electric-magnetic duality (Rev #3) in nLab

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 XX. Its higher parallel transport is the action functional for the sigma-model of (n−1)(n-1)-dimensional objects ((n−1)(n-1)-branes) propagating in XX.

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

The curvature F ∇∈Ω n+1(X)F_\nabla \in \Omega^{n+1}(X) is a closed (n+1)(n+1)-form. The condition that its image ⋆F ∇\star F_\nabla under the Hodge star operator is itself closed

d dR⋆F ∇=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 ⋆F ∇\star F_\nabla itself is the curvature (d−(n+1))(d-(n+1))-form of a circle (d−(n+1)−1)(d-(n+1)-1)-bundle with connection ∇˜\tilde \nabla, where d=dimXd = dim X is the dimension of XX.

If such ∇˜\tilde \nabla exists, its higher parallel transport is the gauge interaction action functional for (d−n−3)(d-n-3)-dimensional objects propagating on XX.

In the special case of ordinary electromagnetism with n=1n=1 and d=4d = 4 we have that electrically charged 0-dimensional particles couple to ∇\nabla and magnetically charged (4−(1+1)−2)=0(4-(1+1)-2) = 0-dimensional particles couple to ∇˜\tilde \nabla.

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

Generalizations

For d=4d= 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 heterotic string theory one considers 1-dimensional objects in d=10d=10-dimensional spaces electrically charged (under the Kalb-Ramond field). Their magnetic duals are 5-dimensional objects (fivebranes), studied in dual heterotic string theory.

• S-duality

  • electro-magnetic 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)