One might also expect that Φ?,W1 and Φ?,W2 define equivalent full Fus (V)-representations iff Wi are equivalent V-modules; we however find that if W1 and W2 are projective modules of the same extension block then also Φ?,W1 and Φ?,W2 define equivalent Fusfull(V)-representations. We show in Section 4 that all the blocks appearing in Rasmussen’s block-diagonalization of the Grothendieck matrices for Wp, come from these ΦU,W . We have not completed the analysis for the 3 × 3 blocks appearing in his decomposition of the tensor ring for Wp. We expect though that the analogous statement holds for the tensor and Grothendieck matrices for any strongly-finite VOA, as well as the indecomposable subrepresentations of Fusfull(V). Together, this would provide the full analogue of (V2) in the strongly-finite context.

7 The big question of course is to relate these open Hopf link operators with the modularity of pseudo-trace functions. It is tempting to guess that for projective V-modules M, there is a natural isomorphism between End(M) and the space of pseudo-traces on M. Now suppose the category C contains a subcategory P that is a tensor ideal closed under retract. Further, we want this subcategory to be semi-simple. Incidentally, the following discussion can easily be generalized to tensor categories with infinitely many simple objects as well as to more than just one such ideal. The key requirement is that we also want P to be generated by an ambidextrous element so that there exists a modified trace t on P. This idea of modified trace has been developed by Geer, Patureau-Mirand and collaborators [GKP1, CGP, GKP2, GPT, GPV]. If W is not simple, then its endomorphism ring is not one-dimensional (but since C is (1) (n) finite, it will be finite-dimensional). Fix a basis {xw , . . . , xw } of the endomorphism ring of W . The map ΦU,W can then be expanded as

n X (i) (i) ΦU,W = ΦU,W xw , i=1 and these coeﬃcients satisfy

n X ijk (i) (j) X X (k) aW ΦU,W ΦV,W = NUV ΦX,W i,j=1 X∈C for every k = 1, . . . , n. Here the structure constants of the endomorphism ring of W and the tensor ring of C are deﬁned via

n (i) (j) X ijk (k) M X xw ◦ xw = aW xw ,U ⊗ V = NUV X. k=1 X∈C

How can we relate all this to modular properties of a C2-cofinite VOA? Let’s take the modified trace of W assuming that W is in P. Then we define the logarithmic Hopf link invariant to be n n (j) xw (j) X (i) (i) (j) X (i) (i) (j) SU,W := tW ΦU,W xw = ΦU,W tw x ◦ x = ΦU,W tw Φ1,W x ◦ x . i=1 i=1 Here 1 is the identity in C. In other words, the logarithmic Hopf link invariants relate to our expansion coefficients. A really nice situation for the VOA case would be if the pseudo-trace c L0− (j) trW q 24 x transforms under τ 7→ −1/τ into a linear combination of true characters and the expansion coefficients (the S-matrix coefficients) have a very direct relation (equality up to overall normalization) to the logarithmic Hopf link invariants. In this article, we will see that this really nice situation works for Wp. Since the triplet algebra Wp is the only fairly well-understood strongly-finite VOA that is not strongly-rational, this is the natural example to test our ideas and conjectures. gr Although Mod (Wp) is braided and rigid, the braiding is not explicitly known so that we have to take an indirect route, through the related quantum group Uq(sl2).

8 In a nutshell, we have the following results. In section three, we compute the open Hopf link invariants ΦV,W of Uq(sl2) in two ways: first following the strategy of [CGP], which uses as input the known tensor products; and secondly using the ideas on logarithmic knot invariants of Murakami and Nagatomo [Mur, MN]. The result is given in Theorem 3.8. The logarithmic Hopf link invariants give us a Verlinde formula for certain structure constants of the tensor ring. This is done in subsection 3.5. Although this doesn’t directly yield all tensor coefficients, Theorem 4.3 shows that the logarithmic Hopf link invariants together with little additional information completely determine the tensor ring. In Section 4, we compare to the modular S-matrix of the Wp triplet algebra and find perfect agreement with the logarithmic Hopf link invariants, see Theorem 4.4. In particular, logarithmic Hopf link invariants with nilpotent endomorphism insertion correspond to S-matrix coefficients involving pseudo-trace functions, while those with identity insertion correspond to the weight zero part of the characters. We also compare there the Jordan blocks of [R] with the open Hopf link operators, for Wp. We thus see that the logarithmic Hopf link invariants correspond to the S-matrix of an SL(2, Z)-action, analogous to the Hopf link invariants in (semi-simple) modular tensor categories. It would be interesting future work to connect this to Lyubashenko [Ly1] as well as giving an independent proof of the SL(2, Z)-action using logarithmic Hopf link invariants if possible. A more important observation is the nice matching of S-matrix coefficients, especially that S-matrix coefficients involving the pseudo-trace functions relate to logarithmic Hopf link invariants with nilpotent endomorphism insertion. We believe that this is no coincidence and our hope is that one can prove a theorem as above for C2-cofinite VOAs whose tensor category is rigid and which have an ideal possessing a modified trace. In Section 5 we give several general results...

1.4 Questions and optimistic guesses

Let V be a strongly-finite VOA, i.e. a simple C2-cofinite VOA of CFT-type with V isomorphic to V∨. Based on the preceding remarks, one could hope that the following are true, at least in broadstrokes, though surely some details will have to be modified. It is perhaps premature to make precise conjectures, but we propose the following as a way to help guide future work.

Question. We need more examples of C2-coﬁnite VOAs! Most fundamental is the question as to what replaces the role of modular tensor category for strongly-ﬁnite VOAs.

Deﬁnition 1.1. A log-modular tensor category C is a ﬁnite tensor category which is ribbon, whose double is isomorphic to the Deligne product C ⊗ Copp.

These terms are defined in Section 2.1. Modular tensor categories C are precisely the ribbon fusion categories whose double is isomorphic to C⊗Copp, and a finite tensor category is the natural nonsemi-simple generalization of a fusion category, so this is the natural definition. A log-modular tensor category is meant to be the category of V-modules for some strongly-finite VOA, so the definition is conjectural in that sense and should be tweaked if necessary as we learn more.

9 This is not obvious; for example, Miyamoto [Miy3] proposes rigidity should be replaced for general C2-cofinite VOAs with a notion he calls ‘semi-rigidity’. In any case, there should be several superficially different but equivalent definitions; what we want is to make connection with the deep work of Lyubashenko [Ly1, Ly2].

Conjecture 1.1. The category Modgr(V) of V-modules is a log-modular category.

Some authors have conjectured that any modular tensor category is realized by a strongly-rational VOA. Analogously, one should ask if any log-modular tensor category is realized by a strongly-finite VOA (we don’t want to elevate this to the status of a conjecture yet). In Section 2.1 we define the ‘semi-simplification’ of a category, by quotienting by the negligible morphisms and restricting to the full subcategory generated by the simple modules.

Conjecture 1.2. The semi-simpliﬁcation of category Modgr(V) is a modular tensor category, whose Grothendieck ring is the quotient of the Grothendieck ring of Modgr(V) by the image of the V-modules which are both simple and projective.

There are several weaker statements, which we prove in Section 5 assuming e.g. rigidity. We collect some of these in the next two conjectures.

Conjecture 1.3. If M is simple then End(M) = CId. If M is indecomposable then all elements of rad(End(M)) are nilpotent and End(M) = CId + rad(End(M)). The projective V-modules form an ideal in Modgr(V), closed under tensor products and taking contragredient.

Conjecture 1.4. Suppose V is strongly-ﬁnite but not strongly-rational. Then: (a) V is not projective as a V-module. (b) V has inﬁnitely many families of indecomposable V-modules, each parametrized by complex numbers. ∼ ∼ i (c) for any grading-restricted weak V-module Y , Pi ⊗Y = Y ⊗Pi = ⊕j,kNk,j∨ hY : MjiPk; (d) the categorical dimension of any projective V-module P is 0.

Most exciting is the question as to what replaces Verlinde’s formula. We propose:

m Conjecture 1.5. (a) S block-diagonalizes all tensor resp. Grothendieck matrices NMj , NPj m −1 m (j) (j) resp. [N ]j: S NM S = ⊕jBM where for each j, M 7→ BM is an indecomposable representation of the tensor ring (similarly for the Grothendieck matrices) (...this seems wrong...). (j) (b) Each indecomposable subrepresentation M 7→ BM of the Grothendieck ring representation is equivalent to a submodule of the open Hopf link invariant M 7→ ΦM,Pj for some projective cover Pj, each occurring with multiplicity exactly 1. Each indecomposable subrepresentation of the regular representation of the full tensor ring Fusfull(V) occurs in Fusfull(V) with multiplicity 1. (c) There is an isomorphism between the space of pseudo-traces and ambidextrous... such that...

10 We do not know which indecomposable subrepresentations should appear in the regular representation of the full tensor ring Fusfull(V) or even the subring Fussimp(V) generated by the simple V-modules Mj and their projective covers Pj — even for V = Wp this question is mysterious. It is a theorem for Wp that open Hopf link operators do not suﬃce for Fusfull(V) or Fussimp(V). This interesting question should be explored. Of course, continua of indecomposables will be needed in general. We expect that the space spanned by the characters of V possesses more structure than it is usually credited. The character of a simple module Recall the slash operator

aτ + b a b F | γ(τ, v) = (cτ + d)−kF , v , γ = . k cτ + d c d

Conjecture 1.6. (a) The C-span ChV of the 1-point functions FM (τ, v), as M runs over (0) (1) (N) all simple V-modules, has a grading ChV = ChV ⊕ ChV ⊕ · · · ⊕ ChV , where for any (k) v ∈ V with L[0]v = nv, the space of all F (?, v) for F ∈ ChV is an SL(2, Z)-module for the slash operator |k+n, and this SL(2,Z)-module structure is independent of v. (b) If M is the projective cover of a simple module, then its 1-point function FM lies in (0) (0) ChV . If M is simple with nilpotency m, then its 1-point function FM lies in ChV ⊕ (m) · · · ⊕ ChV . Pm (k) (c) k=0 dim ChV will equal the number of simple V-modules with nilpotency degree ≤ m. (doesn’t seem write)

SF2 already should give a counterexample to this conjecture... We should work out SF2 explicitly. The known space of pseudo-trace is 11-dim, while the modular closure of characters is 6-dimensional. It’ll be good to see the additional pseudo-traces, and see why they’re there. Question. Relation of Runkel’s approach to Miyamoto’s. What is dimension of space of 1-point functions for Wp and SFd? After all, the dimension of Miyamoto’s space can be bigger than expected, as we explain for symplectic fermions. Could Runkel give a submodule? Miyamoto’s pseudo- traces recover the formal space of 1-point functions on torus, whose definition is copied from Zhu, but maybe in the semi-simple case Zhu’s definition should be chopped down? One of the things to gain from the categorical picture is that it is at present much easier to construct examples of modular tensor categories than it is to construct rational VOAs. We expect the analogue to hold for strongly-finite VOAs:

Conjecture 1.7. (a) Any ﬁnite tensor category (in the sense of Etingof–Ostrik [EO]) can be realized by endomorphisms on some algebra, as sketched in Section 5.5 below. (b) The centre (or quantum double) construction applied to a ﬁnite tensor category yields a log-modular tensor category (as in Conjecture 1.1).

It is a theorem that any unitary fusion category can be realized by ∗-endomorphisms on some ∗-algebra; the generalization to nonunitary fusion categories is the theme of [EG2]. It is a theorem that the centre construction applied to any fusion category is a modular tensor category. Conjecture 1.6 is the obvious generalization to the ﬁnite but nonsemi-simple setting. It is known (Thm. 3.34 of [EO]) that the double of a ﬁnite tensor

11 category is a finite tensor category. Of course most log-modular tensor categories won’t be centres of finite tensor categories, just as most modular tensor categories are not centres or doubles of fusion categories. Just as we expect that any modular tensor category is the category Modord(V) of ordinary V-modules for some strongly-rational VOA V, we would expect that any log-modular tensor category (at least once that term is defined correctly!) is the category Modgr(V) for some strongly-rational VOA V.

Acknowledgements: We would like to thank J¨urgenFuchs for many explanations of Lyubashenko’s work on modularity within nonsemi-simple tensor categories. TC also thanks Shashank Kanade for many discussions on related topics. TG thanks the Physics Department at Karlstad University for a very pleasant and stimulating work environment. Our research is supported in part by NSERC.

2 Background material 2.1 Braided tensor categories We use the books [T], [EGNO] as references on tensor categories. Some features for nonsemi-simple ones are included in [CGP, GKP1]. Let C be our tensor category. We assume it to be strict so that we don’t have to worry about associativity isomorphisms. The category is called rigid if for each object M in the category, there is a dual M ∗ ∗ ∗ and morphisms bV ∈ Hom(1,V ⊗ V ) (the co-evaluation) and dV ∈ Hom(V ⊗ V, 1) (the evaluation) such that

(IdV ⊗ dV ) ◦ (bV ⊗ IdV ) = IdV , (dV ⊗ IdV ∗ ) ◦ (IdV ∗ ⊗ bV ) = IdV ∗

Rigidity is required for the categorical trace and dimension, as we’ll see shortly. In the vertex operator algebra language, a morphism is an intertwiner and an object is a module. Given any modules V,W in the category, the braiding cV,W is an intertwiner in Hom(V ⊗ W, W ⊗ V ) satisfying

cU,V ⊗W = (IdV ⊗ cU,W ) ◦ (cU,V ⊗ IdW ) , cU⊗V,W = (cU,W ⊗ IdV ) ◦ (IdU ⊗ cV,W ) (4) and also for any intertwiners f ∈ Hom(V,V 0), g ∈ Hom(W, W 0),

(g ⊗ f) ◦ cV,W = cV 0,W 0 ◦ (f ⊗ g) . (5)

This property is called naturality of braiding. It implies cV,1 = c1,V = 1 as well as the Yang-Baxter equation

(IdW ⊗ cU,V ) ◦ (cU,W ⊗ IdV ) ◦ (IdU ⊗ cV,W ) = (cV,W ⊗ IdU ) ◦ (IdV ⊗ cU,W ) ◦ (cU,V ⊗ IdW ) .

If the category C is in addition additive (i.e. it has direct sums), then (U ⊕ V ) ⊗ W resp. W ⊗ (U ⊕ V ) are isomorphic with U ⊗ W ⊕ V ⊗ W resp. W ⊗ U ⊕ W ⊗ V , and using these isomorphisms the naturality (5) of the braiding implies that we can make the identiﬁcations cU⊕V,W = cU,W ⊕ cV,W , cW,U⊕V = cW,U ⊕ cW,V . (6)

12 In a (semi-simple) modular tensor category, the braiding is needed for the topological S-matrix, as we’ll see shortly. Given any module V in our category, the twist θV ∈ Hom(V,V ) satisﬁes, for any module V and intertwiner f ∈ Hom(V,V ),

θV ⊗W = cW,V ◦ cV,W ◦ (θV ⊗ θW ) , θV ◦ f = f ◦ θV

This implies θ1 = 1. The twist directly gives us the (diagonal) topological T -matrix in a modular tensor category: for any simple object V , θV is a number and TV,V = θV . Deﬁne morphisms 0 ∗ bV := (IdV ∗ ⊗ θV ) ◦ cV,V ∗ ◦ bV ∈ Hom (1,V ⊗ V ) and 0 ∗ dV := dV ◦ cV,V ∗ ◦ (θV ⊗ IdV ∗ ) ∈ Hom (V ⊗ V , 1) . Given any intertwiner f ∈ Hom(V,V ), the (categorical) trace is deﬁned by

0 tr(f) = dV ◦ (f ⊗ IdV ∗ ) ◦ bV ∈ Hom(1, 1),

We can identify Hom(1, 1) with C. The trace satisfies tr(f ◦ g) = tr(g ◦ f) and tr(f ⊗ g) = tr(f)tr(g). We define the (categorical) dimension to be dim(V ) = tr(IdV ). It will be positive if the category is unitary. Computations are greatly simplified using a graphical calculus, some basic notation is

0 0 V W. bV bV dV dV cV,W

Details on the graphical calculus are found in the textbooks [T, K, BK]. By a finite tensor category [EO] we mean an abelian rigid tensor category over C, where morphism spaces are finite-dimensional, there are only finitely many simple objects, each of which has a projective cover, every object has finite length in the sense of Jordan– Hölder,and the endomorphism algebra of the tensor unit is C. By a fusion category (see e.g. [EGNO]) we mean a finite tensor category which is in addition semi-simple. By a ribbon category [T] we mean a strict braided tensor category equipped with left duality ∨ X and a twist ΘX ∈ End(X). m In a braided fusion category, the categorical S-matrix is defined by SU,V = tr(cU,V ◦ cV,U ) ∈ C, for any simple modules U, V . Its graphical representation is the Hopf link

V W.

