Existence of a supersymmetric massless ground state of the \( \mathrm{SU} (N)\) matrix model globally on its valleys
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Publication:2048100
DOI10.1007/JHEP05(2021)281zbMATH Open1466.81082arXiv2102.00886OpenAlexW3167271607MaRDI QIDQ2048100
Author name not available (Why is that?)
Publication date: 4 August 2021
Published in: (Search for Journal in Brave)
Abstract: In this work we consider the existence and uniqueness of the ground state of the regularized Hamiltonian of the Supermembrane in dimensions and 11, or equivalently the Matrix Model. That is, the 0+1 reduction of the 10-dimensional Super Yang-Mills Hamiltonian. This ground state problem is associated with the solutions of the inner and outer Dirichlet problems for this operator, and their subsequent smooth patching (glueing) into a single state. We have discussed properties of the inner problem in a previous work, therefore we now investigate the outer Dirichlet problem for the Hamiltonian operator. We establish existence and uniqueness on unbounded valleys defined in terms of the bosonic potential. These are precisely those regions where the bosonic part of the potential is less than a given value , which we set to be arbitrary. The problem is well posed, since these valleys are preserved by the action of the constraint. We first show that their Lebesgue measure is finite, subject to restrictions on in terms of . We then use this analysis to determine a bound on the fermionic potential which yields the coercive property of the energy form. It is from this, that we derive the existence and uniqueness of the solution. As a by-product of our argumentation, we show that the Hamiltonian, restricted to the valleys, has spectrum purely discrete with finite multiplicity. Remarkably, this is in contrast to the case of the unrestricted space, where it is well known that the spectrum comprises a continuous segment. We discuss the relation of our work with the general ground state problem and the question of confinement in models with strong interactions.
Full work available at URL: https://arxiv.org/abs/2102.00886
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