The topological string partition function Z(λ, t,t) = exp(λ 2g−2 F g (t,t)) is calculated on a compact Calabi-Yau M . The F g (t,t) fulfill the holomorphic anomaly equations, which imply that Ψ = Z transforms as a wave function on the symplectic space H 3 (M, Z). This defines it everywhere in the moduli space M(M ) along with preferred local coordinates. Modular properties of the sections F g as well as local constraints from the 4d effective action allow us to fix Z to a large extent. Currently with a newly found gap condition at the conifold, regularity at the orbifold and the most naive bounds from Castelnuovo's theory, we can provide the boundary data, which specify Z, e.g. up to genus 51 for the quintic.Coupling topological matter to topological gravity is a key problem in string theory. Conceptually most relevant is the topological matter sector of the critical string as it arises e.g. in Calabi-Yau compactifications. Topological string theory on non-compact Calabi-Yau manifolds such as O(−3) → P 2 is essentially solved either by localization 1 -[1] or large N-techniques [2] and has intriguing connections to Chern-Simons theory [3], open-closed string duality [4], matrix models [5], integrable hierarchies of non-critical string theory [6] and 2d Yang-Mills theory [7].However, while local Calabi-Yau manifolds are suitable to study gauge theories and more exotic field theories in 4d and specific couplings to gravity, none of the techniques above extends to compact Calabi-Yau spaces, which are relevant for important questions in 4d quantum gravity concerning e.g. the properties of 4d black holes [10] and the wave function in mini superspace [11].Moreover, while the genus dependence is encoded in the Chern-Simons and matrix model approaches in a superior fashion by the 1 N 2 -expansion, the moduli dependence on the parameter t is reconstructed locally and in a holomorphic limit, typically by sums over partitions. This yields an algorithm, which grows exponentially in the world-sheet degree or the space-time instanton number.As the total F g (t,t) are modular invariant sections over the moduli space M(M), they must be generated by a ring of almost holomorphic modular forms. This solves the dependence on the moduli in the most effective way. In the following we will show that space-time modularity, the holomorphic anomaly equations of Bershadsky, Cecotti, Ooguri and Vafa, as well as boundary conditions at various boundary components of the moduli space, solve the theory very efficiently.For compact (and non-compact) Calabi-Yau spaces mirror symmetry is proven at genus zero. The modular properties that we need are also established at genus zero. Moreover it has been argued recently that the holomorphic anomaly recursions follow from categorical mirror symmetry [8,9]. To establish mirror symmetry at higher genus, one needs merely to prove that the same boundary data fix the F g (t,t) in the A-and the B-model.• Further conditions are provided by the regularity of the F (g) at orbifold points in M(M). These condition...

Four-dimensional supersymmetric SU(N) Yang-Mills theory on a sphere has highly charged baryon-like states built from anti-symmetric combinations of the adjoint scalars. We show that these states, which are equivalently described as holes in a free fermi sea of a reduced matrix model, are D-branes. Their excitations are stringlike and effectively realize Dirichlet and Neumann boundary conditions in various directions. The low energy brane dynamics should realize an emergent gauge theory that is local on a new space. We show that the Gauss' Law associated to this emergent gauge symmetry appears from combinatorial identities relating the stringy excitations. Although these excitations are not BPS, they can be near-BPS and we can hope to study them in perturbation theory. Accordingly, we show that the Chan-Paton factors expected for strings propagating on multiple branes arise dynamically, allowing the emergent gauge symmetry to be non-Abelian.

We use mirror symmetry, the refined holomorphic anomaly equation and modularity properties of elliptic singularities to calculate the refined BPS invariants of stable pairs on non-compact Calabi-Yau manifolds, based on del Pezzo surfaces and elliptic surfaces, in particular the half K3. The BPS numbers contribute naturally to the five-dimensional N =1 supersymmetric index of M-theory, but they can be also interpreted in terms of the superconformal index in six dimensions and upon dimensional reduction the generating functions count N = 2 Seiberg-Witten gauge theory instantons in four dimensions. Using the M/F-theory uplift the additional information encoded in the spin content can be used in an essential way to obtain information about BPS states in physical systems associated to small instantons, tensionless strings, gauge symmetry enhancement in F-theory by [p, q]-strings as well as M-strings.

We use the holomorphic anomaly equation to solve the gravitational corrections to Seiberg-Witten theory and a two-cut matrix model, which is related by the Dijkgraaf-Vafa conjecture to the topological B-model on a local Calabi-Yau manifold. In both cases we construct propagators that give a recursive solution in the genus modulo a holomorphic ambiguity. In the case of Seiberg-Witten theory the gravitational corrections can be expressed in closed form as quasimodular functions of Γ(2). In the matrix model we fix the holomorphic ambiguity up to genus two. The latter result establishes the Dijkgraaf-Vafa conjecture at that genus and yields a new method for solving the matrix model at fixed genus in closed form in terms of generalized hypergeometric functions.

We give an interpretation of the Ω deformed B-model that leads naturally to the generalized holomorphic anomaly equations. Direct integration of the latter calculates topological amplitudes of four dimensional rigid N = 2 theories explicitly in general Ω-backgrounds in terms of modular forms. These amplitudes encode the refined BPS spectrum as well as new gravitational couplings in the effective action of N = 2 supersymmetric theories. The rigid N = 2 field theories we focus on are the conformal rank one N = 2 Seiberg-Witten theories. The failure of holomorphicity is milder in the conformal cases, but fixing the holomorphic ambiguity is only possible upon mass deformation. Our formalism applies irrespectively of whether a Lagrangian formulation exists. In the class of rigid N = 2 theories arising from compactifications on local Calabi-Yau manifolds, we consider the theory of local P 2 . We calculate motivic Donaldson-Thomas invariants for this geometry and make predictions for generalized Gromov-Witten invariants at the orbifold point.