Split States, Entropy Enigmas, Holes and Halos

TL;DR

This work develops a refined, modular-informed framework for counting BPS degeneracies of D-brane bound states on compact Calabi–Yau threefolds, connecting D4–D2–D0 partitions to D6–anti-D6 bound states via a fareytail expansion and wall-crossing formalism. It introduces an extreme-polar-state program with a controlled cutoff to realize a factorized OSV-like structure ${\\cal Z}_{\\rm BH} \\\sim {\\cal Z}_{\\rm top} \, \\\overline{\\cal Z}_{\\rm top}$ in suitable regimes, while identifying a nontrivial measure factor and the critical role of swing states in delimiting validity. The paper provides explicit tests in quivers and large-radius geometries, derives generating functions tied to DT/GV invariants, and discusses the implications for OSV, M-theory interpretations, and potential nonperturbative completions of the topological string. Its results illuminate how background moduli, bound-state spectra, and modular structures interplay to shape black-hole entropy counting in a controlled, geometrically grounded setting. The findings also delineate precise future directions, including rigorous proofs of split-flow and extreme-polar conjectures and a deeper understanding of the core/corona dichotomy in the DT/PT landscape.

Abstract

We investigate degeneracies of BPS states of D-branes on compact Calabi-Yau manifolds. We develop a factorization formula for BPS indices using attractor flow trees associated to multicentered black hole bound states. This enables us to study background dependence of the BPS spectrum, to compute explicitly exact indices of various nontrivial D-brane systems, and to clarify the subtle relation of Donaldson-Thomas invariants to BPS indices of stable D6-D2-D0 states, realized in supergravity as "hole halos". We introduce a convergent generating function for D4 indices in the large CY volume limit, and prove it can be written as a modular average of its polar part, generalizing the fareytail expansion of the elliptic genus. We show polar states are "split" D6-anti-D6 bound states, and that the partition function factorizes accordingly, leading to a refined version of the OSV conjecture. This differs from the original conjecture in several aspects. In particular we obtain a nontrivial measure factor g_{top}^{-2} e^{-K} and find factorization requires a cutoff. We show that the main factor determining the cutoff and therefore the error is the existence of "swing states" -- D6 states which exist at large radius but do not form stable D6-anti-D6 bound states. We point out a likely breakdown of the OSV conjecture at small g_{top} (in the large background CY volume limit), due to the surprising phenomenon that for sufficiently large background Kahler moduli, a charge N Q supporting single centered black holes of entropy ~ N^2 S(Q) also admits two-centered BPS black hole realizations whose entropy grows like N^3 at large N.