We build upon the prior works of Arkani et al. [J. High Energy Phys. 05 (2018) 096], Banerjee et al. [J. High Energy Phys. 08 (2019) 067], and Raman [arXiv:1906.02985] to study tree-level planar amplitudes for a massless scalar field theory with polynomial interactions. Focusing on a specific example, in which the interaction is given by λ 3 ϕ 3 þ λ 4 ϕ 4 , we show that a specific convex realization of a simple polytope known as the accordiohedron in kinematic space is the positive geometry for this theory. As in the previous cases, there is a unique planar scattering form in kinematic space, associated to each positive geometry which yields planar scattering amplitudes.
In [1], two of the present authors along with P. Raman attempted to extend the Amplituhedron program for scalar field theories [2] to quartic scalar interactions. In this paper we develop various aspects of this proposal. Using recent seminal results in Representation theory [3,4], we show that projectivity of scattering forms and existence of kinematic space associahedron completely capture planar amplitudes of quartic interaction. We generalise the results of [1] and show that for any n-particle amplitude, the positive geometry associated to the projective scattering form is a convex realisation of Stokes polytope which can be naturally embedded inside one of the ABHY associahedra defined in [2,5]. For a special class of Stokes polytopes with hyper-cubic topology, we show that they have a canonical convex realisation in kinematic space as boundaries of kinematic space associahedra.We then use these kinematic space geometric constructions to write world-sheet forms for φ 4 theory which are forms of lower rank on the CHY moduli space. We argue that just as in the case of bi-adjoint φ 3 scalar amplitudes, scattering equations can be used as diffeomorphisms between certain n−4
We present a covariant phase space construction of hamiltonian generators of asymptotic symmetries with "Dirichlet" boundary conditions in de Sitter spacetime, extending a previous study of Jäger. We show that the de Sitter charges so defined are identical to those of Ashtekar, Bonga, and Kesavan (ABK). We then present a comparison of ABK charges with other notions of de Sitter charges. We compare ABK charges with counterterm charges, showing that they differ only by a constant offset, which is determined in terms of the boundary metric alone.We also compare ABK charges with charges defined by Kelly and Marolf at spatial infinity of de sitter spacetime. When the formalisms can be compared, we show that the two definitions agree. Finally, we express Kerr-de Sitter metrics in four and five dimensions in an appropriate Fefferman-Graham form.
In the last few years, there has been significant interest in understanding the stationary comparison version of the first law of black hole mechanics in the vielbein formulation of gravity. Several authors have pointed out that to discuss the first law in the vielbein formulation one must extend the Iyer–Wald Noether charge formalism appropriately. Jacobson and Mohd (2015 Phys. Rev. D
92 124010) and Prabhu (2017 Class. Quantum Grav.
34 035011) formulated such a generalisation for symmetry under combined spacetime diffeomorphisms and local Lorentz transformations. In this paper, we apply and appropriately adapt their formalism to four-dimensional gravity coupled to a Majorana field and to a Rarita–Schwinger field. We explore the first law of black hole mechanics and the construction of the Lorentz-diffeomorphism Noether charges in the presence of fermionic fields, relevant for simple supergravity.
The aim of these Lectures is to provide a brief overview of the subject of asymptotic symmetries of gauge and gravity theories in asymptotically flat spacetimes as background material for celestial holography.
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