Recent developments in Constructive Field Theory
Date: March 1315, 2018
Location: Columbia University, Mathematics
building, rm 407.
This miniworkshop 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
NamGyu Kang
Antti Kupiainen
Kalle Kytola
Nikolai Makarov
Minjae Park
Eveliina Peltola
Scott Sheffield
Hao Shen
Tom Spencer
Xin Sun
Fredrik Viklund
Apply
Please fill this
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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

9:009:45

breakfast

breakfast 
breakfast 
9:4510:45

Abdesselam

Chandra

Sheffield

10:4511

break

break 
break 
1112

Brydges

Shen

Kytola

122

lunch break

lunch break 
lunch break 
23

Chatterjee

Kupiainen

Ganguly

33:15

break 
break 
break 
3:154:15

Kang

Hongler

Peltola

Abstracts and slides
 Abdelmalek Abdesselam
Spacedependent 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 spacedependent 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
longrange 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
groupbased attempts at rigorously proving conformal
invariance. The key idea is to feed the spacedependent
renormalization group spacedependent 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
YangMills for probabilists
[slides]
I will give a short introduction to lattice gauge theories and
Euclidean YangMills 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 YangMills theory for a long time. We will review
the recent work of Chatterjee who rigorously established a
gaugestring duality in a class of such models and discuss
attempts at analyzing the associated string trajectories.
 NamGyu 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 EguchiOoguri's version of
Ward's equation on a torus from the pseudoaddition 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 nonrigorous) 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 singlevalued 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
YangMills 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 MourratWeber'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 nonAbelian YangMills theories. We address issues
regarding Wilson’s lattice regularization, dynamical gauge
fixing, renormalization, Ward identities, and construction of
dynamical loop and string observables.