Contents

# Contents

## Idea

The notion of an inertia orbifold is closely related to that of a free loop space/free loop stack of an orbifold. As such, one may expect there to be an action of the circle group $S^1$ on inertia orbifolds, corresponding to “rotation of loops”. An explicit component-based modification of the inertia orbifold construction which builds in an $S^1$-action of sorts (akin to cyclic loop spaces/cyclic loop stacks) was highlighted in Huan 18 (Def. below), following Ganter 07, Def. 2.3 (and, apparently, following suggestions by Charles Rezk).

The orbifold K-theory of these modified inertia orbifolds (“quasi-elliptic cohomology”) is closely related to equivariant elliptic cohomology at the Tate curve of the original orbifold (see also at Tate K-theory).

## Definition

In the following

###### Definition

For $g \in G$ any element, with $C_g \subset G$ denoting its centralizer, write

$\Lambda_g \;\coloneqq\; \big( C_g \times \mathbb{R} \big) \, \big/ \, \underset{ \simeq \mathbb{Z} }{ \underbrace{ \langle (g^{-1},1) \rangle }}$

for the Lie group which is the quotient group of the direct product group of $G$ with the additive Lie group of real numbers by the subgroup (isomorphic to the natural numbers) which is generated from the pair $(g,-1) \in G \times \mathbb{R}$.

Hence this sits in a short exact sequence of Lie groups of this form:

$1 \to \mathbb{Z} \xhookrightarrow{ \; 1 \mapsto (g^{-1},1) \; } C_g \times \mathbb{R} \overset{\;p_g\;}{\twoheadrightarrow} \Lambda_g \to 1 \,.$

###### Remark

For $g \in G$ the group action of $G$ on $X$ restricts to an action of the centralizer $G$ on the fixed locus $X^g = X^{\langle g\rangle}$:

$C_g \times X^g \xrightarrow{ (-) \cdot (-) } X^g \,.$

Moreover, since $g$ itself (a) commutes with all element in $C_g$ and (b) has trivial action on $X^g$, this lifts to an action of $\Lambda_g$ (Def. )

(1)$\array{ \Lambda_g \times X^g &\xrightarrow{\;\;\;}& X^g \\ ( [h,r], x ) &\mapsto& h \cdot x \,. }$

The following definition modifies the skeletal presentation of inertia orbifolds:

###### Definition

Write

$\Lambda_{S^1} \mathcal{X} \;\coloneqq\; \underset{[g] \in ConjCl(G)}{\coprod} X^g \!\sslash\! \Lambda_g$

for the orbifold which is the disjoint union over the conjugacy classes $[g]$ of $G$ of the global quotient orbifolds of the fixed loci $X^g = X^{\langle g\rangle}$ by the group action (1) of the group from Def. .

(Huan 18, Def. 2.14, review in Dove 19, p. 62)

###### Remark

A similar definition is obtained by restricting Ganter 07, Def. 2.3 to constant loops and to $k = 1$, which yields

$\simeq \underset{[g] \in ConjCl(G)}{\coprod} X^g \!\sslash\! (C_g/\langle g\rangle)$

## Properties

### Relation to inertia orbifold

The canonical group homomorphism (via Def. )

$\array{ C_g &\xhookrightarrow{\;\;}& \Lambda_g &\xrightarrow{\;\;}& C_g/\langle g \rangle \\ h &\mapsto& (h,0) \\ && [h,n] &\mapsto& [h] }$

induce canonical morphism from the plain inertia orbifold to Huan’s (Def. ) and Ganter’s orbifolds (Def. ):

$\array{ \Lambda\mathcal{X} &\xrightarrow{\;\;\;}& \Lambda_{S^1} \mathcal{X} &\xrightarrow{\;\;\;}& \cdots \,. }$

The notion is highlighted in:

following

followiong, in turn, a similar construction in:

and motivated (as made explicit on p. 63 of Dove 19) by the “rotation condition” on Tate K-theory, due to

following Ganter 07, Def. 3.1.

Streamlined review is in: