Four-algebraic extension of the IIB matrix model

We make a four-algebraic extension of the IIB matrix model. The extension can be made by any Lie 4-algebra. The four-algebraic model has the same supersymmetry as the IIB matrix model, and hence as type IIB superstring theory. The four-algebraic model contains twelve bosonic matrices; two of these will be identified with two extra dimensions that characterize F-theory. We construct a Lie 4-algebra that incorporates $u(N)$ Lie algebra and analyze the model explicitly by choosing it. We have three phases in the model with that specific algebra. In the first phase, it reduces to the original IIB matrix model. In the second phase, it reduces to a simple supersymmetric model. In the third phase, it reduces to a model that describes only the dynamics of the two matrices representing the torus.