nLab
electric-magnetic duality

Context

Physics

physics, mathematical physics, philosophy of physics

Surveys, textbooks and lecture notes


theory (physics), model (physics)

experiment, measurement, computable physics

Differential cohomology

Duality

Langlands correspondence

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 (n1)(n-1)-dimensional objects ((n1)(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 dRF =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 (dn3)(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 dn3d-n-3-dimensional objects coupling to ˜\tilde \nabla the magnetic duals of the (n1)(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

duality in physics, duality in string theory

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)