Volume-Preserving Diffeomorphism and Nambu-Poisson Bracket . 7

From previous subsection, we can understand the noncommutative gauge fields theory can be described by Moyal product. If we focus on the large NS-NS B field case, theory is handled by Poisson bracket, which is the generator of Area-Preserving Diffeomorphism (APD). In general. for higher ranks field background, they need a general Poisson bracket, which is called Nambu-Poisson bracket [17–21], to be the generator of Volume-Preserving Diffeomorphism (VPD).

To understand the reason why VPD emerges, we can think in following way. The original worldvolume theory has diffeomorphism symmetry:

x^a→ ´x^a = ´x^a(x^a). (1.18)

When we consider the theory in large field background, the original diffeomorphism sym-metry will be broken by background field, the remaining symsym-metry is volume-preserving diffeomorphism. The n-dimensional volume-preserving diffeomorphism is the reduced symmetry of n-dimensional general coordinate diffeomorphism, which is described by (infinitesimal transformation):

x^a→ ´x^a = ´x^a(x^a) = x^a+ κ^a, ∂aκ^a= 0. (1.19) where a = 0, 1, ..., n − 1. The n-dimensional volume-preserving diffeomorphism can be understood as that this transformation parameter κ^a has additional constraint as shown

in (1.19). To see how this constraint gives rise to the volume-preserving, we can investi-gate the Jacobian of coordinate transformation. For example, we can find the Jacobian of coordinate transformation for n=2 reads

ǫ^ab∂_ax´⁰∂_bx´¹ ={´x⁰, ´x¹}. (1.20) We consider the coordinate transformation in (1.19), after simple calculation, we can find:

{´x⁰, ´x¹} = 1 + ∂aκ^a+ O(κ²). (1.21) Therefore we can see the constraint ∂aκ^a = 0 makes area-preserving. Moreover, higher ranked volume-preserving transformation can be generated by generalize Nambu-Poisson bracket, defined by

{f¹, f2,· · · , fⁿ} ≡ ǫ^a1a2···an∂a1f1∂a2f2· · · ∂^anfn. (1.22) Hence, we can define the VPD transformation as follows

δΛ1,...,Λ_n−1x^a ={Λ1, ..., Λ_n−1, x^a} = ǫ^{a1···an−1a}∂a1Λ1· · · ∂a_n−1Λ_n−1 ≡ κ^a. (1.23) In the special case of APD (n=2):

δΛx^a={Λ, x^a}pb = κ^a. (1.24) This is the simplest case in this kind of symmetry transformation. We can see the Nambu-Poisson bracket is the generator of VPD.

Now we can ask the next question; what is the field theory with VPD? In fact, we already saw the example of field theory with APD in previous subsection. We can find the symmetry transformation is generated by Poisson bracket. We will see more examples in next three chapters.

1.3 A Review of M Theory

In order to understand more nonperturbative effect of superstring theory, people start to study the M theory. M theory is the complete picture of string theory. The five different perturbative string theories and eleven dimensional supergravity theory can be under-stood as the different descriptions of M theory. For example, M theory can be underunder-stood as strong coupling limit of type IIA superstring theory in one higher dimension. Hence, the low energy effective theory of M theory is eleven dimensional supergravity theory.

From the eleven-dimensional superalgebra analysis, there are two kind high-dimensional central charges [22]. They are carried by M2-brane and M5-brane, which are the extended objects 2-brane and 5-brane in eleven dimensions. Following the analysis of 2-brane and 5-brane soliton solutions in eleven-dimensional supergravity theory, we can know the field contents of the effective worldvolume theory of M2-brane and M5-brane. The action of effective field theory for single M2-brane and M5-brane is well known. The M2-brane ef-fective action is the generalized Nambu-Goto action. The efef-fective action of M5-brane is more difficult because it involves the self-dual two-form gauge potential [23–28]. Recently, there are several interesting papers about self-dual gauge theory [29, 30].

The theory also have a background form fields as string does, and it is the three form field background. The M2-brane couple electrically to the 3-form field. As the research of D-brane in NS-NS B field background, people try to generalize the research into M theory.

For example, people [31] tried to study the quantization of open membrane in large C-field background, and they found similar noncommutative behavior as in the open string case because quantization processes naturally adopts Poisson structure. Moreover, people [32]

calculated the scattering amplitudes of open membrane in large C-field background which can be described by Nambu-Poisson algebra (or VPD gauge symmetry). It gives the candidate of generalization of Poisson bracket and Moyal product. To study the M theory in large C-filed background helps us understand the way to generalize Moyal product, which gives the way to quantize string. Hence, people try to apply these researches to understand how to quantize membrane. To study M5 in large C-field background has more interesting physic phenomena. This kind theory includes the self-dual two form, the non-abelian gauge algebra and new action form which is different from PST M5 action [25]. We will discuss this topic in next chapter.

Chapter 2

M5 in Large C-Field Background

Recently, Bagger, Lambert and Gustavsson imposed the Lie-3 algebra into Basu-Harvey BPS system to construct the theory of multiple membranes [33–36]. It is called BLG model, which describes the multiple M2-branes system. In this articles, we will not give any more detail of BLG theory. Latter, people [1, 2] started to impose the Nambu-Poisson structure into the three internal dimensions of BLG model, then they found the new description of single M5-brane theory. We denote it as Nambu-Poisson (NP) M5 theory.

2.1 Nambu-Poisson M5 Theory

The Nambu-Poisson algebra is an infinite dimensional Lie-3 algebra, which is used to de-scribe the algebra in BLG model. People [1,2] consider the additional three internal space dimensions (N ) with 3-dimensional volume-preserving diffeomorphism, which define the space of Nambu-Poisson bracket. Moreover, the worldvolume of multiple M2-branes (M) and the 3 internal dimensions N together can be identified as the worldvolume of M5 theory (M × N ).

These processes will divide the worldvolumes dimensions of M5 into two parts:

{x^µ; y^˙µ} = {x⁰, x¹, x²; y^˙1, y^˙2, y^˙3}. (2.1) Here, the coordinate x^µ label the direction on M, which are the longitudinal directions of M2-branes. Another coordinates y^˙µ label the internal directions on N , which are the space of the volume-preserving diffeomorphism. Roughly speaking, this effective M5 description have 3-dimensional VPD, which is the main characteristic of theory in large C

field background. Hence, the Nambu-Poisson structure in M5 theory is the first evidence of M5 in large C field background.

The fields contents of NP M5 theory are self-dual two form (b_{µ ˙µ}, b_{˙µ ˙ν}), five scalar fields X^I, and chiral Majorana fermion Ψ. The two form gauge fields are self-dual, so the degree of freedom (DOF) of two form fields are ⁶₂ = 3 and we do not need b_µν in this theory. The five scalar fields are the DOF of M5-brane on the transverse directions.

The Majorana fermion is reduction from 11 dimension which satisfies the 6-dimension chirality condition Γ⁷Ψ = Ψ. Hence the DOF of fermion are ¹₂2^[112] = 16. The one half of fermion DOF¹ (¹⁶₂ ) are equal to the bosonic DOF (3 + 5) in NP M5 theory, which is a result of supersymmetry.

In next section, we will give the full action to describe the dynamics of these fields.

We will not give all the details about how to derive the action from the BLG theory.

The main process of action calculation is to replace the Lie-3 bracket by Nambu-Poisson bracket: [•, •, •] → g²{•, •, •}. The other details can be found in the papers [1, 2].