Recent developments in Constructive Field Theory

Date: March 13-15, 2018
Location: Columbia University, Mathematics building, rm 407.

This mini-workshop will center on recent progress on topics such as Stochastic Quantization, discrete complex analysis and graphical methods, local sets, gauge theories, ...

We expect partial support to be available for travel and accommodation for accepted participants.

Organizers: Julien Dubedat, Fredrik Viklund

The workshop is supported by the National Science Foundation (DMS 1308476)

Confirmed participants

Abdelmalek Abdesselam
Juhan Aru
David Brydges
Ajay Chandra
Sourav Chatterjee
Julien Dubedat
Bertrand Duplantier
Julien Fageot
Hugo Falconet
Shirshendu Ganguly
Christophe Garban
Yu Gu
Clement Hongler
John Imbrie
Nam-Gyu Kang
Antti Kupiainen
Kalle Kytola
Nikolai Makarov
Minjae Park
Eveliina Peltola
Scott Sheffield
Hao Shen
Tom Spencer
Xin Sun
Fredrik Viklund

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Schedule

Talks in Math 407; breakfast and coffee breaks in Math 508.

	Tue Mar 13	Wed Mar 14	Thu Mar 15
9:00-9:45	breakfast	breakfast	breakfast
9:45-10:45	Abdesselam	Chandra	Sheffield
10:45-11	break	break	break
11-12	Brydges	Shen	Kytola
12-2	lunch break	lunch break	lunch break
2-3	Chatterjee	Kupiainen	Ganguly
3-3:15	break	break	break
3:15-4:15	Kang	Hongler	Peltola

Abstracts and slides

• Abdelmalek Abdesselam
  Space-dependent renormalization group and anomalous dimensions in a hierarchical model for 3d CFT
  [slides]
  
  An outstanding problem in the area of rigorous renormalization group theory is to develop a Wilsonian formalism which can handle space-dependent couplings in the Euclidean setting. I will present such a method in the simpler hierarchical model case and explain how this allowed us to prove a 46 years old prediction by Wilson regarding the anomalous scaling of the square/energy field in a hierarchical ferromagnet. This is joint work with Ajay Chandra and Gianluca Guadagni. The model we studied is a hierarchical version of a 3d ferromagnet with long-range interactions in a range of parameters which puts it slightly below the upper critical dimension. We constructed the scaling limit for the joint law of the elementary/spin field together with the square/energy field as well as all mixed correlation functions. The Euclidean version of this scaling limit is conjectured to be a conformal field theory in three dimensions according to recent work by physicists in the conformal bootstrap program. There is a natural analogue of conformal invariance in the hierarchical model which thus provides an ideal testing ground for renormalization group-based attempts at rigorously proving conformal invariance. The key idea is to feed the space-dependent renormalization group space-dependent ultraviolet cutoffs. If time permits, I will also mention emerging connections to the AdS/CFT correspondence.
  
  
• David Brydges
  The Lace expansion for $(\varphi^{2})^{2}$
  [slides]
  
  Akira Sakai has shown that a convergent lace expansion exists for the Ising and $\varphi^4$ models. He uses the current representation for the Ising model to convert the system to a percolation. In work with \emph{Tyler Helmuth} and \emph{Mark Holmes} we give a different expansion based on the Symanzik local time isomorphism.  This expansion exists for $|\varphi|^{4} $, $O (n)$ models and the continuous time lattice Edwards model ($n=0$), but we can only prove convergence for $n=0,1,2$ because the GHS inequalities are not known to hold for $n>2$. As in all other lace expansions, for convergence a small parameter is required.  Thus the method gives information on critical exponents for the listed models in high dimensions, or for finite but sufficiently long range coupling.
  
  
• Ajay Chandra
  Renormalization in Regularity Structures
  
  The inception of the theory of regularity structures transformed the study of singular SPDE by generalizing the notion of "Taylor expansion" and classical notions of regularity in a way flexible enough to accommodate the renormalization.
  In the years since then our understanding of how to implement renormalization in regularity structures has developed rapidly. I will sketch the overall framework of the theory of regularity structures and then summarize these recent developments in order to give an idea of the current state of the theory.
  
  
• Sourav Chatterjee
  Yang-Mills for probabilists
  [slides]
  
  I will give a short introduction to lattice gauge theories and Euclidean Yang-Mills theories aimed at probabilists. Probabilistic formulations of some of the central problems will be discussed, along with some recent results if time permits.
  
