Schedule for: 23w5068 - Complex Lagrangians, Mirror Symmetry, and Quantization

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

BIRS Lounge, PDC 2nd floor

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

We consider the problem of reconstructing a holomorphic symplectic manifold from a (complex) Lagrangian submanifold. In many cases this is impossible, but there are some interesting situations where it is possible. For instance, for large genus a general curve is not contained in a $K3$ surface, but a curve contained in a $K3$ surface will uniquely determine the surface (Mukai, Arbarello-Bruno-Sernesi, Feyzbakhsh). In higher dimensions, a holomorphic symplectic manifold fibred by Jacobians of curves is conjectured to be a Beauville-Mukai system, i.e., to come from a complete linear system of curves in a $K3$ surface. Then the Torelli Theorem and the results for $K3$ curves imply that the holomorphic symplectic manifold is uniquely determined by a single Jacobian fibre. Similarly for Hitchin systems there is evidence that a single spectral curve suffices to reconstruct the entire integrable system, via Eynard-Orantin topological recursion (Baraglia-Huang), and efforts have been made to extend this to other integrable systems (Chaimanowong-Norbury-Swaddle-Tavakol).

This is an introductory talk about symplectic duality and Coulomb branches. $$ $$ In the 21st century, there has been a great interest in the study of symplectic resolutions, such as cotangent bundles of flag varieties, hypertoric varieties, quiver varieties, and affine Grassmannian slices. Mathematicians, especially Braden-Licata-Proudfoot-Webster, and physicists observed that these spaces come in dual pairs: this phenomenon is known as 3d mirror symmetry or symplectic duality. In physics, these dual pairs come from Higgs and Coulomb branches of $3d$ supersymmetric field theories. In a remarkable 2016 paper, Braverman-Finkelberg-Nakajima gave a mathematical definition of the Coulomb branch associated to a $3d$ gauge theory. We will discuss all these developments, as well as recent progress building on the work of BFN. We will particularly study the Coulomb branches associated to quiver gauge theories: these are known as generalized affine Grassmannian slices.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

Depending on the interests of participants, the organizers are planning the following topics to be discussed: $$ $$ (1) In Search of Mirror Partners for Hitchin Systems. Coordinators: Eric Boulter (Saskatchewan) and Alex Cruz Morales (Bogota); (2) Cluster Algebras, $\tau$-Functions, and Topological Recursion. Coordinators: Elba Garcia-Failde (Sorbonne) and Dani Kaufman (Copenhagen); (3) Symplectic Topology and $3d$ Mirror Symmetry. Coordinators: Johanna Bimmermann (Bochum) and Orsola Capovilla-Searle (Davis). $$ $$ Other possibilities include Autumn School type short series of lectures by experts on specific topics, such as (4) Quiver Varieties, Hyperpolygons, and Moduli, by Steve Rayan (Saskatchewan).

We prove a correspondence between Donaldson-Thomas invariants of quivers with potential having trivial attractor invariants and genus zero punctured Gromov-Witten invariants of holomorphic symplectic cluster varieties. The proof relies on the comparison of the stability scattering diagram for the wall-crossing of Donaldson-Thomas invariants with a scattering diagram for punctured Gromov-Witten invariants via tropical geometry.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

This talk is motivated by a small difference in two actions which leads to the need for defects in quantized character stacks. $$ $$ We will start by studying quasicoherent sheaves on character stacks, then showing how skein categories give with quantization. With that context in hand we will dive into two actions - one giving the recursion relations of the colored Jones and the other induced by inclusion of the boundary of a 3-manifold. Finally, we'll explain how defects explain the relationship between the actions. $$ $$ Familiarity with skeins and stacks will not be assumed.

