# ${\cal N}{=}4$ Supersymmetric $d=1$ Sigma Models on Group Manifolds

## TL;DR

This paper constructs explicit ${
m N}=4$ supersymmetric sigma models on group manifolds U(2) and SU(3) using harmonic superfield methods, revealing their symmetries and relations to known multiplets.

## Contribution

It provides new off-shell superfield actions for ${
m N}=4$ sigma models on U(2) and SU(3) group manifolds, with detailed symmetry and component analysis.

## Key findings

- Superfield actions are bilinear in superfields.
- Symmetries are enlarged to product groups U(2)$\times$SU(2) and SU(3)$\times$U(2).
- Connections to the nonlinear multiplet $({\bf 3, 4, 1})$ are established.

## Abstract

We construct manifestly ${\cal N}=4$ supersymmetric off-shell superfield actions for the HKT $d=1$ sigma models on the group manifolds U(2) and SU(3), using the harmonic $d=1$ approach. The underlying $({\bf 4, 4, 0})$ and $({\bf 4, 4, 0})\oplus({\bf 4, 4, 0})$ multiplets are described, respectively, by one and two harmonic analytic superfields $q^+$ satisfying the appropriate nonlinear harmonic constraints. The invariant actions in both cases are bilinear in the superfields. We present the corresponding superfield realizations of the U(2) and SU(3) isometries and show that in fact they are enlarged to the products U(2)$\times$SU(2) and SU(3)$\times$U(2). We prove the corresponding invariances at both the superfield and component levels and present the bosonic $d=1$ sigma model actions, as integral over $t$ in the U(2) case and over $t$ and ${\rm SU}(2)$ harmonics in the SU(3) case. In the U(2) case we also give a detailed comparison with the general harmonic approach to HKT models and establish a correspondence with a particular action of the off-shell nonlinear multiplet $({\bf 3, 4, 1})$. A possible way of generalizing U(2) model to the matrix U($2n$) case is suggested.