Abstract:
I will show how the problem of pairs of mutually unbiased bases can be interpreted as an intersection of two Lagrangian varieties in a product of  coadjoint orbits. Using a symplectic embedding of the cotangent bundle to one of these Lagrangian varieties into the product, we realize the intersection as critical points of a potential, which is a Laurent polynomial. We describe the mirror dual symplectic toric variety via the Birkoff polytope of double stochastic matrices and explain its relation to hyperkahler rotation.