Superconformal symmetry and maximal supergravity in various dimensions

TL;DR

The work constructs and interrelates unitary (super)conformal representations across 3D, 4D, and 6D using twistorial oscillators, showing that 4D ${ m N}=8$ supergravity fields fit into the CPT-self-conjugate doubleton of ${ m SU}(2,2|8)$ and that 3D maximal supergravity fits into the ${ m OSp}(16|4,R)$ supersingleton. It further builds a six-dimensional ultra-short $(4,0)$ conformal graviton supermultiplet of ${ m OSp}(8^*|8)$, and demonstrates its reduction to 4D content, suggesting a chiral 6D maximal supergravity analogue whose field content reduces to ${ m N}=8$ supergravity in four dimensions. The paper discusses the viability and constraints for an interacting $(4,0)$ theory in 6D, including higher-spin no-go theorems, and connects these constructions to potential counterterm structure and AdS/CFT perspectives. Overall, the oscillator formalism provides a unifying framework for how conformal/superconformal representations organize maximal supergravity across dimensions and motivates future exploration of non-metric 6D theories and their dimensional reductions.

Abstract

In this paper we explore the relation between conformal superalgebras with 64 supercharges and maximal supergravity theories in three, four and six dimensions using twistorial oscillator techniques. The massless fields of N=8 supergravity in four dimensions were shown to fit into a CPT-self-conjugate doubleton supermultiplet of the conformal superalgebra SU(2,2|8) a long time ago. We show that the fields of maximal supergravity in three dimensions can similarly be fitted into the super singleton multiplet of the conformal superalgebra OSp(16|4,R), which is related to the doubleton supermultiplet of SU(2,2|8) by dimensional reduction. Moreover, we construct the ultra-short supermultiplet of the six-dimensional conformal superalgebra OSp(8*|8) and show that its component fields can be organized in an on-shell superfield. The ultra-short OSp(8*|8) multiplet reduces to the doubleton supermultiplet of SU(2,2|8) upon dimensional reduction. We discuss the possibility of a chiral maximal (4,0) six-dimensional supergravity theory with USp(8) R-symmetry that reduces to maximal supergravity in four dimensions and is different from six-dimensional (2,2) maximal supergravity, whose fields cannot be fitted into a unitary supermultiplet of a simple conformal superalgebra. Such an interacting theory would be the gravitational analog of the (2,0) theory.