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Bootstat 2021: Conformal bootstrap and statistical models
from
Monday, May 3, 2021 (9:00 AM)
to
Friday, May 28, 2021 (6:00 PM)
Monday, May 3, 2021
9:30 AM
IPA director's welcome

Denis ULLMO
(Institut Pascal)
IPA director's welcome
Denis ULLMO
(Institut Pascal)
9:30 AM  9:45 AM
10:00 AM
Introduction to the numerical conformal bootstrap

Alessandro Vichi
(University of Pisa)
Introduction to the numerical conformal bootstrap
(1. Status of the conformal bootstrap)
Alessandro Vichi
(University of Pisa)
10:00 AM  11:00 AM
Lecture 1:  Introduction  Conformal symmetry in $D\geq 2$ dimensions  Unitarity/reflection positivity Lecture 2:  Operator Product Expansion and conformal blocks  Crossing equations Lecture 3:  Recasting crossing symmetry as a convex optimisation problem (main logic) Lecture 4:  Applications of the numerical bootstrap: the Ising model in $D=2,3$ dimensions
11:00 AM
Question time
Question time
11:00 AM  11:30 AM
11:30 AM
Introduction to the numerical conformal bootstrap

Alessandro Vichi
(University of Pisa)
Introduction to the numerical conformal bootstrap
Alessandro Vichi
(University of Pisa)
11:30 AM  12:30 PM
12:30 PM
Question time
Question time
12:30 PM  1:00 PM
Tuesday, May 4, 2021
10:00 AM
Introduction to the numerical conformal bootstrap

Alessandro Vichi
(University of Pisa)
Introduction to the numerical conformal bootstrap
Alessandro Vichi
(University of Pisa)
10:00 AM  11:00 AM
11:00 AM
Question time
Question time
11:00 AM  11:30 AM
11:30 AM
Introduction to the numerical conformal bootstrap

Alessandro Vichi
(University of Pisa)
Introduction to the numerical conformal bootstrap
Alessandro Vichi
(University of Pisa)
11:30 AM  12:30 PM
12:30 PM
Question time
Question time
12:30 PM  1:00 PM
3:00 PM
Numerical bootcamp

Marten Reehorst
(Institute des Hautes Études Scientifique)
Numerical bootcamp
Marten Reehorst
(Institute des Hautes Études Scientifique)
3:00 PM  4:00 PM
This tutorial will allow participants to get started with the numerical bootstrap. We will show how the famous Ising model kink can be found from scratch, in a simplified workflow, using only Mathematica without the need for any external tools. This simplified workflow will help highlight some of the conceptual issues involved. Next, we will introduce the various numerical tools that are used in actual stateoftheart computations. We will show how to use these tools, how to interpret their output, and how to put it all together to set up and run a real bootstrap problem. The tutorial will take the form of guided exercises which will be introduced and discussed in the planned sessions. Active participation in solving the exercises is recommended. Support on the installation and use of the tools used in the numerical bootstrap will also be provided continuously on the #numericalbootcamp Slack channel.
4:00 PM
Question time
Question time
4:00 PM  4:30 PM
Wednesday, May 5, 2021
11:00 AM
Nonrelativistic conformal symmetries

Malte Henkel
(LPCT, Univ. Lorraine Nancy, France)
Nonrelativistic conformal symmetries
(2. Statistical physics targets)
Malte Henkel
(LPCT, Univ. Lorraine Nancy, France)
11:00 AM  12:00 PM
I shall try to give an introduction and physical motivation why to study the nonequilibrium relaxational dynamics in systems either at or below the equilibrium critical point. I shall discuss the main properties of the Schroedinger group, why it predicts the form of response functions and how this compares to models in nonequilibirum statistical mechanics.
12:00 PM
Question time
Question time
12:00 PM  12:30 PM
3:00 PM
Applications of the numerical conformal bootstrap

David Poland
(Yale University)
Applications of the numerical conformal bootstrap
(1. Status of the conformal bootstrap)
David Poland
(Yale University)
3:00 PM  4:00 PM
I’ll discuss various applications of modern numerical bootstrap methods, focusing primarily on 3d CFTs. Along the way I’ll describe various algorithms and software tools that have been recently introduced to help tackle largescale bootstrap problems.
4:00 PM
Question time
Question time
4:00 PM  4:30 PM
Thursday, May 6, 2021
10:00 AM
The navigator function : sailing through the infeasible sea in the conformal bootstrap

