Abstracts & Recordings
Lecture Series 1: Equivalence between the probabilistic and the bootstrap approach to Liouville theory
In these three lectures, we will review the mathematical derivation of the full equivalence between the probabilistic and the so-called bootstrap approach to Liouville conformal field theory. The equivalence consists of three fundamental steps which are:
Derivation of the DOZZ formula for the three point correlation on the sphere. Before, we will review the probabilistic construction using the Gaussian Free Field (Vincent Vargas);
(recording)Spectral resolution for the Liouville Hamiltonian (Colin Guillarmou);
(recording, talk starts at 1:03:00) (slides)Segal's axioms and construction of conformal blocks using step 2 (Antti Kupiainen).
(recording, talk starts at 2:02:00) (slides)
Lecture Series 2: Integrability of SLE and CLE via conformal welding and LCFT
Xin Sun: Two types of integrability in Liouville quantum gravity
There are two major resources of integrability in Liouville quantum gravity: conformal field theory and random planar maps decorated with statistical physics models. I will explain at a high level how these two types of integrability are compatible, and how conformal welding of random surfaces allows us to blend them to obtain exact results on Liouville conformal field theory, mating of trees, Schramm-Loewner evolution, and conformal loop ensemble.
(recording) (notes)
Nina Holden: Conformal welding of Liouville quantum gravity disks and an application to the integrability of SLE
We will present two new conformal welding results for Liouville quantum gravity disks. The SLE curves arising in these conformal welding results is the SLEκ(ρ1;ρ2) and the SLE loop, respectively. We derive an exact formula for the law of a conformal derivative associated with the SLEκ(ρ1;ρ2) by using the first conformal welding result along with exact formulas from LCFT and for planar Brownian motion.
(recording) (slides)
Morris Ang: Integrability of the conformal loop ensemble
For 8/3 < κ < 8, the conformal loop ensemble CLEκ is a canonical random ensemble of loops which is conformally invariant in law, and whose loops locally look like Schramm-Loewner evolution with parameter κ. It describes the scaling limits of the Ising model, percolation, and other models. When κ ≤ 4 the loops are simple curves. In this regime we compute the three-point correlation function of CLEκ on the sphere, and show it agrees with the imaginary DOZZ formula of Zamolodchikov (2005). We also obtain the expression of the (properly normalized) probability that three points are on the same CLE loop in terms of the DOZZ formula. The analogous quantity for three points on the same cluster was previously conjectured by Delfino and Viti. To our best knowledge our formula has not been predicted in the physics literature. Our arguments depend on couplings of CLE with Liouville quantum gravity and the integrability of Liouville conformal field theory. Based on joint work with Xin Sun.
(recording) (slides)
Speakers
Federico Camia: Conformal field theory from Brownian loops
The Brownian loop soup is a conformally invariant Poissonian ensemble of loops in two dimensions, with intensity measure proportional to the unique (up to a multiplicative constant) conformally invariant measure on simple planar loops. In this talk, I will discuss several “operators” that compute statistical properties of the loop soup and behave like conformal primary fields in conformal field theory. For some of these fields one can compute the four-point function explicitly and perform the Virasoro conformal block expansion, which reveals the existence of an infinite number of new primary fields. Among these, we are able to identify all the scalar (spin zero) fields in terms of the operators mentioned above. When the loop soup is restricted to the upper half-plane, we also identify the boundary stress-energy tensor of the theory. The picture that starts to emerge is that of a rich conformal field theory with novel features such as primary fields whose conformal dimensions depend periodically on a real parameter. The tools involved in the analysis include the O(n) model, the full scaling limit of critical percolation, and the Schramm-Loewner Evolution. The talk is based on recent joint work with Alberto Gandolfi, Valentino Foit and Matthew Kleban.
(recording) (slides)
Baptiste Cerclé: Towards integrability of Toda conformal field theories
Toda conformal field theories are a family of two-dimensional conformal field theories indexed by semi-simple and complex Lie algebras. One of their features is that, in addition to conformal symmetry, they enjoy an extended level of symmetry encoded by W-algebras. Besides, they can be defined via a path integral similar to the one of Liouville theory for which they provide a natural generalization.
