Quantization in Representation Theory, Derived Algebraic Geometry, and Gauge Theory

16 Sept 2024, 09:00 → 20 Sept 2024, 18:30 Europe/Zurich

SRS

Andrea Appel (Università di Parma), Francesco Sala (Università di Pisa), Giovanni Felder, Lara Bossinger (Universidad Nacional Autónoma de México), Mauro Porta (Université de Strasbourg), Olivier Schiffmann (Université Paris-Saclay)

Quantization is the mathematical tool to understand the transition between classical and quantum mechanics. Intuitively, classical mechanics is the limit of quantum mechanics as the quantum parameter tends to zero and, conversely, quantization is “an algorithm by which a quantum system corresponds to a classical dynamical one” (F.A. Berezin).

 

The ultimate goal of this workshop is to provide a survey of the most recent trends in mathematics and physics revolving around this idea. We will focus in particular on novel research directions in representation theory and moduli spaces aiming at a unified view built on new inputs from derived algebraic geometry. The workshop will pivot on the following main topics. 

• Quantum Groups, in particular Yangians and quantum loop algebras, and their role in mathematical physics and in algebraic geometry as highlighted in the recent work following Costello-Yamazaki-Witten, Maulik-Okounkov, and Schiffmann-Vasserot.
• Coulomb and Higgs Branches, in the approach developed by Braverman-Finkelberg-Nakajima, and their various applications ranging from affine Grassmannians and cluster theory to symplectic duality and shifted Yangians.
• Deformation Quantization, studied through the lenses of Derived Algebraic Geometry, as in the recent work by Calaque, Safronov, et al.

The above influential research directions are currently having a strong broad impact in geometry, algebra, and mathematical physics. Their originality crucially resides in the use of new methods originally developed in derived algebraic geometry. The latter can eventually be regarded as the right framework for the solution of many problems arising in quantization theory.

 

The workshop is by invitation only.

 

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Speakers.

 

Léa Bittmann (Université de Strasbourg)
Bozec Tristan (Université Angers) 
Li Wei (Institute of Theoretical Physics, Chinese Academy of Sciences)
Tudor Dimofte (University of Edinburgh)
Sachin Gautam (The Ohio State University)
Iva Halacheva (Northeastern University)
Benjamin Hennion (Université de Paris-Saclay)
David Hernandez (Université Paris Cité)
Hiraku Nakajima (Kavli IPMU, the University of Tokyo)
Gabriele Rembado (Université de Montpellier)
Sarah Scherotzke (University of Luxembourg)
Claudia Scheimbauer (TU Munchen)
Valerio Toledano Laredo (Northeastern University)
Ben Webster (University of Waterloo)
Masahito Yamazaki (Kavli IPMU, the University of Tokyo)
Yaping Yang (University of Melbourne)

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Organizers.

Andrea Appel (Università di Parma)
Lara Bossinger (Universidad Nacional Autónoma de México)
Giovanni Felder (ETH)
Mauro Porta (Université de Strasbourg)
Francesco Sala (Università di Pisa)
Olivier Schiffmann (Université de Paris Saclay)

 

Monday 16 September

09:10 Welcome

1 Monoidal Jantzen Filtrations and quantization of Grothendieck rings

We introduce a monoidal analogue of Jantzen filtrations in the framework of monoidal categories with generic braidings. It leads to a deformation of the multiplication of the Grothendieck ring. We conjecture, and we prove in many remarkable situations, that this deformation is associative so that our construction yields a quantization of the Grothendieck ring. This is a joint work with Ryo Fujita.

Speaker: David Hernandez (Université Paris Cité)

10:15 Coffee break

2 The derived critical locus and applications to COHAs

In a first part I will explain how the -1 shifted symplectic structure on the derived critical locus matches the one induced (thanks to Brav--Dyckerhoff) by a 3-Calabi--Yau one on the deformed completion (as defined by Ginzburg and more generally Keller) of a finite type dg category. This is part of a joint work with Damien Calaque and Sarah
Scherotzke.
In a second part I will try to explain how this result induces various examples of COHAs, following constructions of Kinjo, Park, Masuda, Safronov.