By a modular tensor category [T] we mean a fusion category which is ribbon, with invertible S m. The categorical S- and T -matrices deﬁne a projective representation of SL(2, Z). It m m m satisﬁes SU,V = SV,U , S1,V = dim(V ), as well as Verlinde’s formula

X W m −1 m m NU,V SW,X = (dim X) SU,X SV,X W

13 Ideally (e.g. when the VOA is strongly-rational), the categorical S- and T -matrices will agree up to scalar factors with the modular S- and T -matrices defined through the VOA characters. Return for now to a finite tensor category C. Assume it is spherical. By a negligible morphism f ∈ HomC(U, V ) we mean one for which the categorical trace TrC(gf) = 0 for neg all g ∈ HomC(V,U). The negligible morphisms form a subspace IC (U, V ) of HomC(U, V ), closed under taking duals, arbitrary compositions, as well as arbitrary tensor products. Consider the category C whose objects are the same as those of C, but whose Hom- neg spaces are HomC(U, V )/IC (U, V ). This category was originally defined in [BW]; see also p.236 of [EGNO]. If U, V are indecomposable in C and f ∈ HomC(U, V ) is not an isomorphism, then f is negligible. Then C is a spherical tensor category, whose simple objects are precisely the indecomposable objects of C with nonzero categorical dimension (the indecomposables with dimension 0 are in C isomorphic to the 0-object). Moreover, two simple objects in C are isomorphic iff they are isomorphic indecomposables in C. So C will generally have infinitely many inequivalent simples, but it is semi-simple in the sense that every object in C is a direct sum of simples. By the semi-simplification Css of C, we mean the full subcategory of C generated by the simples of C. Then Css will also be a semi-simple spherical tensor category. If in C the tensor product of simples is always a direct sum of simples and projectives (as it is for Wp and the symplectic fermions), then Css will be a fusion category, whose Grothendieck ring is the quotient of that of C by the modules in C which are both simple and projective. If C is ribbon, so is both C and Css.

Theorem 2.1. Note though that the semi-simpliﬁcation of a nonsemi-simple ﬁnite tensor category C can never be unitary, as some quantum-dimensions must be negative in order that the projectives of C have 0 quantum-dimension.

2.2 Categories and 1-point functions of strongly-finite VOAs By a weak module M of a VOA V we mean the usual definition of a V-module, except that L need not act semi-simply. Rather, M = ` M where M is a (possibly infinite- 0 r∈C r r k dimensional) generalized eigenspace of L0 with eigenvalue r — i.e. (L0 − rId) (Mr) = 0 for some k = k(r). We call M an ordinary V-module if L0 acts semi-simply. We say a weak module M is grading-restricted if all spaces Mr are finite-dimensional, and also gr there is some l = lM ∈ R such that Mr = 0 whenever Re(r) < l. Write Mod (V) for the category of all grading-restricted weak V-modules. The morphisms are linear maps f : M → N such that f(YM (u, z)) = YN (u, z)f for all u ∈ V, i.e. f(unw) = unf(w) for all u ∈ V, w ∈ M, and n ∈ Z. For example, L0 is always in the centre of End(W). A regular VOA is one in which any weak V-module is a direct sum of simple ordinary V-modules. Given any algebra A (e.g. a VOA), an A-module P is called projective if for every A- modules M,N and surjective A-morphism f : M → N and any A-morphism h : P → N, there is an A-morphism h0 : P → M such that h = f ◦ h0. An A-morphism h : P → M is called a projective cover of an A-module M if P is a projective A-module and any A-morphism g : N → P for which h ◦ g is surjective, is itself surjective.

Theorem 2.2. Let V be a strongly-ﬁnite VOA. Then:

gr (i) Mod (V) is a C-linear abelian braided tensor category;

14 (ii) Modgr(V) has finitely many simple modules (up to isomorphism), all of which are ordinary modules; (iii) all Hom-spaces in Modgr(V) are finite-dimensional; (iv) any grading-restricted weak V-module has finite Jordan–Hölderlength (hence is Ar- tinian and Noetherian);

(v) any simple V-module Mi has a projective cover Pi. (vi) the contragredient M 7→ M ∨ defines a contravariant endofunctor from Modgr(V). Thus Modgr(V) is a finite tensor category in the sense of [EO], apart from possibly rigidity. Theorem 2.2 is proved in [H3] (though parts were known earlier). In particular, part (i) is Thm.4.11 there (see also [HLZ] and references therein). Part (ii) is part of Thm.3.24 and Thm. 3.12. Part (iii) is Thm. 3.23. Part (iv) is Corollary 3.16. Part (v) is Thm. 3.23 of [H3] and Proposition 10 of [Miy3]. Part (vi) is explicitly given in Proposition 10.4 of [TW]. Using Theorem 2.1 and Proposition 6-3 of [NT], we can identify the VOA category Modgr(V) with the category of finite-dimensional modules of an explicit finite-dimensional algebra:

Theorem 2.3. Let V be strongly-finite and write P = P0 ⊕ · · · ⊕ Pn for the direct sum of projective covers of simple V-modules. Then AV := EndV (P ) is a finite-dimensional gr fin associative algebra, and Mod (V) is equivalent as an abelian category to Mod (AV ), where V-module M corresponds to the right AV -module HomV (P,M). Equivalence as abelian categories means we ignore tensor products and duals, but the equivalence preserves simples, projectives, indecomposables, composition series, etc. We explore consequences of this in Section 5.1 below. The module P = P0 ⊕ · · · ⊕ Pn is called a progenerator. If we have in addition rigidity, then Modgr(V) is in fact a finite tensor category, and we can say more. In particular, Proposition 2.7 of [EO] says any finite tensor category can be regarded as the tensor category of modules of a weak quasi-Hopf algebra. The simplest example of Theorem 2.3 is a strongly-rational V, whose corresponding algebra AV will be a direct sum of C’s, one for each simple V-module. For a more interesting example, take V to be the triplet algebra Wp. Then from the computations 2 p−1 in Section 6 of [NT], we obtain that AWp is the direct sum C ⊕ A8 of algebras, where ± ± ± ± ∓ ± ∓ A8 is the 8-dimensional algebra with basis {e , τ1 , τ2 , τ1 τ2 = τ2 τ1 } and products ± ± ± ± ∓ ± ± ± ± ∓ s t ± ∓ ± ∓ e τi = τi = τi e , e e = e , e e = 0, and all τi τj = 0 except for τ1 τ2 = τ2 τ1 . ± The two copies of C correspond to the simple projective Wp-modules Xp . Each copy of + − A8 has two simple modules, corresponding to Xi ,Xp−i for some 1 ≤ i < p. Let V be strongly-finite. Write Ek(τ) for the Eisenstein series of SL(2, Z), normalized to have constant term −Bk/k!. Write m = C[E4(τ),E6(τ)] for the space of (holomorphic) modular forms of SL(2, Z). Recall the subspace Uq(V) of V ⊗C m defined e.g. in [Z]. Definition 2.4. By a 1-point function on the torus for V is meant a function

F : V ⊗C m × H → C satisfying:

(i) for u(τ) ∈ V ⊗C m, F (u(τ), τ) is holomorphic for τ ∈ H; P P (ii) F ( i ui ⊗ fi(τ), τ) = i fi(τ)F (ui, τ) for all fi(τ) ∈ m and ui ∈ V;

15 (iii) F (u, τ) = 0 for u ∈ Oq(V); (iv) for all u ∈ V,

∞ 1 d X F (L u, τ) = F (u, τ) + E (τ) F (L u, τ) . [−2] 2πi dτ 2k [2k−2] k=1

Let C(V) denote the space of all 1-point functions. It is easy to show (see Theorem 5.2 in [Miy1]) that C(V) carries an action of SL(2, Z) through Möbiustransformations on τ ∈ H as usual. The main result (Theorem 5.5) of [Miy1] is that C(V) is finite-dimensional, and spanned by the pseudo-trace functions SM,φ(u, τ), where M is a ‘generalized Verma module interlocked with a symmetric linear functional φ’ of the mth Zhu algebra Am(V) for m sufficiently large. These functions SM,φ(u, τ) are quite difficult to compute in practise, but [AN] revised [Miy1], and this version is both simpler and fits much more nicely with our approach. We describe their work shortly. The background underlying both [Miy1, AN] is the following. Choose any indecomposable V-module M, and let Mn (n = 0, 1,...) denote the generalized L0-eigenspace of eigenvalue h + n (h is the conformal weight of M). On each subspace Mn, L0 − c/24 acts k as (n + h − c/24)Id + Lnil where Lnil = 0 for some k independent of n. We obtain for the usual character

∞ ∞ k−1 X X X (2πiτ)j F (v, τ) = Tr| (o(v)qL0−c/24) = qn+h−c/24 Tr| (o(v) Lj ) . (7) M Mn j! Mn nil n=0 n=0 j=0

Now, the conformal vector ω of V lies in the centre of the nth Zhu algebra An(V) for any n n, and on ⊕k=0Mn we have o(u ∗n v) = o(u) ◦ o(v) (Theorem 3.2 of [DLM1]) where ∗n is the product in An(V), so on Mn we have o(v) ◦ L0 = o(v ∗n ω) = o(ω ∗n v) = L0 ◦ o(v). Therefore on Mn the nilpotent operator Lnil also commutes with all operators o(v). This j k k k means for any j > 0 and any v ∈ V,(o(v) Lnil) = o(v) Lnil = 0. But the trace of a nilpotent operator is always nilpotent, so (7) collapses to

∞ X n+h−c/24 FM (v, τ) = Tr|Mn (o(v))q . n=0

Thus the ordinary trace can never see the nilpotent part of L0, and FM (v, τ) = FN (v, τ) whenever V-modules M,N have the same composition factors. Nor can we obtain any more 1-point functions if we insert an endomorphism f of V-module M: by definition, f will commute with all zero-modes and hence with L0, so f ◦ o(v) ◦ Lnil is still nilpotent. What we need is a way to generalize the trace of Mn-endomorphisms so that the terms j o(v)Lnil with j > 0 can contribute. There is a classical situation where this happens. Let A be a finite-dimensional associative algebra over C. A linear functional φ : A → C is called symmetric if φ(ab) = φ(ba) for all a, b ∈ A. Let SLF (A) denote the space of all symmetric linear functionals on A — it can be naturally identified with the dual space (A/[A, A])∗. Now suppose W is a finitely- generated projective A-module. Then an A-coordinate system of W consists of finitely P many ui ∈ W and the same number of fi ∈ HomA(W, A) such that w = i fi(w)ui

16 for any w ∈ W . Given an A-coordinate system, we can associate any endomorphism α ∈ EndA(W ) with a matrix [α] whose ijth entry is [α]ij = fi(α(uj)) ∈ A. Fix any symmetric linear functional φ ∈ SLF (A) and A-coordinate system {ui, fi}, and deﬁne φ the pseudo-trace TrW : EndA(W ) → C by

φ X TrW (α) = φ(Tr([α])) = φ (fi(α(ui))) . i

The pseudo-trace φW is independent of the choice of A-coordinate system, and lies in φ φ SLF (EndA(W )). It satisfies TrW (α ◦ β) = TrV (β ◦ α) for any α ∈ HomA(V,W ), β ∈ HomA(W, V ). To apply this generalized notion of trace to our VOA setting, we need to find a finite- dimensional associative algebra A and V-modules M for which the generalized eigenspaces Mn are projective A-modules. In [Miy1], A is the nth Zhu algebras An(V) for n sufficiently large, and M are certain projective V-modules. [AN] makes a different choice. In particular, let M be a V-module for a strongly-finite VOA V. It is elementary that M and in fact all of its generalized eigenspaces Mn are modules over the (finite-dimensional associative) algebra EndV (M). Suppose M is projective over some subalgebra B of EndV (M). Then each subspace Mn will also be projective over B. For any φ ∈ SLF (B), define the pseudo-trace function

∞ k−1 X X (2πiτ)j F B,φ(v, τ) = qr+n−c/24 Trφ (o(v) Lj ) M j! Mn nil n=0 j=0 is a 1-point function, where k is as above. The point is Theorem 4.3.4 of [AN], which says B,φ that these FM (v, τ) are 1-point functions. Quantum field theory teaches that correlation functions should arise from field insertions. These generalized traces can be interpreted presumably as inserting a defect, but the defect would have to break the conformal symmetry. The explanation for why it is so difficult to obtain these 1-point functions here is that the factorization of the full CFT into chiral halves here is much more subtle than in RCFT — the true QFT is the full CFT. The price we pay for the bulk space not being factorizable in the RCFT sense, is the need to introduce these generalized traces (or something equivalent).

2.3 Open Hopf links and modiﬁed traces

Let C be any rigid braided tensor category. Denote associativity isomorphisms by AX,Y,Z : X ⊗ (Y ⊗ Z) → (X ⊗ Y ) ⊗ Z. Let f be an endomorphism of V ⊗ W in C. Then the left partial trace is

−1 −1 ptrL (f) := (dV ⊗IdW )◦AV ∗,V,W ◦(IdV ∗ ⊗f)◦(IdV ∗ ⊗(ψV ⊗IdW ))◦AV ∗,V ∗∗,W ◦(bV ∗ ⊗IdW ) which is an element of End(W ). What replaces here the Hopf link invariant of modular tensor categories, are the open and logarithmic Hopf link invariants. Deﬁnition 2.5. Let V,W be in C, then the open Hopf link operator is

ΦV,W : = ptrL (cV,W ◦ cW,V ) ∈ End(W ) .

17 The name general Hopf link is used in [CGP]. Pictorially the open Hopf link operator is given above in (3). The following theorem is a direct consequence of the graphical calculus and is well-known.

Theorem 2.6. If C is strict, then for any W ∈ C, the map

Φ ? ,W : Obj(C) → End (W ) ,V 7→ ΦV,W is a representation of the tensor ring.

Proof: Linearity under addition is trivial. The ring homomorphism property is then the following identity

U U ΦU⊗V,W = U ⊗ V = V = = ΦU,W ◦ ΦV,W . V

W W W

Here, the third equality holds because of naturality of braiding, and the second one is (4) together with IdU⊗W = IdU ⊗ IdW .

In order to prove this statement for not necessarily strict categories one has to translate the argument to some commutative diagrams.

Theorem 2.7. For any W ∈ C, with not necessarily strict C the map

Φ ? ,W : Obj(C) → End (W ) ,V 7→ ΦV,W is a representation of the tensor ring.

Proof: We ﬁrst introduce a few short-hand notations. Let P and Q be any two parenthe- sized products of the same objects A, B, C, . . . in the same order. Construct isomorphisms from P to Q just using associativity isomorphisms (and their inverses), then any two such isomorphisms coincide by the associativity axiom of monoidal categories. We denote this isomorphism by α. Further identity morphisms will be surpressed for readability. All squares of the following diagram commute due to naturality of braiding, while the triangle commutes because of last paragraphs discussion.

