# nLab electric-magnetic duality

## Surveys, textbooks and lecture notes

#### Differential cohomology

differential cohomology

duality

# Contents

## Idea

Electric-magnetic duality is a lift of Hodge duality from de Rham cohomology to ordinary differential cohomology.

## Description

Consider a circle n-bundle with connection $\nabla$ on a space $X$. Its higher parallel transport is the action functional for the sigma-model of $\left(n-1\right)$-dimensional objects ($\left(n-1\right)$-branes) propagating in $X$.

For $n=1$ this is the coupling of the electromagnetic field to particles. For $n=2$ this is the coupling of the Kalb-Ramond field to strings.

The curvature ${F}_{\nabla }\in {\Omega }^{n+1}\left(X\right)$ is a closed $\left(n+1\right)$-form. The condition that its image $\star {F}_{\nabla }$ under the Hodge star operator is itself closed

${d}_{\mathrm{dR}}\star {F}_{\nabla }=0$d_{dR} \star F_\nabla = 0

is the Euler-Lagrange equation for the standard (abelian Yang-Mills theory-action functional on the space of circle n-bundle with connection.

If this is the case, it makes sense to ask if $\star {F}_{\nabla }$ itself is the curvature $\left(d-\left(n+1\right)\right)$-form of a circle $\left(d-\left(n+1\right)-1\right)$-bundle with connection $\stackrel{˜}{\nabla }$, where $d=\mathrm{dim}X$ is the dimension of $X$.

If such $\stackrel{˜}{\nabla }$ exists, its higher parallel transport is the gauge interaction action functional for $\left(d-n-3\right)$-dimensional objects propagating on $X$.

In the special case of ordinary electromagnetism with $n=1$ and $d=4$ we have that electrically charged 0-dimensional particles couple to $\nabla$ and magnetically charged $\left(4-\left(1+1\right)-2\right)=0$-dimensional particles couple to $\stackrel{˜}{\nabla }$.

In analogy to this case one calls generally the $d-n-3$-dimensional objects coupling to $\stackrel{˜}{\nabla }$ the magnetic duals of the $\left(n-1\right)$-dimensional objects coupling to $\nabla$.

## Generalizations

For $d=4$ EM-duality is the special abelian case of S-duality for Yang-Mills theory. Witten and Kapustin argued that this is governed by the geometric Langlands correspondence.

## References

An exposition of the relation to geometric Langlands duality is given in

• Edward Frenkel, What Do Fermat’s Last Theorem and Electro-magnetic Duality Have in Common? KITP talk 2011 (web)

Revised on January 10, 2013 17:26:40 by Urs Schreiber (89.204.153.52)