Ning Su
(University of Pisa)
The navigator function : sailing through the infeasible sea in the conformal bootstrap
(1. Status of the conformal bootstrap)
Ning Su
(University of Pisa)
10:00 AM  11:00 AM
Conformal bootstrap is shown to be very useful for studying the 3D Ising, SuperIsing, O(N) models. Can we bootstrap more and more CFTs that are relevant to condensed matter and statistical physics? In general, for more complicated CFTs, we expect that one has to bootstrap a large system and carve out theory spaces with many unknown parameters. The traditional numerical bootstrap method searches the theory space by repeatedly checking whether points are allowed or excluded. However this is inadequate for many applications. In this talk, I will discuss a new method that involves a continuous "navigator" function which is negative in the allowed region and positive in the excluded region. With the navigator function, one can very effectively move through the theory space and navigate towards the CFT island. This method outperforms the old method and can do many new things. I will discuss potential applications in statistical physics. This talk is based on [arXiv:2104.09518](https://arxiv.org/abs/2104.09518).
11:00 AM
Question time
Question time
11:00 AM  11:30 AM
3:00 PM
Critical geometry approach to threedimensional percolation

Alessandro Galvani
(SISSA)
Critical geometry approach to threedimensional percolation
Alessandro Galvani
(SISSA)
3:00 PM  4:00 PM
I will describe a theory of bounded critical phenomena based on a geometric approach, introduced in the article [arXiv:1904.08919](https://arxiv.org/abs/1904.08919): a curved metric, conformal to the euclidean one, is added to a bounded domain, with the requirement of constant curvature, to enforce homogeneity. This leads to the socalled Yamabe equation, which is then modified, with the introduction of a fractional Laplacian, to account for the anomalous dimension of the fields. Solving this equation provides a pointdependent scale for the system, which can be used to determine onepoint and twopoint spin correlations functions. After briefly reviewing results for the Ising and XY models, we compare the Yamabe predictions with numerical simulations of continuum percolation in three dimensions, and we present a highprecision estimate of its anomalous dimension $\eta$.
4:00 PM
Question time
Question time
4:00 PM  4:30 PM
Friday, May 7, 2021
10:00 AM
Critical Ising model in varying dimension by conformal bootstrap

Andrea Cappelli
(INFN)
Critical Ising model in varying dimension by conformal bootstrap
(1. Status of the conformal bootstrap)
Andrea Cappelli
(INFN)
10:00 AM  11:00 AM
The singlecorrelator conformal bootstrap is solved numerically for several values of dimension $4 > d > 2$ using SDPB and Extremal Functional methods. Critical exponents and other conformal data of lowlying states are obtained over the entire range of dimensions with up to fourdecimal precision and then compared with several existing results. The conformal dimensions of leadingtwist fields are also determined up to high spin, and their ddependence shows how the conformal states rearrange themselves around $d = 2.2$ for matching the Virasoro conformal blocks in the $d = 2$ limit. The decoupling of states at the Ising point is studied for $3 > d > 2$ and the vanishing of one structure constant at $d = 3$ is found to persist till $d = 2$ where it corresponds to a Virasoro nullvector condition. (See the article [arXiv:1811.07751][1]) [1]: https://arxiv.org/abs/1811.07751
11:00 AM
Question time
Question time
11:00 AM  11:30 AM
3:00 PM
Numerical bootcamp

Marten Reehorst
(Institute des Hautes Études Scientifique)
Numerical bootcamp
Marten Reehorst
(Institute des Hautes Études Scientifique)
3:00 PM  4:00 PM
4:00 PM
Question time
Question time
4:00 PM  4:30 PM
Saturday, May 8, 2021
Sunday, May 9, 2021
Monday, May 10, 2021
10:00 AM
Numerical bootcamp

Marten Reehorst
(Institute des Hautes Études Scientifique)
Numerical bootcamp
Marten Reehorst
(Institute des Hautes Études Scientifique)
10:00 AM  11:00 AM
11:00 AM
Question time
Question time
11:00 AM  11:30 AM
11:30 AM
Analytically solving the Ising model in $2 + \epsilon$ dimensions

