On the Spectrum of PP-Wave Matrix Theory

Home , Perturbation theory

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1. Introduction 1

2. The Model and its Quantization 2

3. Supersymmetry Algebra and Structure of the Spectrum 4

4. The Perturbative Energy Spectrum and Protected States 8

5. Conclusions and Outlook 12

A. The Supersymmetry Algebra 13

B. Details of the Perturbative Calculation 14

1. Introduction

The precise microscopic degrees of freedom of M-theory remain elusive even six years after its discovery [1]. The most promising candidate for such a description today is given by the large N limit of matrix theory [2], the maximally supersymmetric U(N) gauge quantum mechanics [3, 4] which is intimately connected to the quantum supermembrane [5]. The study of this seemingly simple model has been plagued, however, by its nonlinearity and the existence of ﬂat directions in the potential leading to a continuous spectrum [6]. Recently Berenstein, Maldacena and Nastase [7] realized that for eleven dimensional supergravity on the maximally supersymmetric parallel-plane (pp)-wave background [8]

9 3 9 2 + i 2 µ2 a 2 µ2 a′ 2 + 2 ds = 2 dx dx− + (dx ) (x ) + (x ) (dx ) − − 9 36 Xi=1 Xa=1 aX′=4 F123+ = µ (1.1) the corresponding matrix theory is - opposed to its AdS S7 and AdS S4 cousins 4 × 7 × [9] - rather simple. In this model the ﬂat background matrix model is augmented by bosonic and fermionic mass terms with a scale set by µ along with a bosonic cubic interaction in the SO(3) sector. As observed in [7] the mass terms remove the ﬂat directions of the usual matrix theory potential and render its spectrum discrete.

–1– In fact the introduction of the mass parameter µ into the model opens up a new perturbative window of pp-wave matrix theory for µ 1 which we shall study in ≫ this paper. First steps in this direction have been undertaken in [10]. Related work on the pp-wave matrix theory and supermembrane may be found in [11]. We begin our analysis after a careful statement of the model and its quantization with the supersymmetry algebra, which due to the non-rigidness of the associated supersymmetry variations displays some unusual features. The supersymmetry algebra groups the energy eigenstates into multiplets, whose members do not have degenerate energy eigenvalues any more, but differ by fixed amounts of energy in great similarity to the representation of supersymmetry in AdS spaces1. The Hamiltonian naturally splits into a free and an interacting piece in the limit µ 1, of which the ≫ free piece is given by a supersymmetric oscillator system with vanishing groundstate energy protected from perturbative corrections. We then go on to study the leading order energy shifts of the first excited states in perturbation theory and find some surprises. In particular we uncover a multiplet which does not receive any perturbative corrections to its energy eigenvalues in leading order perturbation theory. We argue that this result holds true to all orders. Motivated by additional perturbative evidence, we are led to conjecture the existence of an infinite series of protected states in the full pp-wave matrix model. Finally we end with some concluding remarks.

2. The Model and its Quantization

Our conventions are as follows: Xi and θα denote Hermitian N N matrices, rs rs × i = 1,..., 9 are the transverse vector indices which split into a = 1, 2, 3 and a′ = 4,..., 9. Moreover for the SO(9) Majorana spinors we work with a charge conjugation i ijkl ij ijk matrix equaling unity, the Dirac matrices γαβ,γαβ are symmetric and γαβ,γαβ are antisymmetric running over the spinor indices α, β =1,..., 16. It is useful to perform a rescaling of t τ/(2R) of the time variable of the matrix model proposed in [7] → where R denotes the radius of the compactiﬁed direction in the DLCQ picture. With the help of this rescaling all parameters of the matrix quantum mechanics are cast into the single mass parameter m = µ/(2R). Then the pp-wave matrix model of [7] takes the simple form

S = Sﬂat + SM (2.1) where

1 i 2 1 i j 2 i Sﬂat = dτ Tr 2 (DX ) iθDθ + 4 [X ,X ] + θγi [X , θ] Z h − i ′ 1 m 2 a 2 1 m 2 a 2 m m a b c SM = dτ Tr 2 ( 3 ) (X ) 2 ( 6 ) (X ) + 4 iθγ123θ + 3 iǫabc X X X Z h− − i 1For a recent review see e.g. [12] and references therein.

–2– and the covariant derivative is given by D = ∂ i[ω, ]. It is invariant under O τ O − O the 16+16 linearly and non-linearly realized supersymmetries δXi = 2 θγiǫ(τ) ′ i 1 i j m a m a ′ δθ = iDX γi + 2 [X ,X ] γij + 3 iX γaγ123 6 iX γa γ123 ǫ(τ)+ η(τ) h − i δω = 2 θǫ(τ) (2.2) with m m γ123 τ γ123 τ ǫ(τ)= e− 12 ǫ0 η(τ)= e 4 η0 (2.3) Note the non-rigid character of the supersymmetry transformations: The supersymmetry parameters depend explicitly on time. This is the reason why the supercharge will be shown to not commute with the Hamiltonian in the sequel. The cleanest way to derive this model is to start from the supermembrane action in AdS S7 4 × and AdS S4 backgrounds given in [9] and consider the pp-wave limit of the su- 7 × perspace geometry along with the standard κ gauge fixing condition (Γ+θ = 0) for the fermions. The resulting membrane model may then be discretized in the usual fashion [5] by approximating the group of area preserving diffeomorphisms by U(N) in the limit N . The outcome of this analysis is the model (2.1). This derivation →∞ is spelled out in detail in [10]. It is straightforward to go to a Hamiltonian description of the system. We choose the gauge ω = 0 and find that the resulting Hamiltonian may be split into a free and an interacting piece

