Transversally Elliptic Complex and Cohomological Field Theory

Abstract: This work is a continuation of our previous paper arXiv:1812.06473 where we have constructed ${\cal N}=2$ supersymmetric Yang-Mills theory on 4D manifolds with a Killing vector field with isolated fixed points. In this work we expand on the mathematical aspects of the theory, with a particular focus on its nature as a cohomological field theory. The well-known Donaldson-Witten theory is a twisted version of ${\cal N}=2$ SYM and can also be constructed using the Atiyah-Jeffrey construction. This theory is concerned with the moduli space of anti-self-dual gauge connections, with a deformation theory controlled by an elliptic complex. More generally, supersymmetry requires considering configurations that look like either instantons or anti-instantons around fixed points, which we call flipping instantons. The flipping instantons of our 4D ${\cal N}=2$ theory are derived from the 5D contact instantons. The novelty is that their deformation theory is controlled by a transversally elliptic complex, which we demonstrate here. We repeat the Atiyah-Jeffrey construction in the equivariant setting and arrive at the Lagrangian (an equivariant Euler class in the relevant field space) that was also obtained from our previous work arXiv:1812.06473. We show that the transversal ellipticity of the deformation complex is crucial for the non-degeneracy of the Lagrangian and the calculability of the theory. Our construction is valid on a large class of quasi toric 4 manifolds.