Hexagons and correlators in the fishnet theory

109

Separation of variables (SoV) is a special property of integrable models which ensures that the wavefunction has a very simple factorised form in a specially designed basis. Even though the factorisation of the wavefunction was recently established for higher rank models by two of the authors and G. Sizov, the measure for the scalar product was not known beyond the case of rank one symmetry. In this paper we show how this measure can be found, bypassing an explicit SoV construction. A key new observation is that the measure for spin chains in a highest-weight infinite dimensional representation of slpN q couples Qfunctions at different nesting levels in a non-symmetric fashion. We also managed to express a large number of form factors as ratios of determinants in our new approach. We expect our method to be applicable in a much wider setup including the problem of computing correlators in integrable CFTs such as the fishnet theory, N " 4 SYM and the ABJM model.

“…recently also considered in[66] 22. see also the recent results of[70] for a 3-point function expressed in terms of TBA solutions, suggesting additional fruitful connections.…”

mentioning

confidence: 67%

Separation of variables and scalar products at any rank

Cavaglià

Gromov

Levkovich-Maslyuk

2019

109

“…Not much is yet done in this direction for the full χCFT, apart from the ABA approach of [19] to long operators L 1 and the one-loop study of [30], as well as the results of the current paper on the shortest exchange operators. As concerns the study of the structure constants, the first all-loop results for multi-magnon operators in bi-scalar fishnet CFT have been obtained in the very recent paper [54]. The generalization to four-point functions and to more complicated operators, and to the full χCFT, will necessitate a considerable new insight into integrability properties of these models.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

Generalized fishnets and exact four-point correlators in chiral CFT4

Казаков

Olivucci

Preti

2019

102

We study the Feynman graph structure and compute certain exact four-point correlation functions in chiral CFT 4 proposed byÖ. Gürdogan and one of the authors as a double scaling limit of γ-deformed N = 4 SYM theory. We give full description of bulk behavior of large Feynman graphs: it shows a generalized "dynamical fishnet" structure, with a dynamical exchange of bosonic and Yukawa couplings. We compute certain fourpoint correlators in the full chiral CFT 4 , generalizing recent results for a particular onecoupling version of this theory -the bi-scalar "fishnet" CFT. We sum up exactly the corresponding Feynman diagrams, including both bosonic and fermionic loops, by Bethe-Salpeter method. This provides explicit OPE data for various twist-2 operators with spin, showing a rich analytic structure, both in coordinate and coupling spaces.

“…The model itself allows for a number of generalisations which would preserve integrability. First, one can introduce non-zero AdS radius already at the classical level and also consider a generaliza- 14 The fishchain vertex may also be a useful playground for understanding wrapping corrections in the Hexagon program for correlation functions, [21,22]. tion of our model on a sphere.…”

Section: Discussionmentioning

confidence: 99%

Quantum fishchain in AdS5

Gromov

Sever

2019

115

In our previous paper we derived the holographic dual of the planar fishnet CFT in four dimensions. The dual model becomes classical in the strongly coupled regime of the CFT and takes the form of an integrable chain of particles in five dimensions. Here we study the theory at the quantum level. By applying the canonical quantization procedure with constraints, we show that the model describes a quantum chain of particles propagating in AdS 5 . We prove the duality at the full quantum level in the u(1) sector and reproduce exactly the spectrum for the cases when it is known analytically.In particular, the graph building operator is self-adjoint with respect to this norm. Given the wave function ϕ O we can compute any correlation function of the operator O. For example, the two point function is given by5) 1 Here we have suppressed double trace interactions which are not relevant non-perturbatively [5, 10]. 2 Not to be confused with the 't Hooft coupling of the parent N = 4 SYM theory, λ. 3 When taking the conjugate of the wave function in (2.4), one does not conjugate the 't Hooft coupling ξ 2 . This relation is a straight forward generalization of the J = 2 case considered in [3] to any length.12 The extra polynomial factors in T's can be removed by the gauge transformation g(u) = e πu 2 Γ(−iu). Under the gauge transformationT1(u),T 1 (u) → u J ,T 6 (u) →T 6 (u)/u J andT4(u) →T4(u)/ u 2 + 1 4ξ 2 J whereas T 4 (u) does not change. The Q-operator transforms as Q(u) → g(u)Q(u).