Tentative Schedule: Week 1 (June 1–5)

Course 1: Liouville Conformal Field Theory, by Rémi Rhodes, Colin Guillarmou, Antti Kupiainen

Course 2: Yang–Mills Theory: Random Surface Approach, by Sky Cao

Schedule

Monday
6/1
09:00–10:00
Registration / Opening / Photo
10:00–11:00
Course 1
11:00–12:00
Course 1
12:00–14:00
Lunch
14:00–15:00
Course 2
15:00–16:00
Course 2
16:00–16:30
Tea break
Tuesday
6/2
09:00–10:00
Course 1
10:00–11:00
Course 1
11:00–11:30
Tea break
11:30–12:15
Talk
Raoul Santachiara
12:15–14:00
Lunch
14:00–15:00
Course 2
15:00–16:00
Course 2
16:00–16:30
Tea break
Wednesday
6/3
09:00–09:45
Talk
Jacopo Borga
10:00–10:45
Talk
Semyon Klevtsov
10:45–11:15
Tea break
11:15–12:00
Talk
Nikolay Barashkov
12:00–14:00
Lunch
14:00–16:00
Informal Talk
Nikita Nekrasov
16:00-16:30
Tea break
Thursday
6/4
09:30–10:30
Course 2
10:30–11:00
Tea break
11:00–12:00
Course 2
12:00–14:00
Lunch
14:00–15:00
Course 1
15:00–16:00
Course 1
16:00–16:30
Tea break
16:30–17:30
Poster Session
18:00–20:00
Banquet
Friday
6/5
09:00–09:45
Talk
Denis Bernard
10:00–10:45
Talk
Hao Wu
10:45–11:15
Tea break
11:15–12:00
Talk
Guillaume Baverez
12:00
Lunch
CourseTalk / DiscussionTea breakLunch / BanquetPoster Session

Mini-Courses

Course 1: Liouville Conformal Field Theory

Lecture 1 (Rémi Rhodes): Probabilistic construction of Liouville CFT

This mini course will be a gentle introduction to the Liouville CFT. I will present the motivations and main context, the basic probabilistic ingredients, namely the Gaussian Free Field and The Gaussian multiplicative chaos, and then the construction of the Liouville path integral and main properties. This will serve as a preparation for Colin Guillarmou’s talk about the Segal axioms for the Liouville CFT and Antti’s Kupiainen’s talk about the H3 Wess-Zumino-Witten’s model and the correspondence with the Liouville model.

Lecture 2 (Colin Guillarmou): Segal Axioms for the probabilistic construction of Liouville CFT

We explain how to decompose the correlation functions of Liouville CFT into compositions of “Segal amplitudes”, and how this leads to a conformal bootstrap formula in the general setting of surfaces with marked points.

Segal amplitudes are operators on a Hilbert space associated to the Gaussian Free Field on the unit circle, which carries two commuting representations of Virasoro algebra with central charge 1 + 6Q² > 25.

The course follows Rémi Rhodes’s course on the probabilistic construction of Liouville CFT using Gaussian Multiplicative Chaos theory.

Lecture 3 (Antti Kupiainen): Probabilistic sigma models

In a probabilistic formulation of quantum field theory, a sigma model is a theory of a random field defined on space, a manifold in general, and taking values in a Riemannian manifold M.

On a two-dimensional space, sigma models describe statistical mechanics systems, important examples being the XY model (M = S¹), the Heisenberg model (M = S²), and the Wess–Zumino–Witten model (M = a semisimple Lie group). They also play a prominent role in string theory, M being the space where strings move.

I will briefly discuss the problems for a rigorous analysis of sigma models and then explain the construction of a sigma model where M is a three-dimensional hyperbolic space, giving rise to a conformal field theory. This theory has an interesting connection to Liouville CFT, which in turn has applications to integrable systems and geometry.

Course 2 （Sky Cao）: Yang–Mills Theory: Random Surface Approach

In this mini-course, I will aim to give a fairly detailed exposition of the surface sum representation of Wilson loop expectations in lattice Yang–Mills, based on a recent paper of Cao–Park–Sheffield (2025). This turns out to be an exercise in random matrix theory.

In addition to the derivation of the surface sums, I will discuss the “surface-theoretic” proof of the Master loop equation, which ultimately follows from a recursion satisfied by unitary matrix integrals. Time permitting, I will discuss some ways to use the Master loop equation to derive results about lattice Yang–Mills.

