In this paper we consider systems of quantum particles in the $4d$ Euclidean space which enjoy conformal symmetry. The algebraic relations for conformal-invariant combinations of positions and momenta are used to construct a solution of the Yang-Baxter equation in the unitary irreducibile representations of the principal series $Δ=2+iν$ for any left/right spins $\ell,\dot{\ell}$ of the particles. Such relations are interpreted in the language of Feynman diagrams as integral \emph{star-triangle} identites between propagators of a conformal field theory. We prove the quantum integrability of a spin chain whose $k$-th site hosts a particle in the representation $(Δ_k,\ell_k, \dot{ \ell}_k)$ of the conformal group, realizing a spinning and inhomogeneous version of the quantum magnet used to describe the spectrum of the bi-scalar Fishnet theories. For the special choice of particles in the scalar $(1,0,0)$ and fermionic $(3/2,1,0)$ representation the transfer matrices of the model are Bethe-Salpeter kernels for the double-scaling limit of specific two-point correlators in the $γ$-deformed $\mathcal{N}=4$ and $\mathcal{N}=2$ supersymmetric theories.
We address the question of perturbative consistency in the scalar fishnet models presented by Caetano, Gurdogan and Kazakov\cite{Gurdogan:2015csr, Caetano:2016ydc}. We argue that their 3-dimensional $ϕ^{6}$ fishnet model becomes perturbatively stable under renormalization in the large $N$ limit, in contrast to what happens in their 4-dimensional $ϕ^{4}$ fishnet model, in which double trace terms are known to be generated by the RG flow. We point out that there is a direct way to modify this second theory that protects it from such corrections. Additionally, we observe that the 6-dimensional $ϕ…
We introduce bi-fermion fishnet theories, a class of models describing integrable sectors of four-dimensional gauge theories with non-maximal supersymmetry. Bi-fermion theories are characterized by a single complex scalar field and two Weyl fermions interacting only via chiral Yukawa couplings. The latter generate oriented Feynman diagrams forming hexagonal lattices, whose fishnet structure signals an underlying integrability that we exploit to compute anomalous dimensions of BMN-vacuum operators. Furthermore, we investigate Lunin-Maldacena deformations of $\mathcal{N}=2$ superconformal field …
We compute explicitly the two-dimensional version of Basso-Dixon type integrals for the planar four-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.
We consider four-point integrals arising in the planar limit of the conformal "fishnet" theory in four dimensions. They define a two-parameter family of higher-loop Feynman integrals, which extend the series of ladder integrals and were argued, based on integrability and analyticity, to admit matrix-model-like integral and determinantal representations. In this paper, we prove the equivalence of all these representations using exact summation and integration techniques. We then analyze the large-order behaviour, corresponding to the thermodynamic limit of a large fishnet graph. The saddle-poin…
We consider the continuum limit of 4d planar fishnet diagrams using integrable spin chain methods borrowed from the $\mathcal{N}=4$ 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 suggest…
In this paper we consider a conformal invariant chain of $L$ sites in the unitary irreducible representations of the group $SO(1,5)$. The $k$-th site of the chain is defined by a scaling dimension $Δ_k$ and spin numbers $\frac{\ell_k}{2}$, $\frac{\dot{\ell}_k}{2}$. The model with open and fixed boundaries is shown to be integrable at the quantum level and its spectrum and eigenfunctions are obtained by separation of variables. The transfer matrices of the chain are graph-builder operators for the spinning and inhomogeneous generalization of squared-lattice "fishnet" integrals on the disk. As s…
Four-dimensional conformal fishnet theory is an integrable scalar theory which arises as a double scaling limit of $γ$-deformed maximally supersymmetric Yang-Mills. We give a perturbative reformulation of $γ$-deformed super-Yang-Mills theory in twistor space, and implement the double scaling limit to obtain a twistor description of conformal fishnet theory. The conformal fishnet theory retains an abelian gauge symmetry on twistor space which is absent in space-time, allowing us to obtain cohomological formulae for scattering amplitudes that manifest conformal invariance. We study various class…
We discuss the properties of scattering amplitudes in a conformal bi-scalar fishnet theory that previously appeared in the study of integrable deformations of $\mathcal N=4$ SYM. In distinction with the latter theory, the scattering amplitudes in the conformal fishnet theory do not suffer from infrared divergences and possess enhanced symmetries. We demonstrate that for the simplest four-particle scattering the single- and double-trace partial amplitudes can be computed for arbitrary coupling in the leading color approximation. We also review the invariance of single-trace components of multi-…
