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arXiv:1104.2474v1 [hep-th] 13 Apr 2011

Yangians, S-matrices and AdS/CFT

Alessandro Torrielli

Department of Mathematics, University of York,

Heslington, York, YO10 5DD

United Kingdom

alessandro.torrielli@york.ac.uk

Abstract.

dimensional quantum group (specifically, Yangian) symmetry lying behind the inte-

grability of the AdS/CFT spectral problem. In passing, the chance is taken to give a

concise anthology of basic facts concerning Yangians and integrable systems, and to

store a series of remarks, observations and proofs the author has collected in a five-

year span of research on the subject. We hope this exercise will be useful for future

attempts to study Yangians in field and string theories, with or without supersymme-

try1.

This review is meant to be an account of the properties of the infinite-

Keywords.

AdS/CFT, Integrable Systems, Exact S-matrices, Quantum Groups,

Yangians, Lie Superalgebras, Representation Theory

1This is an author-created, un-copyedited version of an article (invited topical review) accepted

for publication in Journal of Physics A: Mathematical and Theoretical. IOP Publishing Ltd is not

responsible for any errors or omissions in this version of the manuscript or any version derived from it.

The definitive publisher authenticated version will be available online at [details to follow].

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1 Introduction

“What makes you think that the theory will still be integrable?”

“Unlimited optimism.”

(M. Staudacher, replying to A. A. Migdal at the Itzykson Meeting, Paris, 2007)

Gauge theories play a dominant role in our current understanding of the nature of

fundamental interactions at very short distances. A prominent example of such a the-

ory is the Standard Model of elementary particles, which is remarkably successful in

describing the physics up to the currently available energy scale. This description is,

however, to a significant extent restricted to the perturbative regime. The derivation

of analytical results when the coupling constant is large is an extremely challeng-

ing task. This represents an obstacle to the complete understanding of interesting

nonperturbative phenomena, like, for instance, confinement.

The revolutionary discovery of integrable structures in Quantum Chromodynam-

ics (QCD) [1], and, more recently, in planar N = 4 Supersymmetric Yang-Mills

(SYM) theory and AdS/CFT [2], has changed this situation2.

For a Hamiltonian system with 2n-dimensional phase space, complete integrabil-

ity stands for the existence of n independent integrals of motion, written as integrals

of local densities, in involution (i.e. Poisson-commuting with each other). One of

these integrals of motion is the Hamiltonian itself, while the other ones are some-

times referred to as higher Hamiltonians. According to the Liouville-Arnold theo-

rem, the equations of motion can then be solved by quadratures. This means that

there exists a set of canonical coordinates (‘action-angle’) such that the action vari-

2According to the AdS/CFT correspondence [3–7], the scaling dimension of gauge-invariant com-

posite operators should match the energy of the corresponding closed string states. In particular, wewill

be focusing our attention on string states with large values of some spin or angular momentum quantum

number Q, corresponding to composite operators containing a large (order Q) number of fields. The

energy of these states / dimension of these operators can be expressed as E = Q+ε(Q,λ), with ε going

to zero at weak ’t Hooft coupling λ ≡ g2

reduces to the bare dimension Q (see, for instance, [8]). The anomalous dimension ε is a dynamical

quantity which should interpolate between the two sides of the correspondence, and which will be our

main object of interest [9].

YMN (gYMbeing the Yang-Mills coupling) where the dimension

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ables (momenta) are constants of motion, and the angles (coordinates) are linear in

time and parameterize a torus. For a field theory, the number of degrees of freedom

is normally infinite, and one associates integrability with the existence of an infinite

number of independent local conserved charges in involution. In scattering theory,

integrability implies pure reshuffling of momenta (‘diffractionless’ scattering). In

general, flavour degrees of freedom can be transformed in a complicated way during

the scattering. One has ‘transmission’ if the flavours are unchanged, ‘reflection’ if

they are exchanged. We recommend [10–12] for classical references on integrable

systems (see also the excellent [13]).

A link with the Yang-Mills Millennium prize problem3has been also advertised.

The situation in AdS/CFT is quite peculiar because of conformal invariance. More-

over, ’t Hooft’s limit N → ∞, with λ = g2

butions, according to the standard argument that the action for such configurations

scales in this limit asN

one single interacting four-dimensional gauge theory in this special limit will be im-

portant for progress in the Yang-Mills problem as well. For a relatively recent report,

underlying the potential role of AdS/CFT and integrability, see [14].

The N = 4 theory is a quantum conformal field theory (CFT). The information

on its spectrum is encoded in the short-distance power-law behavior of (2-point) cor-

relators of composite operators. In determining this behavior for all operators of the

theory one encounters a non-trivial operator-mixing, which makes the calculations

notoriously difficult. The observation of [2] is that, in the planar limit, the problem

translates into the equivalent problem of finding the spectrum of certain spin-chain

Hamiltonians. This spectrum consists ofspin-wave excitations and their bound states,

and the dynamics (S-matrix4) describing their scattering turns out to be completely

integrable [15,16]. Planarity is probably a crucial ingredient for the appearance of

YMN fixed, suppresses instanton contri-

λ[finite]. However, one hopes that the understanding of even

3For any compact gauge group G, one is to show that quantum Yang-Mills theory on R4exists and

has a mass gap ∆ > 0 (i.e. the lightest particle has strictly positive mass squared).

