Topological string amplitudes, complete intersection Calabi-Yau spaces and threshold corrections

We present the most complete list of mirror pairs of Calabi-Yau completeintersections in toric ambient varieties and develop the methods to solve thetopological string and to calculate higher genus amplitudes on these compactCalabi-Yau spaces. These symplectic invariants are used to remove redundanciesin examples. The construction of the B-model propagators leads to compatibilityconditions, which constrain multi-parameter mirror maps. For K3 fiberedCalabi-Yau spaces without reducible fibers we find closed formulas for allgenus contributions in the fiber direction from the geometry of the fibration.If the heterotic dual to this geometry is known, the higher genus invariantscan be identified with the degeneracies of BPS states contributing togravitational threshold corrections and all genus checks on string duality inthe perturbative regime are accomplished. We find, however, that the BPSdegeneracies do not uniquely fix the non-perturbative completion of theheterotic string. For these geometries we can write the topological partitionfunction in terms of the Donaldson-Thomas invariants and we perform anon-trivial check of S-duality in topological strings. We further investigatetransitions via collapsing D5 del Pezzo surfaces and the occurrence of free Z2quotients that lead to a new class of heterotic duals.Comment: 117 pages, 1 Postscript figur