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Advances in High Energy Physics
VolumeΒ 2010, Article IDΒ 471238, 133 pages
Review Article

On String Integrability: A Journey through the Two-Dimensional Hidden Symmetries in the AdS/CFT Dualities

Nordic Institute for Theoretical Physics (NORDITA), Roslagstullsbacken 23, 10691 Stockholm, Sweden

Received 28 February 2010; Accepted 9 March 2010

Academic Editor: CarlosΒ Nunez

Copyright Β© 2010 Valentina Giangreco Marotta Puletti. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


One of the main topics in the modern String Theory are the AdS/CFT dualities. Proving such conjectures is extremely difficult since the gauge and string theory perturbative regimes do not overlap. In this perspective, the discovery of infinitely many conserved charges, that is, the integrability, in the planar AdS/CFT has allowed us to reach immense progresses in understanding and confirming the duality. We review the fundamental concepts and properties of integrability in two-dimensional 𝜎-models and in the AdS/CFT context. The first part is focused on the AdS5/CFT4 duality, especially the classical and quantum integrability of the type IIB superstring on AdS5Γ—S5 which is discussed in both pure spinor and Green-Schwarz formulations. The second part is dedicated to the AdS4/CFT3 duality with particular attention to the type IIA superstring on AdS4×ℂ𝑃3 and its integrability. This review is based on the author's PhD thesis discussed at Uppsala University the 21st September 2009.

1. Introduction: Motivations, Overview, and Outline

In 1997, Maldacena conjectured that type IIB superstrings on AdS5Γ—S5 describe the same physics of the supersymmetric SU(𝑁) Yang-Mills theory in four dimensions [1] (AdS5/CFT4). The background where the string lives (AdS5Γ—S5) is built of a five-dimensional anti-De Sitter space (AdS), a space with constant negative curvature, times a five-dimensional sphere (S), cf. Figure 1. In 2008, Aharony et al. proposed the existence of a further gauge/gravity duality between a theory of M2-branes in eleven dimensions and a certain three-dimensional gauge theory [2] (AdS4/CFT3). The eleven-dimensional M2-theory can be effectively described by type IIA superstrings when the string coupling constant is very small. For a reason that will be clear later, I will consider only the type IIA as the gravitational dual in the AdS4/CFT3 correspondence, but the reader should keep in mind that this is just a particular regime of the full correspondence. The background where the type IIA strings live is a four-dimensional anti-De Sitter space times a six-dimensional projective space (ℂ𝑃3).1

Figure 1: AdS5Γ—S5. The five-dimensional anti-De Sitter space is represented as a hyperboloid on the right hand side, while the five-dimensional sphere is drawn on the left hand side.

The conformal field theories contained in the AdS/CFT dualities, namely, 𝒩=4 super Yang-Mills (SYM) in the AdS5/CFT4 case and the supersymmetric 𝒩=6 Chern-Simons (CS) theory in the AdS4/CFT3 case, are rather difficult to solve. A general approach to quantum field theory is to compute quantities such as cross-sections, scattering amplitudes, and correlation functions. In particular, for conformal field theories the correlation functions are constrained by the conformal symmetry.2 For a certain class of operators (the conformal primary operators) their two-point function has a characteristic behavior: in the configuration space it is an inverse power function of the distance. The specific behavior, namely, the specific power (the so-called scaling dimension) depends on the nature of the operators and of the theory we are considering. It reflects how this operator transforms under conformal symmetry, in particular for the scaling dimension it reflects how the conformal primary operator transforms under the action of the dilatation operator. At high energy, the scaling dimensions acquire quantum corrections, that is, the anomalous dimension.3 In conformal field theories, the anomalous dimension encodes the physical information about the behavior of the operators under the renormalization process. I will expand this point in Section 2. For the moment it is enough to note that collecting the spectrum of the correlation functions, namely, the spectrum of the anomalous dimensions, gives an outstanding insight of the theory. However, in general it is a very hard task to reach such a knowledge for a quantum field theory.

For this purpose the gauge/string dualities can play a decisive role. Let me explain why. Both correspondences are strong/weak-coupling dualities: the strongly coupled gauge theory corresponds to a free noninteracting string and vice versa fully quantum strings are equivalent to weakly interacting particles. The two perturbative regimes on the string and on the gauge theory side do not overlap. Technical difficulties usually prevent to depart from such regimes. This implies that it is incredibly difficult to compare directly observable computed on the string and on the gauge theory side, and thus to prove the dualities. However, there is a positive aspect of such a weak/strong-coupling duality: in this way it is possible to reach the nonperturbative gauge theory once we acquire enough knowledge of the classical string theory.

Ironically, we are moving on a circle. In 1968, String Theory has been developed with the purposes to explain the strong nuclear interactions. Thus it started as a theory for particle physics. With the advent of the Quantum Chromo Dynamics (QCD), namely, the quantum field theory describing strong nuclear forces, String Theory was abandoned and only later in 1974 it has been realized that the theory necessarily contained gravity. The AdS/CFT dualities give us the possibility to reach a better insight and knowledge of SYM (and hopefully of the CS theory) by means of String Theory. In this sense, String Theory is turning back to a particle physics theory. In this scenario the long-term and ambitious hope is that also QCD might have a dual string description which might give us a deeper theoretical understanding of its nonperturbative regime.

At this point I will mostly refer to the AdS5/CFT4 correspondence, I will explicitly comment on the new-born duality at the end of the section. On one side of the correspondence, the AdS5Γ—S5 type IIB string is described by a quantum two-dimensional 𝜎-model in a very nontrivial background. On the other side, we have a quantum field theory, the SYM theory, which is also a rather complicated model. Some simplifications come from considering the planar limit, namely, when in the gauge theory the number of colors 𝑁 of the gluons is very large, or equivalently in the string theory when one does not consider higher-genus world-sheet. In this limit both gauge and string theories show their integrable structure, which turns out to be an incredible tool to explore the duality.

What does β€œintegrable” mean? We could interpret such a word as β€œsolvable” in a first approximation. However, this definition is not precise enough and slightly unsatisfactory. Integrable theories posses infinitely many (local and nonlocal) conserved charges which allow one to solve completely the model. Such charges generalize the energy and momentum conservation which is present in all the physical phenomena as, for example, the particle scatterings. Among all the integrable theories, those which live in two-dimensions are very special: in this case, the infinite set of charges manifests its presence by severely constraining the dynamics of the model through selection rules and through the factorization, cf. Section 3. In order to fix the ideas, let me consider the scattering of 𝑛 particles in two-dimensions. The above statement means that for an integrable two-dimensional field theory, a general 𝑛-particle scattering will be reduced to a sequence of two-particle scattering. The set of necessary information to solve the model is then restricted in a dramatic way: we only need to solve the two-body problem to have access to the full model! This is indeed the ultimate power of integrability.

The impressing result (which has been historically the starting point of the exploit of integrability in the AdS/CFT context) has been the discovery of a relation between the SYM gauge theory and certain spin chain models. In 2002, Minahan and Zarembo understood that the single trace operators (which are the only relevant ones in the planar limit) could be represented as spin chains [3]: each field in the trace becomes a spin in the chain. This is not only a pictorial representation: the equivalence is concretely extended also to the dilatation operator whose eigenvalues are the anomalous dimensions and to the spin chain Hamiltonian. The key-point is that such a spin chain Hamiltonian is integrable, β€œsolvable.” On the gravity side, the integrability of the AdS5Γ—S5 type IIB string has been rigorously proved only at classical level, which, in general, does not imply that the infinite conserved charges survive at quantum level. However, the assumption of an exact quantum integrability on both sides of AdS5/CFT4 has allowed one to reach enormous progresses in testing and in investigating the duality, thanks to the S-matrix program and to the entire Bethe Ansatz machinery, whose construction relies on such a hypothesis. Nowadays, nobody doubts about the existence of integrable structures underlying the gauge and the gravity side of the AdS5/CFT4 correspondence. There have been numerous and reliable manifestations, even though indirect. Despite of such remarkable developments one essentially assumes that the AdS5Γ—S5 type IIB superstring theory is quantum integrable.4 And on general ground, proving integrability at quantum level is a very hard task as much as proving the correspondence itself. For this reason, there have been very few direct checks of quantum integrability in the string theory side. These are the main motivations of the present work: give some direct and explicit evidence for the quantum integrability of the AdS superstring.

For the β€œyounger” AdS4/CFT3 duality, valuable results have been already obtained, cf. Section 7. It is very natural to ask whether and when it is possible to expect the existence of similar infinite symmetries also in this case. Considering the impressing history of the last ten years in AdS5/CFT4, one would like to reach analogous results also in this second gauge/string duality. Probably understanding which are the differences between these two dualities might provide another perspective of how we should think about the gauge/string dualities and their infinite β€œhidden” symmetries.

In Section 2, I will briefly introduce the AdS5/CFT4 correspondence and the 𝒩=4 SYM theory. It contains also a description of the symmetry algebra, 𝔭𝔰𝔲(2,2∣4), which controls the duality. I will also explain the crucial relation between the anomalous dimension and the spin chain systems as well as the Bethe Ansatz equations for a subsector of the full 𝔭𝔰𝔲(2,2∣4) algebra.
Section 3 is dedicated to two-dimensional integrable field theories, in particular to some prototypes for our string theory, such as the Principal Chiral Models and the Coset 𝜎-models. I will explain the definition of integrability in the first-order formalism approach as well as its dynamical implications for a two-dimensional integrable theory. I will stress the importance of the distinction between classical and quantum integrability.
In Section 4, I will review the type IIB string theory on AdS5Γ—S5: starting from the Green-Schwarz formalism, the Metsaev-Tseytlin formulation of the theory based on a coset approach and finally its classical integrability.
In Section 5, it is presented an alternative formulation of the type IIB AdS5Γ—S5 superstring based on the Berkovits formalism, also called Pure Spinor formalism, and I will focus on its relation with integrability topics, such as the construction of the BRST charges, the finiteness of the monodromy matrix and of its path deformation.
In Section 6, I will come back to the Green-Schwarz formalism and discuss some important limits of the AdS5Γ—S5 string theory such as the plane wave limit (also called BMN limit) and the near-flat-space limit. I will present the Arutyunov-Frolov-Staudacher dressing phase, sketch the construction of the world-sheet scattering matrix, also in the near-flat-space limit, and finally, I will illustrate its factorization.
Section 7 is entirely based on the AdS4/CFT3 duality. I will retrace certain fundamental results of the AdS5/CFT4 correspondence in the new context, with a special attention to the near-BMN corrections of string theory.
In the appendices, some complementary material is reported. In the first appendix, notation and conventions are summarized. The second one contains the full all-loop Bethe Ansatz equations. The third one is devoted to the pure spinor formalism, in particular the results concerning the operator product expansion for the matter and Lorentz ghost currents are listed. The fourth appendix contains an example showing the three-body S-matrix factorization. Finally, in the last one, the geometrical set-up for the AdS4/CFT3 is reported.

Note Added
This work is a shortened and revised version of the author’s PhD thesis, submitted to Uppsala University, Uppsala. It is based on the papers [4–7].

2. The AdS5/CFT4 Duality

The first part of this section is an introduction to the AdS5/CFT4 correspondence, based on the original works which are cited in the main text, and on the following reviews [8–10]. For the introductory part dedicated to the 𝒩=4 SYM and to the Coordinate Bethe Ansatz, I mainly refer to Minahan's review [11], Plefka's review [12], and Faddeev's review [13] and by Dorey at RTN Winter School (2008) [14]. Finally, I find very useful also the PhD theses written by Beisert [15] and Okamura [16].

2.1. Introduction

The Maldacena correspondence [1, 17, 18] conjectures an exact duality between the type IIB superstring theory on the curved space AdS5Γ—S5 and 𝒩=4 super Yang-Mills (SYM) theory on the flat four-dimensional space ℝ3,1 with gauge group SU(𝑁). In order to briefly illustrate the content of the duality, we will start by recalling all the parameters which are present in both theories.

The geometrical background in which the string lives is supported by a self-dual Ramond-Ramond (RR) five-form 𝐹5. In particular, the flux through the sphere is quantized, namely, it is an integer 𝑁, multiple of the unit flux. Both the sphere and the anti-De Sitter space have the same radius 𝑅: 𝑑𝑠2IIB=𝑅2𝑑𝑠2AdS5+𝑅2𝑑𝑠2S5,(2.1) where 𝑑𝑠2AdS5 and 𝑑𝑠2S5 are the unit metric in AdS5 and S5, respectively. The string coupling constant is 𝑔𝑠 and the effective string tension is 𝑇=𝑅2/2πœ‹π›Όξ…ž with π›Όξ…ž=𝑙2𝑠. The string theory side thus has two parameters:5𝑇,𝑔𝑠.

On the other side, SYM is a gauge theory with gauge group SU(𝑁), thus 𝑁 is the number of colors. The theory is maximally supersymmetric, namely, it contains the maximal number of global supersymmetries which are allowed in four dimensions (𝒩=4) [19, 20]. Another important aspect is that SYM is scale invariant at classical and quantum level, which means that the coupling constant 𝑔YM is not renormalized [21–25]. The theory contains two parameters, that is, 𝑁 and 𝑔YM. One can introduce the ’t Hooft coupling constant πœ†=𝑔2YM𝑁. Notice that πœ† is a continuous parameter. Summarizing, the gauge theory side has two parameters, we choose πœ† and 𝑁.

The correspondence states an identification between the coupling constants in the two theories, that is, 𝑔2YM=4πœ‹π‘”π‘ ,𝑇=βˆšπœ†2πœ‹(2.2) (or in terms of πœ†: 𝑔𝑠=πœ†/4πœ‹π‘), and between the observables, that is, between the string energy and the scaling dimension for local operators: 𝐸(πœ†,𝑁)=Ξ”(πœ†,𝑁).(2.3) The conjecture is valid for any value of the coupling constant πœ† and for any value of 𝑁6.

We can consider certain limits of the full general AdS5/CFT4 duality, which are simpler to be treated but still extremely interesting.

Let us consider the limit where 𝑁 is very large and πœ† is kept fixed, namely, 𝑔YMβ†’0 [26]. In this limit, 𝑁 is a continuous parameter and the gauge theory admits a 1/𝑁-expansion. In the large-𝑁 regime (also called the ’t Hooft limit) of the SYM theory only the planar diagrams survive, namely, all the Feynman diagrams whose topology is a sphere. The corresponding gravity dual is a free string propagating in a nontrivial background (AdS5Γ—S5). The string is noninteracting since now 𝑔𝑠→0 and the tension 𝑇 is kept fixed, cf. (2.2). Notice that even though we are suppressing 𝑔𝑠-corrections, so that the string is a free string on a curved background, it is still described by a nonlinear sigma model whose target-space geometry is AdS5Γ—S5. This is a highly nontrivial quantum field theory: the string can have quantum fluctuations which are described by an π›Όξ…ž-expansion.

Furthermore, we can also vary the smooth parameter πœ† between the strong-coupling regime (πœ†β‰«1) and the weak-coupling regime (πœ†β‰ͺ1). In the first case the gauge theory is strongly coupled, while the gravity dual can be effectively described by type IIB supergravity. Indeed, the radius of the background is very large (𝑅=πœ†1/4𝑙𝑠), thus the string is in a classical regime (𝑇≫1).

Conversely, when πœ† takes very small values (πœ†β‰ͺ1), the gauge theory can be treated with a perturbative analysis, while the background where the string lives is highly curved. The string is still free, but now the quantum effects become important (i.e., 𝑇β‰ͺ1).

For what we have learned above, the Maldacena duality is also called a weak/strong-coupling correspondence. This is an incredibly powerful feature, since it allows one to reach strong coupling regimes through perturbative computations in the dual description. At the same time, proving such a correspondence becomes an extremely ambitious task, simply because it is hard to directly compare the relevant quantities. For a summary about the different regimes and parameters we refer the reader to Table 1.

Table 1: Summary of the contents and parameters involved in AdS5/CFT4 duality.

We will only deal with the planar AdS/CFT, since it is in this regime that both theories have integrable structures. In particular, we are interested in the strong coupling regime (πœ†β‰«1), since the string theory side is reachable perturbatively (1/βˆšπœ† expansion) in the large ’t Hooft coupling limit (cf. Table 1). The present work is mainly devoted to this sector.

If the two theories are dual, then they should have the same symmetries. This is the theme of the next section, after a more detailed introduction to 𝒩=4 SYM theory.

2.2. 𝒩=4 Super Yang-Mills Theory in 4d

As already mentioned, the 𝒩=4 super Yang-Mills theory in four dimensions [19, 20] is a maximally supersymmetric and superconformal gauge theory. The theory is scale invariant at classical and quantum level and the 𝛽-function is believed to vanish to all orders in perturbation theory as well as nonperturbatively [21–25]. The action can be derived by dimensional reduction from the corresponding 𝒩=1SU(𝑁) gauge theory in ten dimensions:β„’YM=1𝑔210Trξ‚€βˆ’12𝐹𝑀𝑁𝐹𝑀𝑁+π‘–πœ“Ξ“π‘€π·π‘€πœ“ξ‚.(2.4)𝐷𝑀 is the covariant derivative, 𝐷𝑀=πœ•π‘€βˆ’π‘–π΄π‘€, where 𝐴𝑀 is the gauge field with 𝑀 the SO(9,1) Lorentz index, 𝑀=0,1,…,9, and 𝐹𝑀𝑁 the corresponding field strength, which is given by 𝐹𝑀𝑁=πœ•π‘€π΄π‘βˆ’πœ•π‘π΄π‘€βˆ’π‘–[𝐴𝑀,𝐴𝑁]. The matter content πœ“ is a ten-dimensional Majorana-Weyl spinor. The gauge group is SU(𝑁) and the fields 𝐴𝑀 and πœ“ transform in the adjoint representation of SU(𝑁).

By dimensionally reducing the action (2.4), the ten-dimensional Lorentz group SO(9,1) is broken to SO(3,1)Γ—SO(6), where the first group is the Lorentz group in four dimensions and the second one remains as a residual global symmetry (R-symmetry). Correspondingly, the Lorentz index splits in two sets: 𝑀=(πœ‡,𝐼), where πœ‡=0,1,2,3 and 𝐼=1,…,6. We need to require that the fields do not depend on the transverse coordinates 𝐼. Hence, the gauge field 𝐴𝑀 gives rise to a set of six scalars πœ™πΌ and to four gauge fields π΄πœ‡. Also the fermions split in two sets of four complex Weyl fermions πœ“π‘Ž,𝛼 and πœ“π‘Ž,̇𝛼 in four dimensions, where π‘Ž=1,…,4 is an SO(6)β‰…SU(4) spinor index and 𝛼,̇𝛼=1,2 are both SU(2) indices.

The final action for 𝒩=4 SYM in four dimensions is β„’YM=1𝑔2YMTrξ‚€βˆ’12πΉπœ‡πœˆπΉπœ‡πœˆβˆ’ξ€·π·πœ‡πœ™πΌξ€Έ2+12ξ€Ίπœ™πΌ,πœ™π½ξ€»2+π‘–πœ“Ξ“πœ‡π·πœ‡πœ“+πœ“Ξ“πΌξ€Ίπœ™πΌ,πœ“ξ€»ξ‚.(2.5)

2.3. The Algebra

We have already stressed that the theory has an SU(𝑁) gauge symmetry, thus the gauge fields are 𝔰𝔲(𝑁)-valued, and they also carry an index 𝑖=1,…,𝑁2βˆ’1, which is not explicit in the formulas above.

The conformal group in four dimensions is7SO(4,2)β‰…SU(2,2). The generators for the conformal algebra 𝔰𝔬(4,2) are the Lorentz transformation generators, which consist of three boosts and three rotations π‘€πœ‡πœˆ, the four translation generators π‘ƒπœ‡, coming from the PoincarΓ© symmetry, the four special conformal transformation generators πΎπœ‡, and the dilatation generator 𝐷. Hence in total we have fifteen generators.

The theory is also invariant under the β„›-symmetry, which plays the role of an internal flavor symmetry which can rotate the supercharges and the scalar fields. The β„›-symmetry group is SO(6)β‰…SU(4) and it is spanned by fifteen generators, 𝑅𝐼𝐽.

The supersymmetry charges π‘„π‘Žπ›Ό, π‘„π‘ŽΜ‡π›Ό, which transform under R-symmetry in the four-dimensional representations of SU(4) (πŸ’ and πŸ’, resp.), commute with the PoincarΓ© generators π‘ƒπœ‡. They do not commute with the special conformal transformation generators πΎπœ‡. However, their commutation relations give rise to a new set of supercharges. We denote this new set of supercharges with π‘†π‘Žπ›Ό and π‘†π‘ŽΜ‡π›Ό. They transform in the πŸ’ and πŸ’ representation of SU(4). Thus we have in total 32 real fermionic generators.

The SO(4,2)Γ—SO(6) bosonic symmetry groups and the supersymmetries merge in a unique superconformal group SU(2,2∣4). Actually, due to the vanishing of central charge for SYM, the final symmetry group is PSU(2,2∣4), where P denotes the fact that we are removing ad hoc the identity generators which can appear in the commutators. Notice that in supersymmetric theories usually the anticommutators between the supercharges 𝑄 and 𝑆 give an operator which commutes with all the rest, the so-called central charge.

The relevant relations are 𝐷,π‘ƒπœ‡ξ€»=βˆ’π‘–π‘ƒπœ‡,𝐷,πΎπœ‡ξ€»=π‘–πΎπœ‡,ξ€Ίπ‘ƒπœ‡,πΎπœˆξ€»=2π‘–ξ€·π‘€πœ‡πœˆβˆ’πœ‚πœ‡πœˆπ·ξ€Έ,ξ€Ίπ‘€πœ‡πœˆ,π‘ƒπœ†ξ€»=π‘–ξ€·πœ‚πœ†πœˆπ‘ƒπœ‡βˆ’πœ‚πœ‡πœ†π‘ƒπœˆξ€Έ,ξ€Ίπ‘€πœ‡πœˆ,πΎπœ†ξ€»=π‘–ξ€·πœ‚πœ†πœˆπΎπœ‡βˆ’πœ‚πœ‡πœ†πΎπœˆξ€Έ,ξ€Ίπ‘€πœ‡πœˆ,π‘€πœ†πœŒξ€»=βˆ’π‘–πœ‚πœ‡πœ†π‘€πœˆπœŒ+cycl.perm.ξ‚»π‘„π‘Žπ›Ό,𝑄𝑏̇𝛼=π›Ύπœ‡π›ΌΜ‡π›Όπ›Ώπ‘Žπ‘π‘ƒπœ‡,ξ‚†π‘†π‘Žπ›Ό,𝑆𝑏̇𝛼=π›Ύπœ‡π›ΌΜ‡π›Όπ›Ώπ‘Žπ‘πΎπœ‡,𝐷,π‘„π‘Žπ›Όξ€»=βˆ’π‘–2π‘„π‘Žπ›Ό,𝐷,π‘„π‘ŽΜ‡π›Όξ‚„=βˆ’π‘–2π‘„π‘ŽΜ‡π›Ό,𝐷,π‘†π‘Žπ›Όξ‚„=𝑖2π‘†π‘Žπ›Ό,𝐷,π‘†π‘ŽΜ‡π›Όξ‚„=𝑖2π‘†π‘ŽΜ‡π›Ό,ξ€ΊπΎπœ‡,π‘„π‘Žπ›Όξ€»=πœŽπœ‡π›ΌΜ‡π›Όπœ–Μ‡π›ΌΜ‡β€Œπ›½π‘†π‘ŽΜ‡β€Œπ›½,ξ‚ƒπΎπœ‡,π‘„π‘ŽΜ‡π›Όξ‚„=πœŽπœ‡π›ΌΜ‡π›Όπœ–π›Όπ›½π‘†π‘Žπ›½,ξ‚ƒπ‘ƒπœ‡,π‘†π‘Žπ›Όξ‚„=ξ€·πœŽπœ‡ξ€Έπ›ΌΜ‡π›Όπœ–Μ‡π›ΌΜ‡β€Œπ›½π‘„π‘ŽΜ‡β€Œπ›½,ξ‚ƒπ‘ƒπœ‡,π‘†π‘ŽΜ‡π›Όξ‚„=ξ€·πœŽπœ‡ξ€Έπ›ΌΜ‡π›Όπœ–π›Όπ›½π‘„π‘Žπ›½,ξ€Ίπ‘€πœ‡πœˆ,π‘„π‘Žπ›Όξ€»=π‘–πœŽπœ‡πœˆπ›Όπ›½πœ–π›½π›Ύπ‘„π‘Žπ›Ύ,ξ‚ƒπ‘€πœ‡πœˆ,π‘„π‘ŽΜ‡π›Όξ‚„=π‘–πœŽπœ‡πœˆΜ‡π›ΌΜ‡β€Œπ›½πœ–Μ‡β€Œπ›½Μ‡π›Ύπ‘„π‘ŽΜ‡π›Ύ,ξ‚ƒπ‘€πœ‡πœˆ,π‘†π‘Žπ›Όξ‚„=π‘–πœŽπœ‡πœˆπ›Όπ›½πœ–π›½π›Ύπ‘†π‘Žπ›Ύ,ξ‚ƒπ‘€πœ‡πœˆ,π‘†π‘ŽΜ‡π›Όξ‚„=π‘–πœŽπœ‡πœˆΜ‡π›ΌΜ‡β€Œπ›½πœ–Μ‡β€Œπ›½Μ‡π›Ύπ‘†π‘ŽΜ‡π›Ύ,ξ‚†π‘„π‘Žπ›Ό,𝑆𝑏𝛽=βˆ’π‘–πœ–π›Όπ›½ξ€·πœŽπΌπ½ξ€Έπ‘Žπ‘π‘…πΌπ½+πœŽπœ‡πœˆπ›Όπ›½π›Ώπ‘Žπ‘π‘€πœ‡πœˆβˆ’πœ–π›Όπ›½π›Ώπ‘Žπ‘π·,ξ‚†π‘„π‘ŽΜ‡π›Ό,π‘†π‘Μ‡β€Œπ›½ξ‚‡=βˆ’π‘–πœ–Μ‡π›ΌΜ‡β€Œπ›½ξ€·πœŽπΌπ½ξ€Έπ‘Žπ‘π‘…πΌπ½+πœŽπœ‡πœˆΜ‡π›ΌΜ‡β€Œπ›½π›Ώπ‘Žπ‘π‘€πœ‡πœˆβˆ’πœ–Μ‡π›ΌΜ‡β€Œπ›½π›Ώπ‘Žπ‘π·.(2.6) The matrices πœŽπœ‡π›ΌΜ‡π›Ό are the Dirac 2Γ—2 matrices and (𝜎𝐼𝐽)π‘Žπ‘ are the antisymmetric product of the Dirac 4Γ—4 matrices.

