Two-dimensional conformal field theory arose in both condensed matter physics and string theory and is being formulated, constructed and studied mathematically. It is now a deep, rich and beautiful mathematical theory and has already led to new ideas, surprising results and solutions of many mathematical problems in mathematics and physics. On the other hand, various recent developments in different areas of mathematics and physics can be recognized to be related when the language of tensor categories is used as a unifying concept. One example is modular tensor categories which appear naturally in the study of conformal field theories and also in other related algebraic and analytic structures. Further study of the connection between conformal field theories and tensor categories will benefit the developments of both conformal field theory and the theory of tensor categories.
Two dimensional conformal field theories and tensor categories are mathematical theories studied intensely in recent years by both mathematicians and physicists. It is believed that many important phenomena observed in nature and in laboratory, such as critical phenomena, quantum Hall effects, disorder phenomena and so on, can be described by these theories. More importantly, a deep understanding of these theories might lead to the development of new technologies, for example, new technology for the design of chips for what people call quantum computers, which, if constructed, are computers running much much faster than the computers we have now. This award provides support for mathematicians and physicists, including graduate students, junior faculty, women and members of underrepresented minority groups, in the United States to attend the workshop ``Conformal field theories and tensor categories'' from June 14 to June 18, 2011, at the Beijing International Center for Mathematical Research in Beijing. The purpose of this workshop is to bring together researchers in the United States, China and other countries working on conformal field theories, tensor categories and related topics to discuss new results, exchange ideas and initiate new collaborations and research projects. In particular, one of the main goals of the workshop is to make a wealth of mathematical results and concepts that are ready to use accessible to a larger part of the theoretical physics community and to find new applications of these results. It is believed that mathematicians and physicists in the United States, China and other countries can all benefit a lot from communicating their results and from initiating new collaborations and research projects.
Two-dimensional conformal field theory has its origins in condensed matter physics and string theory and has many exciting applications in mathematics and physics, including, but not limited to, algebra, number theory, combinatorics, topology, geometry, critical phenomena, quantum Hall systems, disorder systems, quantum computing and string theory, and is expected to lead to many more. In the study of two-dimensional conformal field theories, one algebraic structure called tensor categories plays an important role. Tensor categories described certain quasiparticles called nonabelian anyons expected to appear in quantum Hall effects and are the starting point of quantum computing, which, if successful, will lead to a revolution in technology depending on computers. In the workshop "Conformal field theories and tensor categories" supported by this grant, mathematicians and physicists working on the two-dimensional conformal field theory and tensor categories were brought together to communicate their researches on these same problems from different perspectives and to stimulate new ideas and collaborations. There were five overview talks and sixteen invited talks. The participants benefitted greatly from these talks that are aimed at people with diverse backgrounds, from communicating results between the various disciplines and from the attempts to understand them in the framework of conformal field theories and tensor categories. These attempts gave rise to many questions during and after the talks, and, even more importantly, also resulted in numerous lively private discussions among the participants. Between the time when the idea to organize a workshop on this subject was born and today, in many of the fields named above there has been important, sometimes even spectacular, progress. Much of this progress was presented at the workshop. In a few cases the results were actually presented at the workshop for the first time, and more frequently it was at least for the first time to an audience of such a varied background. As examples, we mention Fendley's work on finding holomorphic operators in lattice models, Kawahigashi's work on the operator algebra approach to the N=2 chiral superconformal field theories, Tsuchiya's proposal to the solution of the problem of the Kazhdan-Lusztig correspondence for the bradied tensor category of modules for triplet W-algebras, and Wen's and his collaborators' work on the classifications of one- and two-dimensional topological order. In surprisingly many cases, the progress involved the transfer of ideas and/or techniques from other fields. Conformal field theories and tensor categories have proven to provide most valuable frameworks and tools in this process. In our opinion, an important contribution of this workshop with long term impact is the fact that it has further established conformal field theories and tensor categories as fundamental structures behind a number of mathematical and physical theories. In fact, this workshop has been so far the most prominent meeting point of the communities working on mathematical conformal field theories and on topological orders and lattice models. The workshop had some training impact on students and young researchers. Students and postdocs benefitted greatly from the five overview talks and other research talks and from talking directly to the leading experts in different areas. Some young researchers were invited to present aspects of their work in the workshop. Some graduate students have influenced by the talks by and discussions with experts in the workshop so that they started to work on problems in this area.
