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
The (classical) Langlands program is a modern extension of some ideas of Kronecker and Weber in number theory, relating automorphic forms and Galois representations, and pioneered by Robert Langlands.
There is now a deep analogue for function fields, the geometric Langlands program pioneered by Beilinson and Drinfeld. See there for more details.
A good survey is in
(which however, despite the title, has nothing in it that is particularly related to physics).
See also