On the singlet projector and the monodromy relation for psu(2, 2|4) spin chains and reduction to subsectors

As a step toward uncovering the relation between the weak and the strong coupling regimes of the $\mathcal{N}=4$ super Yang-Mills theory beyond the specral level, we have developed in a previous paper [arXiv:1410.8533] a novel group theoretic interpretation of the Wick contraction of the fields, which allowed us to compute a much more general class of three-point functions in the SU(2) sector, as in the case of strong coupling [arXiv:1312.3727], directly in terms of the determinant representation of the partial domain wall partition funciton. Furthermore, we derived a non-trivial identity for the three point functions with monodromy operators inserted, being the discrete counterpart of the global monodromy condition which played such a crucial role in the computation at strong coupling. In this companion paper, we shall extend our study to the entire ${\rm psu}(2,2|4)$ sector and obtain several important generalizations. They include in particular (i) the manifestly conformally covariant construction, from the basic principle, of the singlet-projection operator for performing the Wick contraction and (ii) the derivation of the monodromy relation for the case of the so-called"harmonic R-matrix", as well as for the usual fundamental R-matrtix. The former case, which is new and has features rather different from the latter, is expected to have important applications. We also describe how the form of the monodromy relation is modified as ${\rm psu}(2,2|4)$ is reduced to its subsectors.


Introduction
Although the idea and the use of AdS/CFT [1][2][3] have been expanded into enormous varieties of directions, still the understanding of the essence of the dynamical mechanism of this remarkable duality remains as one of the most important unsolved problems. For such a fundamental task, it is best to study deeply the prototypical example which has been studied most vigorously, namely the duality between the N = 4 super Yang-Mills theory in 4 dimensions and the type IIB string theory in AdS 5 × S 5 .
In this context, undoubtedly the pivotal work which pointed the way to analyze this strong/weak duality was the discovery of Minahan and Zarembo [4] that the one-loop dilatation operator for the gauge invariant composite operators made out of six scalars of super Yang-Mills theory takes exactly the form of the Hamiltonian of the integrable SO(6) spin chain system. This made (a part of) the integrability structure of the super Yang-Mills theory quite explicit, at least as far as the spectrum of the composite operators are concerned. Two years later, the power and the relevance of integrability was revealed also on the strong coupling side by the works [5,6]. These works opened up the way to use integrability as an extremely powerful tool, not relying on the structure of supersymmetry and suitable for studying the dynamical aspects. Various results obtained along this way up to around 2010 are summarized in the review [7]. Subsequent developments can be classified into several categories. One is the more sophisticated way of computing the spectrum of composite operators, even at finite coupling.
The reader should be referred to the most advanced approach [8], [9] and the references therein. Advancements in another category are the computation and understanding of other observables, such as the scattering amplitudes (for reviews, see [10][11][12][13] ) and the Wilson loops (see for example [14,15] and references therein).
In the realm of the gauge-invariant composite operators, properties beyond the spectrum level have been vigorously pursued. In particular, study of the three-point functions, the main interest behind the present work, have been advanced using the power of "integrability". Below let us briefly summarize the highlights of the recent developments in this category both at weak and strong couplings.
At weak coupling, a systematic procedure called "tailoring" has been developed [16][17][18][19][20], which essentially reduces the computations to those of certain scalar products of spin chain states. To actually bring them into a tractable form a technical improvement was needed [21], and then a special class of three-point functions for non-BPS operators have been expressed explicitly in terms of Slavnov determinants [22]. Furthermore, the semiclassical limit of such three-point functions with large charges were successfully evaluated in a remarkably compact form [18,[23][24][25].
On the other hand, at strong coupling, the lack of the method for quantization of strings in curved space-time only permits one to deal with the semi-classical limit of large charges. Even in such a situation, the lack of the knowledge of the appropriate vertex operators and of the saddle point configuration made the complete computation quite difficult [27,28]. Finally, applying the idea of the state-operator correspondence and with the use of the finite-gap integration method, the computation was rendered possible for the GKP string [29] and for the more interesting case of three-point functions of rather general class in the SU(2) sector, which includes the very special ones computed by the tailoring technique at weak coupling [30]. Now in order to understand the connection between the weak and the strong coupling regimes, it is important to find the common structures in the two regimes. Since the super Yang-Mills theory at weak coupling is completely well-defined, it would be better to start by re-analyzing the three-point functions on this side of the duality.
In a previous communication [31], we have developed two novel viewpoints and applied them explicitly to a class of three point functions in the SU (2) sector which are much more general than had been treated by the tailoring procedure. Let us summarize these two ideas as (I) and (II) below: (I) One is the group theoretic reinterpretation of the Wick contraction of basic fields as a singlet projection of a tensor product of two fields.
(Ia) In the case of the SU(2) sector, one can apply this idea with respect to the more refined SU(2) L × SU(2) R structure present in that sector. This feature can be succinctly referred to as "double spin-chain" and it leads to the factorization of the left and the right sector and simplifies various formulas. This formalism allowed us to study a class of three-point functions of operators built upon more general spin-chain vacua than the special configuration discussed so far in the literature. This formulation makes the correspondence with the strong coupling computation done in [30] quite apparent.
(Ib) Another conspicuous advantage of the new interpretation of the Wick contraction is that one can avoid the scalar products of off-shell states which appear in the tailoring prescription. Emergence of such an object required the trick [21] to turn one of the off-shell states into an on-shell state in order to write it in terms of the Slavnov determinant. In contrast, in our formulation one can directly obtain the expression in terms of the partial domain wall partition functions, and for a certain class of correlators it can be readily expressed as a determinant even for two off-shell states.
(II) The second new idea formulated explicitly in [31] is so-called the monodromy relation, which can be obtained by inserting the monodromy matrices Ω's inside the two-point or the three-point functions and using the unitarity and the crossing relations. This produces a relation between correlation functions of different operators and hence acts like the Schwinger-Dyson equation. In particular, in the special limit where the spectral parameter u goes to ∞, it reduces to the Ward identity for SU(2) L and SU(2) R . Moreover, for the three-point function in the semiclassical limit, where the operators carry large quantum numbers, it takes the classical relation Ω 1 Ω 2 Ω 3 = 1, which is precisely of the form of the monodromy relation that follows from the integrability of the string theory, which played such a crucial role in the computation at strong coupling [27,30]. Thus, its super Yang-Mills counterpart should also be considered as a major part of the concept of "integrability" beyond the spectral level. Now the main purpose of the present paper is to extend these two main ideas explicitly to the full psu(2,2|4) sector and discuss how various general formulas are modified and simplified when we reduce them to the various subsectors. As for the monodromy relation (II), for the sake of clarity of presentation, we shall mainly concentrate on the case of the two-point functions. However, the extension to the three-point functions of our interest is straightforward, as was demonstrated in the case of SU(2) sector in [31], and the form of the result will also be briefly presented.
As far as the basic ideas (I) and (II) sketched above are concerned, similar ideas on the Wick contraction and the monodromy relations have also been discussed independently by [32] 1 . Their work was based largely on the work by [33], which observed and utilized certain similarity of the Wick-contracting operator to the string field theory (SFT) vertex in the spirit of the string bit formulation. In this fashion, the work of [32] discussed already the full psu(2, 2|4) sector making use of the similar vertex, as well as the same oscillator representations and some associated basic formulas, as [33].
