Supersymmetry Anomalies Research Articles

WZW in the lightlike directions

Dimensional reduction of the M5 brane on a Lorentzian manifold along a lightlike direction results in a five-dimensional gauge theory, which can be reformulated covariantly in six dimensions, where one puts the Lie derivatives along the lightlike direction of all fields to zero up to a gauge equivalence as constraints. We find gauge and supersymmetry anomalies for certain Lorentzian six-manifolds. We cancel these anomalies by adding a certain WZW theory in the 2d space that is spanned by two lightlike directions.

Deformations of supergravity and supersymmetry anomalies

We present a BRST analysis of supersymmetry anomalies of mathcal{N} = 1 supersymmetric quantum field theories with anomalous R symmetry. To this end, we consider the coupling of the matter theory to classical mathcal{N} = 1 new minimal supergravity. We point out that a supersymmetry anomaly cocycle associated to the U(1)R field does exist for this theory. It is non-trivial in the space of supergravity fields (and ghosts), but it becomes BRST-exact in the functional space that includes antifields. Equivalently, the U(1)R supersymmetry anomaly cocycle vanishes “on-shell”. It is therefore removable. However, to remove it — precisely because it is not trivial in the smaller space of fields — one needs to deform the supergravity BRST operator. This deformation is triggered, at first order in the anomaly coefficient, by a local operator S1 of ghost number 1. We give a cohomological characterization of S1 and compute it in full detail. At higher orders in the anomaly coefficient, we expect a priori that further deformations of the BRST rules are necessary.

Supersymmetry anomalies and the Wess–Zumino model in a supergravity background

We briefly recall the procedure for computing the Ward identities in the presence of a regulator which violates the symmetry being considered. We compute the first nontrivial correction to the supersymmetry Ward identity of the Wess–Zumino model in the presence of background supergravity using dimensional regularization. We find that the result can be removed using a finite local counter term and so there is no supersymmetry anomaly.

Supersymmetry anomaly in the superconformal Wess-Zumino model

We present a comprehensive analysis of supersymmetry anomalies in the free and massless Wess-Zumino (WZ) model in perturbation theory. At the classical level the model possesses mathcal{N} = 1 superconformal symmetry, which is partially broken by quantum anomalies. The form of the anomalies and the part of the symmetry they break depend on the multiplet of conserved currents used. It was previously shown that the R-symmetry anomaly of the conformal current multiplet induces an anomaly in Q-supersymmetry, which appears first in 4-point functions. Here we confirm this result by an explicit 1-loop computation using a supersymmetric Pauli-Villars regulator.The conformal current multiplet does not exist in the regulated theory because the regulator breaks conformal invariance, R-symmetry and S-supersymmetry explicitly. The minimal massive multiplet is the Ferrara-Zumino (FZ) one and the supersymmetry preserved by the regulator is a specific field dependent combination of Q- and S- supersymmetry of the conformal multiplet. While this supersymmetry is non anomalous, conformal invariance, R-symmetry and the original Q- and S-supersymmetries are explicitly broken by finite contact terms, both in the regulated and renormalized theories.A conformal current multiplet does exist for the renormalized theory and may be obtained from the FZ multiplet by a set of finite local counterterms that eliminate the explicit symmetry breaking, thus restoring superconformal invariance up to anomalies. However, this necessarily renders both Q- and S-supersymmetries anomalous, as is manifest starting at 4-point functions of conformal multiplet currents. The paper contains a detailed discussion of a number of issues and subtleties related to Ward identities that may be useful in a wider context.

The action with manifest E7 type symmetry

We generalize Cremmer-Julia 1st order action of mathcal{N}=8 supergravity with manifest E7(7) symmetry to cases of mathcal{N}=6 with manifest SO∗(12) and mathcal{N}=5 with manifest SU(1, 5) duality symmetries. These U dualities belong to groups of type E7 which do not admit a symmetric bilinear invariant for vector fields. Therefore the 2d order classical action derived from the one with manifest E7 type duality has a ghost vector field which, under appropriate boundary conditions, decouples. We show that when classical mathcal{N}ge 5 supergravitiesaredeformedbyacandidateUVdivergencetheghostfielddoesnot decouple. Therefore we argue that U duality and supersymmetry suggest an explanation of the mysterious cancellation of UV infinities at L = 4, mathcal{N}=5 in d=4. The same reasoning implies that, in absence of duality and supersymmetry anomalies, which still require a better understanding, mathcal{N}ge 5 perturbative supergravities may be UV finite at higher-loops.

