GLSM's for gerbes (and other toric stacks)

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 check 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 CFT's 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 CFT's, involving fields valued in roots of unity.Comment: 43 pages, LaTeX, 3 figures; v2: typos fixe

Schematic homotopy types and non-abelian Hodge theory

In this work we use Hodge theoretic methods to study homotopy types of complex projective manifolds with arbitrary fundamental groups. The main tool we use is the \textit{schematization functor} $X \mapsto (X\otimes \mathbb{C})^{sch}$, introduced by the third author as a substitute for the rationalization functor in homotopy theory in the case of non-simply connected spaces. Our main result is the construction of a \textit{Hodge decomposition} on $(X\otimes\mathbb{C})^{sch}$. This Hodge decomposition is encoded in an action of the discrete group $\mathbb{C}^{\times \delta}$ on the object $(X\otimes \mathbb{C})^{sch}$ and is shown to recover the usual Hodge decomposition on cohomology, the Hodge filtration on the pro-algebraic fundamental group as defined by C.Simpson, and in the simply connected case, the Hodge decomposition on the complexified homotopy groups as defined by J.Morgan and R. Hain. This Hodge decomposition is shown to satisfy a purity property with respect to a weight filtration, generalizing the fact that the higher homotopy groups of a simply connected projective manifold have natural mixed Hodge structures. As a first application we construct a new family of examples of homotopy types which are not realizable as complex projective manifolds. Our second application is a formality theorem for the schematization of a complex projective manifold. Finally, we present conditions on a complex projective manifold $X$ under which the image of the Hurewitz morphism of $\pi_{i}(X) \to H_{i}(X)$ is a sub-Hodge structure.Comment: 57 pages. This new version has been globally reorganized and includes additional results and applications. Minor correction