However, as far as the result (I) for the psu(2, 2|4) is concerned, the exponential form of the vertex written down by [33] was guessed by an analogy with the delta-function overlap in SFT and unfortunately was not psu(2, 2|4) singlet. The work of [32], which was based on [33], modified certain parts of the exponent and checked that it is a singlet projector a posteriori. However, there are two points that one wishes to improve on.
One is the understanding of why the singlet projector is of a simple exponential form, which was assumed in the work of [33] and hence in [32]. Such a form may be natural as the oscillator description of the δ-function overlap familiar in SFT context is indeed exponential. However, the analogy should be taken with care. For one thing the discrete indices of the oscillators for the string case are the Fourier mode numbers, whereas the similar indices in the super Yang-Mills case designate the location along the spin chain.
Furthermore, in the case of the string the parts to be identified are rather homogeneous and hence it is natural to employ the (oscillator representation of) a delta-function to connect them. On the other hand, in the case of the psu(2, 2|4) spin chain the adjacent "string bits" can be quite different and the analogy to the delta function overlap is not intuitively obvious. In fact it is a simple exercise to construct the singlet state in the case of the spin j representation of SU (2) and confirm that it does not take an exponential form.
Therefore the surest way to obtain the desired vertex which effects the Wick contraction is to construct the most general singlet projector systematically in the space of tensor product of two spin chains. We shall show that the singlet projector exists for su (2, 2|4) (as well as its restriction psu(2, 2|4)) but not for u(2, 2|4) and, strictly speaking, for each sector of the representation of the su(2, 2|4) with a definite central charge C, the singlet vertex is not of a simple exponential form. However, provided that one is interested only in a sector with one definite value of the central charge 2 , one is allowed to use the simple exponential form, which is much more tractable. (We shall further elaborate on this later.) In this connection, we shall also explain in an appendix how the simple non-exponential singlet projector constructed for the SU(2) subsector in our previous work can be obtained from the general exponential projector for psu(2, 2|4).
The second point is that one wishes to improve the situation that the singlet projector of [32] is not manifestly conformally invariant, which is not useful for the treatment of the computation of the correlation functions of the local composite operators. In the present work we shall construct the version of the singlet projector which is manifestly conformally invariant and hence much simpler to use.
To explain what we mean by this, it is instructive to recall the following basic facts.
In constructing the representations of the superalgebra u(2, 2|4), there are basically two different schemes, depending on which maximal bosonic subgroups of the supergroup U(2, 2|4) to make use of: Their difference resides in the choice of the subgroups in the SO(4, 2) part. The one, which we shall call E-scheme, makes use of the compact subgroups with the AdS energy E being diagonal. On the other hand, in the scheme to be called D-scheme, the dilatation generator D is diagonal and the rest of the subgroup chosen in SO(4, 2) is the non-compact Lorentz group SL(2, C) × SL(2, C). Therefore, the D-scheme is manifestly conformal covariant.
It is well-known and fully discussed in [34] that these two schemes are connected by a non-unitary similarity transformation generated by the operator U = e (π/4)( In the treatment of [33], and hence [32], the oscillators appropriate for the E-scheme are used as basic building blocks for the generators of u(2, 2|4) and the relevant vertex operators. Since the D-scheme is more natural for the main purpose of computing the correlation functions of the basic super Yang-Mills fields, they transformed various quantities to that scheme by the similarity transformation using the operator U . However, since U does not map an individual component group, such as SU(2) L , in the E-scheme to a definite component group, such as SL(2,C), in the D-scheme 3 , the mapping does not make the description manifestly conformally covariant.
In our construction, to be described fully in section 2, we will stick to the D-scheme throughout, by using the oscillators which transform covariantly under the maximal subgroups shown in (1.2). This will make the entire description quite transparent without the need of the operator U .
Let us next turn to the psu(2, 2|4) version of the monodromy relation (II). There are two natural types of monodromy matrices depending on the choice of the auxiliary space.
One is the simpler and the fundamental one, for which the auxiliary space is taken to be C 4|4 . For this case, the derivation of the monodromy relation is a straightforward extension of the one for the SU(2) sector given in our previous work [31] and agrees with the description given in [32]. Another type is the monodromy relation associated with the so-called harmonic R-matrix, for which the structure of the auxiliary space is the same as that of the physical quantum space [35][36][37][38]. This case may be useful for obtaining local conserved quantities as well as for the study of scattering amplitudes [39][40][41][42][43][44][45][46]. In the present work, we shall derive the monodromy relation for this more complicated case as well, which was not discussed in [32]. As was demonstrated in [31], the monodromy relation for the three-point functions can be straightforwardly derived once that for the two-point functions is established, in this article we shall concentrate on the case of twopoint functions.
Now the monodromy relations for the entire psu(2, 2|4) sector is practically too complicated to analyze at present. In this sense, it is of interest to look first at such relations for simpler subsectors. This has already been done for the SU(2) sector in our previous work. In the present work, we shall first sketch how this result can be rederived by the reduction of the relations for the psu(2, 2|4) sector and then apply similar techniques to the non-compact SL(2) subsector with much more detailed expositions. Under such reductions we shall see that certain non-trivial shifts in the spectral parameters are produced.
It should be stated that all the discussions in this paper are at the tree level. It will be an important future task to extend some of the basic concepts to the loop level.
Having explained the essence of the new findings of the present work, let us briefly summarize the organization of the rest of this article.
In section 2, we start with a review (section 2.1), where we present the representation of the generators of u(2, 2|4) in terms of the oscillators, which transform covariantly under the maximal subgroups U(1) D × SL(2, C) × SL(2, C) of SO(4, 2) (the D-scheme choice discussed above) and U(1) J × SU(2) L × SU(2) R of SO (6)  In section 3, we derive explicitly the formulas for the monodromy relations for the correlation functions in the psu(2, 2|4) spin chain systems, first in the case of the fundamental R-matrix and then in the case of the harmonic R-matrix, which is more involved (some of the details are relegated to Appendix C.).
In section 4, we explain how the monodromy relations for the psu(2, 2|4) can be reduced to the ones for the subsectors. In particular, we study the case of the compact SU (2) subsector and the non-compact SL (2) subsector and see that the reduction produces certain shifts in the spectral parameters.
In the final discussion section (section 5), we shall summarize the essential ideas and methods employed to obtain the new results in this work and discuss how they should be utilized to try to capture the principles through which to relate the super Yang-Mills theory and the string theory in AdS spaces.
As already indicated, three appendices (including Appendix A where we list all the generators of u(2, 2|4) in the D-scheme notation for convenience) are provided to supplement the discussions given in the main text.
2 Conformally covariant oscillator description of psu(2, 2|4) and the singlet projector for the contraction of basic fields We begin by constructing the singlet projector for the full psu(2, 2|4) sector from the first principle, with which one can efficiently perform the Wick contraction between the basic fields of the N = 4 super Yang-Mills fields. As already emphasized in our previous work [31], the use of this object is quite natural and versatile in computing fairly general class of correlation functions of gauge-invariant composite operators made out of SYM fields, at least at the tree level and possibly at the higher loop levels.

