Transversally Elliptic Complex and Cohomological Field Theory

Abstract

This work is a continuation of our previous paper arXiv:1812.06473 where we have constructed 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 N = 2 SYM and can also be constructed using the Atiyah-Jeffrey construction [1]. 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-isntantons around fixed points, which we call flipping instantons. The flipping instantons of our 4D 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.

Twisting with a Flip (The Art of Pestunization)

“…We summarize the content of each section of the paper. Many formal aspects of our construction are only mentioned in this work and will be discussed in the follow up [25].…”

“…In section 6 we summarize the paper, point out open problems and we also provide a short outline of our follow up work [25]. The paper has many appendices where we collect a summary of our conventions and a number of technical results.…”

“…As we will see below in some examples this equation may have a unique solution in every flux sector but we could not show that this is the case in general. A large class of cases is considered in detail in [25].…”

“…It is worth commenting on the relation with Pestun's treatment of the localization locus. On a round S 4 and with the choices made above we can make use of the special property (133) and formula (25) to rewrite the bosonic terms in the localization Lagrangian as…”

“…In [1] Pestun uses the theory of transversely elliptic operators (correspondingly there are notions of transversely elliptic complex and transversely elliptic PDEs) in order to calculate the 1-loop determinants arising from localization. It is one of the goals of this paper (and its follow up [25]) to show that supersymmetry is closely related to such transversely elliptic problems.…”