Direct integration of the topological string

We present a new method to solve the holomorphic anomaly equations governingthe free energies of type B topological strings. The method is based on directintegration with respect to the non-holomorphic dependence of the amplitudes,and relies on the interplay between non-holomorphicity and modularityproperties of the topological string amplitudes. We develop a formalism validfor any Calabi-Yau manifold and we study in detail two examples, providingclosed expressions for the amplitudes at low genus, as well as a discussion ofthe boundary conditions that fix the holomorphic ambiguity. The first exampleis the non-compact Calabi-Yau underlying Seiberg-Witten theory and itsgravitational corrections. The second example is the Enriques Calabi-Yau, whichwe solve in full generality up to genus six. We discuss various aspects of thismodel: we obtain a new method to generate holomorphic automorphic forms on theEnriques moduli space, we write down a new product formula for the fiberamplitudes at all genus, and we analyze in detail the field theory limit. Thisallows us to uncover the modularity properties of SU(2), N=2 super Yang-Millstheory with four massless hypermultiplets.Comment: 75 pages, 3 figure