Wall Crossing and M-theory
Mina Aganagic, Hirosi Ooguri, Cumrun Vafa, Masahito Yamazaki
TL;DR
This work develops a M-theory-based framework to count BPS bound states of D6–D2–D0 branes on a Calabi–Yau 3-fold X without compact 4-cycles, showing that the BPS spectrum forms a free Fock space whose generators are M2-BPS states. The authors derive a chamber-dependent relation ${ m Z}_{ m BPS} = { m Z}_{ m top}^2|_{ m chamber}$, connecting D-brane counting to the topological string partition function and GV/DT invariants across all Kähler chambers. They apply the formalism to explicit geometries (resolved conifold, generalized conifolds, and a non-toric example), reproducing DT, noncommutative DT, and PT results in appropriate chambers and identifying the wall structure. Moreover, they derive the Denef–Moore semi-primitive wall-crossing formula within this M-theory setup, providing a unifying explanation for how topological string data governs D6–D2–D0 degeneracies and their jumps across walls. The framework thus offers a robust, geometry-independent method for BPS counting in Calabi–Yau backgrounds with no compact 4-cycles and clarifies the interplay between wall crossing and topological string theory.
Abstract
We study BPS bound states of D0 and D2 branes on a single D6 brane wrapping a Calabi-Yau 3-fold X. When X has no compact 4-cyles, the BPS bound states are organized into a free field Fock space, whose generators correspond to BPS states of spinning M2 branes in M-theory compactified down to 5 dimensions by a Calabi-Yau 3-fold X. The generating function of the D-brane bound states is expressed as a reduction of the square of the topological string partition function, in all chambers of the Kahler moduli space.