In a recent paper with Kathlén Kohn, Ragni Piene, Kristian Ranestad, Felix Rydell, Boris Shapiro, Miruna-Stefana Sorea, and Simon Telen, we conjectured, based on numerical computation, that a general quartic plane curve is the adjoint of 864 heptagons in the plane. The adjoint curve of a polygon is the numerator of its canonical function appearing in the context of positive geometry. The goal of the talk is to explain the context of this result and give a proof of this number via intersection theory. This is joint work with Daniele Agostini, Daniel Plaumann, and Jannik Wesner.
N = 4 super Yang-Mills is the most symmetric of all four-dimensional quantum field theories. The simplest local operators in this theory are the so-called half-BPS operators. Correlators of these operators play a key role in the AdS/CFT correspondence and impact many key areas in current theoretical physics. In a certain limit, they relate to amplitudes through the so-called correlator/amplitude duality. On the other hand amplitudes in N=4 can be computed from the amplituhedron, a geometrical object whose boundaries encode the singularity structure of the amplitude. In this talk, I will review recent progress in building a geometrical analogue of the amplituhedron for correlators, named the correlahedron, by studying the correlator/amplitude duality. The talk is based on arXiv:2106.09372.
This talk will consist of two parts. In the first part I will review how to compute n-point scattering amplitudes in the $\phi^3$ theory from three different approaches. The first approach is using Feynman diagrams, which is the standard way to compute these observables in physics. Then I will explain how the tropical Grassmannian, TropG(2,n), is closely related to the space of the Feynman diagrams for this theory, and I will propose a formula as an integral over TropG(2,n) to compute first $\phi^3$ and then $\phi^p$ scattering amplitudes, for any p>2. The third approach is the CHY formula, which is an integral over the moduli space of n points on $CP^1$. In the second part of the talk, motivated by the isomorphism between the Grassmannians G(2,n) and G(n-2,n), I will explain how to generalize the CHY formula to an integral over the space of n points on higher dimensional projective spaces $CP^{k-1}$, thus providing a natural generalization of the notion of scattering amplitudes. I will also point out connections with higher dimensional tropical Grassmannians and will comment on some of the features of these new objects.
Scattering amplitudes in string theory are computed by integrals over moduli spaces of curves. At genus-zero, we can write the amplitudes of closed strings as a bilinear combination of amplitudes of open strings -- a fact physicists refer to as "the double copy."
I will recount the genus-zero story of the double copy with a view towards a genus-one realization.
