Quantum field theory (QFT) is omnipresent in the physical world. It is a powerful formalism which describes both the interactions of the elementary constituents of matter and the effective behavior of large systems, like magnets or superconductors. More recently, quantum field theory concepts, like decimation and renromalization have found new applications to the study of biological systems or machine learning. A most surprising feature of QFT is universality: water at the critical point and a magnet at the Curie temperature are described by surprisingly similar equations. Deepening our understanding of QFT will not only provide us with a new understanding of physical phenomena, but also it is today our best bet to understand why machine learning actually works.
We have a good grasp of weakly coupled QFT, but in many instances one needs to deal with strongly coupled systems. Understanding such models has been an ongoing research endeavor for the past 50 years. A particularly fruitful approach to strongly interacting QFTs are the so called large N theories in which the fundamental building block, the field, is not a scalar but a vector, a matrix or a higher order tensor. The number of components of the field is counted by a parameter, N, which can be varied. The vector and matrix case have been studied since the '70s and led to numerous advances in conformal field theory (CFT), string theory etc. . Their success relies on the "1/N expansion": the models significantly simplify when the number of components N is take to be large.
The tensor case has been less studied and took off only about 10 years ago when the PI and collaborators discovered the 1/N expansion for random tensors and the corresponding melonic large N limit. Interest in the field increased significantly after E. Witten noted that such models are similar to the SYK model but do not require quenching and I. Klebanov and collaborators pioneered the study of tensor field theories. This research project studied the implications of the tensor 1/N expansion both for random tensors and tensor field theories. Objective 1 explored and extended the melonic universality class and lead to unexpected connections to quantum information theory and free probability. Objective 2 explored tensor field theories and melonic conformal field theories. Objective 3 applied such theories and lead to the introduction of an asymptotically free tensor field theory ideal for the study of strongly coupled systems and pioneered the study of quantum mechanical models with long range interactions.
Among the achievements of this project we count the discovery of new universality classes of critical phenomena, the identification of an asymptotically free tensor field theory suited for the study of strongly interacting physics, the formulation of a new criterion for entanglement detection in D-parttite quantum systems and the identification of the correct generalization of free cumulants for random tensors.
