Dual Superconformal Symmetry Research Articles

Celestial dual superconformal symmetry, MHV amplitudes and differential equations

Celestial and momentum space amplitudes for massless particles are related to each other by a change of basis provided by the Mellin transform. Therefore properties of celestial amplitudes have counterparts in momentum space amplitudes and vice versa. In this paper, we study the celestial avatar of dual superconformal symmetry of mathcal{N} = 4 Yang-Mills theory. We also analyze various differential equations known to be satisfied by celestial n-point tree-level MHV amplitudes and identify their momentum space origins.

6D dual superconformal algebra

We construct and study the 6D dual superconformal algebra. Our construction is inspired by the dual superconformal symmetry of massless 4D SYM and extends the previous construction of the enhanced dual conformal algebra for 6D SYM to the full 6D dual superconformal algebra for chiral theories. We formulate constraints in 6D spinor helicity formalism and find all generators of the 6D dual superconformal algebra. Next, we check that they agree with the dual superconformal generators of known 3D and 4D theories. We show that it is possible to significantly simplify the form of generators and compactly write the dual superconformal algebra using superindices. Finally, we work out some examples of algebra invariants.

Scattering amplitudes in mathcal{N} = 3 supersymmetric SU(N) Chern-Simons-matter theory at large N

We study mathcal{N} = 3 supersymmetric Chern-Simons-matter theory coupled to matter in the fundamental representation of SU(N). In the ’t Hooft large N limit, we compute the exact 2 → 2 scattering amplitudes of the fundamental scalar superfields to all orders in the ’t Hooft coupling λ. Our computations are presented in mathcal{N} = 1 superspace and make significant use of the residual SO(2)R symmetry in order to solve for the exact four-point correlator of the scalar superfields. By taking the on-shell limit, we are able to extract the exact 2 → 2 scattering amplitudes of bosons/fermions in the symmetric, anti-symmetric and adjoint channels of scattering. We find that the scattering amplitude of the mathcal{N} = 3 theory in the planar limit is tree-level exact to all orders in the ’t Hooft coupling λ. The result is consistent with the conjectured bosonization duality and is expected to have enhanced symmetry structures such as dual superconformal symmetry and Yangian symmetry.

Dual superconformal symmetry of mathcal{N} = 2 Chern-Simons theory with fundamental matter at large N

Dual conformal symmetry and Yangian symmetry are symmetries of amplitudes that have aided the study of scattering amplitudes in highly supersymmetric theories like mathcal{N} = 4 SYM and ABJM. However, in general such symmetries are absent from the theories with lesser or no supersymmetry. In this paper, we show that the tree level 2 → 2 scattering amplitude in the 3d mathcal{N} = 2 Chern-Simons theory coupled to a fundamental chiral multiplet is dual superconformal invariant. In the ’t Hooft large N limit, the 2 → 2 scattering amplitude in this theory has been shown to be tree-level exact in non-anyonic channels, while having only an overall multiplicative coupling dependent renormalisation in the anyonic channel. Therefore, the dual superconformal symmetry that we demonstrate in this paper is all loop exact. This is unlike the previously studied highly supersymmetric theories where dual superconformal symmetry is anomalous at loop levels.Furthermore, we reverse the argument to study the extent to which dual superconformal invariance fixes the scattering amplitude in an mathcal{N} = 2 supersymmetric theory. We demonstrate that requiring the dual superconformal invariance completely fixes the momentum dependence of the 2 → 2 amplitude, while the coupling constant dependence remain unfixed. Further, we use a combination of parity invariance, unitarity and self-duality of the amplitude to constrain the coupling dependence of scattering amplitude.

Heptagons from the Steinmann cluster bootstrap

We reformulate the heptagon cluster bootstrap to take advantage of the Steinmann relations, which require certain double discontinuities of any amplitude to vanish. These constraints vastly reduce the number of functions needed to bootstrap seven-point amplitudes in planar mathcal{N} = 4 supersymmetric Yang-Mills theory, making higher-loop contributions to these amplitudes more computationally accessible. In particular, dual superconformal symmetry and well-defined collinear limits suffice to determine uniquely the symbols of the three-loop NMHV and four-loop MHV seven-point amplitudes. We also show that at three loops, relaxing the dual superconformal left(overline{Q}right) relations and imposing dihedral symmetry (and for NMHV the absence of spurious poles) leaves only a single ambiguity in the heptagon amplitudes. These results point to a strong tension between the collinear properties of the amplitudes and the Steinmann relations.

