Continuum limit of fishnet graphs and AdS sigma model

Abstract

We consider the continuum limit of 4d planar fishnet diagrams using integrable spin chain methods borrowed from the N = 4 Super-Yang-Mills theory. These techniques give us control on the scaling dimensions of single-trace operators for all values of the coupling constant in the fishnet theory. We use them to study the thermodynamical limit of the BMN operator corresponding to the spin chain ferromagnetic vacuum. We find that its scaling dimension exhibits a critical behaviour when the coupling constant approaches Zamolodchikov's critical coupling. Analysis close to that point suggests that the continuum limit of the fishnet graphs is controlled by the two-dimensional AdS 5 non-linear sigma model. More generally, we present evidence that the fishnet diagrams define an integrable lattice regularization of the AdS 5 model. A system of massless TBA equations is derived for the tachyon energy by dualizing the TBA equations of the weakly coupled planar N = 4 SYM theory.arXiv:1806.04105v1 [hep-th]

Generalized fishnets and exact four-point correlators in chiral CFT4

“…It would be also interesting to understand the behavior of large Feynman graphs in the full χCFT, in-line with the early results of [20] and the recent observations of [57] for the fishnet reduction of the χCFT. In particular, if the σ-model interpretation of the latter paper can be generalized to the full χCFT, it could be a big step in the explicit construction of the AdS dual of this chiral CFT, if such one exists at all after the double scaling limit of γ-deformed N = 4 SYM theory.…”

The one-loop spectral problem of strongly twisted $$ \mathcal{N} $$ = 4 Super Yang-Mills theory

“…3. The Jordan blocks are not all of size 11 2 × 2, they do become arbitrarily large with increasing L. We demonstrate this by an explicit example below in (3.15). It is unclear to us how to determine these sizes, in general.…”

“…On the other hand, oddly, the two showcase instances where integrability can be rigorously proved in N =4 SYM are obscured in the double-scaled deformed models. At strong coupling, the interpretation in terms of an integrable string sigma model is missing, see, however, [11]. And at weak coupling, the a priori much simplified situation is also somewhat fuzzy.…”

Exact scattering amplitudes in conformal fishnet theory

“…At strong coupling, the leading Regge singularity scales as O( ξ 2 ). It would be interesting to reproduce the same behaviour using the dual description of the conformal fishnet theory [37].…”