A Homotopy Theory of Orbispaces

TL;DR

This paper develops a homotopy theory framework for orbispaces, aiming to mathematically formalize the stringy aspects of orbifolds that are significant in string theory and related mathematical fields.

Contribution

It introduces a homotopy-theoretic approach to orbispaces, providing a new mathematical foundation for understanding their stringy properties.

Findings

Establishes a homotopy theory for orbispaces

Connects mathematical structures with string theory concepts

Provides tools for analyzing orbifold singularities

Abstract

In 1985, physicists Dixon, Harvey, Vafa and Witten studied string theories on Calabi-Yau orbifolds (cf. [DHVW]). An interesting discovery in their paper was the prediction that a certain physicist's Euler number of the orbifold must be equal to the Euler number of any of its crepant resolutions. This was soon related to the so called McKay correspondence in mathematics (cf. [McK]). Later developments include stringy Hodge numbers (cf. [Z], [BD]), mirror symmetry of Calabi-Yau orbifolds (cf. [Ro]), and most recently the Gromov-Witten invariants of symplectic orbifolds (cf. [CR1-2]). One common feature of these studies is that certain contributions from singularities, which are called ``twisted sectors'' in physics, have to be properly incorporated. This is called the ``stringy aspect'' of an orbifold (cf. [R]). This paper makes an effort to understand the stringy aspect of orbifolds in the realm of ``traditional mathematics''.