Bound state transformation walls

TL;DR

This work introduces bound state transformation walls (BST walls) in 4D ${\cal N}=2$ supergravity to resolve paradoxes that arise when adiabatically moving moduli takes BPS bound states from marginal to anti-marginal stability without energy nonconservation. BST walls describe two distinct phenomena, recombination and conjugation, which alter the attractor flow-tree realization of a bound state while preserving the BPS index; conjugation is tied to monodromy around singular loci and triggers Fermi flips and fadeouts of halo contributions, whereas recombination re-clusters constituents without changing the overall index. The authors develop an attractor-flow-tree framework to locate BST walls, analyze halo states, and derive constraints on the spectrum, including massless vectormultiplet sectors and conifold-type singularities. They illustrate the framework with concrete examples (conifold-like singularities, the FHSV model, and extremal transitions) and discuss how the index and spin content remain continuous across BST walls, while the underlying Hilbert spaces reorganize through monodromy and halo dynamics. The results yield new consistency conditions on BPS spectra, connect to Kontsevich–Soibelman wall-crossing ideas in a companion paper, and illuminate how massless sectors at finite distance shape general wall-crossing phenomena in string/M-theory compactifications.

Abstract

In four dimensional N=2 supergravity theories, BPS bound states near marginal stability are described by configurations of widely separated constituents with nearly parallel central charges. When the vacuum moduli can be dialed adiabatically until the central charges become anti -parallel, a paradox arises. We show that this paradox is always resolved by the existence of "bound state transformation walls" across which the nature of the bound state changes, although the index does not jump. We find that there are two distinct phenomena that can take place on these walls, which we call recombination and conjugation. The latter is associated to the presence of singularities at finite distance in moduli space. Consistency of conjugation and wall-crossing rules near these singularities leads to new constraints on the BPS spectrum. Singular loci supporting massless vector bosons are particularly subtle in this respect. We argue that the spectrum at such loci necessarily contains massless magnetic monopoles, and that bound states around them transform by intricate hybrids of conjugation and recombination.

Figures (16)

• Figure 1: BPS bound states appear to be adiabatically transportable from marginal stability to anti-marginal stability keeping $R>0$, violating conservation of energy.
• Figure 2: Recombination: Constituents rearrange themselves into different clusters. The example represents a family of configurations with $A$ tightly bound to $B$ evolving into a family with $A$ tightly bound to $C$, and a family with $C$ tightly bound to $B$. The corresponding attractor flow tree evolves from an $((A,B),C)$ tree to a $((C,A),B)$ tree plus a $((B,C),A)$ tree. At the transition point, the flow tree has two 3-valent vertices coalescing into a 4-valent vertex. The recombination wall is the blue line with the asterisk next to it.
• Figure 4: Elevation: The initially BPS-saturated minimum of the interaction potential $V(r)$ gets lifted, and the bound state becomes classically non-BPS. The corresponding flow trees are shown on the right. The blue line with the asterisk is the elevation wall. It corresponds to a critical attractor flow hitting a locus in (a suitable finite cover of) moduli space where the mass of one of the constituent particles vanishes but there is no charge monodromy around it, as is the case for example if an ${\cal N}=4$ vector multiplet becomes massless. The tree on the right is shown in grey because it does not represent an actual BPS flow tree, since the split occurs on an anti-marginal stability wall.
• Figure 5: Location of the ${\cal S}(\Gamma_1, \Gamma_2)$ wall.
• Figure 6: Charge $\Gamma_1$ is realized as a bound state of $\Gamma_3+ \Gamma_4$. The dashed line is the recombination wall $RW$ between $(\Gamma_2,(\Gamma_3,\Gamma_4))$ and $(\Gamma_4,(\Gamma_2,\Gamma_3))$+$(\Gamma_3,(\Gamma_2,\Gamma_4))$.