Algebraic Supersymmetry: A case study

TL;DR

This paper constructs a supersymmetric quantum field model within a C*-algebra framework using a novel resolvent algebra, demonstrating the existence of supersymmetric KMS functionals and connecting to noncommutative geometry.

Contribution

It introduces a new C*-algebraic approach to supersymmetry using the resolvent algebra, enabling the construction of supersymmetric KMS functionals.

Findings

A meaningful C*-algebraic model for supersymmetry is developed.

Supersymmetric KMS functionals are constructed and proven to be supersymmetric.

Chern characters and cyclic cocycles are derived within the model.

Abstract

The treatment of supersymmetry is known to cause difficulties in the C*-algebraic framework of relativistic quantum field theory; several no-go theorems indicate that super-derivations and super-KMS functionals must be quite singular objects in a C*-algebraic setting. In order to clarify the situation, a simple supersymmetric chiral field theory of a free Fermi and Bose field defined on $\R$ is analyzed. It is shown that a meaningful C*-version of this model can be based on the tensor product of a CAR-algebra and a novel version of a CCR-algebra, the "resolvent algebra". The elements of this resolvent algebra serve as mollifiers for the super-derivation. Within this model, unbounded (yet locally bounded) graded KMS-functionals are constructed and proven to be supersymmetric. From these KMS-functionals, Chern characters are obtained by generalizing formulae of Kastler and of Jaffe, Lesniewski and Osterwalder. The characters are used to define cyclic cocycles in the sense of Connes' noncommutative geometry which are "locally entire".