DV and WDVV

Abstract

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...

Instantons in Non-Critical Strings From the Two-Matrix Model

“…This implies the following identities, which have been derived in [50] (for more explicit derivation see [52], [53])…”

Gauge theories as matrix models

“…The dependence on smooth periods (2.18) is far more transcendental and, on quasiclassical level, the knowledge is basically exhausted by the gradient formulas (2.21) and their consequences, like residue formulas, the WDVV equations, etc (see e.g. [38,23,25]). …”

Матричные Модели, Комплексная Геометрия И Интегрируемые Системы. I

“…Голоморфные дифференциалы. Выведем формулы для третьих производ-ных препотенциала F, следуя способу, предложенному Кричевером [22] и явно при-мененному в работе [26]. Заметим сначала, что производные элементов T ij матрицы периодов (4.10) могут быть выражены через интеграл по границе ∂Σ разрезанной римановой поверхности Σ:…”

“…Это можно просто проигнорировать, если требуются только формулы вычетов для третьих производных (к рассмотрению которых мы перейдем в п. 4.5), например для решения уравнений Виттена-Дийкграафа-Верлинде-Верлинде [26]. Простейший способ обойти эти слож-ности -рассматривать пару отмеченных точек ∞ и ∞ − как вырожденную "ручку".…”

“…Точки ветвления в z-плоскости определяются нулями дифференциала dz, или уравнением Fz = 0. Рассматривая простейший невырожденный случай кривой (5.22) 26) легко убедиться, что всего существует девять точек ветвления в z-плоскости, не считая бесконечной точки z = ∞ (естественно, к тому же выводу приводит изучение формулы Кардано или теорем об индексе, см. ниже).…”