Superconformal Ward Identities Research Articles

Supersymmetric Casimir energy on N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 conformal supergravity backgrounds

We provide a first principles derivation of the supersymmetric Casimir energy of N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 SCFTs in four dimensions using the supercharge algebra on general conformal supergravity backgrounds that admit Killing spinors. The superconformal Ward identities imply that there exists a continuous family of conserved R-currents on supersymmetric backgrounds, as well as a continuous family of conserved currents for each conformal Killing vector. These continuous families interpolate between the consistent and covariant R-current and energy-momentum tensor. The resulting Casimir energy, therefore, depends on two continuous parameters corresponding to the choice of conserved currents used to define the energy and R-charge. This ambiguity is in addition to any possible scheme dependence due to local terms in the effective action. As an application, we evaluate the general expression for the supersymmetric Casimir energy we obtain on a family of backgrounds with the cylinder topology ℝ × S3 and admitting a single Majorana supercharge. Our result is a direct consequence of the supersymmetry algebra, yet it resembles more known expressions for the non-supersymmetric Casimir energy on such backgrounds and differs from the supersymmetric Casimir energy obtained from the zero temperature limit of supersymmetric partition functions. We defer a thorough analysis of the relation between these results to future work.

On the Kähler-Hodge structure of superconformal manifolds

We show that conformal manifolds in d ≥ 3 conformal field theories with at least 4 supercharges are Kähler-Hodge, thus extending to 3d mathcal{N} = 2 and 4d mathcal{N} = 1 similar results previously derived for 4d mathcal{N} = 2 and mathcal{N} = 4 and various types of 2d SCFTs. Conformal manifolds in SCFTs are equipped with a holomorphic line bundle ℒ, which encodes the operator mixing of supercharges under marginal deformations. Using conformal perturbation theory and superconformal Ward identities, we compute the curvature of ℒ at a generic point on the conformal manifold. We show that the Kähler form of the Zamolodchikov metric is proportional to the first Chern class of ℒ, with a constant of proportionality given by the two-point function coefficient of the stress tensor, CT. In cases where certain additional conditions about the nature of singular points on the conformal manifold hold, this implies a quantization condition for the total volume of the conformal manifold.

Multipoint correlators on the supersymmetric Wilson line defect CFT

We study multipoint correlators of protected scalars on the Maldacena-Wilson line in mathcal{N} = 4 SYM. Working at weak coupling in the planar limit, we derive an explicit recursion relation that captures next-to-leading order correlators with an arbitrary number of insertions of the fundamental scalar field. By pinching fundamental scalars together, we can build composite protected operators with higher values of the R-charge. Our result then encompasses arbitrary n-point correlators of protected operators with arbitrary weight. As a demonstration of our method, we give explicit formulae for correlators with up to six points. Using these results we observe that all our correlators are annihilated by a special class of differential operators. We conjecture that these differential operators are non-perturbative constraints and can be considered a multipoint extension of the superconformal Ward identities satisfied by four-point functions.

Rebooting quarter-BPS operators in mathcal{N} = 4 super Yang-Mills

We start a systematic study of quarter-BPS operators in four-dimensional mathcal{N} = 4 Super Yang-Mills with gauge group SU(N) making use of recently developed tools in conformal field theory. We adapt the technology of embedding space tensor structures in four dimensions to the problem of computing R-symmetry tensor structures, and we use the underlying chiral algebra to obtain the superconformal Ward identities. This allows us to fix the protected part of the four-point correlators, up to few ambiguities. As applications, we use the Lorentzian inversion formula to study the leading order OPE data in the large N supergravity limit and we make contact with the OPE limit of the five-point function of half-BPS operators.

Bootstrapping holographic defect correlators in mathcal{N} = 4 super Yang-Mills

We study two-point functions of single-trace half-BPS operators in the presence of a supersymmetric Wilson line in mathcal{N} = 4 SYM. We use modern analytic bootstrap in the form of an inversion formula, that reconstructs the CFT data starting from a single discontinuity of the correlator. In the planar strong coupling limit only a finite number of conformal blocks contributes to the discontinuity, which allows us to obtain elegant closed-form expressions for two-point functions of single-trace operators mathcal{O} J of weight J = 2, 3, 4. Our bootstrap approach guarantees that the correlator satisfies all the expected properties: it has the correct discontinuity, it satisfies the superconformal Ward identities, and it has a sensible expansion in both defect and bulk OPEs. The result is also consistent with available results coming from localization. The method is completely algorithmic and can be implemented to calculate correlators of arbitrary weight.

