# Connes fusion of the free fermions on the circle

Date: Fri, Jun 11, 2021

Location: Zoom, Online

Conference: CMS Scientific Session on Quantum Mathematics

Subject: Mathematics, Physics, Condensed Matter and Statistical Mechanics, Quantum Physics, Quantum Information, Quantum Computing

Class: Scientific

### Abstract:

A conformal net on $S^1$ is an assignment $\mathcal{A}:\left\{\textrm{open subsets of } S^1\right\} \to \left\{\mbox{von Neumann algebras acting on } \mathcal{F}\right\}$, which satisfies a slew of axioms motivated by quantum field theory. In this talk, I will consider the free fermionic conformal net. In this case, the Hilbert space $\mathcal{F}$ is the Fock space generated by the positive energy modes of square-integrable spinors on the circle $๐ฟ^2(๐^1,\mathbb{S})$; and the von Neumann algebras are Clifford algebras generated by those elements of $๐ฟ^2(๐^1,\mathbb{S})$ whose support lies in $๐ผ\subset ๐^1$. After going over this construction, I will argue that given an open interval $๐ผ\subset ๐^1$, one can equip $\mathcal{F}$ with the structure of $\mathcal{A}(I)-\mathcal{A}(I)$-bimodule. I will then outline the construction of a canonical isomorphism of bimodules $\mathcal{F}\boxtimes_{\mathcal{A}(I_\_)}\mathcal{F}\to\mathcal{F}$ where $\boxtimes_{\mathcal{A}(I_\_)}$ stands for the Connes fusion product over the algebra assigned to the lower semi-circle $I_\_$. If time permits, I will discuss some (anticipated) applications of this isomorphism, for example in string geometry, or in the construction of the free fermion extended topological field theory.