nLab magnetic charge

Surveys, textbooks and lecture notes

Differential cohomology

differential cohomology

Contents

Idea

Maxwell’s equations for the electromagnetic field $F \in \Omega^2(X)$ on a $d$-dimensional spacetime $X$ in the presence of electric current $j_{el} \in \Omega^{d-1}(X)$ read

$d F = 0$
$d \star F = j_{el} \,.$

Formally this suggests an immediate generalization where a conserved current called the magnetic current $j_{mag} \in \Omega^{3}(X)$ is introduced and the first of the equations above is replaced by

$d F = j_{mag} \,.$

The corresponding charge is the magnetic charge.

• net magnetic charge has not been observed in nature.

The other thing is a more conceptual problem: the Dirac quantization condition says that the $2$-form $F$ is not entirely arbitrary, but constrained to be the characteristic curvature $2$-form of a degree-$2$ cocycle in ordinary differential cohomology, for instance the curvature $2$-form of a $U(1)$-principal bundle with connection. But this necessarily implies that $d F = 0$.

Indeed, to circumvent dealing with this problem Dirac, in his original argument, has considered removing from $X$ the support of any magnetic charge density.

It was Dan Freed in (Freed) who discussed that the global description of the electromagnetic field does make sense even in the presence of electric current if one generalizes the model of a degree-$2$ differential cocycle to that of a twisted cocycle.

The magnetic current $3$-form $j_{mag}$ is then realized as the curvature characteristic $3$-form of a degree-$3$ cocycle in ordinary differential cohomology $\hat j_{mag}$, the electromagnetic field $F$ is the curvature characteristic $2$-form of a degree-$2$ twisted differential cocycle $\hat F$ and the equation

$d F = j_{mag}$

expresses the twisting of $\hat F$ by $\hat j_{mag}$ at the level of curvature forms.

This means that Dirac almost found bundle gerbes already in 1931, had he not discussed only the neighbourhood of magnetic monopoles.

Magnetic charge anomaly

Short of an experimental detection of magnetic monopoles the above considerations are of little practical relevance for ordinary electromagnetism. In their (straightforward) generalization to higher abelian gauge theory they do, however, serve to provide a more complete conceptual picture that provides the conceptual bases for effects such as the Green–Schwarz mechanism.

Namely from the very fact that in the presence of magnetic charge the gauge field $\hat F$ is a twisted cocycle, it follows that the standard action functional for electrically charged quantum particles – the one whose consideration led Dirac to his Dirac quantization condition – in general has a quantum anomaly.

To see this, consider the following.

In the simplest case we consider a single electrically charged particle with worldline $\gamma : [0,1] \to X$ propagating inside a single patch $U_i \to X$ of some cover over which the electromagnetic field is represented by a Deligne cocycle with $1$-form $A_i \in \Omega^1(U_i)$. Then then action functional for the coupling of the particle to the background electromagnetic field is

$(\hat F, \gamma) \mapsto \exp( -2 \pi i \int_\gamma A_i) \in U(1) \,.$

This may be rewritten by introducing the corresponding distributional $(d-1)$-form current $(j_{el})_i \in \Omega^{d-1}(U_i)$ as

$(\hat F, \gamma) \mapsto \exp( -2 \pi i \int_{U_i} (j_{el})_i \wedge A_i) \in U(1) \,.$

This integrand may be recognized as the local connection $d$-form of the cup product of the Deligne $(d-1)$-cocycle $\hat {j_{el}}$ and the Deligne $2$-cocycle $\hat F$. This way

$(\hat F, \gamma) \mapsto \exp( -2 \pi i \int_X \hat{j_{el}} \cdot \hat F) \in U(1) \,,$

where now the integral including the prefactor of $- 2 \pi i$ denotes push-forward in cohomology along $X \to {*}$, and we canonically identify degree-$0$ Deligne cohomology of the point with $U(1)$.

So far this is the formulation in differential cohomology of the ordinary action functional for the coupling of electric charges $\hat {j_{el}}$ to the electromagnetic field $\hat F$.

But in this formulation one can discuss what happens to this when a non-vanishing magnetic current $\hat {j_{mag}}$ is present. In that case $\hat F$ is no longer a plain $2$-cocycle, but a twisted $2$-cocycle. Accordingly, the above push-forward to the point produces not a $U(1)$-valued function of $(\hat F, \hat {j_{el}})$, but a “twisted function”, i.e. a section of a line bundle.

Since $\hat F$ is twisted by $\hat {j_{mag}}$, this line bundle is the $2$-cocycle

$\exp( -2 \pi i \int_X \hat{j_{el}} \cdot \hat {j_{mag}}) \,.$

(…discussion of how this is a $2$-cocycle on configuration space goes here…)

Comparing with the discussion at quantum anomaly, it follows that the non-triviality of this $2$-cocycle is the higher gauge theory anomaly of the ordinary action functional describing the coupling of electric charges to the electromagnetic field in the presense of magnetic charges.

This phenomenon has traditionally been known somewhat implicitly in the context of the Green–Schwarz mechanism.

References

Magnetic charge for general compact Lie groups as gauge groups was first discussed in

• François Englert, P. Windey, Quantization Condition For ‘t Hooft Monopoles In Compact Simple Lie Groups, Phys. Rev. D14 (1976) 2728-2731.

In

• P. Goddard, J. Nuyts, and David Olive, Gauge Theories And Magnetic Charge, Nucl. Phys. B125 (1977) 1-28.

it was first noticed that when electric charge takes values in the weight lattice of the gauge group, then magnetic charge takes values in the lattice of what is now called the Langlands dual group. (This led to Montonen-Olive’s S-duality conjecture).

The interpretation of magnetic charge in terms of degree-3 cocycles in ordinary differential cohomology is in the introduction-section of

Last revised on October 16, 2017 at 06:26:03. See the history of this page for a list of all contributions to it.