(Ir)regular singularities and Quantum Field Theory
Titles and Abstracts
Vladimir Bazhanov (Australian National University): "On the scaling behaviour of the alternating spin chain"
In this talk I will report the results of the study of a 1D integrable
alternating spin chain whose critical behaviour is governed by a CFT
possessing a continuous spectrum of scaling dimensions. I will review
both analytical and numerical approaches to analyzing the spectrum of
low energy excitations of the model. It turns out that the computation
of the density of Bethe states of the continuous theory can be reduced
to the calculation of the connection coefficients for a certain class of
differential equations whose monodromy properties are similar to those
of the conventional confluent hypergeometric equation. The finite size
corrections to the scaling are also discussed.
Carl M. Bender (Washington University in St. Louis): "Nonlinear eigenvalue problems and PT symmetry"
We generalize the concept of a differential-equation eigenvalue problem
from linear to nonlinear differential equations. For nonlinear differential
equations the eigenfunctions are separatrix curves and the eigenvalues are
the initial conditions that give rise to separatrix curves. The Painlevé
transcendents serve as examples of nonlinear eigenvalue problems, and
nonlinear semiclassical techniques are devised to calculate the large-n
behavior of the nth eigenvalue. This behavior is found by reducing the
Painlevé equation to the linear Schrödinger equation associated with a
non-Hermitian PT-symmetric Hamiltonian. The notion of a nonlinear
differential-equation eigenvalue problem extends beyond the Painlevé
transcendents to huge classes of nonlinear differential equations. Some of
these equations exhibit new and interesting eigenvalue behavior such as
spectra having fine structure.
Philip Boalch (Université Paris-Sud): "TQFT approach to meromorphic connections---Fission via irregular singularities"
I'll review the TQFT (quasi-Hamiltonian) approach to symplectic moduli spaces of irregular meromorphic connections on Riemann surfaces. Whilst the generic case was done in 2002 (arXiv:0203161), the general case was completed relatively recently in work with D. Yamakawa (arXiv:1512.08091). This generalises the approach to symplectic moduli spaces of flat connections of Alekseev-Malkin-Meinrenken, which corresponds to the tame (regular singular/Fuchsian) case. Their (quasi-classical) fusion operation is complemented by a new operation, fission, breaking the structure group. This involves a clean topological description of the Stokes phenomenon, generalising Stokes' original approach, as emphasised (and illustrated) in arXiv:1903.12612.
Guiseppe De Nittis (Universidad Católica de Chile): "Quantum Butterflies: Filling the white with colors"
Quantum Butterflies are spectra of certain self-adjoint operators in the noncommutative torus. These operators can be realized in "concrete" Hilbert spaces in different ways producing distinct differential/difference equations like the Harper equation, the Hofstadter equation or the (almost-)Mathieu equation. All these models share the same spectrum which is given by the celebrated Hofstadter butterfly. However, there are finer topological properties which complete the spectral picture and allow to distinguish the various models. These topological invariants can be coded with colors producing the colorful version of the Quantum Butterflies.
Patrick Dorey (University of Durham): "Analytic continuation of TBA equation"
Clare Dunning (University of Kent): "Wronskian orthogonal polynomials"
Some properties of Wronskian orthogonal polynomials are
studied, motivated by the connection with the ODE/IM correspondence for
the massless sine-Gordon model at its free fermion point.
Alexandre Emerenko (Purdue University): "PT-symmetric eigenvalues for homogeneous potentials"
Alba Grassi (Stony Brook University): "Fredholm determinants from Topological String Theory"
In this talk I will review some aspects of the TS/ST correspondence and its ability to compute Fredholm determinants of quantum mechanical operators.
As an application I will focus on the example of (modified) Mathieu equation and connect our results with the TBA approach of Zamolodchikov.
David Hernandez (Université Paris-Diderot): "Spectra of quantum KdV Hamiltonians, Langlands duality and category O
(joint work with Edward Frenkel)"
We prove a system of relations in the Grothendieck ring of the category O
of representations of the Borel subalgebra of an untwisted quantum
This system was discovered by Masoero-Raimondo-Valeri who established that
solutions of this system can be attached to certain affine opers
attached to the Langlands dual affine Kac-Moody algebra.
Together with results of Bazhanov-Lukyanov-Zamolodchikov which enable
one to associate quantum
KdV Hamiltonians to representations from the category O, this provides evidences
for a conjecture of Frenkel-Feigin linking the spectra of quantum KdV
Hamiltonians and Langlands
dual affine opers, as an instance of the ODE/IM correspondence.
Supported by the European Research Council under the European Union's
Framework Programme H2020
with ERC Grant Agreement number 647353 Qaffine.
Oleg Lisovyi (Université de Tours): "(Irregular) conformal blocks and Painlevé functions"
Abstract: Regular 4-point Virasoro conformal blocks with central charge
c=1 are known to describe the general solution of Painlevé VI equation.
I will discuss algebraic construction of irregular conformal blocks
(with generic c) in different channels and explain how they can be used
to derive connection formulas for Painlevé V functions.
Sergey Lukyanov (Rutgers University): "Bethe state norms for the Heisenberg spin chain in the scaling limit"
In this talk the norms of the Bethe states for the spin 1/2
Heisenberg chain in the critical regime are discussed. A set of
conjectures will be formulated concerning the scaling behavior of
the norms, which were obtained from a combination of analytic
techniques based on the ODE/IQFT correspondence and numerical analysis.
