AEI 2003-068

Precision Spectroscopy of AdS/CFT

N. Beisert,
S. Frolov^{*}^{*}*Also at Steklov Mathematical Institute, Moscow.,
M. Staudacher
and A.A. Tseytlin^{†}^{†}†Also at Lebedev Physics Institute, Moscow.

Max-Planck-Institut für Gravitationsphysik, Albert-Einstein-Institut

Am Mühlenberg 1, D-14476 Golm, Germany

Department of Physics, The Ohio State University

Columbus, OH 43210, USA

Blackett Laboratory, Imperial College

London, SW7 2BZ, U.K.

Abstract

We extend recent remarkable progress in the comparison of the dynamical energy spectrum of rotating closed strings in and the scaling weights of the corresponding non-near-BPS operators in planar supersymmetric gauge theory. On the string side the computations are feasible, using semiclassical methods, if angular momentum quantum numbers are large. This results in a prediction of gauge theory anomalous dimensions to all orders in the ‘t Hooft coupling . On the gauge side the direct computation of these dimensions is feasible, using a recently discovered relation to integrable (super) spin chains, provided one considers the lowest order in . This one-loop computation then predicts the small-tension limit of the string spectrum for all (i.e. small or large) quantum numbers. In the overlapping window of large quantum numbers and small effective string tension, the string theory and gauge theory results are found to match in a mathematically highly non-trivial fashion. In particular, we compare energies of states with (i) two large angular momenta in , and (ii) one large angular momentum in and each, and show that the solutions are related by an analytic continuation. Finally, numerical evidence is presented on the gauge side that the agreement persists also at higher (two) loop order.## 1 Introduction

It is believed that free type IIB superstring theory on the background is exactly dual to planar supersymmetric quantum gauge theory [1, 2, 3]. Here

(1.1) |

and is the ’t Hooft coupling constant. Since the exact quantization of string theory in this curved background is not yet understood, most of the results on the string side of the duality obtained until a year and a half ago were in the (classical) supergravity limit of infinite string tension , which corresponds, via the “effective string tension” identification

(1.2) |

to the strong coupling limit on the planar gauge theory side. Since in string theory little could be done at finite and , while in gauge theory little could be done at finite , until recently the perception was that any dynamical test of the AdS/CFT correspondence should be very hard to perform. A notable exception were some successful studies of four-point functions (involving BPS operators on the gauge side and supergravity correlators on the string side) where some dynamical modes appear in intermediate channels [4, 5, 6]. This situation has dramatically improved due to new ideas and techniques on both sides of the correspondence, which were largely influenced by the seminal work of [7] (which, in turn, was based on [8, 9]). Very recent progress points towards the exciting prospect that the free string alias planar gauge theory is integrable and thus might be exactly solvable.

On the string theory side, it was understood that in the case when some of the quantum numbers of the string states become large, the string sigma model can be efficiently treated by semi-classical methods [10, 11] (see also [12] and [13, 14, 15]). It was then suggested [16, 17] that a novel possibility for a quantitative comparison with SYM theory in non-BPS sectors appears when one considers classical solutions describing closed strings rotating in several directions in the product space with the metric

(1.3) |

Here is the global time, which, together with
the 5 angles ,
correspond to the obvious “linear” isometries
of the metric, i.e. are related to the 3+3 Cartan generators
of the bosonic isometry group.
Rotating strings can thus carry the 2+3 angular momentum charges (spins)
, while is
associated with the energy .
Once such classical solutions representing string states with several
charges are found [11, 16, 17, 18, 19],
one may evaluate the energy as a function of the spins and :
.
A remarkable feature of string solutions in
is that their energy grows, for large charges,
linearly with the charges [20, 10, 11].
Corrections
to subleading terms in the classical energy
can then be computed using the standard semiclassical (inverse string
tension) expansion [11, 17].
For certain string states with large total spin on for which
^{1}^{1}1Here means that , where are arbitrary
constants which can be numerically large, small or even zero.
Then we can set up a power counting scheme in and .
While we keep all orders of , we systematically
drop terms of subleading orders in .

