Matrix Model on Z-Orbifold

Six dimensional compactification of the type IIA matrix model on the Zorbifold is studied. Introducing a Z3 symmetry properly on the three mirror images of fields in the N -body system of the supersymmetric D0 particles, the action of the Matrix model compactified on the Z-orbifold is obtained. Although N=1 supersymmetry is explicitly demonstrated in the Matrix model, a naive counting of the number of bosonic and fermionic physical degrees of freedom may suggest the existence of N=2 supersymmetry. Introduction Recent development of the string theory makes it possible to investigate the strong coupling regime of the theory as well as the inter-relationship between various string models (or phases) by the duality. To establish the duality (exchange of “electricity” and “magnetism”) it is necessary to have a “magnetic” object of string, in addition to the original ”electric” string. Once it is recognized by Witten [1] that the dynamics of these extended “magnetic” objects, D-branes [2], is described by the dimensionally reduced Supersymetric Yang-Mills Theory (SYM), the non-perturbative study of string becomes more familiar and more realistic. In particular, the dynamics of the simplest pointlike D0 branes in 11 dimensions may play the fundamental role in such a study [3]. Since the coordinates of a gas, consisting ofN D0-branes, are represented by N×N hermitian matrices, its dynamics is given in terms of the SYM theory dimensionally reduced to one temporal dimension, that is the quantum mechanics of the SUSY Matrix model. Hereafter, we refer this model simply as the Matrix model. If the Matrix model is quantized in the light-cone gauge, the model reproduces the discrete version of the super-membrane theory, since the discretized row and column indices σ and ρ of the Matrix, X(t)σρ, naturally become the coordiantes of a membrane, X(t, σ, ρ) [4]. Compactification of the Matrix model is an interesting topic, in particular when the model is located on the orbifold space, i.e., the space of torus divided by its discrete symmetry. First, since the 11d type IIA (yet-unknown) M-theory compactified on a simplest orbifold space, S/Z2, is shown to be connected with the 10d E(8) × E(8) heterotic string theory [5], from which we obtain the well-known E(6) grand unified theory, after compactification of 6d space by Calabi-Yau manifold [6]. The Matrix model describing this (yet-unknown) M-theory may be the type IIA Matrix model. Then, the compactification of this type IIA Matrix model on the orbifold space S/Z2 is demonstrated explicitly by [7], [8], [9], [10], and others. Second, when we intend to obtain “realisitic” 4d string models, by compactifying the extra six dimensions in the 10d heterotic string models, the orbifold compactification is a good method of breaking the supersymmetries. Since the orbifold space does not differ so much from the torus compactification, we can explicitly estimate various physical quantities (amplitudes) in this orbifold compactification, without much difficulty compared with the torus compactification. 2 Among the orbifold compactifications, we have found various realistic models with three generations, having extra U(1) gauge groups and N=1 supersymmetry [11], [12], [13]. In this paper, we study the second problem, that is the 6 dimensional compactification of type IIA Matrix model by the Z-orbifold [14], the most fundamental space among the various orbifold spaces. The model is obtained from the 3N×3N Matrix model, by imposing the Z3 symmetry of the Z-orbifold. It is explicitly checked that there exists the N=1 supersymmetry in the obtained model, but the fields contents of the model suggests the N=2 structure, and therefore there may exist an additional “supersymmetry” in this model. By generalizing our analysis to the more elaborate orbifold spaces, we may find the more realisitic analysis of string model on the basis of the standard model with three generations of quarks and leptons. This work was presented by one of the authors (A. M.) at the JPS meetings held at Niigata University in September 2000 and at Okinawa Kokusai University in September 2001. Matrix Model on Z-Orbifold We start with the type IIA Matrix model, i.e. the 10 dimensional Super YangMills theory (SYM) dimensionally reduced to the single temporal dimension. The action of this model is written by SM = Tr ∫ dτ( 1 2R (DτX ) − R 4 [X , X ] +ΘDτΘ+ iRΘ ΓI [X I ,Θ]). (1) Here, I, J = 0, . . . , 9 label the Minkowski space-time indices, R is the radius of the compactified 11-th dimension, ΓI is the 10d gamma matrix, and X I and Θ are 3N × 3N hermitian matrices, representing the gauge fields and the gaugino fields of the original SYM, respectively. The gaugino fields Θ are given by the Majorana-Weyl spinors, having eight degrees of freedom on the mass shell. In the terminology of the 3N -body D0 brane system, X and Θ are, respectively, the bosonic and fermionic coordinates of the D0 branes. The Dτ is the covariant derivative defined by Dτ ≡ ∂τ − i[Aτ , ]. (2) Starting with this action, we study the action of the Matrix model compactified on the Z-orbifold of T/Z3. For this purpose, it is convenient to use the complex 3 notations for the six spacial dimensions which we are going to compactify. By introducing complex coordinates Zi, (i = 1, 2, 3), Zi ≡ X 2i+2 + iX, (3) the torus lattice T of the Z-orbifold is defined by Zi ≃ Zi + r ≃ Zi + re , (4) where r is the compactification size of the six spacial dimensions. This lattice has a Z3 symmetry, under Zi ≃ e Zi, (5) for i = 1, . . . , 3, so that we can divide the torus lattice by this Z3 symmetry, and obtain the Z-orbifold. Denoting uncompactified and compactified coordinates by X // and Zi, respectively (μ = 0, . . . , 3 and i = 1, . . . , 3), where the complex coordinates Zi are assumed to satisfy the condition of the torus lattice (4). We have to impose furthermore the Z3 symmetry (5) on the bosonic as well as the fermionic coordinates. Under the Z3 symmetry, three points (or three mirror images) on the torus are identified. Corresponding to this identification we have to prepare three copies (or three mirror images) of the fields, and identify them by the Z3 symmetry up to the complex phases. Therefore, we start from the 3N×3N matrices for the N -body system of the D0 branes. Now the Z3 invariance we impose on the bosonic and fermionic fields are given as follows: X // = MX // M , (6) Zi = αiMZiM , (7) Θ = α̂MΘM , (8) where the matrix M is the generator of the Z3 transformation on the fields, and it satisfys M = 13N×3N . The complex phases αj and α̂ for the bosonic and fermionic fields respectively appearing under the Z3 transformation also satisfy M 3 = 1 and (αj) 3 = (α̂) = 1. Then, M , αj and α̂ can be written as follows: