nLab
inner local system

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Higher geometry

Cohomology

cohomology

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Special notions

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Idea

Given an orbifold 𝒳\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 HGH\subset G-fixed loci in the case that 𝒳\mathcal{X} is presented as a global quotient orbifold XGX \sslash G).

These inner local systems make an appearance in the twisted equivariant Chern character of the orbifold K-theory of 𝒳\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

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 13:08:46. See the history of this page for a list of all contributions to it.