Zariski 3-Algebra Model of M-Theory

We review on Zariski 3-algebra model of M-theory. The model is obtained by Zariski quantization of a semi-light-cone supermembrane action. The model has manifest 1   supersymmetry in eleven dimensions and its relation to the supermembrane action is clear.

In this paper, we review one of the models, called Zariski 3-algebra model of M-theory.This model has manifest 1   supersymmetry in eleven dimensions and the relation to the supermembrane action is clear.We start with the fact found in [32] that the supermembrane action in a semi-light-cone gauge is a gauge theory based on a 3-algebra that is generated by the Nambu-Poisson bracket [13,14].The gauge theory's thirty-two supersymmetries form the 1   supersymmetry algebra in eleven dimensions.By performing the Zariski quantization, the action is second quantized and we obtain Zariski 3-algebra model of M-theory.

Supermembrane Action in a Semi-Light-Cone Gauge
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.The fundamental degrees of freedom in M-theory are supermembranes.The covariant supermembrane action in M-theory [35] is given by The action is also invariant under the -symmetry transformations, where If we fix the -symmetry (4) of the action by taking a semi-light-cone gauge [32]  012 , (6)     we obtain a semi-light-cone supermembrane action, where In [32], it is shown under an approximation up to the quadratic order in .
 and but exactly in , , , 2 4 , , where and , , 3, ,10 Majorana-Weyl fermion satisfying (6). is a Levi-Civita symbol in three dimensions and  is a cosmological constant.( 8) is invariant under 16 dynamical supersymmetry transformations, where .These supersymmetries close into gauge transformations on-shell, where gauge parameters are given by 0 and  are equations of motions of  A  and  , respectively, where (10) implies that a commutation relation between the dynamical supersymmetry transformations is up to the equations of motions and the gauge transforma-tions.This action is invariant under a translation, I  are constants.where The action is also invariant under 16 kinematical supersymmetry transformations , and the other fields are not transformed.
is a constant and satisfy 012  .and  should come from sixteen components of thirty-two     supersymmetry consists of remaining 16 target-space supersymmetries and transmuted 16 -symmetries in the semi-light-cone gauge [32,36,37].

   
A commutation relation between the kinematical supersymmetry transformations is given by  supersymmetry parameters in eleven dimensions, corresponding to eigen values of , respectively.This 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  is a translation. If we change a basis of the supersymmetry transformations as We obtain These supersymmetry transformations are summarised as

Zariski Quantization
In this section, we review the Zariski Quantization and apply it for the semi-light-cone supermembrane action (8).In [34], it is shown that the Zariski quantization is a second quantization and the Zariski quantized action reduces to the supermembrane action if the fields are restricted to one-body states.First, we define elements of linear spaces by where the basis where a is a real (complex) number.The coefficients   Y u  are functions over 3-dimensional spaces.Summation is defined naturally as linear spaces.
The quantum Zariski product is defined as Any polynomial can be decomposed uniquely as , , where is a real (complex) number and are irreducible normalized polynomials. where where N S   1, 2, , N  is the permutation group of  is the Moyal product defined by   where and run from 1 to 2.
We define derivatives on by derivatives with One can show that the quantum Zariski product is Abelian, associative and distributive, and the derivative is commutative and satisfies the Leibniz rule [34].
We define the Zariski quantized Nambu-Poisson bracket by where .By definition, the bracket is skew-symmetric.By using the above properties, one can show that it satisfies the Leibniz rule and the fundamental identity; for any .Thus, the Zariski quantized Nambu-Poisson bracket has the same Nambu-Poisson structure as the original Nambu-Poison bracket.
We define a metric for by where is a real (complex) number and is a normalized polynomial, whose monomial of the highest total degree has coefficient 1.
This metric is invariant under a gauge transformation generated by the Zariski quantized Nambu-Poisson bracket [34] as By performing the Zariski quantization of the supermembrane action in a semi-light-cone gauge (8), we obtain The Zariski quantization preserves the supersymmetries of the semi-light-cone supermembrane theory, because the quantum Zariski product is Abelian, associative and distributive, and admits a commutative derivative satisfying the Leibniz rule.

Conclusion
complex coefficients.The summation is taken over all the polynomials of two valuables  1 2 u satisfies au u Z aZ 

Zariski 3 -
algebra model of M-theory has manifest  supersymmetry in eleven dimensions because Zariski quantization preserves the supersymmetry of the supermembrane action in the semi-light-cone gauge.The relation between the model and the supermembrane action is clear: If the fields are restricted to one-body states, the model reduces to the supermembrane action.