Oscillator representation of the generators of u(2,2|4) in the D-scheme
In the case of the SU(2) subsector discussed in our previous paper, the construction of the singlet projector was nothing but the elementary problem of forming a singlet state out of two spin 1/2 particles, once we regard the SU(2) spin chain as a double-chain associated with the two distinct SU(2) groups 4 , SU(2) L and SU(2) R , acting on the chain.
In the case of the full psu(2, 2|4) spin chain, however, the structure of the algebra and its representation are sufficiently involved to render the general construction non-trivial.
Luckily, as our aim is to be able to perform the Wick contraction of only the basic SYM fields, we may restrict ourselves to the singleton representation, which can be realized by a minimal set of oscillators [47,48]. However, before introducing the oscillators, we must recall that there are basically two different bases for the representations of the u(2, 2|4) algebra, depending on which maximal subgroups of the conformal group SO(2,4) are used and the properties of the oscillators depend on such bases. Let us describe and compare them in some detail below following [33].

E-scheme and the D-scheme
As already mentioned in the introduction, we shall call these two schemes E-scheme and D-scheme, where "E" and "D" stand for the energy and the dilatation respectively, for which the subgroups taken are shown below: It should be clear that these two groups, belonging to SU(4), are quite different from the SU(2) L × SU(2) R groups which will appear as a part of the maximal subgroups of SO (2,4) in the E-scheme described below in (2.1).
For the E-scheme, the maximal subgroups are all compact, including the SO(2) E factor, the eigenvalue of which is identified with the AdS energy 5 . Thus, in the context of AdS/CFT, this scheme is useful in describing the states and their spectra on the gravity/string side. On the other hand, for the D-scheme, the maximal subgroups are all non-compact, consisting of the dilatation and the Lorentz groups. As the interpretation of SO (2,4) as the conformal group in four dimensions is manifest in this scheme, D-scheme is more natural in discussing the correlation functions in the SYM theory. Accordingly, the set of oscillators used in these two schemes are different, each set transforming covariantly under the respective maximal subgroups.
Before introducing them and discussing their difference, it is useful to first recall how the SO(2,4) algebra and its representations are described according to these two schemes.
From the point of view of the conformal algebra in four dimensions, the commutation relations of the generators of SO (2,4) are given by where µ , ν = 0, 1, 2, 3 and the metric signature is taken to be η µν = diag(−1, 1, 1, 1).
M µν , P µ , K µ and D are, respectively, the Lorentz, the momentum, the special conformal and the dilatation generators. This set of commutation relations can be expressed more compactly as given respectively by where m, n, l = 1, 2, 3. Obviously, L m and R m commute with E and hence carry zero energy. The rest of the generators of SO(2,4) carry either positive or negative energy and thus the generators of the entire algebra are decomposed in the following fashion: (2.14) Let |e, j L , j R be the simultaneous eigenstate of the energy E and the third components L 3 and R 3 of SU(2) L and SU(2) R respectively with the eigenvalues denoted by e, j L and j R . Namely The physically relevant unitary positive energy representations are built upon the lowest weight state among the set { |e, j L , j R }, which is annihilated by all the energy-lowering generators belonging to E − . We denote them by L ij , where the i and j are actually the spinor indices of SU(2) L and SU(2) R respectively and run from 1 to 2. Therefore we have four annihilation operators in total and the lowest weight state is characterized by By acting onto this vacuum the four raising operators belonging to E + , which we denote by L ij , one obtains unitary representations in the E-scheme.
Next consider the algebra so(2, 4) from the D-scheme point of view. In this scheme, the generators of the maximal subgroups are given by Here, M m and N m denote the generators of the Lorentz group SL(2, C)×SL (2, C). In this scheme, as is apparent from the commutation relations (2.7), P µ and K µ are, respectively the raising or lowering operators. Hence, the decomposition of the conformal algebra so(2, 4) is of the structure so (2,4) From the point of view of CFT in four dimensions, which is directly expressed in the D-scheme, the multiplets of operators are built upon the conformal primaries placed at the origin x µ = 0. They carry definite dilatation charges, belong to the definite Lorentz representations, and are annihilated by the lowering operators K µ . Using the state-operator correspondence, such a primary state, denoted by |∆, j M ,j N with ∆ and (j M ,j N ) being the dilatation charge and the Lorentz spins, is characterized by Then the module is built up by the descendants generated by the multiplicative actions of the raising operators P µ . It should be emphasized that such a representation relevant for discussing the correlation functions is non-unitary, since the anti-hermitian operator −iD has real eigenvalues. Now let us give a brief description of the relation between the E-scheme used in [32,33] and the D-scheme to be employed exclusively in this work. It is well-known by the work of [34] that there exists a non-unitary similarity transformation between the generators of these two schemes. The correspondence between E = J −1,0 and −iD = −iJ −1,4 indicates that such a transformation should rotate the non-compact 0-th direction into the compact 4-th direction and indeed it is effected by the operator Explicit transformations are given by As already mentioned, for the purpose of discussing the CFT correlation functions, the Dscheme is much more natural and if one starts from the E-scheme description as in [32,33], one must necessarily manipulate with the operator U in the intermediate step. Also, in the oscillator representations of the generators, to be elaborated below, the D-scheme oscillators always keep the conformal covariance manifest as opposed to those in the Escheme. We shall make the comparison more explicit later.

Oscillator representation in the D-scheme
Having argued the advantage of the D-scheme for our purpose, let us now introduce appropriate oscillators for this scheme and express the generators as their quadratic combinations.
For this purpose, it is useful to rewrite first the generators of the SO(2,4) algebra using the dotted and the undotted spinor indices of the Lorentz group. We will adopt the following conventions for the conversions of vectors and the tensors: where the Lorentz sigma matrices are defined in terms of the Paul matrices in the following Our convention for the epsilon tensor, with which the indices are raised and lowered, will be 12 = 12 = − 12 = − 12 = 1.
With this preparation, it is now quite natural to introduce two sets of bosonic oscillators, undotted and dotted, which transform under SL(2, C) and SL(2, C) respectively In terms of these oscillators, the conformal generators can be expressed rather simply as
(2.36) 6 Recall that −iD has positive real eigenvalues in our convention.
Essentially the same oscillator representation was given in [49,50]. We shall follow [50] with slight changes of signs and conventions. Now to construct the SU(4) R-symmetry generators, we introduce four sets of fermionic oscillators satisfying the anti-commutation relations Then the SU(4) generators can be constructed as It is easy to check that under R a b the oscillatorsξ b and ξ a transform under the fundamental and anti-fundamental representations. As the SU(4) indices of any generator will be carried by these fundamental oscillators, this guarantees that a generator J c (J c ) having a lower (upper) index transforms as a fundamental (anti-fundamental), i.e.
The remaining generators, namely the fermionic supersymmetry and superconformal generators, are expressed in a very simple way where the transformation properties are directly expressed by those of the constituent oscillators:

Central charge and hyper charge
The 30 bosonic and 32 fermionic generators constructed in terms of the oscillators above constitute the generators of the psu(2, 2|4). Actually, they do not close under (anti-) commutation. The closure requires the operator called the central charge given by As the name indicates, C commutes with all the generators of psu(2, 2|4) and hence it takes a constant value for an irreducible representation. In particular, as we shall describe shortly, for the basic fields of the N = 4 SYM (i.e. for the field strength multiplet) of our interest, C vanishes. Thus in this sector, we can neglect this operator. Another additional operator of interest is the so-called the hypercharge operator 7 given by This is essentially the fermion number operator. One notices that B does not appear in all the (anti-) commutation relations of psu(2,2|4) and thus it can be regarded as an outer automorphism of psu(2,2|4). By adding B and C to psu(2,2|4), we obtain the closed algebra called u(2,2|4).
The generators of u(2,2|4) can be expressed succinctly in terms of the oscillators as One can check that ζ's satisfy the graded commutator of the form where |A| is 1 for fermions and 0 for bosons. Hereafter, for simplicity, all the commutators should be interpreted as graded commutator as above. Then, the graded commutators between the generators of u(2,2|4) are neatly summarized in the following form: It is useful to write down the elements J A B of u(2,2|4) in a matrix form in the following way: From this one can see that the central charge and the hypercharge are related to the trace and supertrace in the following way As we shall see in section 2.2, the singlet projector we shall construct will be valid for the su(2,2|4) algebra as well as for psu(2,2|4), where the generatorsĴ A B of the former is obtained from u(2,2|4) by imposing the supertraceless condition aŝ In particular this gives A (−1) |A|Ĵ A A = 0, which tells us that the hypercharge B is completely removed from su(2,2|4).

Oscillator vacuum and the representations of the fundamental SYM fields
We now move on to the oscillator representation for the fundamental fields which appear in N = 4 SYM. For this purpose, we define the Fock vacuum |0 to be the state annihilated by all the annihilation operators: To be more precise, |0 is a tensor product of two vacua, one for the bosonic oscillators and the other for the fermionic ones. Namely, From the form of C given in (2.42) it is clear that they all carry C = 0. Also, it is easy to check that these oscillator expressions of the fields carry the correct Lorentz and R-symmetry quantum numbers.
In addition to these fundamental fields, we need to express their derivatives. The field at the general position x is obtained by the action of the translation operator e iP ·x as From the oscillator representation P αβ = λ αλβ , we see that the derivatives of a field can be expressed as where ∂ αβ = ∂/∂xβ α and xα β := x µ (σ µ )α β and we have used P ·x = P αβ xβ α and ∂ αβ e iP ·x = iP αβ e iP ·x . Notice that the spinor indices (α, γ, . . .) and (β,δ, . . .) are symmetrized as the bosonic oscillators λ mutually commute. Also note that we can replace some combinations of the (covariant) derivatives by appropriate fields without derivatives using the equations of motion and the Bianchi identities. For example, we can set ∂ αβ ∂β α φ ∝ 2φ and αβ ∂ αα ψ a β to zero due to the free equations of motion 8 . As a result, we can express fields with derivatives by expressions where all the spinor indices are totally symmetrized. Therefore, the independent fields with derivatives are simply generated by acting P αβ = λ αλβ on the oscillator representations for the fundamental fields (2.55)-(2.59). Since P αβ commutes with the central charge, these states with derivatives are still within the subspace with vanishing central charge.

