It has been conjectured recently that the field theory limit of the topological string partition functions, including all higher genus contributions, for the family of CY3-folds giving rise to $\mathcal{N}=2$ 4D $SU(N)$ gauge theory via geometric engineering can be obtained from gauge instanton calculus. We verify this surprising conjecture by calculating the partition functions for such local CYs using diagrammatic techniques inspired by geometric transitions. Determining the Gopakumar-Vafa invariants for these geometries to all orders in the fiber wrappings allows us to take the field theory limit.

We define a physically reasonable mass for an asymptotically Robertson--Walker manifold which is uniquely defined in the case of a normalized representation.

Symmetries of trigonometric integrable two-dimensional statistical face models are considered. The corresponding symmetry operators on the Hilbert space of states of the quantum version of these models define a weak $*$-Hopf algebra isomorphic to the Ocneanu double triangle algebra.

In this paper, we will discuss gauged linear sigma model descriptions of toric stacks. Toric stacks have a simple description in terms of (symplectic, GIT) ${\bf C}^{\times}$ quotients of homogeneous coordinates, in exactly the same form as toric varieties. We describe the physics of the gauged linear sigma models that formally coincide with the mathematical description of toric stacks and check that physical predictions of those gauged linear sigma models exactly match the corresponding stacks. We also see in examples that when a given toric stack has multiple presentations in a form accessible as a gauged linear sigma model, that the IR physics of those different presentations matches, so that the IR physics is presentation-independent, making it reasonable to associate CFTs to stacks, not just presentations of stacks. We discuss mirror symmetry for stacks, using Morrison-Plesser-Hori-Vafa techniques to compute mirrors explicitly, and also find a natural generalization of Batyrev's mirror conjecture. In the process of studying mirror symmetry, we find some new abstract CFTs, involving fields valued in roots of unity.

We use noncommutative topology to study T-duality for principal torus bundles with H-flux. We characterize precisely when there is a "classical" T-dual, i.e., a dual bundle with dual H-flux, and when the T-dual must be "nonclassical," that is, a continuous field of noncommutative tori. The duality comes with an isomorphism of twisted $K$-theories, required for matching of D-brane charges, just as in the classical case. The isomorphism of twisted cohomology which one gets in the classical case is replaced in the nonclassical case by an isomorphism of twisted cyclic homology. An important part of the paper contains a detailed analysis of the classifying space for topological T-duality, as well as the T-duality group and its action. The issue of possible nonuniqueness of T-duals can be studied via the action of the T-duality group.