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
CRG: Quantum Topology and its Applications
Abstract:
A conformal net on S1 is an assignment A:{open subsets of S1}→{von Neumann algebras acting on F}, 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 F is the Fock space generated by the positive energy modes of square-integrable spinors on the circle ?2(?1,S); and the von Neumann algebras are Clifford algebras generated by those elements of ?2(?1,S) whose support lies in ?⊂?1. After going over this construction, I will argue that given an open interval ?⊂?1, one can equip F with the structure of A(I)−A(I)-bimodule. I will then outline the construction of a canonical isomorphism of bimodules F⊠A(I_)F→F where ⊠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.