(1.4) |

it turns out that string sigma model loop corrections to the energy are suppressed by powers of (see [17, 18, 19]). In these cases the leading, , contribution to the energy is given already by the classical string expression, i.e. one does not even need to quantize the string sigma model. Furthermore, as follows from the string action, classically, all charges (including the energy) appear only in the combination . It was shown in [16, 17, 18, 19] that the classical energy admits an expansion in powers of the small parameter . The upshot of the semiclassical string analysis is then that in the limit (1.4) the string-state energy is given by

(1.5) |

where are some functions of the spin ratios , and stands for quantum string sigma model corrections.

Considering string states represented by the classical solutions with several charges has the added advantage that it helps significantly in identifying the corresponding gauge theory operators. As is well known, supersymmetric gauge theory is superconformally invariant, and the bosonic subgroup of the full superconformal group is . The energy of a string state is expected to correspond to the scaling dimension of the associated conformal operator on the gauge theory side:

(1.6) |

The dimension is, in general, a non-trivial function of the ’t Hooft coupling . It is not yet known how to compute exactly, except for supersymmetric BPS operators (for which dimensions are protected, i.e. independent of ) and for certain near-BPS operators with large charge [7, 21]. In principle, one can compute the dimension perturbatively at small as an eigenvalue of the matrix of anomalous dimensions. Obtaining and diagonalizing this matrix is a task where the complexity increases exponentially with the number of constituent fields. At this point, a comparison to string theory energies may appear almost hopeless, since, on the string side, the total charge is required to be large.

Fortunately, it was recently discovered that planar
SYM theory is *integrable* at the one-loop level
[22, 23].
We can therefore make use of the Bethe ansatz for a
corresponding spin chain model to obtain *directly* the
eigenvalues of the matrix of anomalous dimensions.
This observation proves to be especially useful in the
(“thermodynamic”) limit , i.e. for a very long spin chain, where
the algebraic
Bethe equations are approximated by integral equations.
For a large number of fields , the dimension
appears to have a loop expansion equivalent
to the one in (1.5),
^{2}^{2}2For the term the dependence
can be read off from the thermodynamic limit of the Bethe ansatz.
For higher-loops,
there are some numerical indications for this dependence,
but so far there is no
general proof (which, perhaps, may be given
using maximal supersymmetry of the theory).

(1.7) |

Again, the coefficients are functions of the spin ratios .

The string semiclassical expression (1.5), while formally valid for , is actually exact, since, as was mentioned above, all sigma model corrections are suppressed by . Assuming the conditions (1.4) are satisfied, one should be able to compare directly the classical term in the string energy (1.5) to the scaling dimension in gauge theory (1.7), and to show that

(1.8) |

Such a comparison at order was indeed successfully carried out in [24, 18, 19], where a spectacular agreement between the string theory and gauge theory results for the energy or dimension was found for several two-spin string states represented by circular and folded closed strings rotating in .

The contents of the present paper is the following.
In
Section 2 we review the results of the semi-classical
computation of the energy of folded strings rotating in
two planes on , the “” solution [18].
We also review the results of the Bethe ansatz
calculations of the anomalous dimensions of the corresponding
gauge theory operators
[24].
We then present a full analytic *proof* that in the
region of large quantum numbers the relevant terms
in the string energy and the gauge operator dimension match,
i.e that and
are indeed the same as functions of the spin ratio.
This goes beyond
the previous “experimental” evidence of matching series
expansions.
The central part
of the present paper is Section 3,
where we show that the “”
state represented by the string rotating in
two planes in can be analytically
continued, in both string and gauge theory, to
an “” state represented by a string
rotating in just one plane in , but having
also one large spin in [11].
On the gauge theory side, this requires the use of a
recently constructed [23] supersymmetric extension
of the above Bethe ansatz.
As a result, we find the agreement between the string
theory and gauge theory
expressions of the energy/dimension
also for the “” solution.
In Section 4 we
study the possibility to check the matching (1.8)
beyond the leading order .
Using the expression [25] for the
gauge theory two-loop dilatation operator,
we present numerical evidence
that the matching
between the string theory and gauge theory results in the case of
the state extends to
at least to the (two-loop) level.
Section 5 contains some concluding remarks,
The Appendices
contain some general remarks and technical details.
In particular, in Appendix B we explain the relation
between the and string solutions,
and in Appendix C we discuss the solution
of the Bethe ansatz system
of equations for the spin chain which appeared in
Section 3 in connection with the case.
In Appendix D we compare circular strings
on with a different “imaginary” solution of the Bethe equations.
Finally, in Appendix E we consider the dependence of
string energy on the ratio of two spins.

