Abstract

In this introductory review we discuss dynamical tests of the
$AdS5\times S5$
string/
$\mathcal{N}=4$
Super Yang–Mills duality. After a brief introduction to AdS/CFT, we argue that semiclassical string energiesyield information on the quantum spectrum of the string in the limit of large angular momenta on the
${S}^{5}$
. The energies of the folded and circular spinning string solutions rotating on a
${S}^{3}$
within the
${S}^{5}$
arederived, which yield all-loop predictions for the dual gauge theory scaling dimensions. These follow fromthe eigenvalues of the dilatation operator of
$\mathcal{N}=4$
Super Yang–Mills in a minimal SU(2) subsector, and wedisplay its reformulation in terms of a Heisenberg
$s=1/2$
spin chain along with the coordinate Bethe ansatzfor its explicit diagonalization. In order to make contact to the spinning string energies, we then study thethermodynamic limit of the one-loop gauge theory Bethe equations and demonstrate the matching with thefolded and closed string result at this loop order. Finally, the known gauge theory results at higher-looporders are reviewed and the associated long-range spin chain Bethe ansatz is introduced, leading to anasymptotic all-loop conjecture for the gauge theory Bethe equations. This uncovers discrepancies at thethree-loop order between gauge theory scaling dimensions and string theory energies and the implicationsof this are discussed. Along the way, we comment on further developments and generalizations of thesubject and point to the relevant literature. 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
$\lambda :={g}^{2}YMN$
(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\to \infty $
with
$\lambda $
fixed), for example, takes the pictorial form

with suitable coefficients
${c}_{g,l}$
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
${g}_{S}$
.

$$\begin{array}{c}F={N}^{2}+1+\frac{1}{{N}^{2}}+\cdots ={\sum}_{g=0}^{\infty}\frac{1}{{N}^{2g-2}}{\sum}_{l=0}^{\infty}{c}_{g,l}{\lambda}^{l}\end{array}$$ | (1) |

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 (
$Ad{S}_{5}\times {S}^{5}$
) space-time background with the maximally supersymmetric Yang–Mills theory with gauge group SU(
$N$
) (
$\mathcal{N}=4$
SYM) in four dimensions. The
$\mathcal{N}=4$
Super Yang–Mills model is a quantum conformal field theory, as its
$\beta $
-function vanishes exactly. The string model is controlled by two parameters: the string coupling constant
$gS$
and the “effective” string tension
${R}^{2}/{\alpha}^{\prime}$
, where
$R$
is the common radius of the
$Ad{S}_{5}$
and
${S}_{5}$
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
$\lambda :={g}^{2}YMN$
. 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
$\lambda $
. This, however, does not alter the genus expansion and its interpretation in form of string worldsheets.

$$\begin{array}{c}\frac{4\pi \lambda}{N}=gS\sqrt{\lambda}=\frac{{R}^{2}}{{\alpha}^{\prime}}.\end{array}$$ | (2) |

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
$AdS5\times S5$
string, which we denote schematically as
$|{\mathcal{O}}_{A}\rangle $
– with
$A$
being a multi-index – with (suitable) composite gauge theory operators of the form
${\mathcal{O}}_{A}=Tr({\phi}_{{i}_{1}}{\phi}_{{i}_{2}}\dots {\phi}_{{i}_{n}})$
, where
$({\phi}_{i}{)}_{ab}$
are the elementary fields of
$\mathcal{N}=4$
SYM (and their covariant derivatives) in the adjoint representation of SU(
$N$
), i.e.
$N\times N$
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
$\Delta $
of the dual gauge theory operator, which in turn is determined from the two point function of the conformal field theory^{
$\text{1}$
}

with
$\Delta (\lambda ,1/N)\stackrel{!}{=}E({R}^{2}/{\alpha}^{\prime},gS)$
.

$$\begin{array}{c}\langle {\mathcal{O}}_{A}\left(x\right){\mathcal{O}}_{B}\left(y\right)\rangle =\frac{M{\delta}_{A,B}}{(x-y{)}^{2{\Delta}_{A}(\lambda ,\frac{1}{N})}}\iff {\mathcal{\mathscr{H}}}_{String}|{\mathcal{O}}_{A}\rangle ={E}_{A}(\frac{{R}^{2}}{{\alpha}^{\prime}},gS\left)\right|{\mathcal{O}}_{A}\rangle \end{array}$$ | (3) |

