In a remarkable recent work [1], the amplituhedron program was extended to the realm of nonsupersymmetric scattering amplitudes. In particular it was shown that for tree-level planar diagrams in massless φ 3 theory (and its close cousin, bi-adjoint φ 3 theory) a polytope known as the associahedron sits inside the kinematic space and is the amplituhedron for the theory. Precisely as in the case of amplituhedron, it was shown that scattering amplitude can be obtained from the canonical form associated to the Associahedron. Combinatorial and geometric properties of associahedron naturally encode properties like locality and unitarity of (tree level) scattering amplitudes. In this paper we attempt to extend this program to planar amplitudes in massless φ 4 theory. We show that tree-level planar amplitudes in this theory can be obtained from geometry of objects known as the Stokes polytope which sits naturally inside the kinematic space. As in the case of associahedron we show that the canonical form on these Stokes polytopes can be used to compute scattering amplitudes for quartic interactions. However unlike associahedron, Stokes polytope of a given dimension is not unique and as we show, one must sum over all of them to obtain the complete scattering amplitude.Not all Stokes polytopes contribute equally and we argue that the corresponding weights depend on purely combinatorial properties of the Stokes polytopes. As in the case of φ 3 theory, we show how factorization of Stokes polytope implies unitarity and locality of the amplitudes.

We propose a method to holographically compute the conformal partial waves in any decomposition of correlation functions of primary operators in conformal field theories using open Wilson network operators in the holographic gravitational dual. The Wilson operators are the gravitational ones where gravity is written as a gauge theory in the first order Hilbert-Palatini formalism. We apply this method to compute the global conformal blocks and partial waves in 2d CFTs reproducing many of the known results.

Starting with the seminal work of Arkani-Hamed et al [1], in [2], the "Amplituhedron program" was extended to analyzing (planar) amplitudes in massless φ 4 theory. In this paper we show that the program can be further extended to include φ p (p > 4) interactions. We show that tree-level planar amplitudes in these theories can be obtained from geometry of polytopes called accordiohedron which naturally sits inside kinematic space. As in the case of quartic interactions the accordiohedron of a given dimension is not unique, and we show that a weighted sum of residues of the canonical form on these polytopes can be used to compute scattering amplitudes. We finally provide a prescription to compute the weights and demonstrate how it works in various examples.1

We explore the correspondence between geometric function theory (GFT) and quantum field theory (QFT). The crossing symmetric dispersion relation provides the necessary tool to examine the connection between GFT, QFT, and effective field theories (EFTs), enabling us to connect with the crossing-symmetric EFT-hedron. Several existing mathematical bounds on the Taylor coefficients of Typically Real functions are summarized and shown to be of enormous use in bounding Wilson coefficients in the context of 2-2 scattering. We prove that two-sided bounds on Wilson coefficients are guaranteed to exist quite generally for the fully crossing symmetric situation. Numerical implementation of the GFT constraints (Bieberbach-Rogosinski inequalities) is straightforward and allows a systematic exploration. A comparison of our findings obtained using GFT techniques and other results in the literature is made. We study both the three-channel as well as the two-channel crossing-symmetric cases, the latter having some crucial differences. We also consider bound state poles as well as massless poles in EFTs. Finally, we consider nonlinear constraints arising from the positivity of certain Toeplitz determinants, which occur in the trigonometric moment problem.