Matrix Realization
It is natural to reorganize the 𝔰𝔲(2,2∣4) generators as 8Γ—8 supermatrices: 𝑀=βŽ›βŽœβŽœβŽπ‘ƒπœ‡,πΎπœ‡,πΏπœ‡πœˆ,π·π‘„π›Όπ‘Ž,π‘†Μ‡π›Όπ‘Žπ‘†π‘Žπ›Ό,π‘„π‘ŽΜ‡π›Όπ‘…πΌπ½βŽžβŽŸβŽŸβŽ .(2.7) On the diagonal blocks we have the generators for two bosonic subsectors, 𝔰𝔲(2,2) and 𝔰𝔲(4), while on the off-diagonal blocks we have the fermionic generators. The superalgebra is realized by two conditions which naturally generalize the 𝔰𝔲(𝑛,π‘š) algebra. First, the supertrace8 of the matrix (A) vanishes. Second it satisfies a reality condition π»π‘€β€ βˆ’π‘€π»=0,(2.8) where 𝐻=βŽ›βŽœβŽπ›Ύ5001⎞⎟⎠.(2.9) The 4Γ—4 matrix 𝛾5 appears in the above condition because 𝛾5 realizes the Hermitian conjugation in the SU(2,2)β‰…SO(4,2) sector.
Actually, we want to consider the 𝔭𝔰𝔲(2,2∣4) algebra. The 8Γ—8𝔰𝔲(2,2∣4) identity matrix trivially satisfies both properties of tracelessness and of Hermicity. This means that even though such a matrix is not among our set of initial generators of the 𝔰𝔲(2,2∣4) algebra, at some point it will appear as a product of some commutators. This is analogous to what we have discussed above, where the anticommutator between 𝑄 and 𝑆 might have a term proportional to the unit matrix. In the SYM, the central charge is zero, thus we would like to remove the unit matrix. We therefore mod out the 𝔲(1) factor ad hoc. This is indeed the meaning of the 𝔭 in 𝔭𝔰𝔲(2,2∣4). Note that such an algebra cannot be realized in terms of matrices.
The total rank for the PSU(2,2∣4) supergroup is 7. The unitary representation is labelled by the quantum numbers for the bosonic subgroup. This means that the fields of 𝒩=4 SYM, or better, local gauge invariant operators, and the states of the AdS5Γ—S5 string are characterized by 6 charges, which are the Casimirs of the groupξ€·Ξ”=𝐸,𝑆1,𝑆2,𝐽1,𝐽2,𝐽3ξ€Έ.(2.10) The equality for the first charge is really the expression of the AdS/CFT correspondence. Let us see in more detail what these quantum numbers are. Coming from the SU(2,2) sector, since SO(1,1)Γ—SO(3,1)βŠ‚SO(4,2), we have the dilatation operator eigenvalue Ξ” (or the string energy 𝐸), which can take continuous values, and the two spin eigenvalues 𝑆1,𝑆2, which can have half-integer values, and which are the charges related to the Lorentz rotations in SO(3,1). Notice that Ξ” and 𝐸 depend on the coupling constant πœ†, cf. (2.3). The other sector SU(4)β‰…SO(6) contributes with the β€œspins” 𝐽1,𝐽2,𝐽3, which characterize how the scalars can be rotated.

The String Side
The isometry group of AdS5Γ—S5 is SO(4,2)Γ—SO(6), which is nothing but the bosonic sector of PSU(2,2,∣4). Thus on the string side the bosonic symmetries are realized as isometries of the background where the string lives. The superstring also contains fermionic degrees of freedom which will mix the two bosonic sectors corresponding to AdS5 and S5. The string spectrum is labelled by the charges (2.10). In principle one can also have winding numbers to characterize the string state, in addition to (2.10). The string energy 𝐸 is the charge corresponding to global time translation in AdS5, while 𝑆1,𝑆2 correspond to the Cartan generators of rotations in AdS5. The last three charges correspond to Cartan generators for S5 rotations, since the five-dimensional sphere can be embedded in ℝ6, so we have three planes the rotations.

2.4. Anomalous Dimension and Spin Chains

In a conformal field theory the correlation functions between local gauge invariant operators contain most of the relevant dynamical information. There is a special class of local operators, the (super) conformal primary operators, whose correlators are fixed by conformal symmetry. In particular, these are the operators annihilated by the special conformal generators 𝐾 and by the supercharges 𝑆, that is, 𝐾π’ͺ=0 and 𝑆π’ͺ=0. Thus, representations corresponding to primary operators are classified by how the dilatation operator 𝐷 and the Lorentz transformation generators 𝑀 act on π’ͺ, that is, by the 3-tuplet (Ξ”,𝑆1,𝑆2): 𝐷π’ͺ=Ξ”π’ͺ,𝑀π’ͺ=Σ𝑆1,𝑆2π’ͺ,(2.11) where Ξ” is the scaling dimension, namely, the dilatation operator eigenvalues, and Σ𝑆1,𝑆2 tells us how the operator π’ͺ transforms under Lorentz transformations. Since the special conformal transformation generator 𝐾 lowers the dimension by 1 and the supercharge 𝑆 by 1/2, cf. (2.6), in a unitary field theory the primary operators correspond to those operators with lowest dimension. They are also called highest-weight states. All the other operators in the same multiplet can be obtained by applying iteratively the translation operator 𝑃 and the supercharges 𝑄 (descendant conformal operators).

The correlation functions of primary operators are highly restricted by the invariance under conformal transformations, and they are of the form ⟨π’ͺπ‘š(π‘₯)π’ͺ𝑛(𝑦)⟩=πΆπ›Ώπ‘šπ‘›||π‘₯βˆ’π‘¦||2Ξ”.(2.12) In the scaling dimension there are actually two contributions: Ξ”=Ξ”0+𝛾.(2.13)Ξ”0 is the classical dimension and 𝛾 is the so-called anomalous dimension. It is in general a nontrivial function of the coupling constant πœ†. It appears once one starts to consider quantum corrections, since in general the correlators will receive quantum corrections from their free field theory values.

When we move from the classical to the quantum field theory we also need to face the problem of renormalization. In general in quantum field theory the renormalization is multiplicative. The operators are redefined by a field strength function 𝑍 according to π’ͺπ‘š=π‘π‘›π‘šπ’ͺ𝑛,0,(2.14) where the subscript 0 denotes the bare operator, and 𝑍 depends on the physical scale πœ‡ (typically π‘βˆΌπœ‡π›Ύ). As an example, we can consider the correlators in (2.12). Applying the Callan-Symanzik equation, recalling that the 𝛽-function vanishes and defining the so-called mixing matrix Ξ“ as Ξ“π‘˜π‘š=ξ“π‘›ξ€·π‘βˆ’1ξ€Έπ‘›π‘šπœ•π‘π‘˜π‘›πœ•logπœ‡,(2.15) we see that when the operator Ξ“ acts on a basis {π’ͺπ‘š}, then the corresponding eigenvalues are indeed the anomalous dimensions π›Ύπ‘š: Ξ“π’ͺπ‘š=π›Ύπ‘šπ’ͺπ‘š.(2.16) Hence, Ξ“ provides the quantum correction to the scaling operator 𝐷, that is, 𝐷=𝐷0+Ξ“.

2.4.1. The Coordinate Bethe Ansatz for the 𝔰𝔲(2) Sector

In this section, I will sketch the Coordinate Bethe Ansatz, also called Asymptotic Bethe Equations (ABE), for the bosonic closed SU(2) subsector, as the title suggested, in order to get feeling of why such techniques are so important. The ABE are the basic connection between integrability, SYM theory, spin chain, and the S-matrix.

As pointed out in the previous section, a lot of the relevant physical information are contained in the anomalous dimension of a certain class of gauge invariant operators. The fact that the operators are gauge invariant means that we have to contract the SU(𝑁) indices. This can be done by taking the trace. In general, we can have multitrace operators. However, in the planar limit (π‘β†’βˆž) the gauge invariant operators which survive are the single trace ones. Thus from now on, we are only dealing with single trace local operators (and with their anomalous dimension).

The incredible upshot of this section will be that the mixing matrix (2.15) is the Hamiltonian of an integrable (1+1) dimensional spin chain! There are two important points in the last sentence. First, it means that the eigenvalues of the mixing matrix are the eigenvalues of a spin chain Hamiltonian, namely, the corresponding anomalous dimensions are nothing but the solutions of the SchrΓΆdinger equation of certain spin chain Hamiltonians. I cannot say whether it is easier to compute 𝛾, or to solve some quantum mechanical system such as a one-dimensional spin chain. But here it enters the second keyword used: integrable. The spin system has an infinite set of conserved charges, all commuting with the Hamiltonian (which is just one of the charges), which allows us to solve the model itself. In concrete terms, this means that we can compute the energies of the spin chain, namely, the anomalous dimension (of a certain class) of 𝒩=4 SYM operators! Here the advantage is not purely conceptual but also practical: we can exploit and/or export in a string theory context some methods and techniques usually used in the condensed matter physics, for example. And this is what we will see in a moment.

We have just claimed that the anomalous dimensions (for a certain class of operators) can be computed via spin chain picture. We have to make this statement more precise. In particular, we need to specify when and how it is true. In order to illustrate how integrability enters in the gauge theory side, and its amazing implications, I have chosen to review in detail the simplest example: the closed bosonic SU(2) subsector of SO(6). Historically, the connection between SYM gauge theory and spin chain was discovered by Minahan and Zarembo for the scalar SO(6) sector of the planar PSU(2,2∣4) group [3]. This has been the starting point for all the integrability machinery in AdS/CFT.9

The scalar fields πœ™πΌ with 𝐼=1,…,6 can be rearranged in a complex basis. For example, we can write 𝑍=πœ™1+π‘–πœ™2,π‘Š=πœ™3+π‘–πœ™4,π‘Œ=πœ™5+π‘–πœ™6.(2.17) The three complex fields 𝑍,π‘Š, and π‘Œ generate SU(4). The SU(2) subgroup is constructed by considering two of the three complex scalars. For example, we can take the fields 𝑍 and π‘Š. We are considering gauge invariant operators of the type π’ͺ(π‘₯)=Tr(π‘Šπ‘π‘Šπ‘Šπ‘π‘Šπ‘Šπ‘Šπ‘Šπ‘π‘π‘Šπ‘Š)|π‘₯+β‹―,(2.18) where the dots indicate permutations of the fields and the subscript on the right hand side stresses the fact that these fields are all evaluated in the point π‘₯. If one identifies the fields in the following way 𝑍=↑,π‘Š=↓,(2.19) then the operator π’ͺ in (2.18) can be represented by a spin chain. In particular, for the operator (2.18), the corresponding spin chain is represented in Figure 2. If we have 𝐿 fields sitting in the trace of the operator π’ͺ, it means that we are considering a spin chain of length 𝐿, with 𝐿 sites. Each site has assigned a spin, up or down, according to the identification (2.19).

Figure 2: Example of a spin chain. The β€œup” arrow represents the field 𝑍, while the down spin is represented by the field π‘Š.

At one-loop the dilation operator for gauge invariant local operators which are 𝔰𝔲(2) multiplets that can be identified with the Hamiltonian of a Heisenberg spin chain, also denoted as an 𝑋𝑋𝑋1/2 spin chain. Note that this is a quantum mechanics system.

The identification between the Heisenberg spin chain Hamiltonian and the SU(2) one-loop dilatation operator can be seen by an explicit computation of such an operator [3]. In particular, one has that Ξ“(1)=πœ†8πœ‹2𝐿𝑙=1𝐻𝑙,𝑙+1,(2.20) where 𝐻𝑙,𝑙+1 is the operator acting on the sites 𝑙 and 𝑙+1, explicitly: 𝐻=πœ†8πœ‹2𝐿𝑙=1𝐻𝑙,𝑙+1=πœ†8πœ‹2𝐿𝑙=1ξ€·I𝑙,𝑙+1βˆ’π‘ƒπ‘™,𝑙+1ξ€Έ=πœ†16πœ‹2𝐿𝑙=1ξ€·I𝑙,𝑙+1βˆ’βƒ—πœŽπ‘™β‹…βƒ—πœŽπ‘™+1ξ€Έ,(2.21) where 𝑃𝑙,𝑙+1=(1/2)(I𝑙,𝑙+1+βƒ—πœŽπ‘™β‹…βƒ—πœŽπ‘™+1) is the permutation operator. The one-loop order is mirrored by the fact that the Hamiltonian only acts on the sites which are nearest neighbors. The identity operator 𝐼𝑙,𝑙+1 leaves the spins invariant, while the permutation operator 𝑃𝑙,𝑙+1 exchanges the two spins.

We want to compute the spectrum. This means that we want to solve the Schrâdinger equation 𝐻|Ψ⟩=𝐸|Ψ⟩. |Ψ⟩ will be some operators of the type (2.18), and the energy will give us the one-loop anomalous dimension for such operator. The standard approach would require us to list all the 2𝐿 states and then, after evaluating the Hamiltonian on such a basis, we should diagonalize it. This is doable for a very short spin chain, not in general for any value 𝐿. The brute force here does not help, and indeed there are smarter ways as the one found by Bethe in 1931 [27].