H = H0 + HINT (2.4) where

′ 1 i 2 1 m 2 a 2 1 m 2 a 2 m H0 = Tr 2 (P ) + 2 ( 3 ) (X ) + 2 ( 6 ) (X ) 4 iθγ123θ h − i m a b c 1 i j 2 i HINT = Tr 3 iǫabc X X X 4 [X ,X ] θγi[X , θ] h− − − i to be augmented by the gauge constraint = [P i,Xi] i θ, θ =0 . (2.5) G − { } As we shall show for m 1 the interacting piece of the Hamiltonian H is sup- ≫ INT pressed and can be treated perturbatively. Let us now turn to the quantization of the pp-wave matrix theory. The canonical (anti)-commutation relations for the matrix operators are given by [P i ,Xj ]= i δij δ δ θα , θβ = 1 δαβ δ δ (2.6) rs tu − st ru { rs tu} 2 st ru where the factor of 1/2 for the fermions arises from the Dirac procedure of treating α α the constraint Pθ + iθ = 0 properly. In view of H0 in (2.4) it is natural to introduce the creation and annihilation operators in the bosonic sector as

′ ′ ′ aa = 3 (P a i m Xa) ba = 3 (P a i m Xa ) (2.7) q 2m − 3 q m − 6

–3– reﬂecting the 3+6 split of the masses. They obey the standard commutation relations

′ a b ab a′ b a′b′ [ars, atu† ]= δ δst δru [brs, btu† ]= δ δst δru . (2.8)

Using these relations the bosonic part of the free Hamiltonian H0 takes the form

′ ′ B m a a m a a 2 H0 = Tr 3 a† a + 6 b† b + m N . (2.9) h i In the fermionic sector we complexify the real spinor matrices θrs via

1 θ± = Π± θ where Π± = (½ iγ ) (2.10) rs rs 2 ± 123 which yields the anticommutation relations

+ α β 1 + + α + β α β θ , θ− = (Π ) δ δ θ , θ =0= θ− , θ− . (2.11) { rs tu } 2 αβ st ru { rs tu } { rs tu }

Note the chirality property of the complexiﬁed fermions (iγ123) θrs± = θrs± as well as 2 + + ± (Π±) = Π± and Π Π− = 0. As θ = θ + θ− the fermionic term in H0 is now given by F m m + α α 2 H = i Tr [ θγ θ ]= Tr[ θ θ− ] m N (2.12) 0 − 4 123 2 − canceling precisely the zero point energy of the bosonic sector. The zero-energy groundstate of the resulting free Hamiltonian

′ ′ m a a m a a m + α α H0 = Tr 3 a† a + 6 b† b + 2 θ θ− (2.13) h i is denoted by 0 and is annihilated by | i a a′ α a 0 =0 b 0 =0 θ− 0 =0 . (2.14) rs | i rs | i rs | i Physical states are required to be gauge invariant due to the gauge constraint (2.5). + They are given by traces over words in the creation operators a†, b† and θ , viz.

a a′ + α b b′ + β Tr[...a† ...b† ...θ ...] ... Tr[...a† ...b† ...θ ...] 0 . (2.15) | i In this paper we shall be interested in the spectrum of the full U(N) Hamiltonian

H = H0 + HINT. Clearly then the problem factorizes into the trivial free U(1) sector a a′ + α spanned by the excitation operators of wordlength one, Tr[a† ], Tr[b† ], Tr[θ ], and a complicated interacting SU(N) sector spanned by excitation operators of wordlength two and larger.

3. Supersymmetry Algebra and Structure of the Spectrum

The derivation of the supersymmetry algebra is straightforward. One has two supercharges Qα and qα associated with the non-linearly and linearly realized supersymmetries of (2.2). Their form follows from the operator relations

m i i γ123τ δX = 2i[Q ǫ(τ)+ q η(τ),X ] ǫ(τ)= e− 12 ǫ0 m γ123τ δθα = 2i[Q ǫ(τ)+ q η(τ), θα] η(τ)= e 4 η0

–4– One then deduces the two supercharges

′ i i i j m a m a ′ γ Qα = Tr [P γi 2 [X ,X ] γij + 3 X γaγ123 + 6 X γa γ123 ] α θγ (3.1) h − i qα = Tr [ θα ] . (3.2)

Note that qα only acts in the free U(1) sector of the model. We relegate the explicit evaluation of the supersymmetry algebra into appendix A. One ﬁnds in the interacting SU(N) sector of the model

′ ′ m ab m a b i Q , Q = δ H L (γ γ ) + L (γ ′ ′ γ ) + iTr(X )(γ ) { α β} αβ − 6 ab 123 αβ 12 a b 123 αβ G i αβ mi [H, Q ] = (Qγ ) + Tr(θ ) (3.3) α 12 123 α αG Compared to the Minkowski background superalgebra we thus see the emergence of ′ ′ additional terms coupling to the angular momentum operators Lab and La b in the anti-commutator of two supercharges, given by

