Fusion of Positive Energy Representations of LSpin(2n)

Abstract

Building upon the Jones-Wassermann program of studying Conformal Field Theory using operator algebraic tools, and the work of A. Wassermann on the loop group of LSU(n) (Invent. Math. 133 (1998), 467-538), we give a solution to the problem of fusion for the loop group of Spin(2n). Our approach relies on the use of A. Connes' tensor product of bimodules over a von Neumann algebra to define a multiplicative operation (Connes fusion) on the (integrable) positive energy representations of a given level. The notion of bimodules arises by restricting these representations to loops with support contained in an interval I of the circle or its complement. We study the corresponding Grothendieck ring and show that fusion with the vector representation is given by the Verlinde rules. The computation rests on 1) the solution of a 6-parameter family of Knizhnik-Zamolodchikhov equations and the determination of its monodromy, 2) the explicit construction of the primary fields of the theory, which allows to prove that they define operator-valued distributions and 3) the algebraic theory of superselection sectors developed by Doplicher-Haag-Roberts.

arXiv comment157 pages, no figures. This is the text of my 1997 PhD thesis

Cited by in corpus (20)

• Conformal Subnets and Intermediate Subfactors
• Solitons in Affine and Permutation Orbifolds
• Local Nature of Coset Models
• Conformal nets and local field theory
• Fusion and positivity in chiral conformal field theory
• Strong additivity and conformal nets
• Matrix product approximations to conformal field theories
• Conformal Covariance Subalgebras
• Loop Groups and QNEC
• Neveu-Schwarz and operators algebras III: Subfactors and Connes fusion
• Neveu-Schwarz and operators algebras II: Unitary series and characters
• Neveu-Schwarz and operators algebras I: Vertex operators superalgebras
• The classification of chiral WZW models by $H_{+}^4(BG,\mathbb{Z})$
• Regular vertex operator subalgebras and compressions of intertwining operators
• Casimir Operators and Monodromy Representations of Generalised Braid Groups
• A Family of Projective Representations of the Thompson Group and Lifting Problems
• On Affine Orbifold Nets Associated with Outer Automorphisms
• Unitary coboundary weak quasi-Hopf algebras in the Kazhdan-Lusztig-Finkelberg theorem
• Conformal Nets Associated with Lattices and Their Orbifolds
• Fuchsian systems for Dotsenko-Fateev multipoint correlation functions and similar integrals of hypergeometric type