Paul Heslop

MHV amplitudes in N=4 super Yang-Mills and Wilson loops

Abstract It is a remarkable fact that MHV amplitudes in maximally supersymmetric Yang–Mills theory at arbitrary loop order can be written as the product of the tree amplitude with the same helicity configuration and a universal, helicity-blind function of the kinematic invariants. In this note we show how for one-loop MHV amplitudes with an arbitrary number of external legs this universal function can be derived using Wilson loops. Our result is in precise agreement with the known expression for the infinite sequence of MHV amplitudes in N = 4 super-Yang–Mills. In the four-point case, we are able to reproduce the expression of the amplitude to all orders in the dimensional regularisation parameter ϵ. This prescription disentangles cleanly infrared divergences and finite terms, and leads to an intriguing one-to-one mapping between certain Wilson loop diagrams and (finite) two-mass easy box functions.

Note on dual superconformal symmetry of the N=4 super Yang-Mills S matrix

We present a supersymmetric recursion relation for tree-level scattering amplitudes in N=4 super Yang-Mills. Using this recursion relation, we prove that the tree-level S matrix of the maximally supersymmetric theory is covariant under dual superconformal transformations. We further analyze the consequences that the transformation properties of the trees under this symmetry have on those of the loops. In particular, we show that the coefficients of the expansion of generic one-loop amplitudes in a basis of pseudoconformally invariant scalar box functions transform covariantly under dual superconformal symmetry, and in exactly the same way as the corresponding tree-level amplitudes.

Diagonal multi-matrix correlators and BPS operators in N=4 SYM

We present a complete basis of multi-trace multi-matrix operators that has a diagonal two point function for the free matrix field theory at finite N. This generalises to multiple matrices the single matrix diagonalisation by Schur polynomials. Crucially, it involves intertwining the gauge group U(N) and the global symmetry group U(M) with Clebsch-Gordan coefficients of symmetric groups Sn. When applied to = 4 super Yang-Mills we consider the U(3) subgroup of the full symmetry group. The diagonalisation allows the description of a dual basis to multi-traces, which permits the characterisation of the metric on operators transforming in short representations at weak coupling. This gives a framework for the comparison of quarter and eighth-BPS giant gravitons of AdS5 × S5 spacetime to gauge invariant operators of the dual = 4 SYM.

Two-loop polygon Wilson loops in N = 4 SYM

We compute for the first time the two-loop corrections to arbitrary n-gon lightlike Wilson loops in = 4 supersymmetric Yang-Mills theory, using efficient numerical methods. The calculation is motivated by the remarkable agreement between the finite part of planar six-point MHV amplitudes and hexagon Wilson loops which has been observed at two loops. At n = 6 we confirm that the ABDK/BDS ansatz must be corrected by adding a remainder function, which depends only on conformally invariant ratios of kinematic variables. We numerically compute remainder functions for n = 7,8 and verify dual conformal invariance. Furthermore, we study simple and multiple collinear limits of the Wilson loop remainder functions and demonstrate that they have precisely the form required by the collinear factorisation of the corresponding two-loop n-point amplitudes. The number of distinct diagram topologies contributing to the n-gon Wilson loops does not increase with n, and there is a fixed number of ``master integrals, which we have computed. Thus we have essentially computed general polygon Wilson loops, and if the correspondence with amplitudes continues to hold, all planar n-point two-loop MHV amplitudes in the = 4 theory.

The super-correlator/super-amplitude duality: Part II

We extend the recently discovered duality between MHV amplitudes and the light-cone limit of correlation functions of a particular type of local scalar operators to generic nonMHV amplitudes in planar N = 4 SYM theory. We consider the natural generalization of the bosonic correlators to super-correlators of stress-tensor multiplets and show, in a number of examples, that their light-cone limit exactly reproduces the square of the matching super-amplitudes. Our correlators are computed at Born level. If all of their points form a light-like polygon, the correlator is dual to the tree-level amplitude. If a subset of points are not on the polygon but are integrated over, they become Lagrangian insertions generating the loop corrections to the correlator. In this case the duality with amplitudes holds at the level of the integrand. We build up the superspace formalism needed to formulate the duality and present the explicit example of the n−point NMHV tree amplitude as the dual of the lowest nilpotent level in the correlator. Unité de Recherche Associée au CNRS URA 2306 Laboratoire d’Annecy-le-Vieux de Physique Théorique, UMR 5108

Superfield representations of superconformal groups

Representations of four-dimensional superconformal groups are constructed as fields on many different superspaces, including super Minkowski space, chiral superspace, harmonic superspace and analytic superspace. Any unitary irreducible representation can be given as a field on any one of these spaces if we include fields which transform under supergroups. In particular, on analytic superspaces, the fields are unconstrained. One can obtain all representations of the N = 4 complex superconformal group PSL(4|4) with integer dilation weight from copies of the Maxwell multiplet on (4, 2, 2) analytic superspace. This construction is compared with the oscillator construction and it is shown that there is a natural correspondence between the oscillator construction of superconformal representations and those carried by superfields on analytic superspace.Representations of four dimensional superconformal groups are constructed as fields on many different superspaces, including super Minkowski space, chiral superspace, harmonic superspace and analytic superspace. Any unitary irreducible representation can be given as a field on any one of these spaces if we include fields which transform under supergroups. In particular, on analytic superspaces, the fields are unconstrained. One can obtain all representations of the N=4 complex superconformal group

Diagonal free field matrix correlators, global symmetries and giant gravitons

PSL(4|4)

Integral invariants in N = 4 SYM and the effective action for coincident D-branes

with integer dilation weight from copies of the Maxwell multiplet on

OPEs and 3-point correlators of protected operators in N=4 SYM

(4,2,2)

Bootstrapping correlation functions in N=4 SYM

analytic superspace. This construction is compared with the oscillator construction and it is shown that there is a natural correspondence between the oscillator construction of superconformal representations and those carried by superfields on analytic superspace.