Wenliang Li
(Okinawa Institute of Science and Technology Graduate University)
Analytically solving the Ising model in $2 + \epsilon$ dimensions
Wenliang Li
(Okinawa Institute of Science and Technology Graduate University)
11:30 AM  12:30 PM
The analytical studies of the $d>2$ Ising model are usually based on the $\epsilon$ expansion around $d=4$. Since the $d=2$ Ising model is solvable, it would be interesting to deform the $d=2$ exact solution to $d=2+\epsilon$ dimensions. Some strong coupling features may be seen more clearly. I will discuss some attempts using the analytic conformal bootstrap.
12:30 PM
Question time
Question time
12:30 PM  1:00 PM
3:00 PM
Random critical points: Anderson transitions, multifractality, and conformal symmetry

Ilya Gruzberg
(Ohio State University)
Random critical points: Anderson transitions, multifractality, and conformal symmetry
(2. Statistical physics targets)
Ilya Gruzberg
(Ohio State University)
3:00 PM  4:00 PM
I will start with a brief overview of the basic issues related to disordered systems (ensembles, distributions, disorder averages), and introduce examples of disordered systems (random magnets, disordered electronic systems), and basic techniques to get to a field theory description (replicas and SUSY). Then I will focus (perhaps exclusively, depending on the available time) on Anderson transitions and multifractality (MF) of wave functions. I will talk about MF spectra and their properties, and how to compute them in some simple models (Dirac fermion in a random gauge potential). I will also talk about sigma model and how the symmetry of the target space implies symmetries of the MF spectra. Finally, I plan to discuss Suslov's proposal for parabolicity of MF spectra, how it fails in general, and how it can be demonstrated for the IQH transition using CFT (based on [an article](https://arxiv.org/abs/1612.04109) by Bondesan, Wieczorek and Zirnbauer). If time allows, I plan to discuss mappings of network models to statmech models for classes C and A, and relate to geometric disorder, SLE, etc.  Lecture 1. Introduction. General remarks about disorder. Random Hamiltonians, random observables. Statistical treatment, quenched averages. First example: classical spin systems. Replica method. Harris criterion. Two dimensions: c=0 and logarithmic CFTs.  Lecture 2. Second example: quantum spin systems and SYK models. Third example: geometric critical phenomena. Percolation, random cluster, and random loop models. Stochastic geometry. Two dimensions: stochastic conformal geometry, SLE and conformal restriction.  Lecture 3. Fourth example: disordered electronic systems, Anderson localization, Anderson transitions. Supersymmetry method and nonlinear sigma models. AltlandZirnbauer classification. Multifractality of critical wave functions: basic properties and formalism. Multifractal dimensions and field theory.  Lecture 4. Symmetries of multifractal spectra. Generalized multifractal observables. Two dimensions: application of 2D CFT to multifractality. Exact parabolicity of multifractal spectra.
4:00 PM
Question time
Question time
4:00 PM  4:30 PM
4:30 PM
Random critical points: Anderson transitions, multifractality, and conformal symmetry

Ilya Gruzberg
(Ohio State University)
Random critical points: Anderson transitions, multifractality, and conformal symmetry
Ilya Gruzberg
(Ohio State University)
4:30 PM  5:30 PM
5:30 PM
Question time
Question time
5:30 PM  6:00 PM
Tuesday, May 11, 2021
10:00 AM
Upper critical dimension of the 3state Potts model

Shai Chester
(Weizmann Institute)
Upper critical dimension of the 3state Potts model
(2. Statistical physics targets)
Shai Chester
(Weizmann Institute)
10:00 AM  11:00 AM
The 3state Potts model has a unitary second order phase transition in spacetime dimension d=2, which is defined by an exactly solvable minimal model called the critical Potts model. There is also another 2d minimal model called the tricritical Potts model which has the same symmetries but one extra relevant operator. As d increases, these two fixed points are expected to merge and go off into the complex plane, so that the unitary second order phase transition disappears. We find nonperturbative evidence for this scenario from the numerical conformal bootstrap, which suggests that the critical Potts model disappears for d<3.
11:00 AM
Question time
Question time
11:00 AM  11:30 AM
3:00 PM
Random critical points: Anderson transitions, multifractality, and conformal symmetry

