The Betti numbers of the moduli space of stable sheaves of rank 3 on P2

TL;DR

This paper computes the Betti numbers of the moduli space of stable sheaves of rank 3 on P2 and its blow-up, using wall-crossing and expressing results via modular and theta functions.

Contribution

It provides explicit generating functions for Betti numbers of these moduli spaces, connecting algebraic geometry with physics-inspired flow trees.

Findings

Betti numbers for the moduli space on P2 derived from blow-up calculations.

Generating functions expressed in terms of modular and indefinite theta functions.

Wall-crossing techniques used to compute Betti numbers on the blow-up.

Abstract

This article computes the generating functions of the Betti numbers of the moduli space of stable sheaves of rank 3 on the projective plane P2 and its blow-up. Wall-crossing is used to obtain the Betti numbers on the blow-up. These can be derived equivalently using flow trees, which appear in the physics of BPS-states. The Betti numbers for P2 follow from those for the blow-up by the blow-up formula. The generating functions are expressed in terms of modular functions and indefinite theta functions.