Observables in Topological Yang–mills Theories

Abstract

Using topological Yang–Mills theory as example, we discuss the definition and determination of observables in topological field theories (of Witten-type) within the superspace formulation proposed by Horne. This approach to the equivariant cohomology leads to a set of bi-descent equations involving the BRST and supersymmetry operators as well as the exterior derivative. This allows us to find the most general solution to the problem of equivariant cohomology, and to recover the Donaldson–Witten polynomials when choosing a Wess–Zumino-type gauge.

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

“…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 final result of this supersymmetrization is an action free from the supersymmetric charges. Then, this formulation for = 3 dimensions is not convenient to our construction, because as we have seen above, we had to make a new definition of , compared with (14), in order that the action acquire the Chern-Simons form. For that reason, we choose to work with a BF-type formulation, which yields the same expected results without, however, the need of changing the form of the basic elements of the formulation, for example, to accommodate the supersymmetrization process.…”

“…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.…”

Superspace gauge fixing of topological Yang-Mills theories

“…Our purpose in this section is to review and develop a superspace formalism describing topological theories such as Wittens's theory described in section 2 and generalizations of it for more than one supersymmetric generators and for any space-time dimension, starting from the formalism described in [2,4,7].…”