Anomalies, Conformal Manifolds, and Spheres

Abstract

The two-point function of exactly marginal operators leads to a universal contribution to the trace anomaly in even dimensions. We study aspects of this trace anomaly, emphasizing its interpretation as a sigma model, whose target space M is the space of conformal field theories (a.k.a. the conformal manifold). When the underlying quantum field theory is supersymmetric, this sigma model has to be appropriately supersymmetrized. As examples, we consider in some detail N=(2,2) and N=(0,2) supersymmetric theories in d=2 and N=2 supersymmetric theories in d=4. This reasoning leads to new information about the conformal manifolds of these theories, for example, we show that the manifold is Kahler-Hodge and we further argue that it has vanishing Kahler class. For N=(2,2) theories in d=2 and N=2 theories in d=4 we also show that the relation between the sphere partition function and the Kahler potential of M follows immediately from the appropriate sigma models that we construct. Along the way we find several examples of potential trace anomalies that obey the Wess-Zumino consistency conditions, but can be ruled out by a more detailed analysis.

harvmac, 38 pages; references added and small clarifications

References in corpus (17)

• Symmetries and Strings in Field Theory and Gravity
• A Quartic Conformally Covariant Differential Operator for Arbitrary Pseudo-Riemannian Manifolds (Summary)
• A topologically twisted index for three-dimensional supersymmetric theories
• Sphere Partition Functions and the Zamolodchikov Metric
• Comments on the Fayet-Iliopoulos Term in Field Theory and Supergravity
• The equivariant A-twist and gauged linear sigma models on the two-sphere
• New supersymmetric higher-derivative couplings: Full N=2 superspace does not count!
• $t{t}^{\ast }$ equations, localization and exact chiral rings in 4d N=2 SCFTs
• On exact correlation functions in SU(N) $\mathcal{N}=2$ superconformal QCD
• Exact correlation functions in SU(2) N=2 superconformal QCD
• Comments on N=(2,2) Supersymmetry on Two-Manifolds
• Structures on the Conformal Manifold in Six Dimensional Theories
• A challenge to the $a$-theorem in six dimensions
• N=4 Supersymmetric AdS5 Vacua and their Moduli Spaces
• Accidents in (0,2) Landau-Ginzburg theories
• Compact Conformal Manifolds
• Superspace formulation of the local RG equation

Cited by in corpus (83)

