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Léonard Ferdinand: Stochastic quantization for a tensorial non-local field theory

14th June, 2023

I will introduce quartic melonic tensor field theories, a class of field theories built using a non-local quartic interaction term. These resemble the more well-known φd4 models but behave differently with regards to power-counting and the structure of their divergences. In particular, these models are conjectured to be non-trivial in their critical dimension, in contrast with φ44. I will then report on recent joint work with Ajay Chandra where we use stochastic analysis methods to construct the φ24 and φ34 analogues of these models.

Vincent Rivasseau: Random Tensors, Loop Vertex Representation and Cumulants

14th June, 2023

First of all, I briefly discuss the tensor track, a particular approach to quantum gravity based on random tensors. Then I review the Loop Vertex Representation (LVR), which combines combinatorial tools such as the BKAR formula, selective Gaussian integration, and the Fuss-Catalan generating function. In the last part I apply this LVR to obtain convergent expansions for the cumulants of matrix models.

Sabine Harribey: Extraordinary Interfaces and Boundaries in (4 − ε)-dimensional O(N) models

14th June, 2023

The critical O(N) models are one of the most thoroughly studied classes of conformal field theories (CFTs) in three dimensions. Indeed, there exists a variety of approximation methods applicable to them, such as epsilon expansion, large N expansion or conformal bootstrap. It is then of interest to study the 3-dimensional O(N) models with interfaces (co-dimension 1 defects), as well as on spaces with boundaries. In particular, we are interested in "extraordinary" critical interfaces and boundaries which break the O(N) symmetry to O(N −1). Such models were studied recently with a D-dimensional bulk and surface defects quadratic in the fields. In this talk, we will adopt a different approach and study defects that are always of co-dimension 1. More precisely, I will present the renormalization group analysis for a quartic O(N) model in 4 − ε dimensions with cubic interactions on an interface. For sufficiently large N, we find stable IR fixed points with purely imaginary cubic couplings. I will also review the special case N = 1 corresponding to a boundary Yang-Lee model.

Alexander Hock: Non-perturbative results of a just-renormalisable model

14th June, 2023

We consider the scalar φ4 model on the 4-dimensional noncommutative Moyal space. This is the critical dimension where the model becomes just-renormalisable. At the self-dual point, this model breaks down to a matrix model, where the noncommutativity of the underlying space is related to the size N of the matrix. Assuming a formal expansion in 1/N, the Dyson-Schwinger equations (after applying Ward identities) decouple which leads to (non-)linear integral equations at each order in 1/N. We will present and discuss from different perspectives the leading order (genus g = 0) result of the 2-point function, which is a resummation of infinitely many Feynman diagrams. We will also discussion the Hopf-algebraic renormalision of this model in the sense of Connes-Kreimer, which has the same complexitiy as an ordinary just-renormalisable QFT.

Ilya Chevyrev: Invariant measure and universality of the 2D Yang-Mills Langevin dynamic, II

13th June, 2023

In this talk, I will present a recent work on the invariance of the 2D Yang-Mills measure for its Langevin dynamic. The Langevin dynamic both in 2D and 3D had previously been constructed in joint work with Chandra-Hairer-Shen, but it was an open problem to show the existence of an invariant measure even in 2D. In establishing this invariance, we follow Bourgain’s invariant measure argument by taking lattice approximations, but with several twists. An important one, which I will focus on, is that the approximating invariant measures require gauge-fixing, which we achieve by developing a rough version of Uhlenbeck compactness combined with rough path estimates of random walks. I will also present several corollaries of our main result, including a representation of the YM measure as a perturbation of the Gaussian free field, and a new universality result for its discrete approximations.

