The initial idea of a string/gauge duality is due to ’t Hooft [115], who realized that the perturbative
expansion of SU() gauge field theory in the large
limit can be reinterpreted as a genus expansion of
discretized two-dimensional surfaces built from the field theory Feynman diagrams. Here
counts the
genus of the Feynman diagram, while the ’t Hooft coupling
(with
denoting the gauge
theory coupling constant) enumerates quantum loops. The genus expansion of the free energy
of a
SU(
) gauge theory in the ’t Hooft limit (
with
fixed), for example, takes the pictorial
form
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 ()
space-time background with the maximally supersymmetric Yang-Mills theory with gauge group SU(
)
(
SYM) in four dimensions. The
Super Yang-Mills model is a quantum conformal field
theory, as its
-function vanishes exactly. The string model is controlled by two parameters: the string
coupling constant
and the “effective” string tension
, where
is the common radius of the
and
geometries. The gauge theory, on the other hand, is parameterized by the rank
of the gauge group and the coupling constant
, or equivalently, the ’t Hooft coupling
. According to the AdS/CFT proposal, these two sets of parameters are to be identified as
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
string, which we
denote schematically as
- with
being a multi-index - with (suitable) composite gauge
theory operators of the form
, where
are the elementary fields
of
SYM (and their covariant derivatives) in the adjoint representation of SU(
),
i.e.
hermitian matrices. The energy eigenvalue
of a string state, with respect to time in
global coordinates, is conjectured to be equal to the scaling dimension
of the dual gauge
theory operator, which in turn is determined from the two point function of the conformal field
theory1
Clearly, there is little hope of determining either the all genus (all orders in ) string spectrum, or
the complete
dependence of the gauge theory scaling dimensions
. But the identification of the
planar gauge theory with the free (
) string seems feasible and fascinating: Free
string theory should give the exact all-loop gauge theory scaling dimensions in the large
limit! Unfortunately though, our knowledge of the string spectrum in curved backgrounds,
even in such a highly symmetric one as
, 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
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
by virtue of (2
). On the gauge theory side, one has control only in
the perturbative regime where
, which is perfectly incompatible with the accessible
supergravity regime
. 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 language)
become large in a correlated fashion as
. 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
on a great circle of the
space. In the limit of
with
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
for the simplest two string oscillator mode excitation. The key point here is the
emergence of the effective gauge theory loop counting parameter
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/
SYM duality could
be extended to the interacting string (
) 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
-model loops are suppressed by inverse powers of the total angular momentum
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
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
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
string or
equivalently the all-loop scaling dimensions of planar
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
string corresponding to folded and
circular string configurations propagating in a
subspace, with the
lying within
the
. 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
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 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].
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