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PUBLIC VIVA-VOCE NOTIFICATION Date and Time : Tuesday, 30 November 2021, 11.00 am Mathematical aspects of gravitational physics and scattering amplitudes Aneesh P B Chennai Mathematical Institute. 30-11-21 Abstract This thesis is split into two parts. Part I 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 Jager. 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, where one must extend the Iyer-Wald Noether charge formalism appropriately. Jacobson and Mohd, and Prabhu formulated such a generalisation for symmetry under combined spacetime diffeomorphisms and local Lorentz transformations. 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. Part II We build upon the prior works of N. Arkani-Hamed et al., P. Banerjee et al., P. Raman et al., to study tree-level planar amplitudes for a massless scalar field theory with polynomial interactions. Focusing on a specific example, where the interaction is given by ð3 ð3 + ð4 ð4 , 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. P. Banerjee et al., attempted to extend the Amplituhedron program for scalar field theories to quartic scalar interactions. We develop various aspects of this proposal. Using recent seminal results in Representation theory, 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 P. Banerjee et al. 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 N. Arkani-Hamed et al., V. Bazier-Matte et al.. 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 ð4 theory which are forms of lower rank on the CHY moduli space. We argue that just as in the case of bi-adjoint ð3 scalar amplitudes, scattering equations can be used as diffeomorphisms between certain (n−4)/2 forms on the world-sheet and (n−4)/2 forms on ABHY associahedron that generate quartic amplitudes. All are invited to attend the viva-voce examination.
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