Lij = Tr( P i Xj P j Xi + i θγijθ) (3.4) − 2 and obeying [Lij, Lkl]= i ( δjk Lil + δil Ljk δik Ljl δjl Lik ). Despite the appear- − − − ance of these angular momentum operators the old argument for the zero-energy groundstate still goes through: a maximally supersymmetric state (being annihilated by all the Qα) will have zero energy and be a SO(3) and SO(6) singlet. Hence the vanishing energy of the groundstate 0 for m is protected from pertur- | i → ∞ bative corrections and constitutes the unique groundstate of the interacting model. Moreover all excitations will have strictly positive energy. We see in (3.3) that the supercharges do not commute with the Hamiltonian, which simply states that superpartners do not have the same mass in this model. This effect is induced by the time dependent supersymmetry parameter, and the m coefficient 12 can be easily inferred by computing the difference of bosonic and fermionic masses m m = m = m m . This phenomenon is analogous to the 3 − 4 12 4 − 6 situation for representations of supersymmetry in AdS spaces. The remaining (anti)-commutators read

q , q = N δ { α β} 2 αβ 2 m + m + Qα, qβ = 3 Tr[(a/ Π )αβ +(a/ † Π−)αβ]+ 3 Tr[(b/ Π )αβ +(b/ † Π−)αβ] { } q q [H, q ] = mi (qγ ) (3.5) α − 4 123 α where the bosonic matrix ladder operators of (2.7) appear in the second line. For the study of the spectrum of the pp-wave matrix model it is useful to perform a chirality split of the dynamical supercharges Qα according to

1 Q± = Π± Q where Π± = (½ iγ ) . (3.6) 2 ± 123

–5– The supersymmetry algebra then takes the compact form

′ ′ + + m ab + m a b + Q , Q− = (Π ) H + i L (Π γ ) i L (Π γ ′ ′ ) { α β } αβ 6 ab αβ − 12 a b αβ Q±, Q± = 0 { α β } m [H, Q±] = Q± (3.7) α ∓ 12 α where we have dropped the terms proportional to the gauge constraints for trans- G parency, as they do not aﬀect gauge invariant states. It is instructive to spell out the form of Q± explicitly in terms of oscillators

′ m a ′ + 2m a i i j Q− = 3 Tr [ b γa θ ]+ 3 Tr [ a† γa θ−] 2 Tr ( [X ,X ] γij θ− ) q q − ′ + 2m a + m a ′ i i j + Q = 3 Tr [ a γa θ ]+ 3 Tr [ b† γa θ−] 2 Tr ( [X ,X ] γij θ ) (3.8) q q − i ( ) a ( ) a′ where X is given in terms of the oscillators a † and b † through (2.7) and we + have (Q )† = Q−. Note that the free theory supersymmetry algebra generated by

Q0± takes the same form as (3.7), with H replaced by H0 and Q0± given by dropping the commutator terms in (3.8). Clearly now from (3.7) one observes that Q+ lowers and Q− raises the energy eigenvalue of a state by m/12. As there are 8 raising and 8 lowering operators a generic long multiplet will contain 256 states spread over 9 “floors” of equal energies and spanning an energy range from its smallest value E to 2m E + 3 . The simplest long multiplet is built upon a SO(3) and SO(6) singlet on the ”ground floor” and the entire multiplet has 256 states in total. It is straightforward to find out how the states of such a multiplet are grouped into irreducible representations of SO(3) SO(6) on each floor. The result reads × Floor SO(3) SO(6) reps × 8 (1,1) 7 (2, 4) 6 (1, 10) (3, 6) 5 (2, 20) (4,4) (3.9) 4 (1, 20′) (3,15) (5,1) 3 (2,20) (4, 4) 2 (1,10) (3,6) 1 (2,4) 0 (1,1) The energy differences within one multiplet are fixed, however the lowest energy eigenvalue E of a generic multiplet may only be computed approximately in perturbation theory. Let us now study the first excited states of the interacting SU(N) sector, i.e. excitations of wordlength two, which are decomposed into irreducible representations

–6– as follows State SO(3) SO(6) rep Energy × a′ a′ m a′a′ = Tr[b† b† ] 0 (1, 1) 3 | i (a′ b′) | i m a′b′ = Tr[b† b† ] 0 (1, 20′) 3 | i a b′ | i m aa′ B = Tr[a† b† ] 0 (3, 6) 2 | i + aa′ +| i m aa′ F = Tr[θ γ θ ] 0 (3, 6) 2 | i + a′b′c′ +| i m a′b′c′ = Tr[θ γ θ ] 0 (1, 10) 2 | i a a | i 2m (3.10) aa = Tr[a† a† ] 0 (1, 1) 3 | i (a b) | i 2m ab = Tr[a† a† ] 0 (5, 1) | i | i 3

+ 5m a′a′; α = Tr[b/ †θα ] 0 (2, 4) 12 | i (a′ b′|) i+ 5m a′b′; α = Tr[γ b† θα ] 0 (2, 20) 12 | i + | i 7m aa; α = Tr[a/ †θα ] 0 (2, 4) 12 | i (a b)| +i 7m ab; α = Tr[γ a† θ ] 0 (4, 4) | i α | i 12 where the bifermion states are restricted to an odd number of SO(6) vector indices, a consequence of the chirality property θ+ = Π+θ+. In our notation (ij) refers to totally symmetrized indices without the trace part. Note that the states aa′ and | iB aa′ are degenerate in mass and SO(3) SO(6) representation and could potentially | iF × mix. Let us see how these states ﬁt into multiplets of the free superalgebra generated by Q0±. As the free supercharges Q0± preserve the wordlength it comes as no surprise, that the above states may be grouped into two multiplets. Also due to

