Measure of the potential valleys of the supermembrane theory

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.

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.

Introduction
The supermembrane theory was derived in [3]. Its SU(N) regularization was introduced in [16] and in [9,11] the SU(N) regularized Hamiltonian in the light cone gauge was obtained. The zero mode eigenfunction can be described in terms of the D = 11 supergravity multiplet, however the existence of the ground state of the Hamiltonian requires a proof of the existence of a unique nontrivial eigenfunction for the nonzero modes. Moreover, in order to be identified with the 11D supergravity multiplet, it must be invariant under SO (9). The existence of this ground state is still an ellusive open problem. For an (incomplete) list of contributions towards its solution, mainly in asymptotic regimes, c.f. [16,15,13,19,17,18,12].
In 11D supermembrane theory, the zero modes associated with the center of mass and the non-zero modes associated with the internal excitations, decouple. The groundstate of the Hamiltonian with zero eigenvalue and their associated eigenfunction can be described in terms of the D = 11 supergravity multiplet once it is proven the existence for the non-zero modes of a unique nontrivial eigenfunction invariant under the R-symmetry SO(9) [9]. The SU(N) regularized Hamiltonian for nonzero modes coincides with the Hamiltonian of the BFSS matrix model, [1]. This Hamiltonian was first obtained as the 0 + 1 reduction of the 10D Super Yang Mills [7,14]. Hence, the existence of the ground state for this matrix model, turns out to be exactly the missing step in the proof of the existence of the ground state for the D = 11 supermembrane. In [8] a prescription to compute the index of non-Fredholm operators was presented. Although a precise definition of the domain of the non-Fredholm operator ( which certainly is not the whole Hilbert space) is not given and hence the definition of the trace involved in the evaluation of the index is not precise, they introduce a prescription, which they claim, allows to compute an index of the non-Fredholm operator. Using this result they claim to obtain a unique ground state for the Hamiltonian of the BFSS matrix model with gauge group SU (2).
In this paper we emphasise the bounds on the ground state wave function on the flat directions, which is absent in the previous works. Although we consider the SU(2) case, our approach indicates a new way to analyse the SU(N), N going to infinity, model. This is the relevant gauge group for the D=11 Supermembrane.
Due to its complexity, a natural approach for the solution of this problem is to divide it into three parts, [5,4,6]. Firstly, determine the existence and uniqueness of a solution for the Dirichlet problem on a bounded region of arbitrary radius. Secondly, determine the existence and uniqueness of the solution for the Dirichlet problem on the unbounded complementary region. Thirdly, determine if both these solutions match with one another and can be smoothly patched into a single solution of the full problem. The overall state will then be the ground state of the Hamiltonian of the non-zero modes of the D = 11 supermembrane. In [6] (also [5,4]) we settled the first step. The present work is about the second step.
Our proof of existence and uniqueness for bounded regions, relied on two fundamental properties of the Hamiltonian: i) its supersymmetric structure as H = {Q, Q † } and ii) the polynomial form of the potential expression as a function of the bosonic coordinates. We combined these two properties with iii) the Rellich-Kondrashov compact embedding theorem. Then the existence and uniqueness followed from ellipticity and the Lax-Milgram theorem for strongly coercive sesquilinear forms.
The Rellich-Kondrashov compact embedding theorem holds true for every bounded region of R n , but it might fail in general on unbounded regions. For the second step, one requires an estimate for the contribution of the potential to the mean value of the Hamiltonian, taking into account that this potential is unbounded from below along the sub-varieties where the bosonic potential vanishes. An estimate must therefore be obtained on the "valleys", denoted by Ω below, surounding these sub-varieties. In the complement of Ω, the bosonic potential is the dominant part of the potential, it is strictly positive and it tends to infinity at infinity. In these "good" regions, the existence and uniqueness of the solution to the Dirichlet problem follows from general arguments, similar to those used for non-relativistic Schrödinger operators. We expect that the ground state (if it exists) should extend along Ω and decay rapidly to zero in the complement of Ω.
Hard work, however, has to be conducted in the interior of Ω. That is, to show the existence and uniqueness of the solution to the Dirichlet problem on Ω minus a ball of finite radius. In order to achieve this goal, we devote this letter to establishing that the Rellich-Kondrashov compact embedding theorem holds true on Ω for D ≥ 5 on Sobolev spaces defined following [2]. Concretely, we show that the measure (Lebesgue measure) of the unbounded set is finite and decays at infinity for any D ≥ 5. See lemma 2. This includes, for the bosonic potential, the important cases of the D = 7 and D = 11 supermembrane.
Consequently, with respect to properties i), ii) and iii), arguments analogous to i) and iii) can be made on Ω. Property ii) is not valid in Ω, but it might be possible to consider an estimate of the fermionic contribution to the mean value of the Hamiltonian which allows a different version of coercivity. We hope to report on this eventually.

