{'fraction_non_alphanumeric': 0.042404589673235224, 'fraction_numerical': 0.017460713394861563, 'mean_word_length': 4.512373968835931, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 6, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Classification of the BPS states in Bagger-Lambert Theory

We classify, in a group theoretical manner, the BPS configurations in the multiple M2-brane theory recently proposed by Bagger and Lambert. We present three types of BPS equations preserving various fractions of supersymmetries: in the first type we have constant fields and the interactions are purely algebraic in nature; in the second type the equations are invariant under spatial rotation SO(2), and the fields can be time-dependent; in the third class the equations are invariant under boost SO(1,1) and provide the eleven-dimensional generalizations of the Nahm equations. The BPS equations for different number of supersymmetries exhibit the division algebra structures: octonion, quarternion or complex.


Introduction
In a series of recent papers [1], Bagger and Lambert (BL) have constructed a three-dimensional, interacting superconformal gauge theory of multiple M2-branes. The action is maximally supersymmetric with 16 ordinary supersymmetries, and it has been verified that the theory is indeed superconformal with 16 conformal supercharges in [2]. In the quest for the final form of the theory, as usual, it was supersymmetry that provided crucial guiding lights. The work was initiated as an attempt to incorporate Basu and Harvey's generalized Nahm equation -which was a proposal to describe M2-branes ending on an M5-brane [3]-in the full supersymmetric M2-brane action. Their analysis revealed a novel algebraic structure, namely the 3-algebra, which is also investigated independently by Gustavsson [4]. Since the discovery, the multiple M2-brane theory of Bagger and Lambert has attracted an enormous degree of attention . One might expect that, given this genuine superconformal field theory, M-theory is now about to unveil its mysterious and fundamental features.
In the present paper, we set out to classify the BPS states, or the BPS equations of the BL theory using a group theoretical consideration. Apparently the theory of our interest has the Lorentz group SO(1, 2) and the R-symmetry group SO (8). Instead of providing the full and thorough survey of possible BPS equations, we focus mainly on two different types of BPS equations with different number of supersymmetries, and classify them completely. The first class is completely Lorentz invariant, and the other is invariant under the spatial rotation.
In the first type, the BPS equations are given purely in terms of the three-algebra commutators and independent of the three-dimensional worldvolume coordinates. Thus the corresponding nontrivial configurations possess infinite energy, typically corresponding to BPS objects of infinite size. Previously known analogous algebraic soultions include the longitudinal M5-brane in M-theory matrix model which is realized in terms of Heisenberg algebra or large N matrices [36].
In the other type the equations are SO(2) rotation invariant, and the fields can be time-dependent. A technical reason why we focus on the two classes is that in these cases, fully utilizing the SO(8) triality we are able to classify the BPS equations completely.
In addition to the two classes, there is another possibility to obtain third type of BPS equations via simple tensor product. Namely one can obtain various generalizations of the Nahm equations which are invariant under the boost SO(1, 1) ⊂ SO(1, 2). Our BPS equations manifest the division algebra structures: octonion, quarternion or complex. In the paper we will mainly focus on the BPS equations themselves.
Our results hold for both the finite and infinite dimensional three-algebras. Note however that the Lorentz invariant BPS equations can have nontrivial solutions only for infinite dimensional three-algebras. The specific solutions and the physical interpretation will be presented in a separate publication [37].
The organization of the present paper is as follows. Sec.2 is for preliminaries. We first discuss the general features of the 'supersymmetric projection matrices' and review how to derive the corresponding BPS equations for a given projection matrix. We also explain the relevant symmetries. Then we classify the projection matrices for the SO(1, 2), SO (2)  Note added: While this paper is being finished, Ref. [38] appears in ArXiv which partially overlaps with our work, as it discusses the BPS equations of the form: D y X I = 1 3! C IJKL [X J , X K , X L ]. In the present paper, we explicitly spell the coefficients C IJKL and classify various BPS equations.

Preliminaries
The multiple M2-brane theory has 8 real scalar fields X I , I = 1, 2, · · · , 8 and a 16 component Majorana spinor Ψ. The supersymmetry transformation of the fermions in the Bagger-Lambert theory assumes the form: where all the variables are three-algebra valued and we set The bracket [X I , X J , X K ] denotes the three-algebra product which is trilinear and totally antisymmetric.
Note also that in contrast to the original convention [1] we let I = 1, 2, · · · , 8 and take µ ≡ 0, 9, 10 directions as for the M2-brane worldvolume for convenience to present the BPS equations later, The supersymmetry parameter is real and subject to the SO(1, 2) projection condition: which is consistent with the opposite projection property, Γ txy Ψ = −Ψ. Since the product of all the eleven-dimensional gamma matrices leads to the 32×32 identity matrix Γ txy123···8 = 1, the above SO (1,2) projection condition coincides with the chirality condition of SO (8),

Supersymmetry projection matrix -general
In general for supersymmetric theories, the supersymmetry projection matrix Ω can be defined in terms of the commuting, real, orthonormal supersymmetry parameters ε 1 , ε 2 , · · · , ε N , satisfying Ω † = Ω 2 = Ω. Here N denotes the number of the preserved supersymmetries, Naturally the eigenvalues of the projection matrices are either zero or one.
When the supersymmetry transformation of fermions takes the form δΨ = Fε where F denotes a bosonic quantity contracted with gamma matrices as in (2.1), the general strategy to obtain the BPS equations is as follows [40]: 1. Expand the projection matrix Ω in terms of the gamma matrix product basis.
2. Perform the matrix product FΩ and reexpress it in terms of the gamma matrix product basis.