## 2 Strings rotating on

Let us start with a discussion of a particular state corresponding to a folded string rotating in two planes on the five-sphere. This folded solution should have minimal value of the energy for given values of the spins. Our aim will be to demonstrate the equivalence between the leading correction to the classical string-theory energy and the one-loop gauge theory anomalous dimension at the functional level, i.e. going beyond particular expansions and limits considered previously in [24, 18]. The solution in question [18] has the following non-zero coordinates in (1): and satisfies a 1-d sine-Gordon equation in . The string is stretched in with the maximal value (we refer to Appendices A and B for details on the string solutions). The classical energy and the angular momenta of the rotating string may be written as

(2.1) |

Here and from now on, curly letters correspond to charges rescaled by the inverse effective string tension, . The parameters of the solution are related via the conformal gauge constraint and the closed string periodicity condition (we shall consider single-fold solution). Solving these conditions one may express the energy as a function of the spins, . The expression for the energy can then be found as a parametric solution of the following system of two transcendental equations (see Appendix B)

(2.2) |

where the auxiliary parameter is the modulus of the elliptic integrals and of the first and second kind, respectively (their standard definitions can be found in the appendices in (B.7),(C.11)). Eliminating , one finds the energy as a function of , the ratio and the string tension .

Assuming that is large, i.e. that the condition (1.4) is satisfied, one can expand the solution for the energy in powers of the total spin (cf. (1.5))

(2.3) |

where are functions of the spin ratio. It is a non-trivial observation that the string energy admits [18] such an expansion which then looks like a perturbative expansion in . Moreover, quantum string sigma model corrections to are suppressed if [17, 18].

Turning attention to the gauge theory side, the natural operators carrying the same charges are of the general form

(2.4) |

where and are two of the three complex scalars of the
super YM model. The dots indicate that one has to consider
all possible orderings of fields and fields
inside the trace: only very specific linear combinations of these
composite fields possess a definite scaling dimension ,
i.e. are eigenoperators of the anomalous dimension matrix.
These particular two-spin scalar operators do
not mix with any other local operators that contain
other types of factors (fermions, field
strengths, derivatives) [25].
The relation (1.6)
between the string state energy and dimension of the corresponding
gauge theory operator then predicts
that there should exist an operator^{3}^{3}3Since the folded string
solution happens to have lowest energy for given charges, the
corresponding operator
should have the lowest dimension in this class of operators.
of the form
(2.4) whose exact^{4}^{4}4By exact we mean
not only all-order in
but also non-perturbative:
one does not expect instanton effects to be relevant
in the planar gauge theory.
scaling dimension is given, for large , by the solution of
eqs.(2.2). That means, in particular, that
eqs.(2.2) derived from classical
string theory predict the anomalous dimension
of this operator to any order in perturbation theory in !

Can we test this highly non-trivial prediction by a direct one-loop computation in the gauge theory? In the case where is large, doing this from scratch by Feynman diagram techniques is a formidable task due to the large number of possible field orderings (one needs to diagonalize the anomalous dimension matrix whose size grows exponentially with ). What helps is the crucial observation of ref. [22] that the one-loop anomalous dimension matrix for the operators of the two-scalar type (2.4) can be related to a Hamiltonian of an integrable Heisenberg spin chain (XXX model), i.e. its eigenvalues can be found by solving the Bethe ansatz equations of the spin chain. The upshot of the Bethe ansatz procedure [24] is that the system of equations diagonalizing the one-loop anomalous dimension matrix (for any, small or large, values of ) is given by

(2.5) |

where again (we assume ). This is an algebraic system of equations involving the auxiliary parameters , the so-called Bethe roots. We need to find the roots subject to the condition that no two roots coincide and the further constraint

(2.6) |

which ensures that no momentum flows around the cyclic trace.
This yields the one-loop planar anomalous
dimensions for the
operators (2.4).
Here, we will restrict consideration
to symmetric solutions,
i.e. if is a root then must be a root as well,
this automatically solves the momentum constraint (2.6).^{5}^{5}5
For a highest weight state of the desired representation of ,
no roots at infinity are allowed.