We review the quantum spectral curve (QSC) formalism for anomalous dimensions of planar ${\cal\ N}=4$ SYM, including its $γ$-deformation. Leaving aside its derivation, we concentrate on formulation of the "final product" in its most general form: a minimal set of assumptions about the algebraic structure and the analyticity of the $Q$-system -- the full system of Baxter $Q$-functions of the underlying integrable model. The algebraic structure of the $Q$-system is entirely based on (super)symmetry of the model and is efficiently described by Wronskian formulas for $Q$-functions organized into t…
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 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 N=4 SYM - the fishnet 4D CFT. We develop the functional SoV formalism in this theory, which allows us to compute non-perturbatively some nont…
We study integrability of fishnet-type Feynman graphs arising in planar four-dimensional bi-scalar chiral theory recently proposed in arXiv:1512.06704 as a special double scaling limit of gamma-deformed $\mathcal{N}=4$ SYM theory. We show that the transfer matrix "building" the fishnet graphs emerges from the $R-$matrix of non-compact conformal $SU(2,2)$ Heisenberg spin chain with spins belonging to principal series representations of the four-dimensional conformal group. We demonstrate explicitly a relationship between this integrable spin chain and the Quantum Spectral Curve (QSC) of $\mathc…
We propose a $D$-dimensional generalization of $4D$ bi-scalar conformal quantum field theory recently introduced by Gürdogan and one of the authors as a strong-twist double scaling limit of $γ$-deformed $\mathcal{N}=4$ SYM theory. Similarly to the $4D$ case, this D-dimensional CFT is also dominated by "fishnet" Feynman graphs and is integrable in the planar limit. The dynamics of these graphs is described by the integrable conformal $SO(D+1,1)$ spin chain. In $2D$ it is the analogue of L. Lipatov's $SL(2,\mathbb{C})$ spin chain for the Regge limit of $QCD$, but with the spins $s=1/4$ instead o…
A general class of rotating closed string solutions in AdS_5 x S^5 is shown to be described by a Neumann-Rosochatius one-dimensional integrable system. The latter represents an oscillator on a sphere or a hyperboloid with an additional ``centrifugal'' potential. We expect that the reduction of the AdS_5 x S^5 sigma model to the Neumann-Rosochatius system should have further generalizations and should be useful for uncovering new relations between integrable structures on the two sides of the AdS/CFT duality. We find, in particular, new circular rotating string solutions with two AdS_5 and thre…
Integrable spinning extension of a free particle on 2-sphere is constructed in which spin degrees of freedom are represented by a 3-vector obeying the Bianchi type-V algebra. Generalizations involving a scalar potential giving rise to two quadratic constants of the motion, or external field of the Dirac monopole, or the motion on the group manifold of SU(2) are built. A link to the model of a relativistic spinning particle propagating on the near horizon 7d Myers-Perry black hole background is considered. Implications of the construction in this work for the D(2,1;a) superconformal mechanics a…
In this introductory review we discuss dynamical tests of the AdS_5 x S^5 string/N=4 super Yang-Mills duality. After a brief introduction to AdS/CFT we argue that semiclassical string energies yield information on the quantum spectrum of the string in the limit of large angular momenta on the S^5. The energies of the folded and circular spinning string solutions rotating on a S^3 within the S^5 are derived, which yield all loop predictions for the dual gauge theory scaling dimensions. These follow from the eigenvalues of the dilatation operator of N=4 super Yang-Mills in a minimal SU(2) subsec…
We show that solitonic solutions of the classical string action on the AdS_5 x S^5 background that carry charges (spins) of the Cartan subalgebra of the global symmetry group can be classified in terms of periodic solutions of the Neumann integrable system. We derive equations which determine the energy of these solitons as a function of spins. In the limit of large spins J, the first subleading 1/J coefficient in the expansion of the string energy is expected to be non-renormalised to all orders in the inverse string tension expansion and thus can be directly compared to the 1-loop anomalous …
In this letter, we study the open spinning strings and their SYM duals. A new class of folded open spinning strings is found. At planar one-loop level in SYM, by solving the thermodynamic limit of the Bethe ansatz equations for an integrable open spin chain, we find good agreement with string theory predictions for energies of both circular and folded two-spin solutions. A universal relation between the open and closed spinning strings is verified in the spin chain approach.
We derive the coherent state representation of the integrable spin chain Hamiltonian with symmetry group SL(2,R). By passing to the continuum limit, we find a spin chain sigma model describing a string moving on the hyperboloid SL(2,R)/U(1). The same sigma model is found by considering strings rotating with large angular momentum in AdS_5xS^5. The spinning strings are identified with semiclassical coherent states built out of SL(2,R) spin chain states.
