Planar diagrams and Calabi-Yau spaces

Large $N$ geometric transitions and the Dijkgraaf-Vafa conjecture suggest a deep relationship between the sum over planar diagrams and Calabi-Yau threefolds. We explore this correspondence in details, explaining how to construct the Calabi-Yau for a large class of $M$-matrix models, and how the geometry encodes the correlators. We engineer in particular two-matrix theories with potentials $W(X,Y)$ that reduce to arbitrary functions in the commutative limit. We apply the method to calculate all correlators tr$X^p$ and tr$Y^p$ in models of the form $W(X,Y)=V(X)+U(Y)-XY$ and $W(X,Y)=V(X)+YU(Y^{2})+XY^{2}$. The solution of the latter example was not known, but when $U$ is a constant we are able to solve the loop equations, finding a precise match with the geometric approach. We also discuss special geometry in multi-matrix models, and we derive an important property, the entanglement of eigenvalues, governing the expansion around classical vacua for which the matrices do not commute.

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Published 1 January 2003