Matrix models vs. Seiberg–Witten/Whitham theories

Abstract

We discuss the relation between matrix models and the Seiberg-Witten type (SW) theories, recently proposed by Dijkgraaf and Vafa. In particular, we prove that the partition function of the Hermitean onematrix model in the planar (large N ) limit coincides with the prepotential of the corresponding SW theory. This partition function is the logarithm of a Whitham τ -function. The corresponding Whitham hierarchy is explicitly constructed. The double-point problem is solved. 1.It is well known that partition functions of matrix models are τ -functions of integrable hierarchies of the Toda type [1]. In the specific double scaling limit, these τ -functions become τ -functions of various reduction of the KP hierarchy [2]. If one makes the simplest, large-N (planar) limit, the partition function becomes the τ -function of the dispersionless Toda hierarchy, which in turn becomes the τ -function of the dispersionless (reductions of) KP hierarchy [3] after performing the continuum limit (which basically means working nearby a singularity of the partition function). All these dispersionless hierarchies are just Whitham equations over trivial solutions to integrable (Toda, KP) hierarchies.When solving matrix models, most attention was paid to one-cut solutions where the limiting eigenvalue distribution spans one interval on the real axis [4]. The results on multi-cut solutions [5] were few [6,7,8]. Recently, Dijkgraaf and Vafa proposed [9] the new insight on the multi-cut large-N limit of matrix models. Namely, they associated this limit with a Riemann surface and some related SW system. Its prepotential, which we prove here to be the logarithm of the large-N partition function, is typically associated with the logarithm of some Whitham τ -functions [10,11]. This hints that the matrix matrix model in the large N limit of multi-cut type describes the Whitham system over a non-trivial, finite-gap solution to integrable (Toda, KP) hierarchy. In particular, this solution passes to a finite-gap solution of (reductions of) the KP hierarchy in the continuum limit.In this paper, we restrict ourselves with the simplest example of the Hermitean one-matrix model. We show that coefficients of the potential of the model gives rise to Whitham flows and manifestly construct this Whitham system. In fact, the authors of [9] associated the N = 1 SUSY gauge theory studied in [12] with the SW system related to the multi-cut planar limit of matrix models. From the point of view of N = 1 SUSY theory, these coefficients must be identified with couplings in the tree superpotential, while the SW moduli are associated with v.e.v.'s of the gluino condensates. This gives an interpretation of the results of [12] in the Whitham hierarchy terms. *

Unoriented strings, loop equations, andN=1superpotentials from matrix models

“…We have calculated the Feynman diagrams for several higher orders and higher vertices and confirmed this relationship in those cases 6 . It would be nice to have a purely combinatorial proof of (5.42), that would not rely on the loop equation.…”

“…The conjecture has since been extended to a number of other cases [2,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26] and has been derived from SU(N) gauge theory [27,28,29].…”

BGWM as second constituent of complex matrix model

“…From technical point of view the road is now open for search of two different generalizations: to Dijkgraaf-Vafa models [50,51], which are not fully specified by the Virasoro constraints alone and rely upon intriguing and under-developed theory of check-operators [49], and to more interesting unitary-matrix models with Itzykson-Zuber measures and further to Kazakov-Migdal multi-matrix models [34]- [37], important both for Yang-Mills theory and for the theory of integer partitions. Putting all these very different problems into the same context, moreover, underlined by the well established theory of free fields on Riemann surfaces [60], is a challenging and a promising perspective.…”

“…If other time or many times are shifted, then arbitrariness is unavoidable, see [49] for its full description. If partly fixed and parameterized by several arbitrary variables, it provides the family of Dijkgraaf-Vafa non-Gaussian partition functions [48,50,51]. 4 Gaussian phase.…”

“…There are interesting situations, when the choice of P m is not adjusted to the shift in this way: the best known example is provided by Dijkgraaf-Vafa partition functions [50,51], where projector is the same P −1 as in Gaussian model, but the shift ∆Ĵ H (z) = dW (z) is generated by polynomial W (z) of degree higher than two.…”