Conserved Currents, Consistency Relations and Operator Product Expansions in the Conformally Invariant O(N) Vector Model

Abstract

We discuss conserved currents and operator product expansions (OPE's) in the context of a $O(N)$ invariant conformal field theory. Using OPE's we find explicit expressions for the first few terms in suitable short-distance limits for various four-point functions involving the fundamental $N$-component scalar field ${\varphi }^{\alpha }(x)$, $\alpha =1,2,..,N$. We propose an alternative evaluation of these four-point functions based on graphical expansions. Requiring consistency of the algebraic and graphical treatments of the four-point functions we obtain the values of the dynamical parameters in either a free theory of $N$ massless fields or a non-trivial conformally invariant $O(N)$ vector model in $2<d<4$, up to next-to-leading order in a $1/N$ expansion. Our approach suggests an interesting duality property of the critical $O(N)$ invariant theory. Also, solving our consistency relations we obtain the next-to-leading order in $1/N$ correction for $C_T$ which corresponds to the normalisation of the energy momentum tensor two-point function.

45 pages Latex + 2 uuencoded PostScript figures, DAMTP 94/12

Cited by in corpus (113)

• AdS Dual of the Critical O(N) Vector Model
• The Conformal Bootstrap: Theory, Numerical Techniques, and Applications
• Solving the 3D Ising Model with the Conformal Bootstrap
• Conformal Four Point Functions and the Operator Product Expansion
• Solving the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents
• Nonperturbative Formulas for Central Functions of Supersymmetric Gauge Theories
• Bootstrapping the O(N) Vector Models
• Holography in 4D (Super) Higher Spin Theories and a Test via Cubic Scalar Couplings
• Conserved Currents and the Energy Momentum Tensor in Conformally Invariant Theories for General Dimensions
• Correlation Functions of Large N Chern-Simons-Matter Theories and Bosonization in Three Dimensions
• A Mellin space approach to the conformal bootstrap
• Bootstrapping the O(N) Archipelago
• Conformal Bootstrap in Mellin Space
• Critical $O(N)$ Models in $6-\epsilon$ Dimensions
• Quartic AdS Interactions in Higher-Spin Gravity from Conformal Field Theory
• Holography of the N=1 Higher-Spin Theory on AdS4
• Conformal Field Theories in Fractional Dimensions
• Operator Product Expansion of the Lowest Weight CPOs in N=4 SYM_4 at Strong Coupling
• From Free Fields to $AdS$
• The Operator Product Expansion of N=4 SYM and the 4-point Functions of Supergravity
• A Study of Quantum Field Theories in AdS at Finite Coupling
• SL(2,Z) Action on Three-Dimensional CFTs and Holography
• From Free Fields to $AdS$ -- III
• Scattering in Anti-de Sitter Space and Operator Product Expansion
• Universal conductivity and central charges
• Three Loop Analysis of the Critical $O(N)$ Models in $6-\epsilon$ Dimensions
• On Higher Spin Gauge Theory and the Critical O(N) Model
• Evaluating the AdS dual of the critical O(N) vector model
• Conformal Partial Wave Expansions for N=4 Chiral Four Point Functions
• Bootstrapping $O(N)$ Vector Models in $4<d<6$
• 3-D Interacting CFTs and Generalized Higgs Phenomenon in Higher Spin Theories on AdS
• A universal feature of CFT Renyi entropy
• Fractal dimensions of self-avoiding walks and Ising high-temperature graphs in 3D conformal bootstrap
• The critical O(N) CFT: Methods and conformal data
• Eikonalization of Conformal Blocks
• Seeking Fixed Points in Multiple Coupling Scalar Theories in the $\epsilon$ Expansion
• The $O(N)$ Model in $4<d<6$: Instantons and Complex CFTs
• Mellin space bootstrap for global symmetry
• Notes on Spinning Operators in Fermionic CFT
• Spinning Mellin Bootstrap: Conformal Partial Waves, Crossing Kernels and Applications
• Bootstrapping Mixed Correlators in the Five Dimensional Critical O(N) Models
• On $C_J$ and $C_T$ in Conformal QED
• Implications of N=1 Superconformal Symmetry for Chiral Fields
• Shadows, currents and AdS fields
• The analytic structure of conformal blocks and the generalized Wilson-Fisher fixed points
• Theory of higher spin tensor currents and central charges
• Gauge invariant two-point vertices of shadow fields, AdS/CFT, and conformal fields
