3-Algebras in String Theory

In this chapter, we review 3-algebras that appear as fundamental properties of string theory. 3-algebra is a generalization of Lie algebra; it is defined by a tri-linear bracket instead of by a bi-linear bracket, and satisfies fundamental identity, which is a generalization of Jacobi iden‐ tity [1], [2], [3]. We consider 3-algebras equipped with invariant metrics in order to apply them to physics.


Introduction
In this chapter, we review 3-algebras that appear as fundamental properties of string theory. 3-algebra is a generalization of Lie algebra; it is defined by a tri-linear bracket instead of by a bi-linear bracket, and satisfies fundamental identity, which is a generalization of Jacobi identity [1], [2], [3]. We consider 3-algebras equipped with invariant metrics in order to apply them to physics.
It has been expected that there exists M-theory, which unifies string theories. In M-theory, some structures of 3-algebras were found recently. First, it was found that by using u(N ) ⊕ u(N ) Hermitian 3-algebra, we can describe a low energy effective action of N coincident supermembranes [4], [5], [6], [7], [8], which are fundamental objects in M-theory.
Moreover, recent studies have indicated that there also exist structures of 3-algebras in the Green-Schwartz supermembrane action, which defines full perturbative dynamics of a supermembrane. It had not been clear whether the total supermembrane action including fermions has structures of 3-algebras, whereas the bosonic part of the action can be described by using a tri-linear bracket, called Nambu bracket [23], [24], which is a generalization of Poisson bracket. If we fix to a light-cone gauge, the total action can be described by using Poisson bracket, that is, only structures of Lie algebra are left in this gauge [25]. However, it was shown under an approximation that the total action can be described by Nambu bracket if we fix to a semi-light-cone gauge [26]. In this gauge, the eleven dimensional space-time of M-theory is manifest in the supermembrane action, whereas only ten dimensional part is manifest in the light-cone gauge.
The BFSS matrix theory is conjectured to describe an infinite momentum frame (IMF) limit of M-theory [27] and many evidences were found. The action of the BFSS matrix theory can be obtained by replacing Poisson bracket with a finite dimensional Lie algebra's bracket in the supermembrane action in the light-cone gauge. Because of this structure, only variables that represent the ten dimensional part of the eleven-dimensional space-time are manifest in the BFSS matrix theory. Recently, 3-algebra models of M-theory were proposed [26], [28], [29], by replacing Nambu bracket with finite dimensional 3-algebras' brackets in an action that is shown, by using an approximation, to be equivalent to the semi-light-cone supermembrane action. All the variables that represent the eleven dimensional space-time are manifest in these models. It was shown that if the DLCQ limit of the 3-algebra models of Mtheory is taken, they reduce to the BFSS matrix theory [26], [28], as they should [30], [31], [32], [33], [34], [35].

Definition and classification of metric Hermitian 3-algebra
In this section, we will define and classify the Hermitian 3-algebras equipped with invariant metrics.

General structure of metric Hermitian 3-algebra
The metric Hermitian 3-algebra is a map V × V × V → V defined by ( x, y, z ) ↦ x, y; z , where the 3-bracket is complex linear in the first two entries, whereas complex anti-linear in the last entry, equipped with a metric < x, y > , satisfying the following properties: the fundamental identity and the anti-symmetry x, y; z = -y, x; z (id4) for Linear Algebra The Hermitian 3-algebra generates a symmetry, whose generators D(x, y) are defined by From (▭), one can show that D(x, y) form a Lie algebra, There is an one-to-one correspondence between the metric Hermitian 3-algebra and a class of metric complex super Lie algebras [19]. Such a class satisfies the following conditions among complex super Lie algebras S = S 0 ⊕ S 1 , where S 0 and S 1 are even and odd parts, respectively. S 1 is decomposed as The super Lie bracket satisfies From the metric Hermitian 3-algebra, we obtain the class of the metric complex super Lie algebra in the following way. The elements in S 0 , V , and V are defined by (▭), (▭), and (▭), respectively. The algebra is defined by (▭) and x, y : = 0 x , ȳ : = 0 (id12) One can show that this algebra satisfies the super Jacobi identity and (▭)-(▭) as in [19].
Inversely, from the class of the metric complex super Lie algebra, we obtain the metric Hermitian 3-algebra by where α is an arbitrary constant. One can also show that this algebra satisfies (▭)-(▭) for (▭) as in [19].
Real sl(m, n) is defined by where h 1 and h 2 are m × m and n × n anti-Hermite matrices and c is an n × m arbitrary complex matrix. Complex sl(m, n) is a complexification of real sl(m, n), given by where α, β, γ, and δ are m × m, n × m, m × n, and n × n complex matrices that satisfy As a result, we obtain the u(m) ⊕ u(n) Hermitian 3-algebra, where x, y, and z are arbitrary n × m complex matrices. This algebra was originally constructed in [8].
Next, we will construct the sp(2n) ⊕ u(1) Hermitian 3-algebra from C(n + 1). Complex The elements are given by where α is a complex number, a is an arbitrary n × n complex matrix, b and c are n × n complex symmetric matrices, and x 1 , x 2 , y 1 and y 2 are n × 1 complex matrices. (▭) is rewritten as for 3-Algebras in String Theory http://dx.doi.org/10.5772/46480 where w 1 and w 2 are given by As a result, we obtain the sp(2n) ⊕ u(1) Hermitian 3-algebra, , where x 1 , x 2 , y 1 , y 2 , z 1 , and z 2 are n-vectors and

3-algebra model of M-theory
In this section, we review the fact that the supermembrane action in a semi-light-cone gauge can be described by Nambu bracket, where structures of 3-algebra are manifest. The 3-algebra Models of M-theory are defined based on the semi-light-cone supermembrane action. We also review that the models reduce to the BFSS matrix theory in the DLCQ limit.

Supermembrane and 3-algebra model of M-theory
The fundamental degrees of freedom in M-theory are supermembranes. The action of the covariant supermembrane action in M-theory [36] is given by This action is invariant under dynamical supertransformations, These transformations form the = 1 supersymmetry algebra in eleven dimensions, The action is also invariant under the κ-symmetry transformations, If we fix the κ-symmetry (▭) of the action by taking a semi-light-cone gauge [26]Advantages of a semi-light-cone gauges against a light-cone gauge are shown in [37], [38], [39] we obtain a semi-light-cone supermembrane action, In [26], it is shown under an approximation up to the quadratic order in ∂ α X μ and ∂ α Ψ but exactly in X I , that this action is equivalent to the continuum action of the 3-algebra model of M-theory, where Γ 012 = -. These supersymmetries close into gauge transformations on-shell, where gauge parameters are given by (▭) implies that a commutation relation between the dynamical supersymmetry transformations is up to the equations of motions and the gauge transformations.
This action is invariant under a translation, where η I are constants.
The action is also invariant under 16 kinematical supersymmetry transformations and the other fields are not transformed. ˜ is a constant and satisfy Γ 012 ˜= ˜. ˜ and should come from sixteen components of thirty-two = 1 supersymmetry parameters in eleven dimensions, corresponding to eigen values ± 1 of Γ 012 , respectively. This = 1 supersymmetry consists of remaining 16 target-space supersymmetries and transmuted 16 κ-symmetries in the semi-light-cone gauge [26], [25], [40].
A commutation relation between the kinematical supersymmetry transformations is given by A commutator of dynamical supersymmetry transformations and kinematical ones acts as where the commutator that acts on the other fields vanishes. Thus, the commutation relation is given by where δ η is a translation.
If we change a basis of the supersymmetry transformations as we obtain These thirty-two supersymmetry transformations are summarised as Δ = ( δ ' , δ ' ) and (▭) implies the = 1 supersymmetry algebra in eleven dimensions,

Lie 3-algebra models of M-theory
In this and next subsection, we perform the second quantization on the continuum action of the 3-algebra model of M-theory: By replacing the Nambu-Poisson bracket in the action (▭) with brackets of finite-dimensional 3-algebras, Lie and Hermitian 3-algebras, we obtain the Lie and Hermitian 3-algebra models of M-theory [26], [28], respectively. In this section, we review the Lie 3-algebra model.
If we replace the Nambu-Poisson bracket in the action (▭) with a completely antisymmetric real 3-algebra's bracket [21], [22], we obtain the Lie 3-algebra model of M-theory [26], [28], We have deleted the cosmological constant Λ, which corresponds to an operator ordering ambiguity, as usual as in the case of other matrix models [27], [41].
This model can be obtained formally by a dimensional reduction of the = 8 BLG model [4], [5], [6], The formal relations between the Lie (Hermitian) 3-algebra models of M-theory and the = 8 ( = 6) BLG models are analogous to the relation among the = 4 super Yang-Mills in four dimensions, the BFSS matrix theory [27], and the IIB matrix model [41]. They are completely different theories although they are related to each others by dimensional reductions. In the same way, the 3-algebra models of M-theory and the BLG models are completely different theories.
The fields in the action (▭) are spanned by the Lie 3-algebra T a as X I = X a I T a , Ψ = Ψ a T a and A μ = A ab μ T a ⊗ T b , where I = 3, ⋯ , 10 and μ = 0, 1, 2. < > represents a metric for the 3algebra. Ψ is a Majorana spinor of SO(1,10) that satisfies Γ 012 Ψ = Ψ. E μνλ is a Levi-Civita symbol in three-dimensions.
We do not choose 4 algebra because its degrees of freedom are just four. We need an algebra with arbitrary dimensions N, which is taken to infinity to define M-theory. Here we choose the most simple indefinite metric Lie 3-algebra, so called the Lorentzian Lie 3-algebra associated with u(N ) Lie algebra, where a = -1, 0, i (i = 1, ⋯ , N 2 ). T i are generators of u(N ). A metric is defined by a symmetric bilinear form, and the other components are 0. The action is decomposed as