A requiem forAdS4×CP3fermionic self-T duality

Strong evidence for dual superconformal symmetry in $\mathcal{N} = 6$ superconformal Chern-Simons theory has fueled expectations that the AdS/CFT dual geometry $AdS_4 \times \mathbb{C} P^3$ is self-dual under T-duality. We revisit the problem to identify commuting bosonic and fermionic isometries in a systematic fashion and show that fermionic T-duality, a symmetry originally proposed by Berkovits & Maldacena, inevitably leads to a singularity in the dilaton transformation. We show that TsT deformations commute with fermionic T-duality and comment on T-duality in the corresponding sigma model. Our results rule out self-duality based on fermionic T-duality for $AdS_4 \times \mathbb{C} P^3$ or its TsT deformations, but leave the door open for new possibilities.

One-loop soft theorems via dual superconformal symmetry

We study soft theorems at one loop in planar N=4 super Yang-Mills theory through finite order in the infrared regulator and to subleading order in the soft parameter {\delta}. In particular, we derive a universal constraint from dual superconformal symmetry, which we use to bootstrap subleading log {\delta} behaviour. Moreover, we determine the complete infrared-finite subleading soft contribution of n-point MHV amplitudes using momentum twistors. Finally, we compute the subleading log {\delta} behaviour of one-loop NMHV ratio functions at six and seven points, finding that universality holds within but not between helicity sectors.

Bootstrapping correlation functions in N = 4 $$ \mathcal{N}=4 $$ SYM

We describe a new approach to computing the chiral part of correlation functions of stress-tensor supermultiplets in N=4 SYM that relies on symmetries, analytic properties and the structure of the OPE only. We demonstrate that the correlation functions are given by a linear combination of chiral N=4 superconformal invariants accompanied by coefficient functions depending on the space-time coordinates only. We present the explicit construction of these invariants and show that the six-point correlation function is fixed in the Born approximation up to four constant coefficients by its symmetries. In addition, the known asymptotic structure of the correlation function in the light-like limit fixes unambiguously these coefficients up to an overall normalization. We demonstrate that the same approach can be applied to obtain a representation for the six-point NMHV amplitude that is free from any auxiliary gauge fixing parameters, does not involve spurious poles and manifests half of the dual superconformal symmetry.

From six to four and more: massless and massive maximal super Yang-Mills amplitudes in 6d and 4d and their hidden symmetries

A self-consistent exposition of the theory of tree-level superamplitudes of the 4d N=4 and 6d N=(1,1) maximally supersymmetric Yang-Mills theories is provided. In 4d we work in non-chiral superspace and construct the superconformal and dual superconformal symmetry generators of the N=4 SYM theory using the non-chiral BCFW recursion to prove the latter. In 6d we provide a complete derivation of the standard and hidden symmetries of the tree-level superamplitudes of N=(1,1) SYM theory, again using the BCFW recursion to prove the dual conformal symmetry. Furthermore, we demonstrate that compact analytical formulae for tree-superamplitudes in N=(1,1) SYM can be obtained from a numerical implementation of the supersymmetric BCFW recursion relation. We derive compact manifestly dual conformal representations of the five- and six-point superamplitudes as well as arbitrary multiplicity formulae valid for certain classes of superamplitudes related to ultra-helicity-violating massive amplitudes in 4d. We study massive tree superamplitudes on the Coulomb branch of the N=4 SYM theory from dimensional reduction of the massless superamplitudes of the six-dimensional N=(1,1) SYM theory. We exploit this correspondence to construct the super-Poincare and enhanced dual conformal symmetries of massive tree superamplitudes in N=4 SYM theory which are shown to close into a finite dimensional algebra of Yangian type. Finally, we address the fascinating possibility of uplifting massless 4d superamplitudes to 6d massless superamplitudes proposed by Huang. We confirm the uplift for multiplicities up to eight but show that finding the uplift is highly non-trivial and in fact not of a practical use for multiplicities larger than five.

