The B-model topological string theory on a Calabi-Yau threefold X has a symmetry group Γ, generated by monodromies of the periods of X. This acts on the topological string wave function in a natural way, governed by the quantum mechanics of the phase space. We show that, depending on the choice of polarization, the genus g topological string amplitude is either a holomorphic quasi-modular form or an almost holomorphic modular form of weight 0 under Γ. Moreover, at each genus, certain combinations of genus g amplitudes are both modular and holomorphic. We illustrate this for the local CalabiYau manifolds giving rise to Seiberg-Witten gauge theories in four dimensions and local IP 2 and IP 1 ×IP 1 . As a byproduct, we also obtain a simple way of relating the topological string amplitudes near different points in the moduli space, which we use to give predictions for Gromov-Witten invariants of the orbifold C 3 /Z Z 3 .

We propose a complete, new formalism to compute unambiguously B-model open and closed amplitudes in local Calabi-Yau geometries, including the mirrors of toric manifolds. The formalism is based on the recursive solution of matrix models recently proposed by Eynard and Orantin. The resulting amplitudes are non-perturbative in both the closed and the open moduli. The formalism can then be used to study stringy phase transitions in the open/closed moduli space. At large radius, this formalism may be seen as a mirror formalism to the topological vertex, but it is also valid in other phases in the moduli space. We develop the formalism in general and provide an extensive number of checks, including a test at the orbifold point of A p fibrations, where the amplitudes compute the 't Hooft expansion of Wilson loops in lens spaces. We also use our formalism to predict the disk amplitude for the orbifold C 3 /Z 3 .

Candelas and Font i n troduced the notion of a`top' as half of a three dimensional re exive polytope and noticed that Dynkin diagrams of enhanced gauge groups in string theory can be read o from them. We classify all tops satisfying a generalized de nition as a lattice polytope with one facet containing the origin and the other facets at distance one from the origin. These objects torically encode the local geometry of a degeneration of an elliptic bration. We give a prescription for assigning an a ne, possibly twisted Kac-Moody algebra to any such top (and more generally to any elliptic bration structure) in a precise way that involves the lengths of simple roots and the coe cients of null roots. Tops related to twisted Kac-Moody algebras can be used to construct string compacti cations with reduced rank of the gauge group.e-print archive: http://lanl.arXiv.org/abs/hep-th/0303218 206 A ne Kac-Moody algebras, CHL strings and the classi cation of tops

We propose a new, conjectural recursion solution for Hurwitz numbers at all genera. This conjecture is based on recent progress in solving type B topological string theory on the mirrors of toric Calabi-Yau manifolds, which we briefly review to provide some background for our conjecture. We show in particular how this B-model solution, combined with mirror symmetry for the one-leg, framed topological vertex, leads to a recursion relation for Hodge integrals with three Hodge class insertions. Our conjecture in Hurwitz theory follows from this recursion for the framed vertex in the limit of infinite framing.

We prove that the topological recursion reconstructs the WKB expansion of a quantum curve for all spectral curves whose Newton polygons have no interior point (and that are smooth as affine curves). This includes nearly all previously known cases in the literature, and many more; in particular, it includes many quantum curves of order greater than two. We also explore the connection between the choice of ordering in the quantization of the spectral curve and the choice of integration divisor to reconstruct the WKB expansion.Comment: 68 pages, 9 figures. v2: published version (improved presentation