Formulation of the problem
Before establishing our main current contribution, let us summarize the formulation of the problem. We follow the seminal work [9]. The D = 11 supermembrane is described in terms of the membrane coordinates X m and fermionic coordinates θ α , transforming as a Majorana spinor on the target space. Both fields are scalar under worldvolume transformations. When the theory is formulated in the Light Cone Gauge the residual symmetries are global supersymmetry, the R-symmetry SO(9) and a gauge symmetry, the area preserving diffeomorphisms on the base manifold.
The fields of the Hamiltonian and the wavefunction are decomposed according to the symmetry group SO(9) in such a way that the Majorana spinor is expressed in terms of the linear representations of the subgroup SO(7) × U(1) ⊂ SO (9).
The bosonic coordinates X M are decomposed as (X m , Z, Z). Where X m for m = 1, . . . , 7 are the components of a SO (7) vector, and Z, Z are the complex scalars which transform under U(1). The corresponding bosonic canonical momenta is accordingly decomposed as a SO (7) vector of components P m and a complex U(1) momentum P and its conjugate P: P M = (P m , P, P) where P = 1 √ 2 (P 8 − iP 9 ) and P = 1 √ 2 (P 8 + iP 9 ).
Denoting by λ α the invariant SO (7) spinor of the operator associated to the fermionic coordinates. We can express it in terms of an eight component complex spinor θ ± eigenstate of γ 9 , for γ 9 θ ± = ±θ ± , such that where λ † is the fermionic canonical conjugate momentum to λ.
Once the theory is regularized by means of the group SU(N), the field operators are labeled by an SU(N) index A and they transform in the adjoint representation of the group.
The realization of the wavefunctions is formulated in terms of the 2 8(N 2 −1) an irreducible representation of the Clifford algebra span by (λ † + λ) and i(λ † − λ) in the fermion Fock space. The Hilbert space of physical states consists of the wavefunctions which takes values in the fermion Fock space.
Once it is shown that the zero mode states transform under SO(9) as a [(44 ⊕ 84) bos ⊕ 128 fer ] representation which corresponds to the massless D = 11 supergravity supermultiplet, the construction of the ground state wave function reduces to finding a nontrivial solution to HΨ = 0 where H = 1 2 M and Ψ ≡ Ψ non−zero . The latter is required to be a singlet under SO(9) and M is the mass operator of the supermembrane. The Hamiltonian associated to the the regularized mass operator of the supermembrane [9] is ).
The generators of the local SU(N) symmetry are From the supersymmetric algebra, it follows that the Hamiltonian can be express in terms of the supercharges as H = {Q α , Q † α } for the physical subspace of solutions, given by the kernel of the first class constraint ϕ A of the theory, that is The supercharges associated to modes invariant under SO(7) × U(1) are given explicitly in [9] as The corresponding superalgebra satisfies, [9] These must annihilate the physical states. The Hamiltonian H is a positive operator which annihilates Ψ, on the physical subspace, if and only if Ψ is a singlet under supersymmetry 1 . In such a case, Q α Ψ = 0 and Q † α Ψ = 0. The latter ensures that the wavefunction is massless, however it does not guarantee that the ground-state wave function is the corresponding supermultiplet associated to supergravity. For this, Ψ must also become a singlet under SO (9). The spectrum of H in L 2 (R n ) is continuous [10], comprising the segment [0, ∞).
The previous supersymmetric structure implies the following. This is the property i) described above. The current interest is the case Σ = Ω.
The argument is the same as in [6]. If u satisfies Qu = Q † u = 0, then u is analytic in Σ, since the potential is analytic in x. Hence Qu = Q † u = 0 also on the boundary ∂Σ. Then using the explicit expression of Q and Q † , we obtain that the normal derivative of u on ∂Σ is also zero. We thus have u = 0 and ∂ n u = 0 simultaneously on ∂Σ. By virtue of the Cauchy-Kowaleski theorem on ∂Σ, u = 0 in a neighborhood of ∂Σ. Since u is analytic we conclude that u = 0 in Σ.

