The Massive Supermembrane on a Knot

TL;DR

This paper develops a Hamiltonian formulation of a massive supermembrane on a punctured torus, revealing discrete spectra and novel mass terms due to non-trivial compactification and puncture effects in 11D supermembrane theory.

Contribution

It introduces a new Hamiltonian formulation for the supermembrane on a punctured torus, showing how non-trivial compactification leads to mass terms and a discrete spectrum.

Findings

Mass terms dominate the bosonic potential.

Spectrum of the supermembrane is discrete with finite multiplicity.

Punctures induce a cosmological term in the Hamiltonian.

Abstract

We obtain the Hamiltonian formulation of the 11D Supermembrane theory non-trivially compactified on a twice-punctured torus times a 9D Minkowski space-time. It corresponds to a M2-brane formulated in 11D space with ten non-compact dimensions. The critical points like the poles and the zeros of the fields describing the embedding of the Supermembrane in the target space are treated rigorously. The non-trivial compactification generates non-trivial mass terms appearing in the bosonic potential, which dominate the full supersymmetric potential and should render the spectrum of the (regularized) Supermembrane discrete with finite multiplicity. The behaviour of the fields around the punctures generates a cosmological term in the Hamiltonian of the theory. The massive supermembrane can also be seen as a nontrivial uplift of a supermembrane torus bundle with parabolic monodromy in $M_9\times T^2$. The moduli of the theory is the one associated with the punctured torus, hence it keeps all the nontriviality of the torus moduli even after the decompactification process to ten noncompact dimensions. The formulation of the theory on a punctured torus bundle is characterized by the $(1,1)-Knots$ associated with the monodromies.