We develop Floer theory of Lagrangian torus fibers in compact symplectic toric orbifolds. We first classify holomorphic orbi-discs with boundary on Lagrangian torus fibers. We show that there exists a class of basic discs such that we have one-to-one correspondences between a) smooth basic discs and facets of the moment polytope, and b) between basic orbi-discs and twisted sectors of the toric orbifold. We show that there is a smooth Lagrangian Floer theory of these torus fibers, which has a bulk-deformation by fundamental classes of twisted sectors of the toric orbifold. We show by several examples that such bulk-deformation can be used to illustrate the very rigid Hamiltonian geometry of orbifolds. We define its potential and bulk-deformed potential, and develop the notion of leading order potential. We study leading term equations analogous to the case of toric manifolds by Fukaya, Oh, Ohta and Ono.

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Abstract. We classify holomorphic as well as algebraic torus equivariant principal G-bundles over a nonsingular toric variety X, where G is a complex linear algebraic group. It is shown that any such bundle over an affine, nonsingular toric variety admits a trivialization in equivariant sense. We also obtain some splitting results.

The purpose of this paper is to show how the methods of motivic integration of Kontsevich, Denef-Loeser (Invent. Math. 135 (1999) 201-232 and Compositio Math. 131 (2002) 267-290) and Looijenga (Astérisque 276 (2002) 267-297) can be adapted to prove the McKay-Ruan correspondence, a generalization of the McKay-Reid correspondence to orbifolds that are not necessarily global quotients.

We identify the twisted sectors of a compact simplicial toric variety. We do the same for a generic nondegenerate CalabiYau hypersurface of an n-dimensional simplicial Fano toric variety and then explicitly compute h 1,1 orb and h n−2,1 orb for the hypersurface. We give applications to the orbifold string theory conjecture and orbifold mirror symmetry.