nLab twisted equivariant differential K-theory

Idea

The notion of twisted equivariant differential K-theory is meant to be the joint generalization of all three variations of topological K-theory given by

1. twisted K-theory

2. equivariant K-theory

3. differential K-theory

hence the joint generalization of

1. twisted equivariant K-theory

2. twisted differential K-theory

3. equivariant differential K-theory.

It is this joint generalization that is, ultimately, required by

1. the K-theory classification of D-brane charge,

   conjectured to apply to non-perturbative type II string theory;

2. the K-theory classification of topological phases of matter,

   conjectured to apply in topological condensed matter theory.

References

The construction of full twisted & equivariant & differential cohomology theories had long been left open. It does follow as an immediate consequence of the twisted & equivariant Chern-Dold character and as such is discussed in:

• Hisham Sati, Urs Schreiber, Twisted Equivariant Differential non-abelian generalized cohomology (in preparation)

The twisted + equivariant + flat differential sector of K-theory is more readily described:

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