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
${\mathcal{O}}_{A}\left(x\right)$
and
$|{\mathcal{O}}_{A}\rangle $
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
$\Delta $
. But the identification of the planar gauge theory with the free (
$gS=0$
) string seems feasible and fascinating: Free
$AdS5\times S5$
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
$AdS5\times S5$
, 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
$AdS5\times S5$
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
$\sqrt{\lambda}\gg 1$
by virtue of ( 2 ). On the gauge theory side, one has control only in the perturbative regime where
$\lambda \ll 1$
, which is perfectly incompatible with the accessible supergravity regime
$\sqrt{\lambda}\gg 1$
. 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
$AdS5\times S5$
language) become large in a correlated fashion as
$N\to \infty $
. 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
${S}^{5}$
space. In the limit of
$J\to \infty $
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
${\Delta}_{n}=J+2\sqrt{1+\lambda {n}^{2}/{J}^{2}}$
for the simplest two string oscillator mode excitation. The key point here is the emergence of the effective gauge theory loop counting parameter
$\lambda /{J}^{2}$
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/
$\mathcal{N}=4$
SYM duality could be extended to the interacting string (
${g}_{S}\ne 0$
) 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
$\sigma $
-model loops are suppressed by inverse powers of the total angular momentum
$J$
on the five sphere^{
$\text{2}$
}
. 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
$\lambda $
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
$AdS5\times S5$
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
$AdS5\times S5$
string or equivalently the all-loop scaling dimensions of planar
$\mathcal{N}=4$
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
$AdS5\times S5$
string corresponding to folded and circular string configurations propagating in a
$\mathbb{R}\times {S}^{3}$
subspace, with the
${S}^{3}$
lying within the
${S}^{5}$
. 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
$XX{X}_{1/2}$
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
$\mathbb{R}\times {S}^{3}$
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] .

^{
$\text{1}$
}
For simplicity we take
${\mathcal{O}}_{A}\left(x\right)$
to be a scalar operator here.

^{
$\text{2}$
}
This suppression only occurs if at least one angular momentum on the five sphere becomes large. If this is not the case, e.g. for a spinning string purely on the
$Ad{S}_{5}$
[69] , quantum corrections are not suppressed by inverse powers of the spin on
$Ad{S}_{5}$
[60] .

2 The Setup

With the embedding coordinates
${X}^{m}(\tau ,\sigma )$
and
${Y}^{m}(\tau ,\sigma )$
the Polyakov action of the
$AdS5\times S5$
string in conformal gauge (
${\eta}_{ab}=diag(-1,1)$
) takes the form (
$m,n=1,2,3,4,5$
)

where we have suppressed the fermionic terms in the action, as they will not be relevant in our discussion of classical solutions (the full fermionic action is stated in [89, 100] ). A natural choice of coordinates for the
$AdS5\times S5$
space (“global coordinates”) is^{
$\text{3}$
}

Moreover, we have directly written the string action with the help of the effective string tension
$\sqrt{\lambda}={R}^{2}/{\alpha}^{\prime}$
of ( 2 ). It is helpful to picture the
$Ad{S}_{5}$
space-time as a bulk cylinder with a four-dimensional boundary of the form
$\mathbb{R}\times {S}^{3}$
(see Figure 2 ).
A consequence of the conformal gauge choice are the Virasoro constraints, which take the form

where the dot refers to
${\partial}_{\tau}$
and the prime to
${\partial}_{\sigma}$
derivatives. Of course the contractions in the above are to be performed with the metrics of ( 5 ).

