$\beta$-deformation for matrix model of M-theory

A new class of deformation of the matrix model of M-theory is considered. The deformation is analogous to the so-called $\b$-deformation of $D=3+1$, $\mN=4$ Super Yang-Mills theory, which preserves the conformal symmetry. It is shown that the deformed matrix model can be considered as a matrix model of M-theory on a certain curved background in eleven-dimensional supergravity, under a scaling limit involving the deformation parameter and $N$ (the size of the matrices). The background belongs to the so-called pp-wave type metric with a non-constant four-form flux depending linearly on transverse coordinates. Some stable solutions of the deformed model are studied, which correspond to membranes with the torus topology. In particular, it is found that apparently distinct configurations of membranes, having different winding numbers, are indistinguishable in the matrix model. Simultaneous introduction of both $\b$-deformation and mass-deformation is also considered, and, in particular, a situation is found in which the stable membrane configuration interpolates between a torus and a sphere, depending on the values of the deformation parameters.