4We take a chance and clarify that, whenever we will be talking of S-matrices in this review, it will

always be referred to the two-dimensional scattering of excitations in the integrable models effectively

describing the SYM spectral problem in various regimes (spin-chain, sigma model). Never will we be

talking of a spacetime SYM S-matrix (also because, in that case, conformal invariance would be an

obstacle to the definition of asymptotics states).

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integrability. It would be overwhelming to give here a comprehensive list of the rele-

vant references. They can be found in many of the available reviews (just to mention

some of the most recent ones, see [9,17–20]).

The result strictly applies to infinitely long chains, which are related to gauge

theory operators composed of an infinite number of fields. When the spin-chains are

of finite length, certain corrections occur that go under the name of ‘wrapping effects’

[21–23], since the range of the interactions exceeds the length of the spin-chain.

Recently [24,25], these effects have been shown to be calculable for very specific

operators and at the first few significant orders in perturbation theory, by techniques

of finite-volume integrability5. The first confirmation that one has obtained from

these impressive results is that the ingredients used in the mirror theory approach

[27], i.e. the mirror bound states, are all one needs to sum over in order to reproduce

the field theory result. In other words, no excitation is missing.

The technology developed so far has been impressive, see for instance [28–37].

Both gauge perturbation theory for short operators and string perturbation theory in

the form of L¨ uscher corrections have proceeded to a tremendous degree of sophis-

tication. A very convincing matching has been shown6. This remarkable result has

strengthened the expectation that the entire planar sector of the theory may in fact

be integrable, and accessible via the so-called Thermodynamic Bethe Ansatz (TBA)

method. The latter consists in obtaining a set of master equations, whose solutions

encode the spectral data of the theory. This program has the potential of providing a

set of exact analytic results for an interacting four-dimensional quantum field theory,

and, with it, a new insight in our understanding of strongly-coupled nonperturbative

phenomena in gauge theories. Once more, the study of two-dimensional models is

showing its power in modelling our understanding of four-dimensional theories (cf.

[42], Introduction, lines 37-58). Currently, a remarkable effort is being put into the

construction and test of such a TBA system of equations [43–45].

5These techniques involve the use of the so-called L¨ uscher corrections. Such corrections do not

assume integrability, but, if the theory is integrable, they are expected to complete to a set of exact

integral equations for the spectrum (see also [26]).

6Notably, the issue concerning some mismatches [38], which were still announced to affect the

strong coupling regime, has very recently been resolved [39–41].

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Despite the progress obtained, several fundamental questions are still left unan-

swered. First of all, a systematic way of taking into account the above-mentioned

wrapping corrections has not yet been provided, due to their highly complicated na-

ture [22]. Furthermore, no rigorous proof of integrability is yet available, and the

quantum Hamiltonian of the system is not known in closed form, but only to a cer-

tain order in perturbation theory. Instead, so far the approach has been (in the philos-

ophy of the inverse scattering method) to assume integrability and S-matrix factor-

ization, deduce the entire integrable structure, and a posteriori check the validity of

the assumptions (see also [46]). However, with long-range Hamiltonians (as the one

emerging from gauge perturbation theory actually is) even setting up an asymptotic

scattering theory is problematic, and it is still a challenge to rigorously prove the inte-

grability of the asymptotic problem. Perhaps, with the help of the algebraic methods

we are going to describe in this review, the knowledge of the complete Hamiltonian

will eventually become accessible7. The full algebraic structure is still, in many re-

spects, mysterious, and higher correlation functions of the theory are just starting to

be explored from the point of view of integrability. Three-point functions8are still

quite a virgin territory, and it is still unclear if the power of integrability will provide

a systematic way of computing them. When appropriately normalized, these three-

point functions scale as the two-point functions in the planar limit, and one would like

to compute them with spin-chain techniques. In this respect, the universal R-matrix

of quantum groups has been used in the past [48] to encode the braiding relations of

quantum field multiplets in an integrable 1+1-dimensional QFT, thereby extending

“off-shell” the “on-shell” quantum-group symmetry of the S-matrix. Along the same

lines, correlation functions and form factors9could be studied with the help of the

universal R-matrix.

Not fully understood is also the nature of certain fascinating dualities that have

7The so-called ‘dressing phase’ (see formula (74) and subsequent text) is essential for the Hamilto-

nian. In [47], the presence of this phase has been connected to boosts and general twist transformations

for the long-range spin-chain, see also section 3.1 and references therein.

8Because of quantum conformal invariance, one-, two- and three-point functions contain all the

information one needs.

9Form factors are matrix elements of field operators. They satisfy algebraic relations, called form-

factor axioms [49,50], depending locally on the fields and their sectors.

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