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It is known that three-body contact interactions in one-dimensional$n(\geq3)$-body problems of nonidentical particles can be topologicallynontrivial: they are all classified by unitary irreducible representations ofthe pure twin group $PT_{n}$. It was, however, unknown how such interactionsare described in the Hamiltonian formalism. In this paper, we studytopologically nontrivial three-body contact interactions from the viewpoint ofthe path integral. Focusing on spinless particles, we construct an$n(n-1)(n-2)/3!$-parameter family of $n$-body Hamiltonians that corresponds toone particular one-dimensional unitary representation of $PT_{n}$. TheseHamiltonians are written in terms of background Abelian gauge fields thatdescribe infinitely-thin magnetic fluxes in the $n$-body configuration space.

Topologically Nontrivial Three-Body Contact Interaction in One Dimension