Novel construction and the monodromy relation for three-point functions at weak coupling

Abstract

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-chain 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 significant 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 Ω 1 Ω 2 Ω 3 = 1 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 arXiv:1312.3727. This structure may provide a key to the understanding of the notion of "integrability" beyond the spectral level.

Fermionic correlators from integrability

“…There are different embeddings of su(1|1) in the full superconformal group and they can be conveniently parametrized through some polarization vectors. This has a resemblance with the studies made in the su(2) sector presented in [18].…”

One-point functions in defect CFT and integrability

“…Bethe-state overlaps are playing an important rôle in the gauge/string integrability. The three-point functions in the N = 4 SYM at weak coupling can be expressed as generalized overlaps of Bethe states [31][32][33][34][35][36][37][38][39][40][41][42][43] and can be rendered into a compact determinant form [44][45][46][47][48], which is particularly useful in the semiclassical thermodynamic limit [33,[49][50][51][52]. An interesting question is whether the one-point overlap (2.21) also admits a determinant representation.…”

Structure constants of operators on the Wilson loop from integrability

“…The tailoring method is based on these previous papers. 4 Some related works for the three-point function are done in [23][24][25][26][27][28].…”