One-Magnon Sector
Let us choose a vacuum of the type ||0βŸ©β‰‘||β†‘β†‘β‹―β†‘β†‘β‹―β†‘βŸ©,(2.22) and consider an infinite long spin chain, that is, πΏβ†’βˆž. The vacuum has all spins up and it is annihilated by the Hamiltonian (2.21). The choice of the vacuum breaks the initial SU(2) symmetry to a U(1) symmetry. Consider now the state with one excitation, namely, with an impurity in the spin chain |π‘₯βŸ©β‰‘|||||↑↑⋯↑↓ξ‚₯π‘₯↑⋯↑.(2.23) The excitation, called a magnon, is sitting in the site π‘₯ of the spin chain. The wave function is ||Ψ⟩=βˆžξ“π‘₯=βˆ’βˆžΞ¨(π‘₯)|π‘₯⟩.(2.24) By computing the action of the Hamiltonian 𝐻 on |Ψ⟩, one obtains 𝐻||Ψ⟩=βˆžξ“π‘₯=βˆ’βˆžΞ¨(π‘₯)ξ€·|π‘₯βŸ©βˆ’||π‘₯+1βŸ©βˆ’||π‘₯βˆ’1βŸ©ξ€Έ=βˆžξ“π‘₯=βˆ’βˆž(2Ξ¨(π‘₯)βˆ’Ξ¨(π‘₯+1)βˆ’Ξ¨(π‘₯+1))|π‘₯⟩.(2.25) Let us make an ansatz for the wave-function. Choosing Ξ¨(π‘₯)=𝑒𝑖𝑝π‘₯,π‘βˆˆβ„,(2.26) then the SchrΓΆdinger equation for the one-impurity state reads 𝐻||Ψ⟩=βˆžξ“π‘₯=βˆ’βˆžπ‘’π‘–π‘π‘₯ξ€·2βˆ’π‘’π‘–π‘βˆ’π‘’βˆ’π‘–π‘ξ€Έ|π‘₯⟩.(2.27) This means that the energy for the one magnon state is 𝐸(𝑝)=πœ†8πœ‹2ξ€·2βˆ’π‘’π‘–π‘βˆ’π‘’βˆ’π‘–π‘ξ€Έ=πœ†2πœ‹2sin2𝑝2.(2.28) This is nothing but a plane wave along the spin chain.
The spin chain is a discrete system. There is a well-defined length scale, which is given by the lattice size, and the momentum is confined in a region of definite length, typically the interval [βˆ’πœ‹,πœ‹] (the first Brillouin zone). An infinite chain might be obtained by considering a chain of length 𝐿 and assume periodicity. Thus we need to impose a periodic boundary condition on the magnon wave function, which meansΞ¨(π‘₯+𝐿)=Ξ¨(π‘₯)βŸΉπ‘’π‘–π‘πΏ=1βŸΉπ‘π‘›=2πœ‹π‘›πΏ,π‘›βˆˆβ„€.(2.29) These are the Coordinate Bethe equations for the one-magnon sector. They are the periodicity conditions of the spin chain.10
Leaving the spin chain picture, and going back to the gauge theory, the operator π’ͺ in (2.18) is not only periodic but cyclic (due to the trace). For the single magnon, this implies that the excited spin must be symmetrized over all the sites of the chain. Thus the total energy vanishes11. Indeed, operators of the kindπ’ͺ=Tr(β‹―π‘π‘π‘π‘Šπ‘π‘β‹―)(2.30) are chiral primary operators: their dimension is protected and one can see that the cyclicity of the trace means that the total momentum vanishes, which is another way of saying that the energy is zero, cf. (2.28).
Thus there is no operator in SYM that corresponds to the single magnon state. This is actually true for all sectors, since it follows from the cyclicity of the trace.

Two-Magnon Sectors
Consider now a state with two excitations, namely, two spins down: ||π‘₯<π‘¦βŸ©=|||||↑⋯↑↓ξ‚₯π‘₯↑⋯↑↑↓ξ‚₯𝑦↑⋯,||Ψ⟩=βˆžξ“π‘₯<𝑦=βˆ’βˆžΞ¨(π‘₯,𝑦)||π‘₯<π‘¦βŸ©.(2.31) The Hamiltonian (2.21) is short-ranged, thus when π‘₯+1<𝑦 it proceeds as before for the single magnon state, just that in this case the energy 𝐸 would be the sum of two magnon dispersion relations. The problem starts when π‘₯+1=𝑦, namely, in the contact terms. In this case the ScrΓΆdinger equation for the wave-function gives 2Ξ¨(π‘₯,π‘₯+1)βˆ’Ξ¨(π‘₯βˆ’1,π‘₯+1)βˆ’Ξ¨(π‘₯,π‘₯+2)=0.(2.32) It is clear that a wave function given by a simple sum of the two single magnon states as in (2.26) does not diagonalize the Hamiltonian (2.21), but β€œalmost.” Using the following ansatz:12Ξ¨(π‘₯,𝑦)=𝑒𝑖𝑝π‘₯+π‘–π‘žπ‘¦βˆ’π‘–(𝛿/2)+π‘’π‘–π‘žπ‘₯+𝑖𝑝𝑦+𝑖(𝛿/2),π‘₯<𝑦,(2.33) and imposing that it diagonalizes the Hamiltonian, one finds the value for the phase shift 𝛿 that solves the equation, namely, 𝑒𝑖𝛿(𝑝,π‘ž)=βˆ’1βˆ’2π‘’π‘–π‘ž+𝑒𝑖𝑝+π‘–π‘ž1βˆ’2𝑒𝑖𝑝+π‘’π‘–π‘ž+πœ„π‘=βˆ’cot𝑝/2βˆ’cotπ‘ž/2βˆ’2𝑖cot𝑝/2βˆ’cotπ‘ž/2+2𝑖.(2.34) For this phase shift the total energy is just the sum of two single magnon dispersion relations (trivially the ansatz (2.33) with the phase shift given by (2.34) solves the case with π‘₯+1<𝑦). What does this phase shift represent? This is the shift experienced by the magnon once it passes through the other excitation, namely, when it scatters a magnon of momentum π‘ž. Hence, 𝑆(𝑝,π‘ž)≑𝑒𝑖𝛿(𝑝,π‘ž) is nothing but the corresponding scattering-matrix.
We still have to impose the periodic boundary conditions on the wave functions:13Ξ¨(0,𝑦)=Ξ¨(𝐿,𝑦),(2.35) which, after substituting the phase shift (2.34) in (2.33), gives 𝑒𝑖𝑝𝐿=π‘’βˆ’π‘–π›Ώ(𝑝,π‘ž)=𝑒𝑖𝛿(π‘ž,𝑝)=𝑆(π‘ž,𝑝),π‘’π‘–π‘žπΏ=𝑒𝑖𝛿(𝑝,π‘ž)=𝑆(𝑝,π‘ž).(2.36) Again, these are the Coordinate Bethe equations for the 𝔰𝔲(2) sector with two magnons.
Finally, we need to impose the cyclicity condition, that is, 𝑝+π‘ž=0, which means that the Bethe equations (2.36) are solved for𝑝=2πœ‹π‘›πΏβˆ’1=βˆ’π‘ž.(2.37) The energy becomes 𝐸=𝐸(𝑝)+𝐸(π‘ž)=πœ†πœ‹2sin2ξ‚€πœ‹π‘›πΏβˆ’1.(2.38) May be the reader is more familiar to the Bethe equations expressed in terms of the rapidities, also called Bethe roots ,14 namely, introducing π‘’π‘˜=12cotπ‘π‘˜2,(2.39) and using 𝑝=βˆ’π‘ž, the phase shift reads 𝑒𝑖𝛿(𝑒,βˆ’π‘’)=𝑆(𝑒,βˆ’π‘’)=βˆ’π‘’βˆ’(𝑖/2)𝑒+(𝑖/2).(2.40)

𝐾 Magnon Sectors
The results of the previous section can be generalized to any number of magnons 𝐾 (with 𝐾<𝐿). The Bethe equations for general 𝐾 are π‘’π‘–π‘π‘˜πΏ=πΎξ‘π‘—β‰ π‘˜π‘’βˆ’π‘–π›Ώ(π‘π‘˜,𝑝𝑗)=πΎξ‘π‘—β‰ π‘˜π‘†ξ€·π‘π‘—,π‘π‘˜ξ€Έ.(2.41) The energy is a sum of 𝐾 single particle energies 𝐸=πΎξ“π‘˜=1πΈπ‘˜=πœ†2πœ‹2πΎξ“π‘˜=1sin2π‘π‘˜2,(2.42) and the cyclicity condition is πΎξ‘π‘˜=1π‘’π‘–π‘π‘˜πΏ=1.(2.43) In terms of the rapidities (2.39) all these conditions take the maybe more common form of 1=ξ‚΅π‘’π‘˜+(𝑖/2)π‘’π‘˜βˆ’(𝑖/2)𝐿𝐾𝑗=1,π‘˜β‰ π‘—π‘’π‘—βˆ’π‘’π‘˜+π‘–π‘’π‘—βˆ’π‘’π‘˜βˆ’π‘–,(Betheequations),𝐸=πΎξ“π‘˜=1βŽ›βŽœβŽœβŽπ‘–π‘’π‘˜+𝑖2ξ‚βˆ’π‘–π‘’π‘˜βˆ’ξ‚€π‘–2ξ‚βŽžβŽŸβŽŸβŽ ,(energy),1=πΎξ‘π‘˜=1π‘’π‘˜+(𝑖/2)π‘’π‘˜βˆ’(𝑖/2),(cyclicity).(2.44) What have we achieved? The remarkable point is that the Hamiltonian of a (1+1)-dimensional spin chain has been diagonalized by means of the 2-body S-matrix 𝑆(𝑝,π‘ž), cf. (2.41). Indeed, in order to know the spectrum of 𝐾 magnons, where 𝐾 is arbitrary, we only need to solve the Bethe equations and to compute the two-body S-matrix. The 𝐾-body problem is then reduced to a 2-body problem, which is an incredible achievement. This does not happen in general. The underlying notion that we are using here is that each magnon goes around the spin chain and scatters only with one magnon each time. This is possible only for integrable spin chains, or in general for integrable models.15 We will come back more extensively on this in the next section.

2.4.2. The Full Planar π‘ƒπ‘†π‘ˆ(2,2∣4) ABE

Here we have shown in details the SU(2) subsector for the fields in the spin 1/2 representation. However, this can be generalized to other representations for the same group, or to other groups (e.g., SU(𝑁)) and also to higher loops. What is really interesting for us, in an AdS/CFT perspective, is that the asymptotic Bethe equations for the full (planar) PSU(2,2∣4) group have been written down. This has been done by Beisert and Staudacher [28]. They are reported in Appendix B.

At the beginning of the section we explained that the Bethe equations are called β€œasymptotic.” β€œAsymptotic” since the Bethe procedure captures the correct behavior of the anomalous dimension only up to πœ†πΏ order for a chain of length 𝐿. After this order, wrapping effects have to be taken into account. They reflect the fact that the chain has a finite size. At the order 𝑛 in perturbation theory, the spin chain Hamiltonian involves interaction up to 𝑛+1 sites: 𝐻𝑙,𝑙+1,…,𝑙+𝑛. If the spin chain has total length 𝐿=𝑛+1, then it is clear that there might be interactions that go over all the spin chain, namely, they wrap the chain.16 At this point the ABE are no longer valid. In order to compute these finite-size effects, one might proceed with different techniques as the LΓΌscher corrections [29, 30],17 the Thermodynamic Bethe Ansatz (TBA) [31], cf. [32–35] for very recent results, and the π‘Œ-system [36]. These topics currently are one of the main area of research in the context of integrability and AdS/CFT, however here we will not face the problem of finite-size effects.18 The explicit one-loop PSU(2,2∣4) spin chain Hamiltonian has been derived by Beisert in [37]. This means that the expression of the one-loop dilatation operator for the 𝒩=4 SYM is known. Increasing the loop order usually makes things (and thus also the dilatation operator) sensibly more complicated, cf., for example, see [38]. Moreover, we do not really need the explicit expression of the Hamiltonian, once one has the Bethe equations. Indeed, nowadays we have from the one-loop [39] to the all loop asymptotic Bethe equations for the planar PSU(2,2∣4) [28].

3. Classical versus Quantum Integrability

The superstring theory on AdS5Γ—S5 can be described by a very special two-dimensional field theory. Indeed, such a theory shows an infinite symmetry algebra. Before discussing such an algebra for the specific case of the superstring we will review other integrable (1+1) field theories, their conserved (local and nonlocal) charges and finally stress the difference between integrability at classical and quantum level.

The discovery of an infinite set of conserved charges in two-dimensional classical 𝜎 models is due to Pohlmeyer [40] and LΓΌscher and Pohlmeyer [41]. A different derivation of the tower of conserved charges has been given by Brezin et al. in [42]. A very useful review is Eichenherr’s paper [43].

3.1. Principal Chiral Model

As a prototype to start our discussion with, we consider the so-called Principal Chiral Model (PCM). The following presentation is mostly based on [44]. The PCM is defined by the following Lagrangian: β„’=1𝛾2Trξ€·πœ•πœ‡π‘”βˆ’1πœ•πœ‡π‘”ξ€Έ,(3.1) where 𝑔 is a group valued map, π‘”βˆΆΞ£β†’πΊ with Ξ£ a two-dimensional manifold and 𝐺 a Lie group. In particular Ξ£ is parameterized by πœŽπœ‡=(𝜏,𝜎). We can think to Ξ£ as the string world-sheet. 𝛾 is a dimensionless coupling constant, the model is conformally invariant. The model (3.1) possesses a G𝐿×G𝑅 global symmetry (simply due to the trace cyclicity) which corresponds to left and right multiplications by a constant matrix, that is, G𝐿×Gπ‘…βˆΆπ‘”β†’π‘”0πΏπ‘”π‘”βˆ’10𝑅. The conserved Noether currents associated to such symmetries are 𝑗𝑅=βˆ’π‘‘π‘”π‘”βˆ’1,𝑗𝐿=+π‘”βˆ’1𝑑𝑔,withπ‘”π‘—πΏπ‘”βˆ’1=βˆ’π‘—π‘….(3.2) These currents are one-forms and they are also called Maurer-Cartan forms (MC-forms). They are nothing but vielbeins; indeed 𝑗(𝐿,𝑅) are 𝔀-valued functions and they span the tangent space for any point 𝑔(𝜏,𝜎) in 𝐺. We can then write 𝑗=π‘—π‘Žπ‘‘π‘Ž=πΈπ‘Žπ‘€π‘‘π‘‹π‘€π‘‘π‘Ž,π‘—π‘…πœ‡=βˆ’πœ•πœ‡π‘”π‘”βˆ’1,π‘—πΏπœ‡=+π‘”βˆ’1πœ•πœ‡π‘”,(3.3) where 𝑋𝑀 denotes the specific parameterization chosen for the 𝑀-dimensional group manifold 𝐺. π‘‘π‘Ž are the generators of the corresponding Lie algebra 𝔀, which obey the standard Lie algebra relations [π‘‘π‘Ž,𝑑𝑏]=π‘“π‘Žπ‘π‘π‘‘π‘.

