Existence of a Supersymmetric Massless Ground State of the SU(N) Matrix Model globally on its Valleys
aa r X i v : . [ h e p - t h ] F e b EXISTENCE OF A SUPERSYMMETRIC MASSLESSGROUND STATE OF THE SU ( N ) MATRIX MODELGLOBALLY ON ITS VALLEYS
LYONELL BOULTON , MAR´IA PILAR GARC´IA DEL MORAL ,AND ALVARO RESTUCCIA Abstract.
In this work we consider the existence and unique-ness of the ground state of the regularized Hamiltonian of the Su-permembrane in dimensions D = 4 , , SU ( N ) Matrix Model. That is, the 0+1 reduction of the 10-dimensional SU ( N ) Super Yang-Mills Hamiltonian. This groundstate problem is associated with the solutions of the inner andouter Dirichlet problems for this operator, and their subsequentsmooth patching (glueing) into a single state. We have discussedproperties of the inner problem in a previous work, therefore wenow investigate the outer Dirichlet problem for the Hamiltonianoperator. We establish existence and uniqueness on unboundedvalleys defined in terms of the bosonic potential. These are pre-cisely those regions where the bosonic part of the potential is lessthan a given value V , which we set to be arbitrary. The problemis well posed, since these valleys are preserved by the action of the SU ( N ) constraint. We first show that their Lebesgue measure isfinite, subject to restrictions on D in terms of N . We then use thisanalysis to determine a bound on the fermionic potential whichyields the coercive property of the energy form. It is from this,that we derive the existence and uniqueness of the solution. As aby-product of our argumentation, we show that the Hamiltonian,restricted to the valleys, has spectrum purely discrete with finitemultiplicity. Remarkably, this is in contrast to the case of the unre-stricted space, where it is well known that the spectrum comprisesa continuous segment. We discuss the relation of our work withthe general ground state problem and the question of confinementin models with strong interactions. Date : 06-01-2021. Introduction
The present paper is devoted to the ground state of the supersym-metric Hamiltonian related to three theories: the regularization of the D = 11 Supermembrane Theory [1], the BFSS Matrix Model [2] andthe reduction of D = 10 Super Yang-Mills to 0 + 1 dimensions [3]. Theexistence of a massless ground state in any of these three instances isan open problem.The relevance of this regime for (Super)Yang-Mills theories in theinfra-red (IR) limit, the so-called slow-mode regime, was highlightedin [4]. In this IR limit, glueball bound states and flux tubes betweenquarks are expected to be formed. It has been suggested that theycan be described in terms of confining strings which corresponds to aNambu-Goto string connecting pairs of quarks at the extremes, subjectto several corrections [5, 6, 7]. See for example [8, 9, 10] or [11, 12]in the context of lattice QCD. In these works, the confining stringsacquire a width and they have also been modelled out in terms ofD2-brane bound states [13]. (Super) membranes are strongly coupled2 + 1 dimensional objects and their regularized description correspondsto the Matrix Models mentioned above. It has also been suggestedthat membranes can be seen as the IR limit of Yang Mills theories[14] and they have also been used in the literature to describe someaspects of QCD [15]. Indeed, in [16] it was shown that the spectrum ofthe bosonic regularized membrane theory has a mass gap given by theinertia moment and as it is well known the spectrum in this regime ispurely discrete for arbitrary N (in particular for N = 3). The existenceof bound states of these theories, realised as eigenvalues embedded inthe continuous spectrum, could be an interesting research direction.Our present goal is to describe the theories, restricted to certain8 × ( N − V . That is,they are determined by imposing a given constraint on the “height”of the bosonic potential term. These valleys extend to infinity withdecreasing width. Constrained along these valleys, we obtain a 10 D Super Yang Mills theory in the slow mode regime or the regularized SU ( N ) supermembrane. I.e. the SU ( N ) Matrix Model confined toa star-shape tubular region. The corresponding Hamiltonian operatorhas a domain determined by the space of wave functions supported onthe valleys, with vanishing boundary condition. Our analysis coversarbitrary rank of the SU ( N ) gauge group, in particular for N = 2 and N = 3, both cases being of interest.The Supermembrane Theory was developed in [17]. The correspond-ing SU ( N ) regularization was introduced in [18] and in [1, 19] the U(N) GROUND STATE 3 SU ( N ) regularized Hamiltonian in the light cone gauge was obtained.The zero mode eigenfunction can be described in terms of the D = 11supergravity multiplet, however, the existence of the ground state ofthe Hamiltonian requires a proof of existence of a unique nontrivialeigenfunction for the nonzero modes. To the best of our knowledge,no complete proof of this fact has been found to this date. Moreover,in order to be identified with the D = 11 supergravity multiplet, ithas to satisfy the additional constraint of being invariant under SO (9).In the context of asymptotic regimes, a reasonably complete list ofcontributions in this subject is [18, 20, 21, 22, 23, 24, 25].In D = 11 Supermembrane Theory, the zero modes associated withthe center of mass and the non-zero modes associated with the internalexcitations, decouple. The ground state of the Hamiltonian with zeroeigenvalue can be described in terms of the D = 11 supergravity mul-tiplet, once the existence of the non-zero modes of a unique nontrivialeigenfunction (with zero eigenvalue) invariant under the R-symmetry SO (9) is proven, [1]. The SU ( N ) regularized Hamiltonian for nonzeromodes coincides with the Hamiltonian of the BFSS Matrix Model, [2].This Hamiltonian was first obtained as the 0+1 reduction of 10 D SuperYang Mills [3, 26].In order to find the ground state of the Hamiltonian, we propose threemain steps, already sketched in [29, 30, 27, 28]. Firstly, determine theexistence and uniqueness of the solution to the Dirichlet problem on abounded region Ω with smooth boundary ∂ Ω. Secondly, determine theexistence and uniqueness of the solution to the Dirichlet problem onthe complementary unbounded region. Thirdly, establish the smoothpatching along ∂ Ω of both these solutions into a single state which, byconstruction, is the full ground state.We developed the first step in [29] for Ω of arbitrary diameter. Ourargument relied on the polynomial form of the bosonic and fermionicpotentials, as well as on the supersymmetric structure of the Hamil-tonian. In the full space, although the Hamiltonian is positive, thepotential becomes negative and arbitrarily large in modulus along cer-tain directions inside the valleys extending to infinity. However, onbounded regions, the potential is bounded (both above and below).Therefore, the Dirichlet form associated to the Hamiltonian restrictedto Ω is coercive. Moreover, the supersymmetric structure of the Hamil-tonian together with other analytic properties of potential, imply thata state ϕ constrained to cancellation by the supersymmetric chargesin Ω and satisfying the homogeneous Dirichlet condition ϕ = 0 on ∂ Ω,can only be the null state of the Hilbert space. In turns, the existenceand uniqueness of the Dirichlet problem on Ω, follows by standardarguments from the theory of elliptic operators. These involve theRellich-Kondrashov Compact Embedding Theorem, the Lax-MilgramTheorem and the Fredholm Theorem. See [29] for more specific details.
