We construct maximal supergravity in five-dimensions by ‘oxidizing’ the four-dimensional 𝒩 = 8 theory. The relevant symmetries, the unitary symplectic group USp(8) and the exceptional group E6, are both presented in light-cone superspace and their connections with SU(8) and E7 highlighted. We explain a procedure to derive higher-point interaction vertices in both the 4- and 5-dimensional supergravity theories using exclusively the exceptional symmetries. Specific forms for the quartic and quintic interaction vertices in light-cone superspace are derived.

The global symmetries in maximally supersymmetric theories of gravity in d >= 4 are shown to have a universal form in light-cone superspace. The procedure for deriving an all order expression for the d = 4 case is also discussed.

We derive cubic interaction vertices for a class of higher-derivative theories involving three arbitrary integer spin fields. This derivation uses the requirement of closure of the Poincaré algebra in four-dimensional flat spacetime. We find two varieties of permitted structures at the cubic level and eliminate one variety, which is proportional to the equations of motion, using suitable field redefinitions. We then consider soft theorems for field theories with these higher-derivative interactions and construct amplitudes in these theories using the inverse-soft approach.

Universal factors associated with the emission of a soft boson in gauge theories and gravity, formulated in the light-cone gauge, are presented. The inverse-soft method, for constructing higher-point amplitudes from lower-point ones, using these factors is reviewed. These ideas are then examined in (light-cone) superspace and applied to both the N=4 super Yang-Mills and N=8 supergravity theories. One highlight is a compact result for the quartic interaction vertex in N=8 supergravity, a crucial ingredient for finiteness analyses.

In General Relativity, the allowed set of diffeomorphisms or gauge transformations at asymptotic infinity forms the BMS group, an infinite-dimensional extension of the Poincaré group. We focus on the structure of the BMS group in two distinct forms of Hamiltonian dynamics - the instant and front forms. Both similarities and differences in these two forms are examined while emphasising the role of non-covariant approaches to symmetries in gravity.

The first truly non-MHV interaction vertices in the light-cone formulation of pure gravity appear at order 6. From a closed form expression, for gravitation in the light-cone gauge, we extract and present all 6-point interaction vertices. We invoke symmetry arguments to explain the structure of these vertices. Symmetry considerations also allow us to place constraints on the structure of all even- and odd-point vertices in the theory. The origin of MHV vertices within this formalism is also discussed.

We analyze the residual gauge freedom in gravity, in four dimensions, in the light-cone gauge, in a formulation where unphysical fields are integrated out. By checking the invariance of the light-cone Hamiltonian, we obtain a set of residual gauge transformations, which satisfy the BMS algebra realized on the two physical fields in the theory. Hence, the BMS algebra appears as a consequence of residual gauge invariance in the bulk and not just at the asymptotic boundary. We highlight the key features of the light-cone BMS algebra and discuss its connection with the quadratic form structure of the Hamiltonian.

Supersymmetric Yang-Mills theory is formulated in six dimensions, without the use of anti-commuting variables. This is achieved using a new Nicolai map, to third order in the coupling constant. This is the second such map in six dimensions and highlights a potential ambiguity in the formalism.

A bstract We analyze possible local extensions of the Poincaré symmetry in light-cone gravity in four dimensions. We use a formalism where we represent the algebra on the two physical degrees of freedom, one with helicity 2 and the other with helicity − 2. The representation is non-linearly realized and one of the light-cone momenta is the Hamiltonian, which is hence a non-linear generator of the algebra. We find that this can be locally realized and the Poincaré algebra extended to the BMS symmetry without any reference to asymptotic limits.

The light-cone Hamiltonians for spin 1 and spin 2 fields, describing both the pure and the maximally supersymmetric theories, may be expressed as quadratic forms. In this paper, we show that this feature extends to light-cone higher spin theories. To first order in the coupling constant, we prove that the higher spin Hamiltonians, with and without supersymmetry, are quadratic forms. Scattering amplitude structures emerge naturally in this framework and we relate the momentum space vertex in a supersymmetric higher spin theory to the corresponding vertex in the $$ \mathcal{N} $$ = 4 Yang-Mills theory.

