BPS algebras, and Yangians

by Ben Davison (Edinburgh)

Thursday 5 Dec 2024, 16:15 → 17:15 Europe/Berlin

H3 (Geomatikum)

Yangians are quantum groups that arise naturally in the study of quantum integrable systems. Typically, the job of defining and describing a quantum group is considered a topic in algebra, and so when one reads about them, the first things one meets are generators and relations, followed by some representation theory of the resulting algebra and some motivation for studying it. Such motivation typically comes with the construction of an R matrix, or a solution to the Yang-Baxter equation, providing a kind of braiding of the tensor category of modules that enables us to build interesting knot invariants, for example. Somewhat more recently, an approach to Yangians that starts with geometry has been pioneered by Maulik and Okounkov. In this approach, the R matrix is built out of pure symplectic geometry of quiver varieties, and the cohomology of these quiver varieties then forms a very natural category of modules for the Yangian-type algebra that is reverse-engineered from this R matrix. While this is conceptually very satisfying, producing algebras that are made out of pure geometric representation theory, this way of doing things leaves fundamental questions unanswered, such as: how big are the graded pieces of these algebras, what are their generators, relations, etc. More recently still, in joint papers with Botta, and with Hennecart and Schlegel Mejia, we have answered most of these questions by relating these new algebras to BPS algebras; these are certain cohomological Hall algebras. In particular, dimension counts are given by refined BPS invariants of certain noncommutative Calabi-Yau threefolds, enabling us to compute them and confirm a conjecture of Okounkov regarding them.