Quantum field theory, renormalisation and stochastic partial differential equations
Monday 22nd October 2018 to Friday 26th October 2018
09:00 to 09:20  Registration  
09:20 to 09:30  Welcome from David Abrahams (Isaac Newton Institute)  
09:30 to 10:30 
Giovanni JonaLasinio (Università degli Studi di Roma La Sapienza) Some questions and remarks on the theory of singular stochastic PDEs
In the last decade there has been important progress in understanding singular stochastic PDEs (SSPDEs). In this talk I will briefly discuss the following points. 1. There are singular stochastic PDEs successfully used by physicists which are still beyond the reach of the present approaches and deserve mathematical investigation. 2. The present theory should be further developed by viewing SSPDEs as infinite dimensional dynamical systems. 3. Certain methods developed long ago for euclidean field theories in infinite volume may be useful in the study of the large time behavior of SSPDEs.

INI 1  
10:30 to 11:00  Morning Coffee  
11:00 to 12:00 
Volker Bach (Johannes GutenbergUniversität Mainz) Beyond the van Hove time scale
Given an arbitrarily large, but fixed, time t >0, we derive approximations for the time evolution of the spinboson model in terms of the propagator generated by a free effective Hamiltonian. Our construction rests on the renormalization group induced by the isospectral FeshbachSchur ap. This is joint work with Jacob Schach Möller and Matthias Westrich.

INI 1  
12:00 to 13:30  Lunch at Churchill College  
13:30 to 14:30 
Abdelmalek Abdesselam (University of Virginia) Pointwise multiplication of random Schwartz distributions with Wilson's operator product expansion
I will present a general theorem for the multiplication of random distributions which is similar in spirit to the construction of local Wick powers of a Gaussian field. However, this theorem is much more general in scope and applies to nonGaussian measures, even without translation invariance and in the presence of anomalous scaling, provided the random fields involved are less singular than white noise. Conjecturally, the construction of the energy field of the 3D Ising scaling limit as a square of the spin field should fall within the purview of the theorem. Our construction involves multiplying mollified distributions followed by suitable additive and multiplicative renormalizations before a proof of almostsure convergence when the mollification is removed. The main tools for the proof are combinatorial estimates on moments. The main hypothesis for the theorem is Wilson's OPE with precise quantitative bounds for pointwise correlations at noncoinciding points. I will also explain how the theorem works on the example of a simple conformal field theory of mean field type, namely, the fractional Gaussian field.

INI 1  
14:30 to 15:30 
Patricia Gonçalves (Instituto Superior Técnico, Lisboa) Nonequilibrium fluctuations for the slow boundary symmetric exclusion
In this talk, I will present the symmetric simple exclusion process in contact with slow stochastic reservoirs which are regulated by a factor $n^{\theta}$, $\theta\geq 0$. I will review the hydrodynamic limit and the goal of my talk is to present the nonequilibrium fluctuations for this model. Depending on the range of the parameter $\theta$ we obtain processes with various boundary conditions. As a consequence of the previous result together with an application of the matrix ansatz method  which gives us information on the stationary measure for the model  we deduce the nonequilibrium stationary fluctuations. The main ingredient to prove these results is the derivation of precise bounds on the twopoint spacetime correlation functions.

INI 1  
15:30 to 16:00  Afternoon Tea  
16:00 to 17:00 
Manfred Salmhofer (Universität Heidelberg) Functional Integrals for BoseFermi Systems 
INI 1  
17:00 to 18:00  Welcome Wine Reception at INI 
09:00 to 10:00 
Giuseppe Da Prato (Scuola Normale Superiore di Pisa) BV functions in separable Hilbert spaces
A probabily m non necessarily Gaussian is given in a separable Hilbert space. We present necessary and sufficient conditions for a function u has a finite total variation with respect to m. Several examples of set of finite perimeters are presented.