+ + Q a′a′ =0 Q a′b′ =0 0 | i 0 | i Q− aa =0 Q− ab = 0 (3.11) 0 | i 0 | i the multiplets begin with the states of energy m/3 and end with the states of energy 2m/3 - they are short multiplets consisting of 5 ﬂoors. The relevant double step ladder operators connecting ﬂoors of bosonic states are

aa′ a′b′c′ 2 floors up: Q0−γ Q0− Q0−γ Q0− + aa′ + + a′b′c′ + 2 floors down: Q0 γ Q0 Q0 γ Q0 (3.12) which follow again from the chirality property of Q0±. Starting with the lightest state of a′a′ one finds the multiplet ”A” | i − aa′ − + aa′ + Q0 γ Q0 Q0 γ Q0 a′a′ ( aa′ F + √2 aa′ B ) ab | i − −→a′b′c′ − | i | i + ←−a′b′c′ + | i Q0 γ Q0 Q0 γ Q0 a′a′ 0 ab (3.13) | i −→ ←− | i consisting of the 24 bosonic states a′a′ , ( aa′ + √2 aa′ ), ab A. Similarly {| i | iF | iB | i} starting with a′b′ as the lightest state the multiplet “B” is obtained | i − aa′ − + aa′ + Q0 γ Q0 Q0 γ Q0 a′b′ ( aa′ 2 √2 aa′ ) aa | i −→ | iF − | iB ←− | i

–7– ′ ′ ′ ′ ′ ′ − a b c − + a b c + Q0 γ Q0 Q0 γ Q0 a′b′ a′b′c′ aa (3.14) | i −→ | i ←− | i made out of the 49 bosonic states a′b′ , ( aa′ 2 √2 aa′ ), a′b′c′ , aa B. We {| i | iF − | iB | i | i} indeed observe a mixing between the two (3, 6) states aa′ and aa′ , which is | iB | iF orthogonal due to the norms

2 2 aa′ bb′ =4 δ δ ′ ′ N aa′ bb′ = δ δ ′ ′ N aa′ bb′ =0 . (3.15) F h | iF ab a b Bh | iB ab a b F h | iB + For the fermionic states it is obvious from Q− a′a′ = a′a′; α and Q aa = aa; α 0 | i | i 0 | i | i that the fermionic states a′a′; α and ab; α belong to the multiplet “A”, whereas | i | i a′b′; α and aa; α belong to “B”. So the level two states make two irreducible | i | i supermultiplets,

A: (1, 1)+(2, 4)+(3, 6)+(4, 4)+(5, 1)

B: (1, 20′)+(2, 20) + [(1, 10) + (3, 6)] + (2, 4)+(1, 1) . (3.16)

We note that both of them can be part of the simplest long multiplet presented in (3.9). In the next section we shall study how the energies of these multiplets get corrected in perturbation theory.

4. The Perturbative Energy Spectrum and Protected States

The supersymmetry algebra derived in the last section implies that the energy of the maximally supersymmetric ground state, which is annihilated by all supercharges, must be exactly zero. Before we embark on the calculation of energy shifts for the excited states of (3.10) let us verify this in leading order perturbation theory 2. The perturbative corrections to the spectrum are organized in an expansion in 1/m2. To consistently work out the leading correction of the groundstate energy it is then necessary to work up to second order in quantum mechanical perturbation theory and to evaluate the expression 1 ∆E0 (1/m2) = 0 HINT X4 0 + 0 HINT X3+Xθ2 HINT X3+Xθ2 0 (4.1) |O h | | | i h | | E H | | i 0 − 0 where the quartic interaction term contributes in ﬁrst order perturbation theory whereas the cubic and the Yukawa term contribute at second order perturbation theory. The ﬁrst term of the right-hand-side of (4.1) is given by the expectation value

∆E1 = 1 0 Tr[Xi,Xj]2 0 = 1 0 Tr[Xi Xj Xi Xj (Xi)2 (Xj)2] 0 (4.2) 0 − 4 h | | i − 2 h | − | i It is useful to maintain a uniﬁed language for the bosonic ladder operators by introducing the objects i i i X = i (˜a† a˜ ) (4.3) − − 2This check was also performed in [10].

–8– which obey 1 ab i j ij ij 3 2 δ 0 [˜a , a˜† ]= M δ δ M = ′ ′ (4.4) rs tu st ru m 0 δa b With the help of these deﬁnitions one straightforwardly evaluates

0 Tr[Xi Xj Xi Xj] 0 = 2N 3 TrM 2 + N (TrM)2 h | | i 0 Tr[(Xi)2 (Xj)2] 0 = N 3 ( TrM 2 + (TrM)2 )+ N TrM 2 (4.5) h | | i yielding the ﬁrst contribution to the energy shift

11 34 ∆E1 = 1 0 Tr[Xi,Xj]2 0 = · N (N 2 1) (4.6) 0 − 4 h | | i 4 m2 − The contributions to (1/m2) at second order perturbation theory come from the O cubic and the Yukawa terms of HINT. For the ﬁrst contribution one has

2 m2 a b c 1 a b c ∆E = 0 Tr[e a˜ a˜ a˜ ] Tr[e a˜† a˜† a˜† ] 0 (4.7) 0 9 h | abc m abc | i as here only a pure bosonic level 3 state is excited the free Hamiltonian in the denominator has been replaced by 3 m . Upon contracting this expression reduces · 3 to 33 ∆E2 = N(N 2 1) (4.8) 0 −4 m2 − Turning to the ﬁnal contribution from the Yukawa coupling we have

