1 Introduction

String theory was initially discovered in an attempt to describe the physics of the strong interactions prior to the advent of gauge field theories and QCD. Today, it has matured to a very promising candidate for a unified quantum theory of gravity and all the other forces of nature. In this interpretation, gauge fields arise as the low energy excitations of fundamental open strings and are therefore derived, non-fundamental objects, just as the theory of gravity itself. Ironically though, advances in our understanding of non-perturbative string theory and of D-branes have led to a resurrection of gauge fields as fundamental objects. Namely, it is now generally believed that string theory in suitable space-time backgrounds can have a dual, holographic description in terms of gauge field theories and thus the question, which of the two is the fundamental one, becomes redundant. This belief builds on a remarkable proposal due to Maldacena [86 ] known as the Anti-de-Sitter/Conformal Field Theory (AdS/CFT) correspondence (for reviews see [1 , 52 ]).

The initial idea of a string/gauge duality is due to ’t Hooft [115 ], who realized that the perturbative expansion of SU( $N$ ) gauge field theory in the large $N$ limit can be reinterpreted as a genus expansion of discretized two-dimensional surfaces built from the field theory Feynman diagrams. Here $1/N$ counts the genus of the Feynman diagram, while the ’t Hooft coupling $c := g2YM N$ (with $gYM$ denoting the gauge theory coupling constant) enumerates quantum loops. The genus expansion of the free energy $F$ of a SU( $N$ ) gauge theory in the ’t Hooft limit ( $N --> oo$ with $c$ fixed), for example, takes the pictorial form

with suitable coefficients $cg,l$ denoting the contributions at genus $g$ and loop order $l$ . Obviously, this $1/N$ expansion resembles the perturbative expansion of a string theory in the string coupling constant $gS$ .

The AdS/CFT correspondence is the first concrete realization of this idea for four-dimensional gauge theories. In its purest form - which shall also be the setting we will be interested in - it identifies the “fundamental” type IIB superstring in a ten-dimensional anti-de-Sitter cross sphere ( $AdS5 × S5$ ) space-time background with the maximally supersymmetric Yang-Mills theory with gauge group SU( $N$ ) ( $N = 4$ SYM) in four dimensions. The $N = 4$ Super Yang-Mills model is a quantum conformal field theory, as its $b$ -function vanishes exactly. The string model is controlled by two parameters: the string coupling constant $gS$ and the “effective” string tension $2 ' R /a$ , where $R$ is the common radius of the $AdS5$ and $S5$ geometries. The gauge theory, on the other hand, is parameterized by the rank $N$ of the gauge group and the coupling constant $gYM$ , or equivalently, the ’t Hooft coupling $c := g2 N YM$ . According to the AdS/CFT proposal, these two sets of parameters are to be identified as

We see that in the AdS/CFT proposal the string coupling constant is not simply given by $1/N$ , but comes with a linear factor in $c$ . This, however, does not alter the genus expansion and its interpretation in form of string worldsheets.

The equations (2 ) relate the coupling constants, but there is also a dictionary between the excitations of the two theories. The correspondence identifies the energy eigenstates of the $5 AdS5 × S$ string, which we denote schematically as $|OA >$ - with $A$ being a multi-index - with (suitable) composite gauge theory operators of the form $OA = Tr(fi1 fi2 ...fin)$ , where $(fi)ab$ are the elementary fields of $N = 4$ SYM (and their covariant derivatives) in the adjoint representation of SU( $N$ ), i.e. $N × N$ hermitian matrices. The energy eigenvalue $E$ of a string state, with respect to time in global coordinates, is conjectured to be equal to the scaling dimension $D$ of the dual gauge theory operator, which in turn is determined from the two point function of the conformal field theory¹

with $D(c, 1/N )=!E(R2/a', g ) S$ . A zeroth order test of the conjecture is then the agreement of the underlying symmetry supergroup PSU(2,2|4) of the two theories, which furnishes the representations under which $OA(x)$ and $|OA >$ transform. This then yields a hint on how one could set up an explicit string state/gauge operator dictionary.

Clearly, there is little hope of determining either the all genus (all orders in $gS$ ) string spectrum, or the complete $1/N$ dependence of the gauge theory scaling dimensions $D$ . But the identification of the planar gauge theory with the free ( $g = 0 S$ ) string seems feasible and fascinating: Free $AdS × S5 5$ string theory should give the exact all-loop gauge theory scaling dimensions in the large $N$ limit! Unfortunately though, our knowledge of the string spectrum in curved backgrounds, even in such a highly symmetric one as $AdS5 × S5$ , remains scarce. Therefore, until very recently, investigations on the string side of the correspondence were limited to the domain of the low energy effective field theory description of $AdS × S5 5$ strings in terms of type IIB supergravity. This, however, is necessarily limited to weakly curved geometries in string units, i.e. to the domain of $V~ -- c » 1$ by virtue of (2 ). On the gauge theory side, one has control only in the perturbative regime where $c « 1$ , which is perfectly incompatible with the accessible supergravity regime $V~ -- c » 1$ . Hence, one is facing a strong/weak coupling duality, in which strongly coupled gauge fields are described by classical supergravity, and weakly coupled gauge fields correspond to strings propagating in a highly curved background geometry. This insight is certainly fascinating, but at the same time strongly hinders any dynamical tests (or even a proof) of the AdS/CFT conjecture in regimes that are not protected by the large amount of symmetry in the problem.