−1 −1 ∗ ψU ◦ bU ◦ rV ∗ (V ∗ ⊗ V ) ⊗ W / ((V ∗ ⊗ (U ∗ ⊗ U)) ⊗ V ) ⊗ W α α α −1 −1 ψ ◦ b ∗ ◦ r U U V ∗ α + V ∗ ⊗ (V ⊗ W ) / (V ∗ ⊗ (U ∗ ⊗ U)) ⊗ (V ⊗ W ) / (V ∗ ⊗ U ∗) ⊗ (U ⊗ (V ⊗ W ))

cV,W cV,W cV,W −1 −1 ψ ◦ b ∗ ◦ r U U V ∗ α V ∗ ⊗ (W ⊗ V ) / (V ∗ ⊗ (U ∗ ⊗ U)) ⊗ (W ⊗ V ) / (V ∗ ⊗ U ∗) ⊗ (U ⊗ (W ⊗ V ))

18 The next diagram is commutative for the exact same reasons as the previous one.

rV ∗ ◦ dU (V ∗ ⊗ V ) ⊗ W o ((V ∗ ⊗ (U ∗ ⊗ U)) ⊗ V ) ⊗ W O O k α α α

rV ∗ ◦ dU α V ∗ ⊗ (V ⊗ W ) o (V ∗ ⊗ (U ∗ ⊗ U)) ⊗ (V ⊗ W ) o (V ∗ ⊗ U ∗) ⊗ (U ⊗ (V ⊗ W )) O O O cW,V cW,V cW,V

rV ∗ ◦ dU α V ∗ ⊗ (W ⊗ V ) o (V ∗ ⊗ (U ∗ ⊗ U)) ⊗ (W ⊗ V ) o (V ∗ ⊗ U ∗) ⊗ (U ⊗ (W ⊗ V ))

Using that b(U⊗V ) is the map

−1 bV rV bU α 1 / (V ⊗ V ∗) / (V ⊗ 1) ⊗ V ∗ / (V ⊗ (U ⊗ U ∗)) ⊗ V ∗ / (V ⊗ U) ⊗ (U ∗ ⊗ V ∗) and dU⊗V is the map

∗ α dU rV ∗ dV (U ⊗ V ) ⊗ (U ⊗ V ) / (V ∗ ⊗ (U ∗ ⊗ U)) ⊗ V / (V ∗ ⊗ 1) ⊗ V / V ∗ ⊗ V / 1 together with naturality of the pivotal structure the open Hopf link operator can be written as −1 −1 ΦU⊗V,W = `W ◦ dV ◦ A ◦ B ◦ C ◦ ψV ◦ bV ∗ ◦ `W , where A = rV ∗ ◦ dU ◦ α ◦ cW,V is given by the upper right path of the second diagram, −1 −1 −1 B = AA,C,B ◦ cA,C ◦ cC,A ◦ AA,C,B and C = cV,W ◦ α ◦ ψU ◦ bU ∗ ◦ rV ∗ is given by the upper right path of the ﬁrst diagram. Commutativity of these two diagrams implies that they coincide with the two lower left paths, i.e.

−1 −1 A = α ◦ cW,V ◦ rV ∗ ◦ dU ◦ α, C = α ◦ ψU ◦ bU ∗ ◦ rV ∗ ◦ cV,W ◦ α. Inserting this in the open Hopf link operator, we get

−1 −1 ΦU⊗V,W = `W ◦ dV ◦ α ◦ cW,V ◦ D ◦ cV,W ◦ α ◦ ψV ◦ bV ∗ ◦ `W ,

19 where D is given by the left path of the following third diagram.

V ∗ ⊗ (W ⊗ V ) −1 −1 rV ∗ `W

t α * (V ∗ ⊗ 1) ⊗ (W ⊗ V ) / V ∗ ⊗ ((1 ⊗ W ) ⊗ V )

−1 −1 ψU ◦ bU∗ ψU ◦ bU∗ α (V ∗ ⊗ (U ∗ ⊗ U)) ⊗ (W ⊗ V ) / V ∗ ⊗ (((U ∗ ⊗ U) ⊗ W ) ⊗ V )

α α α (V ∗ ⊗ U ∗) ⊗ ((U ⊗ W ) ⊗ V ) / V ∗ ⊗ ((U ∗ ⊗ (U ⊗ W )) ⊗ V )

cU,W ◦ cW,U cU,W ◦ cW,U α (V ∗ ⊗ U ∗) ⊗ ((U ⊗ W ) ⊗ V ) / V ∗ ⊗ ((U ∗ ⊗ (U ⊗ W )) ⊗ V )

α α α (V ∗ ⊗ (U ∗ ⊗ U)) ⊗ (W ⊗ V ) / V ∗ ⊗ (((U ∗ ⊗ U) ⊗ W ) ⊗ V )

rV ∗ ◦ dU `W ◦ dU * t V ∗ ⊗ (W ⊗ V )

The triangles commute due to the triangle axiom together with naturality of left and right multiplication `W , rV ∗ , all the squares commute because of naturality of associativity and braidings. The right-hand side is the open Hopf link operator ΦU,W , hence

−1 −1 ΦU⊗V,W = `W ◦ dV ◦ α ◦ cW,V ◦ ΦU,W ◦ cV,W ◦ α ◦ ψV ◦ bV ∗ ◦ `W −1 −1 = `W ◦ dV ◦ α ◦ cW,V ◦ cV,W ◦ α ◦ ψV ◦ bV ∗ ◦ `W ◦ ΦU,W = ΦV,W ◦ ΦU,W , by naturality of braiding, associativity and left-multiplication.

To help in identifying the open Hopf link invariants, we have the following simple but useful fact, which follows immediately from the naturality (5) of the braiding isomorphisms: Lemma 2.8. Hom intertwines the open Hopf link operators: for any U, V, W ∈ C and any f ∈ Hom(V,W ), ΦU,W ◦ f = f ◦ ΦU,V ∈ Hom(V,W ) .

For example we know that ΦU,W , being an endomorphism of W , will restrict to an endomorphism of the socle of W , and will project to an endomorphism of the top of W (recall the socle of a module is the largest semi-simple submodule, while the top is the quotient by the radical); the lemma allows us to extract both of these. In particular, when End(W ) is one-dimensional, ΦU,W equals either 0 or ΦU,M for any simple submodule M. Moreover, if our category C is in addition additive (i.e. has direct sums), then using the ∼ ∼ isomorphisms of (W1 ⊕W2)⊗V = W1 ⊗V ⊕W2 ⊗V and V ⊗(W1 ⊕W2) = V ⊗W1 ⊕V ⊗W2 to write (6), we obtain

ΦV,W1⊕W2 = ΦU,W1 ⊕ ΦV,W2 . (8)

20 Implicit in (8) is the natural embedding of End(W1) ⊕ End(W2) into End(W1 ⊕ W2) (the latter is often larger). (8) means without loss of generality we can restrict to indecomposable W . Let now P be a full subcategory of C that is a right ideal, meaning that for V in P and W in C, both 1. V ⊗ W is in P, and

2. if α : W → V and β : V → W such that β ◦ α = IdW then W in P. Definition 3.1 of [CGP] (see also [GKP1]) is Definition 2.9. A modified trace on P is a family of linear functions

{ tV : End(V ) → C | V ∈ P } satisfying 1. For U in P and W in C and any f in End (U ⊗ W )

tU⊗W (f) = tV (ptrL (f)) .

2. For U, V in P and morphisms f : V → U and g : U → V

tV (g ◦ f) = tU (f ◦ g) .

For example, the categorical trace on the full category C is a (trivial) example of a modified trace. A more interesting example of a right ideal is spanned by the projective modules in C. By Theorem 2.5.1 of [GKP1], a way to get modified traces is through the existence of ambidextrous objects in C. We pursue this strategy in Section 3 below. Definition 2.10. Let V in C, W in P and x in End (W ), then the logarithmic Hopf link invariant is m ;P,x SV,W := tW (ΦV,W ◦ x) = tW (x ◦ ΦV,W ) . The second equality in the definition comes from property 2 of modified trace, and tells us we only need to apply an endomorphism to one side within the trace. When P = C, we will drop it from the superscript. Likewise we will drop the endomorphism x if x is the identity. The logarithmic Hopf link invariants extract some numbers from the operators ΦU,V , but they lose information too. The endomorphisms x can be used to extract more. We discuss partial traces and the more general symmetric linear functionals φ, in Section 5.1. Lemma 2.11. Let V and W in P then

m ;P m ;P SV,W = SW,V . Proof: This follows from combining the properties of the modiﬁed trace, namely

tW (ΦV,W ) = tW (ptrL (cV,W ⊗ cW,V )) = tW ⊗V (cV,W ⊗ cW,V )

= tV ⊗W (cW,V ⊗ cV,W ) = tV (ptrL (cW,V ⊗ cV,W )) = tV (ΦW,V ) .

21 2.4 The triplet algebra Wp

Our main reference on the triplet algebra is [TW]. The triplet models Wp (p ≥ 2 an integer) are a family of VOAs with central charge cp = 1 − 6(p − 1)/p. They are logarithmic and strongly-ﬁnite [AdM]. As mentioned in Section 2.2, this implies they have ﬁnitely many simple (ordinary) modules, whose characters (when augmented by pseudo- traces) form a vector-valued modular function, and which yields a braided tensor category gr Mod (Wp). The tensor product and rigidity for Wp were determined in [TW] (the tensor product had been previously conjectured in [FHST],[GabR]). + − The simple modules are Xs = Λ(s) and Xs = Π(s) for 1 ≤ s ≤ p. The tensor + + − unit (vacuum) is X1 . Their projective covers are Ps = R0(s) and Ps = R1(s), where ± ± Pp = Xp and

+ + + − − − 0 → Ys → Ps → Xs → 0, 0 → Ys → Pp−s → Xp−s → 0

± for 1 ≤ s ≤ p − 1, where Ys denotes the reducible but indecomposable modules + + − − − + 0 → Xs → Ys → 2·Xp−s → 0 , 0 → Xp−s → Ys → 2·Xs → 0 .

± ± We prefer the [TW] notation Xs and Ps . These 4p − 2 irreducible and/or projective ± ± Wp-modules Xs ,Ps are the most important ones. They are closed under tensor product simp and form the subring Fus (Wp). gr [NT] proved the equivalence as abelian categories of Mod (Wp) with the category fin Mod (Uq(sl2)) of ﬁnite-dimensional modules of the restricted quantum group Uq(sl2) πi/p fin at q = e , and [KS] determined all indecomposables in Mod (Uq(sl2)) (although the key lemma, describing pairs of matrices up to simultaneous conjugation, is really due to Frobenius (1890)). Hence:

Theorem 2.12. [TW] The complete list of indecomposable grading-restricted weak Wp- modules, up to equivalence, is: ± (i) for each 1 ≤ j ≤ p and each sign, the simple modules Xj and their projective covers ± ± ± Pj (Pp = Xp ); ± ± ± 1 (ii) for each 1 ≤ j < p, each sign, and each d ≥ 1, Gj,d, Hj,d, and Ij,d(λ), where λ ∈ CP . ± ± [KS] also give their socle series. Note that Ys = Hs,1. As with any strongly-finite VOA, any other Wp-module will be the direct sum of (finitely many) indecomposable modules. By Proposition 6-6 of [NT], all indecomposable Wp-modules are ordinary (i.e. L0 is ± diagonalizable) except for Pj for 1 ≤ j < p, whose L0 has 2 × 2 and 1 × 1 Jordan blocks. ± ± All indecomposables except those Pj have radical=socle; Pj for 1 ≤ j < p has radical ∓ ± ± ± ± ∼ ∓ ± ± Gj,1. Pj also contains a copy of each Ij,1(λ), and Pj /Ij,1(λ) = Ip−j,1(−λ). The Xs ,Ps are self-dual (Proposition 39 of [TW]). By exactness of duality (e.g. Proposition 4.2.9 of ∨ ∨ ∼ ∓ ± ∨ ∼ ∓ ± ∨ ∼ ∓ [EGNO]) we obtain (Gj,d) = Hp−j,d,(Hj,d) = Gp−j,d, and (Ij,d(λ)) = Ip−j,d(−λ). This fin recovers the duals in Mod (Uq(sl2)) of [KS], where rigidity followed from the elementary and independent reason that Uq(sl2) is a finite-dimensional Hopf algebra. For now, turn to the full subcategory relevant to our discussion, whose objects are ± ± finite direct sums of Xs ,Ps for 1 ≤ s ≤ p. We return to the other indecomposables at the end of this subsection.

22 The fusions are completely determined from the following:

0 0 0 0 X1 ⊗ Xs = Xs ,X1 ⊗ Ps = Ps ,  + X2 s = 1 + +  + + X2 ⊗ Xs = Xs−1 ⊕ Xs+1 2 ≤ s < p ,  + Pp−1 s = p  + − P2 ⊕ 2 · Xp s = 1 + +  + + X2 ⊗ Ps = Ps−1 ⊕ Ps+1 2 ≤ s < p − 1 ,  + + Pp−2 ⊕ 2 · Xp s = p − 1 together with associativity and commutativity. For example one can show

0 + + − + Ps ⊗ Pt = 2·Xs ⊗ Pt ⊕ 2·Xp−s ⊗ Pt .

+ + + + + − In W2, X2 ⊗ Ps must be replaced with X2 ⊗ P1 = 2·X2 ⊕ 2·X2 . ± ± As always, the Z-span of the projective modules {Ps }1≤s≤p ∪ {Xp } form a fusion ± ± ideal. The quotient of the tensor subring Z-span{Xs ,Pt } by that ideal is easily seen to be two copies of the sl(2)d fusion ring at level p − 2, more precisely it is isomorphic − to the fusion ring of the rational VOA L1(sl2) ⊗ Lp−2(sl2), where X1 corresponds to + the integrable highest weight module L(Λ0 + Λ1) of sl(2)d 1 and Xs corresponds likewise to the module L((p − 1 − s)Λ0 + (s − 1)Λ1) of sl(2)d p−2 for 1 ≤ s < p. The proof is elementary, obtained by comparing the tensors by generators. In fact, more importantly, gr ss this also persists categorically: the semi-simpliﬁcation (Mod (Wp)) is a modular tensor category, namely the twist of that of sl(2)d p−2 ⊕ sl(2)d 1 by a simple-current of order 2 as discussed in Section 3.2. ± The characters of Xs are 1 s 1 s ch[X+](τ) = θ (τ) + 2θ0 (τ) , ch[X−](τ) = θ (τ) − 2θ0 (τ) s η(τ) p p−s,p p−s,p s η(τ) p s,p s,p where X 2πiτj2/2 2πizj θs,p(τ, z) = e e √ j∈ √s + 2p 2p Z √ √s ∗ is the theta series associated to the coset 2p + L ∈ L /L of the even lattice L = 2pZ and 0 1 ∂ θs,p(τ) = θs,p(τ, 0) , θs,p(τ) = √ θs,p(τ, z) . 2πi 2p∂z z=0 + − The characters of Ps and Pp−s are both

+ − + − ch[Ps ](τ) = ch[Pp−s](τ) = 2ch[Xs ](τ) + 2ch[Xp−s](τ).

Theta functions of an even lattice are vectors of the dual of the Weil representation of its discriminant, hence

1 2p−1 0 1 2p−1 0 θs,p − 1 X − 2πi`s θ`,p (τ) θs,p − τ 1 X − 2πi`s θ`,p (τ) τ = √ e 2p , = −2iτ √ e 2p . η − 1 2p η (τ) η − 1 2p η (τ) τ `=0 τ `=0

23 We deﬁne the pseudo-trace functions as the weight one part of characters (of course the choice of normalization is a convention) θ0 (τ) θ0 (τ) ptr[X+] := −4iτ p−s,p , ptr[X−] := 4iτ s,p . s η (τ) s η (τ) Not all characters are linearly independent and a basis is given by n o + + + ± B := ch[P` ], ch[X` ], ptr[X` ], ch[Xp ] 1 ≤ ` ≤ p − 1 .

