Wess-Zumino Consistency Condition Research Articles

Broken scale invariance and the regularization of a conformal sector in gravity with Wess-Zumino actions

We elaborate on anomaly induced actions of the Wess-Zumino (WZ) form and their relation to the renormalized effective action, which is defined by an ordinary path integral over a conformal sector, in an external gravitational background. In anomaly - induced actions, the issue of scale breaking is usually not addressed, since these actions are obtained only by solving the trace anomaly constraint and are determined by scale invariant functionals. We investigate the changes induced in the structure of such actions once identified in dimensional renormalization (DR) when the ϵ=d−4→0 limit is accompanied by the dimensional reduction (DRed) of the field dependencies. We show that operatorial nonlocal modifications (∼□ϵ) of the counterterms are unnecessary to justify a scale anomaly. In this case, only the ordinary finite subtractions play a critical role in the determination of the scale breaking. This is illustrated for the WZ form of the effective action and its WZ consistency condition, as seen from a renormalization procedure. Logarithmic corrections from finite subtractions are also illustrated in a pure d=4 (cutoff) scheme. The interplay between two renormalization schemes, one based on dimensional regularization (DR) and the second on a cutoff in d=4, illustrates the ambiguities of DR in handling the quantum corrections in a curved background. Therefore, using DR in a curved background, the scale and trace anomalies can both be obtained by counterterms that are Weyl invariant only at d=4.

Weyl invariance, non-compact duality and conformal higher-derivative sigma models

We study a system of n Abelian vector fields coupled to frac{1}{2} n(n+1) complex scalars parametrising the Hermitian symmetric space {textsf{Sp}}(2n, {mathbb {R}})/ {textsf{U}}(n). This model is Weyl invariant and possesses the maximal non-compact duality group {textsf{Sp}}(2n, {mathbb {R}}). Although both symmetries are anomalous in the quantum theory, they should be respected by the logarithmic divergent term (the “induced action”) of the effective action obtained by integrating out the vector fields. We compute this induced action and demonstrate its Weyl and {textsf{Sp}}(2n, {mathbb {R}}) invariance. The resulting conformal higher-derivative sigma -model on {textsf{Sp}}(2n, {mathbb {R}})/ {textsf{U}}(n) is generalised to the cases where the fields take their values in (i) an arbitrary Kähler space; and (ii) an arbitrary Riemannian manifold. In both cases, the sigma -model Lagrangian generates a Weyl anomaly satisfying the Wess–Zumino consistency condition.

Covariantly constant anomalies on conformal manifolds

Operators with integer scaling dimensions in even-dimensional conformal field theories exhibit well-known type-B Weyl anomalies. In general, these anomalies depend non-trivially on exactly marginal couplings. We study the corresponding fully covariantised anomaly functional on conformal manifolds in several examples. We show that a natural consequence of the Wess-Zumino consistency condition is that the anomalies are covariantly constant with respect to the exactly marginal couplings. The argument is general and applies even when the conformal symmetry is spontaneously broken on moduli spaces of vacua.

Anomalies and supersymmetry

We revisit quantum field theory anomalies, emphasizing the interplay with diffeomorphisms and supersymmetry. The Ward identities of the latter induce Noether currents of all continuous symmetries, and we point out how these consistent currents are replaced by their covariant form through the appearance of the Bardeen-Zumino currents, which play a central role in our study. For supersymmetry Ward identities, two systematic methods for solving the Wess-Zumino consistency conditions are discussed: anomaly inflow and anomaly descent. The simplest inflows are from supersymmetric Chern-Simons actions in one dimension higher, which are used to supersymmetrize flavor anomalies in $d=4$ and, for $d=2$ $\mathcal{N}=(p,q)$, flavor anomalies with $p,q\leq 3$ and Lorentz-Weyl anomalies with $p,q\leq 6$. Finally, we extend the BRST algebra and the subsequent descent, a necessity for the diffeomorphism anomaly in retrospect. The same modification computes the supersymmetrized anomalies, and determines the above Chern-Simons actions when these exist.

