Alexander M. Polyakov

Gauge Theory Correlators from Non-Critical String Theory

Abstract We suggest a means of obtaining certain Greens functions in 3+1-dimensional N =4 supersymmetric Yang-Mills theory with a large number of colors via non-critical string theory. The non-critical string theory is related to critical string theory in anti-deSitter background. We introduce a boundary of the anti-deSitter space analogous to a cut-off on the Liouville coordinate of the two-dimensional string theory. Correlation functions of operators in the gauge theory are related to the dependence of the supergravity action on the boundary conditions. From the quadratic terms in supergravity we read off the anomalous dimensions. For operators that couple to massless string states it has been established through absorption calculations that the anomalous dimensions vanish, and we rederive this result. The operators that couple to massive string states at level n acquire anomalous dimensions that grow as 2 ng YM 2N 1/2 for large `t Hooft coupling. This is a new prediction about the strong coupling behavior of large N SYM theory.

Infinite conformal symmetry in two-dimensional quantum field theory

We present an investigation of the massless, two-dimentional, interacting field theories. Their basic property is their invariance under an infinite-dimensional group of conformal (analytic) transformations. It is shown that the local fields forming the operator algebra can be classified according to the irreducible representations of Virasoro algebra, and that the correlation functions are built up of the “conformal blocks” which are completely determined by the conformal invariance. Exactly solvable conformal theories associated with the degenerate representations are analyzed. In these theories the anomalous dimensions are known exactly and the correlation functions satisfy the systems of linear differential equations.

Quantum Geometry of Bosonic Strings

We develop a formalism for computing sums over random surfaces which arise in all problems containing gauge invariance (like QCD, three-dimensional Ising model etc.). These sums are reduced to the exactly solvable quantum theory of the two-dimensional Liouville lagrangian. At D = 26 the string dynamics is that of harmonic oscillators as was predicted earlier by dual theorists, otherwise it is described by the nonlinear integrable theory.

Pseudoparticle Solutions of the Yang-Mills Equations

We find regular solutions of the four dimensional euclidean Yang-Mills equations. The solutions minimize locally the action integrals which is finite in this case. The topological nature of the solutions is discussed.

A semi-classical limit of the gauge/string correspondence

Abstract A world-sheet sigma model approach is applied to string theories dual to four-dimensional gauge theories, and semi-classical soliton solutions representing highly excited string states are identified which correspond to gauge theory operators with relatively small anomalous dimensions. The simplest class of such states are strings on the leading Regge trajectory, with large spin in AdS 5 . These correspond to operators with many covariant derivatives, whose anomalous dimension grows logarithmically with the space–time spin. In the gauge theory, the logarithmic scaling violations are similar to those found in perturbation theory. Other examples of highly excited string states are also considered.

Quark confinement and topology of gauge theories

Abstract The phenomenon of quark confinement is known to be connected with the restoration of apparently broken gauge symmetry. In this paper we focus on a special mechanism which is responsible for such restoration. The major suggestion is that in the treatment of infrared problems certain classical field trajectories are of paramount importance, which trajectories connect seemingly degenerate vacua thereby eliminating the degeneracy. Initially, we demonstrate this mechanism in certain simple non-gauge models. In order of increasing difficulty, we next examine compact quantum electro-dynamics in 2 + 1 space-time dimensions- essentially a variant of unified models of the Georgi-Glashow type. In this case we prove that charge is confined and show that the force between two charges is independent of the distance between them. For small values of the fine structure constant this force is explicitly evaluated. Finally, we turn to the more realistic case of a 3 + 1 dimensional non-Abelian gauge theory, and analyse the contribution of a single pseudoparticle to the correlation functions. It is proposed that the quantum fluctuations of the pseudoparticle are inessential, and that the one-loop approximation is effectively correct even for the large scale pseudoparticles. The emergent conclusion then is that the renormalized Yang-Mills theory is reduced to the problem of evaluating the effects of those configurations which involve many pseudoparticles, due account being taken of the interactions between them. Some aspects of this last problem are also discussed.

Fractal Structure of 2D Quantum Gravity

We resolve renormalization problems, indicated in Ref. 1 and find explicit formulae for the spectrum of anomalous dimensions in 2d—quantum gravity. Comparison with combinatorial approximation of random surfaces and its numerical analyses shows complete agreement with all known facts.

AdS dual of the critical O(N) vector model

Abstract We suggest a general relation between theories of infinite number of higher-spin massless gauge fields in AdS d +1 and large N conformal theories in d dimensions containing N -component vector fields. In particular, we propose that the singlet sector of the well-known critical 3-d O ( N ) model with the ( φ a φ a ) 2 interaction is dual, in the large N limit, to the minimal bosonic theory in AdS 4 containing massless gauge fields of even spin.

Quantum Geometry of Fermionic Strings

The formalism of the previous paper is extended to the case of supersymmetric strings. The effective theory which sums up fermionic surfaces is described by the supersymmetric Liouville equation. At D = 10 effective decoupling of the Liouville dilaton takes place and our theory coincides with the old ones. At D = 3 our theory is equilavent to the three-dimensional Ising model, which is thus reduced to the two-dimensional supersymmetric Liouville theory.

The string dilation and a least coupling principle

Abstract It is pointed out that string-loop modifications of the low-energy matter couplings of the dilation may provide a mechanism for fixing the vacuum expectation value of a massless dilation in a way which is naturally compatible with existing experimental data. Under a certain assumption of universality of the dilation coupling functions, the cosmological evolution of the graviton-dilaton-matter system is shown to drive the dilaton towards values where it decouples from matter (“Least Coupling Principle”). Quantitative estimates are given of the residual strength, at the present cosmological epoch, of the coupling to matter of the dilaton. The existence of a weakly coupled massless dilaton entails a large spectrum of small, but non-zero, observable deviations from general relativity. In particular, our results provide a new motivation for trying to improve by several orders of magnitude the various experimental tests of Einsteins Equivalence Principle (universality of free fall, constancy of the constants, etc.).