This situation has profoundly changed since 2002 by performing studies of the correspondence in novel limits where quantum numbers (such as spins or angular momenta in the geometric $AdS5 × S5$ language) become large in a correlated fashion as $N --> oo$ . This was initiated in the work of Berenstein, Maldacena and Nastase [33 ], who considered the quantum fluctuation expansion of the string around a degenerated point-like configuration, corresponding to a particle rotating with a large angular momentum $J$ on a great circle of the $S5$ space. In the limit of $J --> oo$ with $J 2/N$ held fixed (the “BMN limit”), the geometry seen by the fast moving particle is a gravitational plane-wave, which allows for an exact quantization of the free string in the light-cone-gauge [88 , 90 ]. The resulting string spectrum leads to a formidable prediction for the all-loop scaling dimensions of the dual gauge theory operators in the corresponding limit, i.e. the famous formula $V~ ------------ Dn = J + 2 1 + c n2/J 2$ for the simplest two string oscillator mode excitation. The key point here is the emergence of the effective gauge theory loop counting parameter $c/J 2$ in the BMN limit. By now, these scaling dimensions have been firmly reproduced up to the three-loop order in gauge theory [21 , 12 , 53 ]. This has also led to important structural information for higher (or all-loop) attempts in gauge theory, which maximally employ the uncovered integrable structures to be discussed below. Moreover, the plane wave string theory/ $N = 4$ SYM duality could be extended to the interacting string ( $gS /= 0$ ) respectively non-planar gauge theory regime providing us with the most concrete realization of a string/gauge duality to date (for reviews see [94 , 97 , 104 , 103 ]).

In this review we shall discuss developments beyond the BMN plane-wave correspondence, which employ more general sectors of large quantum numbers in the AdS/CFT duality. The key point from the string perspective is that such a limit can make the semiclassical (in the plane-wave case) or even classical (in the “spinning string” case) computation of the string energies also quantum exact [61 , 62 ], i.e. higher $s$ -model loops are suppressed by inverse powers of the total angular momentum $J$ on the five sphere² . These considerations on the string side then (arguably) yield all-loop predictions for the dual gauge theory. Additionally, the perturbative gauge theoretic studies at the first few orders in $c$ led to the discovery that the spectrum of scaling dimensions of the planar gauge theory is identical to that of an integrable long-range spin chain [92 , 21 , 24 ]. Consistently, the $AdS5 × S5$ string is a classically integrable model [31 ], which has been heavily exploited in the construction of spinning string solutions. This review aims at a more elementary introduction to this very active area of research, which in principle holds the promise of finding the exact quantum spectrum of the $AdS5 × S5$ string or equivalently the all-loop scaling dimensions of planar $N = 4$ Super Yang-Mills. It is intended as a first guide to the field for students and interested “newcomers” and points to the relevant literature for deeper studies. We will discuss the simplest solutions of the $5 AdS5 × S$ string corresponding to folded and circular string configurations propagating in a $R × S3$ subspace, with the $S3$ lying within the $S5$ . On the gauge theory side we will motivate the emergence of the spin chain picture at the leading one-loop order and discuss the emerging Heisenberg $XXX1/2$ model and its diagonalization using the coordinate Bethe ansatz technique. This then enables us to perform a comparison between the classical string predictions in the limit of large angular momenta and the dual thermodynamic limit of the spin chain spectrum. Finally, we turn to higher-loop calculations in the gauge-theory and discuss conjectures for the all-loop form of the Bethe equations, giving rise to a long-range interacting spin chain. Comparison with the obtained string results uncovers a discrepancy from loop order three onwards and the interpretation of this result is also discussed.

A number of more detailed reviews on spinning strings, integrability and spin chains in the AdS/CFT correspondence already exist: Tseytlin’s review [117 ] mostly focuses on the string side of the correspondence, whereas Beisert’s Physics Report [13 ] concentrates primarily on the gauge side. See also Tseytlin’s second review [116 ], on the so-called coherent-state effective action approach, which we will not discuss in this review. Recommended is also the shorter review by Zarembo [119 ] on the SU(2) respectively $R × S3$ subsector, discussing the integrable structure appearing on the classical string - not covered in this review. For a detailed account of the near plane-wave superstring, its quantum spectroscopy and integrability structures see Swanson’s thesis [114 ].