Talks

Denis Bernard

Title: SLE/CFT correspondence

I will review the SLE/CFT correspondence and some of its applications. This correspondence mostly relies on identifying the connection between statistical physics models and martingales for appropriate exploration processes.

Guillaume Baverez

Title: Kac-Moody unitarising measures

The KMUs are a canonical family of measures on the moduli space of flat connections on the disc (they are to loop groups what SLE is to Diff(S1)). I will survey their main properties and discuss some applications, including WZW models and moduli spaces of flat connections on Riemann surfaces with boundary.

Hao Wu

Title: Connection probabilities for loop O(n) models and BPZ equations

Critical loop O(n) models are conjectured to be conformally invariant in the scaling limit. In this talk, we focus on connection probabilities for loop O(n) models in polygons.

Such probabilities can be predicted using two families of solutions to chordal Belavin–Polyakov–Zamolodchikov (BPZ) equations: Coulomb gas integrals and SLE pure partition functions. The conjecture is proved to be true for the critical Ising model, FK-Ising model, percolation, and uniform spanning tree.

Recent progress of radial BPZ equations will also be discussed.

Jacopo Borga

Title: Lattice Yang–Mills theory in the large N limit via random surfaces

Lattice Yang–Mills theories are important models in particle physics. They are defined on the d-dimensional lattice ℤᵈ using a group of matrices of dimension N, and Wilson loop expectations are the fundamental observables of these theories. Recently, Cao, Park, and Sheffield showed that Wilson loop expectations can be expressed as sums over certain embedded bipartite maps of any genus.

Building on this novel approach, we prove in the strongly coupled regime:

  1. A rigorous formula in terms of embedded bipartite planar maps of Wilson loop expectations in the large N limit, in any dimension d.
  2. An exact computation of Wilson loop expectations in the large N limit, in dimension d = 2.

Similar results to the two aforementioned points were previously established by Chatterjee (2019) and Basu & Ganguly (2016), respectively. Our results offer simpler proofs and provide a new perspective on these significant quantities.

This work is a collaboration with Sky Cao and Jasper Shogren-Knaak, and the talk should be thought of as complementing Sky’s mini-course “Yang–Mills Theory: Random Surface Approach”.

Nikita Nekrasov

Title: Beyond Positive Measures: Nonperturbative QFT, Thimbles, and Localization

This will be an informal and speculative discussion of several nonperturbative issues in quantum field theory, with 4d Yang–Mills theory as the goal. I will emphasize the role of phases of functional integrals, Lefschetz thimbles and integration cycles, gauge fixing and the Gribov problem,lessons from equivariant localization and low dimensional lattice models. The goal is not to present a finished theory, but to suggest directions where probabilistic methods may need to go beyond positive measures and engage more directly with complex, oscillating, and geometrically organized path integrals.

Nikolay Barashkov

Title: Φ⁴₃ as a Markov field

Random fields which possess the Markov property have played an important role in the development of constructive field theory. They are related to their relativistic counterparts through Nelson reconstruction.

In this talk I will describe the Markov property of the Φ⁴ measure in three dimensions. We will also discuss the properties of its generator, i.e. the Φ⁴₃ Hamiltonian. This is based on joint work with T. Gunaratnam.

Raoul Santachiara

Title: Non-compactified Imaginary Liouville Field Theory

We focus on the problem of finding a path-integral definition of a CFT that extends compactified imaginary Liouville theory, in particular by providing correlation functions that do not vanish if Coulomb gas neutrality conditions are not satisfied.

Semyon Klevtsov

Title: Quantum Hall states on surfaces: recent progress and open questions

The quantum Hall effect is a striking phenomenon in condensed matter physics, exhibiting exact quantisation in macroscopic electron systems with imprecise characteristics. Its theoretical description leads to the concept of quasiparticles with exotic exchange statistics—neither bosonic nor fermionic—known as anyons, which are of central interest for topological quantum computation.

A key application of conformal field theory is that its conformal blocks furnish trial wave functions for quantum Hall states. Following the pioneering work of Laughlin (1983) and BPZ (1984), the theory of quantum Hall states developed alongside two-dimensional CFT.

I will talk about recent progress in QHE states on surfaces made in joint works with I. Burban, F. Dupont, and D. Zvonkine, and about remaining questions.