3 a 1 + a + ∆E0 = 0 Tr(θ−γa [ã , θ−]) m m Tr( θ γa [ã† , θ ]) 0 −h | 3 + 2 | i ′ 1 ′ ′ a + ′ a + 0 Tr(θ−γa [ã , θ−]) m m Tr( θ γa [ã† , θ ]) 0 (4.9) −h | 6 + 2 | i Note the two different mass channels appearing for the inverse free Hamiltonian in the above. Now one computes

i + i + 2 i j + 0 Tr(θ−γ [ã , θ−])Tr( θ γ [ã† , θ ]) 0 = N (N 1) tr(γ Π−γ Π ) M (4.10) h | i i | i − ij 1 where Π± = 2 (½ iγ123) are the projectors appearing in (2.10). We note that a a + ± a′ a′ + + tr(γ Π−γ Π )=0 and tr(γ Π−γ Π ) = 6trΠ which yields the final result

8 33 ∆E3 = · N (N 2 1) (4.11) 0 − m2 − Summing up the three contributions (4.6),(4.8) and (4.11) we indeed ﬁnd

1 2 3 ∆E0 = ∆E0 + ∆E0 + ∆E0 = 0 (4.12) the vanishing shift of the groundstate energy in leading order perturbation theory.

–9– The computation of the energy shifts for the ﬁrst excited states of (3.10) goes along the same lines, but is technically more involved. We report on the details of this computation in appendix B and simply state the complete result here. By virtue of the supersymmetry algebra (3.7) it is clear that states within one multiplet should receive the same perturbative correction to their energy eigenvalues. This is indeed what one ﬁnds. Taking care of the normalization of states the leading shift in energy for a generic state φ is given by | i 1 1 ∆E φ (1/m2)= φ HINT X4 + HINT X3+Xθ2 HINT X3+Xθ2 φ . | i |O φ φ h | | | E0 H0 | | i h | i − (4.13) For the states of our multiplet “A” one obtains 108 1 ∆E φ (1/m2)= (N ) | i |O m2 − N

for φ h a′a′ , haa′ ( aa′ F + √2 aa′ B ), hab ab (4.14) | i ∈ n | i | i | i | i o introducing suitable polarization tensors h, haa′ and hab. Most interestingly, however, the energy shift for the members of our multiplet “B” precisely cancels!

∆E φ (1/m2)=0 | i |O

′ ′ ˜ ′ ′ ′ ′ for φ ha b a′b′ , haa ( aa′ F 2 √2 aa′ B ), ga b c a′b′c′ , h aa (4.15) | i ∈ n | i | i − | i | i | i o We shall argue that this remains true to all orders in perturbation theory. The crucial input from the representation theory of Lie superalgebras here is that if a multiplet is short its energy is quantized by the symmetry algebra. One simple way of seeing this is to use the fact that the ground-ﬂoor state must be annihilated by a product of less than nine supercharges, since short multiplets do not span all of the 9

floors. One can calculate the norm of a state of the form ( Q− ) Λ using only the i αi | i superalgebra and obtain a set of linear equations involvingQE and the Dynkin labels of SO(3) SO(6) of the ground-floor state Λ . In fact the representation theory of × | i Lie superalgebras is known to some extent, original classifications and first important results are due to Kac [13]. A more detailed discussion of the representation theory of the M-theory pp-wave superalgebra will be presented in a separate publication [14]. Based on this insight, our calculation implies that the multiplet ”A” should combine with other short multiplets of the free theory to make a long multiplet in the interacting theory for which there is no such quantization rule from the symmetry algebra. The multiplet ”B” on the other hand should stay short. Because there are no lighter states in the free theory the ground-floor states (1, 1) and (1, 20′) must remain as the ground floor of the two multiplets also with interactions. For the multiplet ”A” we see from (3.9) that the first missing block is (1, 10) on the second floor with

– 10 – m [a′ b′ c′] E = 2 . In fact this state is provided from the level 3 spectrum: Tr[b† b† b† ] 0 , m | i which also has E = 2 . One can show that the free theory spectrum can provide all the missing blocks of higher floors which are needed to complete “A” into a long multiplet. In order to turn multiplet ”B” into a long multiplet starting from (1, 20′) we would need the states (2, 4) (1, 20′)=(2, 20) + (2, 60) on the first-floor, where × in terms of Dynkin labels the (60) of SO(6) is given by [1, 2, 0]. But here, unlike 5m the situation for the multiplet ”A”, these additional (2, 60) states at E0 = 12 are simply not present in the free spectrum, even if we consider states of higher level. 5m At E0 = 12 there are no other states than the states listed in (3.9). So it turns out that the multiplet ”B” is truly short even in the interacting theory, and its energy is free from corrections to all orders.

