This webinar series is aimed at people interested in analytical aspects of quantum field theory, probability theory, (stochastic) PDE, quantum statistical mechanics, or related topics. It will be held on a monthly basis, see below for the schedule. If you have a suggestion for a speaker or an interesting topic (with several potential speakers), please contact one of the organizers.

I will introduce the topic of classification of short-range entangled quantum states. Physically speaking, these states are ground states of gapped Hamiltonians without any intrinsic topological order (ie. no anyonic excitations, for example), but one can define them in a neat abstract way that avoids such elaborate concepts.
I will try to sketch some open problems, and some results. In particular on the classification of pumping processes of such states, which can be viewed as a generalization of Thouless pumps.

The macroscopic or mesoscopic dynamics of many systems interacting with a random or chaotic environment can be described in terms of singular (i.e. classically ill-posed) stochastic partial differential equations. Typically, such stochastic PDEs depend only on a few parameters and govern the large-scale behavior of a huge number of different microscopic systems. This property is called universality. In the talk, I will discuss the proof of the universality of the macroscopic scaling limit of solutions of a class of parabolic stochastic PDEs with fractional Laplacian, additive noise and polynomial non-linearity. I consider the so-called weakly non-linear regime and not necessarily Gaussian noises which are stationary, centered, sufficiently regular and satisfy some integrability and mixing conditions. The result applies to situations when the singular stochastic PDE obtained in the scaling limit is close to critical and extends some of the existing universality results about the continuous interface growth models and the phase coexistence models whose large scale behavior is governed by the KPZ equation and the dynamical $\Phi^4_3$ model, respectively. The proof uses a novel approach to singular stochastic PDEs based on the renormalization group flow equation. A nice feature of the method is that it covers the full sub-critical (i.e. super-renormalizable) regime, does not use any diagrammatic representation and avoids all combinatorial problems. Based on arXiv:2109.11380.

The XY and the Villain models are models which exhibit the celebrated Kosterlitz-Thouless phase transitions in two dimensions. The spin wave conjecture, originally proposed by Dyson and by Mermin and Wagner, predicts that at low temperature, spin correlations of these models are closely related to Gaussian free fields. I will review the historical background and discuss some recent progress on this conjecture in d>=3. Based on the joint work with Paul Dario (CNRS). Webinar links and passwords and program updates will be distributed via email. If you would like to be added to the mailing list, please sign up for the mailing list at https://lists.uni-leipzig.de/mailman/listinfo/aqfp_announcements.

In this talk I will review recent progress on describing symmetries and renormalisation using the C*-algebraic framework proposed by Buchholz and Fredenhagen in their paper from 2019. The recent progress involves the formulation of renormalisation group and the anomalous unitary quantum Noether theorem. This is based on a join paper with Brunetti, Duetsch and Fredenhagen.Webinar links and passwords and program updates will be distributed via email. If you would like to be added to the mailing list, please sign up for the mailing list at https://lists.uni-leipzig.de/mailman/listinfo/aqfp_announcements.

In this talk, we prove the invariance of the Gibbs measure for the three-dimensional cubic nonlinear wave equation, which is also known as the hyperbolic Φ4 3-model. This result is the hyperbolic counterpart to seminal works on the parabolic Φ4 3-model by Hairer ’14 and Hairer-Matetski ’18. In the first half of this talk, we illustrate Gibbs measures in the context of Hamiltonian ODEs, which serve as toy-models. We also connect our theorem with classical and recent developments in constructive QFT, dispersive PDEs, and stochastic PDEs. In the second half of this talk, we give a non-technical overview of the proof. As part of this overview, we first introduce a caloric representation of the Gibbs measure, which leads to an interplay of both parabolic and hyperbolic theories. Then, we briefly discuss the local dynamics of the cubic nonlinear wave equation, focusing on a hidden cancellation between sextic stochastic objects. This is joint work with Y. Deng, A. Nahmod, and H. Yue.

In this talk I will introduce the idea of stochastic quantization from a mathematical perspective, as a tool to analyze rigorously Euclidean quantum fields. I will show that there are several different "stochastic quantizations” and I will attempt to exemplify common structures which take the form of a stochastic analysis of Euclidean quantum fields. Webinar links and passwords and program updates will be distributed via email. If you would like to be added to the mailing list, please sign up for the mailing list at https://lists.uni-leipzig.de/mailman/listinfo/aqfp_announcements.

