Dynamical Contents of Unconventional Supersymmetry

The Dirac Hamiltonian formalism is applied to a system in $(2+1)$-dimensions consisting of a Dirac field $\psi$ minimally coupled to Chern-Simons $U(1)$ and $SO(2,1)$ connections, $A$ and $\omega$, respectively. This theory is connected to a supersymmetric Chern-Simons form in which the gravitino has been projected out (unconventional supersymmetry) and, in the case of a flat background, corresponds to the low energy limit of graphene. The separation between first-class and second-class constraints is performed explicitly, and both the field equations and gauge symmetries of the Lagrangian formalism are fully recovered. The degrees of freedom of the theory in generic sectors shows that the propagating states correspond to fermionic modes in the background determined by the geometry of the graphene sheet and the nondynamical electromagnetic field. This is shown for the following canonical sectors: i) a conformally invariant generic description where the spinor field and the dreibein are locally rescaled; ii) a specific configuration for the Dirac fermion consistent with its spin, where Weyl symmetry is exchanged by time reparametrizations; iii) the vacuum sector $\psi=0$, which is of interest for perturbation theory. For the latter the analysis is adapted to the case of manifolds with boundary, and the corresponding Dirac brackets together with the centrally extended charge algebra are found. Finally, the $SU(2)$ generalization of the gauge group is briefly treated, yielding analogous conclusions for the degrees of freedom.Comment: 17 pages. Accepted version for publication in JHE

Scattering of Spinning Black Holes from Exponentiated Soft Factors

We provide evidence that the classical scattering of two spinning black holes is controlled by the soft expansion of exchanged gravitons. We show how an exponentiation of Cachazo-Strominger soft factors, acting on massive higher-spin amplitudes, can be used to find spin contributions to the aligned-spin scattering angle, conjecturally extending previously known results to higher orders in spin at one-loop order. The extraction of the classical limit is accomplished via the on-shell leading-singularity method and using massive spinor-helicity variables. The three-point amplitude for arbitrary-spin massive particles minimally coupled to gravity is expressed in an exponential form, and in the infinite-spin limit it matches the effective stress-energy tensor of the linearized Kerr solution. A four-point gravitational Compton amplitude is obtained from an extrapolated soft theorem, equivalent to gluing two exponential three-point amplitudes, and becomes itself an exponential operator. The construction uses these amplitudes to: 1) recover the known tree-level scattering angle at all orders in spin, 2) recover the known one-loop linear-in-spin interaction, 3) match a previous conjectural expression for the one-loop scattering angle at quadratic order in spin, 4) propose new one-loop results through quartic order in spin. These connections link the computation of higher-multipole interactions to the study of deeper orders in the soft expansion.Comment: 29 pages + appendices + refs, 3 figures; v3 minor corrections, journal versio

Towards Gravity From a Color Symmetry

Using tools from color-kinematics duality we propose a holographic construction of gravitational amplitudes, based on a 2d Kac-Moody theory on the celestial sphere. In the $N\to \infty$ limit the gauge group corresponds to $w_{1+\infty}$, due to the $U(N)$ generators enjoying a simple quantum group structure, which is in turn inherited from a twistor fiber over the celestial sphere. We show how four-dimensional momentum-space is emergent in this picture, which connects directly to the so-called kinematic algebra of the tree-level S-Matrix. On the other hand, the framework can be embedded within a celestial CFT to make contact with holographic symmetry algebras previously observed in the soft expansion. Kac-Moody currents play the role of a graviton to all orders in such expansion, and also lead to a natural notion of Goldstone modes for $w_{1+\infty}$. Focusing on MHV amplitudes, main examples are a BCFW type recursion relation and holomorphic three-point amplitudes.Comment: 5 pages + 5 pages ap

Δ\Delta-Algebra and Scattering Amplitudes

In this paper we study an algebra that naturally combines two familiar operations in scattering amplitudes: computations of volumes of polytopes using triangulations and constructions of canonical forms from products of smaller ones. We mainly concentrate on the case of $G(2,n)$ as it controls both general MHV leading singularities and CHY integrands for a variety of theories. This commutative algebra has also appeared in the study of configuration spaces and we called it the $\Delta$-algebra. As a natural application, we generalize the well-known square move. This allows us to generate infinite families of new moves between non-planar on-shell diagrams. We call them sphere moves. Using the $\Delta$-algebra we derive familiar results, such as the KK and BCJ relations, and prove novel formulas for higher-order relations. Finally, we comment on generalizations to $G(k,n)$.Comment: 36+13 page

Scattering Equations: From Projective Spaces to Tropical Grassmannians

We introduce a natural generalization of the scattering equations, which connect the space of Mandelstam invariants to that of points on ${\mathbb{CP}^1}$, to higher-dimensional projective spaces $\mathbb{CP}^{k-1}$. The standard, $k=2$ Mandelstam invariants, $s_{ab}$, are generalized to completely symmetric tensors $\textsf{s}_{a_1a_2\ldots a_k}$ subject to a `massless' condition $\textsf{s}_{a_1a_2\cdots a_{k-2}\,b\,b}=0$ and to `momentum conservation'. The scattering equations are obtained by constructing a potential function and computing its critical points. We mainly concentrate on the $k=3$ case: study solutions and define the generalization of biadjoint scalar amplitudes. We compute all `biadjoint amplitudes' for $(k,n)=(3,6)$ and find a direct connection to the tropical Grassmannian. This leads to the notion of $k=3$ Feynman diagrams. We also find a concrete realization of the new kinematic spaces, which coincides with the spinor-helicity formalism for $k=2$, and provides analytic solutions analogous to the MHV ones.Comment: 27+7 pages. v2: typos corrected. Connection to trop G(3,7) added at end of section 4. Appendix with numerical seeds for all solutions to X(3,6) equations provide

The S-Matrix of Gauge and Gravity Theories and The Two-Black Hole Problem

This thesis is devoted to diverse aspects of scattering amplitudes in gauge theory and gravity including interactions with matter particles. In Part I we focus on the applications of massive scattering amplitudes in gravity to the Black Hole two-body problem. For this we construct a classical limit putting especial emphasis on the multipole expansion of certain massive amplitudes, which we will use to model spinning black holes in a large distance effective regime or particle approximation. In Part II we study scattering amplitudes in six dimensions, and construct a compact formula analogous to the four-dimensional Witten-RSV/rational maps formulation. This provides a supersymmetric extension of moduli space localization formulae such as the CHY integral. We explore the cases of Super Yang-Mills and Maximal Supergravity theories, among others

Mapping International Relations: A Model for Analysis

The main goal of this article is to build up an analytic model that could be used as a tool for mapping contexts of international relations phenomena thus to help observers obtain a broad and detailed view of their area of interest For any study of international relations the context it is of most importance given that this field of research analyses human phenomena of large scale The international relations involve the widest level of human relations because such relations are the result of a large number of interconnected variables Independent of the way of approach in this research area either structural regional domestic or even individual the phenomena observed from international relations perspective are always linked to a context that cannot be ignored In most cases the context itself if well mapped and interpreted - contains most of the answers for questions posed To accomplish the goal presented we will present in a theoretical discussion of analysis models from the field of political science specifically the multiple arenas model from Tsebelis 1990 combined with some concepts of Alisson and Zelikow s Bureaucratic model 1990 Those concepts will be mixed with the two- level game logic from Robert Putnam 198