Fermions with $SU(1,n)$ Spacetime Symmetry

TL;DR

This paper constructs free fermion theories with $SU(1,n)$ symmetry via null reduction from higher-dimensional backgrounds, revealing connections to 6d superconformal theories and exploring various physical regimes.

Contribution

It introduces a novel method to derive lower-dimensional fermion theories with $SU(1,n)$ symmetry from higher-dimensional deformed backgrounds, linking to superconformal field theories.

Findings

Fermion two-point functions are fixed using supersymmetry considerations.

Full higher-dimensional two-point functions are reconstructed through resummation.

Theories exhibit Galilean and Carrollian limits, and satisfy $SU(1,n)$ Ward identities.

Abstract

We construct theories of free fermions in $(2n-1)$-dimensions with $SU(1,n)$ spacetime symmetry from the null reduction of fermions on a $2n$-dimensional $\Omega$-deformed Minkowski background for $n=2$ and $n=3$. These play a role in the 5d $SU(1,3)$-invariant theories that are conjectured to offer a full description of certain 6d superconformal field theories. We find the $(2n-1)$-dimensional manifestation of the supersymmetry of a free $2n$-dimensional boson-fermion system, which we use to fix the fermion two-point functions. It is then shown that the full $2n$-dimensional two-point function can be recovered through resummation. Limits of the theories are considered, and it is observed that both Galilean and Carrollian field theories appear in different regimes. We confirm that the correlation functions obey the $SU(1,n)$ Ward identities and the representations of the fermions under this group are discussed.