Three-nucleon forces. Implementation and applications to atomic nuclei and dense matter

Recent advances in nuclear structure theory have significantly enlarged the accessible part of the nuclear landscape via ab initio many-body calculations. These developments open new ways for microscopic studies of light, medium-mass and heavy nuclei as well as nuclear matter and represent an important step towards a systematic and comprehensive understanding of atomic nuclei across the nuclear chart. While remarkable agreement has been found between different many-body methods for a given nuclear Hamiltonian, the comparison with experiment and the understanding of theoretical uncertainties are still important open questions. The observed discrepancies to experiment indicate deficiencies in presently used nuclear interactions and operators. Chiral effective field theory (EFT) allows to systematically derive contributions to nucleon-nucleon (NN), three-nucleon (3N) and higher-body interactions including estimates of theoretical uncertainties. While the treatment of NN interactions in many-body calculations is well established, the calculation of 3N interactions and their incorporation in ab initio frameworks is still a frontier. This work reviews in detail recent and current developments on the derivation and implementation of improved 3N interactions and provides a comprehensive introduction to fundamental methods for their practical calculation and representation. We further give an overview of novel and established methods that facilitate the inclusion and treatment of 3N interactions in ab initio nuclear structure frameworks and present a selection of the latest calculations of atomic nuclei as well as nuclear matter based on state-of-the-art nuclear NN and 3N interactions derived within chiral EFT. Finally, we discuss ongoing efforts, open questions and future directions.