Microlocal perverse schobers and Radon transform

Abstract: Perverse schobers are a categorification of perverse sheaves, originally proposed by Kapranov and Schechtman. The purpose of this paper is to initiate a microlocal study of perverse schobers. We first categorify $$\operatorname{Perv}(\mathbb{C},R)/\operatorname{Loc}(\mathbb{C})$$, the category of perverse sheaves on a complex line with singular points at $R$, modulo local systems. We use this to propose a general definition for microlocal perverse schobers supported on the open conormal to a germ of a hypersurface, and we conjecture that this is invariant under Radon transform. Here we make the key observation that while the analogous quotient of perverse schobers $$\mathsf{2Perv}(\mathbb{C},R)/\mathsf{2Loc}(\mathbb{C})$$ is a reasonable categorification, the resulting theory fails to be invariant under Radon transform. In fact, our proposed categorification can be recovered by correcting an instance of this failure in a universal manner. We prove Radon invariance when our hypersurface is the curve $y<sup>m=x<sup>n$ in $\mathbb{C}<sup>2_{x,y}$. Along the way, we explain how our theory relates to Fourier transforms of perverse schobers, periodic SODs, and spherical monads.