Appears in collection : Mathematics Inspired by Physics

Moduli spaces $M(\beta; \chi)$ of one-dimensional sheaves on a complex K3 or abelian surface S have a rich and well-studied enumerative geometry. In this work, we prove that the so-called BPS cohomology (or Donaldson-Thomas invariants) of $M(\beta;\chi)$ is independent of $\chi$ --the Euler characteristic-- for any curve class $\beta$. We establish a relative version of this statement, conjectured by Toda in 2019, over the Chow variety of $1$-cycles on $S$. We do this by constructing an action of the cohomological Hall algebra of zero-dimensional sheaves on the BPS Lie algebra of the stack of coherent sheaves on $S$. This is joint work with B. Davison, L. Hennecart, T. Kinjo and E. Vasserot.