Measure of the potential valleys of the supermembrane theory

Abstract

We analyse the measure of the regularized matrix model of the supersymmetric potential valleys, $\Omega$, of the Hamiltonian of non zero modes of supermembrane theory. This is the same as the Hamiltonian of the BFSS matrix model. We find sufficient conditions for this measure to be finite, in terms the spacetime dimension. For $SU(2)$ we show that the measure of $\Omega$ is finite for the regularized supermembrane matrix model when the transverse dimensions in the light cone gauge $\mathrm{D}\ge 5$. This covers the important case of seven and eleven dimensional supermembrane theories, and implies the compact embedding of the Sobolev space $H^{1,2}(\Omega )$ onto $L^2(\Omega )$. The latter is a main step towards the confirmation of the existence and uniqueness of ground state solutions of the outer Dirichlet problem for the Hamiltonian of the $SU(N)$ regularized $\mathrm{D}=11$ supermembrane, and might eventually allow patching with the inner solutions.

Latex, 11 pages, reference added

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• Asymptotic Factorisation of the Ground-State for SU(N)-invariant Supersymmetric Matrix-Models
• Towards the Ground State of the Supermembrane
• The ground state of the D=11 supermembrane and matrix models on compact regions
• Massless ground state for a compact SU(2) matrix model in 4D
• On the groundstate of octonionic matrix models in a ball

Cited by in corpus (2)

• Differential Cohomotopy implies intersecting brane observables via configuration spaces and chord diagrams
• Existence of a Supersymmetric Massless Ground State of the $SU(N)$ Matrix Model globally on its Valleys