Various "vacua" and their relations
It is an elementary exercise in quantum mechanics to construct the singlet state from two spin 1/2 states by forming a suitable combination of the highest and the lowest states. It is a slightly more involved exercise to extend this to the case of the general spin j, but the structure is similar: One combines the states built upon the lowest weight states and those built upon the highest weight states with simple weights. Indeed, up to an overall constant, the singlet state is given by This indicates that for the construction of the singlet state for much more complicated case of psu(2,2|4), the basic idea should be the same and one would combine the Fock states built upon the lowest weight oscillator vacuum |0 , already introduced, with the states built upon the highest weight oscillator vacuum |0 , which should be defined to be annihilated by the creation operators as Just as for |0 given in (2.54), the more precise definition of |0 is From (2.63) it immediately follows that |0 is annihilated by P αβ = λ αλβ and thus the Fock space built on |0 is a highest weight module as opposed to the lowest weight module built on |0 .
There is an essential difference between the bosonic sector and the fermionic sector.
For the bosonic sector, |0 B and |0 B cannot be related by the action of a finite number of oscillators 9 , but for the fermionic sector one can readily identify |0 F =ξ 1ξ2ξ3ξ4 |0 F .
It will turn out, however, that as for the fermionic oscillator Fock space describing the R-symmetry quantum numbers, "vacua" slightly different from |0 and |0 will be more useful and more physical. To introduce them, we rename the fermionic oscillators in the following way so that half of the creation (annihilation) operators are switched to annihilation (creation) operators 10 : We define the state |Z as annihilated by the new annihilation operators c i and d i , while |Z is defined to be annihilated by the new creation operatorsc i andd i .
As states built on the original vacuum |0 , these new "vacua" can be written as Then, it is easy to see that |Z and |Z carry the quantum numbers ( 1 2 , 1 2 ) and (− 1 2 , − 1 2 ) respectively and hence can be identified with, say, φ 1 +iφ 2 and its complex conjugate. The (de-)excitations of these vacua in the SU(2) sector with the quantum numbers ( 1 2 , − 1 2 ) and (− 1 2 , 1 2 ) respectively, which are often denoted by |X and | −X , are given by (2.74) 9 Actually, by using the operator U 2 = exp[ π 2 (P 0 − K 0 )] one can map |0 to |0 and exchange the role of the annihilation and the creation operators. 10 Such a transformation is sometimes called a particle-hole transformation. 11 Of course the choice of SU(2) L × SU(2) R in SU (4) is not unique. We are simply taking a convenient one. Incidentally, our normalization for J ± is J ± = J 1 ± iJ 2 . Now in order to construct the singlet projector in section 2.2, it will turn out to be convenient to define the scalar states similar to the above, except that their bosonic part of the vacuum is switched from |0 B to |0 B . We will place a line over the kets (or the corresponding bra) to denote such scalar states. For example, In this more precise notation, the previously defined |Z and |Z are written as Since overlined scalar states differ only in the bosonic sector, the properties of such states under the action of the fermionic oscillators are exactly the same as the un-overlined ones.
As we will need them later, it should be convenient to list the properties of the bra (or dual) vacua, which evidently follow from those of the ket vacua. 0| and 0 | have the properties 0|λ α = 0|λα = 0|ξ a = 0 , 0|0 = 1 , This means that the dual Fock space is generated either by the action of (µ α ,μα, ξ a ) on 0| or by the action of (λ α ,λα,ξ a ) on 0 |. As for the properties of the scalar bra vacua under the action of the fermionic oscillators, they satisfy and exactly the same equations hold for the overlined bra states Z| and Z |.

Comparison with the E-scheme formulation
Before we start the systematic construction of the singlet projector using these oscillator representations, let us end this subsection with some comments on the difference between the oscillator representation we use and the one employed in [32,33]. They basically work in the E-scheme, where the oscillators are covariant under the compact subgroup shown in (2.1). This in turn means that the representations corresponding to local composite operators are obtained indirectly by the use of the complicated operator U = exp[ π 4 (P 0 − K 0 )]. To be a little more specific, let us display the E-scheme oscillators and the Fock vacuum used in [32,33]. The difference from ours is in the bosonic oscillators, which are given by Then the SO(2,4) generators are expressed as bi-linears of these oscillators as However, as we mentioned in the introduction, any similarity transformation preserves the structure of the algebra and hence, for example, SU(2) L does not become a Lorentz group SL(2, C). This is reflected in the transformation of the oscillators themselves. By using the explicit oscillator representation of U , we easily find, for example, , which is not informative as far as the useful re-interpretation to the D-scheme is concerned. Thus although the E-and the D-schemes are connected by a similarity transformation U , the conformal covariance cannot be made manifest by just such a transformation. Hence for the purpose of dealing with the local composite operators, the use of D-scheme is much more transparent and indeed in what follows we shall never need the operator U .

Singlet condition and its solution
We shall now give a detailed construction of the singlet projector psu 1 12 | for the states in the product of a pair of Hilbert spaces H 1 ⊗ H 2 , which satisfies the defining equation for the singlet projector In order to find the most general singlet projector, we must proceed systematically. As it will become clear below, actually the desired singlet projector satisfying the relation above can be constructed for su(2, 2|4) as well as for psu(2, 2|4), but not for u(2, 2|4).
Recall that the generators J A B of su(2, 2|4) are obtained from the generator of u(2, 2|4), to be tentatively denoted byĴ A B , by making them supertraceless, i.e.
Because of this condition, when we interchange the order of the two conjugate oscillators making up any diagonal generator J A A , constant terms produced from the (anti-)commutation relations precisely cancel. This property will be of crucial importance for the construction of the true singlet projector.
First, let us begin by identifying the building block for the sector involving the oscillator pair λ α and µ α . Since the generators J A B are quadratic in the oscillators, the building block which would realize the relation (2.89) in the above sector should be of the form Now consider a useful combination of generators belongs to su(2, 2|4) and hence the interchange of the order of λ α and µ α does not produce any constant. When we apply On the other hand, when we apply 1 ⊗ J (1) , since Z | is annihilated by µ α , we should use In order for the sum of (2.92) and (2.93) to vanish, we must have n µ = n λ and α = β.
Hence, the form of the oscillator factor should actually be the combination (µ α ⊗ λ α ) nµ α We can apply the same logic to the sectors consisting of other conjugate pairs, namely (μα,λα), (c i , c i ) and (d j , d j ) and find similar conditions. In this way, we find that the necessary form for the singlet projector for su(2, 2|4) can be written as where f (n) at this stage is an arbitrary function and is to be determined by the requirement of the singlet condition. As for the sum over the powers n µα etc, we shall allow them to be arbitrary non-negative integers.
To see what conditions should be satisfied by the function f (n), let us focus first on a simple generator in the (µ, λ) sector of the form J α β = λ α µ β , where α = β. Since λ α is the annihilation operator for the bra state Z|, just as before, we easily get In the third line we have shifted n λα by 1 and interchanged the order of the factors 1 ⊗ λ α and µ β ⊗ 1. Now the action of µ β ⊗ 1 on n| is easily seen to produce the structure 1 ⊗ µ β with an overall minus sign, together with a shift of n λ β by minus one unit in f (n) under the sum. As for the structure of the operator part, combined with the factor (1 ⊗ λ α ) already produced, we get where we have interchanged the order of µ β and λ α to get back J α β without producing any constant since we are considering the case with α = β. Altogether we obtain the Thus the singlet condition demands The general solution of this equation is where g is an arbitrary function except that n λα 's must appear as the sum n λ 1 + n λ 2 .
Repeating similar analyses for all the off-diagonal 12 generators of su(2, 2|4), one obtains the following list of singlet conditions.
For the bosonic generators, we get For the fermionic generators, the conditions are With the hint from the analysis of the bosonic (µ, λ) sector, it is actually easy to write down the most general solution satisfying these equations. The answer is where h(x) is an arbitrary function of one argument. It is important to note that C is precisely the central charge of u(2, 2|4). As such it can be set to a number in an irreducible representation. In particular, the fundamental SYM fields of our interest belong to the sector where C = 0 and h(0) is just an overall constant, which we shall set to unity for simplicity.
Now we must examine the diagonal generators, such as λ α µ α andc i c i , etc. Because an extra constant is produced upon interchanging the order of the oscillators, for example like λ α µ α = µ α λ α − 2, etc. in the process of the manipulation as in (2.98), in general the singlet condition is not satisfied. However, as we already stressed, for the diagonal generators which belong to su(2, 2|4) and psu(2, 2|4) such constants cancel. Therefore, the conditions we obtained for the function f (n) do not change and the singlet projector for the physical SYM states is obtained as 13 Because of the restriction C = 0 in the sum over the n * 's, this expression does not quite take the form of an exponential. However, we can remove the restriction C = 0 in the sum when applying 1 12 | to the physical SYM states, since the extra states with C = 0 produced are orthogonal to C = 0 states and do not contribute to the inner product with the physical states. Thus, with the C = 0 restriction removed, the singlet state above can be written as a simple exponential given by where in the exponent the sum is implied for the repeated indices. If one wishes to perform the Wick contraction in a manifestly symmetric fashion, one can use the form Hereafter, we shall suppress for simplicity the subscript psu and write 1 12 | for psu 1 12 |.

Crossing relations for the oscillators
Before ending this subsection, let us make an important remark on the property of the singlet projector (2.105). Although we have constructed this state by demanding that it be singlet under the generators of psu(2, 2|4) satisfying (2.89), it is easy to see from the process of construction above that actually the singlet projector (2.105) effects the following "crossing relations" for the individual oscillators: 108) Clearly these relations themselves have no group theoretical meaning and appear to be stronger than the singlet condition. It is remarkable that yet they follow from the requirement of the singlet condition and will be quite useful in the computation of the correlation function, as we shall see in the next subsection.