  
• Shirshendu Ganguly
  Lattice gauge theory and string duality
  [slides]
  
  Matrix integrals provide models for many physical systems.  Gaussian models with finitely many matrices are the most well studied. However certain ensembles with growing number of matrices such as lattice gauge theories have been studied in the physics literature as discrete approximations to Euclidean Yang-Mills theory for a long time. We will review the recent work of Chatterjee who rigorously established a gauge-string duality in a class of such models and discuss attempts at analyzing the associated string trajectories.
  
  
• Nam-Gyu Kang
  Calculus of conformal fields on a compact Riemann surface
  [slides]
  
  Thanks to Eguchi and Ooguri, it is known that the insertion of a stress tensor acts (within correlations of fields in the OPE family) according to the Lie derivative operator along a certain vector field and a certain differential operator with respect to the modular parameters. After presenting analytical implementation of conformal field theory on a compact Riemann surface, I explain how to derive Eguchi-Ooguri's version of Ward's equation on a torus from the pseudo-addition theorem for Weierstrass zeta function and present some examples. Joint work with N. Makarov. 
  
  
• Antti Kupiainen
  Renormalization Group and Stochastic PDE’s
  
  I will review a Renormalization Group approach to Stochastic PDE’s driven by space time white noise
  
  
• Kalle Kytola
  Conformal field theory on the lattice: from discrete complex analysis to Virasoro algebra
  [slides]
  
  Conjecturally, critical statistical mechanics in two dimensions can be described by conformal field theories (CFT). The CFT description has in particular lead to exact and correct (albeit mostly non-rigorous) predictions of critical exponents and scaling limit correlation functions in many lattice models. The main ingredient of CFT is the Virasoro algebra, accounting for the effect of infinitesimal conformal transformations on local fields. In this talk we show that an exact Virasoro algebra action exists on the probabilistic local fields of two discrete models: the discrete Gaussian free field and the critical Ising model on the square lattice.
  The talk is based on joint work with Clément Hongler and Fredrik Viklund.
  
  
• Clément Hongler
  Massive Ising Observables
  
  In this talk, I will discuss some new results concerning the massive limit of the Ising model, which arises when approaching criticality as the same time as taking the scaling limit. In particular, the model shows connections with the theory of isomonodromy deformations developed by Sato, Miwa and Jimbo. Joint work with SC Park.
  
  
• Eveliina Peltola
  Conformal blocks in nonrational CFTs with c ≤ 1
  [slides]
  
  I discuss conformal blocks for fields having Kac conformal weights of type h_{1,s} ( or h_{r,1} ). Correlation functions including such fields satisfy PDEs of order s ( or r ). Using a quantum group symmetry, we can show that there exists a unique single-valued correlation function and we may construct such functions explicitly, finding also the structure constants in closed form. It is worthwhile to note that also in two special cases of CFTs with c = 1 and c = -2, explicit formulas for the conformal blocks can be found, and they are very simple.
  The talk is based on joint works with Alex Karrila, Kalle Kytölä (both at Aalto University) and (in progress) work with Steven Flores.
  
  
• Scott Sheffield
  Gauge theory and the three barriers
  [slides]
  
  It is frequently asked why, given all we know about the theory of Liouville quantum gravity, we still have not succeeded in using this knowledge to construct a continuum version of Yang-Mills gauge theory (with or without a proof that it is a limit of discretized theories). It seems that to relate these theories to one another, one has to move from $c \leq 1$ to $c > 1$, from $N= \infty$ to finite $N$, and from Gaussian measures to compact Haar measures (which introduce oscillating signs in the corresponding discrete random surface sums, complicating any effort to describe an appropriate continuum analog). I will discuss some recent work on these puzzles, focusing mostly on efforts to move beyond the $c=1$ barrier.
  
  
• Hao Shen
  Stochastic quantization of gauge theories
  
  "Stochastic quantization” refers to a formulation of quantum field theory as stochastic PDEs. The recent years witnessed interesting progress in understanding solutions of these stochastic PDEs, one of the remarkable examples being Hairer and Mourrat-Weber's results on the Phi^4_3 equation.
  In this talk we will discuss stochastic quantization of gauge theories, with focus on an Abelian example (that is, two dimensional Higgs or scalar QED), but also provide prospects of non-Abelian Yang-Mills theories. We address issues regarding Wilson’s lattice regularization, dynamical gauge fixing, renormalization, Ward identities, and construction of dynamical loop and string observables.