Our main object of study is the positive energy representation theory of the loop group LSO(N) of the compact Lie group SO(N). This should lead to a complete study of the fusion ring of LSO(N) at any level l and will be submitted as a Ph.D. dissertation in Pure Mathematics at the University of Cambridge.
Since 1988, it has been realised that positive energy representations of the loop group LG = C'infinity'(S1, G) of a compact Lie group G can be studied using von Neumann algebras (J1, J2) Their finite dimensional prototype, the representations theory of the classical groups was understood in two stages. First, the irreducible representations were classified, the decomposition of their tensor products studied For loop groups, the natural class of irreducible representation are the (projective unitary) positive energy ones Defining a tensor product or fusion operation is a difficult problem since the naive one does not work There have been various proposed definitions (Bo, KL, S). The von Neumann algebra approach, successfully implemented for the loop group of SU(N) (Wal, Wa2) and the diffeomorphism group of the circle (L) uses Connes' tensor product operation on bimodules over a von Neumann algebra. The work of Wassermann (J2, Wal, Wa2) on the loop group of SU(N) shows that the computation of fusion reduces to understanding the properties of primary fields, the building blocks of Conformal Field Theory. Of particular importance are their analytical properties, when they are viewed as operator valued distributions and their braiding or commutation properties. The latter are related to the monodromy of a 1st order PDE, the Knizhnik-Zamolodchikov equation and can in principle be derived from a detailed analysis of its solutions. Once the braiding and analytical properties are known. the rest is taken care of by the positivity and unitarity structure implied by the underlying operator algebras. Although the computations related to the KZ ODE seem at present unmanageable in complete generality, we have succeeded in computing a sufficient number of specific braiding rules that should entirely determine the algebraic structure of the fusion ring of the loop group of SO(N), our principle object of interest. This was done by reducing the relevant PDE's to a simplified form, until then unnoticed in the literature. Motivated by recent results of Wasserman (Wa2), we are currently trying to relate our solutions to the generalised hypergeometric functions of Aomoto, Gelfand, and Schechtmann-Varehenko (A, GKZ, SV). This might lead to the computation of a greater number of braiding coefficients and to a proof of their symmetry properties predicted by Witten.
What remains to be studied are the analytical properties of the primary fields whose braiding we have already computed. These should follow from a thorough operator algebraic treatment of the fermionic and bosonic constructions of the standard positive energy representations of LSO(N) (GO). Our work should give a proof of the Verlinde rules (V) which predict the structure of the fusion ring of LSO(N) at any given level l. It should also prove that this ring is a braided tensor category of the type required to produce three manifold invariants. It would in that respect link with the work of Turaev and Wenzl on Quantum Invariant Theory (T). Along these lines, it would also relate to the work of Birman-Wenzl (BW) and Wenzl (We) in that we conjecture that the inclusion of type III, factors naturally associated to any positive energy representation of LSO(N)is isomorphic to the tensor product of the hyperfinite III1 factor with the inclusion defined by Wenzl.
(A) K.Aomoto, Gauss-Manin connection of integral of difference products, J. Math. Soc. Japan Vol. 39, No. 2l 1987, 191-208. (BW) J.S.Birman and H.Wenzl, Braids, Link Polynomials and a Nets Algebra, Trans. Am. Math. Soc. 313(1989) 249-273.
(Bow R.E.Borcherds, Vertez algebra, Kac-Moody algebras and the Monster, Proc. Nat. Acad. Sci. U.S.A. 83(1986) 3068-71.
(GKZ) I.M.Gelfand,M.M.Kapranov and A.V.Zelevinsky, Generalised Euler Integrals and A-Hyperpeometnc Functions, Adv. in Matl 84(1990), 255-71. (GO) P.Goddard, D.Olive, Kac-Moody and Virasoro Algebras in Relation to Quantum Physics, International Journal of Modern Physic
A1(1986) 303-414. (J1) V.F.R.Jones, Subfactors and Knots. CBMS, Regional Conference Series in Mathematics Number 80, AMS 1991. (J2) V.F.R. Jones, Fusion en Algebres de con Neumann et groupes de lacets (d 'apres A. Wassermannj, Seminaire N. Bourbaki, to I presented on the 18th of June 1995. (KL) V.Kazhdan and G.Lusztig, Tensor Structures Arising from Amine Lie Algebras IV, Jour. A.M.S. 7(1994), 383-453. (L) T.Loke, Operator Algebras and Conformal Field Theory of the Discrete Series Representations of Diff(Sl). Ph.D. dissertatio University of Cambridge, 1994. (SV) V.Schechtman and A.Varchenko, Arrangements of Hyperplanes and Lie Algebra Cohomology, Invent. Math. 106(1991), 139-194. (S) G.B.Segal, unpublished notes on Conformal Field Theory. (T) V.G.Turaev, Quantum Invariants of Knots and 3-manifolds, de Gruyter Studies in Mathematics 18, Walter de Gruyter 1994. (V) E.Verlinde, Fusion Rules and Modular Transformations in 2D Conformal Field Theory, Nucl. Phys. B300(1988) 360-376.
(Wax) A.J.Wassermann, Operator Algebras and Conformal Field Theory, to appear in the proceedings of the International Congress Mathematicians, Zurich 1994, Birkhauser Verlag.
(Wa2) A.J.Wassermann, Conformal Field Theory and Operator Algebras III: Fusion for Won Neumann Algebras and Loop groups, to I published. (We) H.Wenzl, Quantum Groups and Subfactors of Type B,C and D, Comm. in Math. Phys. 133(1990), 388-432.
Valerio Toledano Laredo, St. John's College, Cambridge CB2 lTP