New Trends in Non-Perturbative Gauge/String Theory and Integrability

27 juin 2022, 09:15 → 1 juil. 2022, 18:00 Europe/Paris

Salle René Baire (4th floor) (Institut de Mathématiques de Bourgogne (IMB))

Fabrizio Nieri, Taro Kimura

Description

The workshop brings together experts working in diverse yet related corners of modern theoretical physics and mathematics inspired by Supersymmetric Gauge/String Theory, so as to promote new interdisciplinary collaborations and trigger new advances in seemingly distant areas. Most of the progress the field has witnessed in recent years was due to the development of exact investigation techniques, rooted in equivariant localization and matrix models. About 20 years ago, these methods led to the discovery of the BPS/CFT correspondence, a broad arena tightening together BPS observables, 2d CFT, operator algebras, quantum integrability and much more. The aim of the workshop is to discuss the latest developments along this vast research line and its ramifications, including the analysis of future perspectives and where it is heading to.

Topics

• Dualities in SUSY gauge theory and string theory

• Low dimensional CFT, TQFT and quantum modularity

• Geometric Langlands duality, KZ equations and connection problems

• Higher-dimensional gauge theories and enumerative geometry

• Stringy quantum algebras

• Quantum integrable many-body systems and spin chains

• SUSY localization, supermatrix models and refined topological strings

 

Invited speakers

H. Aldarak	Dijon
N. Babinet	Dijon
J.E. Bourgine	Melbourne
I. Coman	Amsterdam
M. Dedushenko	SCGP
P. Descombes	Sorbonne
M. Hallnas	Chalmers
N. Haouzi	IAS
S. Hohenegger	Lyon
C. Iossa	SISSA
O. Khlaif	Birmingham
M. Kool	Utrecht
P. Koroteev	Berkeley
B. Le Floch	CNRS/Sorbonne
C. Paletta	Trinity
B. Pioline	CNRS/Sorbonne
S. Purkayastha	Dijon
P. Putrov	ICTP
H. Rosengren	Chalmers
M. Zabzine	Uppsala
Y. Zenkevich	SISSA

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

lundi 27 juin

09:30 Welcome!

1 Diagrammatic Expansion of Non-perturbative Little String Free Energies

Diagrammatic Expansion of Non-perturbative Little String Free Energies
Abstract: In this talk I discuss the non-perturbative free energy of a class of Little String Theories of A-type, which are engineered by N parallel M5-branes on a circle. Exploiting non-perturbative symmetries of these theories, I provide evidence to leading instanton order (from the perspective of the low energy U(N) gauge theory) for a decomposition, which resembles a Feynman diagrammatic expansion: external states are given by expansion coefficients of the N = 1 BPS free energy and a quasi-Jacobi form that governs the BPS-counting of an M5-brane coupling to two M2-branes. The effective coupling functions can be written as a simple combination of two-point functions of a single free scalar field on the torus. To further instanton orders, a decomposition of the free energy in terms of higher point functions with the same external states is still possible but a priori not unique. The loop corrections appearing in this picture can be linked to dihedral graph functions with bivalent vertices, which suggests an interpretation in terms of disconnected graphs.

Orateur: Stefan Hohenegger (Lyon)

Hohenegger.pdf

2 Random partitions and gauge group integrals: recent results

Random matrix integrals have a natural connection with the Schur measure on random partitions. In this talk I briefly highlight recent work in this direction, in particular focusing on a generalization of the Gross-Witten-Wadia model to classical gauge groups.

