nLab
inertia orbifold

Context

(,1)(\infty,1)-Topos Theory

(∞,1)-topos theory

Background

Definitions

Characterization

Morphisms

Extra stuff, structure and property

Models

Constructions

structures in a cohesive (∞,1)-topos

Contents

Idea

Another name for the free loop space object of an orbifold XX (or plain groupoid or smooth groupoid/stack etc.) is inertia orbifold: the smooth groupoid whose objects are automorphisms in XX and whose morphisms are conjugation of automorphisms by morphisms in XX.

Definition

Given a groupoid GG (in the category of sets) with the set of objects G 0G_0 and the set of morphisms G 1G_1, one defines its inertia groupoid as the groupoid whose set SS of objects is the set of loops, i.e. the equalizer of the source and target maps s,t:G 1G 0s,t: G_1\to G_0; and whose set of maps from f:aaf\colon a\to a to g:bbg\colon b\to b consists of the commutative squares with the same vertical maps of the form

a f a u u b g b\array{a &\stackrel{f}\rightarrow& a\\ u \downarrow && \downarrow u\\ b &\stackrel{g}\rightarrow& b }

i.e. of the morphisms u:abu \colon a\to b in G 1G_1 such that u 1gu=fu^{-1}\circ g\circ u = f.

This is isomorphic to the functor category [S 1,G][S^1,G], where S 1S^1 denotes the free groupoid on a single object with a single automorphism (equivalently, the delooping BB\mathbb{Z} of the integers). It is equivalent to the free loop space object of GG in the (2,1)-category of groupoids.

The same construction can be performed for a groupoid internal to any finitely complete category, or more generally whenever the relevant limits exist. If a (differential, topological or algebraic) stack (or, in particular, an orbifold) is represented by a groupoid, then the inertia groupoid of that groupoid represents its inertia stack. In particular, an orbifold corresponds to a Morita equivalence class of a proper étale groupoid. The inertia groupoid ΛG\Lambda G of GG is the Morita equivalence class of the (proper étale) action groupoid for the action of G 1G_1 by conjugation on the subspace SG 1S\subset G_1 of closed loops.

For quantum field theory on orbifolds the inertia orbifold is related to so called twisted sectors of the corresponding QFT. One can also consider more generally twisted multisectors.

Properties

Convolution algebra and Relation to Drinfeld double

At least for a finite group GG, the groupoid convolution algebra of the inertia groupoid of BG\mathbf{B}G is the Drinfeld double of the group convolution algebra of GG.

References

  • Ernesto Lupercio, Bernardo Uribe, Inertia orbifolds, configuration spaces and the ghost loop space, Quarterly Journal of Mathematics 55, Issue 2, pp. 185-201, arxiv/math.AT/0210222; Loop groupoids, gerbes, and twisted sectors on orbifolds, in: Orbifolds in Mathematics and Physics, Madison, WI, 2001, in: Contemp. Math. 310, Amer. Math. Soc., Providence, RI, 2002, pp. 163–184, MR2004c:58043, math.AT/0110207
  • T. Kawasaki, The signature theorem for V-manifolds, Topology 17 (1978), no. 1, 75–83.
  • V. Hinich, Drinfeld double for orbifolds, Contemporary Math. 433, AMS Providence, 2007, 251-265, arXiv:math.QA/0511476
  • L. Dixon, J. A. Harvey, C. Vafa, E. Witten, Strings on orbifolds, Nuclear Phys. B 261 (1985), no. 4, 678–686. MR87k:81104a, doi; Strings on orbifolds. II, Nuclear Phys. B 274 (1986), no. 2, 285–314, MR87k:81104b, doi
  • Jean-Louis Tu, Ping Xu, Chern character for twisted K-theory of orbifolds, Advances in Mathematics 207 (2006) 455–483, pdf (cf. sec. 2.3)

category: Lie theory

Revised on April 8, 2013 18:13:15 by Urs Schreiber (82.113.106.75)