L. BOULTON, M.P. GARC´IA DEL MORAL, AND A. RESTUCCIA
The Rellich-Kondrashov Compact Embedding Theorem is valid forevery bounded region of the Euclidean space, but unfortunately mightfail on unbounded regions. Nonetheless, according to arguments in [32],the convergence of the partition function of 0 + 1 Yang-Mills [33, 34] isrelated to the fact that the Lebesgue measure of the bosonic valleys isfinite. Following the former work, this property is valid for all algebrasconsidered in [33] and [34].By pursuing an alternative approach to that of [32, 33, 34], we es-tablished in [29] a concrete estimate for the Lebesgue measure of thesebosonic valleys for the SU (2) algebra. We then showed that the embed-ding of H into L is compact. Hence the Rellich-Kondrashov Theoremis, once again, valid for these regions. Our argumentation was intrinsicto the specific structure of the bosonic potential. From the estimateand a relevant bound for the fermionic potential, it should follow thatthe Dirichlet form of the complete Hamiltonian is coercive.One of our main purposes below will be to extend this idea onto an SU ( N ) algebra and prove the existence and uniqueness of the state, inthe physical subspace, annihilated by the Hamiltonian of the SU ( N ) D = 11 Supermembrane, restricted to the valleys. Our construction isbased on specific properties of the potential, one of the most importantbeing an ellipsoidal symmetry along axial directions. The results, con-cerning the finiteness of the measure of the valleys, for any N ≥ SU (2) algebra, which depends on other properties of the po-tential, outside a neighbourhood of the origin. We then show that theDirichlet form associated to the SU ( N ) Hamiltonian of the Supermem-brane is coercive. From this, and the compact embedding H ⊂ L onthe valleys, it follows that the solution of the Dirichlet problem existsand is unique. An important result that follows from our analysis isthat the Hamiltonian restricted to the valleys has discrete spectrumwith finite multiplicity.Formally, the potential is dominated by its bosonic component inthe directions “away” from the valleys, so the wave function is confinedalong those directions. Although a rigorous proof of the latter for anycurve reaching infinity is not currently available, it is clear that theanalysis of the Dirichlet problem along the valleys that we currentlyconduct, sheds an important light on the direction to follow in thesecond step of the program mentioned above.The remaining of the paper is structured as follows. In Section 2 weset the scenary by recalling the 0 + 1 Matrix Model formulation of the11 D Supermembrane Theory in the Light Cone Gauge. In Section 3we summarise the bounds we found on the measure of the potentialvalleys associated to the different su ( N ) models. The full details of ourderivations can be found in Appendices A-D. In them, we determine the U(N) GROUND STATE 5 measure of the valleys when all the eigenvalues of a given configuration su ( N ) matrix are different. We include detailed discussions of the cases su (2), su (3), su (4) and general su ( N ) with arbitrary N . In Appendix Cwe show that the Lebesgue measure in all the cases is finite. Thisimplies that only certain dimensions for a given rank of the SU ( N )bosonic potential are allowed. In Section 4 we obtain the associatedbounds for the fermionic potential. Section 5 is devoted to the mainresults. We establish that the spectrum of the Hamiltonians is purelydiscrete and demonstrate the existence and uniqueness of the groundstate of the theory. In Section 6 we present a discussion of the ideaspresented and our conclusions. The gauge transformations that weemployed in order to show that the valleys have finite measure, aredisplayed in the final Appendix D.2. The SU ( N ) Matrix Model
We begin by discussing the 0 + 1 Matrix Model formulation of the11 D Supermembrane Theory in the Light Cone Gauge. The latter alsocorresponds to a 0 + 1 reduction of 10 D Super Yang Mills, known inthe literature as the BFSS Matrix Model [2]. It describes D D = 11 supermembrane is described in terms of the membranecoordinates X m and fermionic coordinates θ α , transforming as a Ma-jorana spinor on the target space. Both fields are scalar under world-volume transformations. When the theory is formulated in the LightCone Gauge the residual symmetries are the global supersymmetry,the R-symmetry SO (9) and a gauge symmetry, the area preservingdiffeomorphisms on the base manifold.Once the theory is regularized by means of the group SU ( N ), thefield operators are labeled by an SU ( N ) index A and they transformin the adjoint representation of the group. The realization of the wave-functions is formulated in terms of the 2 N − irreducible represen-tation of the Clifford algebra. The Hilbert space of physical statesconsists of the wavefunctions which take values in the fermion Fockspace, subject to the SU ( N ) constraint given by the generator of the SU ( N ) invariance.Once it is shown that the zero mode states transform under SO (9)as a [(44 ⊕ bos ⊕ fer ] representation which corresponds to themassless D = 11 supergravity supermultiplet, the construction of theground state wave function reduces to finding a nontrivial solution to H Ψ = 0where H = M and Ψ subject to the SU ( N ) constraint. The latteris required to be a singlet under SO (9) and M is the mass operator ofthe supermembrane. The Hamiltonian associated to the the regularized L. BOULTON, M.P. GARC´IA DEL MORAL, AND A. RESTUCCIA mass operator of the supermembrane [1] is(1) H = 12 M = − ∆ + V B + V F where ∆ = ∂ ∂X iA ∂X Ai V B = f EAB f CDE X Ai X Bj X iC X jD V F = if ABC X Ai λ Bα Γ iαβ ∂∂θ βC . The generators of the local SU ( N ) symmetry are(2) ϕ A = f ABC (cid:16) X Bi ∂ X Ci + θ Bα ∂ θ Cα (cid:17) . From the supersymmetric algebra, it follows that the Hamiltoniancan be express in terms of the supercharges as(3) H = { Q α , Q † α } for the physical subspace of solutions, given by the kernel of the firstclass constraint ϕ A of the theory. That is ϕ A Ψ = 0 . The Hamiltonian H is a positive operator which annihilates Ψ, on thephysical subspace, if and only if Ψ is a singlet under supersymmetry .In such a case,(4) Q α Ψ = 0 and Q † α Ψ = 0 . This result does not hold when the theory is restricted by boundaryconditions, the case that we will analyze below.All this ensures that the wavefunction is massless, however it doesnot guarantee that the ground-state wave function is the correspond-ing supermultiplet associated to supergravity. For this, Ψ must alsobecome a singlet under SO (9). The spectrum of H in L ( R n ) is con-tinuous [35], comprising the segment [0 , ∞ ).The bosonic potential can be recast as(5) V B ( X ) = − d X M,N ≥ T r [ X M , X N ] = 12 d X M,N ≥ T r [ X M , X N ][ X M , X N ] † where X m = X mA T A , considering X mA real coordinates and T A thegenerators of the algebra su ( N ) ; d = D − su ( N ) generators satisfies [ T A , T B ] = if ABC T C with T r ( T A T B ) = δ AB . Ψ , the zero mode wave function, in distinction is a supermultiplet undersupersymmetry. The index A corresponds to a pair of indices ( a , a ) with a i =0 , . . . , dim( su ( N )) , i = 1 , ,
0) associated to the super-membrane center of mass has being excluded.
