BPS invariants of the projective plane and modular forms

BPS invariants capture topological invariants of moduli spaces of
sheaves. After a brief review of BPS invariants of vector bundles over
curves, I'll discuss generating functions of BPS invariants of semi-stable
sheaves on the projective plane and its blow-up. These generating functions
transform as (mock) modular forms, which is a manifestation of the physical
electric-magnetic duality.