nLab Dirac charge quantization




physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes

theory (physics), model (physics)

experiment, measurement, computable physics

\infty-Chern-Weil theory

Differential cohomology



For monopole charges in electromagnetism

If the field of electromagnetism serves as a background gauge field for electrically charged quantum particles it is subject to a quantization condition: Outside the locus of any magnetic charge – for instance a magnetic monopole topological defect – the electromagnetic field must be a connection on a principal U(1) bundle whose first Chern class is the discrete measure for the units of magnetic charge. Equivalently this means that the electromagnetic field is a cocycle in ordinary differential cohomology of degree 2.

In the underlying topological sector (“monopole”/“instantons”-sector) this is integral cohomology in degree-2, whose classifying space is equivalently the infinite complex projective space BU(1)P B U(1) \simeq \mathbb{C}P^\infty:


This goes back to an insight due to Dirac 31. See Heras 18 for traditional elementary review, but see Alvarez 85, Frankel and Mangiarotti-Sardanashvily 00 for exposition of the modern picture in terms of fiber bundles in physics. See Freed 00, Section 2 for review in terms of differential cohomology with outlook to generalization to higher gauge fields in string theory (more on which in the references below).

A closely related phenomenon is the magnetic flux quantization in type II superconductors, see there.

On the locus of the magnetic charge itself the situation is more complex. There the magnetic current is given by a cocycle in ordinary differential cohomology of degree 3 (with compact support) and now the electromagnetic field is a connection on a twisted bundle (Freed 00, Section 2).

For monopole charges in non-abelian Yang-Mills theory

A similar charge quantization condition govers monopoles in SU(2)-Yang-Mills theory, see at moduli space of monopoles. Here the Atiyah-Hitchin charge quantization (Atiyah-Hitchin 88, Theorem 2.10) says that the moduli space of monopoles is the complex-rational 2-Cohomotopy of an asymptotic 2-sphere enclosing the monopoles:


For RR-field D-brane charges in K-theory

See at D-brane charge quantization in K-theory

For the C-field in J-twisted Cohomotopy

See at supergravity C-field – shifted flux quantization condition


See also the references at fiber bundles in physics.

For the electromagnetic field

The original argument for charge quantization of the electromagnetic field:

Further early discussion

Understanding of charge quantization in terms of Čech cohomology:

reviewed in:

  • Guillermo A. González, Cohomologa de Cĕch y Cuantización Topológica de Parámetros Físicos, Revista de la Academia Colombiana de Ciencias Exactas, Físicas y Naturales 47 182 (2023) 51-71 [doi:10.18257/raccefyn.1782, pdf]

General review and further discussion:

See also

Arguments for electric charge quantization from anomaly cancellation in the standard model of particle physics:

  • K. S. Babu and Rabindra N. Mohapatra, Quantization of electric charge from anomaly constraints and a Majorana neutrino, Phys. Rev. D 41 (1990) 271 [doi:10.1103/PhysRevD.41.271]

Consideration of combined magnetic and electric charge quantization in vacuum (cf. duality-symmetric electromagnetism):

For the weak nuclear force field

Discussion of the moduli space of monopoles for SU(2)-Yang-Mills theory (weak nuclear force):

D-brane charge quantization in ordinary cohomology

Discussion of charge quantization of D-branes in (relative) ordinary cohomology:

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:


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):


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:

For the C-field in F-theory

Discussion of shifted C-field flux quantization of the C-field in F-theory:

For the C-field in M-theory

Discussion of shifted C-field flux quantization of the C-field in D=11 supergravity/M-theory:

Discussion in twisted Cohomotopy (“Hypothesis H”):

and in equivariant Cohomotopy:

Last revised on April 26, 2024 at 16:05:38. See the history of this page for a list of all contributions to it.