Wick contraction of the basic fields using the singlet projector
Let us now show that the Wick contraction of the basic fields of the N = 4 super Yang-Mills theory can be computed quite easily by using the singlet projector constructed in the previous subsection. This can be identified as the method of Ward identity already introduced in [33]. However, as we use the D-scheme from the outset, our method is much more direct and simpler, without the need of rather complicated conversion operator U .
Since the singlet structure for the SU(4) part gets extracted as the unique factor abcd , we where I(x, y) ≡ 1 12 |(e iP ·x |0 ⊗ e iP ·y |0 ) . This gives I(x, y) = I(x − y). Next, we use the dilatation operator given by D = (i/2)(λ α µ α +λαμα + 2). Since P · x can be written as λ αλα xα α , the action of λ α µ α in D on e iP ·x |0 gives Evidently, the action ofλαμα on e iP ·x |0 gives exactly the same contribution. In a similar manner, the contribution from the D acting on e iP ·y |0 in the singlet condition relation produces the same result with x µ replaced by y µ . Altogether, the singlet condition with x µ ∂ ∂x µ + y µ ∂ ∂y µ + 2 I(x − y) = 0 .
(2.114) 14 As we shall see below, the singlet conditions produce Ward identities.
The solution is obviously (2.115) Let us now describe how the contraction of the fundamental fermions, i.e. ψ αa (x)ψ bα (y) can be done using the singlet projector. The singlet part for the R-symmetry obviously gives δ b a and hence we have In this case, we may use the crossing relation (2.109) for the oscillators to rewrite the RHS as Therefore, up to an overall normalization, we obtain Likewise, the Wick contraction for the self-dual field strength can be computed, again using the crossing relations for the oscillators, as Normalizations of these two point functions depend of course on the choice of the normalization of the individual fields but once we fix one of them, then the rest can be determined by supersymmetry.
With the demonstrations above, we wish to emphasize that our method of using the conformally covariant D-scheme is quite simple and useful in that the properties of the singlet projector can be directly and effectively utilized.

Monodromy relations for correlation functions in psu(2,2|4) spin chain system
Having constructed the singlet projector in the conformally covariant basis, we shall now generalize the so-called monodromy relations for the correlation functions developed in our previous work [29] for the SU(2) sector to the full psu(2,2|4) sector. Here one must first note the following new features. In the case of the SU(2) sector, the structure of the auxiliary Hilbert space is unequivocally chosen to be identical to that of the quantum Hilbert space, both two dimensional, describing the up and down "spin" states. On the other hand, for psu(2,2|4) there are two appropriate choices for the auxiliary space. To see this, we should recall the properties of the general R-matrix, to be denoted by R ij (u), from which the monodromy matrix is constructed. It is a linear map acting on the tensor , and satisfying the following Yang-Baxter equation: being the same as the infinite dimensional quantum space in structure. We shall call the 15 For the discussion of concepts requiring the Yang-Baxter equation, we must consider u(2,2|4), but not psu(2,2|4), as it is the R-matrix associated with the former which satisfies the Yang-Baxter equation. We shall give more detailed discussion on this point later. 16 The most of the discussion to follow is insensitive to whether we consider u(2,2|4) or its complexified version gl(4|4). Thus, when the description is easier with the complexified version, we shall use gl(4|4) in place of u(2,2|4). corresponding R-matrix as "fundamental" for the former case and "harmonic" for the latter choice.
For the former case, the monodromy matrix is finite dimensional and its components are operators acting on the quantum space. These components satisfy the exchange relations (or Yang-Baxter algebra) coming from the Yang-Baxter equations, and are quite powerful in diagonalizing the transfer matrix in the context of algebraic Bethe ansatz.
It should be noted that a similar finite dimensional monodromy matrix can be defined classically in the strong coupling regime using the flat connections of the string sigma model and can be used to determine the semi-classical spectrum [51]. Further, beyond the spectral problem, the monodromy relation of this type has its counterpart in the computation of the three-point functions in the strong coupling regime [27], [28][29][30] as the triviality of the total monodromy of the form Ω 1 Ω 2 Ω 3 = 1, where Ω i is the local monodromy produced around the i-th vertex operator in the so-called auxiliary linear problem. As explained in [27,30], this seemingly weak relation is disguisingly powerful, as it captures the important global information governing the three-point functions.
Such monodromy relations for the fundamental R-matrix for gl(4|4) can be derived through a procedure similar to the one for the SU(2) case worked out in detail in our previous work [29] and has been discussed in [32]. Besides the purpose of completeness, we shall re-derive these relations below since we shall use the definition of the Lax operator, slightly different from the one used in [32], which is more natural in connection with the strong coupling counterpart.
Next let us briefly describe the characteristics of the monodromy relations we shall derive for the harmonic R-matrix, which are completely new. In this case, the monodromy matrix is no longer finite dimensional since the auxiliary space V a is the same as the infinite dimensional quantum spin-chain Hilbert space V. One of the virtues of considering such a harmonic R-matrix is that, just as in the case of the SU(2) Heisenberg spin chain, the construction of the conserved charges including the Hamiltonian is much easier, since due to the identical structure of V a and V the R-matrix at specific value of the spectral parameter serves as the permutation operator P an . Such an operator is known to be extremely useful in extracting the Hamiltonian (i.e. the dilatation operator). Because of this and other features, the harmonic R-matrix and the related quantities have already found an interesting applications in the computation of the scattering amplitudes from the point of view of integrability [39][40][41][42][43][44][45][46] and are expected to be useful in the realm of the correlation functions as well.
In any case, since the monodromy matrix, constructed out of either "fundamental" or "harmonic" R-matrices, is a generating function of an infinite number of conserved charges, the monodromy relations can be regarded as a collection of "Ward identities" associated with such higher charges, which should characterize the important properties of the correlation functions.
Remarks on the relevance of of u(2,2|4) for the monodromy relation and psu(2,2|4) for the singlet projector Before we begin the discussion of the monodromy relations, let us give some important clarifying remarks on the relevance of the different super algebras for the two topics we discuss in this work and the role of their oscillator representation.
• The monodromy relation, to be discussed below, is deeply rooted in the integrability of the theory and hence it is crucial that the relevant R-matrix and the Lax matrix must satisfy the Yang-Baxter equations and the RLL=LLR equations. A method has long been known [35][36][37][38] that one can construct such an R-matrix and a Lax matrix from a suitable Lie super algebra. In the present case, one can do so for u(2,2|4) algebra but not for su(2,2|4) or psu(2,2|4). This is a general mathematical statement and has nothing to do with a particular oscillator representation nor • On the other hand, when we use the singlet projector to perform the Wick contractions efficiently in the computation of the correlation functions, the projector is a singlet for su(2,2|4) and psu(2,2|4). This notion has nothing to do with the integrability. In fact a singlet projector for u(2,2|4) does not exist at least in the oscillator representation utilized and this point gives a subtle effect in the crossing relation, to be discussed in the next subsection.
Thus, in the monodromy relations for the correlation functions, two different superalgebras are playing their respective role. The monodromy matrices to be inserted are associated with u(2,2|4), while the singlet projector which works as an elegant device in forming the correlation function for the physical SYM fields is valid for psu(2,2|4).

Basic monodromy relation in the case of fundamental Rmatrix
Let us begin with the case of the monodromy relations with the use of the fundamental Rmatrix. We shall first give the definitions and conventions for the fundamental R-matrix and the associated Lax matrix, which are slightly different from the ones used in [32], and then discuss the two important relations, namely the crossing relations and the inversion relations, which will lead immediately to the monodromy relations of interest.

Fundamental R-matrix and Lax operator
Consider the fundamental R-matrix, for which the quantum space is V and the auxiliary space is taken to be C 4|4 . This kind of R-matrix is often called the Lax operator and will be denoted by L an (u), where a and n refer, respectively, to the auxiliary space and the position on the spin chain. It satisfies the important relation called RLL=LLR relation 3) The solution of the above RRR relation turns out to be of the form where η is an arbitrary complex parameter 17 and E A i B is the fundamental representation of gl(4|4) acting non-trivially on V i ∼ = C 4|4 . To check that the R-matrix above actually satisfies the Yang-Baxter it is useful to note that the operator Π ij : serves as the graded permutation operator 18 . For example, ⊗ a) and so on. Then, the Lax operator 17 Although the Yang-Baxter equation holds for arbitrary η, we will later set η = ±i for our interest. 18 To prove this, we should pay attention to the non-trivial gradings between two auxiliary spaces .

satisfying (3.3) is given by
where J A n B 's are the generators of gl(4|4) defined on the n-th site of the spin chain. It is tedious but straightforward to show that the Lax operator indeed satisfies the RLL=LLR relation, by explicitly computing the both sides. In performing this calculation, one should remember that there are no grading relations between the auxiliary space and quantum spaces, which are two independent spaces. Explicitly, this means Differently put, the definition of the product of the Lax operators is not as supermatrices but as usual matrices. Although the choice for the mutual grading between these two spaces is a matter of convention 19 , our choice is a natural one from the point of view of connecting to the strong coupling regime. This is simply because the monodromy matrix at strong coupling is defined by the path ordered exponential of the integral of the flat connection and the multiplication rule for such matrices is the ordinary one. With this convention, the explicit form of the Lax operator is given in terms of the superconformal generators by (3.9) As usual the monodromy matrix is defined as the product of the Lax operators on each site going around the spin chain of length : Ω a (u) := L a1 (u) · · · L a (u) . (3.10) The monodromy matrix so defined satisfies the following relation, since each Lax operator satisfies the RLL=LLR relation: If we write out the above equation for each component, we obtain the so-called Yang-Baxter exchange algebra. In the rest of this subsection, when there is no confusion we drop the indices for the auxiliary space for simplicity. 19 In [32], the authors adopt the convention where non-trivial gradings between the auxiliary space and the quantum space exist. Namely, . Because of this, the definition of the Lax operator they use, i.e. L(u) := u − i/2 − i(−1) |A| E A B ⊗ J B A , is slightly different from ours.

