# Fusion of implementers for spinors on the circle

## TL;DR

This paper develops a mathematical framework for fusing operators on spinors on the circle, using operator algebra and Tomita-Takesaki theory, aiming to model the central extension of loop groups relevant to string geometry.

## Contribution

It constructs a lift of fusion to the central extension of implementers on a Fock space using Tomita-Takesaki theory, advancing the operator-algebraic understanding of loop group extensions.

## Key findings

- Constructed a lift of fusion to the central extension of implementers.
- Used Tomita-Takesaki theory for Clifford-von Neumann algebras.
- Provides a model for the central extension of the loop group of the spin group.

## 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 Banach-Lie 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 Tomita-Takesaki theory for the Clifford-von Neumann algebras of the decomposed space of spinors. Our motivation is to obtain an operator-algebraic 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.