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.

D =7/D =6 heterotic supergravity with gauged R-symmetry

We construct a family of chiral anomaly-free supergravity theories in D =6 starting from D = 7 supergravity with a gauged noncompact R-symmetry, employing a Hořava-Witten bulk-plus-boundary construction. The gauged noncompact R-symmetry yields a positive (de Sitter sign) D = 6 scalar field potential. Classical anomaly inflow which is needed to cancel boundary-field loop anomalies requires careful consideration of the gravitational, gauge, mixed and local supersymmetry anomalies. Coupling of boundary hypermultiplets requires care with the Sp(1) gauge connection required to obtain quaternionic Kähler target manifolds in D = 6. This class of gauged R-symmetry models may be of use as starting points for further compactifications to D = 4 that take advantage of the positive scalar potential, such as those proposed in the scenario of supersymmetry in large extra dimensions.

Consistent momentum space regularization/renormalization of supersymmetric quantum field theories: the three-loop β-function for the Wess-Zumino model

We compute the three loop $\beta$ function of the Wess-Zumino model to motivate implicit regularization (IR) as a consistent and practical momentum-space framework to study supersymmetric quantum field theories. In this framework which works essentially in the physical dimension of the theory we show that ultraviolet are clearly disantangled from infrared divergences. We obtain consistent results which motivate the method as a good choice to study supersymmetry anomalies in quantum field theories.

Extended BRST cohomology, consistent deformations and anomalies of four-dimensional supersymmetric gauge theories

The local cohomology of an extended BRST differential which includes global N=1 supersymmetry and Poincare transformations is completely and explicitly computed in four-dimensional supersymmetric gauge theories with super-Yang-Mills multiplets, chiral matter multiplets and linear multiplets containing 2-form gauge potentials. In particular we determine to first order all N=1 supersymmetric and Poincare invariant consistent deformations of these theories that preserve the N=1 supersymmetry algebra on-shell modulo gauge transformations, and all Poincare invariant candidate gauge and supersymmetry anomalies. When the Yang-Mills gauge group is semisimple and no linear multiplets are present, we find that all such deformations can be constructed from standard superspace integrals and preserve the supersymmetry transformations in a formulation with auxiliary fields, and the candidate anomalies are exhausted by supersymmetric generalizations of the well-known chiral anomalies. In the general case there are additional deformations and candidate anomalies which are relevant especially to the deformation of free theories and the general classification of interaction terms in supersymmetric field theories.

All couplings of minimal six-dimensional supergravity

We describe the complete coupling of (1,0) six-dimensional supergravity to tensor, vector and hypermultiplets. The generalized Green–Schwarz mechanism implies that the resulting theory embodies factorized gauge and supersymmetry anomalies, to be disposed of by fermion loops. Consequently, the low-energy theory is determined by the Wess–Zumino consistency conditions, rather than by the requirement of supersymmetry. As already shown for the case without hypermultiplets, this procedure does not fix a quartic coupling for the gauginos. With respect to these previous results, the inclusion of charged hypermultiplets gives additional terms in the supersymmetry anomaly. We also consider the case in which abelian vectors are present. As in the absence of hypermultiplets, abelian vectors allow additional couplings. Finally, we apply the Pasti–Sorokin–Tonin prescription to this model.

Abelian vectors and self-dual tensors in six-dimensional supergravity

In this note we describe the most general coupling of abelian vector and tensor multiplets to six-dimensional (1,0) supergravity. As was recently pointed out, it is of interest to consider more general Chern-Simons couplings to abelian vectors of the type H r=dB r−1/2 c rabA adA b , with c r matrices that may not be simultaneously diagonalized. We show that these couplings can be related to Green-Schwarz terms of the form B r c r ab F a F b , and how the complete local Lagrangian, that embodies factorized gauge and supersymmetry anomalies (to be disposed of by fermion loops) is uniquely determined by Wess-Zumino consistency conditions, aside from an arbitrary quartic coupling for the gauginos.

Tensor and vector multiplets in six-dimensional supergravity

We construct the complete coupling of (1,0) supergravity in six dimensions to n tensor multiplets, extending previous results to all orders in the fermi fields. We then add couplings to vector multiplets, as dictated by the generalized Green-Schwarz mechanism. The resulting theory embodies factorized gauge and supersymmetry anomalies, to be disposed of by fermion loops, and is determined by corresponding Wess-Zumino consistency conditions, aside from a quartic coupling for the gauginos. The supersymmetry algebra contains a corresponding extension that plays a crucial role for the consistency of the construction. We leave aside gravitational and mixed anomalies, that would only contribute to higher-derivative couplings.

