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Vladimir Kazakov

Publications and source records attributed to Vladimir Kazakov.

At least 37 records · Page 2Linked to original sources
arXiv2019

Basso-Dixon Correlators in Two-Dimensional Fishnet CFT

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.

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arXiv2018

Bi-scalar integrable CFT at any dimension

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 of $s=0$. Generalizing recent $4D$ results of Grabner, Gromov, Korchemsky and one of the authors to any $D$ we compute exactly, at any coupling, a four point correlation function, dominated by the simplest fishnet graphs of cylindric topology, and extract from it exact dimensions of R-charge 2 operators with any spin and some of their OPE structure constants.

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arXiv2018

Exact Correlation Functions in Conformal Fishnet Theory

We compute exactly various 4-point correlation functions of shortest scalar operators in bi-scalar planar four-dimensional "fishnet" CFT. We apply the OPE to extract from these functions the exact expressions for the scaling dimensions and the structure constants of all exchanged operators with an arbitrary Lorentz spin. In particular, we determine the conformal data of the simplest unprotected two-magnon operator analogous to the Konishi operator, as well as of the one-magnon operator. We show that at weak coupling 4-point correlation functions can be systematically expanded in terms of harmonic polylogarithm functions and verify our results by explicit calculation of Feynman graphs at a few orders in the coupling. At strong coupling we obtain that the correlation functions exhibit the scaling behaviour typical for semiclassical description hinting at the existence of the holographic dual.

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arXiv2018

Chiral limit of N = 4 SYM and ABJM and integrable Feynman graphs

We consider a special double scaling limit, recently introduced by two of the authors, combining weak coupling and large imaginary twist, for the $γ$-twisted $\mathcal{N}=4$ SYM theory. We also establish the analogous limit for ABJM theory. The resulting non-gauge chiral 4D and 3D theories of interacting scalars and fermions are integrable in the planar limit. In spite of the breakdown of conformality by double-trace interactions, most of the correlators for local operators of these theories are conformal, with non-trivial anomalous dimensions defined by specific, integrable Feynman diagrams. We discuss the details of this diagrammatics. We construct the doubly-scaled asymptotic Bethe ansatz (ABA) equations for multi-magnon states in these theories. Each entry of the mixing matrix of local conformal operators in the simplest of these theories - the bi-scalar model in 4D and tri-scalar model in 3D - is given by a single Feynman diagram at any given loop order. The related diagrams are in principle computable, up to a few scheme dependent constants, by integrability methods (quantum spectral curve or ABA). These constants should be fixed from direct computations of a few simplest graphs. This integrability-based method is advocated to be able to provide information about some high loop order graphs which are hardly computable by other known methods. We exemplify our approach with specific five-loop graphs.

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arXiv2018

Quantum Spectral Curve of $γ$-twisted ${\cal N}=4$ SYM theory and fishnet CFT

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 the Hasse diagram. When supplemented with analyticity conditions on $Q$-functions, it fixes completely the set of physical solutions for spectrum of an integrable model. First we demonstrate the spectral equations on the example of $gl(N)$ and $gl(K|M)$ Heisenberg (super)spin chains. Supersymmetry $gl(K|M)$ occurs as a "rotation" of the Hasse diagram for a $gl(K+M)$ system. This picture helps us to construct the QSC formalism for spectrum of AdS$_5$/CFT$_4$-duality, with more complicated analyticity constraints on $Q$-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 $γ$-twisted ${\cal\ N}=4$ 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 $Q$-system for integrable non-compact Heisenberg spin chain with conformal, $SU(2,2)$ symmetry. We apply the QSC for derivation of Baxter equation and the quantization condition for particular, "wheel" fishnet graphs, and review numerical and analytic results for them.

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arXiv2017

Strongly γ-deformed N=4 SYM as an integrable CFT

We demonstrate by explicit multi-loop calculation that γ-deformed planar N=4 SYM, supplemented with a set of double-trace counter-terms, has two nontrivial fixed points in the recently proposed double scaling limit, combining vanishing 't Hooft coupling and large imaginary deformation parameter. We provide evidence that, at the fixed points, the theory is described by an integrable non-unitary four-dimensional CFT. We find a closed expression for the four-point correlation function of the simplest protected operators and use it to compute the exact conformal data of operators with arbitrary Lorentz spin. We conjecture that both conformal symmetry and integrability should survive in γ-deformed planar N=4 SYM for arbitrary values of the deformation parameters.