Ilya Gruzberg
(Ohio State University)
Random critical points: Anderson transitions, multifractality, and conformal symmetry
Ilya Gruzberg
(Ohio State University)
3:00 PM  4:00 PM
4:00 PM
Question time
Question time
4:00 PM  4:30 PM
4:30 PM
Random critical points: Anderson transitions, multifractality, and conformal symmetry

Ilya Gruzberg
(Ohio State University)
Random critical points: Anderson transitions, multifractality, and conformal symmetry
Ilya Gruzberg
(Ohio State University)
4:30 PM  5:30 PM
5:30 PM
Question time
Question time
5:30 PM  6:00 PM
Wednesday, May 12, 2021
10:00 AM
Geometrical correlators in 2d CFTs: an introduction with some open problems

Jacopo Viti
(INFN Florence)
Geometrical correlators in 2d CFTs: an introduction with some open problems
(2. Statistical physics targets)
Jacopo Viti
(INFN Florence)
10:00 AM  11:00 AM
11:00 AM
Question time
Question time
11:00 AM  11:30 AM
3:00 PM
Fourpoint cluster connectivities in the 2d critical Qstate Potts model

Yifei He
(ENS Paris)
Fourpoint cluster connectivities in the 2d critical Qstate Potts model
Yifei He
(ENS Paris)
3:00 PM  4:00 PM
A signature example of the 2d geometric phase transition is the critical percolation, and one of the fundamental observables involves correlation functions describing connectivities of a number of points in terms of cluster configurations. Such quantities can be approached in a field theory language through the FortuinKasteleyn formulation of the Qstate Potts model. In this talk I will consider the fourpoint cluster connectivities in the critical model. Connections with the lattice formulation of minimal models are made which provide insights of the structure of the Potts spectrum. This allows to deduce the existence of "interchiral conformal blocks" which can be constructed using the degeneracy in the spectrum. Using these, I will then bootstrap the fourpoint connectivities. The talk is based on [2002.09071](https://arxiv.org/abs/2002.09071) and [2005.07258](https://arxiv.org/abs/2005.07258).
4:00 PM
Question time
Question time
4:00 PM  4:30 PM
Thursday, May 13, 2021
8:00 AM
Holiday
Holiday
8:00 AM  8:00 PM
Friday, May 14, 2021
10:00 AM
Numerical bootcamp

Marten Reehorst
(Institute des Hautes Études Scientifique)
Numerical bootcamp
Marten Reehorst
(Institute des Hautes Études Scientifique)
10:00 AM  11:00 AM
11:00 AM
Question time
Question time
11:00 AM  11:30 AM
3:00 PM
Conformal field theory of the integer quantum Hall transition: a status report

Martin Zirnbauer
(University of Cologne)
Conformal field theory of the integer quantum Hall transition: a status report
(2. Statistical physics targets)
Martin Zirnbauer
(University of Cologne)
3:00 PM  4:00 PM
Of the five 2D strong topological insulator Anderson transitions, all of which should be logarithmic conformal field theories of WessZuminoWitten type, only one is beginning to be understood: the integer quantum Hall transition. The key feature here is that that the global symmetry of the noncritical system undergoes rank reduction (rather than doubling) at criticality, by a novel mechanism of spontaneous symmetry breaking. In this talk, I will review the current state of our understanding.
4:00 PM
Question time
Question time
4:00 PM  4:30 PM
4:30 PM
The 3d Ising CFT spectrum

Slava Rychkov
(IHES & ENS)
The 3d Ising CFT spectrum
Slava Rychkov
(IHES & ENS)
4:30 PM  5:30 PM
More than a 100 scaling operators and their OPE coefficients have been extracted numerically in the 3d Ising CFT by David SimmonsDuffin in [http://arxiv.org/abs/1612.08471](http://arxiv.org/abs/1612.08471). We will discuss the most salient features of this spectrum such as the appearance of “Regge trajectories”. Time permitting, we will also discuss how this spectrum can be used to do conformal perturbation theory about the Ising CFT (or several copies thereof)
5:30 PM
Question time
Question time
5:30 PM  6:00 PM
Saturday, May 15, 2021
Sunday, May 16, 2021
Monday, May 17, 2021
10:00 AM
Interchiral algebra from the affine TemperleyLieb algebra

