Equivariant cohomology distinguishes the geometric structures of toric hyperkähler manifolds

Toric hyperkähler manifolds are the hyperkähler analogue of symplectic toric manifolds. The theory of Bielawski and Dancer tells us that, while a symplectic toric manifold is determined by a Delzant polytope, a toric hyperkähler manifold is determined by a smooth hyperplane arrangement. The purpose of this paper is to show that a toric hyperkähler manifold up to weak hyperhamiltonian $T$-isometry is determined not only by a smooth hyperplane arrangement up to weak linear equivalence but also by its equivariant cohomology $H_T^*(M;\mathbb Z)$ with a point $\hat a$ in $H^2(M;\mathbb R)\setminus\{0\}$ up to weak $H^*(BT;\mathbb Z)$-algebra isomorphism preserving $\hat a$.