Speaker: Tristan Bozec (Université d'Angers)

12:30 Lunch

3 The non-commutative AKSZ construction

We can construct oriented topological extended field theories using the AKSZ construction in derived algebraic geometry. This was first suggested by Panted-Toen-Vaque-Vezzosi and accomplished by Calaque-Hausend-Scheimbaur. The AKSZ construction is a version of the classical AKSZ filed theories in physic. In [CHS] it is given as a symmetric monoidal functor from a higher category of oriented stacks to a higher category of symplectic stacks. We will give a non-commutative version of this construction, which associates to every finite dg category a symmetric monoidal functor with target a higher category of n-Calabi-Yau categories, refining the commutative version of the AKSZ construction.

Speaker: Sarah Scherotzke (University of Luxembourg)

16:45 Coffee break

4 Quantum moduli spaces of meromorphic connections on Riemann surfaces

Let $G$ be a complex reductive Lie/algebraic group. Certain (de Rham) moduli spaces of regular singular meromorphic connections, defined on principal G-bundles over the Riemann sphere, have natural (complex) Poisson/symplectic structures. These can be deformation-quantised, and generically lead to spaces of (co)invariants for tensor products of Verma modules for $\mathfrak{g} = Lie(G)$: in turn, they are related to spaces of (co)vacua for the affine version of g, towards the usual conformal blocks of the Wess--Zumino--Novikov--Witten (WZNW) model in 2d conformal field theory.
Moreover, upon deforming the position of the simple poles of the meromorphic connections in admissible fashion, a Poisson/symplectic braid-group action arises on the (Betti) moduli spaces of monodromy data, viz. the $G$-character varieties of punctured spheres; and this action was later interpreted as the semiclassical limit of the Drinfel'd--Kohno braiding, i.e. precisely the monodromy of the flat vector bundle of WZNW conformal blocks.

In this talk we will aim at a review of part of this story, and then present extensions about irregular singular meromorphic connections.
In alphabetical order, this is past/present work with P. Boalch, D. Calaque, J. Douçot, G. Felder, M. Tamiozzo, and R. Wentworth.

Speaker: Gabriele Rembado (Université de Montpellier)

19:00 Welcome Aperitif

19:30 Dinner

Tuesday 17 September

5 S-dual of Hamiltonian G spaces and relative Langlands duality

Please, consult the presentation material section.

Speaker: Hiraku Nakajima (Kavli IPMU, the University of Tokyo)

nakajima_abstract.pdf

10:15 Coffee break

6 Higher spin representations of the Yangian of $\mathfrak{sl}_2$ and R-matrices

For the Yangian of $\mathfrak{sl}_2$, higher spin representations are tensor products of the evaluation pullback of the $\ell_i+1$-dimensional irreducible representations of $\mathfrak{sl}_2$, where $\ell_i$ are the highest weights. In my talk, I will give a geometric realization of the higher spin representations in terms of the critical cohomology of representations of the quiver with potential of Bykov and Zinn-Justin.  I will also talk about the construction of R-matrices via the lattice model and the weight functions.

This is based on my joint work with Paul Zinn-Justin.

Speaker: Yaping Yang (University of Melbourne)

12:30 Lunch

7 Bethe algebras, cacti, and crystals

The Yangian of a reductive Lie algebra contains a family of maximal commutative subalgebras—the Bethe subalgebras—parametrized by regular elements of the maximal torus. In the case of $\mathfrak{gl}(n)$, it is known that this family extends to a larger one indexed by points of the Deligne-Mumford compactification of $M(0,n+2)$. For any point $C$ in the real locus of this parameter space, and a fixed tame Yangian representation $V$, the Bethe subalgebra $B(C)$ acts on $V$ with simple spectrum. I will discuss the structure of the resulting unramified covering—with fiber over C given by the set of eigenlines for the action of $B(C)$, which can be identified with a collection of Gelfand-Tsetlin keystone patterns carrying a $\mathfrak{gl}(n)$-crystal structure, as well as the monodromy action realized by a type of cactus group. This is joint work with Anfisa Gurenkova and Leonid Rybnikov.