Hao Shen: Invariant measure and universality of the 2D Yang-Mills Langevin dynamic, I

13th June, 2023

In an earlier work with Chandra, Chevyrev and Hairer, we constructed the local solution to the stochastic Yang-Mills equation on 2D torus, which was shown to have gauge covariance property and thus induces a Markov process on a singular space of gauge equivalent classes. In this talk, we discuss a more recent work with Chevyrev, where we consider the Langevin dynamics of a large class of lattice gauge theories on 2D torus, and prove that these discrete dynamics all converge to the same limiting dynamic. A novel step in the argument is a geometric way to identify the limit using Wilson loops. This universality of the dynamics is crucial for obtaining a sequence of important results for 2D Yang-Mills, including for instance the invariance of the 2D Yang-Mills measure for its Langevin dynamic, which will be discussed by Ilya Chevyrev.

Nicolas Perkowski: Energy solutions and generators for singular SPDEs

13th June, 2023

I will discuss how to use tools from Gaussian analysis and operator semigroups together with some commutator estimates to construct Markov semigroups for some singular SPDEs. This yields in particular uniqueness for Goncalves-Jara-Gubinelli type energy solutions. The method applies to some critical equations and, in finite dimensions, even for some supercritical equations. In infinite dimensions we get Markov semigroups for supercritical equations but we lack a uniqueness result for supercritical energy solutions in infinite dimensions. The main SPDE examples where this works are of Burgers type: quadratic, divergence-free nonlinearity and Gaussian quasi-invariant measure.

Martin Hairer: The role of symmetry in renormalisation

13th June, 2023

There are several interesting situations where the solutions to singular SPDEs exhibit a symmetry at a formal level that could in principle be broken by the renormalization procedure required to define them. We’ll discuss a relatively simple argument showing that, in many cases, the renormalization can be chosen in such a way that the symmetry does indeed hold and we’ll apply it to the stochastic quantization of the 3D Yang-Mills theory.

Nikos Zygouras: SPDEs at the critical dimension

13th June, 2023

I will make an overview of the progress on treating SPDEs at the critical dimension, the current status and further challenges. Examples will include stochastic heat equations and a more recent Allen-Cahn.

Yvain Bruned: Convergence of the renormalised model for the generalised KPZ equation via preparation maps

13th June, 2023

In this talk, we will present the convergence of the renormalised model of the generalised KPZ equation via local transformations that are governed by preparation maps. The main idea is an extension of a result on the convergence of a class Feynman diagrams given by Hairer and Quastel. With this extension, one is able to perform local transformations that make appear the renormalisation given for a model defined recursively via preparation maps in the context of Regularity Structures. This approach works both in the discrete and continuous settings and could lead to a general convergence theorem.

Margherita Disertori: The non-linear supersymmetric hyperbolic sigma model on a complete graph with hierarchical interactions

13th June, 2023

Initially introduced as toy model for quantum diffusion, the non-linear supersymmetric hyperbolic sigma model has been attracting much attention in recent years due to its connection to history dependent stochastic processes. In this talk I will present a version of the model with hierarchical interactions. The internal symmetries of the model allow to perform some block-spin renormalization steps exactly. The resulting effective action has renormalized coefficients but no additional interaction terms. I will show the corresponding derivation and some applications.

Luca Fresta: The forward-backward SDE for subcritical Euclidean fermionic field theories

12th June, 2023

In this talk, I will describe a synergy between the renormalization group (RG) in the form of Polchinski's equation and the stochastic quantisation in the form of a forward-backward stochastic differential equation (FBSDE). This approach can be used for constructing subcritical Grassmann Gibbsian measures and is based on controlling the solution of the FBSDE by means of a flow equation with respect to a scale parameter. However, unlike the standard RG approach, we only need to solve Polchinski’s equation in an approximate way, resulting in a great simplification of the analysis.