In this joint work with Tommaso Botta (ETH Zurich) we study the elliptic characteristic classes called stable envelopes introduced by M. Aganagic and A. Okounkov. Stable envelopes measure singularities, they geometrize quantum group representations, and they can be interpreted as monodromy matrices of certain differential or difference equations. We prove that for a rich class of holomorphic symplectic varieties---called bow varieties---their elliptic stable envelopes display a duality inspired by mirror symmetry in $$d=3, \mathcal{N}=4$$ quantum field theories. In the key step of our proof, we ``resolve'' large charge branes to a number of smaller charge branes. This phenomenon turns out to be the geometric counterpart of the algebraic fusion procedure. Along the way we discover more about the rich geometry of bow varieties, such as their Bruhat order and the elliptic Hall algebra structure on their stable envelopes.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

I will explain how the Hitchin metric is asymptotically related to another metric associated with Hitchin fibration. It is a refinement of a work of Laura Fredrickson.

There exists a well-known connection between the Kloosterman sum in number theory, and the Bessel differential equation. This connection was explained by B. Dwork in 1974 by discovering Frobenius structures in the p-adic theory of the Bessel equation. In my talk I will speculate that this connection extends to the quantum differential equations in quantum cohomology of Nakajima varieties. As an example, I will give an explicit conjectural description of the Frobenius structures for the quantum connections of $T^*Gr(k,n)$ and also $Gr(k,n)$. The traces of Frobenius structures are natural finite field analogs of the integral solutions of quantum differential equations known in mirror symmetry. In particular, for $Gr(k,n)$ we arrive at the exact B-model description of quantum connection discovered by Marsh and Rietsch.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

I will explain how the story of immersed Lagrangian Floer theory of Akaho-Joyce is related to the Legendrian contact homology in the immersed exact case. Then explain the relation to the study using generating functions. Finally propose some conjectured relation to micro support of sheaves. This talk is based on a joint work with A. Daemi.

Cluster integrable systems discovered by Goncharov and Kenyon are an approach to a large class of integrable systems, starting from Poncelet's porism discovered 200 years ago, to tops, to Toda lattices and their quantum versions (which are not yet very well understood so far), to lattice models, to Hofstadter's butterfly, and to many others. The cluster approach to those systems interprets their phase spaces either as the space of configurations of flags in an infinite dimensional space, or as the space of pairs consisting of a spectral curve and a line bundle on it. This dual approach allows us to study the systems in detail and to find their classical solutions. $$ $$ However, the quantization of these systems does not go as smoothly as their classical case. We will suggest a quasi-classical approach to those systems using a tool borrowed from number theory, namely the tame symbol. In its simplest incarnation, tame symbol is a multiplicative analogue of the residue. On the other hand it allows us to define the $\tau$-functions for finite gap solutions. We use the tame symbol to formulate the Bohr-Sommerfeld condition on the Lagrangian subvarieties for the integrable systems, thus giving the quasi-classical spectrum with the wave function given by dilogarithms. $$ $$ If time permits, and also in a Thematic Group Discussion session, I will talk about some elementary applications of the tame symbols.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

We meet in TCPL 201, TCPL 202, and TCPL 102/106

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Following a conjecture of Teleman, we relate the equivariant quantum cohomology of a smooth $T$-variety $X$ and the quantum cohomology of a symplectic reduction $X//T$ by Fourier transformation. We will also discuss applications to decomposition of quantum cohomology $\mathcal{D}$-modules for blowups.

I will propose a conjectural statement on the Stokes phenomenon for Painlev\'e $\tau$-function and the topological recursion partition function. Our claim is based on (i) a relation between the topological recursion and the Painlev\'e $\tau$-function, (ii) the exact WKB analysis of the isomonodromic quantum curve. This is based on a joint work with Marcos Marino: arXiv:2307.02080 [hep-th].

We present several expected properties of the holomorphic Floer theory of a holomorphic symplectic manifold. In particular, we propose a conjecture relat- ing holomorphic Floer theory of Hitchin integrable systems and Donaldson-Thomas invariants of non-compact Calabi-Yau 3-folds.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

We meet in TCPL 201, TCPL 202, and TCPL 102/106

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

I will present our recent joint work with A. Komyo, F. Loray and M.-H. Saito. Based on the existence and generic uniqueness of a cyclic vector, I will introduce a normal form for an irregular connection of rank $2$ over a curve. This leads to a generic Darboux chart of irregular de Rham moduli space. I give an example showing that this approach is suitable to study confluence of singular points.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.