Observables in topological Yang-Mills theorieswith extended shift supersymmetry

Abstract

We present a complete classification, at the classical level, of the observables of topological Yang-Mills theories with an extended shift supersymmetry of N generators, in any space-time dimension. The observables are defined as the Yang-Mills BRST cohomology classes of shift supersymmetry invariants. These cohomology classes turn out to be solutions of an N -extension of Witten's equivariant cohomology. This work generalizes results known in the case of shift supersymmetry with a single generator.

Topological Gravity on (D,N)-Shift Superspace Formulation

“…The supercoordinates are parameterized as = ( , ) defined in a superchart of the basis supermanifold (we stress that the space-time dimension is D, while N is an internal label associated with the number of supersymmetries of the model; = 1 corresponds to a simple SUSY, and > 1 stands for an N-extended SUSY) M, where is the Grassmannian coordinates [10][11][12][19][20][21] (the Grassmann Variables of the topological description may be found in [10][11][12], as well as the notations therein. For example, for = 2, Levi-Civita pseudotensor is the antisymmetric metric, 12 = +1 = − 12 , where = and a field or variable transforms as = and…”

“…The superderivatives as superforms are given bŷ= + . An arbitrary superfield, ( , ), is defined by the action of the transformation generated by the derivatives with respect to the Grassmann coordinates [10,11,14], known as the shift operator, and given according to what follows:…”

“…In the formalism of shift SUSY, or topological supersymmetric formalism, it is understood that (simple) = 1 superspace is considered [10,22]. We wish, in the present contribution, to formulate, with the help of the shift SUSY [5,10,11,13,14], gravity for any dimensionality, , and for a general number of SUSY generators, . We shall adopt the Batalin-Vilkovisky (BV) prescription [23] combined with the Blau-Thompson minimal action gauge-fixing [12,24], which fix the Lagrange multipliers associated with the gauge conditions (we propose the following notation for the superfield charges: Ω , where stands for the SUSY number, denotes the ghost number, and indicates the form degree).…”

“…The topological supersymmetrization of Yang-Mills [9] and -theories by adopting the shift formalism [10][11][12][13] provide a viable way to their extension to any superspace dimensions, with the Wess-Zumino supergauge choice [14] used in connection with the Batalin-Vilkovisky prescription. In other words, an alternative construction can be used for the BV diagrams, where the latter no longer accommodate fields but, instead, superfields; they are referred to in the literature as BV superdiagrams.…”

“…Thus, it is possible to formulate an immediate extension to an arbitrary superspace dimension [11]. We think it is possible to proceed seemingly for topological gravity [15], that is, to describe this theory by adopting a topological formulation based on supersymmetry (SUSY, from now on): a supersymmetric topological gravity [16].…”