Abstract. In this paper we construct an explicit geometric model for the group of gerbes over an orbifold X. We show how from its curvature we can obtain its characteristic class in H 3 (X) via Chern-Weil theory. For an arbitrary gerbe L, a twisting L K orb (X) of the orbifold K-theory of X is constructed, and shown to generalize previous twisting by Rosenberg [28]

Abstract. The purpose of this paper is to introduce the notion of loop groupoid LG associated to a groupoid G. After studying the general properties of LG, we show how this notion provides a very natural geometric interpretation for the twisted sectors of an orbifold [7], and for the inner local systems introduced by Ruan [14] by means of a natural generalization of the concept holonomy of a gerbe.

In this paper we study the string topology (à la Chas-Sullivan) of an orbifold. We define the string homology ring product at the level of the free loop space of the classifying space of an orbifold. We study its properties and do some explicit calculations. 55P35; 18D50, 55R35

We define equivariant projective unitary stable bundles as the appropriate twists when defining K‐theory as sections of bundles with fibers the space of Fredholm operators over a Hilbert space. We construct universal equivariant projective unitary stable bundles for the orbit types, and we use a specific model for these local universal spaces in order to glue them to obtain a universal equivariant projective unitary stable bundle for discrete and proper actions. We determine the homotopy type of the universal equivariant projective unitary stable bundle, and we show that the isomorphism classes of equivariant projective unitary stable bundles are classified by the third equivariant integral cohomology group. The results contained in this paper extend and generalize results of Atiyah–Segal.

Let X be a normal projective Q-Gorenstein variety with at worst log-terminal singularities. We prove a formula expressing the total stringy Chern class of a generic complete intersection in X via the total stringy Chern class of X. This formula is motivated by its applications to mirror symmetry for Calabi-Yau complete intersections in toric varieties. We compute stringy Chern classes and give a combinatorial interpretation of the stringy Libgober-Wood identity for arbitrary projective Q-Gorenstein toric varieties. As an application we derive a new com-binatorial identity relating d-dimensional reflexive polytopes to the number 12 in dimension d ≥ 4.