Analysis of the Lebesgue measure of the bosonic valleys
We simplify the proof of our main result below by denoting the bosonic coordinates with X A i , for i = 1, . . . , D and A = 1, 2, 3 the SU(2) index. We will denote a vector of 3 × D components by means of D vectors of 3 components: X i ∈ R 3 , i = 1, . . . , D. We denote with a single bar, | · |, the Euclidean norm on any number of components. The bosonic potential reduces to Below we repeatedly use the following property without further mention. If R is any rotation of R 3 , then For a 0 ≥ 0, let So that Ω = Ω 0 . We denote the (Lebesgue) measure of any of these sets by µ(Ω a 0 ).
Proof. We firstly notice that we can exchange the orders of integration below, because V B is a polynomial in its components. Fix one direction e and consider the change of variables that rotates X 1 to a e. Then where we have denoted by B; C; D the points in R D−1 with components wherẽ Ω a = {(B; C; D) : RHS of (1) < 1, a 2 + |B| 2 + |C| 2 + |D| 2 ≥ a 0 }.
In order to estimate the integrals in (2) we change variables to where B · C ⊥ = B · D ⊥ = 0 so that α = C · B |B| and β = D · B |B| . The potential becomes For a and B fixed, the region is an ellipsoid which contains where k l (D) are constants. Finaly notice that the right hand side is finite for all a 0 > 0 and decreases to 0 as a 0 → ∞, whenever D ≥ 5.
We denote by H p (Ω) and H p 0 (Ω), respectively, the Sobolev spaces H p,2 (Ω) andW p,2 (Ω) in the notation of [2]. A crucial observation here is the fact that these spaces are amenable to patching inner and outer domains in the solution of Dirichlet problems. We recall that H p (Ω) is the Hilbert space arising from restricting to Ω functions in the Sobolev space H p (R n ), the norm being the infimum of the Sobolev norm over all possible extensions. We also recall that H p 0 (Ω) is the completion with respect to this norm of all smooth functions with support a compact subset of Ω. By combining lemma 2 with Theorem 2.8 of [2], we immediately obtain the remarkable property that that both H p (Ω) and H p 0 (Ω) are compactly embedded into L 2 (Ω) for D ≥ 5.

Bounds for the fermionic potential
In order to prove the existence and uniqueness of the solution to the outer Dirichlet problem we need a bound on the contribution of the mean value of the fermionic potential (u, V F u) L 2 (Ω) . We notice that the fermionic potential is linear on the bosonic coordinates. Then |(u, V F u)| L 2 (Ω) ≤ C(u, ρu) L 2 (Ω) for some C > 0, where ρ 2 = |x| 2 ≡ a 2 + |B| 2 + |C| 2 + |D| 2 . From Lemma 2, it is possible to derive the following bound, In turns, we have the following results which follows from similar arguments as those of lemma 2.
As a corollary of this lemma, we obtain that for D > 5, also (3) holds true. We hope to discuss a sharp bound of the form (u, V F u) L 2 (Ω) ≤ k u 2 H 1 (Ω) in future work.

Conclusions
We have shown that the volume of the valleys, the set Ω, is finite when the dimension of the target space on which the supermembrane theory is formulated is greater than or equal to five (transverse) dimensions. This include the important 7 and 11-dimensional supermembranes. Using a framework due to Berger and Schechter we have shown that on Ω, the embeddings of H 1 (Ω) and H 1 0 (Ω) onto L 2 (Ω) are compact. Notice that this property is not related to the zero point energy of the bosonic membrane, a main observation in the argument to conclude that the bosonic membrane has discrete spectrum. In fact this result does not depend on the target space dimension. Furthermore, we have argued with supporting evidence about bounds for the mean value of the fermionic potential. We have then shown properties i) and iii) proposed in the introduction and claim that it is possible to get an appropiate bound for the mean value of the fermionic potential on any wave function in H 1 0 (Ω). The complete proof of the three statements will allow determining the existence and uniqueness of the solution of the outer Dirichlet problem on the valleys of the bosonic potential.

Acknowledgements
AR and MPGM were partially supported by Projects Fondecyt 1161192 (Chile). LB kindly acknowledges support from MINEDUC-UA project code ANT1755. This research was initiated as part of a visit of this author to the Universidad de Antofagasta in April 2018 and completed when he was on a study leave at theČeské Vysoké Učení Technické v Praze in November 2018.