Hypertoric varieties are symplectic algebraic varieties which are controlled by the combinatorics of a hyperplane arrangement, analogous to the way toric varieties are controlled by polytopes or fans. We describe a number of relations that hold between any pair of hypertoric varieties defined by Gale dual arrangements: (1) a duality between deformations of their cohomologies, (2) a linear duality on their degree two equivariant cohomologies for certain torus actions, and (3) a duality relating certain categories of modules over deformations of their rings of regular functions. These results are proved using the combinatorics of the arrangements, but we conjecture that they hold for other pairs of symplectic algebraic varieties which physicists have identified as belonging to dual $d=3$ gauge theories. Joint work with A. Licata, N. Proudfoot and B. Webster.

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