Dual conformal symmetry on the light cone

We study the properties of conformal operators in the SL(2) sector of planar N=4 SYM and its supersymmetric SL(2|2) extension. The correlation functions of these operators and their form factors with respect to asymptotic on-shell states are determined by two different polynomials which can be identified as eigenstates of the dilatation operator in the coordinate and momentum representations, respectively. We argue that, in virtue of integrability of the dilatation operator, the two polynomials satisfy a duality relation – they are proportional to each other upon an appropriate identification of momenta and coordinates. Combined with the conventional N=4 superconformal symmetry, this leads to the dual superconformal symmetry of the dilatation operator. We demonstrate that this symmetry is powerful enough to fix the eigenspectrum of the dilatation operator to the lowest order in the coupling. We use the relation between the one-loop dilatation operator and Heisenberg spin chain to show that, to lowest order in the coupling, the dual symmetry is generated by the Baxter Q-operator in the limit of large spectral parameter.

Review of AdS/CFT Integrability, Chapter V.2: Dual Superconformal Symmetry

Scattering amplitudes in planar N=4 super Yang-Mills theory reveal a remarkable symmetry structure. In addition to the superconformal symmetry of the Lagrangian of the theory, the planar amplitudes exhibit a dual superconformal symmetry. The presence of this additional symmetry imposes strong restrictions on the amplitudes and is connected to a duality relating scattering amplitudes to Wilson loops defined on polygonal light-like contours. The combination of the superconformal and dual superconformal symmetries gives rise to a Yangian, an algebraic structure which is known to be related to the appearance of integrability in other regimes of the theory. We discuss two dual formulations of the symmetry and address the classification of its invariants.

Dual Superconformal Symmetry of the ABJM Theory

We summarize the proof that all tree-level amplitudes of the ABJM theory enjoy dual superconformal symmetry. This note is based on arXiv:1008.0041 [hep-th] and arXiv:1012.5032 [hep-th] .

Tree-level recursion relation and dual superconformal symmetry of the ABJM theory

We propose a recursion relation for tree-level scattering amplitudes in three-dimensional Chern-Simons-matter theories. The recursion relation involves a complex deformation of momenta which generalizes the BCFW-deformation used in higher dimensions. Using background field methods, we show that all tree-level superamplitudes of the ABJM theory vanish for large deformations, establishing the validity of the recursion formula. Furthermore, we use the recursion relation to compute six-point and eight-point component amplitudes and match them with independent computations based on Feynman diagrams or the Grassmannian integral formula. As an application of the recursion relation, we prove that all tree-level amplitudes of the ABJM theory have dual superconformal symmetry. Using generalized unitarity methods, we extend this symmetry to the cut-constructible parts of the loop amplitudes.

Dual superconformal symmetry of $ \mathcal{N} = 6 $ Chern-Simons theory

We demonstrate that the four and six-point tree-level amplitudes of N=6 superconformal Chern-Simons theory (ABJM) enjoy OSp(6|4) dual superconformal symmetry if one enlarges the dual superspace to include three additional Grassmann-even coordinates which correspond to an abelian isometry of CP3. The inclusion of these coordinates enables us to match the nontrivial dual superconformal generators with level-one Yangian generators when acting on on-shell amplitudes. We also discuss some implications of dual conformal symmetry for loop-level amplitudes.

Superconformal invariants for scattering amplitudes in [formula omitted] SYM theory

Recent studies of scattering amplitudes in planar N = 4 SYM theory revealed the existence of a hidden dual superconformal symmetry. Together with the conventional superconformal symmetry it gives rise to powerful restrictions on the planar scattering amplitudes to all loops. We study the general form of the invariants of both symmetries. We first construct an integral representation for the most general dual superconformal invariants and show that it allows a considerable freedom in the choice of the integration measure. We then perform a half-Fourier transform to twistor space, where conventional conformal symmetry is realized locally, derive the resulting conformal Ward identity for the integration measure and show that it admits a unique solution. Thus, the combination of dual and conventional superconformal symmetries, together with invariance under helicity rescalings, completely fixes the form of the invariants. The expressions obtained generalize the known tree and one-loop superconformal invariants and coincide with the recently proposed coefficients of the leading singularities of the scattering amplitudes as contour integrals over Grassmannians.

