Cohomological Localization of $\mathcal N = 2$ Gauge Theories with Matter

Abstract

We construct a large class of gauge theories with extended supersymmetry on four-dimensional manifolds with a Killing vector field and isolated fixed points. We extend previous results limited to super Yang-Mills theory to general N = 2 gauge theories including hypermultiplets. We present a general framework encompassing equivariant Donaldson-Witten theory and Pestun's theory on S 4 as two particular cases. This is achieved by expressing fields in cohomological variables, whose features are dictated by supersymmetry and require a generalized notion of self-duality for two-forms and of chirality for spinors. Finally, we implement localization techniques to compute the exact partition function of the cohomological theories we built up and write the explicit result for manifolds with diverse topologies.

S-duality and supersymmetry on curved manifolds

“…We focus on the N = 2 vector multiplet with non-abelian gauge group (keeping in mind U(N) as the main example). Nevertheless many considerations can be extended to theories with matter, see [25].…”

Conformal field theories on deformed spheres, anomalies, and supersymmetry

“…In this section we study the partition function of N = 2 SCFTs on supersymmetric curved backgrounds. These backgrounds are obtained by coupling the stress-tensor multiplet of N = 2 theories with the gravity (Weyl) multiplet of N = 2 Poincaré (conformal) supergravity [10,11,23]. The supergravity background in Euclidean signature has a metric g µν , a self-dual two-form…”

“…vector multiplet, we use the language of cohomological fields introduced in [10,11]. A key ingredient is the existence of a Killing vector v which is the square of a supersymmetry transformation.…”

“…One can also study a conformal field theory on M in the presence of other background fields for various conserved currents. In particular, if we have an N = 2 superconformal field theory (SCFT), then in order to preserve supersymmetry we must turn on other background fields in the supergravity multiplet when deforming the metric [9][10][11]. This leads to additional contributions to the conformal anomaly, which results in the supersymmetric completion of the Euler density and the (Weyl) 2 terms.…”

“…These background fields are not uniquely determined. Indeed there is a three-parameter family of background fields which preserve supersymmetry on the deformed sphere[18,23,10,11]. In reproducing the background fields in (4.3)-(4.7) we have made a convenient choice of these parameters such that the the background fields are smooth at the poles.…”