Read off the BPS equations from the coefficients of the linearly independent terms.
For example in the Euclidean four-dimensional minimal super Yang-Mills theory, we have two choices for the projection matrix Ω = 1 2 (1 ± γ 1234 ), while F = F ij γ ij . Consequently, noting γ 12 Ω = ∓γ 34 Ω etc., we get F ij γ ij Ω = 2(F ∓ ⋆ F ) i4 γ i4 Ω such that the corresponding BPS equations are the well-known self-dual or anti-self-dual equations F = ±⋆ F . In this way, the complete classifications of the BPS equations in six and eight-dimensional super Yang-Mills as well as the pp-wave M-theory matrix model [41] have been carried out [40,42,43].
The present paper concerns the BPS equations of the Bagger-Lambert theory. Since the elevendimensional spacetime admits Majorana spinors we can set all the gamma matrices and the spinors to be real. In particular, the spatial gamma matrices are symmetric while the temporal gamma matrix is anti-symmetric. Consequently, also from (2.5), the projection matrices of the Bagger-Lambert theory must satisfy where P is the SO(8) chiral projection matrix, The most general form of such projection matrices reads where c, c ′ , c ′′ are constants, Υ 4 , Υ ′ 4 , Υ ′′ 4 are foursome productions of the SO(8) gamma matrices Γ IJKL contracted with self-dual four-forms, and Υ 2 is a twosome production of the SO(8) gamma matrices Γ IJ contracted with a two-form. All together, a priori, there are 3 + 3 × 1 Note that the SO(8) rotation may take only one of {Υ 4 , Υ ′ 4 , Υ ′′ 4 , Υ 2 } into a canonical form. In our choice, the canonical form of a two-form reads while the canonical form of a self-dual four-form reads where we set The former is well known, while the latter is less familiar and we review it in Appendix A. In (2.13) the subscript spatial indices of the gamma matrices are organized such that the three indices after the common 8 are identical to those of the totally anti-symmetric octonionic structure constants [40,44]: (2.14) We say Ω is invariant under SO(2) rotation invariant on xy-plane if [Γ xy , Ω] = 0. When this holds, for a finite angle φ and rotation G = e φΓxy , from the equivalence we note that the corresponding BPS equations are, as a set, invariant under the rotation. Naturally this generalizes to an arbitrary subgroup of SO(1, 2)×SO(8).
In the present paper instead of attempting to solve for the most general projection matrices, we restrict to the cases where Ω assumes the canonical form. Namely we focus on two types of the BPS equations and classify the corresponding BPS equations completely: one is the SO(1, 2) invariant cases i.e.

SO(2) invariant projection matrices
The basic building blocks of all the possible SO(2) invariant projection matrices are the following N = 2 projection matrices (see Appendix B for derivation): where β 1 , β 2 , β 3 denote three independent signs,  (8) rotations, there are five inequivalent SO(2) invariant projection matrices as follows.

SO(1, 1) invariant projection matrices
For SO(1, 1) invariant projection matrices, we have the following N = 1 projection matrices: where α 0 , α 1 , α 2 , α 3 are four independent signs, Sixteen possible N = 1 projection matrices form an orthogonal basis for the SO(1, 1) invariant projection matrices. Generic N = k SO(1, 1) invariant projection matrices may be obtained straightforwardly as a k sum of the above sixteen N = 1 projection matrices. For each sum, we may decompose
where C IJKL is a SO(7) invariant four-form in eight dimensions, defined in terms of the octonionic structure constant (2.14), where J is a complex structure J 2 = −1, J T = −J and hence SU(4)×SO (2) invariant.
We note that summing two N = 2 projection matrices generates one complex structure. Hence in general, summing k > 2 of N = 2 projection matrices will present where J 1 , J 2 , J 3 are three distinct complex structures satisfying the quaternion relations: It is worth to note that the remaining relation of (3.5) i.e. F (3,0) = 0 is fulfilled automatically for each complex structure.

Discussion
In this paper we studied and identified a number of BPS equations for the multiple M2-brane theory proposed recently by Bagger and Lambert. We employed a method which had been successfully applied to several analogous problems. One first constructs the basic projection matrices for the supersymmetry parameters, and then obtain the corresponding BPS equations. Our classifications are complete for SO (1,2) as well as SO (2)
Clearly the SO(1, 2) projection constraint (2.4) coincides with that of SO (8), This is consistent with the fact that the product of all the eleven-dimensional gamma matrices leads to the identity Γ txy123···8 = 1.

(A.4)
The subscript spatial indices of the gamma matrices are organized such that the three indices after the common 8 are identical to those of the totally anti-symmetric octonionic structure constants (2.14). It is straightforward to see that E i forms a representation of the "square" of the octonions on the eightdimensional chiral space, Since they commute each other, they form a maximal set of the mutually commuting traceless symmetric and real matrices of the definite chirality γ (9) E i = E i . In fact, one can construct a SO(8) symmetric and real gamma matrix representation which makes all E i 's be simultaneously diagonal, utilizing the octonionic structure constants: Here ρ I , I = 1, 2, · · · , 8 are 8×8 real matrices given by 4 and c i is a 7×7 real matrix whose j, k component is nothing but the octonionic structure constant c ijk (2.14), while n i is a seven-dimensional unit vector of which the jth component is defined to be δ j i .
In the above choice of Majorana gamma matrix representation, all the E i 's and P are diagonal, .