Timetable:
| Tuesday 10th | Wednesday 11th | Thursday 12th | |
| 9h30 - 10h30 | van Rees | Trevisani | Guillarmou |
| 11h20 - 12h20 | Kravchuk | Runkel | Sun |
| 14h - 15h | van Vliet | Zan | Schomerus |
| 15h50 - 16h50 | Korchemsky | He | Discussion |
Each talk is followed by 20 minutes discussion.
Titles and abstracts:
From conformal correlators to scattering amplitudes
by Balt van Rees [Handwritten notes are available on request to the speaker]
I will review our understanding of conformally covariant four-point functions in higher-dimensional theories. After explaining the kinematics I will discuss crossing symmetry, analytic continuations, and some aspects of the lightcone bootstrap.
I will then focus on the conformal dispersion relation and highlight the importance of understanding the lightcone limit on the second sheet. Finally I will mention how these results can be used to learn about the non-perturbative structure of scattering amplitudes.
Low-energy N-body spectrum in CFT at large spin
by Petr Kravchuk [Handwritten notes are available on request to the speaker]
Based on simple toy models, I will discuss the structure of the low-energy (leading twist) spectrum in CFT at large spin. I will specifically focus on multi-twist operators, which in AdS/CFT correspondence would be dual to N-body states with N>2.
I will show that their spectrum can be understood in terms of the semiclassical limit of an effective quantum-mechanical problem, where the classical phase space turns out to be a positive Grassmannian.
This is based on work with Jeremy Mann.
Bootstrapping the long range Ising model
by Philine van Vliet [Handwritten notes] [Slides]
While the short-range Ising model is well known and well studied, its long-range cousin, the long-range Ising model (LRI), has remained more mysterious. Nevertheless, the critical point of the LRI forms a natural starting point for the study of nonlocal CFTs, and it can be analyzed using a variety of techniques, including the conformal bootstrap.
We interpret the LRI as a conformal defect in an auxiliary, free scalar bulk CFT. We find relations between OPE coefficients and constraints on the spectrum that are derived from requirements of analyticity of the correlators and conformal blocks. We show how these can be used in the conformal bootstrap, and in a perturbative setup, and the power they have. These relations are not limited to the LRI, and I will highlight some other interesting setups.
Solving four-dimensional superconformal Yang-Mills theories using random matrices
by Gregory Korchemsky [Slides]
I will describe a general method to systematically compute a special class of important observables in strongly coupled four-dimensional superconformal gauge theories.
A distinguished feature of these observables is that, in the planar limit and for any 't Hooft coupling, they can be expressed as determinants of certain semi-infinite matrices. Crucially, the same determinants have previously appeared in the context of random matrix theory, where they were computed exactly in terms of a well-known probability distribution known as the Tracy-Widom distribution (or a more general version of it).
The Parisi-Sourlas uplift and infinitely many solvable 4d CFTs
by Emilio Trevisani [Handwritten notes are available on request to the speaker]
Two-dimensional CFT via three-dimensional topological field theory
by Ingo Runkel [Handwritten notes]
Topological defects in quantum field theories can be understood as symmetries whose action needs not be invertible. Such symmetries can be gauged to give new QFTs, and they allow us to describe dualities between QFTs.
A helpful point of view is to rewrite the n-dimensional QFT as living on the boundary of an n+1 dimensional topological field theory, together with a topological boundary condition. This situation is mathematically best understood in the setting of rational 2d CFT and 3d topological field theory. I will present the mathematical ingredients and review some of the results that can be obtained using this setting.
Quantum groups as global symmetries in the continuum
by Bernardo Zan [Handwritten notes]
Quantum groups are algebras which are known to play a role both in lattice models and in two dimensional CFTs. However, while in the case of lattice models they appear as global symmetries, the situation is more subtle in 2d CFTs.
As an example, even if Virasoro minimal models have no quantum group global symmetry, the fusion kernel of Virasoro blocks in these theories contains the 6j symbols of a quantum group. But what would a 2d CFT with a quantum group as a genuine global symmetry look like?
I will answer this in the case of the quantum group Uq(sl2), giving both the general picture and studying a specific example arising from the continuum limit of a lattice model. If time allows, I will also discuss the connection with Virasoro minimal models.
Logarithmic operators in c=0 bulk CFTs
by Yifei He [Slides]
We study Kac operators (e.g. energy density) in percolation and self-avoiding walk bulk CFTs. Their norms can be deduced from the reality of the CFTs at generic c and vanish at c=0 where the operators mix into logarithmic multiplets.
We compute their conformal data, resolving the "c->0 catastrophe" and find that, contrary to previous belief, the four-point correlator of the bulk energy operator at c=0 does not vanish.
Probabilistic construction of Conformal Field Theories
by Colin Guillarmou [Handwritten notes]
I will review a few models of conformal field theories in 2D that can be rigorously constructed from probabilistic methods. The correlation functions are expressed as expected values of certain random variables, and in some cases we can show that these probabilistic expressions are equal to the explicit formulas that were proposed in physics.
We can also recover the whole symmetry algebra in the probabilistic language as well as diagonalize the Hamiltonian of the theory. This allows to prove that the conformal boostrap approach can be made mathematically rigorous, even for non-compact theories. This is based on several joint works with Kupiainen, Rhodes, Vargas and Baverez.
Backbone exponent and annulus crossing probability for planar percolation
by Xin Sun [Slides]
We report the recent derivation of the backbone exponent for 2D percolation, obtained jointly with Nolin, Qian, and Zhuang. The value is a transcendental number, which is a root of an elementary equation.
We also report an exact formula for the probability that there are two disjoint paths of the same color crossing an annulus, obtained jointly with Zhuang. The backbone exponent captures the leading asymptotic, while the other roots of the elementary equation capture the asymptotic of the remaining terms.
Our approach is based on the coupling between Stochastic Loewner Evolution and Liouville Quantum Gravity (LQG), and the integrability of Liouville conformal field theory that governs the LQG surfaces.
Conformal symmetry at finite temperature
by Volker Schomerus [Slides]
We consider conformal field theories on manifolds beyond the usual cylinder R x Sd-1, and in particular on the thermal geometry S1 x Sd-1 with a circle S1 of radius β ~1/T. While correlation functions in the thermal geometry contain the same dynamical information as for zero temperature, it is packaged quite differently.
In complete analogy to the zero temperature case, the dynamical data may be exposed with the help of thermal partial wave expansions. The relevant thermal partial waves turn out to be wave functions of an integrable model that emerges from elliptic Hitchin systems through some degeneration. Equipped with explicit formulas for the Hamiltonians one can construct thermal partial waves and then apply these to decompose simple thermal one-point functions.
This talk is based on joint work with Ilija Buric, Francesco Russo, and Alessandro Vichi. The relation with integrability is inspired by earlier work with Ilija Buric, Sylvain Lacroix, Jeremy Mann, and Lorenzo Quintavalle on conformal partial wave expansion for the cylinder.
