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 interpre
tation 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.
In this article, we shall develop and formulate two novel viewpoints and properties concerning the three-point functions at weak coupling in the SU(2) sector of the N = 4 super Yang-Mills theory. One is a double spin-chain formulation of the spin-cha
in and the associated new interpretation of the operation of Wick contraction. It will be regarded as a skew symmetric pairing which acts as a projection onto a singlet in the entire SO(4) sector, instead of an inner product in the spin-chain Hilbert space. This formalism allows 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. Furthermore, this new viewpoint has the signicant advantage over the conventional method: In the usual tailoring operation, the Wick contraction produces inner products between off-shell Bethe states, which cannot be in general converted into simple expressions. In contrast, our procedure directly produces the so-called partial domain wall partition functions, which can be expressed as determinants. Using this property, we derive simple determinantal representation for a broader class of three-point functions. The second new property uncovered in this work is the non-trivial identity satisfied by the three-point functions with monodromy operators inserted. Generically this relation connects three-point functions of different operators and can be regarded as a kind of Schwinger-Dyson equation. In particular, this identity reduces in the semiclassical limit to the triviality of the product of local monodromies around the vertex operators, which played a crucial role in providing all important global information on the three-point function in the strong coupling regime. This structure may provide a key to the understanding of the notion of integrability beyond the spectral level.
Based on the method of separation of variables due to Sklyanin, we construct a new integral representation for the scalar products of the Bethe states for the SU(2) XXX spin 1/2 chain obeying the periodic boundary condition. Due to the compactness of
the symmetry group, a twist matrix must be introduced at the boundary in order to extract the separated variables properly. Then by deriving the integration measure and the spectrum of the separated variables, we express the inner product of an on-shell and an off-shell Bethe states in terms of a multiple contour integral involving a product of Baxter wave functions. Its form is reminiscent of the integral over the eigenvalues of a matrix model and is expected to be useful in studying the semi-classical limit of the product.
