Open Access
July 2003 Planar diagrams and Calabi-Yau spaces
Frank Ferrari
Adv. Theor. Math. Phys. 7(4): 619-665 (July 2003).


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 trXp and trYp 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.


Download Citation

Frank Ferrari. "Planar diagrams and Calabi-Yau spaces." Adv. Theor. Math. Phys. 7 (4) 619 - 665, July 2003.


Published: July 2003
First available in Project Euclid: 4 April 2005

zbMATH: 1058.81057
MathSciNet: MR2039033

Rights: Copyright © 2003 International Press of Boston

Vol.7 • No. 4 • July 2003
Back to Top