Let us now pause and
compare the string theory
system (2.2)
for the classical energy and the gauge theory system
(2.5) for the one-loop anomalous dimension.
Both systems are parametric, i.e. finding energy/dimension as a function of
spins
involves elimination
of auxiliary parameters.
The string result is
valid for all , but
restricted to large ,
namely, and .
The gauge result is valid for all , but restricted
to lowest order in .
Remarkably, there is a region of joint validity:
large charge *and* first order in !

Extracting the leading-order or “one-loop” term from the string-theory relations (2.2) is straightforward, as discussed in Appendix B. For large one sets and solves the resulting transcendental equation for . One then finds the parametric solution for

(2.7) |

On the gauge side,
one needs to do some work to extract
the lowest-energy state solution [24].^{6}^{6}6It is worth noting that the Bethe equations
give the energies or dimensions for all states
with the same spins, while the minimal-energy string solution
corresponds to the ground state only.
To select a particular solution of the Bethe equations that
should correspond to a specific (folded or circular, with
extra oscillations or without) string solution
requires a number of steps: First, one needs to make certain
“topological” assumptions about the distribution of roots,
which accumulate on lines presciently termed “Bethe strings”.
The possible choices correspond to “folded” and “circular” strings.
Second, one takes the logarithm on both
sides of the Bethe equations. The possible branches correspond
to the various winding modes of the string.
First, to be able to compare with string theory
we need to consider the “thermodynamic” limit of large spins,
i.e. .
The idea is then to assume a condensation of the Bethe roots into
“strings”
and thus to convert the system of algebraic
Bethe equations into
a continuum (integral) equation.
Making an appropriate ansatz for the
Bethe root distribution selects the ground state. Solving the
corresponding
integral equation (see Appendix C for some details) one finds again
a system of two equations
with energy as a parametric solution

(2.8) |

Here the modulus is related to the
endpoints of the “strings” of Bethe roots.
This system looks similar, but superficially not identical to that in
eq.(2.7).
However, if we relate the
auxiliary parameters and by
^{7}^{7}7Note that
(cf. Appendix C), so
is indeed positive. Let us note also that
this relation between the size of the
folded string () and the “length” of the
Bethe “strings” () suggests that
a transformation between the two integrable models
– string sigma model (Neumann system or 1-d sine-Gordon
system that follows from it) and the spin chain –
should involve some kind of a Fourier transform
(Bethe roots are inversely related to effective 1-d momenta).

(2.9) |

one can show, using the elliptic integral modular transform relations

(2.10) |

that the systems (2.7) and (2.8) are, in fact, exactly the same. As a result, their solutions and do become identical!

## 3 Strings rotating on and

Recently, it was shown in [23] that
the complete one-loop planar dilatation operator of SYM
[26] is integrable.^{8}^{8}8Integrability is related to Yangians.
A Yangian structure in the bosonic coset sigma model
was recently shown [27] to have a generalization to
(classical)
supercoset sigma model of [28].
Very recently [29], this structure was
“mapped” to planar gauge theory.
Possibly, this line of thought will
lead to a deeper understanding of the
matching of energies/anomalous dimensions.
To diagonalize
*any* matrix of anomalous dimensions,
the corresponding Bethe ansatz
was written down in
[23].
This enables one to access a much wider class of states
and perform similar comparisons between gauge theory and semiclassical
string theory.

Here we will present a first interesting example of such a novel test: we shall consider the case of only one non-vanishing angular momentum in (), but also one non-zero spin in (). This situation is clearly different from the one discussed in the last section. However, on the string side, the two scenarios are, in fact, mathematically closely related, as we will explain below (see also Appendix B). Is this also true on the gauge side? There the relevant local operators carrying the same charges have the following generic form