The fact that the multiplet “B” built upon the lightest state a′b′ is protected is | i strongly reminiscent of the situation for the chiral primary operators in = 4 super N Yang-Mills theory which do not receive any radiative corrections to their scaling dimensions. These operators are given by symmetric traceless combinations of the six scalar ﬁelds ΦI with I =1,..., 6 of = 4 super Yang-Mills theory, i.e. N A = CA Tr [ΦI1 ΦI2 ... ΦIn ] (4.16) On I1I2...In

A with CI1...In being totally symmetric in its lower indices and any contraction among A the lower indices vanishing, CI1...I...I...In = 0. Let us therefore consider the multiplets built upon the lightest states

′ ′ ′ a a an C = C ′ ′ ′ Tr [b† 1 b† 2 ...b† ] 0 (4.17) | (n)i a1a2...an | i

′ ′ ′ ′ ′ ′ ′ with Ca1a2...an being totally symmetric and traceless, i.e. Ca1...a ...a ...an = 0. These n m states have the mass ·6 in the free theory. Clearly as

Q+ C = 0 (4.18) 0 | (n)i they constitute the lightest state in a multiplet of the free theory. It is tempting to speculate that the energy eigenvalue of these states is protected from perturbative corrections as well. We have computed the energy shifts for these states for n = 3, 4 and 5 in leading order perturbation theory and indeed ﬁnd that they cancel! The explicit contributions are presented in appendix B. Based on this evidence we therefore conjecture that all the states contained in the multiplets built on (4.17) are protected and that their energy eigenvalues are exactly given by the free theory values. The proof of this conjecture should go along the same lines as the arguments presented in the above for the case C , namely due to the absence of the required | (2)i representations in the free theory at higher mass levels. We leave the detailed proof for future work.

– 11 – 5. Conclusions and Outlook

In this paper we have studied the spectrum of the recently found massive matrix quantum mechanics in a pp-wave background and performed a second order perturbation calculation. We uncovered a protected short multiplet of the theory whose energy eigenvalues are now known exactly in the full interacting model. Moreover we conjectured the existence of an infinite series of such protected states. Employ- ing the matrix model conjecture this is a non-trivial statement about the light-cone Hamiltonian of M-theory in a pp-wave background. In the case of the maximally supersymmetric pp-wave solution of type IIB superstring, the precise energy spectrum in the light cone gauge was presented by Metsaev [15]. Our results can be thought of as the M-theory counterpart. Using the relation between the AdS space and the pp-wave, it was argued that the string spectrum in the pp-wave background must be related to the anomalous dimension of the dual CFT operator with large R-charge. It is very tempting to conjecture the same correspondence between the M-theory pp-wave solutions and the superconformal field theories of M2- and M5-branes. An intriguing fact is that the Penrose limit of both AdS S and AdS S lead to 4 × 7 7 × 4 the same pp-wave solution with SO(3) SO(6) symmetry, implying that they share × essentially the same subsector. The M-brane field theories are still largely mysteri- ous but it would be very interesting if we can compare the matrix theory calculation reported here with field theory calculations. There are a number of further interesting open question emerging. For example the protected energy eigenvalues do not depend on N and should therefore survive the large N limiting procedure under which the matrix model approximates the pp- wave supermembrane. What is the picture of these states in the supermembrane theory? Furthermore, what can we learn from these considerations for the notorious flat matrix model in the limit m 0? → Finally, in our work we have exclusively studied the matrix model around the “trivial” vacuum Xi = 0. As discussed in [7, 10] there is a multitude of further maximally supersymmetric vacua in the bosonic SO(3) sector corresponding to fuzzy sphere solutions of the equations of motion. As these vacua are subject to the same superalgebra we expect that similar protected multiplets exist in these sectors of the theory as well.

Acknowledgments

We would like to thank Gleb Arutyunov, Bernard de Wit, Kimyeong Lee, Jeong- Hyuck Park, Soo-Jong Rey and Piljin Yi for useful discussions. NK thanks KIAS for hospitality during the “KIAS Workshop on Strings and Branes” where an earlier version of this paper was presented.

– 12 – A. The Supersymmetry Algebra

The nontrivial pieces of the supersymmetry algebra (3.3) lie in the (anti)-commmutators involving Q . In computing the anticommutator of two supercharges Q , Q we α { α β} shall only focus on the terms proportional to m as we know from the work of [4] how the m independent terms work out. One then has

γ δ sr ut γ δ ut sr Qα, Qβ = [...]α rs [...]β tu θγ , θδ [...]α rs, [...]β tu θδ θγ (A.1) { } { } − h i where we have used the abbreviation

[...] γ = [P i γ i [Xi,Xj] γ + m Xi γ γ ] γ α rs i − 2 ij i i 123 α rs following from (3.1), r,s,t,u denote U(N) matrix indices. In the above we have moreover employed an extended summation convention for the index i where

′ i m a m a m X γ X γ + X γ ′ i i ≡ 3 a 6 a capturing the two diﬀerent SO(3) and SO(6) masses in mi. The terms in (A.1) are then straightforwardly found to have the structure

Q , Q = 1 (m Xi)2 ++ + + +(m independent terms) (A.2) { α β} 2 i B1 B2 F i where

1 k i i k 1 = 2 mi Tr(P X ) γkγ123γi Tr(X P ) γiγ123γk B − − i i k l 2 = 4 mi Tr(X [X ,X ]) γiγ123γkl + γklγ123γi B i i i i = i γαγ (γ γ123)βδ mi (γ γ123)αγ γβδ mi Tr(θδ θγ ) (A.3) F − Now the bosonic contributions and can be shown to be B1 B2 ′ ′ m a b m a b = Tr(P X )(γ γ ) + Tr(P X )(γ ′ ′ γ ) B1 − 3 ab 123 αβ 6 a b 123 αβ = i m ǫ Tr(Xa Xb Xc) δ B2 − 3 abc αβ In order to reduce the fermionic contributions in (A.2) we make use of the Fierz F identity

N 1 jk 1 jkl Tr(θδ θγ)= 4 δδγ + 32 Tr(θγ θ)(γjk)δγ + 96 Tr(θγ θ)(γjkl)δγ which after some algebra gives us the relation

′ ′ m ab m a b m = i Tr(θγ θ)(γ γ ) + i Tr(θγ θ)(γ ′ ′ γ ) i Tr(θγ θ) δ F − 12 ab 123 αβ 24 a b 123 αβ − 4 123 αβ (A.4)

– 13 – We thus see the emergence of the angular momentum operators (3.4) in the algebra coupling to (γabγ123)αβ and (γa′b′ γ123)αβ respectively. Summarizing we then ﬁnd the following anticommutator relation

′ ′ m ab m a b i Q , Q = δ H L (γ γ ) + L (γ ′ ′ γ ) + iTr(X )(γ ) (A.5) { α β} αβ − 6 ab 123 αβ 12 a b 123 αβ G i αβ putting back in the “old” m independent terms computed in [4].