• Ordinary-derivative formulation of conformal totally symmetric arbitrary spin bosonic fields
• On RG Flow of tau_{RR} for Supersymmetric Field Theories in Three-Dimensions
• On critical exponents without Feynman diagrams
• Conformal Bootstrap in Three Dimensions?
• $C_T$ for Non-unitary CFTs in Higher Dimensions
• The simple scheme for the calculation of the anomalous dimensions of composite operators in the 1/N expansion
• Holographic Normal Ordering and Multi-particle States in the AdS/CFT Correspondence
• Monodromy Defects from Hyperbolic Space
• Analytic and Numerical Bootstrap of CFTs with $O(m)\times O(n)$ Global Symmetry in 3D
• On $C_J$ and $C_T$ in the Gross-Neveu and $O(N)$ Models
• Geodesic bulk diagrams on the Bruhat-Tits tree
• Large $N_f$ quantum field theory
• Interactions in Higher-Spin Gravity: a Holographic Perspective
• More on analytic bootstrap for O(N) models
• A plane defect in the 3d O$(N)$ model
• Dynamics of Finite-Temperature CFTs from OPE Inversion Formulas
• Conserved current correlators of conformal field theories in 2+1 dimensions
• Double-Trace Deformations of Conformal Correlations
• Ordinary-derivative formulation of conformal low-spin fields
• $C_T$ for higher derivative conformal fields and anomalies of (1,0) superconformal 6d theories
• An alternative to diagrams for the critical O(N) model: dimensions and structure constants to order $1/N^2$
• On the stability problem in the O(N) nonlinear sigma model
• Anomalous conformal currents, shadow fields and massive AdS fields
• A note on the analyticity of AdS scalar exchange graphs in the crossed channel
• Partial wave expansion and Wightman positivity in conformal field theory
• Holography, Duality and Higher-Spin Theories
• Vector model in various dimensions
• On conformal correlators and blocks with spinors in general dimensions
• On Holography in General Background and the Boundary Effective Action from AdS to dS
• Charting the space of 3D CFTs with a continuous global symmetry
• Thermal one-point functions and single-valued polylogarithms
• Higher Spins in Hyper-Superspace
• Higher Spins in Hyperspace
• More on the cubic versus quartic interaction equivalence in the $O(N)$ model
• Double-Trace Flows and the Swampland
• Finite-Size Effects and Operator Product Expansions in a CFT for d>2
• A universal feature of charged entanglement entropy
• Stress tensor and current correlators of interacting conformal field theories in 2+1 dimensions: Fermionic Dirac matter coupled to U(1) gauge field
• Gauge invariant approach to low-spin anomalous conformal currents and shadow fields
• The Epsilon-Expansion from Conformal Field Theory
• Aspects of the conformal operator product expansion in AdS/CFT correspondence
• Testing the Boson/Fermion Duality on the Three-Sphere
• The free $\sigma$CFTs
• Collusion of Interactions and Disorder at the Superfluid-Insulator Transition: A Dirty 2d Quantum Critical Point
• On (Un)Broken Higher-Spin Symmetry in Vector Models
• Unparticle self-interactions
• Symplectic critical models in $6+\epsilon$ dimensions
• On the Free-Energy of Three-Dimensional CFTs and Polylogarithms
• Singleton deformation of higher-spin theory and the phase structure of the three-dimensional O(N) vector model
• Conformal bounds in three dimensions from entanglement entropy
• Boundary conformal field theory at the extraordinary transition: The layer susceptibility to $O(\epsilon )$
• The background field method and critical vector models
• AdS/CFT in Fractional Dimension and Higher Spin Gravity at One Loop
• Long-range fermions and critical dualities
• Vector boson excitations near deconfined quantum critical points
• CFT data in the Gross-Neveu model
• Interpolating between $a$ and $F$
• Light-cone AdS/CFT-adapted approach to AdS fields/currents, shadows, and conformal fields
• Conformal correlators in the critical $O(N)$ vector model
• Critical long-range vector model in the UV
• Long-Range Vector Models at Large N
• The thermal representation of conformal ladder integrals
• On overlapping Feynman (sub)graphs
• Universal properties of the U(1) current at deconfined quantum critical points: comparison with predictions from gauge/gravity duality
• The Cubic Fixed Point at Large $N$
• Convexity, large charge and the large-N phase diagram of the ${\phi }^4$ theory