(3.1) |

where is a complex
combination of covariant derivatives
(see [16] for a related discussion).
Can we also treat these operators by a Bethe ansatz?
The integrability property of anomalous dimensions of
similar operators was
recently discussed in the literature [30].
In [23] the precise spin chain interpretation
of these operators was proved to lead to
an integrable XXX Heisenberg chain, and
the corresponding Bethe ansatz was obtained.
Here, the derivatives do not represent
sites of the spin chain, in contradistinction to the
fields of (2.4).^{9}^{9}9In fact, the analogy goes the opposite way: should be viewed as an
equivalent of , whereas
are absent in the spin chain.
In other words, each site can now
a priori (i.e. if is sufficiently large) be
in *infinitely* many spin states ,
as compared to only *two*, , in (2.4).
Under this identification plays the role of the number
of excitations and equals the number of spin sites,
i.e. the length of the chain.
The Bethe ansatz equations for the one-loop
anomalous dimensions then read
(we use and to distinguish
the solutions)

(3.2) |

The similarity to the system of equations (2.5) for the previous case, i.e. for the XXX spin chain, is obvious. In fact, the system (2.5) becomes formally equivalent to (3.2) if we make the following replacements in (2.5)

(3.3) |

The large solution of (2.8) can now be analytically continued to the regime where it gives the correct energy for . In fact, the solution of (2.8) was first derived [24] by assuming that and then analytically continued to ! For further details see Appendix C, where we also review the solution and present some further results that were not included in the paper [24].

Let us now turn to the rotating folded string solution [11] which would be expected to correspond to the just discussed gauge theory operators (3.1). This rotating string is stretched in the radial direction of while its center of mass rotates in , it has the following non-zero coordinates in (1): (see Appendices A and B for details). Now the energy and the spin can be viewed as two “charges” in while – as the charge in . This is clearly reminiscent of the previous example where we had two charges in and one charge in , and we have just found evidence on the gauge side that one should actually expect the two solutions to be related by an analytic continuation. Indeed, as explained in Appendices A and B, a beautiful way to see this connection on the string side stems from the close relation between the and metrics in (1).

On the level of the final expressions for the string charges the relation is as follows. The analogue of the parametric system of equations for the energy in the case here is easily found, using the relations in [11] (see Appendix B). We have again , where depend only on the classical parameters and satisfy

(3.4) |

The parameter here is negative definite for a physical folded rotating string solution. The system (3.4) becomes formally equivalent to the one in (2.2) after the following replacements done in (2.2) (we choose the same signs of the charges as in Appendix B)

(3.5) |

and after the analytic continuation from to in the elliptic integrals. A formal relation between the solutions of the two systems (2.2) and (3.4) is then

(3.6) |

In general, this does not imply a direct relation between the energy expressions in the two cases: one needs to perform the analytic continuation and also to invert the expression for . However, in the limit of large charges (the limit we are interested in) one can show that the leading correction to the energy of the solution

(3.7) |

is indeed directly related to (2.7) in the case of the solution with the replacements implied by (3.5) (see Appendix B). In particular, to the leading order in large-charge expansion one has (where in the second equality stands for ), so that Remarkably, this is the same as (the “thermodynamic” limit of) the relation (3.3) found above between the one-loop energy corrections on the gauge theory side. This implies, in particular, that the correspondence between the string theory and gauge theory results for the leading terms in the energy/dimension holds also in the case of the states. i.e. the two functions are related by

## 4 Higher loop corrections

Let us now comment on a generalization of the above results to higher orders in (“higher loops”). First, let us note that on the string side, we have a complete expression for the energies to all orders in which follows from the systems (2.2) and (3.4). In the interaction picture of perturbation theory, the only non-trivial system of equations is the one determining the leading order contribution; all higher-loop terms can be expressed through the leading order modulus . The two-loop energies for the case and for the case are given by (see Appendix B)

(4.1) |

As implied by the relation (3.6), the two expressions are not expected to (and do not) look similar.

Given that integrability and the Bethe ansatz allow us to obtain
the exact *one-loop* energies for infinite length operators, while
string theory gives us an all-loop prediction, it
would be interesting to find *higher-loop* energies in gauge theory
to compare to string theory.
Although the integrability property of the dilatation operator
acting on the states (2.4) seems to be maintained
(at least) at the two-loop level [25],
the corresponding extension of the Bethe ansatz is not yet known.^{10}^{10}10In principal agreement with the string theory result,
one might express higher-loop energies in terms of the one-loop Bethe roots.
However, this would require calculating matrix elements of the higher-loop
dilatation operator between Bethe states – presently a very
non-trivial issue.
Therefore, in order to find higher-loop
anomalous dimensions of the operators (2.4)
we have to rely on numerical methods of diagonalization of
the matrix of anomalous dimensions.
For the states (2.4), this matrix is generated by the planar
dilatation operator [25]

where exchanges the fields at site and . We shall consider the special case of . For given we can collect all operators of the form (2.4) and act on them with the dilatation operator (4) neglecting all non-planar terms. We can then diagonalize the matrix of anomalous dimensions and find the lowest eigenvalue of a state in the representation , see Tab. 1.

