Seed Seminar of Mathematics and Physics
The conformal block expansion of correlation functions is a fundamental tool in the bootstrap program. Block computation can be related to a specific class of harmonic functions on the conformal group. We generalise this concept to the case of thermal conformal field theories and compute one-point blocks for spinning representations in three dimensions. Specifically, we derive a universal Casimir equation and solve it using recursion relations. Finally, we use the blocks in several examples to explore some features of OPE coefficients.
 
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Seed Seminar of Mathematics and Physics
Tensor network renormalization group (RG) is a powerful technique providing great control over the renormalization group flow both numerically and analytically. This talk will focus on the study of fixed points of tensor network RG maps in the context of classical lattice models (e.g., the Ising model). Studying fixed points allows one to retrieve information about phase transitions in a system, which motivates this research. I will first introduce basic tensor network concepts. Then, I will present some rigorous results about high- and low-temperature fixed points of lattice spin models (based on arXiv:2107.11464, 2210.06669, and 2401.04229). Finally, I will discuss the recent development that may grant access to the critical fixed points of lattice systems (based on arXiv:2408.10312).
 
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Seed Seminar of Mathematics and Physics
Anti-de Sitter space acts as an infra-red cutoff for asymptotically free theories, allowing interpolation between a weakly-coupled small-sized regime and a strongly-coupled flat-space regime. I will discuss this interpolation in the context of Yang-Mills theories in AdS from the perspective of boundary conformal theories and its implications for the confinement/deconfinement transition. We find indications that at the transition a singlet scalar operator becomes marginal, destabilizing the deconfined phase existing at a small size and leading to a confined phase that smoothly connects to flat space.
 
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Seed Seminar of Mathematics and Physics
Non-equilibrium dynamics of isolated quantum many-body systems plays an important role in contemporary theoretical physics research, from foundational questions on quantum statistical mechanics to the development of quantum technologies.I will provide a guided tour of a selection of results on thermalization and ergodicity breaking in quantum many-body dynamics, with an eye on mathematically rigorous studies, including (many) conjectures and (few) proven statements.  
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Seed Seminar of Mathematics and Physics
Our understanding of the behaviors appearing in statistical physics has been built firstly on the well-known Ising Model. Here, we will consider a disordered version of the one-dimensional Ising model: we will present and study the ferromagnetic Ising model on a line graph interacting with an external magnetic field, sampled from an i.i.d. distribution. We will be interested in the regime where the intensity of the disorder is fixed and the spin-spin interaction goes to infinity.
We will also introduce a continuous version of the model, which naturally arises from a weak disorder limit. For this continuous model, various quantities (such as the free energy) can be computed explicitely, thus yielding precise information on the typical configurations of the system.
The free energy of the discrete model can easily be expressed as the Lyapunov exponent of a random product of 2×2 matrices, which we estimate using Furstenberg’s theory: we will present recent results on the asymptotics of the free energy, in the regime we consider.
Furthermore, in both the discrete and the continuous models, we will caracterise the behaviour of the system at the level of configurations. In agreement with predictions in the physics literature, we will show that the configurations are typically close to one given configuration, determined by the environment (the external field), thus showing that the disorder is strongly relevant.
Our discussion will concern mainly the critical case, i.e., the case where the disorder is centered, but we may also address the non-critical case.
 
