2019/04/30 by Guido Festuccia, Jian Qiu, Jacob Winding +1
Physics and Astronomy · Mathematics · #hep-th #math.DG
paper · pdf · doi:10.1016/j.geomphys.2020.103786
88 pages, typos corrected, the final version to be published in JGP
arxiv created 2020/06/28 · arxiv updated 2020/07/15
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.