ABJ correlators with weakly broken higher spin symmetry

We consider four-point functions of operators in the stress tensor multiplet of the 3d mathcal{N} = 6 U(N)k× U(N + M)−k or SO(2)2k× USp(2 + 2M)−k ABJ theories in the limit where M and k are taken to infinity while N and λ ∼ M/k are held fixed. In this limit, these theories have weakly broken higher spin symmetry and are holographically dual to mathcal{N} = 6 higher spin gravity on AdS4, where λ is dual to the bulk parity breaking parameter. We use the weakly broken higher spin Ward identities, superconformal Ward identities, and the Lorentzian inversion formula to fully determine the tree level stress tensor multiplet four-point function up to two free parameters. We then use supersymmetric localization to fix both parameters for the ABJ theories in terms of λ, so that our result for the tree level correlator interpolates between the free theory at λ = 0 and a parity invariant interacting theory at λ = 1/2. We compare the CFT data extracted from this correlator to a recent numerical bootstrap conjecture for the exact spectrum of U(1)2M× U(1 + M)−2M ABJ theory (i.e. λ = 1/2 and N = 1), and find good agreement in the higher spin regime.

All Holographic Four-Point Functions in All Maximally Supersymmetric CFTs

We present a constructive derivation of holographic four-point correlators of arbitrary half-BPS operators for all maximally supersymmetric conformal field theories in $d>2$. This includes holographic correlators in 3d ${\cal N}=8$ ABJM theories, 4d ${\cal N}=4$ SYM theory and the 6d ${\cal N}=(2,0)$ theory, dual to tree-level amplitudes in 11D supergravity on $AdS_4 \times S^7$, 10D supergravity on $AdS_5 \times S^5$ and 11D supergravity on $AdS_7 \times S^4$, respectively. We introduce the concept of Maximally R-symmetry Violating (MRV) amplitude, which corresponds to a special configuration in the R-symmetry space. In this limit the amplitude drastically simplifies, but at the same time the entire polar part of the full amplitude can be recovered from this limit. Furthermore, for a specific choice of the polar part, contact terms can be shown to be absent, by using the superconformal Ward identities and the flat space limit.

All Tree-Level Correlators for M Theory on AdS_{7}×S^{4}.

We present a constructive derivation of all four-point tree-level holographic correlators for eleven dimensional supergravity on AdS_{7}×S^{4}. These correlators correspond to four-point functions of arbitrary one-half BPS operators in the six-dimensional (2,0) theory at large central charge. The crucial observation is that the polar part of the correlators in Mellin space is fully captured by a drastically simpler maximally R-symmetry violating amplitude, while the contact part is fully fixed by superconformal Ward identities and the flat space limit.

Defect CFT in the 6d (2,0) theory from M2 brane dynamics in AdS7 \xd7 S4

Surface operators in the 6d (2,0) theory at large N have a holographic description in terms of M2 branes probing the AdS7×S4 M-theory background. The most symmetric, 1/2-BPS, operator is defined over a planar or spherical surface, and it preserves a 2d superconformal group. This includes, in particular, an SO(2, 2) subgroup of d conformal transformations, so that the surface operator may be viewed as a conformal defect in the 6d theory. The dual M2 brane has an AdS3 induced geometry, reflecting the 2d conformal symmetry. Here we use the holographic description to extract the defect CFT data associated to the surface operator. The spectrum of transverse fluctuations of the M2 brane is found to be in one-to-one correspondence with a protected multiplet of operator insertions on the surface, which includes the displacement operator. We compute the one-loop determinants of fluctuations of the M2 brane, and extract the conformal anomaly coefficient of the spherical surface to order N0. We also briefly discuss the RG flow from the non-supersymmetric to the 1/2-BPS defect operator, and its consistency with a “b-theorem” for the defect CFT. Starting with the M2 brane action, we then use AdS3 Witten diagrams to compute the 4-point functions of the elementary bosonic insertions on the surface operator, and extract some of the defect CFT data from the OPE. The 4-point function is shown to satisfy superconformal Ward identities, and we discuss a related sub- sector of “twisted” scalar insertions, whose correlation functions are constrained by the residual superconformal symmetry.