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Seed Seminar of Mathematics and Physics
Hydrodynamics is one of the oldest example of field theories, describing the long-range behaviour of many-body systems and its study remained mostly classical. Nevertheless, near zero temperature, quantum fluctuations grow in importance raising the natural question: is there a consistent quantum picture of a perfect fluid at zero temperature? In this talk, I will review the subtly inequivalent Lagrangians realizations of the perfect fluid through the lens of modern effective field theory (EFT). I will then discuss its particularity, naming the presence of an infinitely dimensional symmetry. As we will see, its main implication is the existence of transverse modes with vanishing dispersion relation at the classical level, and an infinitely degenerate spectrum and UV-IR mixing at the quantum level. This is based on a work with G. Cuomo, E. Firat, B. Henning and R. Rattazzi https://arxiv.org/abs/2412.10344.
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Seed Seminar of Mathematics and Physics
Introduced by Polyakov in the 1980s, Liouville quantum gravity (LQG) is in some sense the canonical model of a random fractal Riemannian surface. LQG can be defined as a path integral over fields corresponding to the Liouville action, or equivalently as a random metric measure space that turns out to describe the scaling limit of a host of two-dimensional discrete objects. In particular, certain discrete conformal embeddings of random planar maps converge to canonical (up to conformal reparametrization) embeddings of LQG surfaces into 2D Euclidean space. Though one might expect these metric embeddings to retain some vestige of conformality, in fact no embedding of an LQG surface into Rn can be quasisymmetric. This generalizes a result of Troscheit in the special case of sqrt(8/3)-LQG (corresponding to uniform random planar maps). Time permitting, I will also discuss future directions in the study of metric embeddability for LQG.
The zoom link is available by subscribing to the mailing list: sympa@listes.math.cnrs.fr
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Seed Seminar of Mathematics and Physics
On large classes of closed even-dimensional Riemannian manifolds M, we construct and study the Copolyharmonic Gaussian Field, i.e. a conformally invariant log-correlated Gaussian field of distributions on M. This random field is defined as the unique centered Gaussian field with covariance kernel given as the resolvent kernel of Graham—Jenne—Mason—Sparling (GJMS) operators of maximal order. The corresponding Gaussian Multiplicative Chaos is a generalization to the 2m-dimensional case of the celebrated Liouville Quantum Gravity measure in dimension two. We study the associated Liouville Brownian motion and random GJMS operator, the higher-dimensional analogues of the 2d Liouville Brownian Motion and of the random Laplacian. Finally, we study the Polyakov–Liouville measure on the space of distributions on M induced by the copolyharmonic Gaussian field, providing explicit conditions for its finiteness and computing the conformal anomaly.J. London Math. Soc. (2) 2024:110, 1-80, joint work with Ronan Herry, Eva Kopfer, Karl-Theodor Sturm.
The zoom link is available by subscribing to the mailing list: sympa@listes.math.cnrs.fr
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Seed Seminar of Mathematics and Physics
I will describe a newly introduced toolbox that connects two areas of Probability Theory: Schramm-Loewner Evolutions (SLE) and Random Matrix Theory. This machinery opens new avenues of research that allow the use of techniques from one field to another. One aspect of this research direction is centered in an interacting particle systems model, namely the Dyson Brownian motion. In the first part of the talk, I will introduce basic ideas of SLE theory, then I will describe the connection with Random Matrix Theory via a first application of our method. I will finish the talk with some open problems that emerge using this newly introduced toolbox. This is a joint work with A. Campbell and K. Luh.
 
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Seed Seminar of Mathematics and Physics
 
We study the volume of rigid loop-O(n) quadrangulations with a boundary of length 2p in the critical non-generic regime. We prove that, as the half-perimeter p goes to infinity, the volume scales in distribution to an explicit random variable. This limiting random variable is described in terms of the multiplicative cascades of Chen, Curien and Maillard, or alternatively (in the dilute case) as the law of the area of a suitable unit-boundary quantum disc, as determined by Ang and Gwynne. Our arguments go through a classification of the map into several regions, where we rule out the contribution of bad regions to be left with a tractable portion of the map. One key observable for this classification is a Markov chain which explores the nested loops around a size-biased vertex pick in the map, making explicit the spinal structure of the discrete multiplicative cascade. This talk is based on joint work with Élie Aïdékon and XingJian Hu (Fudan University).
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Seed Seminar of Mathematics and Physics
In this talk, I will present JT gravity, a model of two-dimensional quantum gravity on constant negatively curved spacetimes,  as a model of random hyperbolic surfaces. By studying the generating function of volumes of random hyperbolic surfaces with defects, i.e. weighted geodesic boundaries, we explore critical regimes where the surfaces develop macroscopic holes. This is reminiscent of the O(N) model for random maps. We analyse the impact of this critical behavior on the density of states of the theory at the boundary, and we present a family of models that interpolate between systems with $sqrt{E}$ and E3/2, which are commonly found in models of JT Gravity coupled to dynamical end-of-the-world and FZZT branes. 
 
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Seed Seminar of Mathematics and Physics
Loop models are a class of statistical lattice models whose correlation functions can be interpreted geometrically, as sums over configurations of non-intersecting loops.
In the critical limit, the observables of these models are described by a Conformal Field Theory (CFT), which is believed to be exactly solvable.
In this talk, we will shortly review what are loop models and present recent results showing that their correlation functions are related to combinatorial objects called combinatorial maps. Then, we will relate the counting of certain maps on the torus to particular classes of maps on the sphere, and explain the CFT interpretation of this mapping.
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