BPS Conditions of Supermembrane on the PP-wave

We study the BPS conditions in the closed supermembranes on the maximally supersymmetric pp-wave background. In particular, the 1/2 and 1/4 BPS states are discussed in detail. Moreover, we comment on the zero-modes in the invariant mass formulae of the theory.


Introduction
The matrix model approach [1] to M-theory seems a great success. In a recent progress, the pp-wave backgrounds [2,3,4] and Penrose limits [5,6] are attractive subjects. It has been shown that the Green-Schwarz (GS) string theory on the pp-wave is exactly-solvable [7,8,9]. Also, the matrix model on the eleven dimensional maximally supersymmetric pp-wave solution [2] has been proposed in Ref. [10] and further consideration has been done [11].
In our previous paper [12], we have studied the supermembrane [13,14,15] on the maximally supersymmetric pp-wave background, and derived the supercharges and associated algebra with the central charges by carefully treating the surface terms by the use of the Dirac bracket procedure.
In this paper, we consider the BPS states from the superalgebra derived in our previous work.
We use triangular decomposition for the supercharge matrix, and derive the BPS conditions by analyzing the rank of the supercharge matrix. In particular, the 1/2 BPS and 1/4 BPS states are investigated in detail. We also find that the zero-modes of the membrane variables do not appear in the BPS conditions.
In section 2, the BPS states are studied by investigating the rank of the supercharge matrix.
In particular, we consider the cases of 1/2 and 1/4 BPS states, concretely. Section 3 is devoted to conclusions and discussions.

Supercharge Matrix and BPS States
In our previous work [12], we have studied supermembranes on the maximally supersymmetric pp-wave and derived the expressions of the supercharges Q + and Q − . We calculated the associated superalgebra. The results are written as follows: where P r (≡ wD τ X r ) and S α (≡ iwψ T α ) are the canonical momenta of X r and ψ, respectively. The zero-modes of P r and X r are defined by and describe the motion of the membrane's center of mass. Also, the Hamiltonian H is expressed by Other quantities in the above algebra are defined by In order to study the BPS states in the supermembrane theory on the pp-wave background, let us construct the supercharge matrix with 32 × 32 components * (2.14) By the use of the formula of the triangular decomposition, which can be applied for an arbitrary matrix, we can decompose the supercharge matrix as follows: Each block part is a matrix with 16 × 16 components. The component matrix "m" in the * Hereafter, we use the expressions of the supercharges redefined by rearranging the factor 1/ √ 2 into the overall factor of the fermion ψ. triangular decomposed expression is given by r,s,t,u=1 where the central charges z rs , z r , z rstu , U J KI ′ J ′ and U I ′ are written as and constraint ϕ is given by Here, we should note on the invariant mass M, which plays an important role in studying the BPS states in the theory. Let us recall that the invariant mass M of a supermembrane is defined by Thus the first two terms of Eq. (2.17) can be replaced with the invariant mass M 2 of a supermembrane. In particular, if we consider 1/2 BPS states of the closed supermembrane in the flat space (i.e., the µ → 0 limit), then BPS condition arises from the requirement that the coefficient of δ γδ equals zero. This condition implies that a mass should be proportional to its charge in the BPS state.
Hereafter, we shall restrict ourselves to the case of the closed membrane.
In flat case the Hamiltonian contains only the momentum zero-modes P r 0 's and not X r s' zero-modes X r 0 's and fermion zero-modes ψ 0 's. However, the invariant mass M 2 includes no zero-modes since the momentum zero-modes are subtracted by definition.
In the pp-wave case, due to the presence of the additional terms in the action (3-point coupling, boson mass terms and fermion mass term), the Hamiltonian includes the zero-modes X r 0 and ψ 0 in addition to the momentum zero-mode P r 0 even if we consider closed membranes. In fact, we will see that these zero-modes are subtracted and do not appear in our considerations for the BPS states. This result might be understood if one notes that the U(1) parts decouple from SU(N) parts even in the pp-wave background as noted in Refs. [10,11].
From now on, we will show that the matrix "m" is independent of the zero-modes. Let us notice the part in "m" proportional to δ γδ . First we will decompose the coordinate X r and momentum P r around their averaged values Then the focusing part can be rewritten as TheH is interpreted as a Hamiltonian where bosonic zero modes X 0 and P 0 are subtracted.
where the zero-modes X r 0 's are completely subtracted. Therefore, we find that the zero-modes in the first square bracket of (2.17) are spurious.
Next, in the similar way we can subtract the zero modes from the U J KI ′ J ′ and U I ′ , and obtain the following expressions Also, the charges z r , z rs and z rstu are rewritten as Thus, it is shown that the zero-modes in square brackets containing z r , U J KI ′ J ′ , U I ′ of (2.17) are also spurious. Moreover, we can easily show that the other terms in (2.17) do not contain any bosonic zero-modes, and so we obtain the drastically simplified result for the matrix "m" as m γδ = 2 H − 9 r,s=1 z rs z rs δ γδ + 2 9 r=1 z r − d 2 σφX r (γ r ) γδ + 9 r,s,t,u=1 [z rs z tu + 2z rstu ] (γ rstu ) γδ Thus, we have proven that the zero-modes of the variables X and P in "m" (2.17) have been subtracted and "m" can be described in terms of oscillation-modes in X and P only. That is, the matrix "m" represents the excited modes of variables. The first three terms in Eq. We can find the BPS states by analyzing the rank of the supercharge matrix "m". The 1/2 and 1/4 BPS cases will be considered in the following subsections. When we consider the BPS states, the fermion parts are omitted in the discussion because we do not take the winding of fermion into account.

