Gauge theories as string theories: the first results

The gauge/string theory duality in curved space is discussed mainly using a non-Abelian conformal $N=4$ supersymmetric gauge theory and the theory of a closed superstring in the $\mathrm{AdS}_5\times\mathrm{S}^5$ metric as an example. It is shown that in the supergravity approximation, string duality yields the characteristics of a strong-coupling gauge theory. For a special shape of the contour, a Wilson loop expression is derived in the classical superstring approximation. The role of the hidden integrability in lower-loop calculations in gauge theory and in different approximations of string theory is discussed. It is demonstrated that in the large quantum-number limit, gauge theory operators can be described in terms of the dual string picture. Examples of metrics providing the dual description of gauge theories with broken conformal symmetry are presented, and formulations of the vacuum structure of such theories in terms of gravity are discussed.