Orateur: Souradeep Purkayastha (IMB)

Purkayastha.pdf

11:30 Coffee break

3 What can little strings teach us about the geometric Langlands program?

In its simplest incarnation, the geometric Langlands program was defined by Beilinson and Drinfeld in the late 90’s as relating, on one side, a flat connection on a Riemann surface, and on the other side, a more sophisticated structure known as a D-module. Since its inception, this conjectured correspondence has been a highly active and fruitful topic of research both for mathematicians and theoretical physicists. In this talk, we will review a generalization of the correspondence known as the quantum q-Langlands program, due to Aganagic-Frenkel-Okounkov, which establishes an isomorphism between q-deformed versions of conformal blocks, for a W-algebra on one side, and a Langlands dual affine Lie algebra on the other side. We will then extend the correspondence, and invoke physical arguments from six-dimensional little string to give a precise mathematical formulation of ramification, or adding punctures on the Riemann surface in the q-Langlands program. We will also comment on the CFT limit; for instance, when the Lie algebra is specialized to be sl(2), one obtains a new (dual) perspective on recent results of Nekrasov and Tsymbaliuk.

Orateur: Nathan Haouzi (IAS)

12:50 Lunch break

4 Shifted quantum groups and matter multiplets in supersymmetric gauge theories

Orateur: Jean-Emile Bourgine (Melbourne)

15:00 Coffee break

15:30 Discussion session

mardi 28 juin

5 Non-semisimple TQFTs and BPS q-series

In my talk I will describe a relation between the 3d Topological Quantum Field Theories (TQFTs) of Blanchet--Costantino--Geer--Patureau-Mirand, constructed from a non-semisimple category of representations of a quantum group, and counting of BPS states in a 6d (2,0) superconformal field theory compactified on a 3-manifold with a topological twist.

Orateur: Pavel Putrov (ICTP)

6 Black hole perturbations from Liouville correlators

Reversing the logic of the bootstrap approach in Liouville CFT we explicitly compute the connection formulae for degenerate conformal blocks. In the semiclassical limit of the theory, this amounts to solving the connection problem of Fuchsian ODEs. Generalizing to irregular insertions we solve as well for various confluences. Concentrating on the Heun equations, we can solve the wave equations of a large class of gravitational backgrounds. Indeed, when the wave equation of a black hole or a microstate is separable, it often reduces to Heun equations, and exact connection formulae give access to several quantities. We consider a 4d Kerr black hole and an AdS5 spherically symmetric black hole and compute relevant objects. Crucially, everything is computed in terms of combinatorial objects exploiting the AGT duality.

Orateur: Cristoforo Iossa (SISSA)

7 Factorizability of the partition function of the supermatrix model

In this talk we will focus on the supermatrix model which can be seen as a specific case of two-matrix models. The former has been introduced long time ago and it has been noticed or suggested that its properties might not drastically differ from ordinary one-matrix model, at least in the planar limit. We want here to present how the partition function of the supermatrix model actually differs from the ordinary one and how its factorized form can lead to further questions in terms of non-planar graphs and perturbative corrections. To illustrate these problems we will make connection with the Fermi gas description of ABJ theory introduced by Marino and Putrov.

Orateur: Nicolas Babinet (IMB)

11:30 Coffee break

8 Insights from the quantum modularity of 3-manifold invariants

A recently proposed class of topological 3-manifold invariants Zˆ[M] which admit series expansions with integer coefficients has been a focal point of much research over the past few years, proving themselves ubiquitous in a wide range of contexts. They were originally defined physically as an index which computes the BPS spectra of certain supersym- metric quantum field theories in three dimensions, and associated to 3-manifolds M through the 3d-3d correspondence. Mathematically, they have also been shown to possess curious number- theoretic features, giving examples of quantum modular forms. After reviewing some of these new developments, here I explore certain higher rank extensions and highlight emerging fea- tures of the corresponding Zˆ invariants, such as nested relations with respect to their rank and a hidden modular structure.

Orateur: Ioana Coman (Amsterdam)

Coman.pdf

12:50 Lunch break

9 From SUSY gauge theories to generalized cohomology to Bethe ansatz

Supersymmetric ground states and supersymmetric boundary conditions in gauge theories with four supercharges are related to equivariant cohomology theories of their spaces of vacua. When we focus on 3d theories compactified on the elliptic curve, the relevant cohomology theory is elliptic, and upon degenerations to 2d and 1d it reduces to K-theory and de Rham cohomology. Via this connection, certain geometric constructions of stable envelopes due to Okounkov and collaborators receive interpretations as supersymmetric interfaces in gauge theories. The linear actions of infinite-dimensional Hopf algebras (Yangians, quantum loop, and elliptic) on the cohomology, constructed via stable envelopes in the literature, are interpreted as actions on the ground states of families of quantum field theories. Some of the generators are realized as interfaces between different QFTs in a family. These structures underlie what is known as the Bethe/Gauge correspondence of Nekrasov-Shatashvili, in which the space of supersymmetric ground states of a family of QFTs is identified with the Hilbert space of a certain integrable model. I will review some of the developments in this area.