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 ${\mathfrak u}(1)$ sector and reproduce exactly the spectrum for the cases when it is known anal…
The sigma model describing closed strings rotating in AdS_3 x S^3 is known to reduce to the one-dimensional Neumann-Rosochatius integrable system. In this article we show that closed spinning strings in AdS_3 x S^3 x T^4 in the presence of NS-NS three-form flux can be described by an extension of the Neumann-Rosochatius system. We consider closed strings rotating with one spin in AdS_3 and two different angular momenta in S^3. For a class of solutions with constant radii we find the dependence of the classical energy on the spin and the angular momenta as an expansion in the square of the 't H…
We study the motion of a classical spinning particle (with spin degrees of freedom described by a vector of Grassmann variables) in higher-dimensional general rotating black hole spacetimes with a cosmological constant. In all dimensions n we exhibit n bosonic functionally independent integrals of spinning particle motion, corresponding to explicit and hidden symmetries generated from the principal conformal Killing--Yano tensor. Moreover, we demonstrate that in 4-, 5-, 6-, and 7-dimensional black hole spacetimes such integrals are in involution, proving the bosonic part of the motion integrab…
We construct nonrelativistic spinning string solutions corresponding to $ SU(1,2|3) $ Spin-Matrix theory (SMT) limit of strings in $ AdS_5 \times S^5 $. Considering various nonrelativistic spinning string configurations both in $ AdS_5 $ as well as $ S^5 $ we obtain corresponding dispersion relations in the strong coupling regime of SMT where the strong coupling ($ \sim \sqrt{\mathfrak{g}} $) corrections near the BPS bound have been estimated in the slow spinning limit of strings in $ AdS_5 $. We generalize our results explicitly by constructing three spin folded string configurations that has…
University of Oxford（牛津大学）；University of Cape Town（开普敦大学）；The National Institute for Theoretical and Computational Sciences（理论与计算科学国家研究所）；Institut National de la Recherche Scientifique（国家科学研究学院）
We apply recently developed integrable spin chain and dilatation operator techniques in order to compute the planar one-loop anomalous dimensions for certain operators containing a large number of scalar fields in N =4 Super Yang-Mills. The first set of operators, belonging to the SO(6) representations [J,L-2J,J], interpolate smoothly between the BMN case of two impurities (J=2) and the extreme case where the number of impurities equals half the total number of fields (J=L/2). The result for this particular [J,0,J] operator is smaller than the anomalous dimension derived by Frolov and Tseytlin…
We demonstrate that the recently found agreement between one-loop scaling dimensions of large dimension operators in N=4 gauge theory and energies of spinning strings on AdS_5 x S^5 extends to the eigenvalues of an infinite number of hidden higher commuting charges. This dynamical agreement is of a mathematically highly intricate and non-trivial nature. In particular, on the gauge side the generating function for the commuting charges is obtained by integrable quantum spin chain techniques from the thermodynamic density distribution function of Bethe roots. On the string side the generating fu…
There is remarkable relation between self-dual Yang-Mills and self-dual Einstein gravity in four Euclidean dimensions. Motivated by this we investigate the Spin(7) and G_2 invariant self-dual Yang-Mills equations in eight and seven Euclidean dimensions and search for their possible analogs in gravitational theories. The reduction of the self-dual Yang-Mills equations to one dimension results into systems of first order differential equations. In particular, the Spin(7)-invariant case gives rise to a 7-dimensional system which is completely integrable. The different solutions are classified in …
A short review of special relativistic dynamics describing a particle acted upon by an arbitrary conservative external force is presented. If the mass of the particle is zero and the force is central then the equations of motion turn out to be completely integrable. A well-known result. Hamiltonian flows on the twistor phase space T are constructed which, for conservative forces and value of the helicity equal to zero, reproduce equations of motion of the classical massless particle. For helicities different from zero the same hamiltonian flows produce equations of motion showing a curious "Zi…
We derive the coherent state representation of the integrable spin chain Hamiltonian with supersymmetry group $SU(1,1|2)$. By the use of a projected Hamiltonian onto bosonic states, we give explicitly the action of the Hamiltonian on $SU(2)\times SL(2)$ coherent states. Passing to the continuous limit, we find that the corresponding bosonic sigma model is the sum of the known SU(2) and SL(2) ones, and thus it gives a string spinning fast on $S^1_{ϕ_1}\times S^1_{ϕ_1}\times S^1_{ϕ_2}$ in $\rm{AdS}_5 \times S^5$. The full sigma model on the supercoset $SU(1,1|2)/SU(1|1)^2$ is given.
Conformal blocks are the central ingredient of the conformal bootstrap programme. We elaborate on our recent observation that uncovered a relation with wave functions of an integrable Calogero-Sutherland Hamiltonian in order to develop a systematic theory of conformal blocks. Our main goal here is to review central ingredients of the Heckman-Opdam theory for scattering states of Calogero-Sutherland models with special emphasis to the relation with scalar 4-point blocks. We will also discuss a number of direct consequences for conformal blocks, including a new series expansion for blocks of arb…
