Split States, Entropy Enigmas, Holes and Halos

TL;DR

This paper explores the structure of BPS states in string theory, developing a factorization formula, analyzing background dependence, and refining the OSV conjecture through detailed mathematical and physical insights.

Contribution

It introduces a new factorization formula for BPS indices, clarifies the relation between Donaldson-Thomas invariants and BPS states, and refines the OSV conjecture with a novel measure factor and cutoff analysis.

Findings

BPS indices can be factorized using attractor flow trees.

Polar states correspond to split D6-anti-D6 bound states.

The refined OSV conjecture includes a nontrivial measure factor and cutoff dependence.

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.