In this talk I shall describe an ongoing joint project with Festuccia, Winding and Zabzine, aiming at generalizing the Donaldson-Witten theory in a natural way. The DW theory computes integrals of certain characteristic classes over the moduli space of anti-self-dual Yang-Mills instantons. We attempt to generalize it in two directions. First, the instanton configuration $F^+=0$ (where $F$ is the Yang-Mills field strength) is generalized to $PF=0$, where $P$ is a projector projecting $F$ to a rank 3 sub-bundle of 2-forms. Second, we add equivariance. In the case of DW theory over compact space, equivariance is not strictly necessary, but for our generalization, it is crucial for having a good moduli problem. Physically, our theory can be obtained from reducing a 5D Super Yang-Mills theory to 4D along a circle fibration, and so has a natural description in 5D terms. But describing it in purely 4D terms is also rewarding, in that it is a an extension of the super Yang-Mills theory over $S^4$, constructed by Pestun. Localization can be likewise carried out leading to interesting special functions and allowing us to explore the geometry of the underlying space.

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