nLab inner local system

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:

• Daniel Freed, Michael Hopkins, Constantin Teleman, Def. 3.6 of: Twisted equivariant K-theory with complex coefficients, Journal of Topology, Volume 1, Issue 1, 2007 (arXiv:math/0206257, doi:10.1112/jtopol/jtm001)

• Jean-Louis Tu, Ping Xu, Prop. 3.9, Def. 3.10 in: Chern character for twisted K-theory of orbifolds, Advances in Mathematics Volume 207, Issue 2, 20 December 2006, Pages 455-483 $[$arXiv:math/0505267, doi:10.1016/j.aim.2005.12.001$]$

(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:

• Hisham Sati, Urs Schreiber, Section 3 of: Anyonic defect branes in TED-K-theory $[$arXiv:2203.11838, exposition pdf$]$