A compositional coalgebraic model of a fragment of fusion calculus

This work is a further step in exploring the labelled transitions and bisimulations of fusion calculi. We follow the approach developed by Turi and Plotkin for lifting transition systems with a syntactic structure to bialgebras and, thus, we provide a compositional model of the fusion calculus with explicit fusions. In such a model, the bisimilarity relation induced by the unique morphism to the final coalgebra coincides with fusion hyperequivalence and it is a congruence with respect to the operations of the calculus. The key novelty in our work is to give an account of explicit fusions through labelled transitions. In this short essay, we focus on a fragment of the fusion calculus without recursion and replication

A compositional coalgebraic model of a fragment of fusion calculus

This work is a further step in exploring the labelled transitions and bisimulations of fusion calculi. We follow the approach developed by Turi and Plotkin for lifting transition systems with a syntactic structure to bialgebras and, thus, we provide a compositional model of the fusion calculus with explicit fusions. In such a model, the bisimilarity relation induced by the unique morphism to the final coalgebra coincides with fusion hyperequivalence and it is a congruence with respect to the operations of the calculus. The key novelty in our work is to give an account of explicit fusions through labelled transitions. In this short essay, we focus on a fragment of the fusion calculus without recursion and replication

The Novel ''Controlled Intermediate Nuclear Fusion'' and its Possible Industrial Realization as Predicted by Hadronic Mechanics and Chemistry

In this note, we propose, apparently for the first time, a new type of controlled nuclear fusion called "intermediate" because occurring at energies intermediate between those of the ''cold'' and ''hot'' fusions, and propose a specific industrial realization. For this purpose: 1) We show that known limitations of quantum mechanics, quantum chemistry and special relativity cause excessive departures from the conditions occurring for all controlled fusions; 2) We outline the covering hadronic mechanics, hadronic chemistry and isorelativity specifically conceived, constructed and verified during the past two decades for new cleans energies and fuels; 3) We identify seven physical laws predicted by the latter disciplines that have to be verified by all controlled nuclear fusions to occur; 4) We review the industrial research conducted to date in the selection of the most promising engineering realization as well as optimization of said seven laws; and 5) We propose with construction details a specific {\it hadronic reactor} (patented and international patents pending), consisting of actual equipment specifically intended for the possible industrial production of the clean energy released by representative cases of controlled intermediate fusions for independent scrutiny by interested colleagues.Comment: 32 pages, 5 figures. Journal of Applied Sciences, in pres

Fusion Algebras of Logarithmic Minimal Models

We present explicit conjectures for the chiral fusion algebras of the logarithmic minimal models LM(p,p') considering Virasoro representations with no enlarged or extended symmetry algebra. The generators of fusion are countably infinite in number but the ensuing fusion rules are quasi-rational in the sense that the fusion of a finite number of representations decomposes into a finite direct sum of representations. The fusion rules are commutative, associative and exhibit an sl(2) structure but require so-called Kac representations which are reducible yet indecomposable representations of rank 1. In particular, the identity of the fundamental fusion algebra is in general a reducible yet indecomposable Kac representation of rank 1. We make detailed comparisons of our fusion rules with the results of Gaberdiel and Kausch for p=1 and with Eberle and Flohr for (p,p')=(2,5) corresponding to the logarithmic Yang-Lee model. In the latter case, we confirm the appearance of indecomposable representations of rank 3. We also find that closure of a fundamental fusion algebra is achieved without the introduction of indecomposable representations of rank higher than 3. The conjectured fusion rules are supported, within our lattice approach, by extensive numerical studies of the associated integrable lattice models. Details of our lattice findings and numerical results will be presented elsewhere. The agreement of our fusion rules with the previous fusion rules lends considerable support for the identification of the logarithmic minimal models LM(p,p') with the augmented c_{p,p'} (minimal) models defined algebraically.Comment: 22 pages, v2: comments adde

The su(2)_{-1/2} WZW model and the beta-gamma system

The bosonic beta-gamma ghost system has long been used in formal constructions of conformal field theory. It has become important in its own right in the last few years, as a building block of field theory approaches to disordered systems, and as a simple representative -- due in part to its underlying su(2)_{-1/2} structure -- of non-unitary conformal field theories. We provide in this paper the first complete, physical, analysis of this beta-gamma system, and uncover a number of striking features. We show in particular that the spectrum involves an infinite number of fields with arbitrarily large negative dimensions. These fields have their origin in a twisted sector of the theory, and have a direct relationship with spectrally flowed representations in the underlying su(2)_{-1/2} theory. We discuss the spectral flow in the context of the operator algebra and fusion rules, and provide a re-interpretation of the modular invariant consistent with the spectrum.Comment: 33 pages, 1 figure, LaTeX, v2: minor revision, references adde

Comments on nonunitary conformal field theories

As is well-known, nonunitary RCFTs are distinguished from unitary ones in a number of ways, two of which are that the vacuum 0 doesn't have minimal conformal weight, and that the vacuum column of the modular S matrix isn't positive. However there is another primary field, call it o, which has minimal weight and has positive S column. We find that often there is a precise and useful relationship, which we call the Galois shuffle, between primary o and the vacuum; among other things this can explain why (like the vacuum) its multiplicity in the full RCFT should be 1. As examples we consider the minimal WSU(N) models. We conclude with some comments on fractional level admissible representations of affine algebras. As an immediate consequence of our analysis, we get the classification of an infinite family of nonunitary WSU(3) minimal models in the bulk.Comment: 24 page

Null vectors of the WBC2WBC_2 algebra

Using the fusion principle of Bauer et al. we give explicit expressions for some null vectors in the highest weight representations of the \bc algebra in two different forms. These null vectors are the generalization of the Virasoro ones described by Benoit and Saint-Aubin and analogues of the $W_3$ ones constructed by Bowcock and Watts. We find connection between quantum Toda models and the fusion method.Comment: 8 pages, LaTeX, ITP Budapest 50

N-point and higher-genus osp(1|2) fusion

We study affine osp(1|2) fusion, the fusion in osp(1|2) conformal field theory, for example. Higher-point and higher-genus fusion is discussed. The fusion multiplicities are characterized as discretized volumes of certain convex polytopes, and are written explicitly as multiple sums measuring those volumes. We extend recent methods developed to treat affine su(2) fusion. They are based on the concept of generalized Berenstein-Zelevinsky triangles and virtual couplings. Higher-point tensor products of finite-dimensional irreducible osp(1|2) representations are also considered. The associated multiplicities are computed and written as multiple sums.Comment: 17 pages, LaTe

Boundary states for WZW models

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.Comment: 34 pages, harvmac (b), 4 eps-figures. One reference added; some minor typos, as well as the $A_2$ embedding into $D_4$, are correcte