Final Activity Report Summary - YANG-MILLS (Large-N Yang-Mills Theory)
The main topic of this project has been the study of Yang-Mills theory, which is the theory of strongly interacting particles, such as the proton. Though there are many indications that the theory is correct, we still do not know how to solve it, except at very high energies. In particular, we are not yet able to quantitatively explain how quarks and gluons bind to form a proton. The analogous problems of how electrons and nuclei bind to form atoms or how the sun and planets form the solar system are, by comparison, much better understood.
A promising approximation method is the 1/N expansion, where N is the number of colours of quarks, which are three in nature. This approach has been successful in several simpler contexts. However, even the starting point of this approximation, large-N Yang-Mills theory is very difficult to solve. The difficulties include dealing with infinitely many matrix degrees of freedom (gluons), a large redundancy in the formulation (gauge invariance), short distance divergences, the presence of bound states and the need for entirely new mathematical methods to deal with these issues. In order to make progress, I studied simplified models for large-N Yang-Mills theory where we can develop techniques to deal with the many complications in partial isolation.
Large-N matrix models have been investigated, where only a finite number of matrix degrees of freedom are retained. The quantities of physical interest are the gluon correlations, which satisfy quantum corrected equations of motion (factorised Schwinger-Dyson equations or loop equations). These equations were obtained and their algebraic and differential properties were studied. In particular, it was shown that there is a limit where they become differential equations and used this to propose a novel expansion parameter and approximation method to solve them. It was also shown that the loop equations are in general under-determined, i.e. are not sufficient to determine all the correlations. Together with L. Akant, it was found that the reason for this is the presence of certain hidden non-anomalous symmetries of the quantum theory. Taking them into account leads to additional equations (Ward identities), which help to cure the under-determinacy of the loop equations.
A toy-model for gluons in baryons was also formulated and solved, consisting of a matrix model coupled to quarks. A third order phase transition was found, separating a two-cut phase where light quarks are strongly coupled to gluons, from a one-cut phase where heavy quarks are weakly coupled to gluons.
Another project concerned the naturalness and ultraviolet problems of 4 dimensional O(4) scalar field theory in the Higgs sector of the standard model of particle physics. The problem here is that despite its simplicity, the standard quartic scalar field theory has bad high energy behaviour and lacks any mechanism to ensure a light Higgs particle. I tried to use scale invariance to ensure light scalar excitations by constructing a line of non-trivial UV fixed points in O(N) scalar field theory in the large-N limit.
A new direction in the research, established during the fellowship, concerns hydrodynamics, the flow of fluids. There are many outstanding problems in this subject including the description of surprisingly stable objects such as hurricanes as well as extremely complex phenomena such as turbulence. Hydrodynamics is also of wide industrial and practical significance such as in aircraft design and weather prediction. As a first step in the mathematical study of fluid mechanics, the flow on a surface was studied, which is simpler than in a three dimensional space, yet provides a first approximation in some contexts such as atmospheric flows on the earth's surface. Geometric features of surface flow were studied and it was shown that it could be formulated as an abelian gauge theory. A somewhat unexpected consequence of this reformulation was the discovery of a 2+1 dimensional dynamical system with an infinite number of conserved quantities.
A promising approximation method is the 1/N expansion, where N is the number of colours of quarks, which are three in nature. This approach has been successful in several simpler contexts. However, even the starting point of this approximation, large-N Yang-Mills theory is very difficult to solve. The difficulties include dealing with infinitely many matrix degrees of freedom (gluons), a large redundancy in the formulation (gauge invariance), short distance divergences, the presence of bound states and the need for entirely new mathematical methods to deal with these issues. In order to make progress, I studied simplified models for large-N Yang-Mills theory where we can develop techniques to deal with the many complications in partial isolation.
Large-N matrix models have been investigated, where only a finite number of matrix degrees of freedom are retained. The quantities of physical interest are the gluon correlations, which satisfy quantum corrected equations of motion (factorised Schwinger-Dyson equations or loop equations). These equations were obtained and their algebraic and differential properties were studied. In particular, it was shown that there is a limit where they become differential equations and used this to propose a novel expansion parameter and approximation method to solve them. It was also shown that the loop equations are in general under-determined, i.e. are not sufficient to determine all the correlations. Together with L. Akant, it was found that the reason for this is the presence of certain hidden non-anomalous symmetries of the quantum theory. Taking them into account leads to additional equations (Ward identities), which help to cure the under-determinacy of the loop equations.
A toy-model for gluons in baryons was also formulated and solved, consisting of a matrix model coupled to quarks. A third order phase transition was found, separating a two-cut phase where light quarks are strongly coupled to gluons, from a one-cut phase where heavy quarks are weakly coupled to gluons.
Another project concerned the naturalness and ultraviolet problems of 4 dimensional O(4) scalar field theory in the Higgs sector of the standard model of particle physics. The problem here is that despite its simplicity, the standard quartic scalar field theory has bad high energy behaviour and lacks any mechanism to ensure a light Higgs particle. I tried to use scale invariance to ensure light scalar excitations by constructing a line of non-trivial UV fixed points in O(N) scalar field theory in the large-N limit.
A new direction in the research, established during the fellowship, concerns hydrodynamics, the flow of fluids. There are many outstanding problems in this subject including the description of surprisingly stable objects such as hurricanes as well as extremely complex phenomena such as turbulence. Hydrodynamics is also of wide industrial and practical significance such as in aircraft design and weather prediction. As a first step in the mathematical study of fluid mechanics, the flow on a surface was studied, which is simpler than in a three dimensional space, yet provides a first approximation in some contexts such as atmospheric flows on the earth's surface. Geometric features of surface flow were studied and it was shown that it could be formulated as an abelian gauge theory. A somewhat unexpected consequence of this reformulation was the discovery of a 2+1 dimensional dynamical system with an infinite number of conserved quantities.