nLab
twisted differential K-theory

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)

Related concepts

• K-theory

• topological K-theory

• twisted K-theory

• differential K-theory

• twisted equivariant K-theory

• twisted ordinary differential cohomology

References

Preliminary discussion included

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

• Alexander Kahle, Alessandro Valentino, T-Duality and Differential K-Theory

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

• Jacques Distler, Dan Freed, Greg Moore, Orientifold Précis in: Hisham Sati, Urs Schreiber (eds.) Mathematical Foundations of Quantum Field and Perturbative String Theory Proceedings of Symposia in Pure Mathematics, AMS (2011) (arXiv:0906.0795, slides)

Actual constructions appear with

• Alan Carey, Jouko Mickelsson, Bai-Ling Wang, Differential Twisted K-theory and Applications, Journal of Geometry and Physics, Volume 59, Issue 5, May 2009, Pages 632-653 (arXiv:0708.3114, doi:10.1016/j.geomphys.2009.02.002)

Review in

• Ulrich Bunke, Thomas Schick, Section 7 in: Differential K-theory: A survey, in: Christian Bär, Joachim Lohkamp, Matthias Schwarz (eds.), Global Differential Geometry, Springer 2012 (doi:10.1007/978-3-642-22842-1)

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

• Ulrich Bunke, Thomas Nikolaus, Twisted differential cohomology (arXiv:1406.3231)

See also:

• Byungdo Park, Geometric models of twisted differential K-theory I, J. Homotopy Relat. Struct. 13, 143–167 (2018) (arXiv:1602.02292, doi:10.1007/s40062-017-0177-z)

More detailed discussion of twisted differential K-theory for KU/KO and its application to K-theory classification of D-brane charge:

• Daniel Grady, Hisham Sati, Ramond-Ramond fields and twisted differential K-theory (arXiv:1903.08843)

• Daniel Grady, Hisham Sati, Twisted differential KO-theory (arXiv:1905.09085)

• Fabio Ferrari Ruffino, Juan Carlos Rocha Barriga, Twisted differential K-characters and D-branes, Nuclear Physics B Volume 960, November 2020, 115169 (arXiv:2009.04223. doi:10.1016/j.nuclphysb.2020.115169)