Ward identities for non-commutative geometry
The Wightman functions of a super-symmetric quantum field theory given by a super-trace functional have a geometric or cohomological interpretation. This was shown in joint work of the authors with \textit{A. Lesniewski} [ibid. 118, No.1, 1-14 (1988; Zbl 0656.58048)]. This property is summarized by the construction of a cocycle \(\tau\) for the \(\partial\)- complex of entire cyclic cohomology, namely a solution to the equation \[ (I.1)\quad \partial \tau =0. \] Here \(\tau\) is a time average of certain Euclidean Wightman functionals, and \(\partial\) is a standard coboundary operator of non-commutative geometry. Furthermore, this natural cohomological interpretation can be generalized to the case when the Wightman functionals \(\tau\) are constructed from a finite-temperature functional satisfying a super-version of the KMS condition of statistical mechanics. In this case the super-trace (associated with finite volume theories) need not exist - only the super-KMS functional has to be defined. In this note we investigate how equation (I.1) has a set of identities as its consequence, which can be interpreted as a symmetry of the Wightman functions. We call these identities ``Ward identities for non-commutative geometry''.
scientific article; zbMATH DE number 4161743