inertia orbifold



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

(∞,1)-topos theory





Extra stuff, structure and property



structures in a cohesive (∞,1)-topos



Inertia orbifold is a particular model for the free loop space object of an orbifold XX (or plain groupoid or smooth groupoid/stack etc.): the smooth groupoid whose objects are automorphisms in XX and whose morphisms are conjugation of automorphisms by morphisms in XX. In fact for each fibered category one can construct another fibered category, its inertia and the inertia stacks are a special case of this construction.


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.


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.


category: Lie theory

Last revised on June 11, 2020 at 10:32:01. See the history of this page for a list of all contributions to it.