Orateur: Mykola Dedushenko (SCGP)

Dedushenko.pdf

15:00 Coffee break

15:30 Discussion session

mercredi 29 juin

10 XYZ correlations and Painlevé VI

Baxter solved the XYZ spin chain in the sense of computing the free energy in the infinite lattice limit. For special parameter values, the chain has an underlying supersymmetry. It is then possible to obtain exact results even for finite size systems. In the special case of the XXZ chain, this is related to very interesting combinatorics (e.g. the alternating-sign-matrix and Razumov-Stroganov ex-conjectures). For the more general XYZ spin chain, less is known. We will describe how nearest neighbour correlations for finite length supersymmetric XYZ spin chains can be computed explicitly in terms of tau functions of Painlevé VI. This is joint work with Christian Hagendorf (Louvain-la-Neuve).

Orateur: Hjalmar Rosengren (Chalmers)

Rosengren.pdf

11 Explicit computations for spin chains and the remarkable operator Co-derivative

In this talk, we are going to show the power of the operator Co-derivative in converting complicated computations into combinatorial relation. To do so, I will consider the twisted spin chain with the underlying Lie algebra gl(n), and show how to obtain the (T,Q) operators using the D-diagrams that arise from using the Co-derivative on the generating series of the characters. Then, we will consider the case when the underlying Lie algebra is so(2n), show how the operator Co-derivative changes in this case and find the (T,Q) operators in the symmetric case.

Orateur: Helal Aldarak (IMB)

AlDarak.pdf

12 Bethe ansatz solution for a new integrable open quantum system

In nature, the interaction of a system with the environment cannot be avoided. If the response of the environment is Markovian, the density matrix will evolve through the Lindblad Master equation: dependent on the Hamiltonian of the system and a jump operator describing the coupling to the environment. In PRL 126.24 (2021): 240403, we gave a partial classification of Yang Baxter Integrable interacting systems, including several new models with interesting features. In this talk, I will focus on one of the models (model B3): I will give the analytical expression of the Non Equilibrium steady states and also its physical properties. I will also show how to solve this model via the nested Algebraic Bethe Ansatz method.

Orateur: Chiara Paletta (Trinity)

Paletta .pdf

11:30 Coffee break

13 Irrelevant deformations by higher-form symmetries

Zamolodchikov and Smirnov discovered how 2d QFTs can be deformed by an antisymmetric combination of two conserved currents. The landmark example is the TTbar deformation constructed from the stress tensor. After explaining which symmetries these deformations preserve, I will explain higher-dimensional generalizations based on higher-form symmetries. This leads to a curious way to couple a compact scalar field to any theory with U(1) flavour symmetry.

Orateur: Bruno Le Floch (Sorbonne)

LeFloch.pdf

12:50 Lunch break

jeudi 30 juin

14 Dendroscopy of BPS states

The spectrum of BPS states in D=4 supersymmetric field theories and string vacua famously jumps across codimension-one walls in vector multiplet moduli space. The Attractor Flow Tree conjecture postulates that the BPS index Ω(γ,z) for given charge γ and moduli z can be reconstructed from the “attractor indices” Ω(γi) counting BPS states of charge γi in their respective attractor chamber, by summing over all possible decompositions γ=Σi γi and over stable decorated rooted flow trees. Physically, flow trees provide a mesoscopic representation of BPS states as nested multi-centered bound states of elementary constituents. I will present a rigorous version of this formula in the context of quiver quantum mechanics, which governs the BPS spectrum in type IIA string theory compactified on certain conical Calabi-Yau threefolds, in the vicinity of orbifold-type points in Kahler moduli space. Moving away from such orbifold points requires generalizing the flow tree formula from the Abelian category of quiver representations to the derived category of the same. I will present recent progress in this direction in the simplest case of local P2, and argue that its global BPS spectrum and flow tree structure can be deduced from a scattering diagram with simple initial data.

