Implications of Conformal Invariance in Field Theories for General Dimensions

Abstract

The requirements of conformal invariance for two and three point functions for general dimension $d$ on flat space are investigated. A compact group theoretic construction of the three point function for arbitrary spin fields is presented and it is applied to various cases involving conserved vector operators and the energy momentum tensor. The restrictions arising from the associated conservation equations are investigated. It is shown that there are, for general $d$, three linearly independent conformal invariant forms for the three point function of the energy momentum tensor, although for $d=3$ there are two and for $d=2$ only one. The form of the three point function is also demonstrated to simplify considerably when all three points lie on a straight line. Using this the coefficients of the conformal invariant point functions are calculated for free scalar and fermion theories in general dimensions and for abelian vector fields when $d=4$. Ward identities relating three and two point functions are also discussed. This requires careful analysis of the singularities in the short distance expansion and the method of differential regularisation is found convenient. For $d=4$ the coefficients appearing in the energy momentum tensor three point function are related to the coefficients of the two possible terms in the trace anomaly for a conformal theory on a curved space background.

46 pages (plain TeX with use of AMS fonts), DAMTP/93-31 (minor corrections and additions)

References in corpus (1)

• Covariant Perturbation Theory (IV). Third Order in the Curvature

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• More on Heavy-Light Bootstrap up to Double-Stress-Tensor
• Grassmann-odd three-point functions of conserved supercurrents in 3D $\mathcal{N}=1$ SCFT
• Non-perturbative unitarity constraints on the ratio of shear viscosity to entropy density in UV complete theories with a gravity dual
• Scale and Conformal Invariance on (A)dS
• A holographic perspective on non-relativistic conformal defects
• Unparticle Self-Interactions at the Large Hadron Collider
• Partition functions and entanglement entropy: Weyl graviton and conformal higher spin fields
• On Scale Versus Conformal Symmetry in Turbulence
• Conformal anomalies and the Einstein Field Equations
• Instantons and Entanglement Entropy
• Approximate Symmetries in $d=4$ CFTs with an Einstein Gravity Dual
• On the Kähler-Hodge structure of superconformal manifolds
• Conformal a-charge, correlation functions and conical defects
• On correlation functions of higher-spin currents in arbitrary dimensions $d>3$
• On the vector conformal models in an arbitrary dimension
• B-type anomaly coefficients for the D3-D5 domain wall
• AdS backgrounds and induced gravity
• The Quaternionic Geometry of 4D Conformal Field Theory
• Correlation Functions of Sp(2n) Invariant Higher-Spin Systems
• Complexity from Spinning Primaries
• Free energy dependence on spatial geometry for (2+1)-dimensional QFTs
• From Schwinger to Wightman: all conformal 3-point functions in momentum space
• Bounds on the Conformal Scale of a Minimally Coupled Dilaton and Multi-Leptonic Signatures at the LHC
• BICEP2 and the Central Charge of Holographic Inflation
• Hitchhiker's Guide to the Swampland: The Cosmologist's Handbook to the string-theoretical Swampland Programme
• Endpoint contributions to excited-state modular Hamiltonians
• Boundary Weyl anomaly of $\mathcal{N}=(2,2)$ superconformal models
• Deconstructing Conformal Blocks in 4D CFT
• Hidden Conformal Symmetry of the Discrete Series Scalars in dS${}_2$
• Higher-Derivative Supergravity, AdS${}_4$ Holography, and Black Holes
• The gravity dual of Lorentzian OPE blocks
• Higher-Curvature Gravity, Black Holes and Holography
• Anomalies from correlation functions in defect conformal field theory
• The Goldstone Equivalence Theorem and AdS/CFT
• Conformal Symmetry and the Three Point Function for the Gravitational Axial Anomaly
• Analytic constraints on the energy-momentum tensor in conformal field theories
• A Surprising Similarity Between Holographic CFTs and a Free Fermion in $(2+1)$ Dimensions
• Ising Cosmology
• Entanglement entropy of local gravitational quenches
• Introduction to Gauge/Gravity Duality (TASI Lectures 2017)
• Lectures on chiral algebras of $\mathcal{N}⩾2$ superconformal field theories
• Area law of connected correlation function in higher dimensional conformal field theory
• Shape dependence of mutual information in the OPE limit: linear responses
• Exact Results in Supersymmetric Gauge Theories
• Harmonic Analysis in d-dimensional Superconformal Field Theory
• Recurrence Relations for Finite-Temperature Correlators via AdS${}_2$/CFT${}_1$
• Spinors and conformal correlators
• Perturbative Methods for Superconformal Quantum Field Theories in String - Gauge Theory Dualities
• Holographic RG Flows from Quasi-Topological Gravity
• Three-point functions of higher-spin supercurrents in 4D $\mathcal{N}=1$ SCFT: general formalism for arbitrary superspins
• Boundary Conformal Field Theory at Large Charge
• Is the EMI model a QFT? An inquiry on the space of allowed entropy functions
• AdS/CFT correspondence via R-current correlation functions revisited
• Undecidability in Spacetime Geometry via the AdS/CFT Correspondence
• Six-dimensional $\mathcal{N}=(2,0)$ Conformal Superspace
• B-type anomaly coefficients for the D3-D7 domain wall
• Superconformal field theory in three dimensions: Correlation functions of conserved currents
• On Gluing CFT Correlators to Higher-Spin Amplitudes in AdS
• Partition functions on squashed seven-spheres and holography
• On the Stress Tensor Light-ray Operator Algebra
• Superspace invariants and correlators in 4d $\mathcal{N}=1$ superconformal field theories
• Regularization of energy-momentum tensor correlators and parity-odd terms
• Disks globally maximize the entanglement entropy in $2+1$ dimensions
• Searching for strongly coupled AdS matter with multi-trace deformations
• Analytic Studies of Fermions in the Conformal Bootstrap
• The Cubic Fixed Point at Large $N$
• Three-point functions of fermionic higher-spin currents in 4D conformal field theory
• Shape Deformations of Charged Rényi Entropies from Holography
• Three-point functions of semiclassical string states and conserved currents in AdS_4 x CP^3
• Three-point functions of higher-spin spinor current multiplets in $\mathcal{N}=1$ superconformal theory
• Bootstrap and collider physics of parity violating conformal field theories in $d=3$
• Three-point current correlation functions as probes of Effective Conformal Theories
• Holographic Wilson Loop One-point Functions in ABJM Theory
• Light cone OPE in a CFT with lowest twist scalar primary
• Constructive approach to solution of the conservation condition for conformal higher spin tree-point correlation function with equal spins
• The Conformal Anomaly Action to Fourth Order (4T) in $d=4$ in Momentum Space
• A new energy bound for Einstein-Scalar theory in AlAdS${}_4$ and holographic bound for deformed CFT${}_3$
• Estimation theory and gravity