Algebraic Supersymmetry: A case study

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”.