The Lagrangian (3.1) can be written in terms of the right and left currents, namely, β„’=βˆ’(1/𝛾2)Tr(π‘—πΏπœ‡π‘—πœ‡πΏ)=βˆ’(1/𝛾2)Tr(π‘—π‘…πœ‡π‘—πœ‡π‘…). The equations of motion following from (3.1) are nothing but the conservation laws for the right and left currents: πœ•πœ‡π‘—πΏπœ‡=πœ•πœ‡π‘—π‘…πœ‡=0.(3.4) Moreover, by construction the currents also satisfy the so-called Maurer-Cartan identities πœ•πœ‡π‘—(𝑅,𝐿)πœˆβˆ’πœ•πœˆπ‘—(𝑅,𝐿)πœ‡+𝑗(𝑅,𝐿)πœ‡,𝑗(𝑅,𝐿)πœˆξ‚„=0.(3.5) Equation (3.5) encodes all the information about the algebraic structure of the model. Also, 𝑗(𝑅,𝐿)πœ‡ can be seen as a two-dimensional gauge field. Then, when one introduces the covariant derivative 𝐷(𝑅,𝐿)πœ‡=πœ•πœ‡+[𝑗(𝑅,𝐿)πœ‡,], the identity (3.5) can be interpreted as a zero-curvature equation. The covariant derivative π·πœ‡ acts on the elements of the Lie algebra 𝔀.

Local and Nonlocal Conserved Charges in PCM
The PCM has two different sets of conserved charges: the local and the nonlocal ones. Both conserved quantities can be obtained from a unique generating functional, the monodromy matrix. They correspond to an expansion of the monodromy matrix around different points,19 and I will discuss these aspects more extensively below.
First consider the following charges:π‘„π‘Ž(0)=ξ€œβˆžβˆ’βˆžπ‘—π‘Žπœ(𝜎)π‘‘πœŽ,π‘„π‘Ž(1)=ξ€œβˆžβˆ’βˆžπ‘—π‘ŽπœŽ(𝜎)π‘‘πœŽβˆ’12π‘“π‘Žπ‘π‘ξ€œβˆžβˆ’βˆžπ‘‘πœŽπ‘—π‘πœ(𝜎)ξ€œπœŽβˆ’βˆžπ‘‘πœŽξ…žπ‘—π‘πœξ€·πœŽξ…žξ€Έ.(3.6) The first one is local, that is, it is an integral of local functions, and it is the global right and left symmetry of the model; while the second one is bilocal. The Poisson brackets between π‘„π‘Ž(0) and π‘„π‘Ž(1) generate a series of charges, π‘„π‘Ž(𝑛), which are conserved and which are integrals of nonlocal functions. Therefore the set of charges generated by π‘„π‘Ž(0) and π‘„π‘Ž(1) are called nonlocal charges. The basic idea is that such charges show certain β€œhidden” symmetries of the two-dimensional model, not the ones directly seen by dynamical point-particles. The conservation laws for π‘„π‘Ž(𝑛) follow directly from the equations of motion (3.4). Note that since the charges π‘„π‘Ž(𝑛) are nonlocal, they will not commute in general, and they will not be additive when acting on some generic multiparticle state. They are fundamental in order to understand the classical and quantum integrability of the model. In particular when it is possible to extend such charges to the quantum level, they generate a quantum group called Yangian, whose structure yields to the factorizability of the S-matrix.
Beside the charges π‘„π‘Ž(𝑛) there are another type of conserved quantities, which are integrals of local functions of the fields. Such charges are additive on (asymptotic) multiparticle states and since they commute this puts severe constraints on the dynamics, as we will discuss in Section 3.3. The basic idea is that such local charges directly generalize the energy-momentum conservation law to higher spin. Indeed, consider the quantities Tr(𝑗(𝑅,𝐿)±𝑗(𝑅,𝐿)Β±), where we have rewritten the currents in the light-cone coordinates π‘₯Β±=𝜎±𝜏. From the equations of motion (3.4) and the Maurer-Cartan identities (3.5) it follows thatπœ•+Tr𝑗(𝑅,𝐿)βˆ’π‘—(𝑅,𝐿)βˆ’ξ€Έ=πœ•βˆ’Tr𝑗(𝑅,𝐿)+𝑗(𝑅,𝐿)+=0.(3.7) This is nothing but the conservation of the PCM energy-momentum tensor. Differentiating the action (3.1) with respect to the two-dimensional (world-sheet) metric π‘”πœ‡πœˆ one has π‘‡πœ‡πœˆ=βˆ’12𝛾2Trξ‚€π‘—πœ‡π‘—πœˆβˆ’12π‘”πœ‡πœˆξ€·π‘—πœ†π‘—πœ†ξ€Έξ‚,(3.8) and in the light-cone coordinates it becomes 𝑇±±=βˆ’(1/2𝛾2)Tr(𝑗±𝑗±). In general, we can extend (3.7) by considering a higher π‘š rank tensor, namely, πœ•+Tr𝑗(𝑅,𝐿)βˆ’ξ€Έπ‘šξ€Έ=πœ•βˆ’Tr𝑗(𝑅,𝐿)+ξ‚π‘šξ‚=0.(3.9) In particular, in order to satisfy (3.9), any higher π‘š-rank tensor should be associated with the invariant and completely symmetric Casimir tensor πΆπ‘Ž1β‹―π‘Žπ‘šπ‘‘π‘Ž1β‹―π‘‘π‘Žπ‘š. Note that, for the case π‘š=2, the invariant tensor is simply the trace of two generators, that is, πΆπ‘Žπ‘βˆπ›Ώπ‘Žπ‘ (multiplied by a constant numerical factor which depends on the particular normalization of the algebra). Then, the conservation laws (3.7) and (3.9) follow, apart from the equations of motion for the currents, also from the algebraic identities which involve the products of symmetric tensors πΆπ‘Ž1β‹―π‘Žπ‘š and the antisymmetric structure constant π‘“π‘Žπ‘π‘. The corresponding charges are then π‘žπ‘ Β±=ξ€œβˆžβˆ’βˆžπ‘‘πœŽπΆπ‘Ž1β‹―π‘Žπ‘šπ‘—π‘Ž1Β±(𝜎)β‹―π‘—π‘Žπ‘šΒ±(𝜎),(3.10) where 𝑠 denotes the Lorentz spin, namely, 𝑠=π‘šβˆ’1. The currents in π‘žπ‘  can be the right or left-invariant ones, they will give the same local conservation laws.

The Lax Pair in PCM
We have seen that we have currents which are conserved and which are flat, cf. (3.4) and (3.5), respectively. At this point, we would like to construct a flat linear combination of the currents 𝑗 themselves. This means that we consider a linear combination with arbitrary coefficients and demand that it should satisfy (3.5): π‘Žπœ‡=π›Όπ‘—πœ‡+π›½πœ–πœ‡πœˆπ‘—πœˆsuchthatπœ•πœ‡π‘Žπœˆβˆ’πœ•πœˆπ‘Žπœ‡+ξ€Ίπ‘Žπœ‡,π‘Žπœˆξ€»=0.(3.11) Since the mixed terms with 𝛼𝛽 are zero, and the terms with the product πœ–πœ– gives a factor βˆ’1, the solution for the coefficients are obtained from the equation 𝛼2βˆ’π›Όβˆ’π›½2=0, explicitly: 𝛽=12sinhπœ†,𝛼=12(1Β±coshπœ†)(3.12) with πœ†βˆˆβ„. This means that there is an entire family of solutions depending on a parameter πœ†, the spectral parameter .20 The zero-curvature equation for the connection π‘Ž encodes all the dynamical informations, such as equations of motion and Maurer-Cartan identities. Note that in general π‘Ž is not conserved, namely, it does not satisfy the equations of motion (3.4).
We now explain why we want such connection π‘Ž. The flatness condition for π‘Ž is associated with a two-dimensional differential system. In particular, for the generic group-valued function π‘ˆ(𝜏,𝜎), the compatibility condition for the differential equationsπœ•π‘ˆπœ•πœ=π‘Žπœ(πœ†)π‘ˆ,πœ•π‘ˆπœ•πœŽ=π‘ŽπœŽ(πœ†)π‘ˆ(3.13) gives (πœ•2π‘ˆ/πœ•πœπœ•πœŽ)=(πœ•2π‘ˆ/πœ•πœŽπœ•πœ), which corresponds to the zero-curvature equation for the connection π‘Ž, (3.11). The system (3.13) is also called the Lax representation, and for this reason, the two components of the connection π‘Ž are called the Lax pair. The system (3.13) is integrable provided that π‘Ž is flat and the solution for π‘ˆ is given by π‘ˆ(π’ž,πœ†)=Pπ‘’βˆ’βˆ«π’žπ‘Ž,(3.14) where P denotes the path-order prescription for the generators contained in π‘Ž and π’ž is a path on the world-sheet Ξ£. For any initial data, or boundary condition π‘ˆ(𝜏0,𝜎0), the system (3.13) has a unique solution given by the operator (3.14). This Wilson line operator, which defines the parallel transport along the path π’ž with the connection π‘Ž, is called the monodromy matrix.
The integrability of the system (3.13) is guaranteed by the fact that the connection has a zero curvature (3.11), namely, the solution (3.14) is independent of path deformations. Let 𝑠 parameterize the path π’ž. A small variation of the contour of integration, πœŽπœ‡(𝑠)β†’πœŽπœ‡(𝑠)+π›ΏπœŽπœ‡(𝑠), produces a variation on the Wilson loop operator according to [45]π›Ώπ›ΏπœŽπœ‡(𝑠)π‘ˆ=Pξ‚΅β„±πœ‡πœˆπ‘‘πœŽπœˆπ‘‘π‘ π‘’βˆ«π’žπ‘Ž(𝑠)ξ‚Ά,(3.15) where β„±πœ‡πœˆ is the field strength for the connection π‘Ž. It is clear that for a flat current, that is, when β„±πœ‡πœˆ=0, such variation vanishes, namely, the Wilson line operator is invariant under continuos path deformations if the connection is flat. This is a key point: From the fact that π‘ˆ cannot be deformed, it follows that it might be the proper generating functional for the conserved charges. Considering paths π’ž of constant time and looking at small deformations of the contours in the 𝜏 direction, then for a flat connection the Wilson line operator will be invariant under variations of these particular paths, namely, under deformations in time. Explicitly: 𝑄(πœ†)=limπœŽβ†’Β±βˆžπ‘ˆξ€·π’ž0;πœ†ξ€Έ=Pπ‘’βˆ’βˆ«βˆžβˆ’βˆžπ‘Ž|𝜏0,(3.16) where it has been stressed that the contour π’ž0 is over surfaces of constant time 𝜏0 and that πœŽβ†’Β±βˆž.21 Thus, summarizing, the conservation of the charges 𝑄(πœ†,𝜏0) is guaranteed by the flatness of π‘Ž (3.11). One can easily differentiate π‘ˆ, and assuming that the currents fall down to zero at infinity and that π‘Ž is flat, one will get a vanishing time derivative for 𝑄(πœ†,𝜏0).
The nonlocal charges which we have discussed above can be obtained as a Taylor expansion around the zero value of the spectral parameter πœ†. Around πœ†=0 the expansion of the flat connection π‘Ž with the minus solution in (3.12) isπ‘Žπœ‡(πœ†)β‰…πœ†2πœ–πœ‡πœˆπ‘—πœˆβˆ’πœ†24π‘—πœ‡+π’ͺξ€·πœ†3ξ€Έ.(3.17) Then defining 𝑄(πœ†)≑1+βˆžξ“π‘›=1(βˆ’1)𝑛𝑛!πœ†π‘›π‘„(π‘›βˆ’1),(3.18) one has at the leading order in πœ† expanding the exponential in (3.16) 𝑄(0)=12ξ€œβˆžβˆ’βˆžπ‘‘πœŽπ‘—πœ(𝜎),𝑄(1)=12ξ€œβˆžβˆ’βˆžπ‘‘πœŽπ‘—πœŽ(𝜎)βˆ’14ξ€œβˆžβˆ’βˆžπ‘‘πœŽξ€œπœŽβˆ’βˆžπ‘‘πœŽξ…žξ€Ίπ‘—πœ(𝜎),π‘—πœξ€·πœŽξ…žξ€Έξ€».(3.19) Apart for an irrelevant numerical factor these charges are the same presented above in (3.6).
Some concrete examples of the PCM are the models with group 𝐺=SU(𝑁) and the O(4)∼SU(2)Γ—SU(2) model. Most relevant for us is the GS type IIB superstring in AdS5Γ—S5 in the light-cone gauge with symmetry group P(SU(2∣2)Γ—SU(2∣2)). This model will be elaborated on in Section 6.

3.2. Coset Model

We now review some other very special two-dimensional 𝜎-models, namely, those defined on a coset space. The presentation closely follows the paper by Bena et al. [46].

