Conference: Quantum Field Theory and Topological Phases via Homotopy Theory and Operator Algebras

This award provides support the conference Quantum Field Theory and Topological Phases via Homotopy Theory and Operator Algebras, taking place 30 June through 11 July 2025 at the Center of Mathematical Sciences and Applications at Harvard University. The conference will bring together researchers from several communities within mathematics and physics who all explore how to model complicated physical systems and how key characterizing properties emerge (which identify the phase of the system, broadly speaking). These tools are crucial in understanding and developing new materials, particularly those with surprising "topological" phases, and in developing quantum computers, but the focus of the conference is to deepen mutual understanding and to share technical tools between communities who speak different technical languages and thus approach these topics in distinct, yet complementary, ways. To foster effective communication, the first week will be devoted to expository lectures by leading figures in this subject, while the second week will focus on talks on current research results. Graduate students and postdoctoral scholars will attend, fostering the next generation of work in this domain. We note that this conference will have a "twin" at the Max Planck Institute for Mathematics in Germany, so that researchers in Europe are part of this dialogue. In more technical terms, Quantum Field Theory (QFT) and Quantum Statistical Mechanics are central to high energy physics and condensed matter physics; they also raise deep questions in mathematics. The application of operator algebras to these areas of physics is well-known. Recent developments indicate that to understand some aspects of QFT properly a further ingredient is needed: homotopy theory and infinity-categories. One such development is the recognition that symmetry in a QFT is better described by a homotopy type rather than a group (so-called generalized symmetries). Another one is the work of Lurie and others on extended Topological Field Theory (TFT) and the Baez-Dolan cobordism hypothesis. Finally, there is a conjecture of Kitaev that invertible phases of matter are classified by homotopy groups of an Omega-spectrum. This workshop will bring together researchers and students approaching this physics using different mathematical techniques: operator algebras, homotopy theory, higher category theory, etc., and will catalyze new interactions between different communities. The workshop will also highlight recent developments and new progress on two outstanding problems: the Kitaev conjecture as well as the long-standing goal of finding a proper mathematical formulation for QFT. The website for the conference is https://cmsa.fas.harvard.edu/event/mpqft25/ This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria. NSF Award ID: 2520126 | Program: 01002627DB NSF RESEARCH & RELATED ACTIVIT | Principal Investigator: Daniel Freed | Institution: Harvard University, CAMBRIDGE, MA | Award Amount: $20,000 View on NSF Award Search: https://www.nsf.gov/awardsearch/show-award/?AWD_ID=2520126 View on Research.gov: https://www.research.gov/awardapi-service/v1/awards/2520126.html