We discuss the AdS/CFT duality from the perspective of integrable systems and establish a direct relationship between the dimension of single trace local operators composed of two types of scalar fields in N = 4 super YangMills and the energy of their dual semiclassical string states in AdS 5 × S 5 . The anomalous dimensions can be computed using a set of Bethe equations, which for "long" operators reduces to a Riemann-Hilbert problem. We develop a unified approach to the long wavelength Bethe equations, the classical ferromagnet and the classical string solutions in the SU (2) sector and present a general solution, governed by complex curves endowed with meromorphic differentials with integer periods. Using this solution we compute the anomalous dimensions of these long operators up to two loops and demonstrate that they agree with string-theory predictions.

The exact Seiberg-Witten (SW) description of the light sector in the N = 2 SUSY 4d Yang-Mills theory [1] is reformulated in terms of integrable systems and appears to be a Gurevich-Pitaevsky (GP) [2] solution to the elliptic Whitham equations. We consider this as an implication that dynamical mechanism behind the SW solution is related to integrable systems on the moduli space of instantons. We emphasize the role of the Whitham theory as a possible substitute of the renormalization-group approach to the construction of low-energy effective actions.

We present new evidence for the conjecture that BPS correlation functions in the N = 2 supersymmetric gauge theories are described by an auxiliary two dimensional conformal field theory. We study deformations of the N = 2 supersymmetric gauge theory by all gauge-invariant chiral operators. We calculate the partition function of the N = 2 theory on R 4 with appropriately twisted boundary conditions. For the U (1) theory with instantons (either noncommutative, or D-instantons, depending on the construction) the partition function has a representation in terms of the theory of free fermions on a sphere, and coincides with the tau-function of the Toda lattice hierarchy. Using this result we prove to all orders in string loop expansion that the effective prepotential (for U (1) with all chiral couplings included) is given by the free energy of the topological string on CP 1 .Gravitational descendants play an important rôle in the gauge fields/string correspondence. The dual string is identified with the little string bound to the fivebrane wrapped on the two-sphere. We also discuss the theory with fundamental matter hypermultiplets.February 2003 † On leave of absence from: ITEP, Moscow, 117259, Russia INTRODUCTIONThe Holy Grail of the theoretical physics is the nonperturbative theory which includes quantum gravity, sometimes called M-theory [1]. The current wisdom says there is no fundamental coupling constant. Whatever (string) perturbation theory is used depends on the particular solution one expands about. The expansion parameter is one of the geometric characteristics of the background. It is obviously interesting to look for simplified string and field theoretic models, which have string loop expansion, and where the string coupling constant has a geometric interpretation. String expansion in gauge theoryLarge N gauge theories are the most popular, and the most elusive models with string representation. In the gauge/string duality [2][3] one matches the connected correlation functions of the gauge theory observables with the partition function of the string theory in the bulk. The closed string dual has 1 N 2 as a string coupling constant. Advances in the studies of the type II string compactifications on Calabi-Yau manifolds led to another class of models, which in the low-energy limit reduce to N = 2 supersymmetric gauge theories, with a novel type of string loop expansion. Namely, certain couplings F g in the low-energy effective action are given by the genus g partition function of the topologically twisted string on Calabi-Yau. The gauge group of the N = 2 theory does not have to be U (N ) with large N . It is determined by the geometry of Calabi-Yau manifold [4][5] [6].The rôle of effective string coupling is played by the vev of the graviphoton field strength [7], which is usually assumed to be constant [8]. Generalized Scherk-Schwarz constructionIn this paper we shall explain that there exists another, natural from the gauge theory point of view, way to flesh out these couplings. The idea is to put the t...

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The free field representation or "bosonization" rule1 for Wess-Zumino-Witten model (WZWM) with arbitrary Kac-Moody algebra and arbitrary central charge is discussed. Energy-momentum tensor, arising from Sugawara construction, is quadratic in the fields. In this way, all known formulae for conformal blocks and correlators may be easily reproduced as certain linear combinations of correlators of these free fields. Generalization to conformal blocks on arbitrary Riemann surfaces is straightforward. However, projection rules in the spirit of Ref. 2 are not specified. The special role of βγ systems is emphasized. From the mathematical point of view, the construction involved represents generators of Kac-Moody (KM) algebra in terms of generators of a Heisenberg one. If WZW Lagrangian is considered as d−1 of Kirillov form on an orbit of KM algebra,3 then the free fields of interest (i.e. generators of the Heisenberg algebra) diagonalize Kirillov form and the action. Reduction of KM algebra within the same construction should naturally lead to arbitrary coset models.

The second derivatives of prepotential with respect to Whitham time-variables in the Seiberg-Witten theory are expressed in terms of Riemann theta-functions. These formulas give a direct transcendental generalization of algebraic ones for the Kontsevich matrix model. In particular case they provide an explicit derivation of the renormalization group (RG) equation proposed recently in papers on the Donaldson theory.

A 1-matrix model is proposed, which nicely interpolates between doublescaling continuum limits of all multimatrix models. The interpolating partition function is always a KP τ -function and always obeys L −1 -constraint and string equation. Therefore this model can be considered as a natural unification of all models of 2d-gravity (string models) with c ≤ 1.

We introduce a new 1-matrix model with arbitrary potential and the matrix-valued background field. Its partition function is a τ -function of KPhierarchy, subjected to a kind of L −1 -constraint. Moreover, partition function behaves smoothly in the limit of infinitely large matrices. If the potential is equal to X K+1 , this partition function becomes a τ -function of K-reduced KP-hierarchy, obeying a set of W K -algebra constraints identical to those conjectured in [1] for double-scaling continuum limit of (K − 1)-matrix model. In the case of K = 2 the statement reduces to the early established [2] relation between Kontsevich model and the ordinary 2d quantum gravity . Kontsevich model with generic potential may be considered as interpolation between all the models of 2d quantum gravity with c < 1 preserving the property of integrability and the analogue of string equation.