A Matrix Model for the Topological String II: The Spectral Curve and Mirror Geometry

In a previous paper, we presented a matrix model reproducing the topological string partition function on an arbitrary given toric Calabi–Yau manifold. Here, we compute the spectral curve of our matrix model and thus provide a matrix model derivation of the large volume limit of the BKMP “remodeling the B-model” conjecture, the claim that Gromov–Witten invariants of any toric Calabi–Yau threefold coincide with the spectral invariants of its mirror curve.     

A Matrix Model for the Topological String I: Deriving the Matrix Model

We construct a matrix model that reproduces the topological string partition function on arbitrary toric Calabi–Yau threefolds. This demonstrates, in accord with the BKMP “remodeling the B-model” conjecture, that Gromov–Witten invariants of any toric Calabi–Yau threefold can be computed in terms of the spectral invariants of a spectral curve. Moreover, it proves that the generating function of Gromov–Witten invariants is a tau function for an integrable hierarchy. In a follow-up paper, we will explicitly construct the spectral curve of our matrix model and argue that it equals the mirror curve of the toric Calabi–Yau manifold.     

The Ω Deformed B-model for Rigid N = 2 Theories

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 ${\mathbb{P}^2}$ . We calculate motivic Donaldson–Thomas invariants for this geometry and make predictions for generalized Gromov–Witten invariants at the orbifold point.