Determination of anomalies in supersymmetric theories

We develop an efficient technique to compute anomalies in supersymmetric theories by combining the so-called nonlocal regularization method and superspace techniques. To illustrate the method we apply it to a four dimensional toy model with potentially anomalous N=1 supersymmetry and prove explicitly that in this model all the candidate supersymmetry anomalies have vanishing coefficients at the one-loop level.

Low-energy analysis of M- and F-theories on Calabi-Yau threefolds

We elucidate the interplay between gauge and supersymmetry anomalies in six-dimensional N = 1 supergravity with generalized couplings between tensor and vector multiplets. We derive the structure of the five-dimensional supergravity resulting from the S 1 reduction of these models and give the constraints on Chem-Simons couplings that follow from duality to M-theory compactified on a Calabi-Yau threefold. The duality is supported only on a restricted class of Calabi-Yau threefolds and requires a special type of intersection form. We derive five-dimensional central-charge formulas and briefly discuss the associated phase transitions. Finally, we exhibit connections with F-theory compactifications on Calabi-Yau manifolds that admit elliptic fibrations. This analysis suggests that F-theory unifies type-IIb three-branes and M-theory five-branes.

Higher spin BRS cohomology of supersymmetric chiral matter inD=4

We examine the BRS cohomology of chiral matter inN=1,D=4 supersymmetry to determine a general form of composite superfield operators which can suffer from supersymmetry anomalies. Composite superfield operators Ψ(a, b) are products of the elementary chiral superfieldsS and\(\bar S\) and the derivative operatorsD α,\(D_\alpha ,\bar D_{\dot \beta } \) and\(\partial _{x\dot \beta } \). Such superfields Ψ(a, b) can be chosen to have “a” symmetrized undotted indices α i and “b” symmetrized dotted indices\(\dot \beta _j \). The result derived here is that each composite superfield Ψ(a,b) is subject to potential supersymmetry anomalies ifa−b is an odd number, which means that Ψ(a,b) is a fermionic superfield.

Supersymmetry is full of holes

There are an infinite number of ghost charge one polynomials ('holes') in the BRS cohomology space of the supersymmetric Yang-Mills theory coupled to a Wess-Zumino chiral multiplet in a real representation of the gauge group. It is probable that there exists no regularisation procedure that preserves both non-Abelian gauge invariance and supersymmetry. Consequently, these holes may give rise to an infinite number of previously unnoticed 'supersymmetry anomalies' in this theory. The new anomalies are predicted to occur in the renormalisation of most of the composite, local, supersymmetric and gauge-invariant spinor operators that can be formed in the theory.

Weil triviality and anomalies in two-dimensional supergravity

Solves the problem of Weil triviality in the superspace formulation of d=2, N=1, supergravity. In this case, Weil triviality expresses the fact that the superspace invariant, used to describe gravitational anomalies, is exact in the space of invariant superforms. The author explicitly constructs the relevant cochain and show how it relates to anomalies over spacetime. Finally, using superspace techniques, he shows that the method of Bonora et al. (1987) agrees with component calculations of Lorentz and local supersymmetry anomalies, for theories with chiral fermions, in d=2.

Supersymmetry anomalies and analytic regularization

The method of analytic regularization in which the number of dimensions is not generalized is shown to preserve the supersymmetry identity relating the anomalies of the supersymmetry current, the trace of the energy–momentum tensor, and the divergence of the axial current at the one-loop level. Explicit counterterms needed for the identity to hold are constructed. The method preserves Dirac and all other tensor algebras in D = 4 space and renders the computation of anomalies straightforward and simple.

Anomaly cancellations in the local supersymmetric Siegel lagrangian

The local supersymmetric Siegel action for two-dimensional chiral bosons and fermions is studied. It is found that the theory has local supersymmetry anomalies as well as gravitational-like anomalies. A modification of the lagrangian yielding anomaly cancellations is obtained.

The relationship between the canonical super-virasoro and functional super-Weyl anomalies

In this letter, recent work relating the various anomalies encountered in the different quantisation approaches, is extended to the superstring. The reparametrization and supersymmetry anomalies of the canonical approach are constructed from the super-Weyl anomaly of the path-integral approach via the descent equations. We also give an exact correspondence between this procedure and operations in canonical field theory.