IPA director's welcome

• Denis ULLMO (Institut Pascal)

Introduction to the numerical conformal bootstrap (1. Status of the conformal bootstrap)

• Alessandro Vichi (University of Pisa)

Lecture 1: - Introduction - Conformal symmetry in $D\ge 2$ 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

Introduction to the numerical conformal bootstrap

• Alessandro Vichi (University of Pisa)

Introduction to the numerical conformal bootstrap

• Alessandro Vichi (University of Pisa)

Introduction to the numerical conformal bootstrap

• Alessandro Vichi (University of Pisa)

Numerical bootcamp

• Marten Reehorst (Institute des Hautes Études Scientifique)

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 state-of-the-art 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 #numerical-bootcamp Slack channel.

Non-relativistic conformal symmetries (2. Statistical physics targets)

• Malte Henkel (LPCT, Univ. Lorraine Nancy, France)

I shall try to give an introduction and physical motivation why to study the non-equilibrium 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 non-equilibirum statistical mechanics.

Applications of the numerical conformal bootstrap (1. Status of the conformal bootstrap)

• David Poland (Yale University)

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 large-scale bootstrap problems.

The navigator function : sailing through the infeasible sea in the conformal bootstrap (1. Status of the conformal bootstrap)

• Ning Su (University of Pisa)

Conformal bootstrap is shown to be very useful for studying the 3D Ising, Super-Ising, 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).

Critical geometry approach to three-dimensional percolation

• Alessandro Galvani (SISSA)

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 so-called 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 point-dependent scale for the system, which can be used to determine one-point and two-point 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 high-precision estimate of its anomalous dimension $\eta$.

Critical Ising model in varying dimension by conformal bootstrap (1. Status of the conformal bootstrap)

• Andrea Cappelli (INFN)

The single-correlator 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 low-lying states are obtained over the entire range of dimensions with up to four-decimal precision and then compared with several existing results. The conformal dimensions of leading-twist fields are also determined up to high spin, and their d-dependence 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 null-vector condition. (See the article [arXiv:1811.07751][1]) [1]: https://arxiv.org/abs/1811.07751

Numerical bootcamp

• Marten Reehorst (Institute des Hautes Études Scientifique)

Numerical bootcamp

• Marten Reehorst (Institute des Hautes Études Scientifique)

Analytically solving the Ising model in $2+ϵ$ dimensions

• Wenliang Li (Okinawa Institute of Science and Technology Graduate University)

The analytical studies of the $d>2$ Ising model are usually based on the $ϵ$ expansion around $d=4$. Since the $d=2$ Ising model is solvable, it would be interesting to deform the $d=2$ exact solution to $d=2+ϵ$ dimensions. Some strong coupling features may be seen more clearly. I will discuss some attempts using the analytic conformal bootstrap.

Random critical points: Anderson transitions, multifractality, and conformal symmetry (2. Statistical physics targets)

• Ilya Gruzberg (Ohio State University)

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 stat-mech 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 non-linear sigma models. Altland-Zirnbauer 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.

Random critical points: Anderson transitions, multifractality, and conformal symmetry

• Ilya Gruzberg (Ohio State University)

Upper critical dimension of the 3-state Potts model (2. Statistical physics targets)

• Shai Chester (Weizmann Institute)

The 3-state 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 non-perturbative evidence for this scenario from the numerical conformal bootstrap, which suggests that the critical Potts model disappears for d<3.

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)

Geometrical correlators in 2d CFTs: an introduction with some open problems (2. Statistical physics targets)

• Jacopo Viti (INFN Florence)

Four-point cluster connectivities in the 2d critical Q-state Potts model

• Yifei He (ENS Paris)

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 Fortuin-Kasteleyn formulation of the Q-state Potts model. In this talk I will consider the four-point 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 four-point connectivities. The talk is based on [2002.09071](https://arxiv.org/abs/2002.09071) and [2005.07258](https://arxiv.org/abs/2005.07258).

Numerical bootcamp

• Marten Reehorst (Institute des Hautes Études Scientifique)

Conformal field theory of the integer quantum Hall transition: a status report (2. Statistical physics targets)

• Martin Zirnbauer (University of Cologne)

Of the five 2D strong topological insulator Anderson transitions, all of which should be logarithmic conformal field theories of Wess-Zumino-Witten 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 non-critical 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.

The 3d Ising CFT spectrum

• Slava Rychkov (IHES & ENS)

More than a 100 scaling operators and their OPE coefficients have been extracted numerically in the 3d Ising CFT by David Simmons-Duffin 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)

Interchiral algebra from the affine Temperley-Lieb algebra

• Azat GAINUTDINOV (CNRS, Institut Denis Poisson, Université de Tours)

The interchiral algebra is an extension of the product of chiral and anti-chiral Virasoro algebras and appears naturally in the scaling limit analysis of certain statistical physics models where the interaction terms are of Temperley-Lieb type. The idea of this discussion is to see how the interchiral algebra generators show up in the scaling limit of the affine Temperley-Lieb operators.

