Approximate treatment of noncommutative curvature in quartic matrix model

Abstract

We study a Hermitian matrix model with the standard quartic potential amended by a $\mathrm{tr}(R{\Phi }^2)$ term for fixed external matrix $R$. This is motivated by a curvature term in the truncated Heisenberg algebra formulation of the Grosse-Wulkenhaar model -- a renormalizable noncommutative field theory. The extra term breaks the unitary symmetry of the action and leads, after perturbative calculation of the unitary integral, to an effective multitrace matrix model. Accompanying the analytical treatment of this multitrace approximation, we also study the model numerically by Monte Carlo simulations. The phase structure of the model is investigated, and a modified phase diagram is identified. We observe a shift of the transition line between the 1-cut and 2-cut phases of the theory that is consistent with the previous numerical simulations and also with the removal of the noncommutative phase in the Grosse-Wulkenhaar model.

minor text changes; Sections 5 and 6 expanded; new data point in Figures 4 and 5; typos in (2.5) and (4.16) corrected; new references added

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