This article gives a complete account of the modular properties and Verlinde formula for conformal field theories based on the affine Kac-Moody algebra sl(2) at an arbitrary admissible level k. Starting from spectral flow and the structure theory of relaxed highest weight modules, characters are computed and modular transformations are derived for every irreducible admissible module. The culmination is the application of a continuous version of the Verlinde formula to deduce non-negative integer structure coefficients which are identified with Grothendieck fusion coefficients. The Grothendieck fusion rules are determined explicitly. These rules reproduce the well-known fusion rules of Koh and Sorba, negative coefficients included, upon quotienting the Grothendieck fusion ring by a certain ideal.
The modular properties of fractional level affine sl(2)-theories and, in particular, the application of the Verlinde formula, have a long and checkered history in conformal field theory. Recent advances in logarithmic conformal field theory have led to the realisation that problems with fractional level models stem from trying to build the theory with an insufficiently rich category of representations. In particular, the appearance of negative fusion coefficients for admissible highest weight representations is now completely understood. Here, the modular story for certain fractional level the…
By exploring the description of chiral blocks in terms of co-invariants, a derivation of the Verlinde formula for WZW models is obtained which is entirely based on the representation theory of affine Lie algebras. In contrast to existing proofs of the Verlinde formula, this approach works universally for all untwisted affine Lie algebras. As a by-product we obtain a homological interpretation of the Verlinde multiplicities as Euler characteristics of complexes built from invariant tensors of finite-dimensional simple Lie algebras. Our results can also be used to compute certain traces of autom…
We study (generalized) discrete symmetries of 2d semisimple TQFTs. These are 2d TQFTs whose fusion rules can be diagonalized. We show that, in this special basis, the 0-form symmetries always act as permutations while 1-form symmetries act by phases. This leads to an explicit description of the gauging of these symmetries. One application of our results is a generalization of the equivariant Verlinde formula to the case of general Lie groups. The generalized formula leads to many predictions for the geometry of Hitchin moduli spaces, which we explicitly check in several cases with low genus an…
We prove Zuber's conjecture establishing connections of the fusion rules of the $su(N)_k$ WZW model of conformal field theory and the intersection form on vanishing cycles of the associated fusion potential.
We present a compact formula computing rational trigonometric sums. E. Verlinde's expression for the dimension of conformal blocks in WZW theory is an example of such a sum. As an application, we show that a formula of Bismut and Labourie for the Riemann-Roch numbers of moduli spaces of flat connections on a Riemann surface coincides with Verlinde's expression.
The generalized Verlinde formulae expressing traces of mapping classes corresponding to automorphisms of certain Riemann surfaces, and the congruence relations on allowed modular representations following from them are presented. The surfaces considered are families of algebraic curves given by suitably chosen equations, the modular curve $\mathcal{X}(11)$, and a factor curve of $\mathcal{X}(8)$. The examples of modular curves illustrate how the study of arithmetic properties of suitable modular representations can be used to gain information on automorphic properties of Riemann surfaces.
A two dimensional gauge theory is canonically associated to every Drinfeld double. For particular doubles, the theory turns out to be e.g. the ordinary Yang-Mills theory, the G/G gauged WZNW model or the Poisson $σ$-model that underlies the Kontsevich quantization formula. We calculate the arbitrary genus partition function of the latter. The result is the $q$-deformation of the ordinary Yang-Mills partition function in the sense that the series over the weights is replaced by the same series over the $q$-weights. For $q$ equal to a root of unity the series acquires the affine Weyl symmetry an…
We find an agreement of equivariant indices of semi-classical homomorphisms between pairwise mirror branes in the GL(2) Higgs moduli space on a Riemann surface. On one side we have the components of the Lagrangian brane of U(1,1) Higgs bundles whose mirror was proposed by Nigel Hitchin to be certain even exterior powers of the hyperholomorphic Dirac bundle on the SL(2) Higgs moduli space. The agreement arises from a mysterious functional equation. This gives strong computational evidence for Hitchin's proposal.
We show that the WZW fusion rings at finite levels form a projective system with respect to the partial ordering provided by divisibility of the height, i.e. the level shifted by a constant. From this projective system we obtain WZW fusion rings in the limit of infinite level. This projective limit constitutes a mathematically well-defined prescription for the `classical limit' of WZW theories which replaces the naive idea of `sending the level to infinity'. The projective limit can be endowed with a natural topology, which plays an important role for studying its structure. The representation…
The $(p_+,p_-)$ singlet algebra is a vertex operator algebra that is strongly generated by a Virasoro field of central charge $1-6(p_+-p_-)^2/p_+p_-$ and a single Virasoro primary field of conformal weight $(2p_+-1)(2p_--1)$. Here, the modular properties of the characters of the uncountably many simple modules of each singlet algebra are investigated and the results used as the input to a continuous analogue of the Verlinde formula to obtain the "fusion rules" of the singlet modules. The effect of the failure of fusion to be exact in general is studied at the level of Verlinde products and the…
It is shown that traces of mapping classes of finite order may be expressed by Verlinde-like formulae. The 3D topological argument is explained, and the resulting trace identities for modular matrix elements are presented.
