We count the Bethe states of quantum integrable models with twisted boundary
conditions using the Witten index of 2d supersymmetric gauge theories. For
multi-component models solvable by the nested Bethe ansatz, the result is a
novel restricted occupancy problem. For the SU(3) spin chain and the t-J model,
we propose formulae for the solution count on singular loci in the space of
twist parameters.
