We use toric geometry to study open string mirror symmetry on compact Calabi-Yau manifolds. For a mirror pair of toric branes on a mirror pair of toric hypersurfaces we derive a canonical hypergeometric system of differential equations, whose solutions determine the open/closed string mirror maps and the partition functions for spheres and discs. We define a linear sigma model for the brane geometry and describe a correspondence between dual toric polyhedra and toric brane geometries. The method is applied to study examples with obstructed and classically unobstructed brane moduli at various points in the deformation space. Computing the instanton expansion at large volume in the flat coordinates on the open/closed deformation space we obtain predictions for enumerative invariants.

We perform a Hodge theoretic study of parameter dependent families of Dbranes on compact Calabi-Yau manifolds in type II and F-theory compactifications. Starting from a geometric Gauss-Manin connection for B-type branes we study the integrability and flatness conditions. The B-model geometry defines an interesting ring structure of operators. For the mirror A-model this indicates the existence of an open-string extension of the so-called A-model connection, whereas the discovered ring structure should be part of the open-string A-model quantum cohomology. We obtain predictions for genuine Ooguri-Vafa invariants for Lagrangian branes on the quintic in P 4 that pass some non-trivial consistency checks. We discuss the lift of the brane compactifications to F-theory on Calabi-Yau four-folds and the effective couplings in the effective supergravity action as determined by the N = 1 special geometry of the open-closed deformation space.

We present a detailed study of D-brane superpotentials depending on several open and closed-string deformations. The relative cohomology group associated with the brane defines a generalized hypergeometric GKZ system which determines the off-shell superpotential and its analytic properties under deformation. Explicit expressions for the N = 1 superpotential for families of type II/F-theory compactifications are obtained for a list of multi-parameter examples. Using the Hodge theoretic approach to open-string mirror symmetry, we obtain new predictions for integral disc invariants in the A model instanton expansion. We study the behavior of the brane vacua under extremal transitions between different Calabi-Yau spaces and observe that the web of Calabi-Yau vacua remains connected for a particular class of branes.1 See ref.[31] for recent progress from matrix factorizations. 2 See also refs. [32,33,34,35] for further examples and discussions.

Using wall-crossing formulae and the theory of mock modular forms we derive a holomorphic anomaly equation for the modified elliptic genus of two M5-branes wrapping a rigid divisor inside a Calabi-Yau manifold. The anomaly originates from restoring modularity of an indefinite theta-function capturing the wall-crossing of BPS invariants associated to D4-D2-D0 brane systems. We show the compatibility of this equation with anomaly equations previously observed in the context of N = 4 topological Yang-Mills theory on È 2 and E-strings obtained from wrapping M5-branes on a del Pezzo surface. The non-holomorphic part is related to the contribution originating from bound-states of singly wrapped M5-branes on the divisor. We show in examples that the information provided by the anomaly is enough to compute the BPS degeneracies for certain charges. We further speculate on a natural extension of the anomaly to higher D4-brane charge.shown in ref. [18]. 3 A similar story showed up in N = 4 topological U (2) SYM theory on È 2 [5], where it was shown that different sectors of the partition function need a nonholomorphic completion which was found earlier in ref. [22] in order to restore S-dualtiy invariance. An anomaly equation describing this non-holomorphicity was expected [5] in the cases where b + 2 (P ) = 1. In these cases holomorphic deformations of the canonical bundle are absent. The non-holomorphic contributions were associated with reducible connections U (n) → U (m) × U (n − m) [5,4]. In ref.[4] this anomaly was furthermore related to an anomaly appearing in the context of E-strings [23]. These strings arise from an M5-brane wrapping a del Pezzo surface B 9 , also called 1 2 K3. The anomaly in this context was related to the fact that n of these strings can form bound-states of m and (n − m) strings. Furthermore, the anomaly could also be related to the one appearing in topological string theory.The anomaly thus follows from the formation of bound-states. Although the holomorphic expansion would not know about the contribution from bound-states, the restoration of duality symmetry forces one to take these contributions into account. The nonholomorphicity can be understood physically as the result of a regularization procedure. The path integral produces objects like theta-functions associated to indefinite quadratic forms which need to be regularized to avoid divergences. This regularization breaks the modular symmetry, restoring the symmetry gives non-holomorphic objects. The general mathematical framework to describe these non-holomorphic completions is the theory of mock modular forms developed by Zwegers in ref. [24]. 4 A mock modular form h(τ ) of weight k is a holomorphic function which becomes modular after the addition of a function g * (τ ), at the cost of losing its holomorphicity. Here, g * (τ ) is constructed from a modular form g(τ ) of weight 2 − k, which is referred to as shadow.Another manifestation of the background dependence of the holomorphic expansions of the topological theories are wall-cross...

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