Speaker: Iva Halacheva (Northeastern University)

16:45 Coffee break

8 Gluing invariants of Donaldson--Thomas type

Donaldson--Thomas invariants are numerical invariants associated to Calabi--Yau varieties. They can be obtained by glueing singularity invariants from local models of a suitable moduli space endowed with a (-1)-shifted symplectic structure.
By studying the moduli of such local models, we will explain how to recover Brav--Bussi--Dupont--Joyce--Szendroi's perverse sheaf categorifying the DT-invariants, as well as a strategy for gluing more evolved singularity invariants, such as matrix factorizations.
This is joint work with M. Robalo and J. Holstein.

Speaker: Benjamin Hennion (Université de Paris-Saclay)

19:30 Dinner

Wednesday 18 September

12:30 Lunch

16:45 Coffee break

9 3d mirror symmetry

I'll give an introduction (or update, for those who've been introduced) to 3d mirror symmetry from the perspective of a mathematician.

Speaker: Ben Webster (University of Waterloo)

Les Diablerets-2024-messy.pdf

Les Diablerets-2024-revised.pdf

18:15 Break

10 Jeffrey-Kirwan Residues and BPS Crystals/Algebras

In this presentation we discuss approaches to enumerative counting problems of BPS states via the Jeffrey-Kirwan formula of supersymmetric indices. This work is based on a paper with Jiakang Bao and Rak-Kyeong Seong, and another paper with Jiakang Bao.

Speaker: Masahito Yamazaki (Kavli IPMU, the University of Tokyo)

19:30 Dinner

Thursday 19 September

11 BPS algebras for 4D N=2 theories and their line defects.

I will first explain the BPS algebras for 4D N=2 theories in terms of cohomological Hall algebras. We conjecture that for a theory whose BPS spectrum admits a quiver description that is 2-acyclic with infinitely-mutable potential, the BPS algebra reduces to spherical shuffle algebra. I will then explain how to study the 1/2-BPS line defects in the theory as the bimodules of its BPS algebra.

Speaker: Li Wei (Institute of Theoretical Physics, Chinese Academy of Sciences)

10:15 Coffee break

12 R-matrices of affine Yangians

Let $\mathfrak{g}$ be an affine Lie algebra and $Y_{\hbar}(\mathfrak{g})$ be the Yangian associated to $\mathfrak{g}$. Unlike its finite counterpart, the affine Yangian is not known to possess a universal $R$--matrix. In particular, one does not immediately have solutions of the quantum Yang--Baxter equation on an appropriate category of representations of the affine Yangian. The sole exception is the Maulik--Okounkov theory, which provides rational solutions to QYBE, on representations coming from the geometry of quiver varieties.

In this talk I will present a construction of two meromorphic $R$--matrices, related by a unitarity relation, for category $\mathcal{O}$ representations of $Y_{\hbar}(\mathfrak{g})$. I will show that our $R$--matrices can be normalized on highest--weight representations in order to obtain rational solutions to QYBE. This talk is based on joint works with Andrea Appel, Valerio Toledano Laredo and Curtis Wendlandt.

Speaker: Sachin Gautam (The Ohio State University)

12:30 Lunch

13 Stokes phenomena, quantum groups and Poisson-Lie groups

Quantum groups have long been known to be related to Conformal Field Theory through the Knizhnik-Zamolodchikov (KZ) equations. This Betti role as natural receptacles of monodromy has been significantly expanded in recent years by including the Casimir equations which are dual to the KZ ones. This has led to a novel construction of quantum groups from the dynamical KZ (DKZ) equations. Unlike their precursors, these have irregular singularities and therefore exhibit Stokes phenomena which describe the discontinuous change of asymptotic of solutions near singular points. In particular, the Stokes matrices of the simplest DKZ equations are R-matrices of the corresponding quantum group.