2πi Set q := e 2p . We compute modular S-transformations

2p−1 0 1 8 X θp−`,p (τ) ptr[X+] − = √ q−(p−`)(p−s) s τ 2p η (τ) `=0 p−1 4 X s = √ q(p−`)(p−s) − q−(p−`)(p−s) ch[X+] − ch[P +] 2p ` 2p ` `=1 p−1 4 X s = √ (−1)`+s+p q`s − q−`s ch[X+] − ch[P +] 2p ` 2p ` `=1 and similarly

2p−1 0 ! 1 1 X s θp−`,p (τ) θp−`,p (τ) ch[X+] − = √ q−(p−`)(p−s) − 4iτ s τ 2p p η (τ) η (τ) `=0 p−1 (−1) p−1 + s−1 − = √ s(−1) ch[Xp ] + s(−1) ch[Xp ]+ 2p3 p−1 X `+s+1 `s −`s + s(−1) q + q ch[P` ] + `=1 p−1 1 X √ (−1)`+s+p+1 q`s − q−`s ptr[X+] 2p ` `=1 Finally the last one is 2p−1 1 2 X θp−`,p (τ) ch[P +] − = √ q−(p−`)(p−s) s τ 2p η (τ) `=0 p−1 (−1) p−1 + s−1 − = √ 2p(−1) ch[Xp ] + 2p(−1) ch[Xp ]+ 2p3 p−1 X `+s+1 `s −`s + 2p(−1) q + q ch[P` ] `=1 We deﬁne the modular S-matrix coeﬃcients via 1 X X ch[V ] − = Sχ ch[W ] + Sχ,p ptr[W ] . τ V,W V,W characters in B pseudotraces in B

24 The triplet algebra Wp has uncountably many indecomposable modules. One way ∼ 2 p−1 to construct these is through the associative algebra AWp = C ⊕ A8 (recall Section 2.2). Clearly, it suﬃces to understand the right A8-modules. Any right A8-module M ± ± ± decomposes as M = M+ ⊕ M− where e |M± = IdM± and e |M∓ = 0. Moreover, τi : M± → M∓ and any α ∈ End(M) obeys α(M±) ⊆ M±. + + − − For example, there are two simple A8-modules: e A8e and e A8e , both 1-dimensional. The only A8-modules M with M− = 0 (resp. M+ = 0) are isomorphic to a direct sum of + + − − ± ± ± copies of e A8e ’s (resp. e A8e ’s). The projective covers of e A8e are e A8, both 4- ± dimensional. The space End(e A8) is spanned by the identity, and the nilpotent operator ∓ ± ± ∓ τ1 τ2 + τ1 τ2 . In Table 1 we give the complete list of isomorphism classes of ﬁnite-dimensional indecomposable right A8-modules, together with their socle series Si and endomorphism ± ring. We give there a matrix realization for each τi . We let Id there denote the d × d identity matrix, and Bλ,d the d × d Jordan canonical block with eigenvalue λ. Here, (Id 0) and (0 Id) denote the d × (d + 1) matrices consisting of the identity Id augmented by a Id 0 column of zeros; while and denote the (d + 1) × d matrices consisting of Id 0 Id k augmented by a row of zeros. We let Nk denote the k-dimensional ring C[x]/(x ) (i.e. the ring generated by an element nilpotent of order k). 0 ⊂ S1 ⊂ S2 ⊂ · · · ⊂ M denotes the socle sequence (S0 = 0 and Sk is the largest submodule of M containing Sk−1 such that Sk/Sk−1 is a direct sum of simple modules).

± ± ∓ ∓ dim(M±) dim(M∓) τ1 τ2 τ1 τ2 socle S1 S2 S2/S1 S3/S2 End(M) ± X 1 0 0 0 0 0 M − − − C ± 1 0 0 0 0 0 0 0 ± ± ∓ ± P 2 2 0 0 1 0 1 1 1 1 X H1 2·X X N2 ± ± ∓ Id (λ), d d Id Bλ,d 0 0 d·X M d·X − Nd λ ∈ C ± ± ∓ Id (∞) d d B0,d Id 0 0 d·X M d·X − Nd ± ± ∓ H d + 1 d (Id 0) (0 Id) 0 0 d·X M (d + 1)·X − C d G± d d + 1 Id 0 0 0 (d + 1)·X± M d·X∓ − d 0 Id C

Table 1. The indecomposable A8-modules Of course the X± are simple and P ± are their projective covers. The generating ± nilpotent endomorphism for I (λ) sends basis elements ui resp. vj of M+ resp. M− to ui−1 resp. vj−1, while u1, v1 are sent to 0. The endomorphism ring for any indecomposable Wp-module can be read oﬀ from the ± ± ∓ ∓ corresponding entry in Table 1. Interpret X there as Xj and X as Xp−j. In particular, ± ± we see that only the Pj and Ij (λ) for 1 ≤ j < p have endomorphism rings consisting of more than scalar multiples of the identity.

2.5 Symplectic fermions + Another class of strongly-ﬁnite VOAs is SFd for d ≥ 1, the even part of the symplectic + + fermions. The VOAs SF1 and W2 coincide, and actually the structure of SFd is inherited + from Wp as SFd is an extension by an abelian intertwining algebra of the d-fold tensor product of W2. SF1 = SF is a vertex operator super algebra. It has just itself as its only simple module, and the projective cover P of it has the form 0 → T → P → T → 0, where T

25 satisﬁes 0 → SF → T → SF → 0 (both non-split). SFd is its d-fold tensor product, it thus has also only one simple module and the projective cover is the tensor product of the projective covers of the components. + ± ± − SFd has exactly 4 simple modules [Ab]: SFd and SF (θ)d, where SFd is the odd part ± of the symplectic fermion vertex operator superalgebra, and SF (θ)d are the even/odd + parts of a twisted module of the symplectic fermions. SF1 is just W2 and its fusion is thus (for , ν ∈ {±})

ν ν ν ν SF1 ⊗ SF1 = SF1 ,SF1 ⊗ SF1 (θ) = SF1 (θ) , ν ν ν ν SF1 (θ) ⊗ SF1 (θ) = P1 ,SF1 ⊗ P1 = P1 , ν + − ν + − SF1 (θ) ⊗ P1 = 2·SF1 (θ) ⊕ 2·SF1 (θ),P1 ⊗ P1 = 2·P1 ⊕ 2·P1 .

± ± ± ± Here P1 are the even/odd part of P . The modules SFd ,SF (θ)d and Pd are the even/odd part of the d-fold tensor product of the corresponding SF -modules, that is they +⊗d are the following SF1 -modules:

d d + M O i − M O i Xd = X1 ,Xd = X1 ,X ∈ {SF,SF (θ),P }. d i=1 d i=1 ∈F2 ∈F2 2=0 2=1

+⊗d Their fusion ring can be deduced from the one of SF1 and is isomorphic to the case of d = 1. For details on this procedure see section 5 of [AA]. However, note that the fusion + ring is not the Klein four group. The symplectic fermions SFd are rigid (Proposition 3.23 of [DR]). The characters of the simple modules are

1 η(2τ)2d ch[SF ±](τ) = ± η(τ)2d , d 2 η(τ)2d 1 η(τ)4d η(τ/2)2d ch[SF ±(θ) ](τ) = ± . d 2 η(2τ)2dη(τ/2)2d η(τ)2d

± The conformal weights are 0, 1, −d/8, (4 − d)/8 respectively. We see that SF (θ)d are ± ± projective in addition to being simple. Abbreviate these characters as χ , χθ . Modularity of the characters of the simple modules is given by:

χ±(τ + 1) = eπid/6χ±(τ) , (−iτ)d 1 χ±(−1/τ) = ± χ+(τ) − χ−(τ) + χ+(τ) − χ−(τ) , 2 2d+1 θ θ ± −πid/12 ± χθ (τ + 1) = ±e χ (τ) , 1 χ±(−1/τ) = ±2d−1 χ+(τ) + χ−(τ) + χ+(τ) + χ−(τ) . θ 2 θ θ

+ ± Two indecomposable but nonsimple SFd -modules SFdd are constructed in [Ab]. They 2d−1 2d 2d ± both have characters 2 η(2τ) /η(τ) and socles SFd . They have Jordan–H¨older ± ∓ (−1)r 2d ± 2d ( r ) length 2d with composition factors SFd ,SFd ⊗ C ,...,SFd ⊗ C ,...,SFd [Ab]. 2d−1 + 2d−1 − Thus their images in the Grothendieck ring are both 2 [SFd ] + 2 [SFd ]. L0 for both these indecomposable modules acts by Jordan blocks of size d + 1 [AN]. [AN] show

26 2d−1 that the dimension of the SL(2, Z)-representation is at least 2 + 3, and conjecture this + is the exact dimension (since SF1 coincides with W2, we know this conjecture is correct for d = 1). They also show that the span of the pseudo-characters (the evaluation of pseudo-trace functions at v = 1) is a proper subspace of all 1-point functions. ± ± + − SFdd should be the projective covers of SFd . Indeed, SFd := SFd ⊕ SFd has a + − structure of a vertex operator superalgebra, while SFdd := SFdd ⊕SFdd is the antisymmetric part of the 2dth power of SF1, and carries a natural structure of a SFd-module.... One can construct indecomposable modules of Loewy length 2n for 1 ≤ n ≤ d inductively and these modules are not just tensor products of indecomposable modules. For this consider two VOAs V, W with indecomposable modules PV ,PW with Loewy diagrams as A B

A PV A B PW B

A B

Here, A respectively B are (not necessarily indecomposable) V- respectively W-modules. Let C = TA ⊗ TB with 0 → A → TA → A → 0 non-split and same replacing A by B. Then A ⊗ B C

A ⊗ B A ⊗ B and C C

A ⊗ B C are both indecomposable modules for V ⊗ W. This construction can be recursively be used for the d-fold tensor product for V = W = SF1 and hence one gets modules of each + Loewy length 2n for n between one and d. SFd modules are then obtained by restricting to the even or odd part of these modules. Constructing modules for tensor products of Wp is similar, but since there not all simple decomposition factors are isomorphic the structure becomes a bit more complicated. An example for the case V = W = Wp is

27 + + − − Xi ⊗ Xi Xp−i ⊗ Xp−i

+ − + − − + − + Xi ⊗ Xp−i Xi ⊗ Xp−i Xp−i ⊗ Xi Xp−i ⊗ Xi

+ + − − Xi ⊗ Xi Xp−i ⊗ Xp−i

This is a Loewy diagram of length two and the solid arrows denote the action of the second factor and the doted ones of the first one. Diagrams of this type have appeared in [CR3] as bulk CFT modules. Symplectic fermions, having automorphism group the symplectic Lie group, allow for many other orbifolds [CL] than just by Z2 and any orbifold of a C2-cofinite VOA by any solvable finite group is itself C2-cofinite [Miy4]. There is a Schur-Weyl duality for simple orbifold modules [DLM2] that generalizes to indecomposable but reducible modules as long as the orbifold group is abelian [CLR]. This means given an indecomposable SFd- module P and a finite abelian group G that is a subgroup of the full automorphism group G Sp(d) of SFd then P decomposes as G ⊗ SF -module as

∼ M P = λi ⊗ Pi i where the direct sum runs over a subset of G-modules λi and the Pi are indecomposable SFd-modules with same type of composition series as P .

3 Open and logarithmic Hopf link invariants for the triplet algebra Wp Crucial ingredients in the story are the open and logarithmic Hopf link invariants. Our strategy for computing these is indirect, using the restricted quantum group Uq(sl2) at a gr 2p-th root of unity q. Conjecturally, the category Mod (Wp) is braided equivalent to the representation category for Uq(sl2) at a 2p-th root of unity. For p = 2 this is a theorem. Although we certainly expect the open and logarithmic Hopf link invariants for Wp to coincide with those of Uq(sl2), this is only a conjecture.

3.1 The restricted quantum group Let q = eπi/p be a primitive 2p-th root of unity. We look at the restricted quantum group Uq(sl2) at this value, over the ﬁeld C. We recall a few facts following Murakami [Mur]. Original references are [FGST2, KS, Su]. [FGST2] originally conjectured that the representation categories of Wp and Uq(sl2) at a 2p-th root of unity are equivalent as braided tensor categories. [NT] proved equivalence as abelian categories, and it is a direct comparison that fusion rings obtained in [TW]

28 and tensor rings [KS] are isomorphic. Braided equivalence is however false [KS] and it is believed that the diﬀerence only lies in diﬀerent associativity isomorphisms. This is proven in the case p = 2 by Gainutdinov and Runkel [GaiR] using [DR]. Although these categories are not the same, they are closely related and we expect that the open Hopf link invariants of greatest concern to us will be the same. We use the standard symbols n n −n {n}q Y n [n]q! {n}q := q − q , [n]q := , [n]q! := [k]q! , := . {1}q k [k]q![n − k]q! k=1 q

Deﬁnition 3.1. The quantum group Uq(sl2) is generated by K,E,F with relations

K − K−1 KEK−1 = q2E,KFK−1 = q−2F,EF − FE = . q − q−1 The Hopf algebra structure is given by the coproduct ∆, the counit and the antipode S deﬁned by

∆(K) = K ⊗ K, ∆(E) = 1 ⊗ E + E ⊗ K, ∆(F ) = K−1 ⊗ F + F ⊗ 1 , (K) = 1 , (E) = (F ) = 0 , S(K) = K−1 ,S(E) = −EK−1 ,S(F ) = −KF.

Let H satisfy K = qH , then the universal R-matrix is

∞ 2n 1 H⊗H X {1}q n(n−1) n n R = q 2 q 2 (E ⊗ F ) . (9) {n} ! n=0 q

Deﬁne highest-weight representations Wm with basis f0, . . . , fm−1 and action of Uq(sl2) as m−1−2i Efi = [i]qfi−1, F fi = [m − i − 1]qfi+1, Kfi = q fi with the convention 0 = f−1 = fm. Deﬁnition 3.2. The restricted quantum group is

p p 2p Uq(sl2) := Uq(sl2)/ E ,F ,K − 1 .

The restricted quantum group Uq(sl2) has not a universal R-matrix, i.e. it is not a quasi-triangular Hopf algebra. However it can be embedded in a quasi-triangular Hopf algebra D with universal R-matrix [KS] (see also [FGST2]). Modules of Uq(sl2) that lift to D modules are called liftable and all modules we are interested in are of this type. The universal R-matrix action of D on liftable modules coincides with (9). We will know construct the modules of interest using [Mur]. The restricted quantum ± group Uq(sl2) has simple objects Us for 1 ≤ s ≤ p as well as indecomposable but re- ± ducible projective modules Rs for 1 ≤ s ≤ p − 1. Direct sums of these modules form a closed subcategory, call it C, of the representation category of Uq(sl2). Note that we chose the letter R for projective to avoid confusion with Wp-algebra modules later on. In the ± ± ± ± literature, the Rs are often denoted by Ps . The Rs together with Up are the indecomposable objects in the ideal P in C, consisting of projective modules. It is generated by

29 + the ambidextrous element Up and hence there exists a modiﬁed trace on P (see Theorem 3.6 which follows from [GKP1]). The corresponding simple triplet modules are denoted ± ± by Xs for 1 ≤ s ≤ p as well as projective ones Ps for 1 ≤ s ≤ p − 1. The Loewy diagram + + of the Rs and Ps are

+ + Us Xs

− + − − + − Up−s Rs Up−s Xp−s Ps Xp−s

+ + Us Xs

− − while those of the Rs and Ps are

− − Us Xs

+ − + + − + Up−s Rs Up−s Xp−s Ps Xp−s .

− − Us Xs

The endomorphism ring of each projective module is spanned by the identity as well as a ± nilpotent one, xs , whose image is the simple socle module of its Loewy diagram. ± ± ± ± A basis of Us is given by un for 0 ≤ n ≤ s − 1, deﬁne us = u−1 = .... then

± s−1−2n ± ± ± ± ± Kun = ±q un , Eun = ±[n]q[s − n]qun−1 , F un = un+1 .

± The Rs are characterized by the preceding Loewy diagrams. These can be translated into the following short exact sequences

∓ ± ± ∓ ± ± 0 → Up−s → Vs → Us → 0 , 0 → Vp−s → Rs → Vs → 0 .

