Algebraic Topology of Calabi-Yau Threefolds in Toric Varieties

We compute the integral homology (including torsion), the topological K-theory, and the Hodge structure on cohomology of Calabi-Yau threefold hypersurfaces and complete intersections in Gorenstein toric Fano varieties. The methods are purely topological

Wilson Loops, Bianchi Constraints and Duality in Abelian Lattice Models

We introduce new modified Abelian lattice models, with inhomogeneous local interactions, in which a sum over topological sectors are included in the defining partition function. The dual models, on lattices with arbitrary topology, are constructed and they are found to contain sums over topological sectors, with modified groups, as in the original model. The role of the sum over sectors is illuminated by deriving the field-strength formulation of the models in an explicitly gauge-invariant manner. The field-strengths are found to satisfy, in addition to the usual local Bianchi constraints, global constraints. We demonstrate that the sum over sectors removes these global constraints and consequently softens the quantization condition on the global charges in the system. Duality is also used to construct mappings between the order and disorder variables in the theory and its dual. A consequence of the duality transformation is that correlators which wrap around non-trivial cycles of the lattice vanish identically. For particular dimensions this mapping allows an explicit expression for arbitrary correlators to be obtained.Comment: LaTeX 30 pages, 6 figures and 2 tables. References updated and connection with earlier work clarified, final version to appear in Nucl. Phys.

A Note on the Equality of Algebraic and Geometric D-Brane Charges in WZW Models

The algebraic definition of charges for symmetry-preserving D-branes in Wess-Zumino-Witten models is shown to coincide with the geometric definition, for all simple Lie groups. The charge group for such branes is computed from the ambiguities inherent in the geometric definition.Comment: 12 pages, fixed typos, added references and a couple of remark

Functorial semi-norms on singular homology and (in)flexible manifolds

A functorial semi-norm on singular homology is a collection of semi-norms on the singular homology groups of spaces such that continuous maps between spaces induce norm-decreasing maps in homology. Functorial semi-norms can be used to give constraints on the possible mapping degrees of maps between oriented manifolds. In this paper, we use information about the degrees of maps between manifolds to construct new functorial semi-norms with interesting properties. In particular, we answer a question of Gromov by providing a functorial semi-norm that takes finite positive values on homology classes of certain simply connected spaces. Our construction relies on the existence of simply connected manifolds that are inflexible in the sense that all their self-maps have degree -1, 0, or 1. The existence of such manifolds was first established by Arkowitz and Lupton; we extend their methods to produce a wide variety of such manifolds.Comment: 37 pages, 1 figure; v2: added some references, corrected some typos; v3: added observation on multiplicative finite functorial semi-norms; v4: corrected a mistake in Corollary 3.2 (the main results are not affected); to appear in AG