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.

Highlights

  • String theory has been spectacularly successful in microscopically reproducing the entropy of certain classes of black holes, in particular of supersymmetric charged black holes

  • The parallel derivation in the D4-D0 picture in string theory [3] is restricted to the same large D0-charge regime

  • We show that the formula for N1 = 1 but arbtirary N2 is most conveniently given in terms of a generating function:

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Summary

Introduction

String theory has been spectacularly successful in microscopically reproducing the entropy of certain classes of black holes, in particular of supersymmetric charged black holes. One of the problems in the first category is the fact that there is still no general derivation for the case with nonzero D6-brane charge Another one is the need to keep the black hole attractor point within some sufficiently small neighborhood of the infinite radius limit, requiring in particular the magnetic D4 charge P to lie within the Kahler cone (excluding in particular the so-called “small” black holes). Since Ztop is divergent and only makes sense as an asymptotic perturbative expansion, it is clear that the conjecture can at most hold approximately It is not clear a priori what the regime of validity should be, nor what the order of the error is, nor even how to define properly the integral (1.1).

Outline
Challenges for a complete proof and unresolved issues
Preliminaries
A fareytail expansion for the D4-D2-D0 partition function
Theta function decomposition
A Rademacher-Jacobi formula
Basic idea
Review of BPS black hole bound states and attractor flow trees
General stationary BPS solutions
Existence criteria and attractor flow trees
Attractor flow trees and the Hilbert space of quantum BPS states
Symmetries
A class of examples
The Entropy Enigma
D6-D0 bound states
Sun-Earth-Moon systems
Note that the charge to which the D0 binds depends on the choice
Scaling solutions
Even more complicated multicentered bound states
Microscopic description
Quiver description of bound states
Geometrical relations between D4 and D6-anti-D6 bound states
P S2 2
Physical derivation
Mathematical tests and applications
Four node quiver without closed loops
Three node quiver with closed loop
Entropy of the three node quiver in the scaling regime
Note that
Harmless Halos and Catastrophic Cores We now focus on the difference
Discussion
Summary of open problems and potential future directions
Full Text
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