The Minimally Immersed 4D Supermembrane

TL;DR

This paper explores the properties of a minimally immersed 4D supermembrane in 11D, revealing a discrete spectrum, spontaneous supersymmetry breaking, and connections to G2 manifolds, advancing understanding of supermembrane compactifications.

Contribution

It constructs the action of the 11D supermembrane with nontrivial central charges on a 7D torus and analyzes its properties, including symmetries, spectrum, and compactification to G2 manifolds.

Findings

Spectrum of the Hamiltonian is discrete with finite multiplicity.

Supersymmetry is spontaneously broken to N=1 in 4D.

Only 7 minimal holomorphic immersions exist on the 7-torus.

Abstract

In this note we summarize some of the properties found in [1], and its relation with [2]. We comment on the construction of the action of the 11D supermembrane with nontrivial central charges minimally immersed on a 7D toroidal manifold is obtained (MIM2).The transverse coordinates to the supermembrane are maps to a 4D Minkowski space-time. The action is invariant under additional symmetries in comparison to the supermembrane on a 11D Minkowski target space. The hamiltonian in the LCG is invariant under conformal transformations on the Riemann surface base manifold. The spectrum of the regularized hamiltonian is discrete with finite multiplicity. Its resolvent is compact. Susy is spontaneously broken, due to the topological central charge condition, to four supersymmetries in 4D, the vacuum belongs to an N=1 supermultiplet. When assuming the target-space to be an isotropic 7-tori, the potential does not contain any flat direction, it is stable on the moduli space of parameters. Moreover due to the discrete symmetries of the hamiltonian, there are only 7 possible minimal holomorphic immersions of the MIM2 on the 7-torus. When these symmetries are identified on the target space, it corresponds to compactify the MIM2 on a orbifold with G2 structure. Once the singularities are resolved it leads to the compactification of the MIM2 on a G2 manifold as shown in [2].