Contrents

cohomology

# Contrents

(…)

## References

The idea that D-branes have Dirac charge quantization in topological K-theory originates in

Discussion of full-blown twisted differential K-theory and its relation to D-brane charge in type II string theory

Discussion of full-blown twisted differential orthogonal K-theory and its relation to D-brane charge in type I string theory (on orientifolds):

But there remain conceptual issues with the proposal that D-brane charge is in K-theory, as highlighted for in

In particular, actual checks of the proposal that D-brane charge is given by K-theory, via concrete computation in boundary conformal field theory, have revealed some subtleties:

• Stefan Fredenhagen, Thomas Quella, Generalised permutation branes, JHEP0511:004, 2005 (arXiv:hep-th/0509153)

It might surprise that despite all the progress that has been made in understanding branes on group manifolds, there are usually not enough D-branes known to explain the whole charge group predicted by (twisted) K-theory.

Further review and discussion of D-brane charge in K-theory includes the following

A textbook account of D-brane charge in (twisted) topological K-theory is

Discussion of D-branes in KK-theory is reviewed in

based on

In particular (BMRS2) discusses the definition and construction of D-brane charge as a generalized index in KK-theory. The discussion there focuses on the untwisted case. Comments on the generalization of this to topologicall non-trivial B-field and hence twisted K-theory is in

Specifically for D-branes in WZW models see

• Peter Bouwknegt, A note on equality of algebraic and geometric D-brane charges in WZW models (pdf)

More on this, with more explicit relation to noncommutative motives, is in

• Snigdhayan Mahanta, Noncommutative correspondence categories, simplicial sets and pro $C^\ast$-algebras (arXiv:0906.5400)

• Snigdhayan Mahanta, Higher nonunital Quillen $K'$-theory, KK-dualities and applications to topological $\mathbb{T}$-duality, Journal of Geometry and Physics, Volume 61, Issue 5 2011, p. 875-889. (pdf)

Discussion of D-brane matrix models taking these K-theoretic effects into account (K-matrix model) is in

• T. Asakawa, S. Sugimoto, S. Terashima, D-branes, Matrix Theory and K-homology, JHEP 0203 (2002) 034 (arXiv:hep-th/0108085)

The proposal that D-brane charge on orbifolds is measured in equivariant K-theory goes back to

but it was pointed out that only a subgroup of equivariant K-theory can be physically relevant in

Further discussion of equivariant K-theory for D-branes on orbifolds includes the following:

Discussion of real K-theory for D-branes on orientifolds includes the following:

The original observation that D-brane charge for orientifolds should be in KR-theory is due to

and was then re-amplified in

With further developments in

Discussion of orbi-orienti-folds using equivariant KO-theory is in

An elaborate proposal for the correct flavour of real equivariant K-theory needed for orientifolds is sketched in

Discussion of the alleged K-theory classification of D-brane charge in relation to the M-theory supergravity C-field is in