nLab
inertia orbifold

Idea

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

Definition

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

$$\begin{array}{ccc}a& \stackrel{f}{\to }& a\\ u\downarrow & & \downarrow u\\ b& \stackrel{g}{\to }& b\end{array}$$\array{a &\stackrel{f}\rightarrow& a\\ u \downarrow && \downarrow u\\ b &\stackrel{g}\rightarrow& b }

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

This is isomorphic to the functor category $[S^1,G]$, where $S^1$ denotes the free groupoid on a single object with a single automorphism (equivalently, the delooping $B\mathbb{Z}$ of the integers). It is equivalent to the free loop space object of $G$ 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 $\Lambda G$ of $G$ is the Morita equivalence class of the (proper étale) action groupoid for the action of $G_1$ by conjugation on the subspace $S\subset G_1$ 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 $G$, the groupoid convolution algebra of the inertia groupoid of $BG$ is the Drinfeld double of the group convolution algebra of $G$.

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