and
A μij T i , T j → b μ . a μ correspond to the target coordinate matrices X μ , whereas b μ are auxiliary fields.
In this action, T -1 mode; X -1 I , Ψ -1 or A -1a μ does not appear, that is they are unphysical modes.
Therefore, the indefinite part of the metric (▭) does not exist in the action and the Lie 3-algebra model of M-theory is ghost-free like a model in [42]. This action can be obtained by a dimensional reduction of the three-dimensional = 8 BLG model [4], [5], [6] with the same 3algebra. The BLG model possesses a ghost mode because of its kinetic terms with indefinite signature. On the other hand, the Lie 3-algebra model of M-theory does not possess a kinetic term because it is defined as a zero-dimensional field theory like the IIB matrix model [41].

This action is invariant under the translation
where η I and η μ belong to u (1). This implies that eigen values of x I and a μ represent an eleven-dimensional space-time.
The action is also invariant under 16 kinematical supersymmetry transformations and the other fields are not transformed. ˜ belong to u(1) and satisfy Γ 012 ˜= ˜. ˜ and should come from sixteen components of thirty-two = 1 supersymmetry parameters in eleven dimensions, corresponding to eigen values ± 1 of Γ 012 , respectively, as in the previous subsection.
A commutation relation between the kinematical supersymmetry transformations is given by The action is invariant under 16 dynamical supersymmetry transformations, where Γ 012 = -. These supersymmetries close into gauge transformations on-shell, where gauge parameters are given by (▭) implies that a commutation relation between the dynamical supersymmetry transformations is up to the equations of motions and the gauge transformations.
The 16 dynamical supersymmetry transformations (▭) are decomposed as and thus a commutator of dynamical supersymmetry transformations and kinematical ones acts as where the commutator that acts on the other fields vanishes. Thus, the commutation relation for physical modes is given by where δ η is a translation.

Hermitian 3-algebra model of M-theory
In this subsection, we study the Hermitian 3-algebra models of M-theory [26]. Especially, we study mostly the model with the u(N ) ⊕ u(N ) Hermitian 3-algebra (▭).
The continuum action (▭) can be rewritten by using the triality of SO (8) and the SU (4) × U (1) decomposition [8], [43], [44] as where fields with a raised A index transform in the 4 of SU(4), whereas those with lowered one transform in the 4. A μba (μ = 0, 1, 2) is an anti-Hermitian gauge field, Z A and Z A are a complex scalar field and its complex conjugate, respectively. ψ A is a fermion field that satisfies and ψ A is its complex conjugate. E μνλ and E ABCD are Levi-Civita symbols in three dimensions and four dimensions, respectively. The potential terms are given by If we replace the Nambu-Poisson bracket with a Hermitian 3-algebra's bracket [19], [20], we obtain the Hermitian 3-algebra model of M-theory [26], where the cosmological constant has been deleted for the same reason as before. The potential terms are given by This matrix model can be obtained formally by a dimensional reduction of the = 6 BLG action [8], which is equivalent to ABJ(M) action [7], [45]The authors of [46], [47], [48], [49] studied matrix models that can be obtained by a dimensional reduction of the ABJM and ABJ gauge theories on S 3 . They showed that the models reproduce the original gauge theories on S 3 in planar limits., The Hermitian 3-algebra models of M-theory are classified into the models with u(m) ⊕ u(n) Hermitian 3-algebra (▭) and sp(2n) ⊕ u(1) Hermitian 3-algebra (▭). In the following, we study the u(N ) ⊕ u(N ) Hermitian 3-algebra model. By substituting the u(N ) ⊕ u(N ) Hermitian 3-algebra (▭) to the action (▭), we obtain S = Tr(-(2π) 2 In the algebra, we have set α = 2π k , where k is an integer representing the Chern-Simons level. We choose k = 1 in order to obtain 16 dynamical supersymmetries. V is given by we obtain an action that is independent of Chern-Simons level: as opposed to three-dimensional Chern-Simons actions.
If we rewrite the gauge fields in the action as A μ where , and { , } are the ordinary commutator and anticommutator, respectively. The u(1) parts of A μ decouple because A μ appear only in commutators in the action. b μ can be regarded as auxiliary fields, and thus A μ correspond to matrices X μ that represents three space-time coordinates in M-theory. Among N × N arbitrary complex matrices Z A , we need to identify matrices X I (I = 3, ⋯ 10) representing the other space coordinates in M-theory, because the model possesses not SO (8) but SU (4) × U (1) symmetry. Our identification is where X I and x I are su(N ) Hermitian matrices and real scalars, respectively. This is analogous to the identification when we compactify ABJM action, which describes N M2 branes, and obtain the action of N D2 branes [50], [7], [51]. We will see that this identification works also in our case. We should note that while the su(N ) part is Hermitian, the u(1) part is anti-Hermitian. That is, an eigen-value distribution of X μ , Z A , and not X I determine the space- time in the Hermitian model. In order to define light-cone coordinates, we need to perform Wick rotation: a 0 → -ia 0 . After the Wick rotation, we obtain where A 0 is a su(N ) Hermitian matrix.

DLCQ Limit of 3-algebra model of M-theory
It was shown that M-theory in a DLCQ limit reduces to the BFSS matrix theory with matrices of finite size [30], [31], [32], [33], [34], [35]. This fact is a strong criterion for a model of Mtheory. In [26], [28], it was shown that the Lie and Hermitian 3-algebra models of M-theory reduce to the BFSS matrix theory with matrices of finite size in the DLCQ limit. In this subsection, we show an outline of the mechanism.
DLCQ limit of M-theory consists of a light-cone compactification, x -≈ x -+ 2πR, where x ± = 1 2 ( x 10 ± x 0 ) , and Lorentz boost in x 10 direction with an infinite momentum. After appropriate scalings of fields [26], [28], we define light-cone coordinate matrices as We integrate out b μ by using their equations of motion.
A matrix compactification [52] on a circle with a radius R imposes the following conditions on Xand the other matrices Y : where U is a unitary matrix. In order to obtain a solution to (▭), we need to take N → ∞ and consider matrices of infinite size [52]. A solution to (▭) is given by Backgrounds Xare in the Lie 3-algebra case, whereas in the Hermitian 3-algebra case. A fluctuation x that represents u(N ) parts of Xand Ỹ is Each x (s) is a n × n matrix, where s is an integer. That is, the (s, t)-th block is given by We make a Fourier transformation, where x(τ) is a n × n matrix in one-dimension and RR = 2π. From (▭)-(▭), the following identities hold: ∑ t x s,t x ' t ,u = 1 x -, x s,t = 1 where tr is a trace over n × n matrices and V = ∑ s 1.
Next, we boost the system in x 10 direction: The DLCQ limit is achieved when T → ∞, where the "novel Higgs mechanism" [50] is realized. In T → ∞, the actions of the 3-algebra models of M-theory reduce to that of the BFSS matrix theory [27] with matrices of finite size, S = 1 where P, Q = 1, 2, ⋯ , 9.

Supersymmetric deformation of Lie 3-algebra model of M-theory
A supersymmetric deformation of the Lie 3-algebra Model of M-theory was studied in [53] (see also [54], [55], [56]). If we add mass terms and a flux term, From this action, we obtain various interesting solutions, including fuzzy sphere solutions [53].

Conclusion
The metric Hermitian 3-algebra corresponds to a class of the super Lie algebra. By using this relation, the metric Hermitian 3-algebras are classified into u(m) ⊕ u(n) and sp(2n) ⊕ u(1) Hermitian 3-algebras.
The Lie and Hermitian 3-algebra models of M-theory are obtained by second quantizations of the supermembrane action in a semi-light-cone gauge. The Lie 3-algebra model possesses manifest = 1 supersymmetry in eleven dimensions. In the DLCQ limit, both the models reduce to the BFSS matrix theory with matrices of finite size as they should.