In this paper, we take up an old thread of development concerning the characterization of supersymmetric theories without any use of anticommuting variables that goes back to one of the authors' very early work [1]. Our special focus here will be on the formulation of supersymmetric Yang-Mills theories, extending previous results beyond $D=4$ dimensions. This perspective is likely to provide new insights into these theories, and in particular the maximally extended $N=4$ theory. As a new result we present a novel derivation of the admissible dimensions for interacting (pure) super-Yang-Mills theories to exist. This article is dedicated to the memory of Peter Freund, amongst many other things an early contributor to supersymmetry, and an author of one of the very first papers on superconformal gauge theories [2]. The final section contains some personal reminiscences of H.N.'s encounters with Peter Freund.

The action integral contains more information than the equations of motion. Since it is an integral, changes of the integration variables occasionally also expose symmetries more easily than working directly with the equations of motion. We have previously shown that there are signs of an extended exceptional symmetry for \( \mathcal{N}=8 \) supergravity in four dimensions. The symmetry is such that the fields used in the Lagrangian are not representations of the symmetry. Instead one has to add representations to obtain a representation of the extended symmetry group. In this paper we discuss an extended symmetry in four-dimensional gravity which is the “Ehlers Symmetry” in three dimensions. It cannot be spanned by the helicity states of four-dimensional gravity but it can be realised once we treat the helicity states just as field variables of the functional integral, which can be changed like variables in any integral. We also explain how this symmetry is inherent in formulations of \( \mathcal{N}=8 \) supergravity in four dimensions through a truncation in the field space to pure gravity. The establishment of these “hidden” symmetries should play an important role in the perturbative behaviour of the quantum theories. Since the method used n this paper is purely algebraic we will not provide any information on the geometric role of these symmetries.

We argue that N = 8 supergravity in four dimensions exhibits an exceptional E8(8) symmetry, enhanced from the known E7(7) invariance. Our procedure to demonstrate this involves dimensional reduction of the N = 8 theory to d = 3, a field redefinition to render the E8(8) invariance manifest, followed by dimensional oxidation back to d = 4.

We show that N = 8 supergravity may possess an even larger symmetry than previously believed. Such an enhanced symmetry is needed to explain why this theory of gravity exhibits ultraviolet behaviour reminiscent of the finite N = 4 Yang-Mills theory. We describe a series of three steps that leads us to this result.

We describe progress in using the field theory of tensionless strings to arrive at a Lagrangian for the six-dimensional \( \mathcal{N}=\left(2,0\right) \) conformal theory. We construct the free part of the theory and propose an ansatz for the cubic vertex in light-cone superspace. By requiring closure of the (2, 0) supersymmetry algebra, we fix the cubic vertex up to two parameters.

We derive the quartic interaction vertex of pure Yang–Mills theory by demanding closure of the light-cone Poincaré algebra in four-dimensional Minkowski spacetime. This calculation explicitly shows why structure constants must satisfy the Jacobi identity. We prove that there is no correction to the spin generator, for spin one, at this order. We comment briefly on higher spin fields in this context.

We argue that \( \mathcal{N}=8 \) supergravity in four dimensions exhibits an exceptional E8(8) symmetry, enhanced from the known E7(7) invariance. Our procedure to demonstrate this involves dimensional reduction of the \( \mathcal{N}=8 \) theory to d = 3, a field redefinition to render the E8(8) invariance manifest, followed by dimensional oxidation back to d = 4.

We review the derivation of light-cone interaction vertices for fermionic and bosonic fields of arbitrary spin. The resulting amplitudes and their factorization properties are discussed. We then show how this symmetry-based approach works for theories with extended supersymmetry like N = 4 Yang-Mills theory and N = 8 supergravity.

The light-cone Hamiltonians describing both pure (\( \mathcal{N} \) = 0) Yang-Mills and \( \mathcal{N} \) = 4 super Yang-Mills may be expressed as quadratic forms. Here, we show that this feature extends to theories of gravity. We demonstrate how the Hamiltonians of both pure gravity and \( \mathcal{N} \) = 8 supergravity, in four dimensions, may be written as quadratic forms. We examine the effect of residual reparametrizations on the Hamiltonian and the resulting quadratic form.