Monodromy relation
Let us now derive the generic monodromy relation. This can be achieved by proving the following two important relations for the Lax operators, called the crossing relation and the inversion relation. They are respectively of the form where the superscript (i) on L (i) n denotes the i-th spin chain. The crossing relation (C) connects the Lax operator defined on the n-th site of a spin chain called 1 to that defined on the − n + 1-th site of another spin chain called 2. To get the feeling for the crossing relation, it suffices to recall that the singlet projector 1 12 | effects the Wick contraction between a field at the n-th site of one spin chain and a field at the − n + 1-th site of another chain. Actually, it is easy to prove it more precisely from the defining property of the singlet 1 12 |. As it was already emphasized in section 2.2, the operator 1 12 | is a singlet projector for su(2,2|4) or psu(2,2|4) but not for u(2,2|4) which is of our concern here. So the operator 1 12 | transforms the generator J A B of u(2,2|4) acting on the first spin chain into the operator −J A B − (−1) |A| δ A B acting on the second spin chain, where the constant piece −(−1) |A| δ A B comes from the (anti-)commutator term. Applying this to the Lax operator (L(u)) A B = uδ A B + η(−1) |B| J A B , one sees that the constant term shifts the spectral parameter by η and we get the crossing relation as shown in (3.12).
The proof of the inversion relation (I), which says that the Lax operator can be inverted for a specific value of the spectral parameter, is slightly more involved. The product of two Lax operators gives (3.14) First look at the last term quadratic in the generators. In general this cannot be simplified further. However, as we are specializing in the oscillator representation, we can write J A B =ζ A ζ B using the oscillators satisfying [ζ A ,ζ B ] = δ B A , as described in (2.44). Therefore we can reduce the product of the generator in the following way: For the first equality, we have used the fact that the constant term (−1) |C| δ C C , which appears from the commutation relation, vanishes. Further, since [ζ C ζ C ,ζ A ] = 1 and the central charge is given by 2C = trJ =ζ C ζ C , we obtain the result above. Hence the product of the Lax operators (3.14) is simplified to Now as we repeatedly emphasized, for the Yang-Mills fields of our interest we can set C = 0 and hence the coefficient in front of (−1) |B| J A B becomes η(u + v − η). Therefore when u + v = η, the RHS of (3.16) becomes uvδ A B = u(η − u)δ A B , which is precisely the RHS of the inversion equation (3.13).
We are now ready to present the generic form of the monodromy relation, which takes the form where F (u) is some function of u, to be given shortly. Both sides of this relation should be understood as acting on a tensor product of states on two Hilbert spaces of the form To show (3.17), we first prove the following relation with the use of the crossing relation (3.12): Focus first on the LHS of (3.18) and consider moving the Lax operator L (1) n (u) at the n-th site in Ω (1) to the left towards 1 12 |. Since the components of the Lax operators on different sites commute in the graded sense, namely we can move L We can now use the crossing relation (C) to replace the Lax operator (L (1) n (u)) AB with (L (2) −n+1 (η − u)) AB and move it back again to the original position. In this process, the sign factors which appear through the exchange of operators exactly cancel with those produced in the previous process and we get Repeating this to all the Lax operators making up the monodromy matrices, we immediately get (3.18). Now apply Ω (2) (u) to the both sides of (3.18) and use the relation ← − Ω (2) (η − u)Ω (2) (u) ∝ 1, with the overall factor which is readily computable using the inversion relation (I). In this way we obtain (3.17) with the function F (u) given by − η)) . Now if we apply (3.17) explicitly to the state |ψ 1 ⊗ |ψ 2 , we obtain the more explicit monodromy relation for the two-point function

(3.24)
Here , denotes the pairing with the singlet, which gives the Wick contraction between two operators. The sign in front arises when we pass the monodromy through the first state |ψ 1 .
At this point, it is of importance to remark that we obtain the usual Ward identities at the leading order in the expansion of the above equation around u = ∞. This is a direct consequence of the su(2,2|4) invariance of the singlet projector.
Once the monodromy relation is obtained for the two-point functions, the one for the three-point functions can be obtained easily, just as was shown explicitly for the SU (2) sector in [31]. The only differences from that case are the form of the prefactor function F 123 (u) and some sign factors due to the superalgebra nature of psu(2,2|4). Thus the monodromy relation for the three-point function takes the form (3.25)

Basic monodromy relation in the case of harmonic R-matrix
We shall now discuss another important version of the R-matrix, called the harmonic Rmatrix, to be denoted by the bold letter R. We shall derive the inversion and the crossing relations for it and finally prove the relations for the correlation functions obtained with the insertion of monodromy matrices constructed out of the harmonic R-matrices.
The term "harmonic" stems from the form of the Hamiltonian (or dilatation ) density first derived in [49], which can be expressed as 20 H 12 = h(J 12 ), where the function h(j) = j k=1 1/k is the so-called harmonic number and J 2 12 is the quadratic Casimir operator. This Hamiltonian is intimately related to the one in the SL(2) subsector and in that context was derived also as the logarithmic derivative of the R-matrix, just as in the case of the SU(2) Heisenberg spin chain.
The harmonic R-matrix is recently applied in the context of the scattering amplitudes for the N = 4 SYM theory, as a tool to construct the building blocks for the deformed Grassmannian formulas characterized as Yangian invariants [39][40][41][42][43][44][45][46]. The spectral parameter can be naturally introduced in the deformed formulas and turned out to serve as a regulator for the IR divergences.
Since this type of R-matrix is less well-known, we shall first give a brief review of the basic facts on the harmonic R-matrix following [40] and then using these properties derive the crossing and inversion relations, which are essential, as in the case of the fundamental R-matrix, in obtaining the monodromy relations we seek. 20 The subscript 12 signifies that the Hamiltonian is restricted to two fields 1 and 2.

Review of the harmonic R-matrix
The harmonic R-matrix R 12 acting on the tensor product of two copies of the Fock space V 1 ⊗ V 2 should satisfy the following RLL=LLR relation This is obtained from the general formula (3.1) by setting Renaming u = u 1 − u 2 , u 2 =ũ and expanding the above equation in powers ofũ, one can show that the harmonic R-matrix satisfies the following two types of equations: To actually find the form of R 12 (u), we will first solve the equation (3.27). For this purpose, it is convenient to introduce the following notations for sets of oscillators: respect to these two gl(2|2) subalgebras in the following form: The explicit form of these generators in terms of α, β-oscillators are given in [40], but we shall not write them down here. Now using the oscillators above, one introduces the following basis of linear operators acting on V 1 ⊗ V 2 , which will be useful for solving the condition (3.27): Hop (12) k,l,m,n = : should be obtained as a linear combination of Hop (12) k,l,m,n of the form k,l,m,n (u)Hop (12) k,l,m,n , (3.34) where N stands for the total number operator defined by . (3.36) Here I := k+l+m+n 2 is an integer since k +n = l +m, and B(x, y) is a natural generalization of the binomial coefficient, whose arguments can be complex. Notice that it satisfies B(x, y) = B(x, x − y) by definition. As already mentioned before, the overall coefficient ρ(u) is determined so that the unitarity condition (3.29) is satisfied. Explicit calculation gives ρ(u) = Γ(u + 1)Γ(1 − u) .
(3.37) 21 Notice that the central charge is given by β . Since we are interested in the representation in which the central charge vanishes, we neglect dependence on this combination.
An important characteristic of the harmonic R-matrix, for which the quantum and the auxiliary spaces are identical just as for the SU(2) Heisenberg spin chain, is that the Rmatrix at u = 0 yields precisely the permutation operator. Because of this fact, through the well-known manipulation, the Hamiltonian can be extracted from the R-matrix simply as a logarithmic derivative. This is summarized as 22