Now we want to calculate [H, Qα]. First from the Jacobi identity one can easily show that [H, Q ]= 1 [ Q , Q , Q ] (A.6) α − 8 { α β} β Now using the above result for Q , Q and the property { α β} ij i ij [L ,Sα]= 2 Sβ(γ )βα (A.7) for any spinor, we obtain mi [H, Q ]= (Qγ ) + Tr(θ ) (A.8) α 12 123 α αG

B. Details of the Perturbative Calculation

In this section we comment on the calculation of energy shifts for the excited states. Naturally these manipulations are algebraically more involved and we have performed them with the help of Mathematica and FORM [16] computer algebra systems. The considered states of (3.10) which are excited by two raising operators are conveniently expressed with the help of the uniﬁed bosonic ladder operators of (4.3) and (4.4) through

i j h = hijTr(ã† a˜† ) 0 , | i ′| i + aa + f = faa′ Tr(θ γ θ ) 0 , | i ′ ′ ′ | i + a b c + g = g ′ ′ ′ Tr(θ γ θ ) 0 (B.1) | i a b c | i The interactions of H respect the SO(3) SO(6) split of the free Hamiltonian, INT × therefore mixing can only appear for the degenerate states aa′ and aa′ . We | iB | iF will deal with this problem at the end of this section and first study the diagonal elements of the interactions in quantum mechanical perturbation theory given by (4.13). The states excited by bosonic oscillators will be dealt with first. For the calculations done in this paper it turns out that the three different interaction terms can be treated separately: the cross term of the Yukawa and the cubic bosonic term does not contribute. So just like the ground state we first consider the first order perturbation of the quartic Yang-Mills interaction, and then the Yukawa and the

– 14 – cubic bosonic term at second order. The results are summarized in a table at the end of this section. First the Yang-Mills quartic interaction is calculated to give 1 h Tr[Xi,Xj]2 h = −4h 1| | 2i (N 5 N 3)((TrM)2 TrM 2)Trh Mh M − − 1 2 +4(N 3 N)(Trh M 2Trh M 2 Trh M 2h M 2) − 1 2 − 1 2 +8(N 3 N)(TrMTrh M 2h M Trh M 3h M) (B.2) − 1 2 − 1 2 From this result one can easily check that there are no off-diagonal overlaps of the pure bosonic states as expected. Note that the normalization constants for the different sets of states have not been taken into account yet so we have to divide the above result by the norm h h =2N 2Tr(hMhM) (B.3) h | i What we have to do now is simply calculate the ratios of traces of the form Tr(hM phM q) which is straightforward. Next we turn to the contributions from the cubic bosonic or Myers term. We write it in terms of raising and lowering operators and substitute appropriate values for E H . Written more explicitly 0 − 0 m2 1 ∆E2 h h = h Tr[e Xa Xb Xc] Tr[e Xa Xb Xc] h h | i 9 h | abc H E abc | i 0 − 0 m a b c a b c = + h Tr[e a˜ a˜ a˜ ] Tr[e a˜† a˜† a˜† ] h 9 h | abc abc | i a b c a b c +3m h Tr[e a˜ a˜ a˜† ] Tr[e a˜† a˜† a˜ ] h h | abc abc | i a b c a b c 3m h Tr[e a˜ a˜† a˜† ] Tr[e a˜† a˜ a˜ ] h − h | abc abc | i m a b c a b c h Tr[e a˜† a˜† a˜† ] Tr[e a˜† a˜ a˜ ] h (B.4) − 9 h | abc abc | i When we evaluate this it turns out that for level 2 states we are interested in here only the first two channels have nonvanishing contributions. The result is summarized as

36Tr (MhMhM)+12m2ǫabcǫdef M (MhM) (MhM) N 2 1 ∆E2 = 3 ad be cf − − Tr(hMhM) Nm 27(N 3 N) − (B.5) − 4m2 where Tr3X means one should take the trace of the 3 dimensional part only, after calculating X as a 9-dimensional matrix. Again the result for diﬀerent states can be found in table 1. Now we can turn to the consideration of the Yukawa terms. For the states with bosonic oscillators only we can easily perform the Wick contraction of the i + j fermionic oscillators in the interaction term. The result contains tr(γ Π γ Π−)Mij,

