Random paths to QFT: New probabilistic approaches to field theory: October 14- November 22, 2024

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Organized by:

• Denis Bernard (ENS, Paris)
• Massimiliano Gubinelli (University of Oxford)
• Antti Kuipianen (University of Helsinki)
• Nikita Nekrasov (SCGP)
• Remi Rhodes (Aix-Marseille University)

In recent years, new probabilistic methods were developed to offer a rigorous approach to constructing Euclidean path integral measures for several interacting quantum field theories, including the Liouville theory in d=2 and the φ4 theory in d=3. Another rigorous approach explores the BPS/CFT correspondence, relating the non-perturbative physics of supersymmetric gauge theories in four dimensions to conformal blocks of some d=2 CFTs and their integrable FT analogues. For example, Alday, Gaiotto and Tachikawa connected the d=2 Liouville theory to the A1-type S-class N=2 supersymmetric theories in d=4. On the gauge theory side, Nekrasov partition functions provide combinatorial expressions for Liouville conformal blocks. On the probabilistic side, Kupiainen, Rhodes and Vargas applied rigorous methods to the derivation of the DOZZ formula for the 3-point function of Liouville theory, which is a one-loop approximation on the d=4 side. Extension to full conformal bootstrap for vertex operator correlation functions and conformal blocks was achieved in subsequent work of those authors together with Guillarmou. The emergence of the combinatorics of gauge theory predicted by Nekrasov remains a challenge. The approach to Liouville theory proposed by Sheffield gives yet another combinatorial perspective, at least in the genus zero case. An approach based on integration by parts allowing to characterize Euclidean quantum field theories in d=2 with exponential interactions, much like the classical Liouville theory, has been recently put forward by De Vecchi, Gubinelli and Turra. Another interesting direction concerns imaginary versions of Liouville theory, which has been recently constructed via probabilistic methods in Guillarmou, Kupiainen, Rhodes, and their relations to minimal models, the minimal string and 3d gravity.

In d=3 new constructions of the φ4  theory were given using SPDE and stochastic quantisation approaches by Hairer, Gubinelli, Hofmanova, Barashkov and others. We will work to see if these rigorous approaches could be combined with the probabilistic bootstrap method to confront the vast number of results from numerical bootstrap in three and four dimensions. Likewise, it would be interesting to connect the probabilistic bootstrap to the form factor bootstrap method developed for the study of integrable QFTs.

The stochastic optimal control approach of Barashkov and Gubinelli is closely connected to Polchinski’s renormalisation group equation. Building on this, Bailleul, Chevyrev and Gubinelli defined, in any number of spacetime dimensions, a formally non-perturbative quantization method applicable also to gauge theories and independent of a path-integral formulation, compatible with Wilson-Polchinski equations whenever path integral formulations exist. Since string theory predicts the existence of  non-trivial conformal field theories with no classical limit and no path integral formulation, this opens up a bridge between different arenas of research, once these probabilistic methods are extended to include tensor fields of higher rank. 

There are a plethora of other topics we would like to address, such as finding stochastic analytic approaches to path integral Lefschetz thimbles and exploring them in the simplest cases of two dimensional sigma models, such as O(N) and CPN-1. QFT practitioners would benefit from incorporating probabilistic methods, potentially extending them to the physically interesting domain: gauge theories in 3 ≤ d ≤4 dimensions.

There will be six mini-courses giving introductions to regularity structure, stochastic quantization, RG and Wilson-Ito fields, combinatorics from non-perturbative gauge theory, probabilistic approaches to quantum Liouville theory, stochastic aspects of Yang-Mills theory, and integrable quantum field theory.