We prove that the quasiclassical tau-function of the multi-support solutions to matrix models, proposed recently by Dijkgraaf and Vafa to be related to the Cachazo-Intrilligator-Vafa superpotentials of the N = 1 supersymmetric Yang-Mills theories, satisfies the Witten-Dijkgraaf-VerlindeVerlinde equations. IntroductionThe Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations [1] in the most general form can be written [2] as system of algebraic relationsfor the third derivativesof some function F(T). Have been appeared first in the context of topological string theories [1], they were rediscovered later on in much larger class of physical theories where the exact answer for a multidimensional theory could be expressed through a single holomorphic function of several complex variables [2,3,4,5,6,7,8].Recently, a new example of similar relations between the superpotentials of N = 1 supersymmetric gauge theories in four dimensions and free energies of matrix models in the planar limit was proposed [9,10]. It has been realized that superpotentials in some N = 1 four-dimensional Yang-Mills theories can be expressed through a single holomorphic function [9] that can be further identified with free energy of the multi-support solutions to matrix models in the planar limit [10]. A natural question which immediately arises in this context is whether these functions -the quasiclassical tau-functions, determined by multi-support solutions to matrix models, satisfy the WDVV equations? In the case of positive answer this is rather important, since multi-support solutions to the matrix models can play § E-mail: chekhov@mi.ras.ru ¶ E-mail: mars@lpi.ac.ru E-mail: mironov@itep.ru; mironov@lpi.ac.ru * * E-mail: dmtrvass@gate.itep.ru 1 a role of "bridge" between topological string theories and Seiberg-Witten theories [11] which give rise to two different classes of solutions to the WDVV equations (see, e.g., [12] and [13,14]).This question was already addressed in [15], where it was shown that the multicut solution to one-matrix model satisfies the WDVV equations. However, this was verified only perturbatively and, what is even more important, for a particular non-canonical! (and rather strange) choice of variables.In this paper, we demonstrate that the quasiclassical tau-function of the multi-support solution satisfies the WDVV equations as a function of canonical variables identified with the periods and residues of the generating meromorphic one-form dS [16]. An exact proof of this statement consists of two steps. The first, most difficult step is to find the residue formula for the third derivatives (2) of the matrix model free energy. Then, using an associative algebra, we immediately prove that free energy of multi-support solution satisfies WDVV equations, upon the number of independent variables is fixed to be equal to the number of critical points in the residue formula.In sect. 2, we define the free energy of the multi-support matrix model in terms of the quasiclassical tau-function [16] along the line of [17,18,19]. In sec...
A novel functorial relationship in perturbative quantum field theory is pointed out that associates Feynman diagrams (FD) having no external line in one theory Th 1 with singlet operators in another one Th 2 having an additional U (N ) symmetry and is illustrated by the case where Th 1 and Th 2 are respectively the rank r − 1 and the rank r complex tensor model. The values of FD in Th 1 agree with the large N limit of the Gaussian average of those operators in Th 2 . The recursive shift in rank by this FD functor converts numbers into vectors, then into matrices, then into rank 3 tensors and so on. This FD functor can straightforwardly act on the d dimensional tensorial quantum field theory (QFT) counterparts as well. In the case of rank 2-rank 3 correspondence, it can be combined with the geometrical pictures of the dual of the original FD, namely, equilateral triangulations (Grothendieck's dessins d'enfant) to form a triality which may be regarded as a bulk-boundary correspondence.
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