Fusion multiplicities as polytope volumes: N-point and higher-genus su(2) fusion

We present the first polytope volume formulas for the multiplicities of affine fusion, the fusion in Wess-Zumino-Witten conformal field theories, for example. Thus, we characterise fusion multiplicities as discretised volumes of certain convex polytopes, and write them explicitly as multiple sums measuring those volumes. We focus on su(2), but discuss higher-point (N>3) and higher-genus fusion in a general way. The method follows that of our previous work on tensor product multiplicities, and so is based on the concepts of generalised Berenstein-Zelevinsky diagrams, and virtual couplings. As a by-product, we also determine necessary and sufficient conditions for non-vanishing higher-point fusion multiplicities. In the limit of large level, these inequalities reduce to very simple non-vanishing conditions for the corresponding tensor product multiplicities. Finally, we find the minimum level at which the higher-point fusion and tensor product multiplicities coincide.Comment: 14 pages, LaTeX, version to be publishe

The Challenge of Integrating Faith-Learning-Living in Teacher Education

Teacher educators from member institutions of the Coalition for Christian Colleges and Universities are currently challenged in an unprecedented way. The challenge is to satisfy increasingly rigorous state and national teacher education standards and to fulfill the commonly held mission of Coalition institutions to integrate faith-learning-living. The research presented in this article traces the long history of integration and presents various theoretical integration models commonly supported by educators at Christian colleges and universities. This article suggests meeting the challenge in part through an original six component integration model with potential value for Christian educators representing various academic disciplines

Man-machine communication - A transparent switchboard for computers

Device uses pattern of transparent contact touch points that are put on cathode ray tube screen. Touch point system compels more precise and unambiguous communication between man and machine than is possible with any other means, and speeds up operation responses

Two-point Functions in Affine SL(N) Current Algebra

In this letter the explicit form of general two-point functions in affine SL(N) current algebra is provided for all representations, integrable or non-integrable. The weight of the conjugate field to a primary field of arbitrary weight is immediately read off.Comment: 9 pages, LaTe

Two-point Functions in Affine Current Algebra and Conjugate Weights

The two-point functions in affine current algebras based on simple Lie algebras are constructed for all representations, integrable or non-integrable. The weight of the conjugate field to a primary field of arbitrary weight is immediately read off.Comment: 9 pages, LaTe

On string backgrounds and (logarithmic) CFT

We discuss the link between string backgrounds and the associated world-sheet CFTs. In the search for new backgrounds and CFTs, Penrose limits and Lie algebra contractions are important tools. The Nappi-Witten construction and the recently discovered logarithmic CFT by Bakas and Sfetsos, are considered as illustrations. We also speculate on possible extensions.Comment: 9 pages, invited contribution to the proceedings on the IX-th Workshop on Mathematical Physics and Applications, Rabat, 23-25 February 200

Jordan cells in logarithmic limits of conformal field theory

It is discussed how a limiting procedure of conformal field theories may result in logarithmic conformal field theories with Jordan cells of arbitrary rank. This extends our work on rank-two Jordan cells. We also consider the limits of certain three-point functions and find that they are compatible with known results. The general construction is illustrated by logarithmic limits of (unitary) minimal models in conformal field theory. Characters of quasi-rational representations are found to emerge as the limits of the associated irreducible Virasoro characters.Comment: 16 pages, v2: discussion of three-point functions and characters included; ref. added, v3: version to be publishe