We present the most complete list of mirror pairs of Calabi-Yau complete intersections in toric ambient varieties and develop the methods to solve the topological string and to calculate higher genus amplitudes on these compact Calabi-Yau spaces. These symplectic invariants are used to remove redundancies in examples. The construction of the B-model propagators leads to compatibility conditions, which constrain multi-parameter mirror maps. For K3 fibered Calabi-Yau spaces without reducible fibers we find closed formulas for all genus contributions in the fiber direction from the geometry of the fibration. If the heterotic dual to this geometry is known, the higher genus invariants can be identified with the degeneracies of BPS states contributing to gravitational threshold corrections and all genus checks on string duality in the perturbative regime are accomplished. We find, however, that the BPS degeneracies do not uniquely fix the non-perturbative completion of the heterotic string. For these geometries we can write the topological partition function in terms of the Donaldson-Thomas invariants and we perform a non-trivial check of S-duality in topological strings. We further investigate transitions via collapsing D 5 del Pezzo surfaces and the occurrence of free Z 2 quotients that lead to a new class of heterotic duals. 1

We discuss the resolution of toroidal orbifolds. For the resulting smooth Calabi-Yau manifolds, we calculate the intersection ring and determine the divisor topologies. In a next step, the orientifold quotients are constructed. e-print archive: http://lanl.arXiv.org/abs/hep-th/0609014

We discuss general properties of moduli stablization in KKLT scenarios in type IIB orientifold compactifications X 6 . In particular, we find conditions for the Kähler potential to allow a KKLT scenario for a manifold X 6 without complex structure moduli, i.e. h (−) (2,1) (X 6 ) = 0. This way, a whole class of type IIB orientifolds with h (−) (2,1) (X 6 ) = 0 is ruled out. This excludes in particular all Z N -and Z N × Z M -orientifolds X 6 with h (2,1) (X 6 ) = 0 for a KKLT scenario. This concerns Z 3 , Z 7 , Z 3 × Z 3 , Z 4 × Z 4 , Z 6 × Z 6 and Z 2 × Z 6 ′ -both at the orbifold point and away from it. Furthermore, we propose a mechanism to stabilize the Kaehler moduli accociated to the odd cohomology H (1,1) − (X 6 ). In the second part of this work we discuss the moduli stabilization of resolved type IIB Z N -or Z N × Z M -orbifold/orientifold compactifications. As examples for the resolved Z 6 and Z 2 × Z 4 orbifolds we fix all moduli through a combination of fluxes and racetrack superpotential.

We study topological string theory on elliptically fibered Calabi-Yau manifolds using mirror symmetry. We compute higher genus topological string amplitudes and express these in terms of polynomials of functions constructed from the special geometry of the deformation spaces. The polynomials are fixed by the holomorphic anomaly equations supplemented by the expected behavior at special loci in moduli space. We further expand the amplitudes in the base moduli of the elliptic fibration and find that the fiber moduli dependence is captured by a finer polynomial structure in terms of the modular forms of the modular group of the elliptic curve. We further find a recursive equation which captures this finer structure and which can be related to the anomaly equations for correlation functions.

The Yau-Zaslow conjecture predicts the genus 0 curve counts of K 3 K3 surfaces in terms of the Dedekind η \eta function. The classical intersection theory of curves in the moduli of K 3 K3 surfaces with Noether-Lefschetz divisors is related to 3-fold Gromov-Witten invariants via the K 3 K3 curve counts. Results by Borcherds and Kudla-Millson determine these classical intersections in terms of vector-valued modular forms. Proven mirror transformations can often be used to calculate the 3-fold invariants which arise. Via a detailed study of the STU model (determining special curves in the moduli of K 3 K3 surfaces), we prove the Yau-Zaslow conjecture for all curve classes on K 3 K3 surfaces. Two modular form identities are required. The first, the Klemm-Lerche-Mayr identity relating hypergeometric series to modular forms after mirror transformation, is proven here. The second, the Harvey-Moore identity, is proven by D. Zagier and presented in the paper.

We study the differential polynomial rings which are defined using the special geometry of the moduli spaces of Calabi-Yau threefolds. The higher genus topological string amplitudes are expressed as polynomials in the generators of these rings, giving them a global description in the moduli space. At particular loci, the amplitudes yield the generating functions of Gromov-Witten invariants. We show that these rings are isomorphic to the rings of quasi modular forms for threefolds with duality groups for which these are known. For the other cases, they provide generalizations thereof. We furthermore study an involution which acts on the quasi modular forms. We interpret it as a duality which exchanges two distinguished expansion loci of the topological string amplitudes in the moduli space. We construct these special polynomial rings and match them with known quasi modular forms for non-compact Calabi-Yau geometries and their mirrors including local P 2 and local del Pezzo geometries with E 5 , E 6 , E 7 and E 8 type singularities. We provide the analogous special polynomial ring for the quintic.

This work is concerned with branes and differential equations for one-parameter Calabi-Yau hypersurfaces in weighted projective spaces. For a certain class of B-branes we derive the inhomogeneous Picard-Fuchs equations satisfied by the brane superpotential. In this way we arrive at a prediction for the real BPS invariants for holomorphic maps of worldsheets with low Euler characteristics, ending on the mirror A-branes.

This work is concerned with branes and differential equations for oneparameter Calabi-Yau hypersurfaces in weighted projective spaces. For a certain class of B-branes, we derive the inhomogeneous Picard-Fuchs equations satisfied by the brane superpotential. In this way, we arrive at a prediction for the real BPS invariants for holomorphic maps of worldsheets with low Euler characteristics, ending on the mirror A-branes.