Exact hypermultiplet dynamics in four dimensions

TL;DR

This paper formulates the most general N=2 supersymmetric hypermultiplet effective action in four dimensions, demonstrating how it incorporates both perturbative and non-perturbative quantum corrections, and relates to known mathematical structures like hyper-Kähler metrics and elliptic curves.

Contribution

It provides a comprehensive N=2 superspace formulation of hypermultiplet dynamics, including explicit calculations of quantum corrections and connections to Seiberg-Witten theory.

Findings

Perturbative corrections described by the Taub-NUT metric.

Non-perturbative corrections encoded by an elliptic curve.

Results agree with three-dimensional Seiberg-Witten theory.

Abstract

We use N=2 harmonic and projective superspaces to formulate the most general `Ansatz' for the SU(2)_R invariant hypermultiplet low-energy effective action (LEEA) in four dimensions, which describes the two-parametric family of the hyper-K"ahler metrics generalizing the Atiyah-Hitchin metric. We then demonstrate in the very explicit and manifestly N=2 supersymmetric way that the (magnetically charged, massive) single hypermultiplet LEEA in the underlying non-abelian N=2 supersymmetric quantum field theory can receive both perturbative (e.g., in the Coulomb branch) and non-perturbative (e.g., in the Higgs branch) quantum corrections. The manifestly N=2 supersymmetric Feynman rules in harmonic superspace can be used to calculate the perturbative corrections described by the Taub-NUT metric. The non-perturbative corrections (due to instantons and anti-instantons) can be encoded in terms of an elliptic curve, which is very reminiscent to the Seiberg-Witten theory. Our four-dimensional results agree with the three-dimensional Seiberg-Witten theory and instanton calculations.