nLab
electric-magnetic duality

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 $(n-1)$-dimensional objects ($(n-1)$-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 $F_{\nabla }\in {\Omega }^{n+1}(X)$ is a closed $(n+1)$-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 $(d-(n+1))$-form of a circle $(d-(n+1)-1)$-bundle with connection $\tilde{\nabla }$, where $d=\dim X$ is the dimension of $X$.

If such $\tilde{\nabla }$ exists, its higher parallel transport is the gauge interaction action functional for $(d-n-3)$-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 $(4-(1+1)-2)=0$-dimensional particles couple to $\tilde{\nabla }$.

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

Examples

• In N=2 D=4 super Yang-Mills theory electric-magnetic duality is studied as Seiberg-Witten theory.

• In heterotic string theory one considers 1-dimensional objects in $d=10$-dimensional spaces electrically charged (under the Kalb-Ramond field). Their magnetic duals are 5-dimensional objects (fivebranes), studied in dual heterotic string theory.

Related concepts

duality in physics, duality in string theory

• S-duality

  • electro-magnetic duality

    • Montonen-Olive duality

  • Seiberg duality

  • geometric Langlands correspondence

  • quantum 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)