Cardy formula for 4d SUSY theories and localization
Abstract
We study 4d \mathcal{N}=1 supersymmetric theories on a compact Euclidean manifold of the form S ^{1} × ℳ_{3}. Partition functions of gauge theories on this background can be computed using localization, and explicit formulas have been derived for different choices of the compact manifold ℳ_{3}. Taking the limit of shrinking S ^{1}, we present a general formula for the limit of the localization integrand, derived by simple effective theory considerations, generalizing the result of [1]. The limit is given in terms of an effective potential for the holonomies around the S ^{1}, whose minima determine the asymptotic behavior of the partition function. If the potential is minimized in the origin, where it vanishes, the partition function has a Cardylike behavior fixed by Tr( R), while a nontrivial minimum gives a shift in the coefficient. In all the examples that we consider, the origin is a minimum if Tr( R) ≤ 0.
 Publication:

Journal of High Energy Physics
 Pub Date:
 April 2017
 DOI:
 10.1007/JHEP04(2017)055
 arXiv:
 arXiv:1611.00380
 Bibcode:
 2017JHEP...04..055D
 Keywords:

 Anomalies in Field and String Theories;
 Conformal Field Theory;
 Supersymmetric Effective Theories;
 Supersymmetric Gauge Theory;
 High Energy Physics  Theory
 EPrint:
 43 pages, v2: references added, changed slightly discussion in section 5