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One-dimensional topological gravity is defined as a Gaussian integral as itspartition function. The Gaussian integral supplies a toy model as a simplerversion of one-matrix model that is well known to provide a description oftwo-dimensional topological gravity. The one-dimensional topological gravityinherits an integrable hierarchy structure as with two-dimensional topologicalgravity, yet it is the Burgers hierarchy rather than the Korteweg--de Vrieshierarchy. Making use of this fact, an extension of the one-dimensionaltopological gravity to an analogue of two-matrix model is investigated and theassociated partition function is shown to consist of a pair of partitionfunctions of one-dimensional topological gravity intertwined via theMoyal--Weyl product, which enables to provide an explicit formula for its freeenergy. The extended system shows a hierarchy structure interpreted as anoncommutative extension of the Burgers hierarchy. The relation tononcommutative U(1) gauge theory is suggested.

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