• Anomalies in the Space of Coupling Constants and Their Dynamical Applications II
• Anomalies in the Space of Coupling Constants and Their Dynamical Applications I
• $\mathcal{N}=4$ Super-Yang-Mills Correlators at Strong Coupling from String Theory and Localization
• Correlation Functions of Coulomb Branch Operators
• Bootstrapping $\mathcal{N}=2$ chiral correlators
• Renormalised CFT 3-point functions of scalars, currents and stress tensors
• Localized magnetic field in the $O(N)$ model
• The Exact Bremsstrahlung Function in N=2 Superconformal Field Theories
• New Modular Invariants in $\mathcal{N}=4$ Super-Yang-Mills Theory
• A limit for large $R$-charge correlators in $\mathcal{N}=2$ theories
• Anomalies of Duality Groups and Extended Conformal Manifolds
• Tackling the SDC in AdS with CFTs
• Large N Correlation Functions in Superconformal Field Theories
• Renormalized entanglement entropy
• Spin Impurities, Wilson Lines and Semiclassics
• Exact Bremsstrahlung and Effective Couplings
• Large-N correlation functions in $\mathcal{N}=2$ superconformal QCD
• Surface Defect, Anomalies and $b$-Extremization
• Extremal Correlators and Random Matrix Theory
• Towards the Holographic Dual of N = 2 SYK
• On Higher-Spin Points and Infinite Distances in Conformal Manifolds
• Conformal manifolds: ODEs from OPEs
• $\mathcal{N}=2$ conformal gauge theories at large R-charge: the $SU(N)$ case
• From 3d duality to 2d duality
• Superconformal duality-invariant models and $\mathcal{N}=4$ SYM effective action
• Anomalies involving the space of couplings and the Zamolodchikov metric
• Supersymmetric localization in two dimensions
• Localization of twisted $\mathcal{N}=(0,2)$ gauged linear sigma models in two dimensions
• An introduction to supersymmetric field theories in curved space
• Shortening Anomalies in Supersymmetric Theories
• On the chiral algebra of Argyres-Douglas theories and S-duality
• On the realization of impossible anomalies
• Species scale, worldsheet CFTs and emergent geometry
• A brief review of the 2d/4d correspondences
• Graviton Scattering and a Sum Rule for the c Anomaly in 4D CFT
• Can we change $c$ in four-dimensional CFTs by exactly marginal deformations?
• On Newton-Cartan local renormalization group and anomalies
• Osborn Equation and Irrelevant Operators
• Moduli Anomalies and Local Terms in the Operator Product Expansion
• A-twisted correlators and Hori dualities
• How a-type anomalies can depend on marginal couplings
• Conformal Janus on Euclidean Sphere
• Conformal Symmetry and Composite Operators in the $O(N{)}^3$ Tensor Field Theory
• Type-B Anomaly Matching and the 6D (2,0) Theory
• Finiteness and the Emergence of Dualities
• (Mis-)Matching Type-B Anomalies on the Higgs Branch
• M5-branes on S^2 x M_4: Nahm's Equations and 4d Topological Sigma-models
• Non-compact duality, super-Weyl invariance and effective actions
• The Geometry of Gauged Linear Sigma Model Correlation Functions
• $\mathcal{N}=1$ Liouville SCFT in Four Dimensions
• Geometry of Conformal Manifolds and the Inversion Formula
• Conformal Contact Terms and Semi-Local Terms
• Categorical Equivalence and the Renormalization Group
• Gromov-Witten invariants and localization
• A CFT Distance Conjecture
• Emitted Radiation and Geometry
• Conformal Data of Fundamental Gauge-Yukawa Theories
• On the Flow of $\square R$ Weyl-Anomaly
• Conformal field theories on deformed spheres, anomalies, and supersymmetry
• Remarks on Berry Connection in QFT, Anomalies, and Applications
• Recent developments in 2d $\mathcal{N}=(2,2)$ supersymmetric gauge theories
• On genus one fibered Calabi-Yau threefolds with 5-sections
• On exact correlation functions of chiral ring operators in $2d$ $\mathcal{N}=(2,2)$ SCFTs via localization
• Conformal field theories and compact curves in moduli spaces
• On the Kähler-Hodge structure of superconformal manifolds
• Covariantly Constant Anomalies on Conformal Manifolds
• On the global moduli of Calabi-Yau threefolds
• Effective actions in supersymmetric gauge theories: heat kernels for non-minimal operators
• Boundary Weyl anomaly of $\mathcal{N}=(2,2)$ superconformal models
• Partition Functions of $\mathcal{N}=1$ Gauge Theories on $S^2\times {\mathbb{R}}_{\epsilon }^2$ and Duality
• Cancelling Weyl anomaly from position dependent coupling
• Weyl invariance, non-compact duality and conformal higher-derivative sigma models
• Nonabelian mirrors and Gromov-Witten invariants
• Correlation functions in massive Landau-Ginzburg orbifolds and tests of dualities
• Cluster-Enriched Yang-Baxter Equation from SUSY Gauge Theories
• The local renormalization of super-Yang-Mills theories
• Mirror Symmetry and Partition Functions
• Six-dimensional $\mathcal{N}=(2,0)$ Conformal Superspace
• Localizing non-linear $\mathcal{N}=(2,2)$ sigma model on $S^2$
• Global aspects of moduli spaces of 2d SCFTs
• Flavor deformations and supersymmetry enhancement in $4d$ $\mathcal{N}=2$ theories
• Supersymmetric localization: $\mathcal{N}=(2,2)$ theories on S${}^2$ and AdS${}_2$
• Revisiting the $k$-theorem with the ANEC