Azat GAINUTDINOV
(CNRS, Institut Denis Poisson, Université de Tours)
Interchiral algebra from the affine TemperleyLieb algebra
Azat GAINUTDINOV
(CNRS, Institut Denis Poisson, Université de Tours)
10:00 AM  11:00 AM
The interchiral algebra is an extension of the product of chiral and antichiral Virasoro algebras and appears naturally in the scaling limit analysis of certain statistical physics models where the interaction terms are of TemperleyLieb type. The idea of this discussion is to see how the interchiral algebra generators show up in the scaling limit of the affine TemperleyLieb operators.
11:00 AM
Question time
Question time
11:00 AM  11:30 AM
11:30 AM
Basics of symmetry in condensed matter physics

Sylvain Ravy
Basics of symmetry in condensed matter physics
Sylvain Ravy
11:30 AM  12:30 PM
12:30 PM
Question time
Question time
12:30 PM  1:00 PM
3:00 PM
Introduction to structural phase transitions

Pierre Toledano
(Université de Picardie)
Introduction to structural phase transitions
(3. Structural phase transitions and other unitary problems)
Pierre Toledano
(Université de Picardie)
3:00 PM  4:00 PM
1. Basic concepts of the Landau theory. Critical behaviour. Symmetry of the order parameter. Secondary order parameters. Specificity of firstorder transitions. Useful rules for applying the theory. Construction of phase diagrams involving phase transitions. 2. Application of the theory to illustrative examples of structural transitions induced by a single irreducible representation. Transitions induced by several order parameters. Order parameter replication. Incommensurate phase transitions. Analysis of the applicability of the Landau theory to structural transitions in real systems. 3. Analysis of the critical behaviour of structural transitions in real systems. Agreement and disagreement with the theoretical predictions. Dependence of the critical behaviour on the effective space dimensionality, the order parameter dimension, the order parameter anisotropy, the range of the anisotropic interactions. Influence of elastic degrees of freedom. Specific critical behaviour of incommensurate structural transitions. Influence of defects on the critical behaviour.
4:00 PM
Question time
Question time
4:00 PM  4:30 PM
4:30 PM
Introduction to structural phase transitions

Pierre Toledano
(Université de Picardie)
Introduction to structural phase transitions
Pierre Toledano
(Université de Picardie)
4:30 PM  5:30 PM
1. Basic concepts of the Landau theory. Critical behaviour. Symmetry of the order parameter. Secondary order parameters. Specificity of firstorder transitions. Useful rules for applying the theory. Construction of phase diagrams involving phase transitions. 2. Application of the theory to illustrative examples of structural transitions induced by a single irreducible representation. Transitions induced by several order parameters. Order parameter replication. Incommensurate phase transitions. Analysis of the applicability of the Landau theory to structural transitions in real systems. 3. Analysis of the critical behaviour of structural transitions in real systems. Agreement and disagreement with the theoretical predictions. Dependence of the critical behaviour on the effective space dimensionality, the order parameter dimension, the order parameter anisotropy, the range of the anisotropic interactions. Influence of elastic degrees of freedom. Specific critical behaviour of incommensurate structural transitions. Influence of defects on the critical behaviour.
5:30 PM
Question time
Question time
5:30 PM  6:00 PM
Tuesday, May 18, 2021
3:00 PM
Introduction to structural phase transitions

Pierre Toledano
(Université de Picardie)
Introduction to structural phase transitions
Pierre Toledano
(Université de Picardie)
3:00 PM  4:00 PM
1. Basic concepts of the Landau theory. Critical behaviour. Symmetry of the order parameter. Secondary order parameters. Specificity of firstorder transitions. Useful rules for applying the theory. Construction of phase diagrams involving phase transitions. 2. Application of the theory to illustrative examples of structural transitions induced by a single irreducible representation. Transitions induced by several order parameters. Order parameter replication. Incommensurate phase transitions. Analysis of the applicability of the Landau theory to structural transitions in real systems. 3. Analysis of the critical behaviour of structural transitions in real systems. Agreement and disagreement with the theoretical predictions. Dependence of the critical behaviour on the effective space dimensionality, the order parameter dimension, the order parameter anisotropy, the range of the anisotropic interactions. Influence of elastic degrees of freedom. Specific critical behaviour of incommensurate structural transitions. Influence of defects on the critical behaviour.
4:00 PM
Question time
Question time
4:00 PM  4:30 PM
Wednesday, May 19, 2021
10:00 AM
Critical behavior near structural phase transitions

