A proposal for the bulk space of the logarithmic -triplet model at central charge zero is made. The construction is based on the idea that one may reconstruct the bulk theory of a rational conformal field theory from its boundary theory. The resulting bulk space is a quotient of the direct sum of projective representations, which is isomorphic, as a vector space, to the direct sum of tensor products of the irreducible representations with their projective covers. As a consistency check of our analysis, we show that the partition function of the bulk theory is modular invariant, and that the boundary state analysis is compatible with the proposed annulus partition functions of this model.
A proposal for the bulk space of the logarithmic -triplet model at central charge zero is made. The construction is based on the idea that one may reconstruct the bulk theory of a rational conformal field theory from its boundary theory. The resulting bulk space is a quotient of the direct sum of projective representations, which is isomorphic, as a vector space, to the direct sum of tensor products of the irreducible representations with their projective covers. As a consistency check of our analysis, we show that the partition function of the bulk theory is modular invariant, and that the boundary state analysis is compatible with the proposed annulus partition functions of this model.
The analogue of the charge-conjugation modular invariant for rational logarithmic conformal field theories is constructed. This is done by reconstructing the bulk spectrum from a simple boundary condition (the analogue of the Cardy `identity brane'). We apply the general method to the c_1,p triplet models and reproduce the previously known bulk theory for p=2 at c=-2. For general p we verify that the resulting partition functions are modular invariant. We also construct the complete set of 2p boundary states, and confirm that the identity brane from which we started indeed exists. As a by-product we obtain a logarithmic version of the Verlinde formula for the c_1,p triplet models.
The analogue of the charge-conjugation modular invariant for rational logarithmic conformal field theories is constructed. This is done by reconstructing the bulk spectrum from a simple boundary condition (the analogue of the Cardy ‘identity brane’). We apply the general method to the c1,p triplet models and reproduce the previously known bulk theory for p = 2 at c = −2. For general p we verify that the resulting partition functions are modular invariant. We also construct the complete set of 2p boundary states, and confirm that the identity brane from which we started indeed exists. As a by-product we obtain a logarithmic version of the Verlinde formula for the c1,p triplet models.
We present a model-independent study of boundary states in the Cardy case that covers all conformal field theories for which the representation category of the chiral algebra is a – not necessarily semisimple – modular tensor category. This class, which we call finite CFTs, includes all rational theories, but goes much beyond these, and in particular comprises many logarithmic conformal field theories. We show that the following two postulates for a Cardy case are compatible beyond rational CFT and lead to a universal description of boundary states that realizes a standard mathematical setup: First, for bulk fields, the pairing of left and right movers is given by (a coend involving) charge conjugation; and second, the boundary conditions are given by the objects of the category of chiral data. For rational theories our proposal reproduces the familiar result for the boundary states of the Cardy case. Further, with the help of sewing we compute annulus amplitudes. Our results show in particular that these possess an interpretation as partition functions, a constraint that for generic finite CFTs is much more restrictive than for rational ones.
We present explicit mathematical structures that allow for the reconstruction of the field content of a full local conformal field theory from its boundary fields. Our framework is the one of modular tensor categories, without requiring semisimplicity, and thus covers in particular finite rigid logarithmic conformal field theories. We assume that the boundary data are described by a pivotal module category over the modular tensor category, which ensures that the algebras of boundary fields are Frobenius algebras. Bulk fields and, more generally, defect fields inserted on defect lines, are given by internal natural transformations between the functors that label the types of defect lines. We use the theory of internal natural transformations to identify candidates for operator products of defect fields (of which there are two types, either along a single defect line, or accompanied by the fusion of two defect lines), and for bulk-boundary OPEs. We show that the so obtained OPEs pass various consistency conditions, including in particular all genus-zero constraints in Lewellen's list.
Universal formulas for the boundary and crosscap coefficients are presented, which are valid for all symmetric simple current modifications of the charge conjugation invariant of any rational conformal field theory. 1 1. Boundaries and crosscaps In this letter we report progress on the problem of finding boundaries and crosscaps for all conformal field theories that can be obtained as simple current invariants of a given rational conformal field theory (RCFT). This line of research was initiated by Cardy [1], who obtained the (bulk symmetry preserving) boundaries in the case where the torus partition function is the charge conjugation invariant. Later, in [2] the corresponding crosscap coefficients were obtained. In a series of subsequent papers [3,4,5,6,7,8,9,10,11,12,13] various more general situations were considered. In particular one may choose a different Klein bottle projection and a different modular invariant partition function (MIPF) χiZij ¯χ j for the bulk theory. The basic data one would like to determine are the set {m} of Ishibashi labels, the set {a} of boundary labels, a matrix Bm,a of boundary coefficients and a vector Γm of crosscap coefficients.
A (1 + 1)D unitary bosonic rational conformal field theory (RCFT) may be organized according to its genus, a tuple (c,C) consisting of its central charge c and a unitary modular tensor category C which describes the (2 + 1)D topological quantum field theory for which its maximally extended chiral algebra forms a holomorphic boundary condition. We establish a number of results pertaining to RCFTs in “small” genera, by which we informally mean genera with the central charge c and the number of primary operators rank(C) both not too large. We start by completely solving the modular bootstrap problem for theories with at most four primary operators. In particular, we characterize, and provide an algorithm which efficiently computes, the function spaces to which the partition function of any bosonic RCFT with rank(C)≤4 must belong. Using this result, and leveraging relationships between RCFTs and holomorphic vertex operator algebras which come from “gluing” and cosets, we rigorously enumerate all bosonic theories in 95 of the 105 genera (c,C) with c ≤ 24 and rank(C)≤4. This includes as (new) special cases the classification of chiral algebras with three primaries and c < 120/7 ∼ 17.14, and the classification of chiral algebras with four primaries and c < 62/3 ∼ 20.67. We then study two applications of our classification. First, by making use of chiral versions of bosonization and fermionization, we obtain the complete list of purely left-moving fermionic RCFTs with c < 23 as a corollary of the results of the previous paragraph. Second, using a (conjectural) concept which we call “symmetry/subalgebra duality,” we precisely relate our bosonic classification to the problem of determining certain generalized global symmetries of holomorphic vertex operator algebras.