\ \ We have recently developed methods for obtaining exact two-point resistance of the complete graph minus $N$ edges. We use these methods to obtain closed formulas of certain trigonometrical sums that arise in connection with one-dimensional lattice, in proving the Scott's conjecture on permanent of Cauchy matrix, and in the perturbative chiral Potts model. The generalized trigonometrical sums of the chiral Potts model are shown to satisfy recursion formulas that are transparent and direct, and differ from those of Gervois and Mehta. By making a change of variables in these recursion formula…
The boundary states for a certain class of WZW models are determined. The models include all modular invariants that are associated to a symmetry of the unextended Dynkin diagram. Explicit formulae for the boundary state coefficients are given in each case, and a number of properties of the corresponding NIM-reps are derived.
Unitary representations of centrally extended mapping class groups $\tilde M_{g,1}, g\geq 1$ are given in terms of a rational Hopf algebra $H$, and a related generalization of the Verlinde formula is presented. Formulae expressing the traces of mapping class group elements in terms of the fusion rules, quantum dimensions and statistics phases are proposed.
We give a derivation of the Verlinde formula for the $G_{k}$ WZW model from Chern-Simons theory, without taking recourse to CFT, by calculating explicitly the partition function $Z_{Σ\times S^{1}}$ of $Σ\times S^{1}$ with an arbitrary number of labelled punctures. By a suitable gauge choice, $Z_{Σ\times S^{1}}$ is reduced to the partition function of an Abelian topological field theory on $Σ$ (a deformation of non-Abelian BF and Yang-Mills theory) whose evaluation is straightforward. This relates the Verlinde formula to the Ray-Singer torsion of $Σ\times S^{1}$. We derive the $G_{k}/G_{k}$ mod…
The super-$σ$-model of dynamics of the super-charged loop in an ambient supermanifold in the presence of worldsheet defects of arbitrary topology is formalised within Gawȩdzki's higher-cohomological approach, drawing inspiration from the precursor arXiv:0808.1419 [hep-th]. A distinguished class of the corresponding backgrounds (supertargets with additional bicategorial supergeometric data), organised into simplicial hierarchies, is considered. To these, configurational (super)symmetry of the bulk field theory is lifted coherently, whereby the notion of a maximally (super)symmetric background, …
Fusion rules for Wess-Zumino-Witten (WZW) models at fractional level can be defined in two ways, with distinct results. The Verlinde formula yields fusion coefficients that can be negative. These signs cancel in coset fusion rules, however. On the other hand, the fusion coefficients calculated from decoupling of singular vectors are non-negative. They produce incorrect coset fusion rules, however, when factorisation is assumed. Here we give two prescriptions that yield the correct coset fusion rules from those found for the WZW models by the decoupling method. We restrict to the Virasoro minim…
To an RCFT corresponds two combinatorial structures: the amplitude of a torus (the 1-loop partition function of a closed string, sometimes called a modular invariant), and a representation of the fusion ring (called a NIM-rep or equivalently a fusion graph, and closely related to the 1-loop partition function of an open string). In this paper we develop some basic theory of NIM-reps, obtain several new NIM-rep classifications, and compare them with the corresponding modular invariant classifications. Among other things, we make the following fairly disturbing observation: there are infinitely …
Motivated by Wakimoto free field realisations, the bosonic ghost system of central charge $c=2$ is studied using a recently proposed formalism for logarithmic conformal field theories. This formalism addresses the modular properties of the theory with the aim being to determine the (Grothendieck) fusion coefficients from a variant of the Verlinde formula. The key insight, in the case of bosonic ghosts, is to introduce a family of parabolic Verma modules which dominate the spectrum of the theory. The results include S-transformation formulae for characters, non-negative integer Verlinde coeffic…
This article aims to review a selection of central topics and examples in logarithmic conformal field theory. It begins with a pure Virasoro example, critical percolation, then continues with a detailed exposition of symplectic fermions, the fractional level WZW model on SL(2;R) at level -1/2 and the WZW model on the Lie supergroup GL(1|1). It concludes with a general discussion of the so-called staggered modules that give these theories their logarithmic structure, before outlining a proposed strategy to understand more general logarithmic conformal field theories. Throughout, the emphasis is…
We present a closed formula for the branching coefficients of an embedding p in g of two finite-dimensional semi-simple Lie algebras. The formula is based on the untwisted affine extension of p. It leads to an alternative proof of a simple algorithm for the computation of branching rules which is an analog of the Racah-Speiser algorithm for tensor products. We present some simple applications and describe how integral representations for branching coefficients can be obtained. In the last part we comment on the relation of our approach to the theory of NIM-reps of the fusion rings of WZW model…
We analyze in detail the equivariant supersymmetry of the $G/G$ model. In spite of the fact that this supersymmetry does not model the infinitesimal action of the group of gauge transformations, localization can be established by standard arguments. The theory localizes onto reducible connections and a careful evaluation of the fixed point contributions leads to an alternative derivation of the Verlinde formula for the $G_{k}$ WZW model. We show that the supersymmetry of the $G/G$ model can be regarded as an infinite dimensional realization of Bismut's theory of equivariant Bott-Chern currents…
