A localization principle for orbifold theories

Abstract

Abstract. In this article, written primarily for physicists and geometers, we survey several manifestations of a general localization principle for orbifold theories such as K-theory, index theory, motivic integration and elliptic genera. OrbifoldsIn this paper we will attempt to explain a general localization principle that appears frequently under several guises in the study of orbifolds. We will begin by reminding the reader what we mean by an orbifold.The most familiar situation in physics is that of an orbifold of the type X = [M/G], where M is a smooth manifold and G is a finite group acting 1 smoothly on M ; namely, we give ourselves a homomorphism G → Diff(M ). We make a point of distinguishing the orbifold X = [M/G] from its quotient space (also called orbit space) X = M/G. As a set, as we know, a point in X is an orbit of the action: that is, a typical element of M/G is Orb(x) = {xg | g ∈ G}.For us an orbifold X = [M/G] is a smooth category 2 (actually a topological groupoid) whose objects are the points of M , X 0 = Obj(X) = M , and we insist on remembering that X 0 = Obj(X) is a smooth manifold. The arrows of this category are X 1 = Mor(X) = M × G again thinking of it as a smooth manifold. A typical arrow in this category is

The loop orbifold of the symmetric product

“…More interestingly, the action of S 1 on LX has as a fixed suborbifold I (X) which is known as the inertia orbifold of X (cf. [6]). …”

“…More interestingly the S 1 action on LX has as a fixed suborbifold I(X) the inertia orbifold of X. In [3] we have argued that orbifold theories often localize to the inertia orbifold. Chas and Sullivan [1] have defined an associative product on the homology of the loop space H * (LM ).…”

“…In this paper we study this product on H * (L[M n /S n ]). Using this product and the localization principle mentioned above [3] we define an associative product (H * (I[M n /S n ]), •).…”

Chen-Ruan Cohomology of cotangent orbifolds and Chas-Sullivan String Topology

“…But more interestingly, they have proved that the action of S 1 on LG has as a fixed suborbifold Λ(G) which is known as the inertia orbifold of G (cf. [14,8,15]).…”