${\cal N}{=}4$ supersymmetric Schwarzian with $D(1,2;\alpha)$ symmetry

TL;DR

This paper constructs a new super-Schwarzian derivative associated with the superalgebra D(1,2;α) by extending the Cartan form approach used for other superconformal groups, revealing a novel component related to automorphism.

Contribution

It introduces a new super-Schwarzian derivative for D(1,2;α) superalgebra using Cartan forms and invariant superspace coordinates, expanding the understanding of superconformal structures.

Findings

New super-Schwarzian component identified for D(1,2;α)

Method extends Cartan form approach to a broader class of superalgebras

Reveals a super-Schwarzian related to automorphism components

Abstract

It was recently demonstrated that super-Schwarzian derivatives can be constructed from the Cartan forms of the super-conformal supergroups $OSp(1|2),SU(1,1|1), OSp(3|2), SU(1,1|2)$. Roughly speaking, the super-Schwarzian is just the component of the corresponding Cartan forms with the lowest dimension. In this paper, we apply the same approach for superalgebra $D(1,2;\alpha)$. The minimal set of constraints we used includes: a) introducing new superspace coordinates the Cartan forms depend on, which are completely invariant with respect to the corresponding group; b) nullifying the form for dilatation. In contrast to the $SU(1,1|2)$ case, the new super-Schwarzian appears to be a $d\theta^{ia}$ component of the form for $su(2)$ automorphism.