On an Algebraic Structure of Dimensionally Reduced Magical Supergravity Theories

TL;DR

This paper explores the algebraic structure underlying magical supergravities in three dimensions, establishing a unified framework that relates their reduced Lagrangians to coset models through specific algebraic relations.

Contribution

It introduces a condition on the quasi-conformal algebra's commutation relations that enables explicit parameterizations of 3d supergravity theories, unifying their algebraic treatment.

Findings

Verified the algebraic condition for $E_{6(+2)}$, confirming its applicability to a specific magical supergravity.

Established a correspondence between the algebraic structure and the 3d reduced Lagrangian formulations.

Provided a unified algebraic approach to all magical supergravity theories in three dimensions.

Abstract

We study an algebraic structure of magical supergravities in three dimensions. We show that if the commutation relations among the generators of the quasi-conformal group in the super-Ehlers decomposition are in a particular form, then one can always find a parameterization of the group element in terms of various 3d bosonic fields that reproduces the 3d reduced Lagrangian of the corresponding magical supergravity. This provides a unified treatment of all the magical supergravity theories in finding explicit relations between the 3d dimensionally reduced Lagrangians and particular coset nonlinear sigma models. We also verify that the commutation relations of $E_{6(+2)}$, the quasi-conformal group for $\mathbb{A}=\mathbb{C}$, indeed satisfy this property, allowing the algebraic interpretation of the structure constants and scalar field functions as was done in the $F_{4(+4)}$ magical supergravity.