U(N) GROUND STATE 7 Lebesgue measure of the valleys
Prescribe a height V . Let(6) K ≡ { X mA : V B ( X ) < V } . We now quote the range of parameters for which the Lebesgue measureof K , denoted as Vol( K ), is finite. In the appendices A-D, we giveprecise details of how to derive these conclusions.Our argumentation depends on the following simple observation,which we use freely and unambiguously throughout the text. Let e V B ( X )be another potential expression, such that(7) e V B ( X ) ≤ V B ( X ) for all matrices X. Denote by(8) e K = { X mA : e V B < V } . If Vol( e K ) is finite, then so is Vol( K ).In appendices B.4 and D we determine, after the evaluation of severalestimates, that a restriction on d for each N so that the Lebesguemeasure of K is finite turns out to be(9) d > N − N − − . This immediately renders the following.
Lemma 1. If (10) d ≥ and N = 24 and N = 33 and N ≥ , then Vol( K ) < ∞ . In the evaluations leading to the aforementioned observation, wecrucially make use of the ellipsoidal symmetry of the bosonic potential.The final result is in agreement with the previous estimates reportedin [32, 33]. In the next section we will invoke some of our explicitestimates from the appendices, in order to determine a precise boundon the fermionic potential. This will then be fundamental for our mainresults, reported in Section 5.4.
Bounds for the fermionic potential
The main point of this section will be to determine an explicit esti-mate for the fermionic potential on any state. From this estimate, wewill show in the next section that the Hamiltonian operator of the Su-permembrane Theory, in the admissible dimensions given by Lemma 1,is coercive in the Fock space on K . To this end let us recall the prop-erties of the valleys discussed in the appendices. The region we defined L. BOULTON, M.P. GARC´IA DEL MORAL, AND A. RESTUCCIA as the valleys is a star-shaped region containing directions extendingto infinity along which the bosonic potential is zero. On those direc-tions the diagonal components of the matrices tend to infinity whilethe non-diagonal ones remain bounded. In the appendices we provethat given a large enough distance from the origin there always existsa SU ( N ) transformation such that the non-diagonal components notonly are bounded but decrease as the inverse of a diagonal component,when this one goes to infinity. We require an explicit expression for V B ( X ), and for that we introduce the next convenient notation. Thecomponents of a diagonal matrix b X are denoted by(11) ia , . . . , ia N where N X i =1 a i = 0 . The other matrices X n , n = 1 , . . . , d −
1, have diagonal components(12) ib n , . . . , ib nN where N X i =1 b ni = 0and upper-non diagonal components(13) z nij , i < j where i, j = 1 , . . . , N. Here a i and b ni are real numbers while z nij are complex numbers. As the X n are anti-hermitean, z nji = − z nij where z denotes complex conjugationfor j > i . We also introduce the vectors b i with components b ni , and z ij with components z nij , n = 1 , . . . , d − || b i || = d − X n =1 ( b ni ) , || z ij || = d − X n =1 z nij z nij , respectively. The products are defined as ( b i · z jk ) = P d − n =1 b ni z njk and( z ij · z kl ) = P d − n =1 z nij z nkl . We denote M ≡ [1 , . . . , d ] and I = { ( i, j ) : i
Let ǫ > and V > be fixed. There exist two positiveconstant C , C > , such that (20) Z K ρ Ψ · Ψ ≤ C Z K Ψ · Ψ + C Z K ∇ Ψ · ∇ Ψ for all Ψ ∈ H ( K ) . Moreover, C can be chosen arbitrarily smallwhenever ǫ is sufficiently large.Proof. Since C ∞ c ( K ) is a dense subspace of H ( K ) and ρ is smooth, itsuffices to find a constant independent of the wave function such thatthe inequality holds true for all Ψ ∈ C ∞ c ( K ).The argument in the rest of the proof follows Poincar´e’s Lemma, us-ing in addition that the transverse components decrease, as V / b ρ ij alongthe valleys extend to infinite. At γ − , Ψ I = 0, because Ψ I ∈ C ∞ c ( e K ). Then, whenever p ∈ K + , eachcomponent Ψ I ( p ) can be written as( ρ ij Ψ I Ψ I ) / = Z C ij γ − d γ (cid:2) ∂ γ ( ρ ij Ψ I Ψ I ) / (cid:3) . Applying the Cauchy-Schwartz Inequality, we get( ρ ij Ψ I Ψ I ) / ≤ ( γ + − γ − ) / (cid:26)Z γ + γ − d γ (cid:2) ∂ γ ( ρ ij Ψ I Ψ I ) / (cid:3) (cid:27) / . Consequently, as ∂ρ ij ∂γ = ρ − ij γ and because of (18), ρ ij Ψ I Ψ I ≤ V / b ρ ij Z γ + γ − d γ (cid:26) γ ρ − ij Ψ I Ψ I + ρ ij [ ∂ γ (Ψ I Ψ I ) / ] (cid:27) . Moreover, γ ρ − ij ≤ b ρ − ij ( γ ρ − ij ) ≤ c ρ ij − hence, after a straightforwardcalculation, ρ ij Ψ I Ψ I ≤ V / ǫ Z γ + γ − d γ Ψ I Ψ I +4 V / " V / ǫ γ + γ − d γ [ ∂ γ (Ψ I Ψ I ) / ] . Considering all derivatives of (Ψ I Ψ I ) / we then get ρ ij Ψ I Ψ I ≤ V / ǫ Z γ + γ − d γ Ψ I Ψ I + 4 V / " V / ǫ γ + γ − d γ ( ∇ Ψ I · ∇ Ψ I )which is valid for each i, j - su (2) sector. We have used the inequality(21) ∇| Ψ I | · ∇| Ψ I | ≤ ∇ Ψ I · ∇ Ψ I with | Ψ I | = (Ψ I Ψ I ) / , which is always valid. We may now integrateon K + , to get(22) Z K + ρ ij Ψ I Ψ I ≤ V ǫ Z K + Ψ I Ψ I + 8 V ǫ " V / ǫ K + ∇ Ψ I · ∇ Ψ I On the other hand, on K − , we have(23) Z K − ρ ij Ψ I Ψ I ≤ e C Z K − Ψ I Ψ I . Since (22) and (23) are valid for each component Ψ I ,(24) Z K ρ ij Ψ I · Ψ I ≤ C Z K Ψ · Ψ + C Z K ∇ Ψ · ∇ Ψwhere C = max (cid:18) V ǫ , e C (cid:19) , C = 8 V ǫ V / ǫ ! For the final claim, it is enough to take ǫ → ∞ to get C →
0. Aftersummation on all the u (2) sectors the constants C and C acquire afactor N ( N − . (cid:3) U(N) GROUND STATE 11
The fact that we can ensure C becomes arbitrarily small is crucialwhen proving the coercivity part of the next result. Note that C couldbe very large and this has no significant effect in the statement. Lemma 3.
