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 `1 (VOAs) V of CFT-type | this means V has L0-grading V = n=0 Vn with V0 = C1 and dim Vn < 1. 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 2 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 2 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 2 V, where o(v) 2 End(M) is the zero-mode. The character of a module is ch[M](τ) = FM (τ; 1). Zhu proved [Z] that for v 2 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 2 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 2 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 equiva- lent 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), tak- ing 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 2 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.