Noncommutativity in two-matrix model extension of one-dimensional topological gravity

One-dimensional topological gravity is defined as a Gaussian integral as its partition function. The Gaussian integral supplies a toy model as a simpler version of one-matrix model that is well known to provide a description of two-dimensional topological gravity. The one-dimensional topological gravity inherits an integrable hierarchy structure as with two-dimensional topological gravity, yet it is the Burgers hierarchy rather than the Korteweg--de Vries hierarchy. Making use of this fact, an extension of the one-dimensional topological gravity to an analogue of two-matrix model is investigated and the associated partition function is shown to consist of a pair of partition functions of one-dimensional topological gravity intertwined via the Moyal--Weyl product, which enables to provide an explicit formula for its free energy. The extended system shows a hierarchy structure interpreted as a noncommutative extension of the Burgers hierarchy. The relation to noncommutative U(1) gauge theory is suggested.