String-nets, quantum loop gases and the sign problem for non-abelian anyons

Hamiltonians giving rise to topological ground states can be constructed explicitly as sums of local operators acting on Hilbert spaces where distinct classical string-net configurations are orthogonal. We show explicitly the connection beteewn string-nets and quantum loop gas models with their non-orthogonal inner product. Also we emphasize the role of the ``sign problem'' for a Hamiltonian in enforcing the topological character of its ground state.