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 −2 top 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 gtop (in the large background CY volume limit), due to the surprising phenomenon that for sufficiently large background Kähler moduli, a charge ΛΓ supporting single centered black holes of entropy ∼ Λ2S(Γ) also admits two-centered BPS black hole realizations whose entropy grows like Λ3 when Λ → ∞.

We study the boundary states of D-branes wrapped around supersymmetric cycles in a general Calabi-Yau manifold. In particular, we show how the geometric data on the cycles are encoded in the boundary states. As an application, we analyze how the mirror symmetry transforms D-branes, and we verify that it is consistent with the conjectured periodicity and the monodromy of the Ramond-Ramond field configuration on a Calabi-Yau manifold. This also enables us to study open string worldsheet instanton corrections and relate them to closed string instanton counting. The cases when the mirror symmetry is realized as T-duality are also discussed. Permanent address: Department of Particle Physics, Weizmann Institute of Science, 76100 Rehovot Israel. 1 Introduction D-branes in type II string theories have been identified as Ramond-Ramond charged BPS states [1]. In the presence of a D-brane, the boundary conditions for open strings are modified in such a way that Dirichlet boundary conditions ...

We construct a cubic field theory which provides all genus amplitudes of the topological A-model for all non-compact toric Calabi-Yau threefolds. The topology of a given Feynman diagram encodes the topology of a fixed Calabi-Yau, with Schwinger parameters playing the role of Kähler classes of the threefold. We interpret this result as an operatorial computation of the amplitudes in the B-model mirror which is the quantum Kodaira-Spencer theory. The only degree of freedom of this theory is an unconventional chiral scalar on a Riemann surface. In this setup we identify the B-branes on the mirror Riemann

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The topological string partition function Z(λ,t, ¯t) = exp(λ 2g−2 Fg(t, ¯t)) is calculated on a compact Calabi-Yau M. The Fg(t, ¯t) fulfill the holomorphic anomaly equations, which imply that Ψ = Z transforms as a wave function on the symplectic space H 3 (M, Z). This defines it everywhere in the moduli space M(M) along with preferred local coordinates. Modular properties of the sections Fg as well as local constraints from the 4d effective action allow us to fix Z to a large extent. Currently with a newly found gap condition at the conifold, regularity at the orbifold and the most naive bounds from Castelnuovo’s theory, we can provide the boundary data, which specify Z, e.g. up to genus 51 for the quintic.

We examine the recently proposed relations between black hole entropy and the topological string in the context of type II/heterotic string dual models. We consider the degeneracies of perturbative heterotic BPS states. In several examples with N = 4 and N = 2 supersymmetry, we show that the macroscopic degeneracy of small black holes agrees to all orders with the microscopic degeneracy, but misses non-perturbative corrections which are computable in the heterotic dual. Using these examples we refine the previous proposals and comment on their domain of validity as well as on the relevance of helicity supertraces.

We study three-dimensional Chern-Simons theory with complex gauge group SL(2,C), which has many interesting connections with three-dimensional quantum gravity and geometry of hyperbolic 3-manifolds. We show that, in the presence of a single knotted Wilson loop in an infinite-dimensional representation of the gauge group, the classical and quantum properties of such theory are described by an algebraic curve called the A-polynomial of a knot. Using this approach, we find some new and rather surprising relations between the A-polynomial, the colored Jones polynomial, and other invariants of hyperbolic 3-manifolds. These relations generalize the volume conjecture and the Melvin-Morton-Rozansky conjecture, and suggest an intriguing connection between the SL(2,C) partition function and the colored Jones polynomial.

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We introduce a notion of topological M-theory and argue that it provides a unification of form theories of gravity in various dimensions. Its classical solutions involve G2 holonomy metrics on 7-manifolds, obtained from a topological action for a 3-form gauge field introduced by Hitchin. We show that by reductions of this 7-dimensional theory one can classically obtain 6-dimensional topological A and B models, the topological sector of loop quantum gravity in 4 dimensions, and Chern-Simons gravity in 3 dimensions. We also find that the 7-dimensional M-theory perspective sheds some light on the fact that the topological string partition function is a wavefunction, as well as on S-duality between the A and B models. The degrees of freedom of the A and B models appear as conjugate variables in the 7-dimensional theory. Finally, from the topological M-theory perspective we find hints of an intriguing holographic link between non-supersymmetric Yang-Mills in