Wall crossing in local Calabi Yau manifolds

TL;DR

The paper investigates how BPS partition functions for D6/D2/D0 bound states on a local Calabi–Yau manifold respond to wall-crossing, revealing that an extra real moduli parameter $\varphi$ must be included in the extended Kahler moduli space to capture all walls of marginal stability. Using a supergravity framework alongside toric Donaldson–Thomas results, the authors derive partition functions in multiple chambers and show that, in appropriate limits, these match Szendrői's DT invariants for the noncommutative conifold and the standard DT invariants at large radius. The analysis exposes both the power and the limits of supergravity methods for BPS counting and wall-crossing, particularly near threshold stability walls where the halo picture and moduli-space geometry become delicate. The work also connects physical wall-crossing phenomena to established DT computations, clarifying the chamber structure of local Calabi–Yau theories and highlighting avenues for intrinsic stability concepts beyond supergravity.

Abstract

We study the BPS states of a D6-brane wrapping the conifold and bound to collections of D2 and D0 branes. We find that in addition to the complexified Kahler parameter of the rigid sphere it is necessary to introduce an extra real parameter to describe BPS partition functions and marginal stability walls. The supergravity approach to BPS state-counting gives a simple derivation of results of Szendroi concerning Donaldson-Thomas theory on the noncommutative conifold. This example also illustrates some interesting limitations on the supergravity approach to BPS state-counting and wall-crossing.