Stability and duality in N=2 supergravity
Jan Manschot
TL;DR
This work tests whether wall-crossing of BPS spectra in 4d $\mathcal{N}=2$ supergravity is compatible with $S$-duality and electric-magnetic duality by analyzing D4-D2-D0 bound states in the large-volume limit of a Calabi-Yau threefold. The author constructs the supergravity partition function $\mathcal{Z}(\tau,C,t)$ and decomposes it into a CFT part $\mathcal{Z}_\mathrm{CFT}$ and a wall-crossing part $\mathcal{Z}_\mathrm{wc}$, introducing a two-centered contribution whose generating function requires a mock Siegel-Narain theta completion $\Psi_{\mu_{1\oplus 2}}^*$. This completed two-centered piece transforms precisely as the CFT elliptic genus under $S$-duality, preserving modularity while showing that the spectral-flow symmetry of the single-center CFT is generically broken by wall-crossing. The results demonstrate that modularity and duality can be maintained in the presence of wall-crossing, yielding a continuous, duality-consistent framework for analyzing entropy enigmas and guiding future refinements to BPS invariants in $\mathcal{N}=2$ gravity.
Abstract
The BPS-spectrum is known to change when moduli cross a wall of marginal stability. This paper tests the compatibility of wall-crossing with S-duality and electric-magnetic duality for N=2 supergravity. To this end, the BPS-spectrum of D4-D2-D0 branes is analyzed in the large volume limit of Calabi-Yau moduli space. Partition functions are presented, which capture the stability of BPS-states corresponding to two constituents with primitive charges and supported on very ample divisors in a compact Calabi-Yau. These functions are `mock modular invariant' and therefore confirm S-duality. Furthermore, wall-crossing preserves electric-magnetic duality, but is shown to break the `spectral flow' symmetry of the N=(4,0) CFT, which captures the degrees of freedom of a single constituent.