Anomalous symmetries end at the boundary

A global symmetry of a quantum field theory is said to have an ’t Hooft anomaly if it cannot be promoted to a local symmetry of a gauged theory. In this paper, we show that the anomaly is also an obstruction to defining symmetric boundary conditions. This applies to Lorentz symmetries with gravitational anomalies as well. For theories with perturbative anomalies, we demonstrate the obstruction by analyzing the Wess-Zumino consistency conditions and current Ward identities in the presence of a boundary. We then recast the problem in terms of symmetry defects and find the same conclusions for anomalies of discrete and orientation-reversing global symmetries, up to the conjecture that global gravitational anomalies, which may not be associated with any diffeomorphism symmetry, also forbid the existence of boundary conditions. This conjecture holds for known gravitational anomalies in D ≤ 3 which allows us to conclude the obstruction result for D ≤ 4.

Consistency of supersymmetric \u2019t Hooft anomalies

We consider recent claims that supersymmetry is anomalous in the presence of a R-symmetry anomaly. We revisit arguments that such an anomaly in supersymmetry can be removed and write down an explicit counterterm that accomplishes it. Removal of the supersymmetry anomaly requires enlarging the corresponding current multiplet. As a consequence the Ward identities for other symmetries that are already anomalous acquire extra terms. This procedure can only be impeded when the choice of current multiplet is forced. We show how Wess-Zumino consistency conditions are modified when the anomaly is removed. Finally we check that the modified Wess-Zumino consistency conditions are satisfied, and supersymmetry unbroken, in an explicit one loop computation using Pauli-Villars regulators. To this end we comment on how to use Pauli-Villars to regulate correlators of components of (super)current multiplets in a manifestly supersymmetric way.

How a-Type Anomalies Can Depend on Marginal Couplings.

Even dimensional defects and boundaries in conformal field theory support type a anomalies on their world volume. We show that the one-point functions of marginal operators, in the presence of defects and boundaries, are anomalous, and that the Wess-Zumino consistency condition relates them to the derivative of the a anomaly with respect to the marginal coupling. We also argue that the constant term F for odd dimensional surfaces can depend on marginal parameters.

Torsion and anomalies in the warped limit of Lifschitz theories

We describe the physics of fermionic Lifschitz theories once the anisotropic scaling exponent is made arbitrarily small. In this limit the system acquires an enhanced (Carrollian) boost symmetry. We show, both through the explicit computation of the path integral Jacobian and through the solution of the Wess-Zumino consistency conditions, that the translation symmetry in the anisotropic direction becomes anomalous. This turns out to be a mixed anomaly between boosts and translations. In a Newton-Cartan formulation of the space-time geometry such anomaly is sourced by torsion. We use these results to give an effective field theory description of the anomalous transport coefficients, which were originally computed through Kubo formulas in [1]. Along the way we provide a link with warped CFTs.

Supersymmetry anomalies in new minimal supergravity

We determine the general structure of quantum anomalies for the R-multiplet of four dimensional mathcal{N} = 1 supersymmetric quantum field theories in the presence of background fields for an arbitrary number of Abelian flavor multiplets. By solving the Wess-Zumino consistency conditions for off-shell new minimal supergravity in four dimensions with an arbitrary number of Abelian vector multiplets, we compute the anomaly in the conservation of the supercurrent to leading non trivial order in the gravitino and vector multiplet fermions. We find that both R-symmetry and flavor anomalies necessarily lead to a supersymmetry anomaly, thus generalizing our earlier results to non superconformal theories with Abelian flavor symmetries. The anomaly in the conservation of the supercurrent leads to an anomalous transformation for the supercurrent under rigid supersymmetry on bosonic backgrounds that admit new minimal Killing spinors. The resulting deformation of the supersymmetry algebra has implications for supersymmetric localization computations on such backgrounds.

