2026 Simons Collaboration on Probabilistic Paths to Quantum Field Theory Annual Meeting

The 2026 annual meeting brought together almost 120 people at the Simons Foundation, and the meeting followed a three-day satellite conference of about 130 participants at the NYU Courant Institute. The annual event had six speakers from the collaboration and two external speakers, while the satellite conference had twelve external speakers. Both events brought together worldwide experts on the collaboration topics for inspiring lectures and lively discussions.

The annual meeting opened with a pedagogical introduction by PI Sourav Chatterjee to quantum field theory in the context of probability theory. Chatterjee gave a broad overview of the mathematical side of quantum field theory, including both known results and open problems in the rigorous theory, and in particular the history and background of QED and QCD, including open problems in constructive Yang–Mills gauge theory.

Chatterjee was followed by collaboration director Scott Sheffield, who continued the overview with additional discussion of the Polyakov conjecture (in 2D) and Yang–Mills gauge theory (in 4D), including simulations of 2D surfaces in 4D (worldsheets of strings) in simple examples, and discussion of open problems.

PI Martin Hairer next provided an overview of the powerful technique of using stochastic partial differential equations (SPDE) as a tool in constructive QFT. Hairer highlighted several exciting recent results in this area, in particular the construction of 2D Yang–Mills–Higgs theory (which hadn’t been constructed rigorously before) using the SPDE framework.

PI Antti Kupiainen highlighted a different technique, conformal field theory, which has historically been a powerful tool in QFT for both mathematicians and physicists. Kupiainen focused on mathematically rigorous developments in the conformal bootstrap, with special attention to Liouville CFT, which plays a central role in the theory of random surfaces. He described a surprising correspondence between Liouville CFT and a Wess–Zumino–Witten sigma model, originally conjectured by Ribault, Teschner, Hikida, and Schomerus and argued by Teschner and Gaiotto to define a “quantum” deformation of the analytic Langlands correspondence of Etingof, Frenkel, and Kazhdan.

Nathan Seiberg of the Institute for Advanced Study concluded the first day with a broad overview of quantum field theory and a forward-looking discussion of the new techniques and frameworks he expects will be necessary to address the major open questions in non-perturbative theoretical physics.

The second day of the annual conference began with a lecture by Xin Sun on the use of conformal field theory, and Liouville theory in particular, to establish precise exponents and other results relevant to 2D percolation.

Igor Klebanov, who is the director of the Simons Collaboration on Confinement and QCD Strings (and a member of our collaboration’s advisory board), gave a brief review of quantum chromodynamics (QCD) and the Confinement problem. He discussed applications of lattice gauge theory (such as numerical results for the low-lying hadron spectra) and several other relevant topics (e.g., quark confinement, the two-flavor Schwinger model, Coleman’s half-asymptotic particles).

PI Yilin Wang concluded the event with discussion of another powerful tool, the “holographic principle,” which looks to encode geometric quantities by conformally invariant quantities on its boundary of one lower dimension, or vice versa. This philosophy has been very successful in relating gravity theory in the bulk to the conformal field theory on the boundary. Wang surveyed the use of this geometric construction to give exact holographic expressions for several actions in QFT, including Liouville action, Loewner energy, and Schwarzian action.

Overall, the annual meeting highlighted remarkable advances in unifying fundamental forces and computational techniques. Lively collaborations and fresh perspectives underscored the field’s vitality. Looking ahead, continued cross-disciplinary efforts promise deeper insights, pushing the boundaries of knowledge and inspiring new breakthroughs.

Sourav Chatterjee
Stanford University

What Is a Probabilistically Defined Quantum Field Theory?
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Sourav Chatterjee will give a broad overview of what it means to rigorously define and construct a quantum field theory, what progress has been made so far, and what remains to be done. Since the subject is vast, Chatterjee will be highly selective, but they will try to present some actual mathematics and emphasize a few concrete examples and frameworks that make the basic ideas feel less mysterious. The talk will end with a brief discussion of the Clay Millennium problem of Yang–Mills existence and the mass gap, and how it fits into the broader quest to build interacting quantum field theories.
 

Martin Hairer
École Polytechnique Fédérale de Lausanne (EPFL)

Stochastic Quantisation of Gauge Theories

Martin Hairer will review Parisi and Wu’s original motivation for introducing the ’stochastic quantisation’ procedure for QFT’s. 40 years after their insight, this approach is finally bearing fruit in a way that they predicted at the time.
 

