Integrability of Conformal Fishnet Theory

Abstract

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 $\text{tr}({\varphi }_1^J)$ where ${\varphi }_1$ 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.

71 pages, 14 figures

References in corpus (6)

• The Quantum Spectral Curve of the ABJM theory
• Gluing Ladders into Fishnets
• New Construction of Eigenstates and Separation of Variables for SU(N) Quantum Spin Chains
• The full Quantum Spectral Curve for $Ad{S}_4/CF{T}_3$
• RG stability of integrable fishnet models
• Noncompact sl(N) spin chains: BGG-resolution, Q-operators and alternating sum representation for finite dimensional transfer matrices

Cited by in corpus (58)

• $d$-dimensional SYK, AdS Loops, and $6j$ Symbols
• Harmonic Analysis and Mean Field Theory
• Strongly γ-deformed N=4 SYM as an integrable CFT
• Quantum Spectral Curve and Structure Constants in N=4 SYM: Cusps in the Ladder Limit
• Bi-scalar integrable CFT at any dimension
• Yangian Symmetry for Bi-Scalar Loop Amplitudes
• Basso-Dixon Correlators in Two-Dimensional Fishnet CFT
• Exact Correlation Functions in Conformal Fishnet Theory
• Exactly solvable magnet of conformal spins in four dimensions
• Hexagonalization of Correlation Functions II: Two-Particle Contributions
• Generalized Fishnets and Exact Four-Point Correlators in Chiral CFT${}_4$
• The Holographic Fishchain
• Classifying integrable spin-1/2 chains with nearest neighbour interactions
• Separation of Variables in AdS/CFT: Functional Approach for the Fishnet CFT
• Thermodynamic Bethe Ansatz for Fishnet CFT
• Continuum limit of fishnet graphs and AdS sigma model
• Quantum Fishchain in $Ad{S}_5$
• Exactly solvable single-trace four point correlators in $\chi$CFT${}_4$
• BFKL Spectrum of N=4 SYM: non-Zero Conformal Spin
• Hexagons and Correlators in the Fishnet Theory
• Surface operators in the 6d $\mathcal{N}=(2,0)$ theory
• Exploring the spectrum of planar $Ad{S}_4/CF{T}_3$ at finite coupling
• Colour-Twist Operators I: Spectrum and Wave Functions
• The Holographic Dual of Strongly γ-deformed N=4 SYM Theory: Derivation, Generalization, Integrability and Discrete Reparametrization Symmetry
• A review of the AdS/CFT Quantum Spectral Curve
• The One-Loop Spectral Problem of Strongly Twisted $\mathcal{N}$=4 Super Yang-Mills Theory
• Solving the Hagedorn temperature of AdS5/CFT4 via the Quantum Spectral Curve: Chemical potentials and deformations
• Mirror channel eigenvectors of the $d$-dimensional fishnets
• Open Fishchain in N=4 Supersymmetric Yang-Mills Theory
• The Loom for General Fishnet CFTs
• Exact scattering amplitudes in conformal fishnet theory
• Conformal Symmetry and Composite Operators in the $O(N{)}^3$ Tensor Field Theory
• Yangian Ward Identities for Fishnet Four-Point Integrals
• Cosmic Wheels: From integrability to the Galois coaction
• The Integrable (Hyper)eclectic Spin Chain
• Observations on BPS observables in 6d
• Hexagonalization of Fishnet integrals I: mirror excitations
• Ladder and zig-zag Feynman diagrams, operator formalism and conformal triangles
• ABJM quantum spectral curve and Mellin transform
• Instability of complex CFTs with operators in the principal series
• Twistor fishnets
• Combinatorial Solution of the Eclectic Spin Chain
• Checkerboard CFT
• Amplitudes in fishnet theories in diverse dimensions and Box ladder diagrams
• Integrability treatment of AdS/CFT orbifolds
• The fate of the Konishi multiplet in the β-deformed Quantum Spectral Curve
• Bootstrability in Defect CFT: Integrated Correlators and Sharper Bounds
• Light-cone limits of large rectangular fishnets
• Loop Operators in Three-Dimensional $\mathcal{N}=2$ Fishnet Theories
• Regge amplitudes in Generalized Fishnet and Chiral Fishnet Theories
• Fishnet four-point integrals: integrable representations and thermodynamic limits
• Correlation functions of determinant operators in conformal fishnet theory
• ABJM quantum spectral curve at twist 1: algorithmic perturbative solution
• Analytic continuation of Bethe energies and application to the thermodynamic limit of the $SL(2,\mathbb{C})$ non-compact spin chains
• A Large Twist Limit for Any Operator
• Yangian Symmetry in Five Dimensions
• BC-type open $SL(2,\mathbb{C})$ spin chain
• The Gelfand-Tsetlin basis for infinite-dimensional representations of $g{l}_n(\mathbb{C})$