Fusion of implementers for spinors on the circle
Abstract
We consider the space of odd spinors on the circle, and a decomposition into spinors supported on either the top or on the bottom half of the circle. If an operator preserves this decomposition, and acts on the bottom half in the same way as a second operator acts on the top half, then the fusion of both operators is a third operator acting on the top half like the first, and on the bottom half like the second. Fusion restricts to the BanachLie group of restricted orthogonal operators, which supports a central extension of implementers on a Fock space. In this article, we construct a lift of fusion to this central extension. Our construction uses TomitaTakesaki theory for the Cliffordvon Neumann algebras of the decomposed space of spinors. Our motivation is to obtain an operatoralgebraic model for the basic central extension of the loop group of the spin group, on which the fusion of implementers induces a fusion product in the sense considered in the context of transgression and string geometry. In upcoming work we will use this model to construct a fusion product on a spinor bundle on the loop space of a string manifold, completing a construction proposed by Stolz and Teichner.
 Publication:

arXiv eprints
 Pub Date:
 May 2019
 DOI:
 10.48550/arXiv.1905.00222
 arXiv:
 arXiv:1905.00222
 Bibcode:
 2019arXiv190500222K
 Keywords:

 Mathematics  Operator Algebras;
 Mathematics  Differential Geometry;
 Primary 53C27;
 22E66;
 22D25;
 Secondary 30H20;
 47C15;
 81R10
 EPrint:
 50 pages