In this note, we discuss the notion of symmetric self-duality of shaded planar algebras, which allows us to lift shadings on subfactor planar algebras to obtain Z/2Z-graded unitary fusion categories. This finishes the proof that there are unitary fusion categories with fusion graphs 4442 and 3333.

We re-examine the constraints imposed by causality and unitarity on the low-energy effective field theory expansion of four-particle scattering amplitudes, exposing a hidden “totally positive” structure strikingly similar to the positive geometries associated with grassmannians and amplituhedra. This forces the infinite tower of higher-dimension operators to lie inside a new geometry we call the “EFT-hedron”. We initiate a systematic investigation of the boundary structure of the EFT-hedron, giving infinitely many linear and non-linear inequalities that must be satisfied by the EFT expansion in any theory. We illustrate the EFT-hedron geometry and constraints in a wide variety of examples, including new consistency conditions on the scattering amplitudes of photons and gravitons in the real world.

We show that the angles between Teichmüller geodesic rays issuing from a common point, defined by using the law of cosines, do not always exist. The proof uses an estimation for the Teichmüller distance on finite dimensional Teichmüller spaces. As a consequence, the Teichmüller space equipped with the Teichmüller metric is not a CAT(k) space for any k∈R . We also discuss some necessary conditions for the existence of angle between the Teichmüller geodesics.

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