The Hamiltonian of the Supermembrane, valued on the su ( N ) algebra, has a Dirichlet form coercive on the Fock space F ( K ) .Proof. Firstly, the bosonic potential can be expressed in terms of thematrices X m , valued on the su ( N ) algebra, where m = 1 , . . . , d, (25) X m = X A X mA T A , Tr T A T B = δ AB . The diagonal components, associated to the diagonal generators canbe re-written, in terms ( a i − a j ) and ( b mk − b ml ) components with abounded Jacobian. The non-diagonal components z mij , coincide withthe corresponding components of the non-diagonal generators T A .Combining the linearity of V F in X with Lemma 2, it follows thatthere exists a constant C > Z K Ψ · V F Ψ ≥ − C Z K ρ Ψ · Ψ ≥ − CC Z K Ψ · Ψ − CC Z ∇ Ψ · ∇ Ψ . Given V , choose ǫ large enough, such that CC <
1. Then Z K ∇ Ψ · ∇ Ψ + Z K Ψ · V B Ψ + Z K Ψ · V F Ψ ≥ (1 − CC ) Z K ∇ Ψ · ∇ Ψ + Z K Ψ · ( V B − CC )Ψ . Consequently, for the Hamiltonian operator H = − ∆ + V B + V F ,(27) (Ψ , H Ψ) L ( K ) ≥ λ (Ψ , Ψ) L ( K ) + b C k Ψ k H ( K ) for all Ψ ∈ F ( K ). Here λ and b C are real constants, satisfying b C > λ is possibly negative but it is bounded from below. This impliesthat the Dirichlet form of the Hamiltonian is coercive on F ( K ) asclaimed. (cid:3) Main results
Let(28) D (Ψ , Φ) = Z K ∇ Ψ · ∇ Φ + Z K Ψ · ( V B + V F )Φbe the Dirichlet form associated to the left hand side of (27). Let(29) ˜ D (Ψ , Φ) = D (Ψ , Φ) − λ (Ψ , Φ) L ( K ) where the parameter λ is as in the proof of Lemma 3. Since(30) | (Ψ , V F Φ) | L ( K ) ≤ | (Ψ − Φ , V F (Ψ − Φ)) L ( K ) | + 12 | (Ψ − i Φ , V F (Ψ − i Φ)) L ( K ) | + | (Ψ , V F Ψ) L ( K ) | + | (Φ , V F Φ) L ( K ) | , it is readily seen that ˜ D (Ψ , Φ) is strongly coercive and bounded in H ( K ). Then, by virtue of the Lax-Milgram Theorem, it follows thatthere exists a bounded operator T : L ( K ) −→ H ( K ) ∩ H ( K ) suchthat for any Ξ ∈ L ( K )(31) ˜ D (Ψ , T Ξ) = (Ψ , Ξ) L ( K ) for all Ψ ∈ H ( K ) . That is T = ( H − λ ) − is a well defined bounded operator. This facthas an important consequence highlighted below.We know that H ( K ) is compactly embedded in L ( K ), [Theorem 3,in Section 3 of [37]]. Hence, the composition of T with the inclusionoperator from H ( K ) ∩ H ( K ) ⊂ H ( K ) into L ( K ), is a compactoperator on L (Ω). In other words, the resolvent T of H at λ , iscompact. This implies two main consequences. Lemma 4.
The Hamiltonian H considered in this paper, with domainin H ( K ) , has a purely discrete spectrum of eigenvalues, each of finitemultiplicity, with no accumulation point other than + ∞ .Proof. This is a direct consequence of the coercivity and the compactembedding, as the resolvent of H becomes a compact operator. (cid:3) Theorem 5.
Given Ξ ∈ L ( K ) , there exists a unique Ψ ∈ H ( K ) suchthat (32) D (Φ , Ψ) = (Φ , Ξ) L ( K ) for all Φ ∈ H ( K ) . Proof.
Since the spectrum of H comprises only isolated eigenvaluesof finite multiplicity, we just have to verify that Ker( H ) = { } . If H e Ψ = 0 for some non-zero e Ψ ∈ H ( K ), then D ( e Ψ , e Ψ) = 0. Hence Q ( e Ψ) = Q † ( e Ψ) = 0. Thus, from Lemma 1 of [36], e Ψ = 0. This isclearly a contradiction. So indeed Ker( H ) = { } and by the FredholmAlternative, the stated result follows. (cid:3) Note that the regularity properties of elliptic operators ensure thatin the above theorem, Ψ ∈ H ( K ) ∩ H ( K ). Consequently Ψ is theunique solution of(33) H Ψ = Ξ in K Ψ ∈ H ( K ) ∩ H ( K ) . Given Θ ∈ H ( K ), set Ξ = − H Θ ∈ L ( K ) and Φ = Ψ + Θ ∈ H ( K ) ∩ H . Then H Φ = 0 in K Φ = Θ on ∂K.
U(N) GROUND STATE 13
That is, Φ is the unique solution to the homogeneous Dirichlet problemassociated with the region K for the Hamiltonian H .If in the above equations we impose the constraint ϕ A Ψ = 0 for all A = 1 , . . . , ( N −
1) and Ξ ∈ H ( K ), then(34) ϕ A Ξ = ϕ A H Ψ = Hϕ A Ψ = 0 . That is, Ξ also satisfies the constraint. Hence, in the search of theground state, there is no loss of generality when imposing the constrainton Ξ. For the potential we have, we know that(35) ϕ A V B ( X, z, z ) = 0for all indices A also. And for all real parameters ξ A (36) ξ A ϕ A X Ci = ξ A f ABC X iB and ξ A ϕ A z C = ξ A f ABC z B . Consider in R n the transformation(37) X Ci X Ci + ξ A ϕ A X Ci , z C z C + ξ A ϕ A z C , z C z C + ξ A ϕ A z C where ξ is an infinitesimal parameter. Then V B ( X Ci + ξ A ϕ A X Ci , z C + ξ A ϕ A z C , z C + ξ A ϕ A z C ) = V ( X, z, z ) + ξ A f ABC (cid:18) X iB ∂∂X Ci + z B ∂∂z C + z B ∂∂z C (cid:19) V B ( X, z, z )+ O ( | ξ | ) . Thus, the constraint generates transformations of coordinates on R n which preserve the value of V B . Furthermore Ψ = 0 on ∂K . Hence(38) Ψ( X + ξϕX, z + ξϕz, z + ξϕz, θ + ξϕθ ) = 0on ∂K , because the fields on the θ expansion are also evaluated at ∂K and by the conditions of equation they are zero. SinceΨ( X + ξϕX, z + ξϕz,z + ξϕz, θ + ξϕθ ) | ∂K =Ψ( X, z, z, θ ) | ∂K + ξϕ Ψ( X, z, z, θ ) | ∂K + O ( | ξ | ) , we then obtain(39) ϕ A Ψ( X, z, z, θ ) = 0 on ∂K.
Moreover, if Ψ ∈ H ( K ) ∩ H ( K ), we get ϕ A Ψ ∈ H ( K ). All thisensures the validity of the following. Lemma 6.
Let Θ ∈ H ( R n ) and ϕ A Θ = 0 for A = 1 , . . . , ( N − . Then the solution Ψ in the context of the weak problem (32) alsosatisfies ϕ A Ψ = 0 .Proof.