Also I will discuss the role of the different Hermitian structures
associated with the integrable structures studied in the series of works
of Bazhanov, Lukyanov and Zamolodchikov in the mid nineties.
Martin T. Luu (University of California Davis): "Feigin-Frenkel image of Witten-Kontsevich points"
In the ODE/IM correspondence for a simple Lie algebra g constructed by
Masoero, Raimondo, and Valeri a certain family of meromorphic
connections plays a crucial role. A special member of this family of
connections turns out to be closely related to the Witten-Kontsevich
point in the phase space of the Drinfeld-Sokolov integrable hierarchy
associated to g. For this special connection we describe an interesting
Langlands duality of Segal-Sugawara operators.
Marcos Mariño (Université de Genève): "A new perspective on resurgence in quantum theory"
The building blocks of quantum observables are collections
of asymptotic series in the coupling constant,
corresponding to perturbative and non-perturbative sectors. According
to the theory of resurgence,
these series are related to each other in a non-trivial way, and these
relations are encoded in the so-called Stokes
automorphisms of the theory. In this seminar we put forward the idea that
knowledge of these automorphisms and of the classical limit might make
it possible to solve the quantum theory at all
values of the coupling. We show how this program can be implemented in
detail in quantum mechanics with polynomial potentials.
In this case, the solution takes the form of a set of coupled integral
equations of the TBA type, which generalizes the famous ODE/IM
correspondence. This implementation uses ideas about BPS states and
their wall-crossing in supersymmetric gauge theories.
Evgeny Mukhin (Purdue University): "Local Hamiltonians for deformed W algebras"
We use representation theory of quantum toroidal algebras to give a uniform construction of all known deformations of W algebras corresponding to Dynkin diagrams of classical types. Moreover, our method produces a number of new examples, covering, in particular, supersymmetric cases. It allows us to construct analogs of local integrals of motions in all cases. We discuss the existence of the affine screening operator commuting with the integrals of motion. The affine screening operators correspond to deformations of affine Cartan matrices of classical types (twisted, untwisted, and supersymmetric). This is a joint work with B. Feigin, M. Jimbo, and I. Vilkoviskiy.
Suresh Nampuri (Universidade de Lisboa): "Counting N=2 black holes and modular forms"
We propose an approximate formula for the degeneracy of large dyonic
1/2 BPS black hole in the N=2 Sen-Vafa STU model
and uncover a rich mathematical structure involving modular forms that
underlie black hole microstate counting.
Based on work with Gabriel Cardoso and Davide Polini.
Stefano Negro (Stony Brook University): "The TT deformation: a gentle introduction"
In this talk I will introduce a particular kind of irrelevant
deformation of 2D QFTs, known as TT
deformation and review some of the results obtained recently on this
subject. The general tone will be pedagogical and intended for an
audience of non-experts. After defining the operator
TT and describing its main properties, I
will present some motivations which justify the interest in these
peculiar deformations. I will then move on to the derivation of
important non-perturbative results and describe the principal features
of the deformed theories. Finally I will display some of the most
recent results and a number of interesting questions that still wait
Andrea Raimondo (Università di Milano-Bicocca): "Opers for higher states of quantum KdV models"
In this talk I will consider the ODE/IM correspondence for
all states of the quantum g-KdV model, where g is a simply laced
affine Kac-Moody algebra. I will show how to construct quantum g-KdV
opers as an explicit realization of a class of opers introduced by
Feigin and Frenkel, which are defined by fixing the singularity
structure at 0 and infinity, and by allowing an arbitrary but finite
number of additional singular terms with trivial monodromy. The
generalized monodromy data of the quantum g-KdV opers satisfy the
Bethe Ansatz equations of the quantum g-KdV model. In the sl2 case,
the opers obtained are equivalent to the Schroedinger operators with
"monster potential" obtained by Bazhanov, Lukyanov and Zamolodchikov
in relation with the higher states of the quantum KdV model. Talk
based on joint work with Davide
Vitaly Tarasov (Purdue University): "Fuchsian equations with three non-apparent singularities"
Roberto Tateo (Università di Torino): "TBA? TBA!"
Miguel Tierz (Universidade de Lisboa): "Large N phase transition and TT deformation of 2d
We study aspects of the TT deformation of 2d
Yang-Mills theory on a Riemann surface, introduced by Conti, Iannella,
Negro and Tateo. We
focus mainly on the large N behavior of the partition function of the
theory on the two-sphere, showing that the Douglas-Kazakov phase
transition persists for a range of values of the deformation
parameter, and that the critical area is lowered. The transition is
still of third order and also induced by instantons, whose
contributions we characterize.
Joint work with Leonardo Santilli,
work in progress.
Benoit Vicedo (University of York): "Non-ultralocality, affine Gaudin models and ODE/IM"
Non-ultralocality is a long-standing open problem in
classical integrable field theory which has precluded the
first-principle quantisation of many important models. I will review a
recent proposal for dealing with this issue, which relates it to the
study of quantum Gaudin models associated with affine Kac-Moody
algebras. Although comparatively little is presently known about
quantum Gaudin models in affine type, the description of the spectrum
of the integrals of motion in finite type quantum Gaudin models is
very well understood and has a deep connection to the geometric
Langlands correspondence. I will go on to review recent progress
towards constructing the local integrals of motion in quantum Gaudin
models in affine type, based on joint work with Sylvain Lacroix and
Charles Young, which draws closely on the analogy with the well
understood case of finite type. I will argue that all these results
suggest there is a close connection between the problem of
non-ultralocality and the ODE/IM correspondence.