The numerical one-loop results for finite are already reasonably close to the string theory prediction. As proposed in [24], we can improve the results by extrapolating to . This is done fitting to the first two terms in the series expansion in

(4.3) |

As was demonstrated in [24], there are two distinct sequences of states for even and odd values of . The even values approach reasonably fast to the desired energies. Here we use to extrapolate to .

The extrapolated values of the dimensions at one-loop and two-loop orders [25] are found to be about off the string theory prediction. These results agree very well and we can clearly confirm that the correspondence works at ! For the three-loop conjecture of [25] (see also [31]) the results are somewhat inconclusive. On the one hand, using the vertex that was constructed assuming that integrability holds at , we get an extrapolation, , which is off the string prediction. On the other hand, the vertex that was matched to near plane-wave string theory results [32] gives an extrapolation, , that is only away. Nevertheless, we expect the three-loop energies to converge rather slowly and the three values of used to extrapolate are clearly not sufficient: In the spin chain picture the three-loop interaction already extends over four lattice sites, and finite size effects, due to the relatively small chain lengths, become more pronounced. An indication for this is that the extrapolation is still away from the input values. The terms which were neglected are expected to have a much stronger influence on the finite values as compared to term. Therefore, a mismatch seems reasonable and we can neither confirm nor rule out any of the conjectured three-loop vertices [31] (or the correspondence at ).

## 5 Conclusions and Outlook

In this paper we demonstrated that spectroscopy is becoming a very precise and versatile tool for establishing the validity of the AdS/CFT duality conjecture on a quantitative, dynamical level. Following the suggestion of [16, 17] and extending the earlier break-through work of [24, 18, 19] we have shown that in the non-BPS sector of two large charges, as in the near-BPS BMN sector with single large charge [7], the duality between the SYM theory and string theory relates perturbative results on both sides of the correspondence and thus can be tested using existing tools.

It should be fairly evident that our derivation of mathematically highly involved energy expressions, such as eqs.(2.7),(2.8), from both string theory and gauge theory constitutes a “physicist’s proof” of the correspondence. We believe that the present work is just the beginning of a much wider unraveling of dynamical details of the AdS/CFT duality. At the end, we expect to gain much insight into superstring theory on curved backgrounds, and into gauge theory at finite coupling.

Our work suggests a large number of further inquiries. The precise interpretation of the circular versus folded string solutions remains somewhat obscure in the Bethe ansatz picture. In particular, it would be important to understand the analog of the string solution for in the Bethe ansatz and thus complete the picture outlined in Appendix E. Furthermore, it seems that the Bethe ansatz allows for very complicated distributions of “Bethe root strings”, involving multi-cut solutions, the role of which is unclear so far on the string theory side.

Another obvious problem is to extend the comparison to include terms by computing (as in [11, 17]) the 1-loop string sigma model correction to the string energy and comparing the result to the leading correction to the “thermodynamic” limit of the XXX Bethe system. It would be interesting also to compute energies of excited string states by expanding the superstring action near the ground-state two-spin solution. In contrast to the BMN case [7], here one expects (from experience with special circular solutions [17]) that there will be many nearby states with the same charges and with energies differing from the ground state energy by order ). terms (these are of course negligible as compared to similar terms in the classical ground-state energy in the limit

It should be relatively straightforward, if laborious to extend the analysis to more than two spins. In string theory this has largely been accomplished for three non-vanishing angular momenta on in [19], but one could try to also include concurrently the two charges. For gauge theory, the corresponding Bethe equations are known [23], but have not yet been analyzed in any generality. Ideally, one would like to understand how to prove these equivalences directly, i.e. without actually solving the classical string sigma model equations and the Bethe equations in the thermodynamic limit.

