From conformal loop ensembles to conformal field theory

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Sometimes, many basic constituents that are interacting amongst each other in simple and understood ways, such as electrons in a metal, molecules in a liquid, or buyers and sellers on the stock market, when present in large numbers, give rise to unexpected results on large scales. This is usually called emergent behaviours , and it is very hard to predict, in general, what such behaviours can be. Many systems of interest to physicists are those with very many constituents that can fluctuate (thermally or quantum mechanically) while interacting amongst not-too-far neighbours. Quite surprisingly, although the interaction is local , it happens in some situations that the constituents form very large groups, chains of very many neighbours, that fluctuate together, as if the groups were new constituents of a new system. These are emergent behaviours. Situations where big groups tend to form are called critical , because then the system is hyper-sensitive to external disturbances: whole groups will react to such disturbances, producing a big, large-distance change. Naturally, these quite surprising emergent collective behaviours are responsible for a wealth of interesting physical phenomena, like the formation of Kondo clouds that change conductive properties of metals with magnetic impurities. It is also tempting, and promise to be fruitful in the future, to make a connection with the emergent behaviours from individual agents in macroeconomics: a small sub-prime market crash gave us an international recession!Physicists came up with a very powerful theory, based on physical principles, that describes the emergent behaviours in critical systems. This is quantum field theory. In fact, one of the great achievements of theoretical physics of the twentieth century is the understanding that all fundamental particles that are observed in current-day experiments can be understood as emerging from a simpler, more symmetrical theory: this is the standard model of quantum field theory. We then have an understanding of such emergent behaviours, but this understanding does not form yet a mathematically coherent whole, neither is it a complete understanding of the emergent collectivities themselves. We understand emergent behaviours through quantum particles and how they scatter, through energy and how it varies locally, and through local probes and how they react to local disturbances. But we often don't know how to relate these various ideas, and how to connect them to, and actually describe, the fluctuating emergent collectivities of constituents.Conformal field theory is a family of models of quantum field theory where the standard elements of our understanding enumerated above are much better developed and connected to each other. They correspond to a small, but very instructive, corner of quantum field theory. About three years ago, mathematicians proposed a family of mathematical measures supposed, and sometimes proved, to describe certain aspects of large-distance behaviours in critical systems, aspects that fall into the corner described by conformal field theory. In some works, I recently emphasized that these measures in fact exactly describe all emergent fluctuating objects in that corner, at least for a wide family of models. My research consists in using these mathematical measures in order to fully connect the emergent collectivities with the powerful structure of conformal field theory. This will give us an entirely new insight into the more subtle way emergent objects behave, and will provide, for the first time, a complete path from underlying many-constituent systems, to quantum field theory.