Set Ξ = − H Θ ∈ H ( K ) as above. Then(40) ϕ A Ξ = H ( ϕ A Ψ) = 0with ϕ A Ψ ∈ H ( K ) ∩ C ∞ ( K ). Hence, for each index A ,(41) Qϕ A Ψ = 0 and Q † ϕ A Ψ = 0
Thus, according to [36, Lemma 1], we conclude that ϕ A Ψ = 0 in K . (cid:3) from this lemma we gather that the homogeneous problem(42) H Φ = 0 in Kϕ A Φ = 0 in K Φ = Θ on ∂K has a unique solution Φ ∈ H ( K ) ∩ H ( K ) for any given Θ ∈ H ( K )satisfying the constraint ϕ A Θ = 0. As we have already noticed thereis no loss of generality by imposing the constraint on Θ.6.
Conclusions
In this work we fully examine the Hamiltonian of the regularized SU ( N ) Supermembrane in eleven dimensions on an unbounded region.The region is naturally connected with the theory, as it is defined bythe set K = { X mA : V B ( X ) < V } . These are the so-called valleysof the bosonic potential. Importantly, on these valleys, a) there aresub-regions extending to infinity where V B vanishes, b) the potentialis dominated by the fermionic sector and c) the full potential is un-bounded from below.It is well known that, despite of a), the bosonic Hamiltonian definedon the unrestricted space has discrete spectrum with finite multiplic-ity. This is in contrast to the well-known fact, shown in [35], thatthe spectrum of the supersymmetric Hamiltonian defined on the unre-stricted space is continuous and comprises the whole segment [0 , ∞ ).One of our main contributions presently is the fact that, remarkably,the supersymmetric Hamiltonian restricted to the valleys and for wave-functions vanishing on the boundary, has discrete spectrum with finitemultiplicity. Notably, and in agreement with the established result,the wavefunctions constructed in [35] for the proof of continuity of thespectrum do not (and must not) vanish on this boundary. Moreover,since these regions are preserved by the action of the SU ( N ) constraint,the formulation of the present SU ( N ) regularization restricted to thevalleys, is both natural and well posed.Our findings suggest several puzzling avenues of further enquiry. Dothe eigenvalues survive as embedded modes inside the continuous spec-trum for the unrestricted space? In the context of (Super) Yang Millstheory, if they survive and the slow mode regime captures relevantaspects of confinement, can these eigenvalues describe glueball bound-states and flux tubes connecting quarks when the theory is properlycompactified to D = 4? Equivalently, in the regularized SU ( N ) de-scription of the Supermembrane compactified to four dimensions, dothese eigenvalues capture aspects of QCD confinement?We also establish the existence and uniqueness of the state which isannihilated by the Hamiltonian on the physical subspace determined by U(N) GROUND STATE 15 the SU ( N ) constraint, satisfying a prescribed boundary condition on ∂K . The proof of this fact has three main ingredients. i) The volumeof K is finite, subject to constraints on d , the number of transversedirections in the light cone gauge. ii) The fermionic potential satisfies acrucial estimate (see Section 4), which renders a coercive Hamiltonian.iii) The embedding H ( K ) ⊂ L ( K ) is compact according to knowresults in [36]. All this is in agreement with the previous findings of[32, 33, 34].Although we consider explicitly the supersymmetric Hamiltonian ofthe SU ( N ) regularized D = 11 Supermembrane, the estimates wefound for the fermionic potential rely only on the linear dependence ofthe bosonic coordinates. We therefore expect that the present findingscould be extrapolated to the SU ( N ) Supermembrane on the other ad-missible dimensions, provided the restrictions of Lemma 1 are fulfilled.The SU ( N ) Matrix Models with N ≥ D ≥ D = 4. However in the context of SuperYang Mills theories, for SU ( N ) gauge groups with N = 2 ,
3, the space-time dimensions must be with D ≥ , N ) formulated in D = 5 , ,
11 only.By Domain Monotonicity, many of our claims extend to a formula-tion of the theory on any reasonably regular region inside the valleys(with suitable boundary and SU ( N ) constraints). For instance star-shaped domains for large enough V . Therefore an asymptotic anal-ysis of the groundstate of the regularized SU ( N ) supermembrane isperhaps possible, by considering a sequence of Dirichlet problems onregions taking V → ∞ . We have shown the existence and uniquenessof the solution of the homogeneous Dirichlet problem and hence the ex-istence and uniqueness of the state annhiliated by the supersymmetricHamiltonian. However, this is generically not annihilated by the su-persymmetric charges, as it is only for a particular boundary conditionthat the corresponding state may be annihilated.The unique state that we have determined by solving the Dirichletproblem, is the minimizer of the norm defined in terms of the super-symmetric charges, namely || ϕ || Q ≡ ( Qϕ, Qϕ ) L ( K ) + ( Q † ϕ, Q † ϕ ) L ( K ) , for a given boundary condition on ∂K . Perhaps it would be possibleto pursue further studies of the (Super) Yang Mills Theory in the slowmode regime, when it is confined to this tubular/star-shaped region. Acknowledgements
AR and MPGM were partially supported by Projects Fondecyt 1161192(Chile). AR was partially supported by MINEDUC-UA project codeANT1855.
Appendix A. Measure of the su (2) sectors in su ( N )In this first appendix we analyze the Lebesgue measure of diago-nal matrices that belong to the Cartan subalgebra of the su ( N ) alge-bra (describing the longitudinal directions along the valleys), when weconsider Matrix Models for different rank of the su ( N ) gauge groups.The potential V B ( X ) is invariant under conjugation by U ∈ su ( N ),(43) X m → U X m U − . We can diagonalize one of the X m matrices, say b X . If the eigenvaluesof b X are all different, then U is determined up to a diagonal matrixacting on the right to U . If there are equal eigenvalues, then U hasadditionally non-diagonal terms undetermined and we can fix some ofthe components of another matrix X . This procedure will be explainedin detail in due course.We characterize the measure b X of the matrices of the Cartan subal-gebra with different rank, in order to obtain inductively the expressionfor arbitrary N . Associated to the su (2) matrix b X (44) b X ≡ (cid:18) ia m − m − ia (cid:19) there is the following measure M ( b X ) = ρ d ρ dΩ, with ρ = a + mm .We can fix m = 0 by a selection of U . Hence the measure can beexpressed as M ( b X ) = a d | a | dΩ . For a su (3) matrix, there are three su (2) sectors(45) i α m − m − i α