In this talk we will review recent progress made towards integrabilty of such theories, relying on a probabilistic formulation of the models and additional materials specific to Toda theories.
This is based on joint works with Huang, Rhodes, Vargas.
(recording) (slides)
Jesper Lykke Jacobsen: Four-point functions in the Fortuin-Kasteleyn cluster model
The determination of four-point correlation functions of two-dimensional lattice models is of fundamental importance in statistical physics. In the limit of an infinite lattice, this question can be formulated in terms of conformal field theory (CFT). For the so-called minimal models the problem was solved more than 30 years ago, by using that the existence of singular states implies that the correlation functions must satisfy certain differential equations. This settles the issue for models defined in terms of local degrees of freedom, such as the Ising and 3-state Potts models. However, for geometrical observables in the Fortuin-Kasteleyn cluster formulation of the Q-state Potts model, for generic values of Q, there is in general no locality and no singular states, and so the question remains open. As a warm-up to solving this problem, we discuss which states propagate in the s-channel of such correlation functions, when the four points are brought together two by two. To this end we combine CFT methods with algebraic and numerical approaches to the lattice model. We then outline work in progress that aims at solving the problem entirely, through an interchiral conformal bootstrap setup that makes contact with time-like Liouville field theory and a number of profound algebraic results.
(recording) (slides)
Matthis Lehmkühler: Liouville Quantum Gravity weighted by Conformal Loop Ensemble nesting statistics
It is conjectured (and partially known) that the scaling limit of certain loop-decorated planar map models is given by a Liouville Quantum Gravity (LQG) surface decorated by an independent Conformal Loop Ensemble (CLE). One interesting construction in this context is to consider the loop-decorated planar map together with some uniformly chosen vertices and reweight the planar map by the number of loops surrounding the points (called the nesting statistic). In the talk, it will be explained what the continuum counterpart to this reweighted object is and we will explain what the analogue to the discrete peeling algorithm for the loop-decorated planar map is in the continuum. Based on joint work with Nina Holden.
(recording)
Eveliina Peltola: On log-CFT for uniform spanning trees and SLE(8)
I discuss the emergence of logarithmic CFT content associated to SLE(8) and non-local observables in the planar uniform spanning tree (UST) model, constructed via scaling limits of Peano curves and their crossing probabilities. In particular, with explicit correlation functions and their fusion thus obtained, one sees that any CFT describing the geometry of UST must be non-unitary (thus not reflection positive). This is of course no surprise - we give a systematic construction directly from the lattice model via its scaling limit, together with immediate relation to SLE(8).
Joint work with Mingchang Liu and Hao Wu.
(recording) (slides)
Guillaume Remy: Integrability of boundary Liouville CFT
Liouville theory is a fundamental example of a conformal field theory (CFT) first introduced in physics by A. Polyakov to describe a canonical random 2d surface. In recent years it has been rigorously studied using probabilistic techniques. In this talk we will study the integrable structure of Liouville CFT on a domain with boundary by proving exact formulas for its structure constants. Our latest result is derived using conformal welding of random surfaces, in relation with the Schramm-Loewner evolutions. We will also discuss an intriguing relation between the boundary three-point function of Liouville CFT and the so-called fusion kernels of conformal blocks. Based on joint work with Morris Ang, Xin Sun and Tunan Zhu.
(recording) (slides)
Raoul Santachiara: Bootstrap solutions for c less than one
We present a panorama of the currently known CFTs that have central charge less than one. Our point of view will be a bootstrap and algebraic point of view. We briefly review some recent applications in statistical models and discuss some open questions in the field.
(recording) (slides)
Yi Sun: Probabilistic construction of conformal blocks for Liouville CFT on the torus
This talk presents a probabilistic construction of 1-point Virasoro conformal blocks on the torus for central charge greater than 25, which are those appearing in Liouville CFT. I will present our construction using Gaussian multiplicative chaos and give a sketch of the proof, which uses the BPZ equations, operator product expansion, and Dotsenko-Fateev type integrals. I will also mention connections to work in progress on modular symmetry for these conformal blocks. Based on joint work with Promit Ghosal, Guillaume Remy, and Xin Sun.
(recording) (slides)