+ + + + + A basis of Rs is given by xj , yj , an , bn for 0 ≤ j ≤ p − s − 1 and 0 ≤ n ≤ s − 1. Set

30 + + + + x−1 = yp−1 = a−1 = as = 0, then the quantum group action is + 2p−s−1−2j + + −s−1−2j + Kxj = q xj , Kyj = q yj , + s−1−2n + + s−1−2n + Kan = q an , Kbn = q bn , ( + + + + −[j]q[p − s − j]qyj−1, 1 ≤ j ≤ p − s − 1 Exj = −[j]q[p − s − j]qxj−1, Eyj = + , as−1, j = 0 ( + + + + + [n]q[s − n]qbn−1 + an−1, 1 ≤ n ≤ s − 1 Ean = [n]q[s − n]qan−1, Ebn = + , xp−s−1, n = 0 ( + + + + xj+1, 0 ≤ j ≤ p − s − 2 F yj = yj+1, F xj = + , a0 , j = p − s − 1 ( + + + + bn+1, 0 ≤ n ≤ s − 2 F an = an+1, F bn = + . y0 , n = s − 1 − − − − − A basis of Rp−s is given by xj , yj , an , bn for 0 ≤ j ≤ p − s − 1 and 0 ≤ n ≤ s − 1. Set − − − − x−1 = a−1 = xp−s = bs = 0, then the quantum group action is − −s−1−2j − − −s−1−2j − Kxj = q xj , Kyj = q yj , − s−1−2n − − −2p+s−1−2n − Kan = q an , Kbn = q bn , ( − − − − as−1, j = 0 Exj = −[j]q[p − s − j]qxj−1, Eyj = − − , −[j]q[p − s − j]qyj−1 + xj−1, else ( − − − − [n]q[s − n]qbn−1, 1 ≤ n ≤ s − 1 Ean = [n]q[s − n]qan−1, Ebn = − , xp−s−1, n = 0 ( − − − − yj+1, 0 ≤ j ≤ p − s − 2 F xj = xj+1, F yj = − , b0 , j = p − s − 1 ( − − − − an+1, 0 ≤ n ≤ s − 2 F bn = bn+1, F an = − . x0 , n = s − 1

In [FGST2] the center of Uq(sl2) is investigated. It consists of the elements es for 0 ≤ s ≤ p ± + − together with ws for 1 ≤ s ≤ p − 1. The es acts as the identity on both Rs and Rp−s − + for 1 ≤ s ≤ p − 1, while e0 is the identity on Up and ep on Up . Further, the element + + ws in the radical is the nilpotent endomorphism of Rs whose non-trivial action is given + + + − − by ws bn = an . The element ws is the nilpotent endomorphism in Rp−s with non-trivial − − − action ws yj = xj . The tensor ring is computed in [Su], it is completely determined by 0 0 0 0 U1 ⊗ Us = Us ,U1 ⊗ Rs = Rs  + U2 s = 1 + +  + + U2 ⊗ Us = Us−1 ⊕ Us+1 2 ≤ s < p  + Rp−1 s = p  + − R2 ⊕ 2 · Up s = 1 + +  + + U2 ⊗ Rs = Rs−1 ⊕ Rs+1 2 ≤ s < p − 1  + + Rp−2 ⊕ 2 · Up s = p − 1

31 together with associativity and commutativity. We will later need

+ + + + + Up ⊗ Up+1−i = Up ⊗ Up−1−i ⊕ Ri . (10)

3.2 Restriction to semi-simpliﬁcation of Uq(sl2) ± We begin with the easy computation of Φ?,V for simple nonprojective V , i.e. for V = Us when 1 ≤ s < p. The argument is as in Lemma 6.7 of [CGP]. The semi-simpliﬁcation yields the fusion ring of the modular tensor category corre- ± sponding to sl(2)d p−2 ⊕ sl(2)d 1. Label the simples of the latter by Us (so the signs label the sl(2)d 1 weights and s the sl(2)d p−2 weights); then the modular S-matrix of the latter is

00 0 aff πss S 0 = √ sin , U ,U s s0 p p for all , 0 ∈ {±1} and 1 ≤ s, s0 < p, where 00 = +1 unless = 0 = − in which case 00 = ± s+1 s+1 −1. The quantum- or categorical dimension of Us is (−1) [s]q = (−1) sin(πs/p)/ sin(π/p). (By contrast, the quantum-dimensions of the projective modules are all 0.) That sign (−1)s+1 means that the semi-simplification cannot be identified with that of the affine algebra, but rather with its conformal flow in the sense of [LM]. Whenever End(V ) is 1-dimensional, it can be canonically identified with C by choosing the basis IdV : i.e. we can write f = hfiIdV for f ∈ End(V ). That is the case for the ± V = Us , of course. Since

M =: S m qdim(Mj) := Mj = 1 j 1j

0 m m we obtain Φ = S 0 /S + . U 0 ,Us U ,U U ,U s s0 s 1 s The open Hopf link operators Φ?,V for the simple nonprojective objects V are

0 s0+1 00 sin(πss /p) ΦU 0 ,U = (−1) , ΦR0 ,U = 0 , s0 s sin(πs/p) i s for all 1 ≤ s < p, 1 ≤ s0 ≤ p, 1 ≤ i < p, and , 0 ∈ {±1}, where 00 is as above. Moreover,

0 m m s0+1 0 sin(πss /p) m m S 0 = S 0 = (−1) ,S 0 = S 0 = 0 . U ,U U ,U R ,U U ,R s0 s s s0 sin(π/p) i s s i

ss aff ss In particular, S 0 = S . Since the Hopf link invariant S of the semi- U ,U U 0 ,U s0 s p−s0 p−s simpliﬁcation is therefore nondegenerate, we then know the semi-simpliﬁcation for Wp is indeed a modular tensor category, as predicted by Conjecture 1.2 above. The quantities

(braiding, co-evaluation etc) in this modular tensor category are those of sl(2)d p−2 ⊕sl(2)d 1, except with simple-currents s 7→ p − s included judiciously. (Indeed, it is elementary to verify that any modular tensor category can be twisted in this way by an order-2 simple-current.)

32 3.3 Open Hopf links for projectives of Uq(sl2)

In this section, we compute the open Hopf link operators of Uq(sl2). The strategy of [CGP] is used — they computed these for the unfolded quantum group.

± − + Lemma 3.3. The open Hopf link operators for Rj at U1 and U2 satisfy

p+j Φ − + = (−1) ej = Φ − − U1 ,Rj U1 ,Rp−j and j −j 2 + Φ + + = − q + q ej − {1}qw = Φ + − . U2 ,Rj j U2 ,Rp−j

Proof: The open Hopf link operator ΦV,W is an element of End(W ) hence Φ + = V,Rj + − aej + bwj . In the ﬁrst case, b must be zero since both E and F act as zero on U1 , while H⊗H − + − a is computed from q on U1 ⊗ Rj and then taking the quantum trace on U1 . In the second case, a is obtained by considering the action of the ribbon element on highest- H⊗H + + weight vectors, in this case we get it by adding the eigenvalues of q on u0 ⊗ a0 and + + + 1−p u1 ⊗a0 . One then needs to take the quantum trace over U2 , that is weighted with K . + + + + The nilpotent part of the ribbon element acts on u0 ⊗ bj−1 as zero and on u1 ⊗ bj−1 as

1 1 1 1 2 (H⊗H) 2 (H⊗H) + + 2 (H⊗H) 2 − 2 (j+1) + + q (F ⊗ E) q (E ⊗ F ) u1 ⊗ bj−1 = q (F ⊗ E) {1}qq u0 ⊗ y0 2 −1 + + = {1}qq u1 ⊗ aj−1 .

− The argument for the case of Rj is analogous, the nilpotent part of the second case is obtained by computing

1 1 1 1 2 (H⊗H) 2 (H⊗H) + − 2 (H⊗H) 2 − 2 (p+j+1) + − q (F ⊗ E) q (E ⊗ F ) u1 ⊗ yj−1 = q (F ⊗ E) {1}qq u0 ⊗ b0 2 −1 + − = {1}qq u1 ⊗ xj−1 .

± − + Lemma 3.4. The open Hopf link operators for Up at U1 and U2 satisfy

p Φ − + = ep , Φ − − = (−1) e0 U1 ,Up U1 ,Up and Φ + + = 2ep , Φ + − = −2e0. U2 ,Up U2 ,Up

± Proof: Since Up are simple, this follows from the action of the Cartan element on highest- ± weight states of Up as above.

Let us for convenience deﬁne

+ 2 + − 2 − xj := {1}q{j}qwj , xj := {1}q{j}qwj , e+ := ep , e− := e0 , then it follows

33 ± Theorem 3.5. Identifying End(Up ) with C in the canonical way, we obtain

j+1 00 i+1 00 Φ 0 = j , Φ 0 = 2p , Uj ,Up Ri ,Up

00 00 0 p + − where = 1 if = + otherwise = . For any Rj ∈ {Rj ,Rp−j}, the open Hopf link operators are

(−1)i+1p+j Φ = ({ij} e + (i − 1){(i + 1)j} − (i + 1){(i − 1)j} x ) , Ui ,Rj q j q q j {j}q i+1 p+j Φ = (−1) 4p cos(πij/p) x . Ri ,Rj j

+ + − − Here xj = xj for Rj = Rj and xj = xj for Rj = Rj .

Proof: This follows by induction from Lemma 3.3 together with the relation

Φ + + + Φ + + = Φ + + ◦ Φ + + , Ui+1,Rj Ui−1,Rj Ui ,Rj U2 ,Rj which holds since the open Hopf link operator deﬁnes a representation of the tensor ring. The second one is deduced from the ﬁrst one using (10) so that Φ + + = Φ + + ◦ Φ + + − Φ + + . Ri ,Rj Up ,Rj Up+1−i,Rj Up−1−i,Rj Further p+1+j + Φ + + = (−1) 2px . Up ,Rj j ± The case for Up is the same argument and the remaining Hopf link invariants follow from − + the previous Lemma, since the tensor ring is generated by U1 and U2 .

± ± a b We can identify aej + bxj ∈ End(Rj ) with the matrix 0 a , if desired. Note that the i+1 ss ss eigenvalues of Φ + equal (−1) S /S = Φ + + . Ui ,Rj ij 1j Ui ,Uj

3.4 The modiﬁed trace for projectives of Uq(sl2) Now let P to be the ideal of projective modules. The following is the same argument as Theorem 5.3 of [CGP].

Theorem 3.6. There is a unique modiﬁed trace on P with the property that

p−1 t + (f) = (−1) hfi , where f = hfi Id + . Up Up

+ + Proof: The ideal of projective modules is generated by Up . The module Up is simple + and since the representation category C is finite, Up is then also absolutely simple. It is its own contragredient dual and the tensor product with itself contains the projective + cover of the identity R1 , see (10) with i = 1. By Lemma 3.1.1 and Theorem 3.1.3 of [GKP1] it is thus a right ambidextrous element so that by Theorem 2.5.1 of that article there is a unique modified trace on P. Up to a scalar this is fixed by t + (f) = hfi. We Up choose this scalar to be (−1)p−1.

34 Theorem 3.7. The modiﬁed trace of Theorem 3.6 satisﬁes

i i −i t − Id − = 1, t + Id + = (−1) q + q = t − Id − , Up Up Ri Ri Rp−i Rp−i + i − t + x = (−1) = t − x . Ri i Rp−i i

Proof: The symmetry of the logarithmic Hopf link invariant gives us

p−1+j + p−1 2p(−1) t + x = t + Φ + + = t + Φ + + = (−1) 2p . Rj j Rj Up ,Rj Up Rj ,Up Similarly

p−1+j + 2p(−1) t − x = t − Φ + − = t + Φ − + = −2p . Rj p−j Rj Up ,Rj Up Rj ,Up

− For the identity on Up we use the same argument to get

p−1 p−1 p(−1) t − Id − = t − Φ + − = t + Φ − + = (−1) p . Up Up Up Up ,Up Up Up ,Up

The remaining ones require the use of the property of the modiﬁed trace on tensor products, t ± Id ± = t ± + Id ± + − t ± + Id ± + Ri Ri Up ⊗Up+1−i Up ⊗Up+1−i Up ⊗Up−1−i Up ⊗Up−1−i = t ± ptrR Id ± + − t ± ptrR Id ± + Up Up ⊗Up+1−i Up Up ⊗Up−1−i + + = t ± Id ± qdim U − qdim U Up Up p+1−i p−1−i (−1)p+i = tU ± IdU ± ({p + 1 − i}q − {p − i − i}q) p p {1}q i+p−1 i −i = t ± Id ± (−1) q + q . Up Up

Here, we used that the quantum dimension is qdim U + = (−1)i+1 {i}q . i {1}q

In the matrix notation introduced around Theorem 3.5, this becomes

p−1 a b i i a b t ± (a) = (∓1) a , t + = (−1) 2a cos(πi/p) + (−1) b = t − . Up Ri 0 a Rp−i 0 a

3.5 Logarithmic Hopf link invariants for Uq(sl2) Continue with the choice of P to be the ideal of projective modules. The logarithmic + − Hopf link invariants of Ri and Rp−i coincide so that we set

;P,x+ ;P,x− S m := S m ;P = S m ;P ,S m ;x := S m i = S m i , V,Ri + − V,Ri + − V,Ri V,Rp−i V,Ri V,Rp−i

± where P is implicit in the logarithmic Hopf link invariants of Ri in order to get nontrivial ± values, and we can simplify xi to x as it is clear which nilpotent endomorphism we must insert. We get the following list:

35 Theorem 3.8. The logarithmic Hopf link invariants for projectives are

m i+j+1 m i+j+1 SRi,Rj = (−1) 4p cos(πij/p) ,S ± = (−1) 2i cos(πij/p) , Ui ,Rj m ;x i+j+1 sin(πij/p) m ;x S + = (−1) = −S − , Ui ,Rj sin(πj/p) Up−i,Rj m p−1 p−1 p−1 S + + = (−1) i , SU − ,U + = (−1) (p − i) ,SR ,U + = (−1) 2p , Ui ,Up p−i p i p m i−1 m i−1 m i−1 S + − = (−1) i , S − − = (−1) (p − i) ,S − = (−1) 2p . Ui ,Up Up−i,Up Ri,Up

+ − for Ri ∈ {Ri ,Rp−i}. Note that the defining properties of the modified trace imply that the logarithmic Hopf link invariant restricted to P must be symmetric, as can be verified explicitly for Wp in − + this theorem. Define for convenience R0 := Up and Rp := Up . Then the (p + 1) × (p + 1)- matrix S m is invertible. Next section we will use this to get a Verlinde formula. Ri,Rj

3.6 An alternative derivation The purpose of this section is two-fold. First, above derivation of open Hopf link operators used the known tensor product of modules of Uq(sl2) and here we would like to illustrate another route that does not require this knowledge. Second, it makes a clear connection to the logarithmic knot invariants of Murakami and Nagatomo [MN, Mur] and thus justiﬁes the name logarithmic Hopf link invariant. In this section, we take q to be generic and consider Uq(sl2). Clearly in the limit q to πi/p e the module Ws is also a module of the restricted quantum group for 1 ≤ s ≤ p−1 and + + as such isomorphic to Us . Murakami [Mur] proves the following. Let Ym be the module + + + + isomorphic to W2p−m ⊕Wm with basis α0 , . . . , α2p−m−1, β0 , . . . , βm−1 and quantum group action  [i] α+ , i ≤ p − m or i ≥ p + 1  q i−1 +  " # Eαi = + 2p − m − i − 1 + [i]qα + β , p − m + 1 ≤ i ≤ p  i−1 p − 1 m−p+i−1  q  [2p − m − i − 1] α+ , i 6= p − m − 1  q i+1 +  " # F αi = + p − 1 + [p]qα + β , i = p − m − 1  i+1 m − 1 0  q + + + + + 2p−m−1−2i + + m−1−2i + Eβi = [i]qβi−1, F βi = [m − i − 1]qβi+1 , Kαi = q αi , Kβi = q βi .

