Speaker
Description
Vertex operator algebras (VOAs) provide a precise mathematical framework for the study of chiral conformal field theories (chiral CFTs). They capture the relevant algebraic properties of chiral conformally invariant fields on the circle. Conformal nets provide another well-developed mathematical approach to chiral CFTs, based on the theory of von Neumann algebras on Hilbert spaces. The first approach is mainly algebraic, while the second is mainly functional analytic. A first general connection between these two approaches to chiral CFTs was described by Kawahigashi, Longo, Weiner, and me in a paper published in 2018 and first posted on the arXiv in 2015. In this paper, it was shown that all suitably well-behaved simple unitary VOAs, called “strongly local” VOAs, give rise to corresponding conformal nets. Moreover, various important families of unitary VOAs were shown to be strongly local, and it was conjectured that every simple unitary VOA is strongly local. In subsequent years, the understanding of the correspondence between VOAs and conformal nets has considerably improved thanks to the work of various researchers. Many new examples of unitary VOAs have been shown to be strongly local, and important results concerning the representation-theoretic aspects of the correspondence have been obtained. On 6 September 2026, OpenAI published more than 700 papers containing mathematical results in many areas of mathematics. One of these papers, entitled “Strong locality for strongly rational unitary vertex operator algebras,” gives a proof of strong locality for all strongly rational unitary VOAs. Moreover, it is shown that, for any such VOA, the corresponding conformal net is completely rational in the sense of Kawahigashi, Longo, and Müger, and that its modular tensor category of representations is equivalent to that of the generating VOA. This remarkable OpenAI result, like many of the other results published on the same day, is currently being examined by the mathematical community and has so far been only partially formalized in Lean. In this talk, I will explain the strong locality problem and its partial solution by OpenAI, together with some of the human results that led to it.