  We extend the formalism of self-consistent Green's function theory to include
three-body interactions and apply it to isotopic chains around oxygen for the
first time. The third-order algebraic diagrammatic construction [ADC(3)]
equations for two-body Hamiltonians can be exploited upon defining
system-dependent one- and two-body interactions coming from the three-body
force, and correspondingly dropping interaction reducible diagrams. This goes
beyond the standard normal ordering truncations recently used in ab-inito
studies. The Koltun sum rule for the total binding energy acquires a correction
due to the added three-body interaction. This formalism is then applied to
study chiral two-nucleon (2N) and three-nucleon forces (3NF) evolved to low
momentum cutoffs. We find that these interactions reproduce the binding
energies of nitrogen, oxygen and fluorine isotopes to great accuracy, providing
clear indication of the predictive power of this approach. All three neutron
driplines are correctly predicted when full 3NF are included. The formalism
introduced also allows to calculate form factors for nucleon transfer on doubly
magic systems.