INI 1  
10:00 to 11:00 
Marcello Porta (Eberhard Karls Universität Tübingen) Edge universality in interacting topological insulators
In the last few years there has been important progress on the rigorous understanding of the stability of gapped topological phases for interacting condensed matter systems. Most of the available results deal with bulk transport, for systems with no boundaries. In this talk, I will consider interacting 2d topological insulators on the cylinder. According to the bulkedge duality, one expects robust gapless edge modes to appear. By now, this has been rigorously understood for a wide class of noninteracting topological insulators; the main limitation of all existing proofs is that they do not extend to interacting systems. In this talk I will discuss the bulkedge duality for a class of interacting 2d topological insulators, including the HaldaneHubbard model and the KaneMeleHubbard model. Our theorems give a precise characterization of edge transport: besides the bulkedge duality, the interacting edge modes satisfy the Haldane relations, connecting the velocities of the edge currents, the edge Drude weights and the edge susceptibilities. The proofs are based on rigorous renormalization group, with key nonperturbative inputs coming from the combination of lattice and emergent Ward identities. Based on joint works with G. Antinucci (Zurich) and V. Mastropietro (Milan).

INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
Slava Rychkov (IHES); (École Normale Supérieure) Walking, Weakly FirstOrder Phase Transitions, and Complex CFTs
Teaser: Most people have heard that the 2d Potts model with Q=5 states has a first order phase transition, but not everyone knows that the correlation length at this phase transition is 2500 lattice spacings.
This is going to be a nonrigorous physics talk. We will give an introduction to "walking RG" behavior in gauge theories and connect it to Type II weak firstorder phase transitions in statistical physics. Despite appearing in very different systems (QCD below the conformal window, the Potts model, deconfined criticality) these two phenomena both imply approximate scale invariance in a range of energies and have the same RG interpretation: a flow passing between pairs of fixed point at complex coupling, dubbed "complex CFTs". Observables of the real walking theory are approximately computable by perturbing the complex CFTs. The general mechanism will be illustrated by a specific and computable example: the twodimensional Qstate Potts model with Q > 4.
Based on http://arxiv.org/abs/1807.11512 and http://arxiv.org/abs/1808.04380

INI 1  
12:30 to 14:00  Lunch at Churchill College  
14:00 to 15:00 
Seiichiro Kusuoka (Okayama University) Invariant measure and flow associated to the Phi4quantum field model on the threedimensional torus
We consider the invariant measure and flow of the Phi4model on the threedimensional torus, which appears in the quantum field theory. By virtue of Hairer's breakthrough, such a nonlinear stochastic partial differential equation became solvable and is studied as a hot topic. In the talk, we apply the paracontrolled calculus and directly construct the global solution and the invariant measure by using the invariant measures of approximation equations and showing the tightness of associated processes. This is a joint work with Sergio Albeverio.

INI 1  
15:00 to 16:00 
Martin Lohmann (University of British Columbia) The critical behavior of $\phi^4_4$
We discuss the approach to the critical point of the $\phi^4$ model in 4 dimensions. One of the major successes of the renormalization group technique has been to explain why this model features logarithmic corrections to the scaling predictions for the blow up of thermodynamic quantities. We review the strategy of the proof in the "symmetric regime" with zero external magnetic field, in which case this is a classic result. We then present the proof of logarithmic corrections to the magnetization as the magnetic field tends to zero. Despite being a central aspect of the model, these have been an open problem until now, probably because technical complications where expected due to the broken symmetry. We have found these concerns to be unfounded, and our proof only needs a single cluster expansion on top of the classic RG construction for the critical point.

INI 1  
16:00 to 16:30  Afternoon Tea 
09:00 to 10:00 
Stefan Hollands (Universität Leipzig); (Universität Leipzig) Perturbative QFT in D = 4
Nonabelian YangMills theories are the key building blocks of the standard model of particle physics. Their renormalization, even at the perturbative level, is a difficult problem because it must be shown that  and in precisely what sense  there exists a renormalization scheme preserving local gauge invariance.In this talk, I outline solutions to this problem (a) in the context of curved Lorentzian spacetimes and (b) within the context of flat Euclidean space. The methods presented are rather different, in that the first is based on a generalization of the EpsteinGlaser method  also called "causal perturbation theory", while the second is based on the so called "flow equation method".The talk will be introductory in nature.

