Algebra of the Infrared, secondary polytopes and perverse schobers

Abstract: This survey paper, based on a talk at the International Congress of Basic Science in Beijing in July 2025, summarizes joint work of the authors with M. Kontsevich [1408.2673] establishing the relation between the ``Algebra of the Infrared" of D. Gaiotto, G. Moore and E. Witten [1506.04087] and the theory of secondary polytopes introduced in the 1990s in the study of higher-dimensional discriminants. It also summarizes subsequent work with L. Soukhanov [2011.00845] where the tunneling data were observed to be similar to linear algebra data describing perverse sheaves on the complex plane except that in the physical context vector spaces are replaced by triangulated categories. The relevant concept here is that of perverse schobers, which are conjectural categorical analogs of perverse sheaves proposed by M. Kapranov and V. Schechtman [1411.2772]. Finally, we sketch a research program of extending these ideas to $4$-dimensional theories and the resurgence formalism.