Connections on bundles
Higher abelian differential cohomology
Higher nonabelian differential cohomology
Application to gauge theory
Electric-magnetic duality is a lift of Hodge duality from de Rham cohomology to ordinary differential cohomology.
Consider a circle n-bundle with connection on a space . Its higher parallel transport is the action functional for the sigma-model of -dimensional objects (-branes) propagating in .
For this is the coupling of the electromagnetic field to particles. For this is the coupling of the Kalb-Ramond field to strings.
The curvature is a closed -form. The condition that its image under the Hodge star operator is itself closed
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 itself is the curvature -form of a circle -bundle with connection , where is the dimension of .
If such exists, its higher parallel transport is the gauge interaction action functional for -dimensional objects propagating on .
In the special case of ordinary electromagnetism with and we have that electrically charged 0-dimensional particles couple to and magnetically charged -dimensional particles couple to .
In analogy to this case one calls generally the -dimensional objects coupling to the magnetic duals of the -dimensional objects coupling to .
For 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.
duality in physics, duality in string theory
It was originally noticed in
- P. Goddard, J. Nuyts, and David Olive, Gauge Theories And Magnetic Charge, Nucl. Phys. B125 (1977) 1-28.
that where electric charge in Yang-Mills theory 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 the electric/magnetic duality conjecture formulation in
According to (Kapustin-Witten 06, pages 3-4) the observaton that the Montonen-Olive dual charge group coincides with the Langlands dual group is due to
See also the references at S-duality.
The insight that the Montonen-Olive duality works more naturally in super Yang-Mills theory is due to
and that it works particularly for N=4 D=4 super Yang-Mills theory is due to
- H. Osborn, Topological Charges For Supersymmetric Gauge Theories And Monopoles Of Spin 1, Phys. Lett. B83 (1979) 321-326.
The observation that the electric/magnetic duality extends to an -action in this case is due to
John Cardy, E. Rabinovici, Phase Structure Of Zp Models In The Presence Of A Theta Parameter, Nucl. Phys. B205 (1982) 1-16;
John Cardy, Duality And The Theta Parameter In Abelian Lattice Models, Nucl. Phys. B205 (1982) 17-26.
A. Shapere and Frank Wilczek, Selfdual Models With Theta Terms, Nucl. Phys. B320 (1989) 669-695.
and specifically the embedding of this into string theory S-duality originates in
- Ashoke Sen, Dyon - Monopole Bound States, Self-Dual Harmonic Forms on the Multi-Monopole Moduli Space, and Invariance in String Theory (arXiv:hep-th/9402032)
The understanding of this -symmetry as a remnant conformal transformation on a 6-dimensional principal 2-bundle-theory – the 6d (2,0)-superconformal QFT – compactified on a torus is described in
The relation of S-duality to geometric Langlands duality was understood in
Exposition of this is in
- Edward Frenkel, What Do Fermat’s Last Theorem and Electro-magnetic Duality Have in Common? KITP talk 2011 (web)