Xiangchan Zhu: A class of singular SPDEs via convex integration

12th June, 2023

In this talk I will talk about our recent work on a class of singular SPDEs via convex integration method. In particular, we establish global-in-time existence and non-uniqueness of probabilistically strong solutions to the three dimensional Navier–Stokes system driven by space-time white noise. In this setting, solutions are expected to have space regularity at most −1/2 − κ for any κ > 0. Consequently, the convective term is ill-defined analytically and probabilistic renormalization is required. Up to now, only local well-posedness has been known. With the help of paracontrolled calculus we decompose the system in a way which makes it amenable to convex integration. By a careful analysis of the regularity of each term, we develop an iterative procedure which yields global non-unique probabilistically strong paracontrolled solutions. Our result applies to any divergence free initial condition in L2 ∪ B−1+κ∞,∞, κ > 0, and implies also non-uniqueness in law. Finally I will show the existence, non-uniqueness, non-Gaussianity and non-unique ergodicity for singular quasi geostrophic equation in the critical and supercritical regime.

Francesco De Vecchi: Non-commutative Lp-spaces and Grassmann stochastic analysis

12th June, 2023

We introduce a theory of non-commutative Lp spaces suitable for non-commutative probability in a non-tracial setting and use it to develop stochastic analysis of Grassmann-valued processes, including martingale inequalities, stochastic integrals with respect to Grassmann Itô processes, Girsanov’s formula and a weak formulation of Grassmann SDEs. We apply this new setting to the construction of several unbounded random variables, including a Grassmann analog of the φ24 Euclidean QFT in a bounded region.

Răzvan Gurău: The small-N series in the zero-dimensional O(N) model

12th June, 2023

I will discuss the application of some constructive field theory inspired techniques to the study of resurgence in the 0-dimensional O(N) model and its small N limit. This is the first step in the program of applying such techniques to fully fledged higher-dimensional quantum field theory.

Rongchan Zhu: Large N limit and 1/N expansion of the observables for O(N) linear sigma model

12th June, 2023

In this talk, we talk about the large N problems for the Wick renormalized linear sigma model, i.e. N-component φ4 model, in two spatial dimensions, using stochastic quantization methods and Dyson–Schwinger equations. We identify the large N limiting law of a collection of Wick renormalized O(N) invariant observables. In particular, under a suitable scaling, the quadratic observables converge in the large N limit to a mean-zero (singular) Gaussian field Q with an explicit covariance; and the observables which are renormalized powers of order 2n converge in the large N limit to suitably renormalized nth powers of Q. Furthermore, we derive the 1/N expansion for the k-point functions of the quadratic observables by employing a graph representation and carefully analysing the order of each graph from Dyson-Schwinger equations. Finally, we obtain the next order stationary dynamics.

Nikolay Barashkov: Gluing and Markov property for φ34 on Cylinders

12th June, 2023

The Markov property is an important property of Random Fields that allows to use them to construct a Quantum Field Theory. It is closely connected to Segal's Axioms, which describe how to assemble Random Fields on a bigger manifold from Fields on smaller pieces. In this talk I will describe how to establish these properties for the φ34 model on cylinders.

Frans Pretorius: Insights into the rotating black hole interior from numerical solutions

31st May, 2023

The last couple of decades has seen tremendous progress in numerical solution of the Einstein field equations for regions of spacetime exterior to black hole horizons. For reasons I will briefly discuss, the corresponding advances have not been of much help for the problem of the black hole interior, in particular for black holes formed from gravitational collapse outside of spherical symmetry.

We are thus still left having to appeal to simplified scenarios to try to gain some insight into this problem. In that regard, I will present results form numerical studies of rotating black holes formed from scalar field collapse in asymptotically Anti de-Sitter spacetime, in 2+1 dimensions, with circular symmetry imposed. Despite the simplicity of this model, the interior exhibits rich phenomenology that I will describe in the talk.

Reinhard Werner: Inductive limits of quantum systems, equilibrium states and dynamics

26th May, 2023

Several limits of quantum theory can be understood from a simple framework in which some of the basic features of these limits can be established by abstract general results. Often the limit may have a very different structure from its approximants, so one cannot simply let some parameter go to a limiting value. We focus on the unified definition of the limit theory with its states and observables, its dynamics and its equilibrium states. Examples that will be mentioned are (1) the mean field with or without tagged particles, (2) The classical limit (ħ → 0), (3) the limit of infinite lattice systems, (4) Some continuum limits/renormalization theories.