Fishnet CFT Research Articles

The loom for general fishnet CFTs

We propose a broad class of d-dimensional conformal field theories of SU(N) adjoint scalar fields generalising the 4d Fishnet CFT (FCFT) discovered by Ö. Gürdogan and one of the authors as a special limit of γ-deformed mathcal{N} = 4 SYM theory. In the planar N → ∞ limit the FCFTs are dominated by the “fishnet” planar Feynman graphs. These graphs are explicitly integrable, as was shown long ago by A. Zamolodchikov. The Zamolodchikov’s construction, based on the dual Baxter lattice (straight lines on the plane intersecting at arbitrary slopes) and the star-triangle identities, can serve as a “loom” for “weaving” the Feynman graphs of these FCFTs, with certain types of propagators, at any d. The Baxter lattice with M different slopes and any number of lines parallel to those, generates an FCFT consisting of M (M – 1) fields and a certain number of chiral vertices of different valences with distinguished couplings. These non-unitary, logarithmic CFTs enjoy certain reality properties for their spectrum due to a symmetry similar to the PT-invariance of non-hermitian hamiltonians proposed by C. Bender and S. Boettcher. We discuss in more detail the theories generated by a loom with M = 2, 3, 4, and the generalisation of the loom FCFTs for spinning fields in 4d.

Separation of variables in AdS/CFT: functional approach for the fishnet CFT

The major simplification in a number of quantum integrable systems is the existence of special coordinates in which the eigenstates take a factorised form. Despite many years of studies, the basis realising the separation of variables (SoV) remains unknown in mathcal{N} = 4 SYM and similar models, even though it is widely believed they are integrable. In this paper we initiate the SoV approach for observables with nontrivial coupling dependence in a close cousin of mathcal{N} = 4 SYM — the fishnet 4D CFT. We develop the functional SoV formalism in this theory, which allows us to compute non-perturbatively some nontrivial observables in a form suitable for numerical evaluation. We present some applications of these methods. In particular, we discuss the possible SoV structure of the one-point correlators in presence of a defect, and write down a SoV-type expression for diagonal OPE coefficients involving an arbitrary state and the Lagrangian density operator. We believe that many of the findings of this paper can be applied in the mathcal{N} = 4 SYM case, as we speculate in the last part of the article.

Amplitudes in fishnet theories in diverse dimensions and box ladder diagrams

We investigate properties of four-point colour ordered scattering amplitudes in D = 6 fishnet CFT. We show that these amplitudes are related via a very simple relation to their D = 4 counterparts previously considered in the literature. Exploiting this relation, we obtain a closed expression for these amplitudes and investigate its behaviour at weak and strong coupling. As a by product of this investigation, we also obtain a generating function for on-shell D = 6 Box ladder diagrams with l rungs.

Spontaneous conformal symmetry breaking in fishnet CFT

Quantum field theories with exact but spontaneously broken conformal invariance have an intriguing feature: their vacuum energy (cosmological constant) is equal to zero. Up to now, the only known ultraviolet complete theories where conformal symmetry can be spontaneously broken were associated with supersymmetry (SUSY), with the most prominent example being the N=4 SUSY Yang-Mills. In this Letter we show that the recently proposed conformal “fishnet” theory supports at the classical level a rich set of flat directions (moduli) along which conformal symmetry is spontaneously broken. We demonstrate that, at least perturbatively, some of these vacua survive in the full quantum theory (in the planar limit, at the leading order of 1/Nc expansion) without any fine tuning. The vacuum energy is equal to zero along these flat directions, providing the first non-SUSY example of a four-dimensional quantum field theory with “natural” breaking of conformal symmetry.

Thermodynamic Bethe Ansatz for Biscalar Conformal Field Theories in Any Dimension.

We present the TBA equations for the exact spectrum of multi-magnon local operators in the D-dimensional bi-scalar fishnet CFT. The mixing matrix of such operators is given in terms of fishnet planar graphs of multiwheel and multispiral type. These graphs probe the two key building blocks of the TBA approach, the magnon dispersion relation and scattering matrix, which we obtain by diagonalizing suitable graph-building operators. We also obtain the dual version of the TBA equations, which relates, in the continuum limit, D-dimensional graphs to two-dimensional sigma models in AdS_{D+1}.

Quantum fishchain in AdS5

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 AdS5. We prove the duality at the full quantum level in the mathfrak{u} (1) sector and reproduce exactly the spectrum for the cases when it is known analytically.

Basso-Dixon correlators in two-dimensional fishnet CFT

We compute explicitly the two-dimensional version of Basso-Dixon type integrals for the planar 4-point correlation functions given by conformal “fishnet” Feynman graphs. These diagrams are represented by a fragment of a regular square lattice of power-like propagators, arising in the recently proposed integrable bi-scalar fishnet CFT. The formula is derived from first principles, using the formalism of separated variables in integrable SL(2, ℂ) spin chain. It is generalized to anisotropic fishnet, with different powers for propagators in two directions of the lattice.

Quantum Spectral Curve of γ-Twisted 𝒩 = 4 SYM Theory and Fishnet CFT

We review the quantum spectral curve (QSC) formalism for the spectrum of anomalous dimensions of [Formula: see text] SYM, including its [Formula: see text]-deformation. Leaving aside its derivation, we concentrate on the formulation of the “final product” in its most general form: a minimal set of assumptions about the algebraic structure and the analyticity of the [Formula: see text]-system — the full system of Baxter [Formula: see text]-functions of the underlying integrable model. The algebraic structure of the [Formula: see text]-system is entirely based on (super)symmetry of the model and is efficiently described by Wronskian formulas for [Formula: see text]-functions organized into the Hasse diagram. When supplemented with analyticity conditions on [Formula: see text]-functions, it fixes completely the set of physical solutions for the spectrum of an integrable model. First, we demonstrate the spectral equations on the example of [Formula: see text] and [Formula: see text] Heisenberg (super)spin chains. Supersymmetry [Formula: see text] occurs as a simple “rotation” of the Hasse diagram for a [Formula: see text] system. Then we apply this method to the spectral problem of [Formula: see text]/CFT4-duality, describing the QSC formalism. The main difference with the spin chains consists in more complicated analyticity constraints on [Formula: see text]-functions which involve an infinitely branching Riemann surface and a set of Riemann–Hilbert conditions. As an example of application of QSC, we consider a special double scaling limit of [Formula: see text]-twisted [Formula: see text] SYM, combining weak coupling and strong imaginary twist. This leads to a new type of non-unitary CFT dominated by particular integrable, and often computable, 4D fishnet Feynman graphs. For the simplest of such models — the bi-scalar theory — the QSC degenerates into the [Formula: see text]-system for integrable non-compact Heisenberg spin chain with conformal, [Formula: see text] symmetry. We describe the QSC derivation of Baxter equation and the quantization condition for particular fishnet graphs — wheel graphs, and review numerical and analytic results for them.