Amnon Aharony
(Tel Aviv University)
Critical behavior near structural phase transitions
(3. Structural phase transitions and other unitary problems)
Amnon Aharony
(Tel Aviv University)
10:00 AM  11:00 AM
SrTiO$_3$ and LaAlO$_3$ undergo structural phase transitions from a cubic trructure to lower symmetry structures. These transitions can be affected by uniaxial stress. Following original work by Alex Mueller, I have worked on these transitions since 1973. Many details depend on whether the 3component cubic problem is dominated by the isotropic or by the cubic fixed point. Most of the past literature assumed that the former dominates. Following discussions with Slava Rychkov, I realized that it is now believed that the latter wins. This talk will discuss the consequences of this change, and the experimental possibilities to test it.
11:00 AM
Question time
Question time
11:00 AM  11:30 AM
3:00 PM
Critical behavior near structural phase transitions

Amnon Aharony
(Tel Aviv University)
Critical behavior near structural phase transitions
Amnon Aharony
(Tel Aviv University)
3:00 PM  4:00 PM
SrTiO$_3$ and LaAlO$_3$ undergo structural phase transitions from a cubic trructure to lower symmetry structures. These transitions can be affected by uniaxial stress. Following original work by Alex Mueller, I have worked on these transitions since 1973. Many details depend on whether the 3component cubic problem is dominated by the isotropic or by the cubic fixed point. Most of the past literature assumed that the former dominates. Following discussions with Slava Rychkov, I realized that it is now believed that the latter wins. This talk will discuss the consequences of this change, and the experimental possibilities to test it.
4:00 PM
Question time
Question time
4:00 PM  4:30 PM
4:30 PM
Conformal bootstrap studies of puzzles in critical phenomena

Andreas Stergiou
(Los Alamos National Laboratory)
Conformal bootstrap studies of puzzles in critical phenomena
(3. Structural phase transitions and other unitary problems)
Andreas Stergiou
(Los Alamos National Laboratory)
4:30 PM  5:30 PM
Renormalization group methods have been used for almost 50 years to obtain results for critical exponents and other conformal field theory (CFT) observables. Agreement with experiment has been good in many cases, but disagreements between theory and experiment that have remained unresolved for decades also exist. In this talk, I will discuss the conformal bootstrap of threedimensional CFTs with O(m) ⨉ O(n), O(m)^n ⋊ S_n and ℤ_2^n ⋊ S_n global symmetries. For some values of the parameters m and n such CFTs describe continuous phase transitions in frustrated magnets and structural phase transitions, where theory and experiment have not yet reached satisfactory agreement. I will show that the conformal bootstrap gives evidence for the existence of previously unknown CFTs with potential relevance to these phase transitions.
5:30 PM
Question time
Question time
5:30 PM  6:00 PM
Thursday, May 20, 2021
10:00 AM
Fourpoint functions for nonunitary geometrical problems in two dimensions

Hubert Saleur
(IPhT Saclay)
Fourpoint functions for nonunitary geometrical problems in two dimensions
Hubert Saleur
(IPhT Saclay)
10:00 AM  11:00 AM
11:00 AM
Question time
Question time
11:00 AM  11:30 AM
Friday, May 21, 2021
10:00 AM
RG Flows in Coupled Replica CFTs

Hirohiko Shimada
(NIT Tsuyama)
RG Flows in Coupled Replica CFTs
Hirohiko Shimada
(NIT Tsuyama)
10:00 AM  11:00 AM
Consider M copies of the qstate Potts models and the O(n) models coupled through the bondbond interaction. Nontrivial IR fixed points exist both in the disordered model (the replica limit $M \to 0$) and in the unitary model (such as q=3 with M=3, 4, 5, ...). Conformal perturbation theory yields the critical exponents in the expansions in (q2) or (1n) around the Mcoupled Ising CFTs in 2d, where the coupling is marginal. In addition, the RG flow generated by the Zamolodchikov Cfunction extracted from the transfer matrix can capture nonperturbative multicritical fixed points at M=0. The Smatrix and Monte Carlo method may also be used to explore the theory space. We also discuss some basic known results for M>2.
11:00 AM
Question time
Question time
11:00 AM  11:30 AM
11:30 AM
Critical behavior near structural phase transitions