– 15 – States Quartic Myers Yukawa Total 891(N 3 N) 27(N 3 N) 864(N 3 N) − − − 0 (1,1) 4m2 4m2 4m2 0 | i 891N 4 351N 2 540 27(−N 4+19N 2 20) − 864(N 3 N) − − aa (1,1) −4m2N − 4m2N 4m2 0 | i 891N 4 405N 2 486 − 27(N 4+N 2 2) − 864(N 3 N) 108(N 2 1) − − − ab (5,1) −4m2N − 4m2N 4m2 m2N | i 891N 4+405N 2 1296 − 27(N 3 N) − 864(N 4 1) 108(N 2 1) − − − a′a′ (1,1) 4m2N − 4m2 4m2N m2N | i 891N 4 27N 2 864 − 27(N 3 N) − 864(N 4 1) − − a′b′ (1,20’) 4−m2N − 4m2 4m2N 0 | i 891N 3 207N 684 −27(N 4+3N 2 4) 432(2− N 4 N 2 1) 36(N 2 1) − − − − aa′ B (3,6) 4−m2N − 4m2N 4m2N m2N | i 891(N 3 N) − 27(N 3 N) −288(3N 4 4N 2+1) 72(N 2 1) − − − − aa′ F (3,6) 4m2 4m2 4m2N m2N | i 891(N 3 N) − 27(N 3 N) − 864(N 3 N) a′b′c′ (1,10) 2− −2 2− 0 | i 4m − 4m − 4m Table 1: The diagonal contributions of the second order perturbation calculation of the energy spectrum according to eq. (4.13). The states are deﬁned in (3.10) and the numbers in the parenthesis represent the associated SO(3) SO(6) representation. Only the states × of represenation (3, 6) receive oﬀ-diagonal contributions, which are evaluated in (B.10). and it is easy to see that only the SO(6) part of the Yukawa term gives nontrivial contributions. So

3 3 a′ + b′ + ∆E h h = h Tr(θ−γ ′ [ã , θ−])Tr(θ γ ′ [ã† , θ ]) h h | i −2mh | a b | i 3 + a′ + b′ h Tr(θ γ ′ [ã† , θ ])Tr(θ−γ ′ [ã , θ−]) h −mh | a b | i 216 3 36 a′ a′ a′ a′ = (N N) h h h NTr(ã† a˜ ) Trã† Trã h − m2 − h | i − m h | − | i 216 144 = (N 3 N) h h (N 3 N)Tr (MhMhM) (B.6) − m2 − h | i − m − 6 where Tr6 means we take the trace of the six dimensional part only. Now for the states excited by fermionic oscillators it is clear that the contribution of the quartic and the Myers term must be the same as the ground state. The Yukawa interaction can be considered as before, computing the different channels separately. For a state Tr(θ+Γ θ+) 0 , we get α αβ β | i 54(4N 4 5N 2 + 1) 18(N 2 1) tr(ΓγaΠ+Γγa) ∆E3 = − + − − m2N m2N tr(Γ2Π+)

′ For Γ = γaa , tr(ΓγaΠ+Γγa) = tr(Γ2Π+), so 72 ∆E3 = 3N 3 4N +1/N (B.7) −m2 − ′ ′ ′ and for Γ = γa b c using tr(ΓγaΠ+Γγa)= 3tr(Γ2Π+), − 216 ∆E3 = N 3 N (B.8) − m2 −

– 16 – State Quartic Myers Yukawa Sum 6 4 2 6 2 6 4 2 297N +432N −729N 9(N −N ) −288N −432N +720N C(3) 4(m/3)2 4(m/3)2 4(m/3)2 0 | i 7 5 3 −7 5 3 7 5 3 99N +588N −111N −576N −3N −12N +15N −96N −576N +96N +576N C(4) (m/3)2 (m/3)2 (m/3)2 0 | i 2 6 4 2 2 6 4 2 2 6 4 2 15N (33N +542N +649N −1224) −15N (N +14N −7N −8) −60N (2N +33N +41N −76) C 2 2 2 0 | (5)i 4(m/3) 4(m/3) (m/3) Table 2: The vanishing of the second order perturbation calculation of the energy shifts of the totally symmetrized SO(6) higher level states deﬁned in eq. (4.17).

Again this result is summarized in the table.

What remains to be done is the computation of the mixing of aa′ and aa′ | iB | iF under perturbation theory. Here it is essential to work with the properly normalized states 1 1 aa′ = aa′ aa′ = aa′ (B.9) | iBnorm N | iB | iFnorm 2 N | iF following from (3.15). For the cross term only the Yukawa interaction piece contributes and one ﬁnds 1 36(N 2 1) ′ ′ aa′ HINT 2 HINT 2 bb′ = √2 − δ δ . (B.10) Bnorm h | |Xθ E H |Xθ | iFnorm m2 N ab a b 0 − 0 Combining this with the result quoted in table 1 one thus obtains the mixing matrix 36(N 2 1) 2 √2 − (B.11) m2 N √2 1 for the normalized states aa′ Fnorm and aa′ Bnorm . It indeed has the eigenvalues 0 and 108 (N 2 1) | i | i − √ m2 N associated to the eigenvectors ( aa′ Fnorm 2 aa′ Bnorm ) and ( aa′ Fnorm + 1 | i − | i | i aa′ ) respectively. These are precisely the combinations appearing in the √2 | iBnorm (3, 6) sector of the free ﬁeld multiplets “A” and “B” as stated in (4.14) and (4.15). Finally we turn to the totally symmetrized SO(6) higher level states C of | (n)i (4.17). The explicit contributions from the three sectors of perturbation theory are stated in table 2 for n =3, 4, 5 which add up to zero. During the calculation we have not distinguished connected and disconnected 3 diagrams, so for each type of interaction the leading correction appears to be ( N ) O m2 for level-two states, however they always add up to zero due to the underlying supersymmetry. We can thus see that the physical coupling constant in a large N N expansion is m2 .

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– 18 –