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Three-nucleon (3N) forces are an indispensable ingredient for accuratefew-body and many-body nuclear structure and reaction theory calculations.While the direct implementation of chiral 3N forces can be technically verychallenging, a simpler approach is given by employing instead amedium-dependent NN interaction V_med that reflects the physics of three-bodyforces at the two-body normal-ordered approximation. We review the derivationand construction of V_med from the chiral 3N interaction atnext-to-next-to-leading order (N2LO), consisting of a long-range$2\pi$-exchange term, a mid-range $1\pi$-exchange component and a short-rangecontact term. Several applications of V_med to the equation of state of coldnuclear and neutron matter, the nucleon single-particle potential in nuclearmatter, and the nuclear quasiparticle interaction are discussed. We alsoexplore differences in using local vs. nonlocal regulating functions on 3Nforces and make direct comparisons to exact results at low order inperturbation theory expansions for the equation of state and single-particlepotential. We end with a discussion and numerical calculation of the in-mediumNN potential V_med from the next-to-next-to-next-to-leading order (N3LO) chiral3N force, which consists of a series of long-range and short-range terms.

Implementing chiral three-body forces in terms of medium-dependent two-body forces