Algebraic Supersymmetry: A Case Study

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 CCRalgebra, the "resolvent algebra". The elements of this resolvent algebra serve as mollifiers for the super-derivation. Within this model, unbounded (yet locally bounded) graded KMSfunctionals 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". the underlying C * -algebra [9]. Second, there is growing evidence that supersymmetric KMSfunctionals, although being unbounded, are locally bounded, in accordance with the heuristic considerations in [3]. And third, although these functionals generically do not have analytic elements in their domain of definition, they can still be used to define local versions of Connes' entire cyclic cocycles by relying on techniques from complex analysis. Based on these insights, a proper C * -algebraic framework for the formulation of supersymmetry and the analysis of its consequences in quantum field theory seems within reach. We hope to return to this problem elsewhere.The plan of our paper is as follows. We will state our results in the body of the paper, and defer almost all the proofs to the appendix. In Sect. 2 we present in the Wightman framework the basic supersymmetry model which we wish to analyze; in Sect. 3 we prepare for its analysis in a C*-setting by considering algebraic mollifying relations for the quantum fields, which leads to the study of the C*-algebras generated by the resolvents of the fields. In Sect. 4 we use these tools to present the C*-algebraic framework of the model. In Sect. 5 we define (unbounded) graded KMSfunctionals on the model and prove basic properties for them, including their supersymmetry invariance and their local boundedness. In Sect. 6 we use these KMS-functionals to define a Chern character formula (generalizing the construction in [15,11]), from which we obtain a (locally) entire cyclic cocycle in the sense of Connes. This can then be taken as input to an index theory for supersymmetric quantum field theories, of the type proposed by Longo [18]. The modelWe begin by presenting here our model in the Wightman framework, which we would like to model in a C*-algebra setting. It is the the simplest example for supersymmetry on noncompact spacetime, in that we have one dimension, one boson and one fermion.

Superconformal nets and noncommutative geometry

“…[BG07]. To conclude the proof of denseness we would like to see that F (f 1 X 1 )F (f 2 X 2 ) is in the weak closure of the * -algebra generated by these operators.…”

Spectral Triples and the Super-Virasoro Algebra

“…This will give a net of quantum algebras on S 1 naturally associated with the net of von Neumann algebras A λ and hence a noncommutative geometric structure [6] associated to superconformal quantum field theories. Inspired by the work of Buchholz and Grundling [2] we shall use resolvents of the smeared Bose fields to exhibit local elements in the domain of δ. Yet the models considered here appear to be more complicated than the free field model considered by them.…”

“…According to the proof of Proposition 5.6, we have local rotation covariance, namely if I and e it I belong to I 0 for all |t| < ε for some ε > 0, then the above equation and the group property of t → e itL λ 0 allow to extend consistently (A λ , H λ , δ λ ) to a rotation covariant net of generalised quantum algebras on S 1 (2) .…”

“…A similar problem has been studied in the free supersymmetric case in [2], where a crucial simplification occurred due to the Weyl commutation relations and the fact that the smeared free Fermi fields are bounded operators.…”

Supergeometry in Locally Covariant Quantum Field Theory

“…There are however some notable exceptions: In [BG06], Buchholz and Grundling address the nontrivial problems of implementing supersymmetry transformations into the C * -algebraic framework of algebraic quantum field theory and constructing super-KMS states. The study of superconformal nets in two spacetime dimensions has been initiated by Capri, Kawahigashi and Longo in [CKL08].…”