Thursday, April 10
09:00-09:50 Registration
09:50-10:00 Opening remarks
Session Chair: Jian Ding, Peking University
10:00-11:00 Nathanaël Berestycki (University of Vienna)
On the Spectral Geometry of Liouville Quantum Gravity
In this talk we will discuss the spectral geometry of the Laplace-Beltrami operator associated to Liouville quantum gravity. In particular we will show that the eigenvalues a.s. obey a Weyl law. This result (joint work with Mo Dick Wong) comes from an analysis of the LQG heat trace, which homogenises despite overwhelming pointwise fluctuations. I will also discuss some conjectures which suggest a connection to "quantum chaos".
11:00-11:30 Morning Tea Break
11:30-12:30 Rémi Rhodes (Aix-Marseille Université)
On the H3 Wess-Zumino-Witten Model and the Liouville Correspondence
Wess-Zumino-Witten (WZW) models are among the most basic and most studied Conformal Field Theories (CFT). They have had a huge influence not only in physics but also in mathematics, in representation theory and geometry. However their rigorous probabilistic construction and analysis starting from the path integral is still missing and all their properties have been obtained algebraically from their postulated affine Lie algebra symmetry. In the path integral description, the field takes values in a Lie Group G or a coset space. This talk will focus on the probabilistic construction of the path integral for the coset SL(2,C)/SU(2), which can be identified with the 3 dimensional hyperbolic space. Then I will discuss a mapping to Liouville theory, initially found by Ribault-Teschner or Schomerus in physics, which was dubbed Quantum Analytic Langlands Correspondence by Teschner and Gaiotto.
12:30-12:40 Group Photo
12:40-14:30 Lunch Break
Session Chair: Xinyi Li, Peking University
14:30-15:30 Jesper Jacobsen (École Normale Supérieure)
Exact Three- and Four-point Correlation Functions in the O(n) and Potts Loop Models
We give an overview of our recent work on geometrically defined bulk correlators in two-dimensional conformally invariant loop models. These correlators correspond to combinatorial maps, describing the connectivities between insertion points. Selected three-point correlators include the probability that three points belong to the same loop, or that two loops come close together in three points, or that an open curve running between two points pass through a third point. While the three-point correlators have a closed expression in terms of certain special functions, the four-point correlators are more involved and require deploying more technology. To determine them we combine the global symmetry of the CFT, the cellular algebra of its lattice discretisation, and interchiral conformal bootstrap. The 235 simplest four-point structure constants are found to be a product of a universal function of conformal dimensions, built from Barnes' double Gamma function, and a polynomial function of loop weights. The polynomial factors, which can be isolated by forming certain amplitude ratios, are retrieved in the lattice models and found to be independent of the size of the lattice, and even independent on whether the model stands at its critical point.
15:30-16:00 Afternoon Tea Break
16:00-17:00 Yves Le Jan (Université Paris-Saclay)
Loops, Trees and Fields on Complete Graphs
After introducing the relations between spanning trees, random loops and bosonic/fermionic fields on general graphs, we focus on the case of the complete graph and derive a few asymptotics as its size increases to infinity.
