Spring Workshop

Program

Titles & Abstracts

Lectures

S. Chatterjee. Timelike Liouville field theory: rigorous results and open problems

Abstract. Liouville field theory, known as Liouville quantum gravity (LQG) in probability theory, has attracted widespread attention in the probability and mathematical physics communities in recent years. In physics, it is sometimes referred to as “spacelike” Liouvillle theory, to distinguish it from its “timelike” counterpart. Timelike Liouville field theory is a far less understood theory, both in physics and in mathematics. The importance of the timelike theory stems from the fact that it is supposed to have a positive curvature semiclassical limit, unlike the spacelike theory, which has a negative curvature semiclassical limit. Einstein’s general relativity gives rise to positive curvature metrics, which motivates the search for more “realistic” theories of quantum gravity that yield positive curvature in the semiclassical limit. The problem with such theories is that they invariably run into the problem of working with random fields with negative variance, which do not belong to the framework of ordinary probability theory. After introducing a detailed background, I will talk about some of my recent work on timelike Liouville theory, which proves some of the conjectured results and develops some rigorous mathematical understanding of the nature of this theory. This includes the development of a theory of Gaussian random variables with negative variance, the proof of the “timelike DOZZ formula” in the charge neutral regime, and recent advances beyond charge neutrality. Since much of the theory still remains poorly understood, I will spend time on open problems and conjectures. 

M. Gubinelli. TBA

Abstract. TBA

M. Schiavina. Hamiltonian approach to gauge field theory

Abstract. A local field theory is usually called "gauge" if it enjoys a symmetry by an infinite dimensional local group. This property is both a technical advantage and a complication depending on what one is interested in studying. In particular, if we consider field theory as a multi-dimensional generalisation of classical mechanics, the presence of such symmetries directly hinders the standard Hamiltonian approach to the mechanical system, which then requires an appropriate extension. It turns out that the Hamiltonian picture is intimately related to boundary conditions / cauchy data for the theory, and this becomes a very rich subject precisely in field theory, when boundaries are generic codimension one submanifolds, often with boundaries of their own. The resulting symplectic geometry of the problem, which is the well known geometric backdrop of Hamiltonian mechanics, requires a significant adaptation in these more general scenarios.

In this series of lectures we will discuss (finite dimensional) symplectic reduction, as a mathematical tool and guideline, and then apply it to Hamiltionian gauge theory, with a focus on the description of "phase space" for these highly symmetric systems. This can be seen as the starting point for rigorous (and possibly nonperturbative!) quantisation of the field theory, following various methods such as geometric, deformation or cohomological quantisation schemes.

Talks

M.S. Adamo. Osterwalder-Schrader reconstructed Wightman fields for unitary full vertex operator algebras 

Abstract. Axiomatic Quantum Field Theory is a research area aiming to give a mathematically rigorous foundation to quantum field theory. One of the first attempts in this direction is given by the Wightman axiomatization, describing observables through quantum fields verifying physically motivated axioms. 

In a series of two papers from 1973 and 1975, Osterwalder and Schrader showed a reconstruction theorem that allows to construct a quantum field theory à la Wightman. Namely, they showed that if a family of Euclidean correlation functions verify a set of Osterwalder-Schrader axioms, then the analytic continuation of such functions produce distributions that verify the Wightman axioms. 

In our work, we construct correlation functions that verify Osterwalder-Schrader axioms that include a regularity condition on the correlation functions, known as linear growth, for a reasonable class of unitary full Vertex Operator Algebras. Since our Euclidean correlation functions are constructed using full Vertex Operator Algebras, I will present our strategy on how to recover Osterwalder-Schrader reconstructed Wightman distributions explicitly. 

Z. Ammari. Gibbs measures for Hamiltonian PDEs: KMS property and completeness 

Abstract. During the past few decades, Gibbs measures have been used in nonlinear PDEs to establish various remarkable results related to almost sure well-posedness and flow properties. The main ingredients are Fourier analysis, the Hamiltonian structure and the measure invariance.   In this talk I will report on some recent contributions obtained in collaboration with Shahnaz Farhat and Vedran Sohinger. In particular, the following aspects will be discussed:

(i) The Kubo-Martin-Schwinger (KMS) property: I will define the KMS equilibrium states for Hamiltonian PDEs and show under certain hypotheses that there exists a unique KMS equilibrium state for such system given by the Gibbs measure.
(ii) Completeness: I will underline a general principle proving that if a Hamiltonian PDE admits a stationary probability measure then the PDE admits  global solutions almost surely.