Orateur: Boris Pioline (Sorbonne)

Pioline.pdf

15 Donaldson-Thomas invariants of toric quivers

A toric localization formula for numerical Donaldson-Thomas invariants, giving the virtual Euler number of moduli spaces which are critical lcus, is known since Graber and Pandharipande. We provide here a refining of this formula for cohomological Donaldson-Thomas invariants, which can be seen as a 'virtual version of Bialinicky Birula decomposition', giving the virtual cohomology of the attracting variety as a shifted sum of the virtual cohomology of the fixed components. We show how this formula gives a refining of the computation of numerical DT invariants of toric quivers by enumerating pyramids provided by Mozgovoy and Reineke.

Orateur: Pierre Descombes (Sorbonne)

16 Testing 3d IR Dualities Using Algebraic Geometry

In this talk I will revisit some well-studied 3d IR dualities including Aharony Duality and that of Giveon-Kutasov. I will also discuss some recently proposed ones and show how one can connect these dualities together. In the second part, I will show how, using algebraic geometry and supersymmetric localization, one can explicitly compute the twisted supersymmetric index on any Riemann surface. One can then test these dualities by matching twisted indices on both sides of the duality. Background Chern-Simons terms play a crucial role in this story.

Orateur: Osama Khlaif (Birmingham)

11:30 Coffee break

17 Equivariant indices and geometry

Orateur: Maxim Zabzine (Uppsala)

12:50 Lunch break

18 The Geometry of Magnificent Four

Motivated by super-Yang–Mills theory on a Calabi–Yau 4-fold, Nekrasov and Piazzalunga assigned weights to r-tuples of solid partitions (4-dimensional piles of boxes) and conjectured a formula for their weighted generating function. We define K-theoretic virtual invariants of Quot schemes of 0-dimensional quotients of Or on affine 4-space by realizing them as zero loci of isotropic sections of orthogonal bundles. Using the Oh–Thomas localization formula, we recover Nekrasov–Piazzalunga’s weights. Applying ideas from Okounkov in the 3-dimensional case, we prove Nekrasov-Piazzalunga’s formula. Joint work with J. Rennemo.

Orateur: Martijn Kool (Utrecht)

Kool.pdf

15:00 Coffee break

15:30 Discussion session

19:00 Social dinner

vendredi 1 juillet

19 Branes and DAHA Representations

Orateur: Peter Koroteev (Berkeley)

20 Refined topological strings and qq-characters

Using the identification of certain topological string amplitudes with R-matrices of Ding-Iohara-Miki algebras we find that qq-charaters can be constructed in a very simple way from degeneration of certain open refined topologcial string amplitudes. We illustrate the construction with several examples.

Orateur: Yegor Zenkevich (SISSA)

11:30 Coffee break

21 Super-Macdonald polynomials: Orthogonality properties and physical interpretations

Macdonald polynomials are two-parameter generalizations of Schur polynomials, with important connections to representation theory, geometry as well as integrable systems. In particular, they provide eigenfunctions of Macdonald-Ruijsenaars difference operators, which define an integrable relativistic quantum many-body system. This talk is focused on the so-called super-Macdonald polynomials, which generalize the Macdonald polynomials to two kinds of variables and formally provide eigenfunctions of the deformed Macdonald-Ruijsenaars operators first introduced by Chalykh and Sergeev and Veselov. I will present somewhat non-standard orthogonality properties of the super-Macdonald polynomials, a related Hilbert space interpretation and raise the question of possible physical interpretations.

Orateur: Martin Hallnäs (Chalmers)

12:50 Lunch break

14:00 Discussion session