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School
The four courses that make up the school will each have 6 lectures and around 3 hours of tutorial time. The school will take place in Durham from 17 to 21 July 2023.
Schedule
Current-Current Deformations
In physics we often try to understand complicated models as perturbations (or deformations) of simpler ones. In this course we will study the consequences of perturbing a 2-dimensional field theory with perturbations that are controlled by the symmetries of the original model itself. This strategy generates the so-called "current-current deformations" that we will analyse in relation to both the conformal invariance and the integrability of the theory, assuming that these two properties are present in the first place. We will discuss a geometrical formulation of these deformations in terms of the O(d,d) group and, if time permits, we will explore generalisations of the construction such as higher derivative corrections to the actions of the deformed models.
Prerequisites: basic knowledge on 2-dimensional integrable models and conformal field theories that will be covered in the pre-school courses.
Riccardo Borsato (IGFAE Santiago de Compostela) works on integrability in AdS/CFT and integrable deformations. He obtained his PhD from Utrecht University in 2015 and later continued his research at Imperial College London, NORDITA and finally Santiago de Compostela, where he holds a stable position.
ODE/IM Correspondence
The course will cover some aspects of the ODE/IM correspondence.
Prerequisites: Some familiarity with conformal field theory will be assumed.
Clare Dunning (University of Kent) works on classical and quantum integrable models. She is a Reader in Applied Mathematics at the University of Kent.
Integrability in Conformal Field Theory and Sigma Models
Non-linear sigma models in 1+1 dimensions are some of the most interesting QFTs, both from the point of view of the mathematical physics involved and especially due to their potential applications in High Energy/String Theory and Condensed Matter Physics. In this lecture series we will discuss one of the best studied models - the so-called 2D Euclidean black hole CFT. The solution of the spectral problem in finite volume will be explained, pointing out the presence of the integrable structures along the way. We will also discuss the density of states of the continuous component in the spectrum, which reproduces the modular invariant partition function proposed in the work of Maldacena, Ooguri and Son and its development in Hanany, Prezas and Troost. Remarkably, the density of states was found by considering a different problem, namely, the scaling limit of a certain 1D integrable, critical spin chain.
Prerequisites: The course will assume only minimal knowledge from participants: basic concepts of QFT that can be found, e.g., in part I of Peskin and Schroder; and some familiarity with 1+1 dimensional Conformal Field Theory (chapters 5 and 6 of the CFT textbook by Francesco, Mathieu and Senechal).
Gleb Kotousov's (Leibniz University Hannover) research interests are evenly divided between the study of first principles quantization of integrable 1+1 dimensional sigma models and the scaling limit of integrable critical 1D spin chains. He obtained his PhD at the end of 2019 jointly from Rutgers University and the Australian National University. He is currently based in Hannover.
Affine Gaudin Models
In this course, we will give an introduction to Affine Gaudin models, which provide a general framework for the systematic construction and study of a large class of integrable two-dimensional field theories. A key role will be played by Kac-Moody currents, which are fields satisfying a particular Poisson bracket. After reviewing these notions, we will discuss in detail the construction of Affine Gaudin models in the language of Hamiltonian field theories. Special emphasis will be placed on their symmetries and conserved charges, including the construction of infinite families of local and non-local Poisson-commuting charges in terms of Kac-Moody currents. Moreover, we will study explicit examples of affine Gaudin models, making the link with the realm of integrable sigma-models. Finally, we will mention various perspectives concerning these theories, including the question of their quantisation.
Prerequisites: We will assume some familiarity with classical integrability in two-dimensional field theories, including the notions of Lax connections, r-matrices and tensorial notations for Poisson brackets. Prior knowledge of the example of the Principal Chiral Model would be helpful but is not a necessary prerequisite. These subjects are discussed in many classical textbooks and reviews available online, as for instance the lecture notes by S. Driezen (see also a video summary here). They will moreover be presented in the pre-school.
Sylvain Lacroix (ETH Zurich) works on integrable two-dimensional field theories and in particular integrable sigma-models. He obtained his PhD in 2018 at the ENS de Lyon and the University of Hertfordshire and continued with a postdoctoral position at the University of Hamburg. He is currently a postdoctoral fellow at ETH Zürich.
TTbar Deformations
This course will provide a brief introduction to the so-called TTbar deformations. This is a universal deformation which can be constructed for any Poincare'-invariant two-dimensional QFT. Despite being irrelevant in the sense of the renormalisation group, it has many intriguing features: it preserves (most) symmetries, including integrability, and it has a deceptively simple action on the spectrum and S-matrix of the original model. We will present these features and discuss some of the possible generalisations of TTbar deformations.
Alessandro Sfondrini (University of Padova) obtained his PhD with honours from Utrecht University in 2014. He has been a Marie-Curie Fellow at Humboldt University in Berlin, a senior researcher at ETH Zurich, a member of the IAS in Princeton, and he is currently an Associate Professor at Padova University. He has worked on applying integrability techniques to the computation of the spectrum and correlation functions in string theory, and on integrable deformations of two-dimensional QFTs and lattice models.
Pre-School
In the week preceding the school there will be two online courses. Each of these will have 2 to 3 lectures and their aim is to provide background material for the school.
Schedule
Monday 10 July, 13:00-15:00 - Some Basics on Classical Integrability (Marc Magro)
Tuesday 11 July 13:00-15:00 - Conformal Field Theory (Roberto Volpato)
Wednesday 12 July 13:00-15:00 - Conformal Field Theory (Roberto Volpato)
Thursday 13 July 13:00-15:00 - Some Basics on Classical Integrability (Marc Magro)
The lectures will be automatically recorded and made available to participants of the school. If you or you know of someone who would like to attend the pre-school, but who is not attending the school, please ask them to contact one of the organisers.
Some Basics on Classical Integrability
Aspects which will be reviewed include: Lax connection, monodromy matrix, tensorial notation, Poisson brackets and classical r-matrices. Examples will include the sine-Gordon theory and the Principal Chiral Model.
Marc Magro (ENS Lyon) studies integrable sigma-models at the classical level. He is Assistant Professor at Ecole Normale Supérieure de Lyon.
Conformal Field Theory
These lectures are a short introduction to the basic principles of conformal field theory (CFT) in two dimensions. We will start with a description of conformal invariance in two dimensions, of the Virasoro algebra, and its representations. We will then study how symmetries constrain the correlation functions in CFTs and discuss the operator product expansion, the radial quantization and the state-operator correspondence. The general theory will be then applied to some simple examples, such as the free boson. If time permits, we will discuss the properties of CFT on a torus, and some more complicated examples of CFT, such as the Wess-Zumino-Witten models.
Prerequisites: We assume familiarity with basic notions in quantum field theory, as can be found in any textbooks or lecture notes on the subject (roughly, the content of Chapter 2 in "Conformal Field Theory", by Di Francesco, Mathieu, Senechal).
Roberto Volpato (University of Padua) got his PhD from the University of Padova in 2008. After postdoctoral positions at ETH in Zurich, at the Albert Einstein Institute in Potsdam, and at SLAC/Stanford University, in 2016 he moved to Padova. He works on two dimensional conformal field theory and string theory.