Basics of symmetry in condensed matter physics

• Sylvain Ravy

Introduction to structural phase transitions (3. Structural phase transitions and other unitary problems)

• Pierre Toledano (Université de Picardie)

1. Basic concepts of the Landau theory. Critical behaviour. Symmetry of the order parameter. Secondary order parameters. Specificity of first-order 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.

Introduction to structural phase transitions

• Pierre Toledano (Université de Picardie)

1. Basic concepts of the Landau theory. Critical behaviour. Symmetry of the order parameter. Secondary order parameters. Specificity of first-order 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.

Introduction to structural phase transitions

• Pierre Toledano (Université de Picardie)

1. Basic concepts of the Landau theory. Critical behaviour. Symmetry of the order parameter. Secondary order parameters. Specificity of first-order 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.

Critical behavior near structural phase transitions (3. Structural phase transitions and other unitary problems)

• Amnon Aharony (Tel Aviv University)

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 3-component 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.

Critical behavior near structural phase transitions

• Amnon Aharony (Tel Aviv University)

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 3-component 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.

Conformal bootstrap studies of puzzles in critical phenomena (3. Structural phase transitions and other unitary problems)

• Andreas Stergiou (Los Alamos National Laboratory)

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 three-dimensional 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.

Four-point functions for non-unitary geometrical problems in two dimensions

• Hubert Saleur (IPhT Saclay)

RG Flows in Coupled Replica CFTs

• Hirohiko Shimada (NIT Tsuyama)

Consider M copies of the q-state Potts models and the O(n) models coupled through the bond-bond interaction. Non-trivial 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 (q-2) or (1-n) around the M-coupled Ising CFTs in 2d, where the coupling is marginal. In addition, the RG flow generated by the Zamolodchikov C-function extracted from the transfer matrix can capture non-perturbative multicritical fixed points at M=0. The S-matrix and Monte Carlo method may also be used to explore the theory space. We also discuss some basic known results for M>2.

Critical behavior near structural phase transitions

• Amnon Aharony (Tel Aviv University)

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 3-component 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.

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)

The crossing equations without positivity: tools and challenges (4. Non-unitary bootstrap methods)

• Marco Meineri (CERN)

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 non-unitary 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.

The crossing equations without positivity: tools and challenges

• Marco Meineri (CERN)

Operator expansions, layer susceptibility and two-point functions in BCFT

• Mykola Shpot (ICMP Lviv)

We found that in boundary conformal field theories, there exists a one-to-one correspondence between the boundary operator expansion of the two-point 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 two-point correlators in BCFTs. To show how it works we derive an explicit expression for the correlation function $⟨{\varphi }_i{\varphi }^i⟩$ of the $O(n)$ model at the extraordinary transition in $4-\epsilon$ dimensional semi-infinite space to order $O(\epsilon )$. The bulk operator product expansion of the two-point 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 $O({\epsilon }^2)$. These agree with the known results both in $\epsilon$ and large-$n$ expansions.

Critical exponents from the Lorentzian inversion formula (4. Non-unitary bootstrap methods)

• Johan Henriksson (University of Pisa)

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 four-point 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 Wilson--Fisher fixed-point in the $ϵ$ expansion of ${\varphi }^4$ theory (Ising CFT). I will also discuss how it can be used to study general ${\varphi }^p$ theories near their upper critical dimensions, including the non-unitary case for odd $p$.

How can the pertubative renormalization group help the bootstrap?

• Kay Wiese (LPENS)

Unitarity versus positivity (4. Non-unitary bootstrap methods)

• Connor Behan (University of Oxford)

The numerical bootstrap finds crossing symmetric four-point functions which have real OPE coefficients in their conformal block expansions. Some of these are four-point functions of unitary CFTs while others are part of non-unitary CFTs that simply require more correlators before we can see the non-unitarity. I will discuss this phenomenon in the context of 2D where the bounds are saturated by non-unitary 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.

Conformal Field Theories near an Edge

• António Antunes (University of Porto)

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 co-dimension 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 non-perturbatively. I will finish by pointing out some open problems that can potentially be addressed in the Bootstrap framework.

Extremal flows for the non-positive bootstrap? (4. Non-unitary bootstrap methods)

• Miguel Fernandes Paulos (Ecole Normale Superieure)

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 so-called Gliozzi method and they can in principle be also used to bootstrap non-unitary 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 non-unitary solutions by continuous changes in some parameter such as the N in O(N) or the spacetime dimension. 2. Directly solving multiple non-linear equations in several variables.