Towards the ground state of the supermembrane

Michishita, Yoji; Trzetrzelewski, Maciej

doi:10.1016/j.nuclphysb.2012.11.013

Cited by 8 publications

(7 citation statements)

References 27 publications

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“…Let us verify the condition (K) for the supersymmetric charges. Assume that (11) Q α ψ = 0 and Q † α ψ = 0 in Ω for ψ ∈ H 2 (Ω) ∩ H 1 0 (Ω). We wish to prove that ψ = 0 in Ω. Regularity at the boundary (Lemma 5), allows us to extend the restriction on ∂ψ ∂ρ ′ arising from (11) smoothly to the boundary.…”

Section: The Ground State Of the D = 11 Supermembranementioning

confidence: 99%

“…Assume that (11) Q α ψ = 0 and Q † α ψ = 0 in Ω for ψ ∈ H 2 (Ω) ∩ H 1 0 (Ω). We wish to prove that ψ = 0 in Ω. Regularity at the boundary (Lemma 5), allows us to extend the restriction on ∂ψ ∂ρ ′ arising from (11) smoothly to the boundary. The conditions Q α ψ| ∂Ω = 0 and Q † α ψ| ∂Ω = 0 for the SU(N) regularized supermembrane found in [8], now evaluated on the boundary where ψ = 0, are…”

Section: The Ground State Of the D = 11 Supermembranementioning

confidence: 99%

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The ground state of the D = 11 supermembrane and matrix models on compact regions

2016

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We establish a general framework for the analysis of boundary value problems of matrix models at zero energy on compact regions. We derive existence and uniqueness of ground state wavefunctions for the mass operator of the D = 11 regularized supermembrane theory, that is the N = 16 supersymmetric SU (N ) matrix model, on balls of finite radius. Our results rely on the structure of the associated Dirichlet form and a factorization in terms of the supersymmetric charges. They also rely on the polynomial structure of the potential and various other supersymmetric properties of the system.Date: 4th May 2016.

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Section: The Ground State Of the D = 11 Supermembranementioning

confidence: 99%

Section: The Ground State Of the D = 11 Supermembranementioning

confidence: 99%

The ground state of the D = 11 supermembrane and matrix models on compact regions

2016

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“…For an (incomplete) list of contributions towards its solution, mainly in asymptotic regimes, c.f. [16,15,13,19,17,18,12].…”

Section: Introductionmentioning

confidence: 99%

Measure of the potential valleys of the supermembrane theory

2019

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We analyse the measure of the regularized matrix model of the supersymmetric potential valleys, Ω, 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 Ω is finite for the regularized supermembrane matrix model when the transverse dimensions in the light cone gauge D ≥ 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 (Ω) onto L 2 (Ω). 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 D = 11 supermembrane, and might eventually allow patching with the inner solutions.

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“…One of these started with [1]. Although the problem remains open several interesting contributions to it have been obtained [1,[20][21][22][23][24][25]. We follow this perspective and prove, for a well-defined region around the valleys of the potential extended to infinity, the existence and uniqueness of the nontrivial state annihilating the Hamiltonian.…”

Section: Introductionmentioning

confidence: 99%

Existence of a supersymmetric massless ground state of the SU(N) matrix model globally on its valleys

Boulton

Moral

Restuccia

2021

J. High Energ. Phys.

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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-dimensional SU(N) 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 V0, which we set to be arbitrary. The problem is well posed, since these valleys are preserved by the action of the SU(N) constraint. We first show that their Lebesgue measure is finite, subject to restrictions on D in terms of N. 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.

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Towards the ground state of the supermembrane

Cited by 8 publications

References 27 publications

The ground state of the D = 11 supermembrane and matrix models on compact regions

The ground state of the D = 11 supermembrane and matrix models on compact regions

Measure of the potential valleys of the supermembrane theory

Existence of a supersymmetric massless ground state of the SU(N) matrix model globally on its valleys

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