Curved Superspaces and Local Supersymmetry in Supermatrix Model (original) (raw)

2006, Progress of Theoretical Physics

In a previous paper, we introduced a new interpretation of matrix models, in which any d-dimensional curved space can be realized in terms of d matrices, and the diffeomorphism and the local Lorentz symmetries are included in the ordinary unitary symmetry of the matrix model. Furthermore, we showed that the Einstein equation is naturally obtained, if we employ the standard form of the action, S = −tr [Aa, A b ][A a , A b ] +• • •. In this paper, we extend this formalism to include supergravity. We show that the supercovariant derivatives on any d-dimensional curved space can be expressed in terms of d supermatrices, and the local supersymmetry can be regarded as a part of the superunitary symmetry. We further show that the Einstein and Rarita-Schwinger equations are compatible with the supermatrix generalization of the standard action. *) Because a covariant derivative introduces a new vector index, it is not an endomorphism. Therefore, it cannot be represented by a set of matrices. *) Ra b and R (a) b represent the same quantity. However, we distinguish them, because a and (a) obey different transformation laws. Specifically, a is transformed by the action of G, while (a) is not. *) Strictly speaking, because Aa is Hermitian, we should introduce the anticommutator { , } in Eq. (2. 10): *) As in §2.2, because Aa and Ψα are Hermitian, we should introduce the commutator and the anticommutator. Here we omit them for simplicity.