Anomalous Supersymmetry.

We show that supersymmetry is anomalous in N=1 superconformal quantum field theories (SCFTs) with an anomalous R symmetry. This anomaly was originally found in holographic SCFTs at strong coupling. Here we show that this anomaly is present, in general, and demonstrate it for the massless superconformal Wess-Zumino model via a one-loop computation. The anomaly appears first in four-point functions of two supercurrents either with two R currents or with an R current and an energy-momentum tensor. In fact, the Wess-Zumino consistency conditions together with the standard R-symmetry anomaly imply the existence of the anomaly. We outline the implications of this anomaly.

Supersymmetry anomalies in mathcal{N}=1 conformal supergravity

We solve the Wess-Zumino consistency conditions of mathcal{N}=1 off-shell conformal supergravity in four dimensions and determine the general form of the superconformal anomalies for arbitrary a and c anomaly coefficients to leading non trivial order in the gravitino. Besides the well known Weyl and R-symmetry anomalies, we compute explicitly the fermionic mathcal{Q} - and mathcal{S} -supersymmetry anomalies. In particular, we show that mathcal{Q} -supersymmetry is anomalous if and only if R-symmetry is anomalous. The mathcal{Q} - and mathcal{S} -supersymmetry anomalies give rise to an anomalous supersymmetry transformation for the supercurrent on curved backgrounds admitting Killing spinors, resulting in a deformed rigid supersymmetry algebra. Our results may have implications for supersymmetric localization and supersymmetry phenomenology. Analogous results are expected to hold in dimensions two and six and for other supergravity theories. The present analysis of the Wess-Zumino consistency conditions reproduces the holographic result of arXiv:1703.0429 and generalizes it to arbitrary a and c anomaly coefficients.

Realization of impossible anomalies

The Wess-Zumino consistency condition allows more exotic forms of anomalies than those we usually encounter. For example in two-dimensional conformal field theories in the curved background with space-time dependent coupling constant $\lambda^i(x)$, a $U(1)$ current could possess anomalous divergence of the form $D^\mu J_\mu = \tilde{c} R + \chi_{ij} \partial^\mu \lambda^i \partial_\mu\lambda_j + \tilde{\chi}_{ij} \epsilon^{\mu\nu} \partial_\mu \lambda^i \partial_\nu \lambda^j + \cdots $. Another example is the CP odd Pontryagin density in four-dimensional Weyl anomaly. We could, however, argue that they are impossible in conformal field theories because they cannot correspond to any (unregularized) conformally invariant correlation functions. We find that this no-go argument may be a red herring. We show that some of these impossible anomalies avoid the no-go argument because they are not primary operators, and the others circumvent it because they are realized as semi-local terms as is the case with the conformally invariant Green-Schwartz mechanism and in the higher dimensional analogue of Liouville or linear dilaton theory.

Connecting holographic Wess-Zumino consistency condition to the holographic anomaly

The Holographic Wess-Zumino (HWZ) consistency condition is shown through a step by step mapping of renormalization group flows to Hamiltonian systems, to lead to the Holographic anomaly. This condition codifies how the energy scale, when treated as the emergent bulk direction in Holographic theories, is put on equal footing as the other directions of the space the field theory inhabits. So, this is a defining feature of theories possessing local Holographic bulk duals. In four dimensional Holographic conformal field theories, the a and c anomaly coefficients are equated, and this is seen as a defining property of theories which possess General Relativity coupled to matter as a dual. Hence, showing how the former consistency condition leads to the latter relation between anomaly coefficients adds evidence to the claim that the HWZ condition is a defining feature of theories possessing local gravity duals.

Anomaly-corrected supersymmetry algebra and supersymmetric holographic renormalization

We present a systematic approach to supersymmetric holographic renormalization for a generic 5D mathcal{N}=2 gauged supergravity theory with matter multiplets, including its fermionic sector, with all gauge fields consistently set to zero. We determine the complete set of supersymmetric local boundary counterterms, including the finite counterterms that parameterize the choice of supersymmetric renormalization scheme. This allows us to derive holographically the superconformal Ward identities of a 4D superconformal field theory on a generic background, including the Weyl and super-Weyl anomalies. Moreover, we show that these anomalies satisfy the Wess-Zumino consistency condition. The super-Weyl anomaly implies that the fermionic operators of the dual field theory, such as the supercurrent, do not transform as tensors under rigid supersymmetry on backgrounds that admit a conformal Killing spinor, and their anticommutator with the conserved supercharge contains anomalous terms. This property is explicitly checked for a toy model. Finally, using the anomalous transformation of the supercurrent, we obtain the anomaly-corrected supersymmetry algebra on curved backgrounds admitting a conformal Killing spinor.

Weyl versus conformal invariance in quantum field theory

We argue that conformal invariance in flat spacetime implies Weyl invariance in a general curved background metric for all unitary theories in spacetime dimensions d ≤ 10. We also study possible curvature corrections to the Weyl transformations of operators, and show that these are absent for operators of sufficiently low dimensionality and spin. We find possible ‘anomalous’ Weyl transformations proportional to the Weyl (Cotton) tensor for d > 3 (d = 3). The arguments are based on algebraic consistency conditions similar to the Wess-Zumino consistency conditions that classify possible local anomalies. The arguments can be straightforwardly extended to larger operator dimensions and higher d with additional algebraic complexity.

Background independent holographic dual to Toverline{T} deformed CFT with large central charge in 2 dimensions

The geometerization of the renormalization group flow triggered by the Toverline{T} deformation of large c conformal field theories in two dimensions is presented. This entails the construction of the off shell Einstein-Hilbert action in three dimensions from said renormalization group flow.The crucial ingredient to this construction will be the encoding of general covariance in the emergent bulk theory in a very particular form of the Wess-Zumino consistency conditions. The utilisation of the local renormalization group, which requires putting the theory under consideration on an arbitrary background geometry, supplemented by the aforementioned covariance condition ensures that the whole construction is background independent.

Conformal anomaly and off-shell extensions of gravity

The gauge dependence of the conformal anomaly for spin 3/2 and spin 2 fields in non-conformal supergravities has been a long standing puzzle. In this Letter we argue that the `correct' gauge choice is the one that follows from requiring all terms that would imply a violation of the Wess-Zumino consistency condition to be absent in the counterterm, because otherwise the usual link between the anomaly and the one-loop divergence becomes invalid. Remarkably, the `good' choice of gauge is the one that confirms our previous result that a complete cancellation of conformal anomalies in D=4 can only be achieved for N-extended (Poincar\'e) supergravities with $N\geq 5$.

Can we change c in four-dimensional CFTs by exactly marginal deformations?

There is no known obstructions, but we have not been aware of any concrete examples, either. The Wess-Zumino consistency condition for the conformal anomaly says that a cannot change but does not say anything about c. In supersymmetric models, both a and c are determined from the triangle t’Hooft anomalies and the unitarity demands that both must be fixed, so the unitary supersymmetric conformal field theories do not admit such a possibility. Given this field theory situation, we construct an effective AdS/CFT model without supersymmetry in which c changes under exactly marginal deformations.

On three-dimensional trace anomaly from holographic local RG

Odd-dimensional quantum field theories (QFTs) can have nonzero trace anomalies if external fields are introduced and some ingredients needed to make Lorentz scalars with appropriate mass dimensions (or weights) are supplied. We have studied a three-dimensional QFT and explicitly computed the trace of the stress tensor using the holographic local renormalization group (RG). We have checked some properties of vector beta functions and the Wess-Zumino consistency condition, however, found the anomalies vanish on fixed points. We clarify what is responsible for the vanishing trace anomalies.

Anomalies, conformal manifolds, and spheres

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.