Monodromy relation
Having reviewed the basic facts on the harmonic R-matrix and displayed its explicit form in terms of the oscillators, we now discuss the monodromy relations involving such R-matrices. As in the case of the fundamental R-matrix, the basic ingredients for the derivation is (i) the inversion relation and (ii) the crossing relation.
The inversion relation is already given in (3.29), together with the computation of the factor ρ(u), shown in (3.37), needed for the normalization.
As for the crossing relation, its form for the harmonic R-matrix turned out to be similar to (but not identical with) the one for the Lax matrix for the case of the fundamental R-matrix shown in (3.12) and is given by Once this is verified, the crossing relation for the product of harmonic R-matrices is easily given by Now define the monodromy matrix as where {|n } is a complete set of states in the auxiliary space satisfying 1 = n |n n|.
Note that the components of the monodromy matrix take values in operators acting on the quantum space. The crossing relation for them follow immediately from (3.41) by taking the matrix element between the states m| and |n and is expressed as As an example for the use of this relation, contract both sides with |O 1 ⊗ |O 2 . We then obtain the monodromy relation for a two-point function of the form where , denotes the Wick contraction via the singlet projector.
Just as in the case of the fundamental R-matrix, the derivation of the monodormy relation for the three-point functions is straightforward. In fact, the prefactor function in this case is trivial and the result takes the simple form: Let us now describe how one can prove the basic crossing relation (3.40). As shown in (3.34), the harmonic R-matrix is made up of the hopping operators (3.33) and the coefficients (3.36). Therefore we need to derive the crossing relations for these two quantities.
Since the manipulations are somewhat involved, we relegate the details to Appendix C and only sketch the procedures and some relevant intermediate results below.
First, using the crossing relations for the oscillators, it is easy to derive the crossing relation for an arbitrary function of the number operators. The result reads In particular, the coefficient A N (2) ) I under such crossing.
Next one can show that the crossing relation for the hopping operator takes the form k,l,m,n = 1 12 |C • Hop Combining these crossing operations, we find that (3.51) In the above expression ofÃ   In Appendix C, we will show that this sum givingÃ Putting this result into (3.50) and summing over k, l, m, n, we find that the RHS of (3.50) becomes 1 12 |R (2) (−u) and this proves the crossing relation (3.40).

Reduction of monodromy relation to subsectors
In the previous section, we have derived the monodromy relation for the full psu (2,2|4) sector both in the case of the fundamental R-matrix and of the harmonic R-matrix.
The former can be obtained by a rather straightforward generalization of the SU(2) case 23 This is a direct consequence from the binomial theorem, namely, (1 + x) α = ∞ k=0 B(α, k)x k . The formula readily follows by considering (1 + x) α+β = (1 + x) α · (1 + x) β . discussed in our previous work [31] and we tried to give a slightly more detailed exposition compared with the result already given in [32]. On the other hand, the case for the harmonic R-matrix is new. Although its derivation turned out to be substantially more involved than the case of the fundamental R-matrix, the symmetric set up for which the quantum and the auxiliary spaces carry identical representation of psu(2,2|4) can be of particular value, as was already indicated in the application to the scattering amplitude.
Also the fact that the Hamiltonian can be obtained simply as the logarithmic derivative of the harmonic R-matrix should find useful applications.
The most important original purpose for formulating the monodromy relations, however, is their possible use as the set of powerful equations which govern, from the integrability perspective, the structures of the correlation functions. Although this idea has not yet been studied explicitly, to perform such an analysis it is natural to begin with the simplest set-ups, namely the cases of important tractable subsectors of the full theory.
In what follows, we shall consider the compact SU(2) sector and the non-compact SL (2) sector as typical examples, and derive the monodromy relations for them from the point of view of a systematic reduction of the general psu(2,2|4) case.

Reduction to the SU(2) subsector
In the case of the SU(2) sector, the monodromy relation was already obtained in our previous work [29]. Therefore the purpose here is to re-derive it through the reduction of the psu(2,2|4) case. To do this, the first step is to identify the generators of SU(2) L and SU(2) R , given in (2.71) and (2.72) in terms of those of u(2,2|4) shown in (2.47). This is simple since SU(2) L × SU(2) R is contained entirely in the R-symmetry group SU(4), and hence the relevant u(2,2|4) generators are W a b =ξ a ξ b = R a b + 1 2 δ b a B given in (2.50). Explicitly, we have It will be important to recognize that the following combinations, B L and B R , of diagonal W a a generators act as central charges for the group SU(2) L × SU(2) R : representation, which in our case of interest is the spin ( 1 2 ) L × ( 1 2 ) R representation. Evaluating B L and B R on any of the states in this representation, say |Z =ξ 3ξ4 |0 , it is easy to obtain B L = B R = 1 2 . With this in mind, let us write down the SU(2) L Lax operators as embedded in that of gl(4|4), given in (3.9). We get Using the relation B L = 1 2 = 1 2 (W 3 3 + W 1 1 ), i.e. W 3 3 + W 1 1 = 1, this can be re-written as Let us now recall the general form of the Lax operator for gl(4|4), which is L(u) A B = uδ A B + η(−1) |B| J A B , and its more refined form in terms of the superconformal generators given in (3.9). The part relevant for the SU (2)  It is important to note that the occurrence of the shift of the spectral parameter is due to the emergence of the extra central charges when a group is restricted to its subgroup and hence is a rather general phenomenon.

Inversion relation
Let us now derive the inversion relation. As in the previous discussion, we shall concentrate on the SU(2) L part. The object to consider is the product of two u(2,2|4) Lax operators given in (3.14), with the indices A, B taken to be the SU(2) indices 24 a, b. This where γ runs over the indices of the auxiliary space other than the SU(2) indices a, b, . . ..
Using the oscillator representation, the second terms can be simplified as where 10) and B R was already defined in (4.3). In the SU(2) sector the quantities Z L and Z R vanish 25 and B R = 1/2, as was already explained. Therefore the factor 2Z L + 2Z R + 2B R is simply unity and we simply obtain Substituting this back into (4.9) and rewriting the spectral parameters in order to express the result in terms of the Lax operator for SU(2) L , we obtain the relation To compare with the result of [31], we must substitute u → −u + i 2 and v → u + i 2 . Then the relation above takes the form (L SU(2) (−u + (i/2))L SU(2) (u + (i/2))) ab = −(u 2 + 1)δ ab , (4.16) which agrees with the equation (5.1) of [31] (with the inhomogeneity parameter θ set to zero).
From this inversion relation, one can easily obtain the monodromy relation, as already described in [29]. So we shall omit this derivation for the SU(2) sector. Instead, in the next subsection, we shall present a derivation of the monodromy relation for the SL (2) sector directly from that for the psu(2,2|4) sector. The result is new and the method can of course be applied to the SU(2) case as well to provide an alternative derivation of a known result given in [31].

Reduction to the SL(2) subsector
Having been warmed up with the reduction to the simplest SU(2) sector, we now perform the reduction to the SL(2) subsector to derive the explicit form of its monodromy relation, which is new. First consider the simple "light-cone" combinations of operators given by They satisfy the following simple set of commutation relations: This shows that the generators of the SL(2) L × SL(2) R can be taken as   With these properties, the relations (4.24) and (4.25) simplify and we can easily embed the SL(2) L Lax operator into that of u(2,2|4) in the following fashion: . (4.28) Here again the shift of u → u − i 2 is due to the effect of the central charge, as in the case of SU (2). Now comparing this with the form of the , we readily find that the SL(2) L Lax operator is embedded in the u(2,2|4) Lax operator as 27 (L SL(2) (u)) ab = (L(u − i/2)) ab , (a, b = 1,1) , η = −i . where γ runs over the all indices of the auxiliary except for those of SL(2), namely a, b = 1,1. Note at this stage the sign of the second term is opposite to the one for SU (2) case given in (4.8). Now as before we can simplify this term quadratic in the generators in the following manner: where we used Z R = 0, B = 1. Thus we get an extra minus sign from this manipulation and hence obtains the same sign for the second term as for the SU(2) case. Now we make an appropriate shift of the spectral parameter and get Thus if we set u+v = η = −i, then since the left hand side of the above equation becomes uvδ ab , we get Note that if we compare (4.36) with (4.15), we can recognize that the function f (u, v) for the SU(2) and the SL(2) cases can be written in a unified manner as where S 2 is the Casimir operator for the respective group.

