Conformal field theory 3 ways: integrable, probabilistic, and supersymmetric

Contribution List

1. Introduction to Liouville field theory - Part 1

Joerg Teschner (DESY)

1/22/24, 9:00 AM

In the the first lecture I will outline the main mathematical features of the bootstrap
approach to Liouville conformal field theory, assembling the Liouville correlation function
from holomorphic functions called conformal blocks. The most important properties
of the conformal blocks will be described. The second lecture will review a construction
of the conformal blocks using an...

2. On the interplay of random curves with CFT and hidden symmetries- Part 1

Eviliina Peltola

1/22/24, 10:30 AM

I will discuss how random curves appearing in critical models, probability theory, and complex geometry are related to evident and hidden algebraic content in CFT.

3. Bicommutant Categories from Conformal Nets, towards constructing fully extended functorial (Segal) chiral CFTs

Nivedita Nivedita

1/22/24, 11:35 AM

Two-dimensional chiral CFTs have three mathematical formulations, namely VOAs, conformal nets and Segal (functorial) CFTs. We are working on the construction of a fully extended 2d chiral functorial field theory given the data of a conformal net. We introduce some ingredients of the target category of Bicommutant Categories (a model for 3-Hilb) as a categorification of the Morita category of...

4. AGT correspondence: a 2d CFT valued in 4d QFTs - Part 1

Bruno Le Floch

1/22/24, 4:30 PM

Starting from a 6d point of view, we will develop the basic dictionary associating 4d N=2 gauge theories to punctured Riemann surfaces, and explain some extensions thereof. As an application, we will explore a rich set of operators (loops, surfaces, walls) that can be added to the story, and how their use led to a conjecture for some braiding kernels in Toda CFT.

5. On the interplay of random curves with CFT and hidden symmetries- Part 2

Eviliina Peltola

1/22/24, 5:45 PM

I will discuss how random curves appearing in critical models, probability theory, and complex geometry are related to evident and hidden algebraic content in CFT.

7. Irreducible representations in Liouville CFT

Guillaume Baverez

1/22/24, 6:45 PM

In the context of the probabilistic construction of Liouville CFT, we prove the irreducibility of highest-weight representations of the Virasoro algebra on the Kac table. Joint with Baojun Wu.

6. Introduction to Liouville field theory

Joerg Teschner (DESY)

1/23/24, 9:00 AM

In the the first lecture I will outline the main mathematical features of the bootstrap
approach to Liouville conformal field theory, assembling the Liouville correlation function
from holomorphic functions called conformal blocks. The most important properties
of the conformal blocks will be described. The second lecture will review a construction
of the conformal blocks using an...

9. Probabilistic approach of Liouville theory

Remi Rhodes (Aix-Marseille university)

1/23/24, 10:30 AM

In these lectures, we will review the probabilistic construction of Liouville CFT. We will explain the Segal axioms for CFTs and how they can be implemented in Liouville CFT with the construction of amplitudes. Gluing amplitudes then allows us to identify the Hamiltonian of the Liouville CFT and to provide a representation of the Virasoro algebra in the Hilbert space of Liouville theory. We...

11. Convergence of Nekrasov's Instanton Partition Functions

Paolo Arnaudo (SISSA)

1/23/24, 11:35 AM

In this talk, we will study the convergence properties of instanton partition functions in four-dimensional N=2 gauge theory with group U(N), also in the presence of matter in the adjoint or (anti)fundamental representation.
The main result is that if the considered theory is conformal its instanton function, seen as a power series in the complexified gauge coupling, has a finite radius of...

14. AGT correspondence: a 2d CFT valued in 4d QFTs

Bruno Le Floch

1/23/24, 5:00 PM

Starting from a 6d point of view, we will develop the basic dictionary associating 4d N=2 gauge theories to punctured Riemann surfaces, and explain some extensions thereof. As an application, we will explore a rich set of operators (loops, surfaces, walls) that can be added to the story, and how their use led to a conjecture for some braiding kernels in Toda CFT.

16. Probabilistic approach of Liouville theory

Remi Rhodes (Aix-Marseille university)

1/23/24, 6:15 PM

In these lectures, we will review the probabilistic construction of Liouville CFT. We will explain the Segal axioms for CFTs and how they can be implemented in Liouville CFT with the construction of amplitudes. Gluing amplitudes then allows us to identify the Hamiltonian of the Liouville CFT and to provide a representation of the Virasoro algebra in the Hilbert space of Liouville theory. We...

8. From Liouville to Painlevé and Heun via CFT/isomonodromy relation

Oleg Lisovyi

1/24/24, 9:00 AM

I will review connections between the problem of construction of linear ordinary differential equations with prescribed monodromy and the 2D conformal field theory. This correspondence leads to a number of conjectures in the theory of Painlevé and Heun equations some of which have already been proven rigorously and some remain open. The two main applications I will focus on are the...