INI 1  
10:00 to 11:00 
Kasia Rejzner (University of York) Renormalized quantum BV operator and observables in gauge theories and gravity
In this talk I will give an overview of the BV quantization, which is a universal framework for constructing models in perturbative QFT (including gauge theories and effective quantum gravity). Using the version of this framework developed by Fredenhagen and myself, one can construct local nets of observable algebras satisfying HaagKastler axioms, in the sense of formal power series in hbar. The crucial role is played by the renormalized quantum BV operator, which is defined abstractly, without the explicit use of the BRST charge.

INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
Martin Hairer (Imperial College London) The RG landscape in 1+1 dimensions
This is joint work with Giuseppe Cannizzaro.

INI 1  
12:30 to 13:30  Lunch at Churchill College  
13:30 to 17:00  Free Afternoon  
19:30 to 22:00  Formal Dinner at Emmanuel College (Old Library) 
09:00 to 10:00 
Ajay Chandra (University of Warwick) Renormalisation in Regularity Structures: Part I
The inception of regularity structures provided a robust deterministic theory that generalized the notion of “Taylor expansion" and classical notions of regularity in a way flexible enough to encode renormalisation  this led to rapid development in the local existence theory for singular stochastic SPDE.
In the years that followed the framework for implementing renormalisation in regularity structures has become much more robust. I will describe these developments with an emphasis on how the stochastic estimates needed in regularity structures can be obtained by using methods from multiscale perturbation theory with a twist.

INI 1  
10:00 to 11:00 
Jeremie Unterberger (Université de Lorraine) The scaling limit of the KPZ equation in space dimension 3 and higher
We study in the present article the KardarParisiZhang (KPZ) equation
$$ \partial_t h(t,x)=\nu\Del h(t,x)+\lambda \nabla h(t,x)^2 +\sqrt{D}\, \eta(t,x), \qquad (t,x)\in{\mathbb{R}}_+\times{\mathbb{R}}^d $$
in $d\ge 3$ dimensions in the perturbative regime, i.e. for $\lambda>0$ small enough and a smooth, bounded, integrable initial condition
$h_0=h(t=0,\cdot)$. The forcing term $\eta$ in the righthand side is
a regularized spacetime white noise. The exponential of $h$  its socalled ColeHopf
transform  is known to satisfy a
linear PDE with multiplicative noise.
We prove a largescale diffusive limit for the solution, in particular a
timeintegrated heatkernel behavior for the covariance in a parabolic scaling.
The proof is based on a rigorous implementation of K. Wilson's renormalization group
scheme. A double cluster/momentumdecoupling expansion allows for perturbative
estimates of the bare resolvent of the ColeHopf linear PDE in the smallfield region where the noise is not too large, following the broad lines of IagolnitzerMagnen. Standard large deviation estimates for $\eta$ make it possible to extend the above estimates to the largefield region. Finally,
we show, by resumming all the byproducts of the expansion, that the solution $h$ may be written in the largescale limit (after a suitable Galilei transformation) as a small perturbation of the solution of the underlying linear EdwardsWilkinson model ($\lambda=0$) with renormalized coefficients $\nu_{eff}=\nu+O(\lambda^2),D_{eff}=D+O(\lambda^2)$.
This is joint work with J. Magnen.

INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
Franco Flandoli (Scuola Normale Superiore di Pisa) A scaling limit from Euler to NavierStokes equations with random perturbation
In the past years there has been intense research on Euler equations with multiplicative transport type noise and NavierStokes equations with additive noise. Each model has its own motivations but apparently there is no link between them. We show that a special scaling limit of the stochastic Euler equations leads to the stochastic NavierStokes equations. Remarkable is the difference of the noises. And the inversion with
respect to usual paradigms which consider Euler equations as limit of NavierStokes equations in special regimes.
This is a joint work with Dejun Luo, Academy of Sciences, Beijing.