πi/p + in the limit q to e this is also a module of Uq(sl2) and as such isomorphic to Rm. We can thus compute open Hopf link operators for generic q and then take the limit. For example

+ + + + + + Φ + + α = X α + x (m, n)β , Φ + + β = X β Ym,Yn k 2p−n k k−p+n k−p+n Ym,Yn k n k where sm −sm q − q [sm]q Xs := χm(s) + χ2p−m(s) and χm(s) = s −s = . q − q [s]q

36 + F commutes with any endomorphism of Yn as e.g. the open Hopf link operator, hence ! ! + + p − 1 + + p − 1 + F Φ + + α = X2p−n [p]qα + β = Φ + + [p]qα + β Ym,Yn p−n−1 p−n n − 1 0 Ym,Yn p−n n − 1 0 q q p − 1 = X [p] α+ + [p] x+(m, n)β+ + X β+ 2p−n q p−n q 0 0 n n − 1 0 q so that p − 1 X − X x+(m, n) = 2p−n n . 0 n − 1 q [p]q The limit q to ζ = eπi/p can be evaluated by l’Hˆopital’srule and we get

p − 1 2p lim x+(m, n) = ζnm + ζ−nm . 0 n − 1 q→ζ ζ χn(1)

The computation of Φ + is similar: Wm,Yn

+ + + + + + Φ + α = Y α + y β , Φ + β = Y β Wm,Yn k 2p−n k k−p+n k−p+n Wm,Yn k n k where Ys := χm(s). Then analogously to before ! ! + + p − 1 + + p − 1 + F Φ + α = Y2p−n [p]qα + β = Φ + [p]qα + β Wm,Yn p−n−1 p−n n − 1 0 Wm,Yn p−n n − 1 0 q q p − 1 = Y [p] α+ + [p] y+(m, n)β+ + Y β+ 2p−n q p−n q 0 0 n n − 1 0 q so that p − 1 Y − Y y+(m, n) = 2p−n n . 0 n − 1 q [p]q The limit q to ζ = eπi/p can be evaluated by l’Hˆopital’srule and we get

p − 1 1 lim y+(m, n) = ((m + 1){n(m − 1)} − (m − 1){n(m + 1)} ) . 0 n − 1 2 q q q→ζ ζ χn(1)

Up to the normalization of the nilpotent endomorphism (which we did not ﬁx) we see agreement with the results of the previous section.

4 Punchlines for Wp This section and the next are the main sections of the paper, and form the basis for the conjectures of Section 1.4.

37 4.1 The logarithmic Verlinde formula X Deﬁne structure constants NUV via

M X U ⊗ V = NUV X, X where the sum is over isomorphism classes of indecomposable X ∈ C, so that

X X NU,V ΦX,W = ΦU,W ◦ ΦV,W . (11) X

Consider first that W is absolutely simple. Then by definition U 7→ ΦU,W is a one- dimensional representation and one defines the Hopf link invariant as in (V1):

m SV,W = trW (ΦV,W ) = trW ⊗V (cV,W ◦ cW,V ) ∈ C .

m m m Now, SV,W = 0 if either the quantum-dimension S1,V or S1,W vanishes. Consider the quotient category by the ideal of objects with vanishing quantum dimension; it is called the semi-simpliﬁcation S (recall Section 1.4). Then in S, (11) becomes

X X SX,W SU,W SV,W NU,V = S1,W S1,W S1,W X where the sum is over all indecomposable objects in S and S is the normalized S m. If this S-matrix is invertible in S then one obtains the standard Verlinde formula S S S−1 X X U,W V,W W,X NU,V = . S1,W W ∈S

(Presumably invertibility can only occur if S is indeed semi-simple....) Let us now assume that a projective module W has endomorphism ring generated by 2 IdW and xW with xW = 0. Then every open Hopf link operator is of the form

ΦU,W = aU,W IdW + bU,W xW for complex numbers aU,W and bU,W . These numbers relate to the tensor structure constants via

X X X X NU,V aX,W = aU,W aV,W , NU,V bX,W = aU,W bV,W + bU,W aV,W (12) X X where the sums are over all indecomposable objects. On the other hand these numbers also relate to the logarithmic Hopf link invariants, since

xW xW xW SU,W = aU,W S1,W + bU,W S1,W ,SU,W = aU,W S1,W , so that they can be expressed in terms of normalized logarithmic Hopf link invariants:

xW xW ! SU,W S1,W SU,W SU,W aU,W = xW , bU,W = xW − xW . (13) S1,W S1,W S1,W S1,W

38 0 4.2 Verlinde (V1) for Wp

In this subsection we review the block-diagonalization of the Grothendieck ring of Wp performed by [PRR], and the block-diagonalization of the tensor ring of Wp performed by [R]. ± Consider first the Grothendieck ring of Wp. It has basis Xs , which we put in the order + − + − + − − X1 ,X1 ,X2 ,X2 ,...,Xp ,Xp . As the tensor ring is generated by the simple-current X1 + and X , it suffices to block-diagonalize [J] := [N ] − and [Y ] := [N ] + . 2 X1 X2 Consider first p = 2. Then 0 1 [Q]−1[J][Q] = diag(1, −1, −1, 1) , [Q]−1[Y ][Q] = diag(2, , −2) , 0 0 for 1 4 0 −1 1 −4 0 −1 [Q] =   . 2 0 4 2  2 0 −1 2 More generally [PRR], there is an invertible matrix [Q] (discussed below) which simultaneously diagonalizes [J] and puts [Y ] into Jordan canonical form. Those matrices [Q]−1[J][Q] and [Q]−1[Y ][Q] fall naturally into two 1 × 1 blocks and p − 1 2 × 2 blocks:

−1 j p−1 p [Q] [J][Q] = diag(1; −I2; ... ;(−1) I2; ... ;(−1) I2;(−1) ) ,

−1 [Q] [Y ][Q] = diag(λ0; Bλ1,2; ... ; Bλj ,2; ... ; Bλp−1,2; λp) , where 1 ≤ j < p, the eigenvalues are λj = 2 cos(πj/p), and we write Ik for the k × k identity matrix and Bλ,k for the canonical k × k Jordan block with eigenvalue λ. Because any other matrix [N ] ± will be a polynomial in [J] and [Y ], [Q] also block- Xs −1 diagonalizes all [N ] ± . The 0th and pth blocks of [Q] [N ] ± [Q] will be the numbers s Xs Xs and (±1)p(−1)s−1s respectively; its jth block, for 1 ≤ j < p, will be 2×2 upper-triangular, with diagonal entries (eigenvalues) ± sin(πjs/p)/ sin(πs/p). Those 2 × 2 blocks will not in general be canonical Jordan blocks, however. Now turn to the tensor ring of Wp. We are interested in the full subcategory spanned by ± ± the irreducible Wp-modules Xs and their projective covers Ps . The corresponding 4p−2 tensor matrices NM , each of size (4p−2)×(4p−2), were put into simultaneous block form in + − + − + − + − [R]. We will put these 4p−2 Wp-modules in order X1 ,X1 ,...,Xp ,Xp ,P1 ,P1 ,...,Pp−1,Pp−1. Again, it suﬃces to block-diagonalize J := N − and Y := N + . X1 X2 Let us consider W2 ﬁrst. A matrix block-diagonalizing all 6 tensor matrices is 1 1 1 0 0 1  1 1 −1 0 0 1    2 0 0 1 0 −2 Q =   . 2 0 0 −1 0 −2   4 0 0 0 1 4  4 0 0 0 −1 4

Then [R]

−1 −1 Q JQ = diag(1, 1, −1, −1, −1, 1) ,Q YQ = diag(2, 0,B0,3, −2) .

39 More generally [R], there is an invertible matrix Q (discussed below) which simultaneously diagonalizes J and puts Y into Jordan canonical form. Those matrices Q−1JQ and Q−1YQ fall naturally into two 1 × 1 blocks and p − 1 pairs of 1 × 1 and 3 × 3 blocks:

−1 j−1 j p−2 p−1 p Q JQ = diag(1; 1, −I3; ... ;(−1) , (−1) I3; ... ;(−1) , (−1) I3;(−1) ) ,

−1 Q YQ = diag(λ0; λ1,Bλ1,3; ... ; λj,Bλj ,3; λp−1,Bλp−1,3; λp) .

Because any other matrix N ± or N ± is a polynomial in J and Y , Q also block- Xs Ps −1 diagonalizes them. The 0th and pth blocks of Q N ± Q will be the numbers s and Xs (±1)p(−1)s−1s respectively; the 1×1 part of its jth block, for 1 ≤ j < p, will be the number (±1)j−1 sin(πjs/p)/ sin(πs/p); the 3 × 3 part of its jth block will be upper-triangular, with diagonal entries (eigenvalues) (±1)j sin(πjs/p)/ sin(πs/p). Again, those 3 × 3 blocks −1 will not in general be canonical Jordan blocks, however. The block structure for Q N ± Q Ps is similar.

0 4.3 Verlinde (V2) for Wp In this subsection we identify the indecomposable representations contained in the regular representation of the tensor ring of Wp, with open Hopf link operators. We listed these last subsection. That decomposition amounts to expressing the regular representations as a direct sum of 1-dimensional, 2-dimensional and 3-dimensional subrepresentations. The Grothendieck ring and tensor ring have the same two 1-dimensional subrepresentations, corresponding to j = 0 and j = p. The j = 0 one is uniquely determined by it − + − p + sending X1 to 1 and X2 to 2; while the j = p one sends X1 to (−1) and X2 to −2. Using Lemma 3.4, we identify these with Φ + and Φ − respectively. ?,Up ?,Up The Grothendieck ring also has a 2-dimensional subrepresentation for each 1 ≤ j < p, − j + which sends X1 to (−1) I2 and X2 to Bλj ,2. By Lemma 3.3, this representation is isomorphic to Φ + or equivalently Φ − . (The one used in Lemma 3.3 is a better ?,Rp−j ?,Rj choice than the one we used here; it is also diﬀerent from that used in [PRR].) − The tensor ring has a 1-dimensional representation for each 1 ≤ j < p, sending X1 to j−1 + (−1) and X2 to λj. This is Φ −j−1 . ?,Up−j What remains is to understand the 3-dimensional representations in the tensor ring.

Theorem 4.1. For any Wp-modules U, M, the map ΦU,M ∈ End(M) decomposes only into 1×1 and 2×2 Jordan blocks. In particular, no Φ?,M can contain a subrepresentation equivalent to any of the 3-dimensional indecomposable tensor ring representations given in the previous section.

Proof: It suﬃces to consider indecomposable M, by (8). M cannot be one of the pro- ± jective modules P , since Φ ± are 1- or 2-dimensional. By Theorem 2.12, this means j ?,Pj 0 that M corresponds to a right A8-module (which we will denote by M ) with either + + − − τ1 = τ2 = 0 identically, or τ1 = τ2 = 0 identically. Assume without loss of generality the latter. Fix any Wp-module U. We will show that αU := ΦU,M ∈ End(M) is a diagonalizable 0 0 matrix. First, M− is an A8-submodule, consisting of the direct sum of d = dim(M−) − − copies of the simple module e A8e . It corresponds to the socle soc(M), which is d

40 − j copies of Xj . Let f : d·X− → M be the corresponding inclusion of Wp-modules; it is in − HomWp (d·X ,M) so we can apply Lemma 2.8 to tell us αU |soc(M) = Φ − . But by j U,d·Xj (8) Φ − = Φ − ⊕ · · · ⊕ Φ − (d times), i.e. the number Φ − times the identity U,d·Xj U,Xj U,Xj U,Xj 0 + operator on soc(M). Note that M/soc(M) is isomorphic to say d copies of Xp−j, where 0 0 d = dim(M+). Because any endomorphism α of M restricts to an endomorphism of the socle, it descends to an endomorphism α on M/soc(M), which we can identify here with 0 0 0 0 the restriction α+ of the corresponding α ∈ End(M ) to M+. Interpreting the projection 0 + f : M → M/soc(M) as living in Hom(M, d ·Xp−j), Lemma 2.8 and (8) tell us α is the 0 number Φ + times the identity operator on M/soc(M). But α is uniquely determined U,Xp−j by α±, so αU is either 0 or (if Φ − = Φ + ) is a scalar multiple of the identity. We U,Xj U,Xp−j see that ΦU,M is diagonalizable.

The same method applied to Φ ± only identiﬁes the diagonal part (i.e. the scalar ?,Pj multiple of the identity), and not the nilpotent part, which is much more subtle. full The full tensor ring Fus (Wp) should be block-diagonalized in the appropriate sense (involving direct integrals). Likewise, not all indecomposable blocks arising in that decomposition will be open Hopf link operators.

0 4.4 Verlinde (V3) for Wp + − Let us for convenience set Rp := Up and R0 := Up . The tensor ring has various ideals. If we forget about arrows, we descend to the Grothendieck ring K[C], but we can also look at the semi-simpliﬁcation, that is quotient by the ideal of projectives. Forgetting about arrows amounts to having the relations

+ − + − 2·Ui ⊕ 2·Up−i = Ri = Rp−i which hold for the logarithmic Hopf link invariants

m m m m 2S + + 2S − = S + = S − Ui ,Rj Up−i,Rj Ri ,Rj Rp−i,Rj and m ;x m ;x m ;x m ;x 2S + + 2S − = S + = S − = 0 . Ui ,Rj Up−i,Rj Ri ,Rj Rp−i,Rj The logarithmic Hopf link invariants can thus capture at most the Grothendieck ring. We will now discuss that under mild additional input one recovers the complete tensor ring. m ;x The S + are the Hopf link invariants of the fusion category of the level p − 2 WZW Ui ,Rj model. This is a rational VOA, and thus the corresponding Verlinde formula determines ± the fusion of the Ui for 1 ≤ i ≤ p modulo the ideal P. Using equations (12) and (13) it is a computation to deduce that the only possibility for tensor product with nonnegative

41 integer coeﬃcients is

0 0 U1 ⊗ Us = Us  + U2 s = 1 + +  + + U2 ⊗ Us = Us−1 ⊕ Us+1 2 ≤ s < p  Rp−1 s = p (14)  − R2 ⊕ 2 · Up s = 1 +  U2 ⊗ Rs = Rs−1 ⊕ Rs+1 2 ≤ s < p − 1 .  + Rp−2 ⊕ 2 · Up s = p − 1 Requiring that fusion of the simple socle modules of the left-hand side is contained in the right-hand side forces

0 0 U1 ⊗ Rs = Rs  + − R2 ⊕ 2 · Up s = 1 (15) + +  + + U2 ⊗ Rs = Rs−1 ⊕ Rs+1 2 ≤ s < p − 1 .  + + Rp−2 ⊕ 2 · Up s = p − 1

+ + − It remains to show that U2 ⊗Up = R1 is impossible. It turns out that this is inconsistent + with Up being its own tensor dual

+ + + Lemma 4.2. We have U2 ⊗ Up = Rp−1.

+ + − Proof: Assume that U2 ⊗ Up = R1 , then using (14) and (15) it inductively follows that

s−1 + + M s − s + Us ⊗ Up = ns0,p·Rs0 ⊕ np·Up s0=1

s s for non-negative integers ns0,p, np. Especially the identity does not appear in the tensor + product and Up is not its own tensor dual.

We can thus conclude

Theorem 4.3. The tensor ring of Uq(sl2) is completely determined from the following + three: the socle series of projective modules; the ambidextrous element Up is its own tensor dual and the complete list of logarithmic Hopf link invariants.

We remark that there is also a Verlinde-type formula for some of the tensor ring structure constants. Namely, the S-matrix restricted to the Ri is the S-matrix of theta functions as we will see in the Wp section. It is especially invertible and hence

p Sx S S−1 U +,R Rj ,R` `k Rk X i ` NVi,Rj = x . S + `=0 U1 ,R`

42 Collecting results, we ﬁnd:

p+1 p−1 + (−1) X m ;x + s + ptr[Xs ](−1/τ) = 4 √ {`}qS + ch[X` ] − ch[P` ] , 2p Us ,R` 2p `=1 p−1 + (−1) m + m − ch[Xs ](−1/τ) = √ S + + ch[Xp ] + S + − ch[Xp ]+ 2p3 Us ,Up Us ,Up

p−1 p+1 p X m + (−1) X m ;x + S + ch[R` ] + √ {`}qS + ptr[X` ] , Us ,R` 2p Us ,R` `=1 `=1 p−1 p−1 + (−1) m + m − X m + ch[Ps ](−1/τ) = √ S + + ch[Xp ] + S + − ch[Xp ] + S + ch[P ] . 3 Rs ,Up Rs ,Up Rs ,R` ` 2p `=1 Hence: Theorem 4.4. The logarithmic Hopf link invariants of the restricted quantum group and S-matrix coeﬃcients satisfy

Sχ,p m ;x Sχ S m Sχ m Sχ m X±,X+ S ± X±,X± U ±,U ± X±,P + S ± P ±,P + S ± s ` = Us ,R` , s p = s p , s ` = Us ,R` , s ` = Rs ,R` . χ,p m ;x χ m χ m χ m S + + S + S + ± S + ± S + + S + S + + S + X1 ,X` U1 ,R` X1 ,Xp U1 ,Up X1 ,P` U1 ,R` P1 ,P` R1 ,R` We thus see that the logarithmic Hopf link invariants correspond to the S-matrix of an SL(2, Z)-action, analogous to the Hopf link invariants in modular tensor categories. 1,p 0,b 2,b 2,p The matrix [Q] of [PRR] has an interpretation as [Q] = (Sr,s | · · · |Sr,s Sr,s | · · · |Sr,s ) in terms of the modular S matrix.

