BPS Algebraic Structures Related to Toric Calabi-Yau Manifolds

Recent studies indicate that the canonical notion of a symmetry in physics as a group might acquire a revision towards some categorification: higher form, refined, hidden, categorical symmetries and other algebro-geometric structures classifying non-perturbative states, operators and defects. In this talk I would like to concentrate on algebraic structures governing BPS spectra in effective quiver field theories describing D-brane systems on toric Calabi-Yau manifolds. The resulting algebraic structure is similar to affine Yangians and generalizes this notion beyond the Dynkin diagram classification towards what is called a quiver Yangian (and respective generalizations to trigonometric, a.k.a. quantum toroidal, and elliptic algebras). I would review some basic elements of this construction as well as some currently open questions. If time permits we will mention relations to integrability problems, constructing stable envelopes, categorification in supersymmetric models, and some puzzles about fresh developments in the theory of toric Calabi-Yau 4-folds.