SURPRISING SYMMETRY OF SCATTERING AMPLITUDES IN GAUGE THEORIES

I review a recent progress in computing scattering amplitudes in strongly coupled gauge theories - a fascinating subject which has been recently boosted by the formulation of the gauge/string duality in maximally supersymmetric Yang-Mills theory. In addition to the conventional symmetry of the underlying Lagrangian, the scattering amplitudes in this theory exhibit a new, dual superconformal symmetry. This symmetry is powerful enough to completely determine the scattering amplitudes for arbitrary coupling in a suitably defined limit.

Symmetries and analytic properties of scattering amplitudes in [formula omitted] SYM theory

In addition to the superconformal symmetry of the underlying Lagrangian, the scattering amplitudes in planar N = 4 super-Yang–Mills theory exhibit a new, dual superconformal symmetry. We address the question of how powerful these symmetries are to completely determine the scattering amplitudes. We use the example of the NMHV superamplitudes to show that the combined action of conventional and dual superconformal symmetries is not sufficient to fix all the freedom in the tree-level amplitudes. We argue that the additional information needed comes from the study of the analytic properties of the amplitudes. The requirement of absence of spurious singularities, together with the correct multi-particle singular behavior, determines the unique linear combination of superinvariants corresponding to the n-particle NMHV superamplitude. The same result can be obtained recursively, by relating the n- and ( n − 1 ) -particle amplitudes in the singular collinear limit. We also formulate constraints on the loop corrections to the superamplitudes, following from the analytic behavior in the above limits. We then show that, at one-loop level, the holomorphic anomaly of the tree amplitudes leads to the breakdown of dual Poincaré supersymmetry (equivalent to ordinary special conformal supersymmetry) of the ratio of the NMHV and MHV superamplitudes, but this anomaly does not affect dual conformal symmetry.

SURPRISING SYMMETRY OF SCATTERING AMPLITUDES IN GAUGE THEORIES

I will review a recent progress in computing scattering amplitudes in strongly coupled gauge theories — a fascinating subject which has been recently boosted by the formulation of the gauge/string duality in maximally supersymmetric Yang–Mills theory. In addition to the conventional symmetry of the underlying Lagrangian, the scattering amplitudes in this theory exhibit a new, dual superconformal symmetry. This symmetry is powerful enough to completely determine the scattering amplitudes for arbitrary coupling in a suitably defined limit.

Dual superconformal symmetry of scattering amplitudes in [formula omitted] super-Yang–Mills theory

We argue that all scattering amplitudes in the maximally supersymmetric N = 4 super-Yang–Mills theory possess a new, dual superconformal symmetry which extends the previously discovered dual conformal symmetry of MHV amplitudes. To reveal this property we formulate the scattering amplitudes as functions on the appropriate dual superspace. Rewritten in this form, all tree-level MHV and next-to-MHV amplitudes exhibit manifest dual superconformal symmetry. We propose a new, compact and Lorentz covariant formula for the tree-level NMHV amplitudes for arbitrary numbers and types of external particles. The dual superconformal symmetry is broken at loop level by infrared divergences. However, we provide evidence that the dual conformal anomaly of the MHV and NMHV superamplitudes is the same and, therefore, their ratio is dual conformally invariant. We show this explicitly for the six-particle amplitudes at one loop. We conjecture that these properties hold for all, MHV and non-MHV, superamplitudes in N = 4 SYM both at weak and at strong coupling.

Exacting 𝒩 = 4 superconformal symmetry

Tree level scattering amplitudes in N = 4 super Yang-Mills theory are al- most, but not exactly invariant under the free action of the N = 4 superconformal algebra. What causes the non-invariance is the holomorphic anomaly at poles where external par- ticles become collinear. In this paper we propose a deformation of the free superconformal representation by contributions which change the number of external legs. This modified classical representation not only makes tree amplitudes fully invariant, but it also leads to additional constraints from symmetry alone mediating between hitherto unrelated ampli- tudes. Moreover, in a constructive approach it appears to fully constrain all tree amplitudes when combined with dual superconformal alias Yangian symmetry.