For a coset space, the map 𝑔(𝜏,𝜎) takes values in the quotient space 𝐺/𝐻. 𝐻 is a 𝐺-subgroup, called isotropy group or stabilizer since it is required to leave invariant the 𝐺 elements. The coset space 𝐺/𝐻 corresponds to the identification 𝑔(𝜏,𝜎)≅𝑔(𝜏,𝜎)β„Ž(𝜏,𝜎),β„Ž(𝜏,𝜎)∈𝐻.(3.20) In some sense we can say that we have β€œhalf” of the global symmetries compared with the PCM of the previous Section 3.1: what is now left is only the invariance under global left multiplication. However, now the subgroup 𝐻 plays the important role of gauge group, since each point in every orbit in the target-space is defined up to a local transformation, that is, a gauge transformation, which does not contain any further physical information. For this reason 𝑔(𝜏,𝜎) is the coset representative. Note that we could have used left-multiplication in (3.20) to identify different 𝑔 and then the remaining global symmetry would have been the right one. The forthcoming arguments then run analogously, with some obvious exchange between the left and right sectors.

It is possible to give a geometric construction for spaces such as ℂ𝑃𝑛=SU(𝑛+1)/(π‘ˆ(1)Γ—SU(𝑛)), AdS𝑛=SO(π‘›βˆ’1,2)/SO(π‘›βˆ’1,1) and 𝑆𝑛=SO(𝑛+1)/SO(𝑛). For example, consider the 𝑛-dimensional sphere 𝑆𝑛 embedded in ℝ𝑛+1. Fixing the north-pole (0,0,…,1) we still can have all the rotations in the 𝑛 transverse directions, namely, SO(𝑛), which leave the north pole fixed and do not change the points on the sphere 𝑆𝑛.

As already seen in the previous section, we can introduce the one-forms π½πœ‡=π½π‘Žπœ‡π‘‘π‘Ž=π‘”βˆ’1πœ•πœ‡π‘”.(3.21) We follow the literature and use capital letters π½πœ‡ for the left-invariant currents and vice versa, small letters π‘—πœ‡ for the conjugated currents, since now the roles played by the two kinds of Maurer-Cartan forms are very different. Indeed, the group 𝐺 acts on the coset representative as a left multiplication 𝑔0, thus the currents π½πœ‡ transform according to π½πœ‡=π‘”βˆ’1πœ•πœ‡π‘”βŸΆξ€·π‘”0π‘”ξ€Έβˆ’1πœ•πœ‡ξ€·π‘”0𝑔=π‘”βˆ’1πœ•πœ‡π‘”,(3.22) since 𝑔0 is constant. Thus the currents are left-invariant, which corresponds to the action of the global symmetry 𝐺. What happens to the MC-forms when we consider the coset identification? This means that an element 𝑔 will be multiplied by an element of the subgroup 𝐻, which now depends on the world-sheet coordinates πœŽπœ‡. Replacing π‘”β†’π‘”β„Ž in 𝐽 we obtain the following transformation π½πœ‡βŸΆβ„Žβˆ’1π‘”βˆ’1πœ•πœ‡π‘”β„Ž+β„Žβˆ’1πœ•πœ‡β„Ž.(3.23) The first term transforms covariantly under a local gauge 𝐻 transformation, but not the second term. Considering the conjugate currents π‘—πœ‡=βˆ’π‘”π½πœ‡π‘”βˆ’1=βˆ’πœ•πœ‡π‘”π‘”βˆ’1,(3.24) we see that they transform covariantly under global left-multiplication: π‘”βŸΆπ‘”0𝑔,π‘—πœ‡βŸΆπ‘”0π‘—πœ‡π‘”βˆ’10.(3.25) For this reason it is important to distinguish between the left and right sectors, since now the two types of currents are not both conserved anymore as it was in the PCM case (3.4), and they transform in different ways under gauge transformations. Obviously, we could have started defining the coset space by a left-multiplication and inverted the role between β€œsmall” and β€œcapital” currents.

The algebra 𝔀 is split in two sectors with respect to the 𝐻-action: 𝔀=π”₯βŠ•π”¨, where 𝔨≑𝔀/π”₯ is the orthogonal complement in 𝔀 with respect to π”₯. As a consequence, also the left-invariant currents undergo the same split, namely, 𝐽=𝐾+𝐻,(3.26) with obvious notation for the various terms. Thus 𝐻 is really a connection, a gauge field, while 𝐾 represents the part of the one-form which transforms covariantly under gauge transformations, that is, β„Žβˆ’1π‘”βˆ’1πœ•π‘”β„Ž in (3.23). Notice that the current π‘˜=βˆ’π‘”πΎπ‘”βˆ’1(3.27) is gauge invariant. Finally, the current π‘—πœ‡ does not have a defined grading, since the rotation with 𝑔 and π‘”βˆ’1 mixes the two sectors π”₯ and 𝔀/π”₯, however one keeps the notation β„Ž and π‘˜ to denote π‘”π»π‘”βˆ’1 and π‘”πΎπ‘”βˆ’1, respectively.

The Lagrangian is as for the PCM (3.1) β„’=1𝛾2Trξ€·πœ•πœ‡π‘”βˆ’1πœ•πœ‡π‘”ξ€Έ=1𝛾2Trξ€·π½πœ‡π½πœ‡ξ€Έ=1𝛾2Trξ€·π‘—πœ‡π‘—πœ‡ξ€Έ.(3.28) Since the two tangent spaces π”₯ and 𝔨 are orthogonal, this leads to the following expression for the β„’β„’=1𝛾2Trξ€·π»πœ‡π»πœ‡+πΎπœ‡πΎπœ‡ξ€Έ.(3.29) The term Tr(𝐴|𝐺/𝐻𝐡|𝐻) vanishes, as it should, since the trace is a bilinear invariant tensor that respects the structure of the space: 𝔨,π”₯ξ€»βŠ†π”¨,ξ€Ίπ”₯,π”₯ξ€»βŠ†π”₯.(3.30) Indeed, the grading 𝔀=π”₯βŠ•π”¨ means that the generators of one set span the tangent space labelled by 𝔨 and the other complementary set generates π”₯, and there is no generator left. Thus, the trace between any two elements spanning orthogonal spaces vanishes, since the trace is nothing but a scalar product in this tangent space.

Since the action (3.29) is gauge invariant, it is clear that one can integrate out the gauge field 𝐻 so that the only remaining contribution to the currents in 𝐺/𝐻 is ℒ𝐺/𝐻=1𝛾2Trξ€·πΎπœ‡πΎπœ‡ξ€Έ,(3.31) which is again manifestly gauge invariant (recall that 𝐾 is covariant under local 𝐻 transformations) and it is naturally defined on the quotient space 𝐺/𝐻.

Again it follows from the equations of motion that the left-invariant currents are conserved; they satisfy the usual identity πœ•πœ‡π½πœˆβˆ’πœ•πœˆπ½πœ‡+[π½πœ‡,𝐽𝜈]=0. As for the PCM, we can construct the flat linear combination π‘Ž. However, in the coset space we need a further requirement: the space should be symmetric, namely, beyond the standard algebraic structure for a coset space (3.30), we need also that [𝔨,𝔨]βŠ†π”₯.(3.32) This is indeed a necessary and sufficient condition for a bosonic coset space to have a Lax representation [47, 48]. Note that other models can still have a Lax representation. The AdS5Γ—S5 superstring case is eloquent in this sense: the bosonic subsector, which is strictly the coset AdS5Γ—S5, is a symmetric space. However, its full supersymmetric generalization is not. The corresponding superstring action is not simply 𝑆𝐺/𝐻 but there is a further contribution of the Wess-Zumino-Witten (WZW) type [49] which allows a Lax pair reformulation [46].

In order to construct a flat connection, let us consider the projections of the Maurer-Cartan identities over π”₯ and 𝔨. Then πœ•πœ‡π½πœˆβˆ’πœ•πœˆπ½πœ‡+[π½πœ‡,𝐽𝜈]=0 gives πœ•πœ‡π»πœˆβˆ’πœ•πœˆπ»πœ‡+ξ€Ίπ»πœ‡,π»πœˆξ€»+ξ€ΊπΎπœ‡,πΎπœˆξ€»=0,πœ•πœ‡πΎπœˆβˆ’πœ•πœˆπΎπœ‡+ξ€Ίπ»πœ‡,πΎπœˆξ€»+ξ€ΊπΎπœ‡,π»πœˆξ€»=0.(3.33) Without the condition (3.32) the commutator [πΎπœ‡,𝐾𝜈] would have contributed to both the differentials, 𝑑𝐻 and 𝑑𝐾.22 Using the following identity πœ•πœ‡π‘™πœˆβˆ’πœ•πœˆπ‘™πœ‡=βˆ’π‘”ξ€·πœ•πœ‡πΏπœˆβˆ’πœ•πœˆπΏπœ‡ξ€Έπ‘”βˆ’1βˆ’ξ€Ίπ‘™πœ‡,π‘—πœˆξ€»βˆ’ξ€Ίπ‘—πœ‡,π‘™πœˆξ€»(3.34) valid for any current 𝐿 and its conjugate 𝑙=βˆ’π‘”πΏπ‘”βˆ’1, one has πœ•πœ‡π‘˜πœˆβˆ’πœ•πœˆπ‘˜πœ‡+2ξ€Ίπ‘˜πœ‡,π‘˜πœˆξ€»=0.(3.35) In this way, the flat connection corresponding to π‘Ž in the PCM is just the gauge-invariant one-form 2π‘˜πœ‡, since it is conserved and it is also flat. Then the construction for the monodromy matrix follows exactly the PCM model in Section 3.1.

3.3. The Magic of (1+1)-Dimensional Theories

Something special happens for two-dimensional field theories which have an infinite amount of conserved higher charges. This is mainly due to the fact that there is only one spatial dimension, and that the charges can be used to reshuffle the amplitudes in scattering processes. The role of integrability in constraining the dynamics of the theory was discovered in the late 1970s and early 1980s by Zamolodchikov and Zamolodchikov [50], LΓΌscher [51], Kulish [52], Parke [53], and by Shankar and Witten [54]. In order to illustrate this point, we start with a two-dimensional theory with an infinite set of charges, which are integrals of local functions and which are diagonal in one-particle states. The charges are of the kind illustrated in Section 3.1.

Let us first introduce some notations and define what we mean by scattering. We denote the particle state with the wave-function |𝐴(πœƒ)⟩, where πœƒ is the rapidity, which is defined for a massive field theory23 as 𝑝+π‘Ž=2π‘šπ‘Žπ‘’πœƒπ‘Ž,π‘βˆ’π‘Ž=2π‘šπ‘Žπ‘’βˆ’πœƒπ‘Ž.(3.36)𝑝+ and π‘βˆ’ are the momenta in the light-cone coordinates.24 Suppose the asymptotic in-state is composed of π‘š particles. We can then write ||in⟩=||π΄π‘Ž1ξ€·πœƒ1ξ€Έβ‹―π΄π‘Žπ‘šξ€·πœƒπ‘šξ€Έξ¬.(3.37) The hypothesis is that the particles are described by wave packets with an approximate position for each momentum (for each rapidity) and that all the interactions are short-ranged (since we are discussing massive field theories) such that the π‘š-particle state can be approximated by a sum of π‘š single-particle states (the wave packets are far enough apart to be considered single particle states). An asymptotic in-state means that sufficiently backwards in time the π‘š particles do not interact. This imposes a certain ordering in the state, since the particle which is traveling faster must be on the left in order to avoid crossing with all other particles, vice versa the slowest particle should be the first on the right, that is, πœƒ1>πœƒ2>β‹―>πœƒπ‘š,forin-states.(3.38) This also implies the reversed ordering for the out-state. Consider as well the asymptotic state containing 𝑛 particles, namely, 𝑛 independent wave packets |out⟩=||𝐴𝑏1ξ€·πœƒ1ξ€Έβ‹―π΄π‘π‘›ξ€·πœƒπ‘›ξ€Έξ¬.(3.39) Now the particles should travel without interacting for future times and the slowest particle should be on the left and the particle moving fastest on the right, namely, in terms of rapidities πœƒ1<πœƒ2<β‹―<πœƒπ‘›,forout-states.(3.40) The letters π‘Ž1,…,π‘Žπ‘š and 𝑏1⋯𝑏𝑛 denote any possible set of quantum numbers characterizing the particles.

The S-matrix or scattering matrix is by definition the mapping relating the in- and out-states, namely, it is defined by ||π΄π‘Ž1ξ€·πœƒ1ξ€Έβ‹―π΄π‘Žπ‘šξ€·πœƒπ‘šξ€Έξ¬=𝑆𝑏1β‹―π‘π‘›π‘Ž1β‹―π‘Žπ‘šξ€·πœƒ1β‹―πœƒπ‘š;πœƒξ…ž1β‹―πœƒξ…žπ‘›ξ€Έ||𝐴𝑏1ξ€·πœƒξ…ž1ξ€Έβ‹―π΄π‘π‘šξ€·πœƒξ…žπ‘›ξ€Έξ¬,(3.41) where it is intended to sum over the indices 𝑏1⋯𝑏𝑛, and over the outgoing rapidities, which are ordered as explained above. We can also introduce the Faddeev-Zamolodchikov (ZF) notation [50, 55] and write each asymptotic state as a sequence of π΄π‘Ž(πœƒ)’s, remembering that they do not commute and they are ordered in increasing or decreasing rapidity for in- or out-state, respectively, according to (3.38) and (3.40). Then one can write the state and the S-matrix element in the following way: π΄π‘Ž1ξ€·πœƒ1ξ€Έβ‹―π΄π‘Žπ‘šξ€·πœƒπ‘šξ€Έ,π΄π‘Ž1ξ€·πœƒ1ξ€Έβ‹―π΄π‘Žπ‘šξ€·πœƒπ‘šξ€Έ=𝑆𝑏1β‹―π‘π‘›π‘Ž1β‹―π‘Žπ‘šξ€·πœƒ1β‹―πœƒπ‘š;πœƒξ…ž1β‹―πœƒξ…žπ‘›ξ€Έπ΄π‘1ξ€·πœƒξ…ž1ξ€Έβ‹―π΄π‘π‘›ξ€·πœƒξ…žπ‘›ξ€Έ.(3.42) The S-matrix is a unitary operator, namely, it should respect the condition (in operator notation) π‘†ξ€·πœƒ1,πœƒ2ξ€Έπ‘†β€ ξ€·πœƒ2,πœƒ1ξ€Έ=πŸ™.(3.43) In general one also requires that the S-matrix is invariant under parity transformation (in our case the discrete symmetry which flips the spatial coordinate 𝜎 to βˆ’πœŽ), time reversal, and charge conjugation. In relativistic quantum field theories the S-matrix turns out to be invariant also under the crossing symmetry, namely, the transformation which exchanges one incoming particle of momentum 𝑝 with an outgoing antiparticle of momentum βˆ’π‘, cf. discussion in Section 6.4.