Amnon Aharony
(Tel Aviv University)
Critical behavior near structural phase transitions
Amnon Aharony
(Tel Aviv University)
11:30 AM  12:30 PM
SrTiO$_3$ and LaAlO$_3$ undergo structural phase transitions from a cubic trructure to lower symmetry structures. These transitions can be affected by uniaxial stress. Following original work by Alex Mueller, I have worked on these transitions since 1973. Many details depend on whether the 3component cubic problem is dominated by the isotropic or by the cubic fixed point. Most of the past literature assumed that the former dominates. Following discussions with Slava Rychkov, I realized that it is now believed that the latter wins. This talk will discuss the consequences of this change, and the experimental possibilities to test it.
12:30 PM
Question time
Question time
12:30 PM  1:00 PM
2:00 PM
The random field Ising model

Marco Picco
(LPTHE)
Silvio Franz
(LPTMS Orsay)
Emilio Trevisani
(École Polytechnique)
The random field Ising model
Marco Picco
(LPTHE)
Silvio Franz
(LPTMS Orsay)
Emilio Trevisani
(École Polytechnique)
2:00 PM  3:00 PM
3:00 PM
Question time
Question time
3:00 PM  3:30 PM
3:30 PM
The random field Ising model

Marco Picco
(LPTHE)
Emilio Trevisani
(École Polytechnique)
Silvio Franz
(LPTMS Orsay)
The random field Ising model
Marco Picco
(LPTHE)
Emilio Trevisani
(École Polytechnique)
Silvio Franz
(LPTMS Orsay)
3:30 PM  4:30 PM
4:30 PM
Question time
Question time
4:30 PM  5:00 PM
Saturday, May 22, 2021
Sunday, May 23, 2021
Monday, May 24, 2021
8:00 AM
Holiday
Holiday
8:00 AM  8:00 PM
Tuesday, May 25, 2021
10:00 AM
The crossing equations without positivity: tools and challenges

Marco Meineri
(CERN)
The crossing equations without positivity: tools and challenges
(4. Nonunitary bootstrap methods)
Marco Meineri
(CERN)
10:00 AM  11:00 AM
After briefly reviewing the role of positivity in the numerical bootstrap, I will focus on cases where positivity is lost. Gliozzi's truncation method will be discussed, together with some of the variations introduced in later works. I will illustrate some strengths and weaknesses of the method, via examples drawn from nonunitary CFTs and unitary CFTs with a boundary. If time permits, I will discuss some analytic consequences of the crossing equations in the presence of boundaries and defects.
11:00 AM
Question time
Question time
11:00 AM  11:30 AM
11:30 AM
The crossing equations without positivity: tools and challenges

Marco Meineri
(CERN)
The crossing equations without positivity: tools and challenges
Marco Meineri
(CERN)
11:30 AM  12:30 PM
12:30 PM
Question time
Question time
12:30 PM  1:00 PM
Wednesday, May 26, 2021
10:00 AM
Operator expansions, layer susceptibility and twopoint functions in BCFT

Mykola Shpot
(ICMP Lviv)
Operator expansions, layer susceptibility and twopoint functions in BCFT
Mykola Shpot
(ICMP Lviv)
10:00 AM  11:00 AM
We found that in boundary conformal field theories, there exists a onetoone correspondence between the boundary operator expansion of the twopoint correlation function and a power series expansion of the layer susceptibility. This general property allows a direct identification of the boundary spectrum and expansion coefficients from the layer susceptibility and opens a new way for efficient calculations of twopoint correlators in BCFTs. To show how it works we derive an explicit expression for the correlation function $\langle\phi_i \phi^i\rangle$ of the $O(n)$ model at the extraordinary transition in $4\varepsilon$ dimensional semiinfinite space to order $O(\varepsilon)$. The bulk operator product expansion of the twopoint function gives access to the spectrum of the bulk CFT. In our example, we obtain the averaged anomalous dimensions of scalar composite operators of the $O(n)$ model to order $O(\varepsilon^2)$. These agree with the known results both in $\varepsilon$ and large$n$ expansions.
11:00 AM
Question time
Question time
11:00 AM  11:30 AM
2:00 PM
Critical exponents from the Lorentzian inversion formula