In this talk I will present a Hamiltonian approach to the scaling limit of the Ising anyon chain, a 1+1-dimensional instance of the classical 2-dimensional Ising model. The scaling limit is constructed using an operator algebraic formulation of the Wilson-Kadanoff renormalization group. At criticality, conformal symmetry is recovered by showing the convergence of the Koo-Saleur formula, approximating the Virasoro generators. If time permits, I will comment on applications to the quantum simulation of conformal field theories.
This is joint work with Tobias J. Osborne and Daniela Cadamuro, and based on previous work with Vincenzo Morinelli, Gerardo Morsella and Yoh Tanimoto.

One of the goals of constructive Quantum Field Theory (QFT) is to provide a convergent algorithm for computing a consistent set of Euclidean correlation functions starting from a given bare action and, next, to reconstruct the corresponding real-time correlations via analytic continuation in the time variable. This program proved successful for constructing several low dimensional toy QFT models but results in 3 and 4 dimensions are still scarce. Until now, not even a consistent construction of infrared QED4 with small electron mass at all orders in renormalized perturbation theory was available, unless a loop regularization scheme was employed or a number of non-gauge-invariant counter-terms were included in the bare action.
In this talk I will describe such a consistent construction, at all orders in renormalized perturbation theory, in a lattice gauge theory model of QED4 with massless electron and no other counter-term than the one for the electron mass. We also prove that, in the presence of an infrared (IR) cutoff on the photon propagator, the model is non-perturbatively well-defined, provided the electron charge is sufficiently small (a priori, non-uniformly in the IR cutoff).
The proof is based on a Wilsonian Renormalization Group (RG) scheme and uses ideas developed in the last decade in the context of lattice gauge theory models of graphene and Weyl semimetals. In particular, we use Ward Identities at each RG step to control the flow of the effective couplings, including the non-gauge-invariant ones produced at intermediate steps by the multiscale procedure, and prove their infrared asymptotic freedom. Time permitting, I will comment on the perspectives opened by this and related works on the full construction of infrared QED4, on the non-perturbative computation of the chiral anomaly and on the spontaneous emergence of Lorentz symmetry. Joint work with Marco Falconi and Vieri Mastropietro.

In this talk I will give an overview over recent results on mean-field spin-glass models with a transversal magnetic field.
For such models both thermodynamic quantities such as the free energy and its fluctuations, as well as spectral and localization properties of eigenvectors are of interest to a diverse list of communities. A full mathematical analysis of properties is completed for the simplest, yet ubiquitous quantum random energy model.
For models with a more complicated (classical) correlation structure such as the Sherrington-Kirkpatrick model, a more qualitative analysis proves the existence of a phase transition.

We consider SPT-phases with on-site finite group G symmetry for two-dimensional Fermion systems.We derive an invariant of the classification.Webinar links and passwords and program updates will be distributed via email. If you would like to be added to the mailing list, please sign up for the mailing list at https://lists.uni-leipzig.de/mailman/listinfo/aqfp_announcements.

We will briefly review Wilson-Kadanoff type renormalization group (RG) maps for Ising spin systems and the lack of progress in proving that there is a non-trivial fixed point for these maps. (These maps are also known as real-space RG transformations.) The Ising model can be written as a tensor network, and RG maps can be defined in the tensor network formalism. Numerical studies of such RG maps have been quite successful at reproducing the known critical behavior in two dimensions. In joint work with Slava Rychkov we proved that in two dimensions for a particular tensor network RG map the high temperature fixed point is locally stable, i.e., there is a neighborhood of the high temperature fixed point such that for an initial tensor in this neighborhood, the iterations of the RG map converge to the high temperature fixed point. We hope that this is a modest first step towards proving the existence of a non-trivial fixed point for a tensor network RG map which would correspond to the critical point of the Ising model. Webinar links and passwords and program updates will be distributed via email. If you would like to be added to the mailing list, please sign up for the mailing list at https://lists.uni-leipzig.de/mailman/listinfo/aqfp_announcements.

I will talk about some recent progress on the problem of constructing 3D Euclidean Yang-Mills theories. This is based on joint work with Sky Cao. Webinar links and passwords and program updates will be distributed via email. If you would like to be added to the mailing list, please sign up for the mailing list at https://lists.uni-leipzig.de/mailman/listinfo/aqfp_announcements.