5 From even binary codes to C2-coﬁnite VOAs

We would like to construct new examples of C2-coﬁnite VOAs that illustrate our ideas. This will be achieved by VOA extensions of tensor products of triplet VOAs. These extensions will be of abelian intertining algebra type where the algebra is associated to even binary codes. We have to review VOA extensions for this following [?, ?, ?, ?].

5.1 VOA extensions and algebras 5.2 VOA extensions from even binary codes n Let C ⊂ F2 be a binary code, that is a ﬁnite-dimensional vector space over F2. Such a code is called even if the number of ones of each code word is even and doubly even if this number is a divisible by four. Fix p1, p2, . . . , pn integers at least two and consider

W = Wp1 ⊗ Wp2 ⊗ · · · ⊗ Wpn and deﬁne π : F2 → {±} by π(0) = + and π(1) = −. We then associate to a vector n x = x1 . . . xn ∈ F2 the W-module

π(x1) π(x2) π(xn) X1(x) := X1 ⊗ X1 ⊗ · · · ⊗ X1 .

43 Deﬁne M A = X1(x). x∈C We want to understand under which conditions A is a VOA.

Theorem 5.1. Let p1, . . . , pn be all even positive integers with pi = pj mod 4 for all 1 ≤ i, j ≤ n. Then A is a VOA. If C is doubly even then pi = pj mod 4 for all 1 ≤ i, j ≤ n is enough to guarante that A is a VOA.

Proof:

5.3 Relating Hopf links to the modular S-matrix 6 Some general observations

We conclude this paper with some observations concerning strongly-ﬁnite VOAs.

6.1 The equivalent ﬁnite-dimensional algebra Theorem 2.3 is useful because the representation theory of ﬁnite-dimensional associative algebras is fairly well understood (see e.g. [ASS] for an accessible introduction). For example:

Proposition 6.1. Suppose V is strongly-finite and let AV be the associated finite-dimensional algebra. Then AV is of finite representation-type iff V is strongly-rational. In particular, if V is not strongly-rational, then V has uncountably many indecomposable modules of arbitrarily high Jordan–Hölderlength.

A finite-dimensional associative algebra A is said to have finite representation-type if it only has finitely many indecomposable modules, otherwise it has infinite representation- type. If A is of infinite representation-type, then there are infinitely many dimensions di for which A has uncountably many indecomposable modules in each dimension di (this is the Second Brauer–Thrall conjecture, proved by Nazarova–Roiter [ASS]). Clearly, a strongly-rational VOA has an algebra AV of finite representation-type, since the only indecomposables are the simples Mi. Suppose then that V is strongly-finite but not strongly-rational. We know [EO] that its Cartan matrix cannot be invertible. But Proposition III.3.10 of [ASS] says this requires AV to have infinite global dimension, which requires AV to have infinite representation-type. This means there are infinitely many dimensions ni where AV has parametrized families of indecomposable modules. Since the finitely many simple AV -modules have bounded dimension, this means the composition series of these indecomposables will grow with the dimensions ni. The same will apply to the corresponding V-modules. If the algebra AV = EndV (⊕iPi) of Theorem 2.3 is to be directly relevant to the C2- cofinite story, it will probably involve the observation that HomV (⊕iPi,M) is a (AV , EndV (M))- bimodule. It is through EndV (M) that Miyamoto’s pseudo-trace functions can be recov- ered.

44 There should be a connection between the pseudo-traces of Miyamoto and the modiﬁed traces of [GKP1] (recall Deﬁnition 2.9).

Proposition 6.2. Given any modified trace {tP }, there is one and only one symmetric linear functional φ ∈ SLF (AV ), such that tP = φP for all projective V-modules P . Con- versely, the pseudo-traces φP of any symmetric linear functional φ ∈ SLF (AV ) necessarily satisfy property 2 of Definition 2.9. Certainly property 1 of Definition 2.9 is not automatically satisfied by a symmetric linear functional (see next paragraph). Given any φ, we get a pseudo-trace φB :

EndAV (B) → C in the usual way for each projective ﬁnite-dimensional AV -module B.

The first statement is now Proposition 2.12(2) of [Br]: take φ to be tAV . To get the second, use Proposition 2.12(1) of [Br] to see φB(f ◦ g) = φC (g ◦ f) whenever B,C are projective and f ∈ HomAV (B,C) and g ∈ HomAV (C,B). Thanks to Theorem 2.3, this means we have a family of φP : EndV (P ) → C for all projective V-modules, satisfying property 2 of Definition 2.9. For example, AV for Wp was given in Section 2.2 above. We find that the space SLF (Wp) has dimension 3p −1. The simple Wp-modules contribute 2p to this dimension, so quotienting away that semi-simple part leaves p − 1 independent symmetric linear (s) + − (s) + − (s) − + functionals φ , one for each pair Ps ,Pp−s. They satisfy φ (τ1 τ2 ) = 1 = φ (τ1 τ2 ), + − i.e. have the same values on the nilpotent parts of Ps and Pp−s, and are 0 on the other 6 basis vectors of A8, as well as vanishing on the two copies of C and the p − 2 other copies of A8 in AWp . All that is special about the φ coming from the unique modified trace of (s) Wp, is that it links together these φ . In fact the ordinary ‘semi-simple’ trace on the projectives vanishes because their quantum-dimensions do, so the only way to see this diagonal=idempotent part of the endomorphism ring is through inserting the nilpotent operators x. This would be avoided if we simply used symmetric linear functionals rather than modified traces.

6.2 Rigidity implies ribbon Theorem 6.3. Let V be strongly-finite. Suppose Modgr(V) is rigid. Then Modgr(V) is a finite ribbon category. Moreover, if M is simple then End(M) = CId. If M is indecomposable then all elements of rad(End(M)) are nilpotent and End(M) = CId + rad(End(M)). A ribbon category is a pivotal braided tensor category with twist. The twist is redundant in a spherical braided category, and is defined by ΘX = (trX ⊗ IdX )(cX,X ) (see e.g. Section 4 of [Mug]), and in a rigid braided category pivotal implies spherical. So it suffices to verify that Modgr(V) is pivotal. By construction, it is manifest that the contragredient of the contragredient of a V-module is itself: (X∨)∨ = X. We need to verify that (f ∨)∨ = f for all morphisms.... Schur’s Lemma for VOAs says that End(M) = CId when M is simple. By Theorem thm:misc, we know a module is finite length; Fitting’s Lemma says that every endomorphism of an indecomposable module of finite length is either invertible or is nilpotent (this trivially implies End(M) = CId + rad(End(M)).). Let M0,...,Mn be the simple V-modules and P0,...,Pn their projective covers. Dis- tinguish as usual between the tensor product M ⊗ N of modules, and the Grothendieck

45 ring. The latter is spanned over Z by the classes [Mi], and any (grading-restricted weak) P V-module M is identified with [M] = ihM : Mii[Mi], where hM : Mii denotes the number of times [Mi] appears in the list of composition factors of M. In the Grothendoeck k P k ring, define the Grothendieck coefficients [N ]ij by [Xi][Xj] = k[N ]ij[Xk]. Corollary 6.4. Let V be strongly-finite but not strongly-rational. Assume Modgr(V) is rigid. Let Proj(V) denote the full subcategory of projective grading-restricted weak V- modules. Then: (a) Proj(V) is an ideal of Modgr(V) closed under taking contragredients; ∼ ∼ i (b) for any grading-restricted weak V-module Y , Pi⊗Y = Y ⊗Pi = ⊕j,kNk,j∨ hY : MjiPk; (c) any indecomposable projective grading-restricted weak V-module has a unique simple submodule; (d) the categorical dimension of any projective module P ∈ Obj(Modgr(V)) is 0;

(e) the n × n matrix with entries hPi : Xji is not invertible. The simple submodule is called its socle. These are Propositions 2.1-2.3 and Theorem 2.16 in [EO], once we have our Theorem 6.3. The matrix hPi : Xji is called the Cartan matrix of the category.

6.3 A test for rationality

Theorem 6.5. Suppose V is strongly-finite. Suppose the 1-point functions FM (τ, v) of the simple modules M are closed under SL(2, Z). Then V is strongly-rational. Suppose V is strongly-finite but not strongly-rational. If the tensor identity V (which is simple by hypothesis) is projective, then so is M = M ⊗ V for any M, so all simple modules are also projective, and V is regular, hence strongly-rational, a contradiction. Thus V cannot be projective as a V-module, and so has a projective cover P (V) 6= V. Then P (V) has some pseudo-trace other than the ordinary trace, which will have nontrivial log q dependence. But Proposition 5.9 of [Miy1] says that pseudo-trace must be a linear combination over C[τ] of the (usual) 1-point functions of simple modules. But the C-span of the latter is closed under e.g. τ 7→ −1/τ, so the psuedo-trace, written as a combination over C[τ], will involve a rational function in τ. But the 1-point functions of simple modules are linearly independent over rational functions in τ. This gives a contradiction. A somewhat related test of rationality is Corollary 17 of [Miy3], which says that when a simple C2-cofinite VOA V of CFT-type, all of whose V-modules are semi-rigid, is projective as a V-module, then V is rational.

6.4 The space spanned by characters Let V be strongly-ﬁnite. Let W be an indecomposable projective logarithmic V-module, ∼ ∼ and suppose there are submodules W1,W2 of W such that W/W1 = W2 and W/W2 = W1, as V-modules. Suppose W1 is a submodule of W2. We call W (W1,W2)-intertwined. This happens for example with W1 = 0,W2 = W , or W1 = soc(W ) and W2 = rad(W ). Then

46 L in each generalized eigenspace W (n) of L0, the operator o(v)q 0 has matrix form

ABC  0 DBt 0 0 A where the ﬁrst row/column refers to the subspace W1, the second row/column refers to any lift to W of W2/W1, and the third row/column refers to any lift to W of W/W2. Note that E can be regarded as an operator on W1. By the pseudo-trace ptrW (n) of this W1 operator we mean Tr E. By a 1-point pseudo-function FW (τ, v) we mean

W1 X L0−c/24 FW (τ, v) = ptrW (n)o(v)q . n These are precisely Miyamoto’s pseudo-trace functions.

Theorem 6.6. If nilpotent degree ≤ 2, then Conjecture 1.5 holds.

6.5 Finite tensor categories and the Leavitt algebra There is by now a substantial industry of realizing fusion categories with endomorphisms. Realizing finite tensor categories with endomorphisms is much more difficult, partly because (for the categories relevant to us) there will be infinitely many continuous families of indecomposables, each of which must be assigned an endomorphism. To illustrate the general idea, it suffices to consider a very simple finite tensor category C. We will generalize the approach of [EG2], which realizes (not necessarily unitary) fusion categories. In turn, [EG2] is modelled on the subfactors method (which only works though for unitary fusion categories). It is known that any unitary fusion category can be modelled in a similar way by endomorphisms in a C∗-algebra. Below, we will try to model the simple category with endomorphisms in some C-algebra A (to be defined shortly). That is, the objects in C are identified with A-endomorphisms, and morphisms X ∈ Hom(f, g) are intertwiners, i.e. X ∈ A for which Xf(x) = g(x)X for all x ∈ A. Composition of morphisms is multiplication in A. The tensor product of objects is composition: f ⊗ g = f ◦ g, while of morphisms is: X ⊗ Y = Xf(Y ) = g(Y )X ∈ Hom(f ◦ f 0, g ◦ g0) when X ∈ Hom(f, g), Y ∈ Hom(f 0, g0). A basic fact about these intertwiners are that for any A-endomorphism h and X ∈ Hom(f, g), h(X) ∈ Hom(h ◦ f, h ◦ g): indeed, h(X) h(f(y)) = h(Xf(Y )) = h(g(Y )X) = h(g(Y )) h(X). Similarly, Hom(f, g) ⊆ Hom(f ◦ h, g ◦ h). Suppose we have two inequivalent simple objects, namely the unit 1 and say A. Then they both must be self-dual: 1∨ = 1 and A∨ = A. Let the tensor product be A ⊗ A =∼ 1. Write P1 and PA for the projective covers. We are interested in the nonsemi-simple case, so the images [P1] and [PA] in the Grothendieck ring must be proportional. But 1 and A are self-dual, so [P1] must contain at least 2 copies of [1] while PA must contain at least 2 copies of [A]. Thus the simplest possible Loewy diagrams are then

47 1 A

A P1 A 1 PA 1

1 A

This situation can be regarded as the simplest possible nonsemi-simple ﬁnite tensor category. The simple object 1 will correspond to the identity IdA. Let ρ be the endomorphism attached to module A, and π1 and πρ the endomorphisms attached to P1 and PA. The ∼ ∼ ∼ tensor relations A ⊗ A = 1, P1 ⊗ A = PA and A ⊗ P1 = PA can be written

−1 −1 ρ(ρ(x)) = uxu , π1(ρ(x)) = vπρ(x)v , πρ(x) = ρ(π1(x)) , ∼ for some invertible u, v ∈ A, where without loss of generality we used A ⊗ P1 = PA to deﬁne the endomorphism πρ. These imply

−1 −1 −1 −1 ρ(πρ(x)) = uπ1(x)u , πρ(ρ(x)) = (v π1(u)) π1(x)(v π1(u)) , ∼ ∼ which recovers the tensor products A ⊗ PA = P1 and PA ⊗ A = P1. ∼ Moreover, P1 ⊗ P1 = 2·P1 ⊕ 2·PA (which holds by Corollary 6.4(b)) becomes

0 0 0 0 π1(π1(x)) = s1π1(x)s1 + s2π1(x)s2 + t1πρ(x)t1 + t2πρ(x)t2 , (16)

0 0 where si, si, tj, tj ∈ A satisfy the Cuntz–Leavitt relations

0 0 0 0 0 0 0 0 0 0 sisi = titi = 1 , sitj = tjsi = sisi0 = titi0 = 0 , s1s1 + s2s2 + t1t1 + t2t2 = 1 . (17)

0 This is the meaning of direct sum here. Think of si, si as projecting out copies of P1, and 0 P tj, tj as projecting out copies of PA. We have freedom to redeﬁne them bys ¯i = k Aiksk, ¯ P 0 P 0 ¯0 P 0 t t tj = k Bjktk,s ¯i = k Ciksk, tj = k Djktk, where AC = I = BD . From (16) we obtain for free

2 2 2 2 X (1) (1)0 X (1) (1)0 X (2) (2)0 X (2) (2)0 πρ(π1(x)) = si π1(x)si + ti πρ(x)ti , π1(πρ(x)) = si π1(x)si + ti πρ(x)ti , i=1 i=1 i=1 i=1 2 2 X (3) (3)0 X (3) (3)0 πρ(πρ(x)) = si π1(x)si + ti πρ(x)ti , i=1 i=1

(1) (1)0 −1 0 (1) (1)0 0 (2) where each 4-tuple {si = ρ(ti)u, si = u ρ(ti), ti = ρ(si), ti = ρ(si)}, {si = (1) (2)0 (1)0 −1 (2) (1) (2)0 (1)0 −1 (3) (3)0 0 −1 −1 (3) vsi , si = si v , ti = vti , ti = ti v }, and {si = uvsi, si = siv u , ti = (3)0 0 −1 −1 uvti, ti = tiv u } separately satisﬁes the Leavitt–Cuntz relations. For example,

