  We introduce a fully antisymmetrized treatment of three-cluster dynamics
within the ab initio framework of the no-core shell model/resonating-group
method (NCSM/RGM). Energy-independent non-local interactions among the three
nuclear fragments are obtained from realistic nucleon-nucleon interactions and
consistent ab initio many-body wave functions of the clusters. The
three-cluster Schr\"odinger equation is solved with bound-state boundary
conditions by means of the hyperspherical-harmonic method on a Lagrange mesh.
We discuss the formalism in detail and give algebraic expressions for systems
of two single nucleons plus a nucleus. Using a soft
similarity-renormalization-group evolved chiral nucleon-nucleon potential, we
apply the method to an $^4$He+$n+n$ description of $^6$He and compare the
results to experiment and to a six-body diagonalization of the Hamiltonian
performed within the harmonic-oscillator expansions of the NCSM. Differences
between the two calculations provide a measure of core ($^4$He) polarization
effects.