P. Duch. Construction of fermionic QFTs via renormalization group flow equation

Abstract. We present a new approach to solving the renormalization group flow equation (the Polchinski equation) in a fully nonperturbative fermionic setting. We reformulate the equation as a well-posed infinite-dimensional PDE on a Banach space and construct solutions directly via a fixed point argument, avoiding the expansion-based techniques used in previous work. The key idea is to combine this PDE perspective with tools from noncommutative probability, in particular through a scale decomposition of the free fermionic field in a filtered noncommutative probability space.

As an application, we construct the Gross–Neveu model and prove convergence of its Schwinger functions as the cutoff is removed. Moreover, we obtain a random field on the noncommutative probability space of the free theory whose moments match these Schwinger functions, yielding a strong coupling between free and interacting fields at criticality. This approach is robust and is expected to extend to a broad class of critical and subcritical quantum field theories. 

R. Fukuizumi. Φ⁴ model associated with the harmonic oscillator 

Abstract. We present recent developments in the mathematical analysis of the stochastic Gross–Pitaevskii equation. In particular, we study the structure of the dynamics with space-time white noise for the equation with a confining potential, including well-posedness and invariant measures, while only briefly indicating its physical motivation from Bose–Einstein condensation near the critical temperature. 

This talk is based on several collaborative works with A. de Bouard, A. Debussche, A. Deya, T. Iwabuchi,  L. Thomann.

A. Giuliani. Renormalization Group for Euclidean non-compact lattice QED in d=4

Abstract. There is a well known difficulty in applying Renormalization Group (RG) to QED4: cut-offs and momentum slicing break gauge invariance explicitly. Ward Identities, which are needed for the very construction of the theory, are typically violated at finite scales, and are recovered only after having removed cut-offs. However, due to the expected triviality of QED4, in view of a non-perturbative construction of the theory, it is desirable to define it with a fixed, although large, ultraviolet cut-off. In this talk I will discuss a RG construction of a lattice gauge-invariant Euclidean non-compact model of QED4, which allows us to compute all the gauge invariant observables as series in a sequence of finitely many running coupling constants (rcc) with bounded coefficients, uniformly in the cut-offs. The flow of the rcc is perturbatively bounded uniformly in the infrared cut-off, in the presence of an ultraviolet cut-off that can be chosen to be exponentially large in the electron charge. Ward Identities are valid even in the presence of finite infrared cut-offs, and this is used to control the flow of a few spurious non-gauge-invariant rcc at intermediate scales, generated by momentum slicing, without any need of introducing unphysical non-gauge-invariant counterterms. Joint work with Simone Fabbri, Marco Falconi, Vieri Mastropietro.

T. Miyao. Toward Finite-Temperature Ordering in a  Lattice Fermion Model 

Abstract. I will discuss a rigorous approach to finite-temperature ordering in a Hubbard-type lattice fermion model. The goal is to understand how an ordered phase can persist at positive temperature in the presence of strong local constraints and nontrivial many-body effects.

A central point is that, in an appropriate regime, the problem can be reduced to an effective description governed by the antiferromagnetic Heisenberg model. I will explain this effective viewpoint and describe how the analysis is further reduced to a comparison between the ordered state and configurations containing defects. The key step is to control the free-energy cost of such defects in a form suitable for a Peierls-type argument. While some expansion methods enter in the background, the emphasis will be on the effective reduction, the defect mechanism, and recent progress toward a complete proof of finite-temperature long-range order.

Y. Tanimoto. Renormalization group according to Balaban-Dimock  

Abstract. We review the fundamental ideas and known results of the renormalization group approach by Balaban and Dimock, with a particular emphasis on the Φ⁴₃ model. We present partial results towards the convergence of the partition functions of the Φ⁴₃ model and discuss briefly possible extensions to other models.