Igor Klebanov
Princeton University

Dimensional Transmutation in Some Models
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The talk will begin with a brief review of Quantum Chromodynamics (QCD) and the Confinement problem. Lattice Gauge Theory (LGT) provides a non-perturbative formulation of QCD, which has led to good numerical results for the low-lying hadron spectra. Excitation spectra of long confining strings are
approximately described by the Nambu-Goto area action.

The 1+1 dimensional gauge theories have served as useful models of quark confinement. Igor Klebanov will revisit the classic Schwinger model and its lattice Hamiltonian formulation. A mass shift between the lattice and continuum definitions of mass, which is motivated by the anomalous chiral symmetry, is shown to lead to improved results. Klebanov will also present the zero-temperature phase diagram of the two-flavor Schwinger model at theta=pi, which exhibits dimensional transmutation and spontaneous breaking of charge conjugation. Coleman’s half-asymptotic particles have an exponentially small mass when the model is strongly coupled.
 

Antti Kupiainen
University of Helsinki

Conformal Field Theory and Path Integrals
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Conformal field theory (CFT) describes universality classes of statistical mechanics systems at the critical temperature of a second order phase transition as well as Quantum field theories of fundamental physics in the limits of small and large length scales. CFTs have special symmetries leading to a rich mathematical structure which has inspired representation theory and geometry. However, a rigorous mathematical foundation for CFT is still lacking. Two major approaches in physics to CFT are the path integral approach and the conformal bootstrap approach. In this talk, Antti Kupiainen will discuss a path integral approach to two dimensional CFT based on probabilistic tools such as the Gaussian multiplicative chaos. Kupiainen will explain how the conformal bootstrap can be approached in the probabilistic setup leading to a complete solution of Liouville CFT, which plays a central role in the theory of random surfaces and how these constructions could be extended to other CFTs, such as sigma models, which have interesting connections to geometry. In particular, the probabilistic construction allows us to prove a surprising correspondence between Liouville CFT and a Wess–Zumino–Witten sigma model originally conjectured by Ribault, Teschner, Hikida, and Schomerus and argued by Teschner and Gaiotto to define a “quantum” deformation of the analytic Langlands correspondence of Etingof, Frenkel, and Kazhdan.
 

Nathan Seiberg
Institute for Advanced Study

What Is Quantum Field Theory?

Seiberg will share his views on the subject.
 

Scott Sheffield
Massachusetts Institute of Technology (MIT)

Yang–Mills Gauge Theory and Random Geometry in 2 and 4 Dimensions
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Many of the basic objects of 2D mathematical physics (such as bond percolation, uniform spanning trees, loop-erased random walks, Ising and other sigma models, Gaussian free fields, etc.) have very natural one-form-level-higher analogs in 4D (plaquette percolation, uniform spanning 2-trees, cycle-spanning branches, Ising and other lattice gauge theories, 1-form Gaussian free fields, etc.). Scott Sheffield will discuss the direct and indirect role that two-dimensional random surfaces play in our understanding of four dimensional models, along with many open problems involved in the rigorous construction of off-critical continuum limits.
 

Xin Sun
Peking University

CFT Perspectives on 2D Percolation and Brownian Motion

Conformal field theory (CFT) has proven to be a powerful framework for studying two-dimensional (2D) Bernoulli percolation at criticality. Classic applications include the derivation of certain critical exponents and crossing probabilities. Yet, the precise nature of the underlying CFT remains elusive and is a subject of active investigation in both the mathematics and physics communities. In this talk, Xin Sun will first provide an overview of this topic and then present recent progress on another fundamental planar fractal “Brownian motion,” revealing that its percolative properties may be governed by an equally rich, and perhaps more enigmatic, CFT.
 

Yilin Wang
Eidgenössische Technische Hochschule (ETH) Zürich

Holography from a Geometric Perspective

The holographic principle looks to encode geometric quantities by conformally invariant quantities on its boundary of one lower dimension, or vice versa. This philosophy has been very successful in relating gravity theory in the bulk to the conformal field theory on the boundary.

From a geometric perspective, a classical construction using Epstein’s truncation, relates the hyperbolic geometry in the bulk to the conformal geometry on its boundary. Yilin Wang will survey how this geometric construction can be applied to give exact holographic expressions for several actions in QFT, including Liouville action, Loewner energy, and Schwarzian action.

To view the all of the lectures please visit: https://www.youtube.com/playlist?list=PLWAzLum_3a18mr_niojMUvvfQldIWyQFQ