Seiberg-Witten Theories on Ellipsoids

TL;DR

The paper extends Seiberg-Witten theories to curved backgrounds by constructing 4D ellipsoids with suitable background fields that preserve rigid supersymmetry. It derives Killing spinors on the ellipsoid, analyzes the square of the SUSY generator, and realizes an Omega-deformed near the poles with ${\epsilon_1}=\ell^{-1}$ and ${\epsilon_2}=\tilde{\ell}^{-1}$. Through SUSY localization, it computes the exact partition function on the ellipsoid, expressing one-loop determinants via the Upsilon function and incorporating Nekrasov's instanton contributions; the result naturally encodes a deformation parameter ${b=(\ell/\tilde{\ell})^{1/2}}$ and supports a Liouville/Toda-type AGT correspondence for general ${b}$. The work provides explicit results for vector and hypermultiplets and demonstrates consistency checks in simple SU(2) setups, highlighting the broader significance for holography between 4D SW theories and 2D CFTs on deformed geometries. This advances the program of exact results on curved spaces and broadens the landscape of backgrounds compatible with rigid supersymmetry and AGT-type relations.

Abstract

We present a set of equations for a 4D Killing spinor which guarantees the Seiberg-Witten theories on a curved background to be supersymmetric. The equations involve an SU(2) gauge field and some auxiliary fields in addition to the metric. Four-dimensional ellipsoids with U(1)xU(1) isometry are shown to admit a supersymmetry if these additional fields are chosen appropriately. We compute the partition function of general Seiberg-Witten theories on ellipsoids, and the result suggests a correspondence with 2D Liouville or Toda correlators with general coupling constant b.