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arXiv2017

Integrability of Conformal Fishnet Theory

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 $\mathcal{N}=4$ SYM. Using QSC and spin chain methods, we construct Baxter equation for $Q-$functions of the conformal spin chain needed for computation of the anomalous dimensions of operators of the type $\text{tr}(ϕ_1^J)$ where $ϕ_1$ is one of the two scalars of the theory. For $J=3$ we derive from QSC a quantization condition that fixes the relevant solution of Baxter equation. The scaling dimensions of the operators only receive contributions from wheel-like graphs. We develop integrability techniques to compute the divergent part of these graphs and use it to present the weak coupling expansion of dimensions to very high orders. Then we apply our exact equations to calculate the anomalous dimensions with $J=3$ to practically unlimited precision at any coupling. These equations also describe an infinite tower of local conformal operators all carrying the same charge $J=3$. The method should be applicable for any $J$ and, in principle, to any local operators of bi-scalar theory. We show that at strong coupling the scaling dimensions can be derived from semiclassical quantization of finite gap solutions describing an integrable system of noncompact $SU(2,2)$ spins. This bears similarities with the classical strings arising in the strongly coupled limit of $\mathcal{N}=4$ SYM.

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arXiv2017

Yangian Symmetry for Fishnet Feynman Graphs

Various classes of fishnet Feynman graphs are shown to feature a Yangian symmetry over the conformal algebra. We explicitly discuss scalar graphs in three, four and six spacetime dimensions as well as the inclusion of fermions in four dimensions. The Yangian symmetry results in novel differential equations for these families of largely unsolved Feynman integrals. Notably, the considered fishnet graphs in three and four dimensions dominate the correlation functions and scattering amplitudes in specific double scaling limits of planar, gamma-twisted N=4 super Yang-Mills or ABJM theory. Consequently, the study of fishnet graphs allows us to get deep insights into the integrability of the planar AdS/CFT correspondence.

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arXiv2017

Yangian Symmetry for Bi-Scalar Loop Amplitudes

We establish an all-loop conformal Yangian symmetry for the full set of planar amplitudes in the recently proposed integrable bi-scalar field theory in four dimensions. This chiral theory is a particular double scaling limit of gamma-twisted weakly coupled N=4 SYM theory. Each amplitude with a certain order of scalar particles is given by a single fishnet Feynman graph of disc topology cut out of a regular square lattice. The Yangian can be realized by the action of a product of Lax operators with a specific sequence of inhomogeneity parameters on the boundary of the disc. Based on this observation, the Yangian generators of level one for generic bi-scalar amplitudes are explicitly constructed. Finally, we comment on the relation to the dual conformal symmetry of these scattering amplitudes.

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arXiv2016

New integrable non-gauge 4D QFTs from strongly deformed planar N=4 SYM

We consider the $γ$-deformed $\mathcal{N}=4$ SYM in the double scaling limit of large imaginary twists and small coupling, which discards the gauge fields and retains only certain Yukawa and scalar interactions with three arbitrary effective couplings. In the 't~Hooft limit, these 4D theories are integrable, with most of the correlators being conformal such that the whole arsenal of AdS/CFT integrability remains applicable. In particular, for one non-zero effective coupling, we obtain a QFT of two complex scalars with a chiral, quartic interaction. The BMN vacuum anomalous dimension is dominated in each non-zero loop order by a single "wheel" graph, in principle computable by integrability. Thus we also provide an explicit conjecture for the periods of double-wheel graphs with an arbitrary number of spokes.

