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.
duality in physics, duality in string theory
An exposition of the relation to geometric Langlands duality is given in