nLab D-brane charge quantization in topological K-theory -- references

D-brane charge quantization in topological K-theory

D-brane charge quantization in topological K-theory

On the conjectural D-brane charge quantization in topological K-theory:

Origin and basics

The idea that D-branes have Dirac charge quantization in topological K-theory originates with the observation that their charge expressed in RR-field flux densities resembles the image of a Chern character:

Further early discussion:

and with emphasis on the full picture of twisted differential K-theory in:

Here:

From Sch 18

Expression of these D-brane K-theory classes via the Atiyah-Hirzebruch spectral sequence:

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)

Understanding the solitonic (non-singular) D-branes and their T-duality in K-theory:

Towards a matrix model taking these K-theoretic effects into account (K-matrix model):

Twisted, equivariant and differential refinement

Discussion of charge quantization in twisted K-theory for the case of non-vanishing B-field:

An elaborate proposal for the correct flavour of equivariant KR-theory needed for orientifolds is sketched 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):

Reviews

Amplification of torsion-charges implied by charge quantization in Ktheory

Review of D-branes charge seen in KK-theory:

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

Conceptual problems

But there remain conceptual issues with the proposal that D-brane charge is in K-theory, as highlighted 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, JHEP 0511:004 (2005) [arXiv:hep-th/0509153, doi:10.1088/1126-6708/2005/11/004]

    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. […] it is fair to say that a satisfactory answer is still missing.

The closest available towards an actual check of the argument for K-theory via open superstring tachyon condensation (Witten 98, Section 3) seems to be

which, however, concludes (on p. 32) with:

It would also be interesting to see if these developments can shed light on the long-speculated relation between string field theory and the K-theoretic description of D-brane charge [[75, 76, 77]]. We leave these questions for future work.

See also

which still lists (on p. 112) among open problems of string field theory:

“Are there topological invariants of the open string star algebra representing D-brane charges?”

For orbifolds in equivariant K-theory

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

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

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

See also

More complete discussion of double dimensional reduction of the supergravity C-field in 11d to the expected B-field and RR-field flux forms in 10d:

Last revised on February 10, 2024 at 16:27:23. See the history of this page for a list of all contributions to it.