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Planar diagrams and Calabi-Yau spaces. (English) Zbl 1058.81057

Summary: 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 \(\text{tr }X^{p}\) and \(\text{tr }Y^{p}\) 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.

MSC:

81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
15B52 Random matrices (algebraic aspects)
15A90 Applications of matrix theory to physics (MSC2000)
32Q25 Calabi-Yau theory (complex-analytic aspects)
81T18 Feynman diagrams
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