In the context of the AdS/CFT correspondence, in Euclidean signature, an important basic fact is the bijection between conformal transformations of the boundary and hyperbolic isometries of the bulk. An infinite regular tree with the graph distance can be seen as a quintessential bare-bones version of a hyperbolic space. It turns out there is a natural way to define analogues of conformal maps on the boundary of such a tree and, quite miraculously, these are in bijection with tree isometries. Moreover, a Euclidean QFT on this boundary is the same as a hierarchical model as considered by Dyson in his study of the long-range Ising model and by Wilson when he introduced the approximate renormalization group recursion. I will try to give a pedagogical introduction to this circle of ideas, and I will discuss a particular model where there is hope to be able to prove conformal invariance from first principles via a rigorous nonperturbative renormalization group approach. Webinar links and passwords and program updates will be distributed via email. If you would like to be added to the mailing list, please sign up for the mailing list at https://lists.uni-leipzig.de/mailman/listinfo/aqfp_announcements.

One usually considers wave equations as evolution equations, i.e. imposes initial data and solves them. Equivalently, one can consider the forward and backward solution operators for the wave equation; these solve an equation $Lu=f$, for say $f$ compactly supported, by demanding that $u$ is supported at points which are reachable by forward, respectively backward, time-like or light-like curves. This property corresponds to causality. But it has been known for a long time that in certain settings, such as Minkowski space, there are other ways of solving wave equations, namely the Feynman and anti-Feynman solution operators (propagators). I will explain a general setup in which all of these propagators are inverses of the wave operator on appropriate function spaces, and also mention positivity properties, and the connection to spectral and scattering theory in Riemannian settings, self-adjointness, as well as to the classical parametrix construction of Duistermaat and Hormander.

Gibbs measures on spaces of functions or distributions play an important role in various contexts in mathematical physics. They can, for example, be viewed as continuous counterparts of classical spin models such as the Ising model, they are an important stepping stone in the rigorous construction of Quantum Field Theories, and they are invariant under the flow of certain dispersive PDEs, permitting to develop a solution theory with random initial data, well below the deterministic regularity threshold. These measures have been constructed and studied, at least since the 60s, but over the last few years there has been renewed interest, partially due to new methods in stochastic analysis, including Hairer’s theory of regularity structures and Gubinelli-Imkeller Perkowski’s theory of paracontrolled distributions. In this talk I will present two independent but complementary results that can be obtained with these new techniques. I will first show how to obtain estimates on samples from of the Euclidean $\phi^4_3$ measure, based on SPDE methods. I will then discuss a new method to show the emergence of phase transitions in the phi^4_3 theory. This is based on joint works with A. Chandra, A. Moinat https://arxiv.org/abs/1910.13854 and A. Chandra, T. Gunaratnam https://arxiv.org/abs/2006.15933 />

I will discuss the quantum error correction (QEC) approach to the AdS/CFT correspondence from an algebraic point of view. I will study exact QEC codes as models of AdS/CFT and connect these models to Longo-Rehren subnets. I do this by proving the existence of a consistent assignment of conditional expectations acting on the boundary theory algebras. I will also discuss shortcomings of these exact codes that will hopefully be fixed by introducing small errors. Webinar links and passwords and program updates will be distributed via email. If you would like to be added to the mailing list, please sign up for the mailing list at https://lists.uni-leipzig.de/mailman/listinfo/aqfp_announcements.

A solution of the Klein-Gordon equation can be viewed as a signal carried by a classical wave packet, or as the wave function of a quantum particle, or as a coherent state in Quantum Field Theory. Our recent work concerns the definition, computation and interpretation of the local entropy of this object and its relation to quantum energy inequalities. The Operator Algebraic approach, in particular the Tomita-Takesaki modular theory, provides a natural framework and powerful methods for our analysis. In this talk, I will discuss part of the general ground for our analysis and some key results, in particular the solution of an old problem in QFT: the description of the modular Hamiltonian associated with a space ball B in the free scalar massive QFT; this sets up the formula for the entropy density of a real wave packet. Webinar links and passwords and program updates will be distributed via email. If you would like to be added to the mailing list, please sign up for the mailing list at https://lists.uni-leipzig.de/mailman/listinfo/aqfp_announcements.

We prove that any deterministic matrix is approximately the identity in the eigenbasis of a large random Wigner matrix W with an optimal error inversely proportional to the square root of the dimension. This verifies a strong form of Quantum Unique Ergodicity with an optimal convergence rate and we also prove Gaussian fluctuations around this convergence after a small spectral averaging. This requires to extend the classical CLT for linear eigenvalue statistics, Tr f(W), to include a deterministic matrix A and we identify three different modes of fluctuation for Tr f(W)A in the entire mesoscpic regime. The key technical tool is a new multi-resolvent local law for Wigner ensemble.