Method of direct reduction from psu(2,2|4) monodromy relation
We now wish to demonstrate that we can derive the monodromy relation for the two-point function in the SL(2) sector more directly from that for psu(2,2|4), with the judicious use of the product relation (4.34) for the Lax operators.
The monodromy relation for psu(2,2|4) is given in (3.23). To reduce it to the SL (2) sector, we set the indices A and C to be those of SL(2), say a and c which take values in {1,1}, and for convenience make a shift of the spectral parameter u → u − i/2. Hereafter we shall employ the notation f − (u) ≡ f (u − i/2) for such a shift for any function f (u).
Then from (3.23) we get 38) or more succinctly, before taking the inner product with |ψ 1 ⊗ |ψ 2 , What we wish to derive from it is the relation involving the SL (2)  This difficulty can be overcome by noting that the formula for the crossing relation for the individual u(2, 2|4) Lax matrices is quite simple and that a certain product of the u(2, 2|4) generators which appear in the product of two Lax matrices can be reduced to a single generator, as was demonstrated already in (4.31). To make use of these properties, we first focus on the leftmost and the rightmost Lax operators forming the two monodromy matrices and make the split Then, the LHS of (4.39) becomes As the operators at different positions all commute, we can move Ω (1)− −1 Ω (2)− −1 in the middle to the left until they hit 1 12 |. Then we can use the monodromy relation (4.38) for the case of length − 1 and write (4.42) as Now substitute the definition L(u) A B = uδ A B + η(−1) |B| J A B for the Lax operators on the RHS and expand. This gives where β stands for indices other than those of SL (2). Further the second term on the RHS can be written out explicitly as where we used (−1) |c| = 1 since c is a bosonic index. Apply both sides now to the singlet projector 1 12 | and use the crossing relation for the generator 1 12 |J which is valid since a = β. Then using the identity (4.31) to the RHS, (4.45) becomes (2) a c and the equation (4.43) can be simplified to Note that on the RHS all the indices have turned into SL (2) indices.
Now we make a slight trick to split the last term of the RHS into identical halves 28 as (2) a c and then hit just one half to 1 12 | to change it into J (1) a c acting on the spin chain 1. Since the generator here is that of u(2, 2|4), in doing so there appears an extra constant term ∝ δ a c coming from the commutator of the oscillators forming this generator. Then we get We shall now show that the terms linear in the generators appearing on the RHS can be absorbed by a judicious shifts of the spectral parameters of the expression L which is the first term on the RHS of (4.44). In fact, one can easily check that (4.44) can be re-expressed as where the factor f 1 (η) is given by f 1 (η) = −η 2 /4. Thus combining with (4.43), the LHS of the original psu(2, 2|4) monodromy relation (4.39), namely As the final step, we now rewrite the quantity 1 12 |F − −1 δ bd in the first term by using the fundamental monodromy relation (4.39) with replaced by − 1. This process is a reversal of the splitting procedure we started out with and inserts the products of the monodromy matrices defined on the RHS of the splitting equations (4.40) and (4.41).
Then, the equation above becomes On the other hand, each Lax operator on the RHS of the relation (4.43) can be interpreted as a special monodromy matrix of length one and hence we can use the monodromy relation to write the RHS as 28 If one wishes, one can actually use a more general split with coefficients α and β satisfying α + β = 1 and follow the same logic to be described below for the 1 2 + 1 2 split. This will lead to more general forms of the SL(2) monodromy relations. Below we shall only describe the simplest split for the sake of clarity.
Thus equating (4.50) and (4.51) and rearranging, we obtain an important relation where the function g 1 (u) is given by The point to be noted here is that the part containing the unrestricted indices in the above relation is (Ω It should now be clear that we can perform this reduction process repeatedly until all the indices become those of SL(2) only. Then, taking η to be −i and identifying the SL(2) Lax operator as upon acting on the state |ψ 1 ⊗ |ψ 2 we obtain the monodromy relation for the genuine SL(2) monodromy matrices inserted as (Ω SL(2) (u − i/2)) ab |ψ 1 , (Ω SL(2) (u + i/2)) bc |ψ 2 = u 2 δ ac |ψ 1 , |ψ 2 .

Discussions
In this paper, we studied the tree-level three-point functions in the entire psu(2, 2|4) sector of N = 4 super Yang-Mills theory from a group theoretic and integrability-based point of view. We in particular developed the manifestly conformally invariant construction of the singlet-projection operator and used it to express the Wick contraction. Unlike the preceding works [32,33], our construction doesn't necessitate the "U -operator" which intertwines two schemes of representations of the superconformal algebra. This property greatly simplifies the analysis and allowed us to derive the monodromy relation for the harmonic R-matrix, as well as for the usual fundamental R-matrix.
The simplicity and the manifest conformal covariance of our construction will surely be of help when analyzing the weak-coupling three-point functions using integrability.
So far, such analysis was performed thoroughly only for a particular class of three-point functions in su(2) [16,31], sl(2) [52], su(3) [53] and su(1|1) [54] sectors. In the forthcoming paper [55], we will use our formalism to study more general three-point functions in the sl(2) sector, which involve more than one non-BPS operators. It would be an interesting future problem to study other sectors, in particular higher-rank sectors based on our construction.
It would also be interesting to study the loop correction in our formulation. For this purpose, a more detailed analysis of the harmonic R-matrix may be useful since the harmonic R-matrix is intimately related to the local conserved charges including the one-loop Hamiltonian. Another avenue of research is to explore the relation with the scattering amplitudes [39][40][41][42][43][44][45][46]. Also in that case, the harmonic R-matrix and the monodromy relation played an important role. It would be interesting if one could make a more direct connection.
Lastly, it would be important to understand the relations with the recently-proposed non-perturbative frameworks for the string vertex [56] and the three-point functions [57].
Understanding such a non-perturbative framework from the perturbative-gauge-theory point of view will lead to deeper understanding of the AdS/CFT correspondence. In this appendix, we briefly explain how this form is indeed obtained from the exponential form.
The SU(2) sector is only a part of the large Hilbert space in which the exponential state belongs. Also, in our previous paper, we were only considering the spin 1/2 representation for SU(2) L and SU(2) R . Thus to get our singlet formula (B.1) from the exponential form, we must project out such a sector from the full exponential projector.
It turns out that to do this appropriately, we must first write out the exponential state for the full SU(4) SO(6) sector which are generated by the fermionic oscillators only. This is given by Since, for each Hilbert space component, A consists of four different fermionic oscillators, the expansion of e A stops at order A 4 . The terms coming from the odd powers of A, 29 Here, since we are only concerned with the SU(2) sector where only the SU(4) oscillators are relevant, we shall denote |Z by |Z for simplicity .
i.e. A and A 3 , are fermionic. When we take inner product with scalar states, they do not contribute. Thus we only need to look at terms of order 1, A 2 and A 4 .
(i) At order 1, we simply get |Z ⊗ |Z . (ii) The next simplest contribution comes from A 4 . Writing this out explicitly, we get To see the meaning of this expression clearly, it is instructive to write down the generators of SU(2) L × SU(2) R in terms of these fermionic oscillators. They are given by From this we see that the RHS of (B.4) can be written as J L − J R − |Z ⊗ J L + J R + |Z . The action of these lowering and raising operators turn |Z into |Z and |Z into |Z , so that we get the simple result Proof of the formula (3.48) This formula can be understood in the following way. For simplicity, we concentrate on the oscillatorsᾱ, α. We first transform the creation operatorsᾱ A 1 by crossing and get 1 12 | 1 k!l!m!n!ᾱ A 1 2 · · ·ᾱ A k 2 · · ·ᾱ B 1 a · · ·ᾱ Bm a · · · α 1 Bm · · · α 1 B 1 · · · α a A k · · · α a A 1 .

(C.3)
Then, we wish to move the annihilation operators α 1 A 1 next to the singlet projector in order to use the crossing formula for them. This can be easily done, but after the crossing, we need to move them back to the original position, which in turn generates extra terms since 30 To be precise, N (1) α transforms as N (1) α → −N (2) α − (−1) |A| δ A A . However, the constant term vanishes as the signs are opposite for the bosonic and fermionic oscillators and hence they exactly cancel with each other in the present case. This is also true for N (1) β .

α 2
A 's do not commute with the creation operatorsᾱ A 2 . As a result, the Kronecker delta δ A B appears, which contracts the indices for the oscillators of the auxiliary spaceᾱ B a , α a A . In this way we find that the number operator of the auxiliary space N (a) α is inserted in the middle of the oscillators. Now when we move the number operator to the left most position, the number operator is shifted by a constant due to the presence of the creation operators on the way. This gives the expression of the form (N (a) α − m + 1) · · · (N (a) α − m + p) ×ᾱ A 1 2 · · ·ᾱ A k−p 2 · · ·ᾱ B 1 a · · ·ᾱ B m−p a · · · α 2 B m−p · · · α 2 B 1 · · · α a A k−p · · · α a A 1 .

(C.4)
By carefully treating the numerical coefficients and performing the same calculation for the oscillatorsβ, β, we find that the crossing formula for the hopping operator is given by (3.48).
Explanation of (3.49) Let us make cautionary remarks for using the crossing relations for the coefficients and the hopping operator already obtained to derive the crossing relation for the harmonic R-matrix given in (3.49). This has to do with the effects due to the order of crossing.