Open BPS wall crossing and M-theory

Consider the degeneracies of BPS bound states of one D6-brane wrapping Calabi-Yau X with D0-branes and D2-branes. When we include D4-branes wrapping Lagrangian cycles in addition, D2-branes can end on them. These give rise to new bound states in the d=2, N=(2,2) theory of the D4-branes. We call these “open” BPS states, in contrast to closed BPS states that arise from D-branes without boundaries. Lifting this to M-theory, we show that the generating function is captured by free Fock space spanned by M2-brane particles ending on M5-branes wrapping the Lagrangian. This implies that the open BPS bound states are counted by the square of the open topological string partition function on X, reduced to the corresponding chamber. Our results give new predictions for open BPS invariants and their wall crossing phenomena when we change the open and closed string moduli. We relate our results to the work of Cecotti and Vafa on wall crossing in the two-dimensional N=(2,2) theories. The findings from the crystal melting model for the open BPS invariants proposed recently fit well with the M-theory predictions.