The three-fold symmetry of planar boron nitride, the III-V analog to graphene, prohibits an electric polarization in its ground state, but this symmetry is broken when the sheet is wrapped to form a BN nanotube. We show that this leads to an electric polarization along the nanotube axis which is controlled by the quantum mechanical boundary conditions on its electronic states around the tube circumference. Thus the macroscopic dipole moment has an intrinsically nonlocal quantum mechanical origin from the wrapped dimension. We formulate this novel phenomenon using the Berry's phase approach and discuss its experimental consequences. 72.40.+w, 78.20.Jq, 61.48.+c, 85.40.Ux Physical properties of materials at the nanoscale can differ dramatically from their bulk counterparts. This is especially evident in the electronic properties, since the quantum behavior of electrons on this scale is sensitive to the size, shape and symmetry of the sample. Recent discovery of carbon nanotubes [1] provides a striking example, where metallic or semiconducting tubes of identical compositions have only slightly different radii [2][3][4]. Layered BN provides a III-V analog to these materials; it can be formed in single and multiwall nanotubes that have the same Bravais lattice as their graphene counterparts, but with inequivalent atomic species on its two sublattices [5,6].Here we show that the broken sublattice symmetry produces a macroscopic electric polarization in BN nanotubes, dependent on their topology. Remarkably, this ground state polarization is an intrinsically nonlocal quantum effect that cannot be described by a classical theory. The sign and size of the longitudinal polarization of the heteropolar tube are controlled by the boundary conditions on its electronic wave functions along its wrapped compact dimensions. We analyze this novel phenomenon by developing a quantum theory of the nanotube polarization in terms of a geometric phase [7][8][9].A natural description of the electronic properties of this system is developed from an expansion of the tight binding Hamiltonian for the π electrons at small wavevectors q around the K and K ′ points at the corners of the Brillouin zone of the 2D hexagonal lattice [10,11]. Introducing index α = ±1 for these points then leads to the long wavelength Hamiltonians [10][11][12]