INI 1  
12:30 to 14:00  Lunch at Churchill College  
14:00 to 15:00 
Martina Hofmanova (Bielefeld University) A PDE construction of the Euclidean $\Phi^4_3$ quantum field theory
We present a selfcontained construction of the Euclidean $\Phi^4$ quantum field theory on $\mathbb{R}^3$ based on PDE arguments. More precisely, we consider an approximation of the stochastic quantization equation on $\mathbb{R}^3$ defined on a periodic lattice of mesh size $\varepsilon$ and side length $M$. We introduce an energy method and prove tightness of the corresponding Gibbs measures as $\varepsilon \rightarrow 0$, $M \rightarrow \infty$. We show that every limit point satisfies reflection positivity, translation invariance and nontriviality (i.e. nonGaussianity). Our argument applies to arbitrary positive coupling constant and also to multicomponent models with $O(N)$ symmetry. Joint work with Massimiliano Gubinelli.

INI 1  
15:00 to 16:00 
Bertrand Duplantier (CEA/Saclay); (IHES) CLE Nesting and Liouville Quantum Gravity
We describe recent advances in the study of SchrammLoewner Evolution (SLE), a canonical model of noncrossing random paths in the plane, and of Liouville Quantum Gravity (LQG), a canonical model of random surfaces in 2D quantum gravity. The latter is expected to be the universal, conformally invariant, continuum limit of random planar maps, as weighted by critical statistical models. SLE multifractal spectra have natural analogues on random planar maps and in LQG. An example is extreme nesting in the Conformal Loop Ensemble (CLE), as derived by Miller, Watson and Wilson, and nesting in the O(n) loop model on a random planar map, as derived recently via combinatorial methods. Their respective large deviations functions are shown to be conjugate of each other, via a continuous KPZ transform inherent to LQG.
Joint work with Gaetan Borot and Jérémie Bouttier.

INI 1  
16:00 to 16:30  Afternoon Tea 
09:00 to 10:00 
Thierry Bodineau (École Polytechnique) Spectral gap for Glauber dynamics of hierarchical spin models
We will present a renormalisation group approach to estimate the decay of the spectral gap of hierarchical models. In particular, we will consider a hierarchical version of the 4dimensional $\Phi_4$ model at the critical point and its approach from the high temperature side, as well as a hierarchical 2dimensional SineGordon model. For these models, we show that the spectral gap decays polynomially like the spectral gap of the dynamics of a free field. This is a joint work with Roland Bauerschmidt.

INI 1  
10:00 to 11:00 
Fabien VignesTourneret (Université Claude Bernard Lyon 1) Constructive Tensor Field Theory through an example
In the last ten years, a new approach to quantum gravity has emerged. Called Tensor Field Theory, it generalizes random matrix models in a straightforward way. This talk will be the occasion of recalling the main motivations for such field theories and to present the stateoftheart of their constructive study.
This is joint work with Vincent Rivasseau.

INI 1  
11:00 to 11:30  Morning Coffee  
11:30 to 12:30 
Yvain Bruned (Imperial College London) Renormalisation in Regularity Structures: Part 2
The amount of computation for solving some singular SPDEs via Regularity Structures is huge and requires a good algebraic framework. In this talk, we will present the main ideas which allow us to automatize the renormalisation of these singular SPDEs and to get some symmetry properties of the solution.

INI 1  
12:30 to 13:30  Lunch at Churchill College  
13:30 to 14:30 
Margherita Disertori (Rheinische FriedrichWilhelmsUniversität Bonn) Supersymmetry and Ward identities: an alternative approach to renormalization.
joint work with T.Spencer and M.Zirnbauer

INI 1  
14:30 to 15:30 
John Imbrie (University of Virginia) FeshbachSchur RG for the Anderson Model
Consider the localization problem for the Anderson model of a quantum particle moving in a random potential. We develop a renormalizationgroup framework based on a sequence of FeshbachSchur maps. Each map produces an effective Hamiltonian on a lowerdimensional space by localizing modes in space and in energy. Randomness in everlarger neighborhoods produces nontrivial eigenvalue movement and separates eigenvalues, making the next step of the RG possible. We discuss a particularly challenging case where the disorder has a discrete distribution.

INI 1 