00 0 0 + − i β m − m i β + − i γ m − m i γ = i ( α − β ) m m − m − i ( α + γ ) m − m − m i ( β + γ ) We may perform a linear change of variables from the original b X A coordinates to the new ones α, β, γ, m ij . Each su (2) sector contributesto the measure as in (44), we then have the following measure associatedto b X ,(46) M ( b X ) = α β γ d | α | d | β | d | γ | δ ( γ − α − β ) . U(N) GROUND STATE 17
We can shift α and β by the same amount λ while γ by − λ and(45) remains invariant. We can always fix this invariance by taking γ − α − β = 0, in agreement with the total number of degrees offreedom of a su (3) matrix. The measure can be expressed in termsof the diagonal components of the su (3) matrix, a ≡ ( α − β ), a ≡− ( α + γ ) = − (2 α + β ), and a ≡ ( β + γ ) . In fact, by defining a ij = a i − a j (47) a = α, a = β, a = α + β. The measure can be then re-expressed as(48) M ( b X ) = | a | | a | | a | d | a | d | a | . In general for su ( N ) there are N ( N − u (2) sectors. Each sector isdefined by two diagonal components, say b i , b j and the non-diagonalcomponent z ij . For example, for su (4) there are six sectors and themeasure can be expressed as(49) M ( b X ) = | a | | a | || a | | a | | a | | a | d | a | d | a | d | a | . The main point to write the measure in this form, is that all the factorsin the bracket will be cancelled from a contribution during the integra-tion procedure in the evaluation of Vol( K ). The above expression canbe generalized to su ( N ) in a straightforward way. Appendix B. Bounds on the measure
Vol( K )We now consider Vol( K ) in the region where all the eigenvalues of b X are different. We denote by N the region where all the differencessatisfy | a ij | > ǫ for all i = j . We find the bounds for three cases: su (2), su (3), su (4) and finally the general expression for su ( N ).B.1. The su (2) case. In this case a = − a , then a − a = 2 a > ǫ .The expression of the potential is(50) 12 V B = 4( a + b ) || z || − | b · z | + || z || − ( z · z )( z · z ) . It follows from (50) that | z | < V ǫ . Define(51) e V B = 8( a + | b | ) || z || − | b · z | . Since || z || − ( z · z )( z · z ) = 4[Re( z ) (Im( z ) − (Re( z ) · Im( z )) ] ≥ e V B ( X ) ≤ V B ( X )for all a, b, z. We decompose z = λb + z ⊥ , where b · z ⊥ = 0 . Then(53) 18 e V B = a | λ | + ( a + || b || ) || z ⊥ || . The set e K ≡ { X : e V B ( X ) < V } , for a and b fixed, is the interior of anellipsoid E described by the coordinates { Re( λ ) , Im( λ ) , Re( z ⊥ ) , Im( z ⊥ ) } . Consequently(54) Vol( e K ∩ N ) = e C Z e K ∩N d a ( d || b || ) (cid:2) a || b || d − ( a ( a + || b || ) d − ) − (cid:3) ≤ C Z e K ∩N d ρρ d − ( ρ d − ) − where ρ = a + || b || and d a d || b | = ρ d ρ d ϕ and we used || b || d − ≤ ρ d − . The factor ( a ( a + || b || ) d − ) − corresponds to the volume of E . In e K ∩ N , ρ > ǫ since 2 | a | > ǫ . Consequently, the above integral isconvergent, provided(55) 2( d − − ( d − > . We therefore conclude that Vol( e K ∩ N ) is finite, and hence Vol( K ∩ N )is also finite for d ≥ d parametrizes transversedimensions. Hence the measure of the valleys of the su (2) bosonicpotential are finite for D ≥
7. Recall that Supermembrane Theory isconsistently defined in 4 , , a in themeasure of the integral is cancelled by a factor a − arising from thevolume of E , associated to the coordinate | λ | . This cancellation occursalso for the su ( N ) potential, as we will see later on.B.2. The su (3) case. The explicit expression for the potential is(56) V B = 12 X m,n (1 , mn (1 , mn + (2 , mn (2 , mn + (3 , mn (3 , mn + X m,n (1 , mn (1 , mn + (2 , mn (2 , mn + (1 , mn (1 , mn where ( i, j ) mn denotes the i, j component of the matrix [ X m , X n ]. Thediagonal components ( i, i ) mn , i = 1 , ,
3, depend solely on the non-diagonal components of X m and X n . The contribution of the matrix b X to (56) is only to the non-diagonal terms and it is(57) 2 | a | || z || + 2 | a | || z || + 2 | a | || z || . There is no contribution of b X to the diagonal terms ( i, i ) in the expres-sion of the potential since b X is diagonal. In the set K ∩ N , || z ij || arebounded, in fact (56) implies(58) || z ij || < V ǫ , i, j = 1 , , . In the case of the su (2) algebra the sum of diagonal terms in (56)correspond to the term in the last two terms on the right in (50). U(N) GROUND STATE 19
The non-diagonal terms of V B can be re-arranged in terms of thethree u (2) sectors(59) V + V + V , where(60) V ij = | a ij | || z ij || + || b ij || || z ij || − | ( b ij ) · z ij | − i [( b ij ) · z ik ]( z ij · z jk ) + i [( b ij ) · z jk ]( z ij · z ik )+ i [( b ij ) · z ik ]( z ij · z jk ) − i [( b ij ) · z jk ]( z ij · z ik )+ || z ik || || z jk || − | z ik · z jk | . for b ij ≡ b i − b j , and i = j = k and i, j, k = 1 , ,
3. We define(61) e V B ≡ V + V + V − ( || z || || z || − | z · z | )( || z || || z || − | z · z | ) − ( || z || || z || − | z · z | ) . Then(62) e V B < V B for all a i , b j , z kl since the terms with brackets in (61) are positive. Wethen define(63) e K = { a i , b j , z kl : e V B < V } . On each u (2) sector we can shift the corresponding z ij , in order tosimplify the expression for V ij .For the three u (2) sectors,(64) z ij = λ i b ij || b ij || + z ⊥ ij , b ij · z ⊥ ij = 0 . Define(65) e z ij = z ⊥ ij + iρ ij { [ b ij · z ik ] z jk − [ b ij · z jk ] z ik } where ρ ij = | a ij | + || b ij || . Then,(66) e V ij = V ij − ( || z ik ) || || z jk || − | z ik · z jk | )= | a ij | | λ i | + ρ ij || e z ⊥ ij || − A ij where(67) A ij = 1 ρ ij || ( b ij · z ik ) z jk − ( b ij · z jk ) z ik || is bounded from above(68) A ij < (cid:18) V ǫ (cid:19) . Consequently, A ij + A ik + A jk is bounded by 12 (cid:0) V ǫ (cid:1) . The measure of e K ∩ N is bounded by the measure of the set b K ∩ N ,with(69) b K = ( a, b, λ, e z : b V B < V + 12 (cid:18) V ǫ (cid:19) ) where(70) b V B ≡ | a | λ + | a | ˆ λ + | a | ˆˆ λ ++ ρ || e z ⊥ || + + ρ || e z ⊥ || + + ρ || e z ⊥ || . For su ( N ), the factor 12 in (69) changes to 2 N ( N − su (2) case, given a ij , ρ ij ≡ | a ij | + || b ij || , the set of points satisfying b V B < b V = V + 12 (cid:0) V ǫ (cid:1) coincides with the interior of an ellipsoid E determined by radii(71) b V (cid:26) | a | , | a | , | a | , ρ , ρ , ρ (cid:27) . Its volume is then(72) Vol( E ) = Π i 3, satisfying r + r + r = 0. Observe that ρ ij = || r i − r j || . Define u ≡ r − r , and u ≡ r − r || r − r || . Then(75) ρ = || u || , ρ = || u ||·|| u || , ρ = || u − u || u || ||·|| u || . The integral (74) can then be expressed as an integral in u and u ,(76) M ( b K ∩ N ) = Z || u || d || u || dω || u || d || u || dω · ( || u || d − · || u || d − · || u − u || u || || d − ) − . U(N) GROUND STATE 21 As noted before, since we are integrating on N , each factor in thedenominator is bounded away from zero. The third power arises fromthe following expressions(77) d | a | d || b || → || u || d || u || dϕ d | a | d || b || → ρ dρ dϕ = || u || || u || d || u || dϕ . The term on the right hand side of (76) factorizes into two integrals,(78) I (3)1 = Z d || u |||| u || − d − and(79) I (3)2 = Z d || u |||| u || − ( d − || u − u || u || || − d − dω ω . So, in order to have a convergent integral, each factor must be finite,and we then require(80) 4( d − − > ⇒ d > ⇒ d ≥ , and(81) 3( d − − > ⇒ d > ⇒ d ≥ , from equations (78) and (79), respectively. If d ≥ 4, then the volumeof the valley for the su (3) algebra is finite. The restriction arising fromthe integration on || u || in (80) is stronger than (81), because of thefactor || u || . This also occurs for the su ( N ) case.B.3. The su (4) case. Following the same procedure as above, we ob-tain the same bound (58) for all the non-diagonal components of thematrices X m . We end up with the integral(82) M ( b K ∩ N ) = Z d | a | d | a | d | a | d || b || d || b || d || b || d Ω d Ω d Ω R R = ( ρ d − ρ d − ρ d − ρ d − ρ d − ρ d − ) − Set, as before, the variables r i for i = 1 , . . . , P i =1 r i = 0.Then(83) ρ ij = || r i − r j || = || ( r i − r ) + ( r − r j ) || . For(84) u = r − r , u = r − r || r − r || and u = r − r || r − r || it follows that,(85) ρ = || u || , ρ = || u || · || u || , ρ = || u || · || u || ,ρ = || u − u || u || || · || u || , ρ = || u − u || u || || · || u || ,ρ = || u − u || · || u || . Although the integral (69) can be performed without using the fol-lowing bound(86) ρ = || r − r || > ǫ, ρ − d − < ǫ − d − , for d > , this not change the restriction on integral dimensions d. We may thendismiss the factor ρ − d − since (69) is bounded by an integral whichfactorizes into M ( b K ∩ N ) = I · I for(87) I = Z d || u |||| u || ( || u || d − ) − and(88) I = ǫ − d − Z d || u || d || u || d Ω d Ω d Ω A with A = || u || − ( d − |||| u || − d − || u − u || u || || − d − || u − u || u || || − d − . If(89) 7( d − − > ⇒ d > ⇒ d ≥ d − − > ⇒ d > ⇒ d ≥ , the integral (69) is convergent. If d ≥ su (4) algebra is finite. There are no divergences arising for factorsgoing to zero, since we are working in the region N .B.4. Bounds in the su ( N ) case. We obtain the same bound (58)for all the non-diagonal components of the matrices X m . The result,concerning the finiteness of the measure, follows directly by dismissingall the terms involving ρ ij for all i ≥ j > i , because all of themare bounded by powers of ǫ . The integral representing the measureof K ∩ N is then bounded by an integral which factorizes into twointegrals. An integral on || u || , with positive powers ( N − N − 2) =2 N − d | a i | d || b i || → ρ i dρ i dϕ i with(92) ρ i → ( || u || i = 2 || u i || · || u || i > . This contributes with ( N − 1) to the exponent of || u || . Followed by(93) dρ i = || u || d || u i || U(N) GROUND STATE 23 for i > 2, which contributes with ( N − 2) to the exponent. Finally,( N − d − 2) arising from the measure factors || b − b i || d − . Thecontribution to the negative powers arises from the integrals on z ij , i < j and the further change of variables ρ ij → || u ij || . Since we areonly considering the pairs, 12 , . . . , N, , . . . , N factors we have apower − [( N − 1) + ( N − d − N − N − d − − ( N − d − − ( N − − ( N − > . That is(95) d > N − N − − . The term(96) 2( N − N − − N = 21 N = 3 < N ≥ . We thus recover the previous results for su (2) in (55), su (3) in (80), su (4) in (89) and obtain the general result. The integral is finite if d ≥ , for su ( N ) whenever N ≥ . The second integral associated to the measure of M N ( b K ∩ N ) isbounded by the integral(97) I = C Z d || u || . . . d || u N || d Ω . . . d Ω N B and, B = Π Ni =3 || u i || − ( d − || u i − u || u || || − d − which is convergent provided(98) 3( d − − > → d ≥ . This occurs for N = 3 and N = 4. We then conclude that, if d ≥ M N ( b K ∩ N ) is finite for the algebra su ( N ) whenever N ≥ 4. From the viewpoint of the Supermembrane Theory taking N to infinity, the restriction D ≥ Appendix C. Gauge transformations Given X ∈ u ( N ), we consider the gauge transformations(99) X ′ = U − XU, U ∈ SU ( N ) . Under (99), Tr X and Tr X † X remain invariant. In particular if X ∈ su ( N ), using the notation introduced in Section 3,(100) N Tr X † X = N Tr X † X − (Tr X )(Tr X † ) = X I [( b ij ) + 2 N | Z ij | ] , is also invariant under (99). In the case under consideration we have m = 1 , . . . , d matrices X m ∈ su ( N ). For each of them (100) remainsinvariant under (99).We consider the unbounded region, P m Tr X m † X m > C . This re-gion decomposes into subsets where at least for one m , say e m ,(101) Tr X e m † X e m > C d . Hence, from (100),(102) X I [( b e mij ) + 2 N | Z e mij | ] > N C d . We perform now a gauge transformation which diagonalize X e m . Then,after (99), we have(103) X I ( b ′ e mij ) > N C d . In order to simplify the notation, from here on we do not use theprime for the new components. Since there are N ( N − ( i, j ) sectors andtaking into consideration (103), for at least one sector ( i, j ), say (1 , N ),we must have,(104) ( b e m N ) > C d ( N − . Note that (104) implies(105) | b e m i | + | b e miN | ≥ | b e m N | > (cid:20) C d ( N − (cid:21) , i = 2 , . . . , N − . Therefore at least ( N − 1) pairs ( i, j ) ∈ I satisfy ( b e mij ) > C d ( N − . Appendix D. The measure of K In the previous appendices we considered the measure of the set K ∩ N ,(106) K ∩ N ≡ { x ∈ k : | a i − a j | > ǫ for all i, j, i < j } . Now we invoke these results and show that also the Lebesgue measureof K is finite. U(N) GROUND STATE 25 D.1. Case su (2) . Consider X m = (cid:18) ib m z m − z m − ib m (cid:19) , for m = 1 , . . . , d. We do not distinguish here b X from the other u (2) matrices. Under thegauge transformation (43), the traces Tr X m X † m are invariant for each m . Hence,(107) P m ≡ 12 Tr X m X m † = ( b m ) + z m z m is invariant.We decompose K into a finite number of subsets, determined bywhether P m satisfies the condition P m ≤ ǫ or the condition P m > ǫ ,for m = 1 , . . . , d. The subset P m ≤ ǫ for all m = 1 , . . . , d has finitemeasure, so we are left with other subsets for which at least for one m ,say e m , P b m > ǫ. We have(108) ( b e m ) + z e m z e m > ǫ . We now perform a gauge transformation