Yang-Mills Theory and Large N Limits.

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The research concerns quantum field theory(QFT), a mathematical framework for systems with many variables subject to random fluctuations. It is the language of microscopic physics as well as the physics of phase transitions(eg. boiling of water), where the randomness is either quantum or thermal. Remarkably, ideas from QFT have also been used to solve several outstanding problems in pure mathematics. QFT methods are also used to analyse turbulent fluid flow and fluctuations in financial markets.Yang-Mills or gauge theories generalise Maxwell's electrodynamics, which describes light and electromagnetism. Quantum Chromodynamics (QCD, a gauge theory with N=3 colours of quarks) describes strong forces between quarks and gluons. However, all strongly interacting sub-atomic particles (hadrons, eg. proton, neutron) are bound states of quarks and gluons. An outstanding mathematical challenge is to deduce from QCD a theory of the bound states. There are two main difficulties: gauge invariance and infinities. Though the theory is written in terms of quarks and gluons, these are not gauge invariant. This is accompanied by the mysterious phenomenon of confinement: the impossibility to remove a quark or gluon from a hadron. Infinities are due to the lack of a mathematical framework (renormalisation) to treat the short distance interactions between infinitely many degrees of freedom in bound states. We develop methods to make progress with these two mathematical difficulties.Yang-Mills theory is acknowledged as a difficult and important problem with deep relations to geometry, algebra and analysis. It appears on the Clay Institute's list of million dollar maths problems. The large-N limit is a promising approximation for Yang-Mills theory. It is a classical limit: fluctuations in gauge invariant observables are small. Though Planck's constant is not assumed small, techniques of classical mechanics are applicable.We aim to understand mathematical structures and solve, either approximately or exactly, some large-N gauge theories such as spherical QCD and QCD on a cylinder. The former models hadrons containing a heavy quark surrounded by light quarks. Gluons are not dynamical, but contribute a coulomb potential, as in an atom. The energy is unbounded below due to infinities. We aim to renormalise the theory so as to study the spectrum of bound states. What makes this model very interesting is that it is conjectured to be asymptotically free, i.e. interactions deviate logarithmically from that of free particles at short distances, as in QCD. In QCD on a cylinder, we assume space is a circle. While quarks have infinitely many degrees of freedom, gluons lack local dynamics, but have global degrees of freedom around the circle. This is the simplest version of QCD, with dynamical gauge fields, that has not yet been solved. We aim to reformulate the theory in terms of gauge invariant variables and study their Poisson algebra in order to better understand how to solve Hamilton's equations in the large-N limit.We also study matrix models of local gluon dynamics: quantum systems with several dynamical hermitian matrices, but basis independent observables. We develop methods to determine the correlations of large-N multi-matrix models. For Euclidean models, it is a method based on symmetries of both action and measure, satisfying a non-abelian Lie algebra. For hamiltonian matrix models, we investigate insights from spin chains and integrable systems.Finally, we study the existence of a line of interacting ultraviolet fixed points in four dimensional O(N) scalar field theory in the large-N limit. This is of interest in the Higgs sector of particle physics, which is based on massive scalar fields with a quartic interaction, but suffers from bad UV behavior and a naturalness problem. A line of UV fixed points would ensure good high energy behavior and naturalness, by leading to a symmetry (scale invariance) when masses are set to zero.