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تسجيل مستخدم جديدMeasure of the potential valleys of the supermembrane theory
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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 $mathrm{D}geq 5$. 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.
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In this work we consider the existence and uniqueness of the ground state of the regularized Hamiltonian of the Supermembrane in dimensions $D= 4,,5,,7$ and 11, or equivalently the $SU(N)$ Matrix Model. That is, the 0+1 reduction of the 10-dimensiona
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