X 1 10 X 1 10 X −1 0 X 0 X 0 X 0 si si + sj sj = ρ(ti)uu ρ(ti)+ ρ(sj)ρ(sj) = ρ( titi+ sjsj) = ρ(1) = 1 . i j i j i j

48 ∼ ∼ ∼ This recovers PA ⊗ P1 = P1 ⊗ PA = PA ⊗ PA = 2·P1 ⊕ 2·PA. Note that (16) permits us 0 2 to interpret these elements si etc in various intertwiner spaces: namely, si ∈ Hom(π1, π1), 0 2 2 2 ti ∈ Hom(π1, πρ), si ∈ Hom(π1, π1), ti ∈ Hom(πρ, π1). Both Hom(1,P1) and Hom(A, PA) will be one-dimensional, spanned by say a (which maps 1 isomorphically onto the socle of P1) and b (defined likewise). Then Hom(P1, 1) 0 and Hom(PA,A) are likewise spanned by say a (a projection of P1 onto 1 with kernel the socle) and b0 (defined likewise). We see that a0 ◦ a = 0 = b0 ◦ b. Use the same symbols a, a0, b, b0 denote the associated intertwiners in A. 0 0 0 The objects P1 := P1/1 and PA := PA/A satisfy 0 → A⊕A → P1 → 1 → 0 and 0 → 1⊕ 0 1 → PA → A → 0. Using these and the exactness of Hom(P1,?) and Hom(PA,?) we obtain 0 ∼ 0 ∼ Hom(P1,P1) = Hom(P1, 1) and Hom(PA,PA) = Hom(PA,A) and hence dim(End(P1)) = 0 0 2 = dim(End(PA)). This means End(π1) = C1+Caa and End(πρ) = C1+Cbb (of course 0 0 ∼ aa and bb both square to 0, from the previous paragraph). Likewise, Hom(P1,PA) = 0 ∼ ∼ 0 ∼ Hom(P1,PA) = Hom(P1, 2·1) and Hom(PA,P1) = Hom(PA,P1) = Hom(PA, 2·A) are both 2-dimensional. 0 0 In fact, a, a , b, b are redundant. To see this, note first that the morphism a ⊗ IdP1 ∈ Hom(1⊗P1,P1 ⊗P1) embeds a copy of P1 into P1 ⊗P1, and we can regard that copy as the 0 one selected out by s1, s1. In other words, since a ⊗ IdP1 corresponds to element a ∈ A, 0 0 0 0 0 we can take a = s1. Then a ∈ Cs2 (since it must lie in Cs1 + Cs2 and satisfy a a = 0), 0 0 −1 and we can fix a = s2. Moreover, a ∈ Hom(ρ, π1ρ) so we can identify v a ∈ Hom(ρ, πρ) with b and likewise b0 with a0v. We want to show that ρ and π1 (and hence πρ) are endomorphisms on the subalge- −1 −1 0 0 bra of A consisting of polynomials in u, u , v, v , si, si, ti, ti. In other words, we want to identify ρ and π1 of these generators. We have some redundancy in most of these generators — e.g. we can multiply v on the left by any x + yaa0 with x 6= 0. To begin with, ρ(u) ∈ Hom(ρ, ρ3), and that intertwiner space is 1-dimensional space, so is spanned by u. Because ρ2(u) = uuu−1 = u, we see ρ(u) = ±u for some sign. Moreover, ρ(s1) = ρ(a) ∈ Hom(ρ ◦ Id, ρ ◦ π1) = Hom(ρ, πρ) = Cb and ρ(a) 6= 0 since ρ is −1 0 invertible, so we can rescale v if necessary so that v s1 = ρ(s1). Likewise, ρ(a ) will be 0 0 0 a scalar multiple of b : ρ(s2) = s2v for some scalar y 6= 0.

The endomorphism IdP1 ⊗ a ∈ Hom(P1 ⊗ 1,P1 ⊗ P1) is another embedding of P1 into P1 ⊗ P1. This image P1 ⊗ 1 intersects our other image 1 ⊗ P1 at 1 ⊗ 1. This means 0 0 s1π1(a) ∈ Hom(π1, π1) is nonzero but noninvertible, so it is a scalar multiple xaa for 0 x 6= 0. Now, the module P1 ⊗ 1 must intersect the module corresponding to s2, s2 in some other module, but any nontrivial submodule of P1 ⊗ 1 must contain 1 ⊗ 1, hence the 0 0 projection s2π1(a) = 0. Likewise, tjπ1(a) = 0. This means

X 0 X 0 2 0 π1(s1) = ( sisi + tjtj)π1(a) = xs1s2 i j

0 0 0 0 0 for some scalar x 6= 0. Likewise, π1(s2) = x s1s2s1 for some scalar x . We can ﬁx some of the arbitrariness in ti by saying the copy of PA in P1 ⊗ P1 selected 0 0 by t1, t1 has a length-2 submodule a quotient of the left P1, while t2, t2 has a length-2 submodule a quotient of the right P1.

49 References

[Ab] T. Abe, A Z2-orbifold of the symplectic fermionic vertex operator superalgebra, Math. Z. 255 (2007), 755–792. [AA] T. Abe and Y. Arike, Intertwining operators and fusion rules for vertex operator algebras arising from symplectic fermions, J. Alg. 373 (2013), 39–64. [AdM] D. Adamovic and A. Milas, On the triplet vertex algebra W(p), Adv. Math. 217 (2008) 2664–2699. [AC] C. Alfes and T. Creutzig, The mock modular data of a family of superalgebras, Proc. Amer. Math. Soc. 142 (2014), 2265-2280. [AnM] G. Anderson and G. Moore, Rationality in conformal field theory. Commun. Math. Phys. 117 (1988), 441–450. [AN] Y. Arike and K. Nagatomo, Some remarks on pseudo-trace functions for orbifold models associated with symplectic fermions. Internat. J. Math. 24 (2013), no. 2, 1350008, 29 pp. [ASS] I. Assem, D. Simson, and A. Skowronski, Elements of the representation theory of associative algebras, Vol. 1 and 2. (Cambridge University Press, 2006). [BK] B. Bakalov and A. Kirillov, Lectures on tensor categories and modular functors, University Lecture Series, 21. American Mathematical Society, (2001). [BW] J. Barrett and B. Westbury, Spherical categories, Adv. Math. 143 (1999), 357– 375. [Br] M. Broué, Higman criterion revisited [CGP] F. Costantino, N. Geer, and B. Patureau-Mirand, Some remarks on the unrolled quantum group of sl(2). J. Pure Appl. Algebra 219 (2015), no. 8, 3238–3262. [C] T. Creutzig, Quantum dimensions in logarithmic CFT, Oberwolfach report No. 16 (2015) 52–54. [CL] T. Creutzig and A. Linshaw, Orbifolds of symplectic fermion algebras, to appear in Transactions of the AMS, arXiv:1404.2686. [CLR] T. Creutzig, A. Linshaw and D. Ridout, in preparation. [CM1] T. Creutzig and A. Milas, False theta functions and the Verlinde formula, Adv. Math. 262 (2014) 520–545. [CM2] T. Creutzig and A. Milas, in preparation. [CMW] T. Creutzig, A. Milas and S. Wood, On regularised quantum dimensions of the singlet vertex operator algebra and false theta functions, arXiv:1411.3282. [CR1] T. Creutzig and D. Ridout, Modular data and Verlinde formulae for fractional level WZW models I, Nucl. Phys. B 865 (2012) 83–114. [CR2] T. Creutzig and D. Ridout, Modular data and Verlinde formulae for fractional level WZW models II, Nucl. Phys. B 875 (2013) 423–458. [CR3] T. Creutzig and D. Ridout, Relating the archetypes of logarithmic conformal field theory, Nucl. Phys. B 872 (2013) 348-391.

50 [CR4] T. Creutzig and D. Ridout, Logarithmic conformal field theory: beyond an introduction, J. Phys. A 46 (2013) 4006. [DR] A. Davydov and I. Runkel, Z/2Z-extensions of Hopf algebra module categories by their base categories, Adv. Math. 247 (2013) 192–265. [DLM1] C.-Y. Dong, H.-S. Li and G. Mason, Vertex operator algebras and associative algebras. J. Algebra 206, (1998), no. 1, 67–96. [DLM2] C. Dong, H. Li, and G. Mason, Compact automorphism groups of vertex operator algebras, Internat. Math. Res. Notices 1996, no. 18, 913-921. [EGNO] P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor categories (American Math. Soc., Providence 2015). [EO] P. Etingof and V. Ostrik, Finite tensor categories, Mosc. Math. J. 4 (2004) 627–654, 782–783. [EG1] D.E. Evans and T. Gannon, The exoticness and realisability of twisted Haagerup- Izumi modular data, Commun. Math. Phys. 307 (2011), 463–512. [EG2] D.E. Evans and T. Gannon, Non-unitary fusion categories and their doubles via endomorphisms, arXiv:1506.03546. [FGST1] B. L. Feigin, A. M. Gainutdinov, A. M. Semikhatov and I. Y. Tipunin, Modular group representations and fusion in logarithmic conformal field theories and in the quantum group center, Commun. Math. Phys. 265 (2006) 47–93. [FGST2] B. L. Feigin, A. M. Gainutdinov, A. M. Semikhatov and I. Y. Tipunin, The Kazhdan-Lusztig correspondence for the representation category of the triplet W-algebra in logarithmic CFT, Theor. Math. Phys. 148 (2006) 1210–1235. [Fi] F. Fiordalisi, Logarithmic intertwining operators and genus-one correlation functions, arXiv:/1602.03250. [FHST] J. Fuchs, S. Hwang, A. M. Semikhatov, and I. Y. Tipunin, Nonsemisimple fusion algebras and the Verlinde formula, Commun. Math. Phys. 247 (2004) 713–742. [FS] J. Fuchs and C. Schweigert, Hopf algebras and finite tensor categories in conformal field theory. Rev. Un. Mat. Argentina 51 (2010), 43–90. [GabR] M. R. Gaberdiel and I. Runkel, From boundary to bulk in logarithmic CFT, J. Phys. A 41 (2008) 075402. [GaiR] A. M. Gainutdinov and I. Runkel, Symplectic fermions and a quasi-Hopf algebra structure on U¯isl(2), arXiv:1503.07695. [GKP1] N. Geer, J. Kujawa, and B. Patureau-Mirand, Ambidextrous objects and trace functions for nonsemisimple categories. Proc. Amer. Math. Soc. 141 (2013), no. 9, 2963–2978. [GKP2] N. Geer, J. Kujawa, and B. Patureau-Mirand, Generalized trace and modified dimension functions on ribbon categories. Selecta Math. (N.S.) 17 (2011), no. 2, 453–504. [GPT] N. Geer, B. Patureau-Mirand, and V. Turaev, Modified quantum dimensions and re-normalized link invariants. Compos. Math. 145 (2009), no. 1, 196–212.

51 [GPV] N. Geer, B. Patureau-Mirand, and A. Virelizier, Traces on ideals in pivotal categories. Quantum Topol. 4 (2013), no. 1, 91–124. [H1] Y.-Z. Huang, Vertex operator algebras and the Verlinde conjecture, Commun. Contemp. Math. 10 (2008) 103–154. [H2] Y.-Z. Huang, Rigidity and modularity of vertex tensor categories, Commun. Contemp. Math. 10 (2008) 871–911. [H3] Y.-Z. Huang, Cofiniteness conditions, projective covers and the logarithmic tensor product theory, J. Pure Appl. Algebra 213 (2009) 458–475. [HL] Y.-Z. Huang and J. Lepowsky, A theory of tensor products for module categories for a vertex operator algebra. IV. J. Pure Appl. Algebra 100 (1995) 1-3, 173–216. [HLZ] Y.-Z. Huang, J. Lepowsky, and L. Zhang, Logarithmic tensor category theory, VIII: Braided tensor category structure on categories of generalized modules for a conformal vertex algebra, arXiv:1110.1931. [K] C. Kassel, Quantum groups. Graduate Texts in Mathematics, 155. Springer- Verlag, New York, 1995. [KL] T. Kerler and V.V. Lyubashenko, Non-semisimple topological quantum field theories for 3-manifolds with corners. Springer Lecture Notes in Mathematics 1765 (Springer Verlag, New York 2001) [KS] H. Kondo and Y. Saito, Indecomposable decomposition of tensor products of modules over the restricted quantum universal enveloping algebra associated to sl2. J. Algebra 330 (2011), 103–129. [LM] R. Laber and G. Mason, C-graded vertex algebras and conformal flow, arXiv:1308.0557. [Ly1] V. Lyubashenko, Modular transformations for tensor categories. J. Pure Appl. Algebra 98 (1995), no. 3, 279–327. [Ly2] V. Lyubashenko, Modular properties of ribbon abelian categories, arXiv:hep- th/9405168. [Mil] A. Milas, Weak modules and logarithmic intertwining operators for vertex operator algebras, In: Recent developments in infinite-dimensional Lie algebras and conformal field theory (Charlottesville, VA, 2000), Contemp. Math., 297, (Amer. Math. Soc., Providence, RI, 2002) pp. 201–225.

[Miy1] M. Miyamoto, Modular invariance of vertex operator algebra satisfying C2- cofiniteness, Duke Math. J. 122 (2004) 51–91. [Miy2] M. Miyamoto, A theory of tensor products for vertex operator algebra satisfying C2-cofiniteness, arXiv:0309350v3 [Miy3] M. Miyamoto, Flatness and semi-rigidity of vertex operator algebras, arXiv:1104.4675. [Miy4] M. Miyamoto, C2 -Cofiniteness of Cyclic-Orbifold Models, Commun. Math. Phys. 335 (2015) 1279–1286. [MS] G. W. Moore and N. Seiberg, Classical and quantum conformal field theory, Commun. Math. Phys. 123 (1989) 177–254.

52 [Mug] M. M¨uger, Tensor categories: a selective tour, Rev. Un. Mat. Argent. 51 (2010) 95–163. [Mur] J. Murakami, From colored Jones invariants to logarithmic invariants. arXiv:1406.1287. [MN] J. Murakami and K. Nagatomo, Logarithmic knot invariants arising from restricted quantum groups. Internat. J. Math. 19 (2008), no. 10, 1203–1213. [NT] K. Nagatomo and A. Tsuchiya, The triplet vertex operator algebra W(p) and the restricted quantum group at root of unity, Adv. Stud. in Pure Math., Amer. Math. Soc. 61 (2011) 1–49. [PRR] P. A. Pearce, J. Rasmussen, and P. Ruelle, Grothendieck ring and Verlinde-like formula for the W-extended logarithmic minimal model WLM(1, p), J. Phys. A43 (2010) 045211, 13pp. [R] J. Rasmussen, Fusion matrices, generalized Verlinde formulas, and partition functions in WLM(1, p), J. Phys. A 43 (2010), no. 10, 105201, 27 pp. [RW1] D. Ridout and S. Wood, Modular transformations and Verlinde formulae for logarithmic (p+, p−)-models, Nucl. Phys. B 880 (2014) 175–202. [RW2] D. Ridout and S. Wood, The Verlinde formula in logarithmic CFT, J. Phys. Conf. Ser. 597 (2015) 1, 012065. [Ru] I. Runkel, A braided monoidal category for free super-bosons, [Sh] K. Shimizu, The monoidal center and the character algebra, arXiv:1504.01178v2.

[Su] R. Suter, Modules over Uq(sl2). Comm. Math. Phys. 163 (1994), no. 2, 359–393. [TW] A. Tsuchiya and S. Wood, The tensor structure on the representation category of the Wp triplet algebra, J. Phys. A 46 (2013), no. 44, 445203, 40 pp. [T] V. G.Turaev, Quantum invariants of knots and 3-manifolds. de Gruyter Studies in Mathematics, 18. Walter de Gruyter & Co., Berlin, 1994. x+588 pp. [V] E. P. Verlinde, Fusion rules and modular transformations in 2D conformal ﬁeld theory, Nucl. Phys. B 300 (1988) 360–376. [Z] Y. Zhu, Modular invariance of characters of vertex operator algebras, J. Amer. Math. Soc. 9 (1996) 237–302.

Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1, Canada. email: [email protected] and [email protected]