Selection Rules
Let us now come back to the local charges π‘žπ‘ Β±. Since they commute with the momentum operator, for a single particle state we have π‘žπ‘ Β±||π΄π‘Ž(πœƒ)⟩=πœ”(𝑠)π‘Žπ‘’Β±π‘ πœƒ||π΄π‘Ž(πœƒ)⟩,(3.44) where πœ”(𝑠)π‘Ž are the corresponding eigenvalues. For 𝑠=0 and 𝑠=1 we can think about them as the energy and the momentum. However, we are assuming that there exists an infinite number of higher rank local conserved charges, namely, we are assuming 𝑠>1. Suppose now we act with the local conserved charges on the in- and out-states. Since the wave packets are well separated and the charges are integrals of local functions, their action on such states is additive, namely, π‘žπ‘ ||π΄π‘Ž1ξ€·πœƒ1ξ€Έβ‹―π΄π‘Žπ‘šξ€·πœƒπ‘šξ€Έξ¬=ξ‚€πœ”(𝑠)π‘Ž1π‘’π‘ πœƒ1+β‹―+πœ”(𝑠)π‘Žπ‘šπ‘’π‘ πœƒπ‘šξ‚||π΄π‘Ž1ξ€·πœƒ1ξ€Έβ‹―π΄π‘Žπ‘šξ€·πœƒπ‘šξ€Έξ¬.(3.45) Again, just to understand, for 𝑠=0 the above relation is the energy conservation condition and for 𝑠=1 the momentum conservation law. Obviously we can write the expression above (3.45) also for outgoing states: π‘žπ‘ ||𝐴𝑏1ξ€·πœƒξ…ž1ξ€Έβ‹―π΄π‘π‘šξ€·πœƒξ…žπ‘šξ€Έξ¬=ξ‚€πœ”(𝑠)𝑏1π‘’π‘ πœƒξ…ž1+β‹―+πœ”(𝑠)π‘π‘šπ‘’π‘ πœƒξ…žπ‘šξ‚||𝐴𝑏1ξ€·πœƒξ…ž1ξ€Έβ‹―π΄π‘π‘šξ€·πœƒξ…žπ‘šξ€Έξ¬.(3.46) The charges are conserved during the entire scattering process and they are diagonalized by asymptotic multiparticle states as stated above (3.45) and (3.46). Then for any π‘šβ†’π‘› scattering amplitude it must be true that πœ”(𝑠)π‘Ž1π‘’π‘ πœƒ1+β‹―+πœ”(𝑠)π‘Žπ‘šπ‘’π‘ πœƒπ‘š=πœ”(𝑠)𝑏1π‘’π‘ πœƒξ…ž1+β‹―+πœ”(𝑠)π‘π‘›π‘’π‘ πœƒξ…žπ‘›(3.47) for all the possible infinite values of 𝑠. Thus there are 𝑠 such equations, with 𝑠 taking infinitely many values. Hence, the only solution for generic values of the incoming momenta is 𝑛=π‘š,πœ”(𝑠)π‘Žπ‘–=πœ”(𝑠)𝑏𝑖,πœƒπ‘–=πœƒξ…žπ‘–,(3.48) with 𝑖=1,…,π‘š. The consequences of the solutions (3.48) are severe for the dynamics of the system.(i)Since 𝑛 must be equal to π‘š this implies that there cannot be processes where the number of particles changes, namely, the number of particles is conserved during the scattering and there cannot be particle production.(ii)The set of incoming momenta, {𝑝𝑖} must be equal to the set of outgoing momenta {π‘ξ…žπ‘–}, or in terms of rapidities {πœƒπ‘–}={πœƒξ…žπ‘–}.However, this does not imply that the sets of quantum numbers before and after the scattering {π‘Žπ‘–} and {𝑏𝑖} should be the same. They can have different values, namely, scatterings which lead to changing flavor are still allowed. There is some subtlety, in the sense that one might find solutions to (3.47) for specific values of the incoming momenta and for π‘›β‰ π‘š. However these values turn out to not be physical [56]. The scatterings which are possible and consistent with the infinite set of charges are the elastic processes.

S-Matrix Factorization
There is still another dynamical constraint which makes the two-dimensional integrability a really powerful tool: the factorizability of the S-matrix. Each wave packet is localized, and we can model it by a gaussian distribution around the position π‘₯𝑖 with momentum 𝑝𝑖. Acting on such a state with an operator of the type π‘’βˆ’πœ„π‘π‘ƒπ‘  shifts the phase factor by a function depending on the momentum:25 in particular, the position is shifted by 𝛿π‘₯𝑖=π‘π‘ π‘π‘ βˆ’1𝑖. When the operator acts on an π‘š-particle state of the type seen before, namely, π‘š times a single particle state, then each localized wave packet is shifted by a different quantity since such shift depends on the wave packet momentum. Then, since the asymptotic states are eigenstates for the higher conserved charges and since such charges commute with the S-matrix, we can use them in order to reshuffle the in- and out-states. Explicitly one can write out||𝑆||in=out||π‘’π‘–π‘π‘ƒπ‘ π‘†π‘’βˆ’π‘–π‘π‘ƒπ‘ ||inξ‚­.(3.49) We can rearrange the wave-packets and make their phase factor change according to their momenta. In order to illustrate the ideas, let us consider the 3β†’3 scattering. At tree level we can have three types of diagrams, cf. Figure 3. The first graph (π‘Ž) visualizes the scattering of three particles at the same point, while the remaining two diagrams, (𝑏) and (𝑐), represent a series of three two-body scatterings. Namely, in the diagram (𝑏), first the particles 2 and 3 meet, collide and then the particle 3 collides further with 1 and then the particle 2 with 1. Of course we can start with the initial scattering between 1 and 2 and proceed analogously, as in Figure 3(c). Now, we use the operator π‘’βˆ’π‘–π‘π‘ƒπ‘  in order to shift the particle positions as in (3.49). However, everything must respect the macro-causality principle, namely, it cannot happen that the particle 1 goes out before that also the particle 3 participates in the scattering. Otherwise, the corresponding amplitude would just vanish.26 Namely, nothing can happen between the slowest incoming particle and the fastest outgoing particle before that all the incoming particles have collided. Now the point is that one can use the higher charges to rearrange the phase shift for the multiparticle state, but indeed the diagrams in Figure 3 only differ by a phase factor. This means that we can use the operators 𝑃𝑠 in order to move the lines 1, 2, and 3 in Figure 3(a), in order to get any of the two other graphs in Figure 3. Hence all the graphs in Figure 3 are equal. This implies that the three-body S-matrix (Figure 3(a)) is equal to a sequence of two-body S-matrices (Figures 3(b) and 3(c)). This is the meaning of the first equality in Figure 4, where what we have discussed for the tree-level is extended to generic 𝑛-loop order. The second equality in Figure 4 represents the Yang-Baxter equations. They are really nontrivial equations, since they fix the flow of indices that we can have in the S-matrix elements. This is something special which can happen in two dimensions. Indeed, we are using the higher charges to reshuffle the incoming particle positions. Hence, if their rapidities differ, they will still meet at some point in space. This is not true for the four-dimensional case, where there are still two dimensions where the incoming particles can completely avoid the scattering. This is the main reason why an integrable theory in 4 dimensions only has a trivial S-matrix, which is stated in the Coleman-Mandula theorem [57]. In one spatial dimension the particles necessary will meet at some point: They run in the same line there is no way to go out.27
Let us pause here and summarize the previous paragraph. In any (1+1)-dimensional theory with infinitely many local conserved charges, any 𝑛→𝑛 process can actually be known since the corresponding S-matrix element is given by a sequence of 𝑛(π‘›βˆ’1)/2 two-body S-matrix elements. In many well-understood theories even if the 2-body S-matrix is computed, it is hopeless to compute the three-particle S-matrix. But now we are saying that we do not need it. We can compute any particle number scattering and the corresponding amplitude will be a product of 2β†’2 scattering amplitudes. Thus, any scattering process involving more than two particles is a sequence of 2 by 2 collisions, which are all elastic and before and after any collision the particles keep on traveling freely.
Until now we have only discussed the local conserved charges since the arguments in order to run need to use the fact that these objects are additive on multiparticle states. However, in [58] Iagolnitzer gave a more general proof for the S-matrix factorization and for the selection rules. The same is done in LΓΌscher’s paper [51] where he proved the relation between nonlocal charges and S-matrix factorization for the 𝑂(𝑛) sigma model. For simplicity and for pedagogical reasons we have chosen to use the local charges to simpler visualize the arguments.

Figure 3: Tree level diagrams for three-body scattering 3β†’3.
Figure 4: Factorization of the three-body S-matrix and Yang-Baxter equation.

Remarks on the AdS5Γ—S5 String World-Sheet S-Matrix
From the discussion above, it is clear that we can use the factorization of the S-matrix and the selection rules (and the Yang-Baxter equations) as a definition for a two-dimensional integrable field theory. It is often really difficult to explicitly construct the (nonlocal and local) charges and usually it is more useful to know the S-matrix elements. This has been studied in [5], where we have explicitly verified the factorization of the one-loop S-matrix for the near-flat-space limit of the type IIB superstring on AdS5Γ—S5. This is equivalent to state the integrability of the model at leading order in perturbation theory. However, this will be explained in more detail in Section 6. Here, we only want to stress once more that these dynamical constraints severely restrict the motion in the phase space. For example, consider the 3β†’3 process. Any scattering amplitude must respect the energy and the momentum conservation laws. In the light-cone coordinates one has that π‘βˆ’π‘+=4π‘š2. Then 𝑝± can be parameterized as 𝑝+=2π‘šπ‘Ž and π‘βˆ’=2π‘š/π‘Ž and the energy-momentum conservation laws become 1π‘Ž+1𝑏+1𝑐=1𝑑+1𝑒+1𝑓,π‘Ž+𝑏+𝑐=𝑑+𝑒+𝑓,(3.50) where the set (π‘Ž,𝑏,𝑐) is for the incoming momenta, which are fixed (it is the external input which we give when we start to run our collision), while (𝑑,𝑒,𝑓) is the set of outgoing momenta, which are constrained to respect the above (3.50). The equations in (3.50) describe two surfaces. Without any further conservation law the outgoing particles could lie in any point along the curve described by the intersection of the two equations. However, since we have a higher charge and we can impose another equation, there are only six valid points in all the phase space! These points correspond to the permutations given by the equation {π‘Ž,𝑏,𝑐}={𝑑,𝑒,𝑓}, see Figure 5. This of course means that we have completely solved the motion. If we have a 4β†’4 scattering then we need a fourth higher charge to fix univocally the points in the phase space, and so on. This is the concrete way how the charges manifest themselves. How to get the extra equation, namely, how the higher charges actually operate on the phase space, will be discussed in Section 6. There we also explain why we want to show the quantum integrability of the AdS superstring.

Figure 5: Three particle phase space.
3.4. Quantum Charges in PCM and Coset Model

Until now the discussion has only been at the classical level. Can we generalize the arguments above to the corresponding quantum field theory in a straightforward way? This question is far from trivial: numerous works in the past years (’70s–’80s) have been devoted to understand when integrability survives at the quantum level. However, also the answer is far from being trivial: for the 𝑂(𝑛) model all the integrability properties survive after quantization [51, 59, 60], which is not the case for the ℂ𝑃𝑛 model [61]. Can we say why? Can we say where and how the troubles are originated? Can we learn something useful for the type IIB string theory? In this section, we will try to partly answer these questions.