$$\begin{array}{c}I=-\frac{\sqrt{\lambda}}{4\pi}\int d\tau d\sigma \left[{G}_{mn}^{\left(Ad{S}_{5}\right)}{\partial}_{a}{X}^{m}{\partial}^{a}{X}^{n}+{G}_{mn}^{\left({S}^{5}\right)}{\partial}_{a}{Y}^{m}{\partial}^{a}{Y}^{n}\right]+\text{fermions},\end{array}$$ | (4) |

$$\begin{array}{ccc}d{s}_{Ad{S}_{5}}^{2}& =& d{\rho}^{2}-{cosh}^{2}\rho d{t}^{2}+{sinh}^{2}\rho (d{\overline{\psi}}^{2}+{cos}^{2}\overline{\psi}d{{\phi}_{1}}^{2}+{sin}^{2}\overline{\psi}d{{\phi}_{2}}^{2})\end{array}$$ |

$$\begin{array}{ccc}d{s}_{{S}^{5}}^{2}& =& d{\gamma}^{2}+{cos}^{2}\gamma d{{\phi}_{3}}^{2}+{sin}^{2}\gamma (d{\psi}^{2}+{cos}^{2}\psi d{{\phi}_{1}}^{2}+{sin}^{2}\psi d{{\phi}_{2}}^{2}).\end{array}$$ | (5) |

$$\begin{array}{c}0\stackrel{!}{=}{\dot{X}}^{m}{X}_{m}^{\prime}+{\dot{Y}}^{p}{Y}_{p}^{\prime}0\stackrel{!}{=}{\dot{X}}^{m}{\dot{X}}_{m}+{\dot{Y}}^{p}{\dot{Y}}_{p}+{{X}^{m}}^{\prime}{X}_{m}^{\prime}+{{Y}^{p}}^{\prime}{Y}_{p}^{\prime},\end{array}$$ | (6) |

As mentioned in the introduction, it is presently unknown how to perform an exact quantization of this model. It is, however, possible to perform a quantum fluctuation expansion in
$1/\sqrt{\lambda}$
. For this one expands around a classical solution of ( 4 ) and integrates out the fluctuations in the path-integral loop order by order. This is the route we will follow. Of course, in doing so, we will only have a patch-wise access to the full spectrum of the theory, with each patch given by the solution expanded around.

The embedding coordinates
${X}^{m}(\tau ,\sigma )$
and
${Y}^{m}(\tau ,\sigma )$
in ( 4 ) are hence given by
${X}^{m}=(\rho ,\overline{\psi},{\phi}_{1},{\phi}_{2},t)$
and
${Y}^{m}=(\gamma ,\psi ,{\phi}_{1},{\phi}_{2},{\phi}_{3})$
.

^{
$\text{3}$
}
This metric arises from a parameterization of the five sphere
${x}_{1}^{2}+{x}_{2}^{2}+{x}_{3}^{2}+\cdots +{x}_{6}^{2}=1$
and anti-de-Sitter space
$-{y}_{-1}^{2}-{y}_{0}^{2}+{y}_{1}^{2}+{y}_{2}^{2}+\cdots {y}_{4}^{2}=-1$
through

$$\begin{array}{ccc}{x}_{1}+i{x}_{2}=sin\gamma cos\psi {e}^{i{\phi}_{1}},& & {x}_{3}+i{x}_{4}=sin\gamma sin\psi {e}^{i{\phi}_{2}},{x}_{5}+i{x}_{6}=cos\gamma {e}^{i{\phi}_{3}};\end{array}$$ |

$$\begin{array}{ccc}{y}_{1}+i{y}_{2}=sinh\rho cos\overline{\psi}{e}^{i{\phi}_{1}},& & {y}_{3}+i{y}_{4}=sinh\rho sin\overline{\psi}{e}^{i{\phi}_{2}},{y}_{-1}+i{y}_{0}=cosh\rho {e}^{it}.\end{array}$$ |

2.1 The rotating point-particle

It is instructive to sketch this procedure by considering the perhaps simplest solution to the equations of motion of ( 4 ): the rotating point-particle on
${S}^{5}$
, which is a degenerated string configuration.

$$\begin{array}{c}t=\kappa \tau ,\rho =0,\gamma =\frac{\pi}{2},{\phi}_{1}=\kappa \tau ,{\phi}_{2}={\phi}_{3}=\psi =0.\end{array}$$ | (7) |