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arXiv2016

T-system on T-hook: Grassmannian Solution and Twisted Quantum Spectral Curve

We propose an efficient grassmannian formalism for solution of bi-linear finite-difference Hirota equation (T-system) on T-shaped lattices related to the space of highest weight representations of $gl(K_1,K_2|M)$ superalgebra. The formalism is inspired by the quantum fusion procedure known from the integrable spin chains and is based on exterior forms of Baxter-like Q-functions. We find a few new interesting relations among the exterior forms of Q-functions and reproduce, using our new formalism, the Wronskian determinant solutions of Hirota equations known in the literature. Then we generalize this construction to the twisted Q-functions and demonstrate the subtleties of untwisting procedure on the examples of rational quantum spin chains with twisted boundary conditions. Using these observations, we generalize the recently discovered, in our paper with N. Gromov, AdS/CFT Quantum Spectral Curve for exact planar spectrum of AdS/CFT duality to the case of arbitrary Cartan twisting of AdS$_5\times$S$^5$ string sigma model. Finally, we successfully probe this formalism by reproducing the energy of gamma-twisted BMN vacuum at single-wrapping orders of weak coupling expansion.

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arXiv2015

Quantum spectral curve for arbitrary state/operator in AdS$_5$/CFT$_4$

We give a derivation of quantum spectral curve (QSC) - a finite set of Riemann-Hilbert equations for exact spectrum of planar N=4 SYM theory proposed in our recent paper Phys.Rev.Lett. 112 (2014). We also generalize this construction to all local single trace operators of the theory, in contrast to the TBA-like approaches worked out only for a limited class of states. We reveal a rich algebraic and analytic structure of the QSC in terms of a so called Q-system -- a finite set of Baxter-like Q-functions. This new point of view on the finite size spectral problem is shown to be completely compatible, though in a far from trivial way, with already known exact equations (analytic Y-system/TBA, or FiNLIE). We use the knowledge of this underlying Q-system to demonstrate how the classical finite gap solutions and the asymptotic Bethe ansatz emerge from our formalism in appropriate limits.

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arXiv2015

QCD Pomeron from AdS/CFT Quantum Spectral Curve

Using the methods of the recently proposed Quantum Spectral Curve (QSC) originating from integrability of ${\cal N}=4$ Super--Yang-Mills theory we analytically continue the scaling dimensions of twist-2 operators and reproduce the so-called pomeron eigenvalue of the Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation. Furthermore, we recovered the Faddeev-Korchemsky Baxter equation for Lipatov's spin chain and also found its generalization for the next-to-leading order in the BFKL scaling. Our results provide a non-trivial test of QSC describing the exact spectrum in planar ${\cal N}=4$ SYM at infinitely many loops for a highly nontrivial non-BPS quantity and also opens a way for a systematic expansion in the BFKL regime.

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arXiv2015

Finite Size Spectrum of SU(N) Principal Chiral Field from Discrete Hirota Dynamics

Using recently proposed method of discrete Hirota dynamics for integrable (1+1)D quantum field theories on a finite space circle of length L, we derive and test numerically a finite system of nonlinear integral equations for the exact spectrum of energies of SU(N)xSU(N) principal chiral field model as functions of m L, where m is the mass scale. We propose a determinant solution of the underlying Y-system, or Hirota equation, in terms of determinants (Wronskians) of NxN matrices parameterized by N-1 functions of the spectral parameter, with the known analytical properties at finite L. Although the method works in principle for any state, the explicit equations are written for states in the U(1) sector only. For N>2, we encounter and clarify a few subtleties in these equations related to the presence of bound states, absent in the previously considered N=2 case. In particular, we solve these equations numerically for a few low-lying states for N=3 in a wide range of m L.

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arXiv2013

Two-point correlator of twist-2 light-ray operators in N=4 SYM in BFKL approximation

We generalize local operators of the leading twist-2 of N=4 SYM theory to the case of complex Lorentz spin j using principal series representation of sl(2,R). We give the direct computation of correlation function of two such non-local operators in the BFKL regime when j -> 1. The correlator appears to have the expected conformal coordinate dependence governed by the anomalous dimension of twist-2 operator in NLO BFKL approximation predicted by Kotikov and Lipatov.

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arXiv2013

Quantum spectral curve for AdS_5/CFT_4

We present a new formalism, alternative to the old TBA-like approach, for solution of the spectral problem of planar N = 4 SYM. It takes a concise form of a non-linear matrix Riemann-Hilbert problem in terms of a few Q-functions. We demonstrate the formalism for two types of observables - local operators at weak coupling and cusped Wilson lines in a near BPS limit.

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