In a parallel development, Boalch constructed the Poisson structure on the dual $G^*$ of a complex reductive group $G$ by using Stokes phenomena for the simplest irregular connection on the trivial $G$-bundle over $\mathbb{P}^1$. This transcendental linearization of $G^*$ is particularly tantalizing in that it is very close in spirit to the above construction of quantum groups.

I will explain how quantum groups arise from the dynamical KZ equations, describe Boalch’s construction, and obtain a precise link between these two uses of Stokes phenomena, by showing that the latter construction can be obtained as a semiclassical limit of the former.

Speaker: Valerio Toledano Laredo (Northeastern University)

2024 09 Les Diablerets.pdf

16:45 Coffee break

14 The Dyck path algebra associated to a surface

The Dyck algebra $\mathbb{A}_{q}$ and the double Dyck path algebra $\mathbb{A}_{q,t}$ were introduced by Carlsson and Mellit as part of their proof of the shuffle conjecture and the latter is known to be related to the type A double affine Hecke algebra. In this talk, we will see how to define a skein theoretic version of Dyck path algebra $\mathbb{A}(\Sigma)$ associated to a surface $\Sigma$. We will focus on the following cases: the disk, the annulus and the torus. These last two give variants of the Dyck and double Dyck path algebra, respectively. By studying these algebras further, we give a basis of $\mathbb{A}(D)$, together with a tableau interpretation, and conjecture one for $\mathbb{A}(A)$. This is based on a joint work in progress with A. Mellit and C. Novarini.

Speaker: Léa Bittmann (Université de Strasbourg)

19:30 Raclette Dinner

Friday 20 September

15 Assembly of constructible factorization algebras

Factorization algebras describe the observables of a perturbative QFT, but also algebraic objects such as ($A_\infty$-)algebras, bimodules, and $E_n$-algebras. As such, they should satisfy certain properties: for instance, it has long been “known” that the assignment taking a stratified manifold to its category of constructible factorization algebras satisfies gluing, i.e., is itself a sheaf. Unfortunately, this and other related facts about factorization algebras have long been “folklore knowledge”, but with no proofs available.
These are crucial ingredients for several constructions in the literature, one of them being higher Morita categories. These are many prominent examples of targets of (possibly relative) functorial field theories e.g. for Turaev-Viro theory and Reshetikin-Turaev theory.

In this talk, I will report on recent work with Eilind Karlsson and Tashi Walde, where we close some of these gaps in the literature, including the aforementioned gluing result. I will of course start with the big picture of why we would like such a result.

Speaker: Claudia Scheimbauer (TU Munchen)

10:00 Coffee break

16 Meromorphic tensor categories and shifted r-matrices

I'll discuss recent work (with Wenjun Niu and Victor Py, to appear) on the representation theory of meromorphic tensor categories, a.k.a. chiral categories. From a physical perspective -- our entry point -- these are categories of line operators in 3d holomorphic-topological theories, such as twists of 3d N=2 gauge theories. In the 3d N=2 examples, one expects their cyclic homology to be related to quantum K-theory. As categories, they look roughly like coherent sheaves or matrix factorizations on loop spaces. I'll explain some physical ways to access the chiral tensor product in such categories, with examples. Then, following the Koszul-duality approach of Costello-Paquette, I'll explain how they may be represented as modules for what roughly looks like a homologically-shifted Yangian: an A-infinity bialgebra with a chiral coproduct, whose Maurer-Cartan element behaves like an r-matrix.

Speaker: Tudor Dimofte (University of Edinburgh)

12:00 Lunch