1/2 BPS States
The 1/2 BPS states can be described by the condition m=0, which means some constraints 2H − 2 It is also possible to obtain the remaining unbroken supercharges under the 1/2 BPS conditions. Note the supercharge matrix can be rewritten as (2.45) We can easily find the unbroken supersymmetry and its charge is given by (2.46)

1/4 BPS States
The 1/4 BPS conditions are that the rank of the matrix "m" is eight. In order to study the rank of "m", we decompose the 16 × 16 SO(9) gamma matrices γ r (r = 1, . . . By the use of the above matrices,Ũ I ′ (γ I ′ γ 123 ) andŨ J KI ′ J ′ (γ J KI ′ J ′ ) can be rewritten as We would like to consider the rank of the matrix "m", but the general analysis is more complicated and difficult. Here we shall present some special solutions concretely.
To begin, we consider only µ-dependent parts by letting µ-independent parts vanish. For simplicity, we impose further conditionsM IJ 0 =MĨ ′J ′ 0 =Ũ J KI ′ J ′ =Ũ 9 = 0. Then the matrix "m" can be written as m = 2µ We can easily find the conditions that the rank of the matrix "m" is eight, those arẽ We may consider such solutions as rotating membranes which are 1/4 BPS. In fact, the superalgebra includes the angular momentum operators, which might be considered as the remnants of the AdS 7 × S 4 or AdS 4 × S 7 backgrounds, and such rotating solutions can exist in the theory.
Furthermore, (2.52) and (2.53) mean the "stringy exclusion principle" which is also a remnant in AdS space physics. That is, configurations with larger angular momenta than the given brane charges cannot exist. The central charges in the superalgebra may also indicate other extended objects only living on the pp-wave, which might be expected as a fuzzy membrane, a giant graviton [10] or other 1/4 BPS states [17] coming from Myers effects [18].
In the above case, only the µ-dependent parts have been studied, but we should note that such a situation can be also realized by taking the large µ limit without imposing certain conditions on µ-independent parts. The large µ limit has been discussed in [11] where the existence of such 1/4 BPS states is stated at least in this limit. Conversely speaking, in our considerations we could avoid the large µ limit by requiring µ-independent parts to satisfy vanishing conditions. Moreover, we comment that it is possible to discuss the 1/4 BPS states in the same way as in the flat space [16] if we considerM IJ 0 =M I ′ J ′ 0 =Ũ I ′ =Ũ J KI ′ J ′ = 0 (i.e., µ = 0). Also, it would be possible to apply the similar considerations forŨ J KI ′ J ′ .
Finally, the unbroken supercharges of the 1/4 BPS states are given by where the minus (−) sign corresponds to the case in (2.52) and the plus (+) one to (2.53), respectively.
in the flat space. For the 1/4 BPS states, we could not present general solutions but a special solution as an example of the 1/4 BPS states, which is peculiar in the pp-wave case.
It is an interesting future work to investigate more general BPS mass formulae systematically.