The correlation function of (1, 1) and (2, 2) supersymmetric theories with Toverline{T} deformation

In the paper, we compute the correlation functions in 2D mathcal{N} = (1, 1) and mathcal{N} = (2, 2) superconformal field theories with Toverline{T} deformation up to the first order of the deformation in terms of perturbation theory. With the help of superconformal Ward identity in mathcal{N} = (1, 1) and mathcal{N} = (2, 2) theories and careful regularization, the correlation functions in the deformed theory can be obtained up to the first order perturbation. This study is the extension from previous bosonic Toverline{T} deformation to the supersymmetric one.

AdS4/CFT3 from weak to strong string coupling

We consider the four-point function of operators in the stress tensor multiplet of the U(N)k× U(N)−k ABJM theory, in the limit where N is taken to infinity while N/k5 is held fixed. In this limit, ABJM theory is holographically dual to type IIA string theory on AdS4× ℂℙ3 at finite string coupling gs∼ (N/k5)1/4. While at leading order in 1/N, the stress tensor multiplet four-point function can be computed from type IIA supergravity, in this work we focus on the first subleading correction, which comes from tree level Witten diagrams with an R4 interaction vertex. Using superconformal Ward identities, bulk locality, and the mass deformed sphere free energy previously computed to all orders in 1/N from supersymmetric localization, we determine this R4 correction as a function of N/k5. Taking its flat space limit, we recover the known R4 contribution to the type IIA S-matrix and reproduce the fact that it only receives perturbative contributions in gs from genus zero and genus one string worldsheets. This is the first check of AdS/CFT at finite gs for local operators. Our result for the four-point correlator interpolates between the large N, large ’t Hooft coupling limit and the large N finite k limit. From the bulk perspective, this is an interpolation between type IIA string theory on AdS4× ℂℙ3 at small string coupling and M-theory on AdS4× S7/ℤk.

Superconformal blocks for mixed 1/2-BPS correlators with SU(2) R-symmetry

For SCFTs with an SU(2) R-symmetry, we determine the superconformal blocks that contribute to the four-point correlation function of a priori distinct half-BPS superconformal primaries as an expansion in terms of the relevant bosonic conformal blocks. This is achieved by using the superconformal Casimir equation and the superconformal Ward identity to fix the coefficients of the bosonic blocks uniquely in a dimension-independent way. In addition we find that many of the resulting coefficients are related through a web of linear transformations of the conformal data.

Super-Weyl anomaly from holography and rigid supersymmetry algebra on a two-sphere

Operators transform anomalously under the symmetry in the presence of quantum anomalies. We study this aspect of the super-Weyl anomaly in $\mathcal N=(1,1)$ superconformal field theories (SCFTs), in the context of AdS/CFT. In particular, we carry out holographic renormalization for $(1,1)$ pure AdS$_3$ supergravity that is supposed to be a gravity dual of the $\mathcal N=(1,1)$ SCFT, and derive holographic superconformal Ward identities with corresponding anomalies. We show that the obtained super-Weyl anomaly of the $\mathcal N=(1,1)$ SCFT induces a quantum correction term in the transformation law of the supercurrent under the rigid supersymmetry. We demonstrate that the correction term does not affect the $\mathcal N=(1,1)$ rigid supersymmetry algebra on two-sphere.

Amplitudes from anomalous superconformal symmetry

We initiate a systematic study of the consequences of (super)conformal symmetry of massless scattering amplitudes. The classical symmetry is potentially broken at the quantum level by infrared and ultraviolet effects. We study its manifestations on the finite hard part of the scattering process. The conformal Ward identities in momentum space are second-order differential equations, difficult to analyze. We prefer to study superconformal symmetry whose generators are first-order in the momenta. Working in a massless mathcal{N} = 1 supersymmetric Wess-Zumino model, we derive on-shell superconformal Ward identities. They contain an anomaly due to collinear regions of loop momenta. It is given by an integral with one loop less than the original graph, with an extra integral over a collinear splitting parameter. We discuss the relation to the holomorphic anomaly that was previously studied in tree-level amplitudes and at the level of unitarity cuts. We derive and solve Ward identities for various scattering processes in the model. We classify the on-shell superamplitudes according to their Grassmann degree, in close analogy with the helicity classification of gluon amplitudes. We focus on MHV-like and NMHV-like amplitudes with up to six external particles, at one and two loops. Interestingly, the superconformal generator acting on the bosonic part of the amplitudes is Witten’s twistor collinearity operator. We find that the first-order differential equations, together with physically motivated boundary conditions, uniquely fix the answer. All the cases considered give rise to uniform weight functions. Our most interesting example is a five-point non-planar hexa-box integral with an off-shell leg. It gives first indications on the function space needed for Higgs plus two jets production at next-to-next-to leading order.