Quantum Nonlocal Charges
Going back to the definition of the nonlocal charges (3.6), one would like to implement such definition at the quantum level. The first trouble which one needs to face is the fact that the currents, and all fields in general, now are promoted to operators. The first term in (3.6) now contains a product of two operators. When the two points where the operators are sitting at get closer and closer the currents can interact and give rise to singularities. In quantum field theory any product of operators is in general not well defined. Also, the second term in (3.6) can get renormalized and in general there will be some field renormalization coefficient which can be divergent.
In order to have a reliable charge definition, it is necessary to slightly modify the expression in (3.6) [51]:π‘„π‘Ž(1)=π‘ξ€œβˆžβˆ’βˆžπ‘—π‘ŽπœŽ(𝜎)π‘‘πœŽβˆ’12π‘“π‘Žπ‘π‘ξ€œβˆžβˆ’βˆžπ‘‘πœŽπ‘—π‘πœ(𝜎)ξ€œπœŽβˆ’βˆžπ‘‘πœŽξ…žπ‘—π‘πœξ€·πœŽξ…žξ€Έ.(3.51) The second step is to compute the short-distance expansion for the current product in (3.51) and see if UV-dangerous terms can come out. This means to compute the operator product expansion (OPE) for the currents: π‘—π‘Žπœ‡(π‘₯βˆ’πœ–)π‘—π‘πœˆ(π‘₯+πœ–)βˆΌξ“π‘˜πΆπ‘˜πœ‡πœˆ(πœ–)π’ͺπ‘Žπ‘π‘˜(π‘₯),πœ–βŸΆ0,(3.52) where the sum π‘˜ denotes the sum over a basis of operators π’ͺπ‘Žπ‘π‘˜. The operators π’ͺπ‘Žπ‘π‘˜ do not depend on the short-distance parameter πœ–, while the coefficients in the expansion πΆπ‘˜πœ‡πœˆ are functions of the coordinates, and thus of πœ–. The problematic terms are linearly (i.e., 1/πœ–) and logarithmically divergent in πœ–. For example, for the PCM by dimensional analysis and since the currents have conformal dimension 1, we can expect an expansion of the type π‘—π‘Žπœ‡(π‘₯βˆ’πœ–)π‘—π‘πœˆ(π‘₯+πœ–)βˆΌπΆπœ†,π‘Žπ‘π‘πœ‡πœˆ(πœ–)π‘—π‘πœ†(π‘₯)+π·πœ†πœŒ,π‘Žπ‘π‘πœ‡πœˆ(πœ–)πœ•πœ†π‘—π‘πœŒ(π‘₯)+β‹―,(3.53) where πΆπœ†πœ‡πœˆ(πœ–) behaves as 1/πœ–, just by dimensional analysis. This gives rise to possible logarithmic terms once one integrates.
For the O(𝑛)𝜎 model, LΓΌscher showed that the quantum charges are well defined, they are conserved quantum mechanically and they force the S-matrix to factorize [51]. The same is not true for the ℂ𝑃𝑛 model, which was investigated by Abdalla et al. in [61]. The ℂ𝑃𝑛 model is classically integrable, however, at quantum level an anomaly appears in the conservation law for the quantum nonlocal charges. As before, one needs to study the short-distance expansion for the currents (3.52) and then plug back the OPE in the quantum nonlocal charge (3.51). The term responsible for the anomaly in the ℂ𝑃𝑛 case is the field strength of the currents, namely, a dimension two operators, whose corresponding coefficient in (3.52) contains logarithmically and linearly divergent terms. (Notice that the supersymmetric ℂ𝑃𝑛 is quantum integrable [62].)
Can we give some kind of rules, about when or whether we could expect an anomaly in the charge conservation laws? For symmetric coset models of the type discussed in Section 3.2 this issue has been addressed in [63]. If one would like to summarize the results of the paper, one could say that the breaking of integrability at quantum level is related to U(1) factor in the denominator of the quotient space, a fact which is confirmed by the ℂ𝑃𝑛 example, where the corresponding field strength gives rise to the anomaly. In some sense in the O(𝑛) model there is not a great variety of operators π’ͺπ‘Žπ‘π‘˜ of dimension 1 and 2 with the proper symmetries required by the model itself in order to be a candidate for the anomaly.28

Remarks on the AdS5Γ—S5 Superstring Case
From all this, one can understand why it is not so trivial to investigate the quantum integrability for two-dimensional 𝜎 model, as, for example, the superstring world-sheet theory. Recall that the supercoset AdS5Γ—S5 is not a symmetric space, thus we cannot extend directly the analysis of [63]. However we can learn much from the ℂ𝑃𝑛 case and with this example in mind we have started to investigate the quantum pure spinor superstring in AdS5Γ—S5 in the papers [4, 7]. In particular, recall the expression for the variation of the monodromy matrix (3.15), the integrability of the model is strictly related to the tensor β„±πœ‡πœˆ, cf. Section 5.

4. Green-Schwarz-Metsaev-Tseytlin Superstring

The section is mainly based on the textbooks by [64, 65] and also on the original papers by Green and Schwarz [66, 67] for the first part. For the second part, I will mainly refer to the work by Metsaev and Tseytlin [49] for the supercoset construction of the action, to the paper by Bena et al. [46] for the classical integrability of the GSMT action, and finally to the reviews written by Zarembo [68] and by Arutyunov and Frolov [69].

4.1. Green-Schwarz Action in Flat Space

In the Green-Schwarz (GS) approach, the target space supersymmetries are manifest and in some sense the superspace coordinates are treated more symmetrically with respect to the Ramond-Neveu-Schwarz (RNS) formalism.29 In string theory, the embedding coordinates π‘‹π‘Ž(𝜏,𝜎) map the world-sheet Ξ£, parameterized by (𝜏,𝜎), into the target space. Now the same concept is generalized to the β€œfermionic embedding coordinates” πœƒπΌ(𝜏,𝜎). These are spinors on the target space and scalars from a world-sheet point of view.

The GS superstring action in a flat background [67] is 𝑆GS,flat=𝑆kin+𝑆WZW=βˆ’14πœ‹π›Όξ…žξ€œπ‘‘2πœŽβˆšβˆ’β„Žβ„Žπœ‡πœˆξ‚€πœ•πœ‡π‘‹π‘Žβˆ’π‘–πœƒπΌΞ“π‘Žπœ•πœ‡πœƒπΌξ‚ξ‚€πœ•πœˆπ‘‹π‘Žβˆ’π‘–πœƒπ½Ξ“π‘Žπœ•πœˆπœƒπ½ξ‚+12πœ‹π›Όξ…žξ€œπ‘‘2πœŽπœ–πœ‡πœˆξ‚€βˆ’π‘–πœ•πœ‡π‘‹π‘ŽπœŽπΌπ½3πœƒπΌΞ“π‘Žπœ•πœˆπœƒπ½+πœƒ1Ξ“π‘Žπœ•πœ‡πœƒ1πœƒ2Ξ“π‘Žπœ•πœˆπœƒ2.(4.1)β„Žπœ‡πœˆ is the world-sheet metric, π‘‹π‘Ž are the ten embedding coordinates in the flat space π‘Ž=0,…,9, and πœƒπΌ with 𝐼=1,2 are the two Majorana-Weyl spinors in ten dimensions,30 with 𝜎𝐼𝐽3=diag(1,βˆ’1). For the specific case of the type IIB superstring, the two fermions have the same chirality, vice versa in type IIA they have opposite chirality, namely, Ξ“11πœƒπΌ=πœƒπΌwith𝐼=1,2typeIIB,Ξ“11πœƒπΌ=(βˆ’1)𝐼+1πœƒ1with𝐼=1,2typeIIA,(4.2) where Ξ“11=Ξ“0Ξ“1β‹―Ξ“9 and Ξ“π‘Ž are the 32Γ—32Ξ“-matrices which satisfy the SO(9,1) Clifford algebra: ξ€½Ξ“π‘Ž,Γ𝑏=2πœ‚π‘Žπ‘withπœ‚π‘Žπ‘=diag(βˆ’1,1,…,1).(4.3) The action (4.1) is essentially built of two terms. The first contribution 𝑆kin is a 𝜎-model (the term symmetric in the world-sheet indices). The second line comes from the Wess-Zumino-Witten (WZW) term, that is, 𝑆WZW (the one antisymmetric in the world-sheet indices). I will give more detail on the two terms at the end of the section.

An important feature of the GS action (4.1), which is valid also in curved backgrounds, is the invariance under a local fermionic symmetry, which is called πœ…-symmetry [67]. Such a symmetry fixes univocally the coefficient in front of the WZW term. The πœ…-symmetry allows one to gauge away half of the fermionic degrees of freedom, leaving only the physical ones. Counting the fermionic degrees of freedom, we start with a Dirac fermion in ten dimensions, namely, with 2𝐷/2=32 components. We impose the Majorana-Weyl condition which removes half of the components, leaving only 16 real fermionic degrees of freedom. Finally we can use the πœ…-symmetry to reduce the spinor components further, namely, to 8. Recalling that we started with two supersymmetries (𝐼=1,2), we have in total 16 real independent fermionic degrees of freedom.31 Furthermore, the action (4.1) is invariant under super-PoincarΓ© transformations and world-sheet reparameterizations.

4.2. Type IIB Superstring on AdS5Γ—S5: GSMT Action

Before getting to the hearth of the discussion about the AdS superstring action, let me first review certain crucial properties of the 𝔭𝔰𝔲(2,2∣4) algebra. In the next paragraph, I will heavily use the results of the two Sections 3.1 and 3.2.

More on the Algebra
A notably property of the 𝔭𝔰𝔲(2,2∣4) algebra is its inner automorphism,32 defined by a map Ξ© which decomposes the algebra in four subsets. Explicitly, we have 𝔭𝔰𝔲(2,2∣4)≑𝔀=𝔀0+𝔀1+𝔀2+𝔀3,(4.4) and the β„€4-grading is generated by the transformation Ξ©, where Ξ©(𝑀)=βˆ’Ξ£π‘€π‘†π‘‡Ξ£βˆ’1.(4.5) Here 𝑀 and 𝑀𝑆𝑇 are 8Γ—8 supermatrices and Ξ£ is the following matrix Ξ£=βŽ›βŽœβŽπ½00𝐽⎞⎟⎠,where𝐽=βŽ›βŽœβŽβˆ’π‘–πœŽ200βˆ’π‘–πœŽ2⎞⎟⎠,(4.6) with 𝜎2 the Pauli matrix. The subsets π”€π‘˜ are the eigenspaces with respect to Ξ©, namely, Ξ©π”€π‘˜=π‘–π‘˜π”€π‘˜. The β„€4-grading respects the bilinear invariants of the algebra, namely, ξ€Ίπ”€π‘š,𝔀𝑛=π”€π‘š+𝑛mod4.(4.7) From the above relation we can see the reason why the supersymmetric extension of AdS5Γ—S5 is not a symmetric space, namely, [𝔀1,𝔀1]=[𝔀3,𝔀3]=𝔀2, cf. (3.32) in Section 3.2. The bilinear invariants can be naturally represented by the supertrace in the algebra space, and we have βŸ¨π‘‡π‘š,π‘‡π‘›βŸ©=0unlessπ‘š+𝑛=0(mod4).(4.8) In particular, the subalgebra 𝔀0 is the invariant locus of the 𝔭𝔰𝔲(2,2∣4) algebra and it is the algebra for the gauge group 𝐻, which in our case is SO(4,1)Γ—SO(5). This is a crucial point from the supercoset construction point of view. 𝔀2 contains all the bosonic generators which are left after modding out the Lorentz generators for 𝔰𝔬(4,1)×𝔰𝔬(5), namely, it contains the translation generators, and it is a ten-dimensional space. Notice that 𝔀2 is not a subalgebra.33 Finally 𝔀1 and 𝔀3 are spanned by the fermionic generators, and the two sectors are related by complex conjugation.
According to the algebra decomposition (4.4), also the currents will respect the β„€4-grading. Denoting with π½π‘šβ‰‘π½|π”€π‘š the projection onto the subalgebra π”€π‘š, then𝐽=𝐽0+𝐽2+𝐽1+𝐽3.(4.9) Notice that 𝐽1 and 𝐽3 are even since they are contracted with the generators and that the gauge-invariant currents 𝑗 mix under the β„€4-grading. In the language of the previous section, 𝐽0 is 𝐻, cf. Section 3.2.

Green-Schwarz-Metsaev-Tseytlin Action
Let me first explain the name for this action. In 1998 Metsaev and Tseytlin constructed the world-sheet action for the type IIB superstring on AdS5Γ—S5 from a geometrical point of view based on a super coset approach [49]. They use the Green-Schwarz (GS) formalism [66, 67], where the target space supersymmetry is manifest. This is due to the fact that the background, curved and with Ramond-Ramond (RR) fluxes, prevents the use of the Ramond-Neveau-Schwarz (RNS) approach, (cf. Section 5).
Recalling how the anti-De Sitter spaces and the spheres are realized:AdS5=SO(4,2)SO(4,1),S5=SO(6)SO(5),(4.10) and that the direct product SO(4,2)Γ—SO(6) is the bosonic sector for the full PSU(2,2∣4), thus the supersymmetric generalization of the above relation is PSU(2,2∣4)SO(4,1)Γ—SO(5)=superξ€·AdS5Γ—S5ξ€Έ.(4.11) In particular, 𝑔 maps the string world-sheet Ξ£ into the supercoset PSU(2,24)/(SO(4,1)Γ—SO(5)). To be more precise, we should say instead of PSU(2,2∣4) its corresponding universal covering. The left-invariant Maurer-Cartan forms are defined in the same way as in (3.21): π½πœ‡=π½π΄πœ‡π‘‡π΄=π‘”βˆ’1πœ•πœ‡π‘”,π½π΄πœ‡=π½π΄ξ‚Šπ‘€πœ•πœ‡π‘ξ‚Šπ‘€,(4.12) where 𝐴 is the 𝔭𝔰𝔲(2,2∣4) algebraic index, 𝑇𝐴 are the corresponding generators, which span the four π”€π‘š as in (4.9), πœ‡ is the world-sheet index, ξ‚Šπ‘€ is the ten-dimensional target space index, and the embedding coordinates are π‘ξ‚Šπ‘€=(𝑋𝑀,πœƒπ›Ό,Μ‚β€ŒπœƒΜ‚π›Ό). Recalling the action for the coset model (3.28) and considering for simplicity only the bosonic sector, then one easily sees that the one-forms π½π΄πœ‡ are indeed nothing but vielbeins, namely,34𝑆𝐺/𝐻=βˆ’βˆšπœ†4πœ‹ξ€œπ‘‘2πœŽβˆšβˆ’β„Žβ„Žπœ‡πœˆSTrξ€·π½πœ‡π½πœˆξ€Έ|𝔀2=βˆ’βˆšπœ†4πœ‹ξ€œπ‘‘2πœŽβˆšβˆ’β„Žβ„Žπœ‡πœˆπ½π΄ξ‚Šπ‘€π½π΅ξπ‘πœ•πœ‡π‘ξ‚Šπ‘€πœ•πœˆπ‘ξπ‘STr𝑇𝐴𝑇𝐡|𝔀