Contents

Ingredients

Concepts

Constructions

Examples

Theorems

cohomology

# Contents

## Idea

Given an orbifold $\mathcal{X}$, an inner local system (Ruan 00, Def. 3.1.6) is a family of local systems (in fact flat connections with coefficients in U(1)) on loci with non-trivial automorphisms (i.e. on $H\subset G$-fixed loci in the case that $\mathcal{X}$ is presented as a global quotient orbifold $X \sslash G$).

These inner local systems make an appearance in the twisted equivariant Chern character of the orbifold K-theory of $\mathcal{X}$, which lands, fixed-locus wise (in fact: automorphism-wise), in the corresponding twisted de Rham cohomology.

## References

The terminology “inner local system” is due to

• Yongbin Ruan, Def. 3.1.6 in: Stringy Geometry and Topology of Orbifolds $[$arXiv:math/0011149$]$

As a contribution to the twist in the twisted equivariant Chern character the concept appears (not under this name) in:

(On the other hand, this inner local system twist seems to be missing in the twisted equivariant Chern character of Mathai &Stevenson 03?)

A possibly more transparent (?) derivation of this subtle twisting by inner local systems, for the simple special case of a global singularity, is offered in:

Created on May 13, 2022 at 17:08:46. See the history of this page for a list of all contributions to it.