Tensor quantum field theories and the large N expansion

This is a project in theoretical particle physics. It addresses questions in the STFC themes
What are the fundamental particles and fields? and What are the fundamental laws and symmetries of physics? Quantum field theory is the framework within which the answers to these questions can be investigated. We are far from knowing all the consistent quantum field theories, especially beyond perturbation theory, and understanding their renormalisation group fixed-point structure and phases. The large N expansion is a method of accessing some kinds of non-perturbative structure; used in model field theories it enables us to establish, for example, the existence of potentially non-trivial conformal field theories which have no known direct Lagrangian formulation.

Tensor field theories, both bosonic and fermionic, have been studied recently in the context of the large N limit in three space-time dimensions where they are typically dominated by so-called melonic diagrams. Their direct product symmetry groups lead to theories with a (very large in some cases) number of coupling constants which in turn leads to a potentially complicated phase diagram with an intricate RG structure consisting of many fixed points and potentially lines of fixed points. They are thus good laboratories for identifying new CFTs. This project will study two aspects of these theories.
The large number of coupling constants in theories containing a scalar field with a multiple-index internal symmetry has been dealt with in some cases by truncating to a self-consistent subset of more manageable size. However it is not known whether this subset is stable against addition of extra directions in the full coupling constant space or indeed whether there are other self-consistent choices which lead to substantially different RG structures. There are at least two approaches to this problem and the project will pursue both. The first is brute force: the large number of degrees of freedom cannot be managed by hand and the calculations have to be automated. However even automated calculations can proliferate rapidly in size and establishing the best basis in which to work using conventional symbolic mathematics systems is already a challenge. A speculative line which may turn out to be more powerful ultimately is the recent emergence of deep learning for symbolic mathematics; at present work in this area is on integration and differentiation but exploring a high dimensional space seems a natural candidate for deep learning. The second approach to the problem is to identify hidden symmetries that forbid the generation of paths into the extra directions, for example by generating the model from some underlying theory in which the symmetry is explicit.