# nLab twisted differential K-theory

Contents

### Context

#### Differential cohomology

differential cohomology

# Contents

## Idea

Twisted differential K-theory is the twisted differential cohomology version K-theory: the combination of differential K-theory and twisted K-theory.

Roughly this is an extension of twisted de Rham cohomology by twisted K-theory via the twisted Chern character map.

To the extent that D-brane charge is classified by K-theory (see there), it is twisted differential K-theory that is relevant: the differential aspect captures the higher gauge field called the RR-field, and the twisted aspects captures the higher gauge field called the B-field, in string theory.

This example of D-brane charge used to be one of the main motivations for finding a definition and construction of twisted differential cohomology theories. Earlier articles on D-brane charge had to assume that such a theory exists (e.g. DFM 09). Actual models now exist (Grady-Sati 19a, Grady-Sati 19b)

## References

Preliminary discussion included

• Alan Carey, Differential twisted K-theory and applications , ESI preprint (pdf)

and specifically the need for twisted differential K-theory for the description of type II string theory backgrounds with their RR-fields and B-fields on orientifolds was highlighted in

A definition and proof of its pertinent properties of twisted differential cohomology in general and of twisted differential K-theory in particular is due to

More detailed discussion of twisted differential K-theory for KU is due to

and case of twisted differential KO-theory is discussed in detail in

Last revised on July 23, 2019 at 02:58:46. See the history of this page for a list of all contributions to it.