Institute of Mathematics


Modul:   MAT074  Talks in mathematical physics

BPS Algebras and Generalised Kac-Moody algebras from 2-Calabi-Yau categories

Talk by Sebastian Schlegel Meija

Date: 27.11.23  Time: 13.15 - 14.15  Room: Y27H25

Associated to a 2-Calabi–Yau (2CY) abelian category – informally, a noncommutative symplectic surface – is its cohomological Hall algebra (CoHA), an algebra with underlying vector space given by the Borel–Moore homology of the moduli of objects in the category. CoHAs were originally defined by Kontsevich and Soibelman as a mathematical incarnations of Harvey and Moore’s algebras of BPS states. Following the approach of Davison and Meinhardt, BPS Lie algebras are supposed to be smaller and manageable objects which “control” the CoHA. Making this precise, I will present a Poincaré–Birkhoff–Witt type theorem for the CoHA of a 2CY category in terms of the BPS Lie algebra.A crucial step is to identify the BPS algebra as (the positive half of) a generalised Kac–Moody Lie algebra modelled on the intersection cohomology of moduli spaces. This could be interpreted as a mathematical incarnation of observations made by Harvey and Moore relating algebras of BPS states to generalised Kac–Moody algebras. Time permitting, I will mention applications to the cohomology of Nakajima quiver varieties. The talk is based on joint work with Ben Davison and Lucien Hennecart.