Supersymmetric M2-branes with Englert Fluxes and the simple group PSL(2,7)

TL;DR

This paper introduces new M2-brane solutions in 11-dimensional supergravity with Englert fluxes, explores their supersymmetry properties, and constructs solutions with discrete symmetries related to the simple group PSL(2,7).

Contribution

It presents a novel class of M2-brane solutions with Englert fluxes, establishes criteria for supersymmetry, and links these solutions to the simple group PSL(2,7) and discrete symmetries.

Findings

Constructed M2-brane solutions with Englert fluxes and N<4 supersymmetry.

Identified PSL(2,7) as a crystallographic group acting on the A7 lattice.

Developed a 4-parameter N=1 solution with non-abelian discrete symmetry G21.

Abstract

A new class is introduced of M2-branes solutions of d=11 supergravity that include internal fluxes obeying Englert equation in 7-dimensions. A simple criterion for the existence of Killing spinors in such backgrounds is established. Englert equation is viewed as the generalization to d=7 of Beltrami equation defined in d=3 and it is trated accordingly. All 2-brane solutions of minimal d=7 supergracity can be uplifted to d=11 and have N > 4 or N = 4 supersymmetry. It is shown that the simple group PSL(2,7) is crystallographic in d=7 having an integral action on the A7 root lattice. By means of this point-group and of the T7 torus obtained quotiening R7 with the A7 root lattice we were able to construct new M2 branes with Englert fluxes and N < 4. In particular we exhibit here an N=1 solution depending on 4-parameters and admitting a large non abelian discrete symmetry, namely G21 = Z3 semidirect product with Z7 = subgroup of PSL(2,7). The dual d=3 field theories have the same symmetries and have complicated non linear interactions.