10. (Quantum) Painleve’ / gauge theory correspondence and Heun connection problem

Alessandro Tanzini

1/24/24, 10:30 AM

We discuss the relation between Painleve’ equations, supersymmetric gauge theories and Liouville CFT. As an application we show how within this circle of ideas one can solve the connection problem for Heun functions in terms of explicit combinatorial formulae arising from supersymmetric localisation. We also present work in progress on the extension of the correspondence to quantum Painleve’.

12. Limiting distributions of Spherical and Spin O(N) models: Appearance of GFF

Aleksandra Korzhenkova

1/24/24, 11:35 AM

Spherical model is a mathematical model of a ferromagnet introduced by Berlin and Kac in 1952 as a rough but analytically convenient modification of the Ising model. Since its inception it has enjoyed considerable popularity among the mathematicians and physicists as an exactly soluble model exhibiting a phase transition. In this talk we will explain its relation to the Gaussian free field in...

13. From Liouville to Painlevé and Heun via CFT/isomonodromy relation

Oleg Lisovyi

1/24/24, 6:00 PM

I will review connections between the problem of construction of linear ordinary differential equations with prescribed monodromy and the 2D conformal field theory. This correspondence leads to a number of conjectures in the theory of Painlevé and Heun equations some of which have already been proven rigorously and some remain open. The two main applications I will focus on are the...

15. Tau function, topological string and spectral theory

Qianyu Hao (Universite de Geneve (CH))

1/24/24, 7:00 PM

Tau function not only is the essential object in the study of integrable system in mordern mathematics but also plays important role in physics; they correspond to the (chiral) conformal blocks. I will talk about our work on an application of tau function to study the physics, which is to prove a special limit of the conjectured topological string/ spectral theory (TS/ST) correspondence. I...

17. Random surfaces and Yang-Mills gauge theory

Scott Sheffield

1/25/24, 9:00 AM

Constructing and understanding the basic properties of Euclidean Yang-Mills theory is a fundamental problem in physics. It is also one of the Clay Institute's famous Millennium Prize problems in mathematics. The basic problem is not hard to understand. You can begin by describing a simple random function from a set of lattice edges to a group of matrices. Then you ask whether you can...

19. AGT in algebraic geometry

Andrei Negut (MIT)

1/25/24, 10:30 AM

The Alday-Gaiotto-Tachikawa correspondence between conformal field theory and 4D gauge theory has a very interesting incarnation in geometric representation theory. Here the objects on the two sides of the correspondence are W-algebras of type gl_r and moduli spaces of rank r sheaves on algebraic surfaces. We prove a q-deformed version of the correspondence, which reveals some higher (more...

20. Affine Yangian approach to integrability in 2D CFT

Alexey Litvinov (Skoltech)

1/25/24, 11:35 AM

I will review a recently developed approach to studying integrability in
CFT based on affine Yangian symmetry. I will focus on the derivation of
Bethe's ansatz equations for the spectrum of integrals of motion.

21. On the anlytical continuation of Liouville field theory path integral

Raoul Santachiara (Paris-Saclay)

1/25/24, 5:15 PM

22. A distant descendant of the six-vertex model

Vladimir Bazhanov

1/25/24, 6:15 PM

In this talk I present a new solution of the star-triangle relation having positive Boltzmann weights. The solution defines an exactly solvable two-dimensional Ising-type (edge interaction) model of statistical mechanics where the local ``spin variables'' can take arbitrary integer values, i.e., the number of possible spin states at each site of the lattice is infinite. There is also an...

23. Quantum algebras, wild curves and Painlevé equations

Marta Mazzocco (Birmingham)

1/26/24, 9:00 AM

24. A probabilistic approach to Toda Conformal Field Theories

Baptiste Cercle (EPFL)

1/26/24, 10:30 AM

Toda Conformal Field Theories form a family of two-dimensional quantum field theories generalizing Liouville theory. One of their features is that they enjoy, in addition to conformal invariance, an enhanced level of symmetry encoded by W-algebras.
In this talk we describe their mathematical definition and study some of their properties. Namely we will explain how the understanding of its...

25. From Segal amplitudes to conformal block

Colin Guillarmou (Paris Saclay and CNRS)

1/26/24, 11:15 AM

We shall explain how to construct conformal blocks on Riemann surfaces for Liouville CFT from the probabilistic representation of Segal amplitudes in Liouville theory and the diagonalisation of the Hamiltonian done using analytic methods.
This is based on several joint works with Baverez, Kupiainen, Rhodes, Vargas.