such that X e m becomes di-agonal. The new b e m which we denote it with the same letter satisfies(109) ( b e m ) > ǫ . The two eigenvalues b e m and − b e m are such that(110) | b e m − ( − ) b e m | = 2 | b e m | > | ǫ | . We may then apply the argument in Appendix B.1. We conclude thatif d ≥ 5, Vol( K ) is finite. Here b e m plays the role of the component a in the notation of Appendix B.D.2. Case su (3) . We consider now the case N = 3. Following appen-dix C, there are at least two pairs, say (1 , 3) and (2 , 3) which satisfy(111) ( b e m ) > C d and ( b e m ) > C d . Using (56), we obtain || z || < dV C and || z || < dV C . The other pairmay satisfy the same inequality or not. In the first case(112) ( b e mij ) > C d for all ( i, j ) ∈ I. In the second case(113) ( b e m ) ≤ C d . In both cases z e mij = 0, for all ( i, j ) ∈ I . In the second case we may have(114) ( b m ) + | z m | ≤ C d for all m ∈ M . That is, all the (1 , 2) sector is bounded, the (1 , 2) sectorhas then finite measure, or for some b m (115) h = ( b b m ) + | z b m | > C d where h ≥ U = A C − C + A 00 0 1 . Here AA + CC = 1, hence det( U ) = 1. Then(117) b ′ b m = b b m · ( AA − CC ) + 2 iz b m CA − iz b m CA. We choose C = Au , with u = − i ( b b m ) ∓ h z b m , which yields(118) b ′ b m = h, z ′ b m = 0 . Under this gauge transformation(119) (cid:18) z ′ e m z ′ e m (cid:19) = (cid:18) A C − C − A (cid:19) (cid:18) z e m z e m (cid:19) , hence z ′ e m = z ′ e m = 0 . We then have, from (100) and (102),(120) ( b ′ e m ) + | z ′ e m | + ( b ′ e m ) + ( b ′ e m ) > C d . But(121) ( b ′ e m ) + | z ′ e m | = ( b e m ) ≤ C d is invariant under the gauge transformation generated by (116). Wethus have, from (120) and (121),(122) ( b ′ e m ) + ( b ′ e m ) > C d . This implies (from the argument in (104), (105) and (121)) that(123) ( b ′ e m ) > C d , ( b ′ e m ) > C d and from (111) and the matrix U we are considering(124) || z ′ || + || z ′ || < dV C → || z ′ || < V ǫ , for large enough ǫ , proportional to C . Consequently, in su (3), theunbounded region P m Tr X m † X m > C is the union of subsets. Ineach one of them there exists two unbounded sectors ( i, j ), which by U(N) GROUND STATE 27 a gauge transformation satisfy ( b e mij ) > C d , z e mij = 0 , ( i, j ) ∈ I for some e m ∈ M . The third ( k, l ) sector either satisfies(125) [( b mkl ) + | z mkl | ] ≤ C d for all m ∈ M ( i.e. it is bounded) or there exists b m such that(126) ( b b mkl ) > C d , z b mkl = 0 . In all cases, we have(127) || z ij || < V ǫ for all ( i, j ) ∈ I , and large enough ǫ .We now consider the expression of the potential V B . We do notdistinguish any diagonal matrix as in Appendix B.2. The quadraticterms on b e mij corresponding to the u (2) sector ( i, j ), are(128) v ij = || b ij || || z ij || − ( b ij · z ij ) . More explicitly we have(129) v ij = | b e mij | || z ij || + X M \ e m | b mij | X M \ e m | z nij | − X M \ e m | b mij z mij | and we obtain an analogue expression to the ones in Appendix B.2.Although there are linear terms on the diagonal components in thepotential, the bound (127) allows to show that || z ij || < V | b e mij | , the wholeargument of finite volume follows identical steps. When a u (2) sectoris bounded, it is always possible to eliminate it from the expression of V B and the calculations for the unbounded sectors, restricted by thebounded one, are as in Appendix B.2. The measure of K for the su (3)algebra is finite subject to the same conditions as before.D.3. Case su ( N ) . Let Y mN ∈ u ( N ) , m ∈ M . Under a gauge transfor-mation(130) Y mN → U − Y mN U, m ∈ M, U ∈ su ( N ) , the traces Tr Y mM and Tr Y m † N Y mN remain invariant. Consequently,(131) P m ≡ X I ( | b mij | + 2 N | z mij | ) , m ∈ M, remains also invariant under (130).We decompose K into a finite number of subsets. the subset forwhich(132) X m P m ≤ C for arbitrary C > 0, has all variables b mij , z mij bounded. On the comple-ment, P m P m > C , there always exists e m for which P e m > C . That is, in all subsets of the complement at least for one index the conditionis satisfied. Y e mN can be diagonalized by a gauge transformation (130).From (131) we obtain(133) X I | b e mij | > C , z e mij = 0 . There are at least ( N − b e mij satisfying(134) | b e mij | > C h , h = 12 N ( N − . Then(135) || z ij || < hV C . In fact, at least one, say b e m , must satisfy | b e m | > C h . Then(136) | b e m i | + | b e m i | > | b e m | > (cid:18) C h (cid:19) / , for all i = 3 , . . . N. This implies (134).Set(137) J ≡ { ( i, j ) : i < j and b e mij satisfy (134) } , which contains N − j = N (138) J = { ( i, N ) : i = 1 , . . . , N − } , so its complement becomes(139) J = { ( i, j ) : i < j, i, j ∈ [1 , . . . , N − } . In this case,(140) U = (cid:18) U 00 1 (cid:19) , U ∈ su ( N − . Under this gauge transformation b e mNN and P N − i | z e miN | remain invariantfor each m . Hence, defining Y mN − ∈ u ( N − N th row and column of Y mN , the traces Tr Y mN − and Tr Y m † N − Y mN − are invariant and so is(141) Q m = X J ( | b mij | + 2( N − | z mij | ) . Moreover, the action of U preserves the condition (134), which inthis case is(142) | b e miN | > C h , z e miN = 0 , || z iN || < hV C . We have then reduced the u ( N ) case to the u ( N − 1) case, satisfying(142). Furthermore, since in the case of u (3) we have shown that each u (2) sector is either bounded or there exists an index e m for each sector U(N) GROUND STATE 29 ( i, j ) satisfying (127), we conclude that the statement of the followinglemma is valid. Lemma 7. For m ∈ M , let Y m ∈ u ( N ) . Let C > be constant. Then,each u (2) sector, b mij , z mij , is either bounded or there exists an index e m ,depending on the sector, for which (143) | b e mij | > C , z e mij = 0 . We now consider the Lebesgue measure of K for the potential, valuedon a su ( N ) algebra. The bounded u (2) sectors are dismissed fromthe potential leaving only those which are unbounded. For each oneof these, there is only one quadratic term on b e mij as in (129) in thepotential, although there is also linear terms on it, we obtain(144) || z ij || < V | b e mij | < V ǫ for large enough ǫ . We then reduce the evaluation of the subsets of K to the particular cases of section B. 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Maxwell Institute for Mathematical Sciences and Department ofMathematics Heriot-Watt University, Edinburgh, EH14 4AS, UnitedKingdom. Email address : [email protected] , Departamento de F´ısica, Universidad de Antofagasta, Aptdo 02800,Chile. Email address ::