Scattering Amplitudes from Superconformal Ward Identities.

We consider finite superamplitudes of N=1 matter, and use superconformal symmetry to derive powerful first-order differential equations for them. Because of on-shell collinear singularities, the Ward identities have an anomaly, which is obtained from lower-loop information. We show that in the five-particle case, the solution to the equations is uniquely fixed by the expected analytic behavior. We apply the method to a nonplanar two-loop five-particle integral.

On Mellin amplitudes in SCFTs with eight supercharges

We extend the Mellin space techniques of [1] for computing holographic four-point correlation functions in maximally superconformal theories to theories with only eight Poincaré supercharges. The one-half BPS operators in these correlators are taken to be the superconformal primary in the mathcal{D}left[kright] multiplet (with k = 2 corresponding to the flavor current multiplet), and transform in the adjoint representation of a flavor group G. Because of the smaller R-symmetry group SU(2), each individual superconformal Ward identity is less powerful. On the other hand, the constraining power is compensated in number by the different flavor channels in the four-point function. As concrete test cases, we study the Seiberg theories in five dimensions and E-string theory in six dimensions at the large N limit. We show that the flavor current multiplet four-point functions are fixed by superconformal symmetry up to two free parameters, which are proportional to the squared OPE coefficients for the flavor current multiplet and the stress tensor multiplet.

Holographic four-point functions in the (2, 0) theory

We revisit the calculation of holographic correlators for eleven-dimensional supergravity on AdS7 × S4. Our methods rely entirely on symmetry and eschew detailed knowledge of the supergravity effective action. By an extension of the position space approach developed in [1, 2] for the AdS5 × S5 background, we compute four-point correlators of one-half BPS operators for identical weights k = 2, 3, 4. The k = 2 case corresponds to the four-point function of the stress-tensor multiplet, which was already known, while the other two cases are new. We also translate the problem in Mellin space, where the solution of the superconformal Ward identity takes a surprisingly simple form. We formulate an algebraic problem, whose (conjecturally unique) solution corresponds to the general one-half BPS four-point function.

How to succeed at holographic correlators without really trying

We give a detailed account of the methods introduced in [1] to calculate holographic four-point correlators in IIB supergravity on AdS5 × S5. Our approach relies entirely on general consistency conditions and maximal supersymmetry. We discuss two related methods, one in position space and the other in Mellin space. The position space method is based on the observation that the holographic four-point correlators of one-half BPS single-trace operators can be written as finite sums of contact Witten diagrams. We demonstrate in several examples that imposing the superconformal Ward identity is sufficient to fix the parameters of this ansatz uniquely, avoiding the need for a detailed knowledge of the supergravity effective action. The Mellin space approach is an “on-shell method” inspired by the close analogy between holographic correlators and flat space scattering amplitudes. We conjecture a compact formula for the four-point correlators of one-half BPS single-trace operators of arbitrary weights. Our general formula has the expected analytic structure, obeys the superconformal Ward identity, satisfies the appropriate asymptotic conditions and reproduces all the previously calculated cases. We believe that these conditions determine it uniquely.

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.

Instanton corrections to twist-two operators

We present the calculation of the leading instanton contribution to the scaling dimensions of twist-two operators with arbitrary spin and to their structure constants in the OPE of two half-BPS operators in mathcal{N}=4 SYM. For spin-two operators we verify that, in agreement with mathcal{N}=4 superconformal Ward identities, the obtained expressions coincide with those for the Konishi operator. For operators with high spin we find that the leading instanton correction vanishes. This arises as the result of a rather involved calculation and requires a better understanding.