# From conformal loop ensembles to conformal field theory

Lead Research Organisation:
King's College London

Department Name: Mathematics

### Abstract

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.

### Planned Impact

The proposed two-year research project lies entirely at the academic level: it is a mathematical project attempting to uncover mathematical structures in, and a deeper understanding of, conformal field theory as a theory for scaling limits, i.e. as a theory for certain types of emergent behaviours in statistical systems. The knowledge created will be of immediate use to academic beneficiaries. If no economic/social impacts are expected in the course or immediately after the research project, it is clear that the general idea of emergent behaviour has deep significance for economy and its meaning in society. Indeed, the idea that many constituents, although essentially independent (excepts for local dependences), can behave, in certain situations, like if they were connected in large groups, must occur in many aspects of the global economy. Perhaps the current macroeconomic system may be seen as being near to criticality (i.e. to some sort of instability), and that emergent collectivities can be seen as playing an important role in, for instance, the recent global economic break-down. Eventually, a better understanding of emergent behaviours, like those occuring in statistical models, will tell us how to produce a better global economic model on which to base our society. Moreover, many other complex systems occur in the real world, with consequences on society and well-being (the earth's atmosphere, the brain); perhaps the ideas of emergent behaviours and of criticality can play a role in these other situations as well. I expect the time scale for the discoveries of, or ideas fostered in, my proposed research to influence, in some way, macroeconomics or complex systems studies, to be of the order of 10 to 20 years. I do not expect to actively search for economical applications of emergent behaviours during the short time which the grant would support, but I will continue publishing papers in high-impact journals, emphasising the wider context of the research being reported both in these papers and in my personal web-page (hosted at my institution), and being generally aware of the literature on, in particular, subjects related to macroeconomy. I wish to emphasise as well that there may be many other economic/sociologic impacts of the present research, of the mathematical techniques developed and of the ideas explored: a basic characteristics of fundamental research is that one cannot predict exactly what will be discovered.

## People |
## ORCID iD |

Benjamin Doyon (Principal Investigator) |

### Publications

Doyon B
(2012)

*Conformal Loop Ensembles and the Stress-Energy Tensor*in Letters in Mathematical Physics
Doyon B
(2012)

*Calculus on manifolds of conformal maps and CFT*in Journal of Physics A: Mathematical and Theoretical
Doyon Benjamin
(2011)

*Factorisation of conformal maps on finitely connected domains*in arXiv e-prints
Doyon Benjamin
(2011)

*Higher conformal variations and the Virasoro vertex operator algebra*in arXiv e-prints
Doyon Benjamin
(2014)

*Random loops and conformal field theory*in JOURNAL OF STATISTICAL MECHANICS-THEORY AND EXPERIMENT
Doyon Benjamin
(2013)

*Hypotrochoids in conformal restriction systems and Virasoro descendants*in JOURNAL OF STATISTICAL MECHANICS-THEORY AND EXPERIMENT
Herdeiro V
(2016)

*Monte Carlo method for critical systems in infinite volume: The planar Ising model.*in Physical review. EDescription | In this research I studied mathematical models where loops in the plane can fluctuate in a random fashion. Such loops, or more generally surfaces in higher dimensions, do arise in physical systems, for instance as boundaries between domains where atoms are in different states. These domain boundaries fluctuate because of temperature, or because of the quantum. I found profound connections between such random loops, the theory of analytic functions, and infinite-dimensional algebras, which give exact predictions. |

Exploitation Route | My findings may be used by mathematical physicists in the study of many-body systems and emerging behaviours. |

Sectors | Other |

Description | Cardiff talk |

Form Of Engagement Activity | A talk or presentation |

Part Of Official Scheme? | No |

Geographic Reach | Local |

Primary Audience | Participants in your research and patient groups |

Results and Impact | I gave an invited seminar talk at Cardiff University about Calculus of Variations of Conformal Maps and CFT. I gave the talk "Calculus of variation of conformal maps and CFT" as an invited seminar talk. This was about a work, under the EPSRC grant, that was still in preparation. Some of my research results were transferred to other researchers in my area |

Year(s) Of Engagement Activity | 2010 |

URL | http://www.mth.kcl.ac.uk/~bdoyon/pub.html |

Description | External member of the selection committee |

Form Of Engagement Activity | A formal working group, expert panel or dialogue |

Part Of Official Scheme? | No |

Geographic Reach | International |

Primary Audience | Participants in your research and patient groups |

Results and Impact | I was asked (and accepted) to be external member of the selection committee for a relatively prestigious position in Mathematical Physics at University of Bourgogne (Dijon, France). The position is "Ma?tre de Conf?rence + Chaire CNRS 5 ans" which is equivalent to a 5-year research fellowship followed by a permanent lectureship. It includes analyzing and reporting on files of international applicants, and conducting interviews. My expenses to go to Dijon (France) are paid by the University of Bourgogne. There are 11 members in the committee, including 6 external members, three of which (including me) are outside of France (Italy, United States, and United Kingdom). Awarding Body - University of Bourgogne, Name of Scheme - Selection Committee for a position in Mathematical Physics The main impact was to allow for a correct and informed decision. |

Year(s) Of Engagement Activity | 2012 |

Description | Laval University talk |

Form Of Engagement Activity | A talk or presentation |

Part Of Official Scheme? | No |

Geographic Reach | Local |

Primary Audience | Other academic audiences (collaborators, peers etc.) |

Results and Impact | I gave a seminar talk at Laval University, Qu?bec (Canada), while I was visiting Prof. Pierre Mathieu. Talk "Calculus of variation of conformal maps and CFT" given to an audience of physics students (undergraduate and PhD), mathematical physicists and experts in CFT. Audience learned about my work. |

Year(s) Of Engagement Activity | 2011 |

URL | http://www.mth.kcl.ac.uk/~bdoyon/pub.html |

Description | Talk Cargèse |

Form Of Engagement Activity | Participation in an activity, workshop or similar |

Part Of Official Scheme? | No |

Geographic Reach | International |

Primary Audience | Other audiences |

Results and Impact | An invited talk was given at the summer school "Summer School, Quantum integrable systems, conformal field theories and stochastic processes", at the Institut d'Études Scientifiques de Cargèse, France. The talk title was "Aspects of conformal field theory from random loops", based on my main results related to the present grant. I would say about 40-50 researcher in this and related areas attended, as well as 5-10 post-doctoral researchers and some PhD students. Many questions were asked, interest was stimulated. |

Year(s) Of Engagement Activity | 2016 |

URL | https://indico.in2p3.fr/event/12461/ |