The biggest challenge clearly is to find out how to extend the calculational power on either side of the correspondence in a way that would allow one to derive results that are not in the overlapping window of large quantum numbers and small effective string tension. On the string theory side, this would require to include quantum (inverse string tension) corrections in the Green-Schwarz supercoset sigma model of [28]. For gauge theory, we would need to understand the proper extension of the Bethe ansatz so as to make it applicable to all orders in Yang-Mills perturbation theory. Maybe integrability will lead the way.

### Acknowledgments

We are grateful to J. Russo and K. Zarembo for useful discussions.
In particular, we thank G. Arutyunov for many useful comments and
help with the discussion in Appendix D.
A.T. is also grateful to the organizers of
Simons workshop in Mathematics and Physics at Stony Brook
for the hospitality at the workshop
during which this paper was completed.
The work of S.F. and A.T. was supported by the DOE grant
DE-FG02-91ER40690. The work of A.T. was also supported in part
by the PPARC SPG 00613 and INTAS 99-1590 grants and the Royal
Society Wolfson award. N.B. dankt der *Studienstiftung des
deutschen Volkes* für die Unterstützung durch ein
Promotionsförderungsstipendium.

## Appendix A Rotating string solutions

Let us make some general observations on 5-spin string solutions in pointing out some relations between different types of solutions via an analytic continuation. The general rotating strings carrying 2+3 charges and the energy are described by the following ansatz [16] (see (1))

(A.1) |

Then the 3+3 obvious integrals of motion are^{11}^{11}11As
discussed in [16, 19], all other
generators (conserved charges)
of except the Cartan ones
,
should vanish in order for the rotating string solution to
represent a semiclassical string state carrying the
corresponding quantum numbers.

(A.2) |

They satisfy

(A.3) |

The second-order equations for

(A.4) |

and

(A.5) |

are decoupled from each other. As explained in [19], the resulting system of equations is completely integrable, being equivalent to a combination of the two Neumann dynamical systems. As a result, there are 2+2 “hidden” integrals of motion, reducing the general problem to solution of two independent systems of two coupled first-order equations, with parameters related through the conformal gauge constraint

(A.6) |

Let us now observe the following symmetry of the above system. The two metrics in (1) are related by the obvious analytic continuation and change of the overall sign, which is equivalent in the present rotational ansatz (A.1) case to

(A.7) |

This transformation maps the system (A.4) into the system (A.5) and also preserves the constraint (A). Thus it formally maps solutions into solutions. Under (A.7) the conserved charges (A.2) (or Cartan generators of ) transform as follows

(A.8) |

We could, of course, assume instead of (A.7) that but then . Note that the transformed solutions may not necessarily have a natural physical interpretation. In order for some two physical solutions to be related by this analytic continuation prescription at least one of them should have a non-vanishing spin (which transforms into the energy of the solution).

One can find also other transformations that map solutions into solutions by combining (A.7) with special (discrete) isometries that do not induce other components of the rotation generators except the above Cartan ones (e.g., interchanging the angular coordinates induces interchanging of the charges in (A.2), etc.). Below we shall consider such an example.

## Appendix B Relation between two-spin solutions

Let us now show that the two previously known two-spin folded string solutions are, in fact, related by the above analytic continuation. Firstly, there is the “” solution [11]

(B.1) |

where the string is stretched in the radial direction of . It rotates () in about its center of mass which in turn moves () along a large circle of . The gauge constraint (A.7) and integrals of motion (A.2) become

(B.2) |

Secondly, we have the “” solution [18] where the string located at the center of is stretched () along a great circle of and rotates () about its center of mass which moves () along an orthogonal great circle of :

(B.3) |

The gauge constraint (A.7) and integrals of motion (A.2) are then

(B.4) |

Following the discussion in Appendix A we
conclude that these two solutions are related
by the following
analytic continuation:
^{12}^{12}12
Note that here .
Choosing instead ,
we would
get ,
so that .

(B.5) |

We can, in fact, directly relate the systems of equations expressing the closed string periodicity condition and definitions of the respective energies and spins and thus relating the three integrals of motion in the two cases (see, respectively, [11] and [18]). In the first case we get the following relations (we introduce the parameter related to in [11] by , , where is the maximal value of the radial coordinate)

(B.6) |

where and are the standard elliptic integrals (see Appendix C) related to the hypergeometric functions used in [11] by