Topologically nontrivial three-body contact interaction in one dimension

It is known that three-body contact interactions in one-dimensional n( ≥ 3)-body problems of nonidentical particles can be topologically nontrivial: they are all classified by unitary irreducible representations of the pure twin group PTn. It was, however, unknown how such interactions are described in the Hamiltonian formalism. In this paper, we study topologically nontrivial three-body contact interactions from the viewpoint of path integral. Focusing on spinless particles, we construct an n(n − 1)(n − 2)/ 3!-parameter family of n-body Hamiltonians that corresponds to one particular one-dimensional unitary representation of PTn. These Hamiltonians are written in terms of background Abelian gauge fields that describe infinitely-thin magnetic fluxes in the n-body configuration space.