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. See 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).

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 K-theory classification of D-brane charge

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 is due to


See also

For the weak nuclear force field

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

For the B-field and RR-field in string theory

Discussion in the broader context of the higher gauge fields in string theory (B-field, RR-field) charge-quantized in generalized cohomology theories (twisted topological K-theory):

For a comprehensive list of literature in this case see at D-brane – Charge quantization in K-theory.

The idea that D-brane charge should be quantized in topological K-theory originates with these articles:

See also at anti-D-brane.

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

See also for instance

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 *C^\ast-algebras (arXiv:0906.5400)

  • Snigdhayan Mahanta, Higher nonunital Quillen KK'-theory, KK-dualities and applications to topological 𝕋\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

See also

For more on this perspective as 10d type II as a self-dual higher gauge theory in the boudnary of a kind of 11-d Chern-Simons theory is in

More complete discussion of the decomposition of the supergravity C-field as one passes from 11d to 10d is in

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 October 19, 2020 at 14:24:24. See the history of this page for a list of all contributions to it.