Johan Henriksson
(University of Pisa)
Critical exponents from the Lorentzian inversion formula
(4. Nonunitary bootstrap methods)
Johan Henriksson
(University of Pisa)
2:00 PM  3:00 PM
The Lorentzian inversion formula is a powerful tool for understanding the dynamical data of conformal field theories, specifically it can be used to extract conformal data of spinning operators from singularities of the fourpoint function in Lorentzian signature. In this lecture I aim to ``demystify'' the inversion formula by giving a concrete and explicit application of it to the WilsonFisher fixedpoint in the $\epsilon$ expansion of $\phi^4$ theory (Ising CFT). I will also discuss how it can be used to study general $\phi^p$ theories near their upper critical dimensions, including the nonunitary case for odd $p$.
3:00 PM
Question time
Question time
3:00 PM  3:30 PM
Thursday, May 27, 2021
2:00 PM
How can the pertubative renormalization group help the bootstrap?

Kay Wiese
(LPENS)
How can the pertubative renormalization group help the bootstrap?
Kay Wiese
(LPENS)
2:00 PM  3:00 PM
3:00 PM
Question time
Question time
3:00 PM  3:30 PM
3:30 PM
Unitarity versus positivity

Connor Behan
(University of Oxford)
Unitarity versus positivity
(4. Nonunitary bootstrap methods)
Connor Behan
(University of Oxford)
3:30 PM  4:30 PM
The numerical bootstrap finds crossing symmetric fourpoint functions which have real OPE coefficients in their conformal block expansions. Some of these are fourpoint functions of unitary CFTs while others are part of nonunitary CFTs that simply require more correlators before we can see the nonunitarity. I will discuss this phenomenon in the context of 2D where the bounds are saturated by nonunitary CFTs which are exactly solvable; the generalized minimal models. The methods will be general enough to prove or disprove positivity when there is an extended chiral algebra as well.
4:30 PM
Question time
Question time
4:30 PM  5:00 PM
Friday, May 28, 2021
10:00 AM
Conformal Field Theories near an Edge

António Antunes
(University of Porto)
Conformal Field Theories near an Edge
António Antunes
(University of Porto)
10:00 AM  11:00 AM
Critical systems with a boundary are important for various phenomenological, theoretical and even experimental applications. In this case, the formalism of Boundary CFT and the Conformal Bootstrap has led to a firm theoretical framework to study this scenario. Less explored is the case of systems with two intersecting boundaries, which form a codimension 2 edge. I will review some old results by Cardy in the framework of the epsilon expansion, and present a recent Bootstrap formulation of this setup which defines this class of systems nonperturbatively. I will finish by pointing out some open problems that can potentially be addressed in the Bootstrap framework.
11:00 AM
Question time
Question time
11:00 AM  11:30 AM
2:00 PM
Extremal flows for the nonpositive bootstrap?

Miguel Fernandes Paulos
(Ecole Normale Superieure)
Extremal flows for the nonpositive bootstrap?
(4. Nonunitary bootstrap methods)
Miguel Fernandes Paulos
(Ecole Normale Superieure)
2:00 PM  3:00 PM
CFT correlators such as in the 3d Ising model, which saturate bootstrap bounds, must have a sparse spectrum of operators. We show that sparsity translates into tight constraints which allow us to bootstrap solutions to (truncated) crossing equations directly, bypassing the traditional numerical bootstrap approach based on optimization and positivity. In practice these constraints amount to a certain modification of the socalled Gliozzi method and they can in principle be also used to bootstrap nonunitary CFTs or solutions to crossing equations without positivity, as can be shown in concrete albeit simple examples. We discuss the prospects and difficulties for doing this more systematically using two approaches: 1. Flowing from unitary to nonunitary solutions by continuous changes in some parameter such as the N in O(N) or the spacetime dimension. 2. Directly solving multiple nonlinear equations in several variables.
3:00 PM
Question time
Question time
3:00 PM  3:30 PM