Diversity of off--Shell Twisted (4,4) Multiplets in SU(2) x SU(2) Harmonic Superspace

TL;DR

This paper classifies four types of twisted ${ m N}=(4,4)$ supermultiplets in 2D harmonic superspace, constructs their sigma model actions, and analyzes their interactions and potentials, revealing restrictions on multiplet interactions and supersymmetry constraints.

Contribution

It introduces a detailed classification of twisted ${ m N}=(4,4)$ supermultiplets in harmonic superspace and constructs their invariant actions, including interaction restrictions and potential formulations.

Findings

Constructed off-shell sigma model actions for four supermultiplet types.

Proved that supersymmetry enforces the splitting of actions for different multiplet pairs.

Identified that interactions occur only through specific mass terms within self-dual pairs.

Abstract

We elaborate on four different types of twisted ${\cal N}=(4,4)$ supermultiplets in the $SU(2) \times SU(2)$, 2D harmonic superspace. In the conventional ${\cal N}=(4,4)$, 2D superspace they are described by the superfields $\hat q^{i a}$, $\hat q^{\und i a}$, $\hat q^{i \und a}$, $\hat q^{\und i \und a}$ subjected to proper differential constraints, $(i, \und i, a, \und a)$ being the doublet indices of four groups SU(2) which form the full R-symmetry group $SO(4)_L\times SO(4)_R$ of ${\cal N}=(4,4)$ supersymmetry. We construct the torsionful off--shell sigma model actions for each type of these multiplets, as well as the corresponding invariant mass terms, in an analytic subspace of the $SU(2) \times SU(2)$ harmonic superspace. As an instructive example, ${\cal N}=(4,4)$ superconformal extension of the $SU(2) \times U(1)$ WZNW sigma model action and its massive deformation are presented for the multiplet $\hat q^{i \und a}$ . We prove that ${\cal N}=(4,4)$ supersymmetry requires the general sigma model action of pair of different multiplets to split into a sum of sigma model actions of each multiplet. This phenomenon also persists if a larger number of non-equivalent multiplets are simultaneously included. We show that different multiplets may interact with each other only through mixed mass terms which can be set up for multiplets belonging to ``self-dual'' pairs $(\hat q^{i a}, \hat q^{\und i \und a})$ and $(\hat q^{\und i a}, \hat q^{i \und a})$ . The multiplets from different pairs cannot interact at all. For a ``self-dual'' pair of the twisted multiplets we give the most general form of the on-shell scalar potential.