physics, mathematical physics, philosophy of physics
theory (physics), model (physics)
experiment, measurement, computable physics
Axiomatizations
Tools
Structural phenomena
Types of quantum field thories
abstract duality: opposite category,
concrete duality: dual object, dualizable object, fully dualizable object, dualizing object
between higher geometry/higher algebra
Langlands duality, geometric Langlands duality, quantum geometric Langlands duality
Electric-magnetic duality is a lift of Hodge duality from de Rham cohomology to ordinary differential cohomology.
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
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 $\nabla$.
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.
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.
electric-magnetic duality of D-branes/RR-fields in type II string theory:
electric charge | magnetic charge |
---|---|
D0-brane | D6-brane |
D1-brane | D5-brane |
D2-brane | D4-brane |
D3-brane | D3-brane |
as imposed by the equations of motion for 11d supergravity (see here). The associated C3-field and C6-field are electric-magnetic duals.
duality in physics, duality in string theory
Detailed review is in
It was originally noticed in
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 observation 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
The observation that the $\mathbb{Z}_2$ electric/magnetic duality extends to an $SL(2,\mathbb{Z})$-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
The understanding of this $SL(2,\mathbb{Z})$-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
Edward Witten, pages 4-5 of Some Comments On String Dynamics, Proceedings of String95 (arXiv:hepth/9507121)
Edward Witten, On S-Duality in Abelian Gauge Theory (arXiv:hep-th/9505186)
Edward Witten, Conformal Field Theory In Four And Six Dimensions (arXiv:0712.0157) An exposition of the relation to geometric Langlands duality is given in
The relation of S-duality to geometric Langlands duality was understood in
Exposition of this is in