Fusion of Positive Energy Representations of LSpin(2n)

TL;DR

This paper advances the understanding of fusion rules for positive energy representations of the loop group Spin(2n) using operator algebraic methods, including Connes' tensor product and solutions to Knizhnik-Zamolodchikhov equations.

Contribution

It provides a new operator algebraic framework for fusion in Spin(2n) loop groups, connecting conformal field theory, von Neumann algebras, and representation theory.

Findings

Fusion with the vector representation follows Verlinde rules.

Solved a 6-parameter family of Knizhnik-Zamolodchikhov equations.

Constructed primary fields as operator-valued distributions.

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.