Weyl semimetals are three-dimensional condensed matter systems characterized by a degenerate Fermi surface, consisting of a pair of `Weyl nodes'. Correspondingly, in the infrared limit, these systems behave effectively as Weyl fermions in 3+1 dimensions. As predicted by Nielsen and Ninomiya in 1983, when exposed to electromagnetic fields these materials are expected to simulate the axial anomaly of QED, by giving rise to a net quasi-particle flow between Weyl nodes.
We consider a class of interacting lattice models for Weyl semimetals and prove that the quadratic response of the quasi-particle flow is universal, and equal to the chiral triangle graph of QED. Universality is the counterpart of the Adler-Bardeen non-renormalization property of the axial anomaly for QED, in a condensed matter setting. Our proof relies on the rigorous Wick rotation for real-time transport coefficients, on constructive bounds for Euclidean ground state correlations, and on lattice Ward Identities.
Joint work with A. Giuliani and V. Mastropietro.

The talk will present the recent proof that in four dimensions the spin fluctuations of Ising-type models at their critical points are Gaussian in their scaling limits (infinite volume, vanishing lattice spacing). Similar statement is proven for the scaling limits of more general PHI^4 fields constructed through a lattice cutoff. The proofs are facilitated by the systems’ random current representation, in which the deviation from Wick's law are expressed in terms of intersection probabilities of random currents with prescribed sources. This approach previously yielded such statements for D>4. Their recent extension to the marginal dimension was enabled by a multiscale analysis of the critical clusters’ intersections.
Joint work with Hugo Duminil-Copin.

Liouville conformal field theory (LCFT) is a family of Conformal field theories which arise in a wide variety of contexts in the physics and the probabilistic literature: SUSY Yang-Mills, the scaling limit of large planar maps, etc...
There are two main and seemingly unrelated approaches to LCFT in the physics literature: one in the Feynman path integral formulation and one in the conformal bootstrap approach. Recently, we constructed rigorously LCFT in the Feynman path integral formulation via probability theory (and more specifically the Gaussian Free Field). In this talk, I will present recent work which shows that both approaches (probabilistic construction of the Feynman path integral and conformal bootstrap) are in fact identical. A key ingredient in our work is the analysis of an infinite dimensional semigroup, the so-called Liouville semigroup.
Based on a series of joint works with C. Guillarmou, F. David, A. Kupiainen and R. Rhodes.

Most QFT axioms are only good to prove theorems but not to compute anything measurable. One exception are the Euclidean Conformal Field Theory (CFT) axioms in d>=3 dimensions, which do lead to surprisingly strong “bootstrap" constraints on scaling dimensions of various conjecturally existing Euclidean CFTs (such as the critical point of the 3d Ising and O(2) models). In this talk I will not discuss the bootstrap as such, but I will explain the Euclidean CFT axioms and will relate them to the Osterwalder-Schrader and Wightman axioms. The OS linear growth condition does not obviously follow from the Euclidean CFT axioms, but fortunately there is a route to Wightman axioms which does not rely on the Glaser-Osterwalder-Schrader construction. Based on work in progress with Petr Kravchuk and Jiaxin Qiao. Webinar links and passwords and program updates will be distributed via email. If you would like to be added to the mailing list, please sign up for the mailing list at https://lists.uni-leipzig.de/mailman/listinfo/aqfp_announcements.

We derive a multiscale generalisation of the Bakry-Emery criterion for a measure to satisfy a Log-Sobolev inequality. Our criterion relies on the Polchinski equation which is well known in renormalisation theory. This multiscale approach implies the usual Bakry-Emery criterion, but we show that it remains effective for measures which are far from log-concave. In particular, it applies to the Glauber and Kawasaki dynamics of the massive continuum sine-Gordon model with $\beta < 6 \pi$ and leads to asymptotically optimal Log-Sobolev inequalities. This is joint work with Roland Bauerschmidt.

In this talk recent results in the quantum theory of interacting Bose gases are reviewed. Use is made of a representation of Bose gases as a kind of scalar field theory resulting from a Hubbard-Stratonovich transformation;
Ginibre’s Brownian loop representation;
an interpolation in the number N of atom species; and
simplifications appearing in various limiting regimes.
The purpose is to study phenomena related to Bose-Einstein condensation and properties of polymer chains (N → 0). Some novel support for the conjecture that λ|φ|^{4}_{d}-theory in d ≥ 4 dimensions is non-interacting will also be described.