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

Beginning on Sunday, October 15 and ending Friday October 20, 2023

All times in Banff, Alberta time, MDT (UTC-6).

Sunday, October 15 | |
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16:00 - 17:30 | Check-in begins at 16:00 on Sunday and is open 24 hours (Front Desk - Professional Development Centre) |

17:30 - 19:30 |
Dinner ↓ A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building. (Vistas Dining Room) |

20:00 - 22:00 |
Informal gathering ↓ BIRS Lounge, PDC 2nd floor (Other (See Description)) |

Monday, October 16 | |
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07:00 - 08:45 |
Breakfast ↓ Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |

08:45 - 09:00 |
Introduction and Welcome by BIRS Staff ↓ A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions. (TCPL 201) |

09:00 - 10:00 |
Justin Sawon: Complex Lagrangians ↓ 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). (TCPL 201) |

10:00 - 10:30 | Coffee Break (TCPL Foyer) |

10:30 - 11:30 |
Joel Kamnitzer: Symplectic Resolutions, Coulomb Branches, and 3d Mirror Symmetry ↓ 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. (TCPL 201) |

11:30 - 13:00 |
Lunch ↓ Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |

14:00 - 14:20 |
Group Photo ↓ 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! (TCPL Foyer) |

14:30 - 15:00 |
Formation of Thematic Discussion Groups ↓ 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). (TCPL 201) |

15:00 - 15:30 | Coffee Break (TCPL Foyer) |

15:25 - 16:25 |
Hulya Arguz: Mirror Symmetry and Cluster Varieties ↓ 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. (TCPL 201) |

16:30 - 17:30 | Monica Kang: Finding Isomorphic Superconformal Field Theories (TCPL 201) |

17:30 - 19:30 |
Dinner ↓ A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building. (Vistas Dining Room) |

20:00 - 21:00 | Ron Donagi: In Search of Mirror Partners for Hitchin Systems. An Introduction (TCPL 201) |

Tuesday, October 17 | |
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07:00 - 08:45 |
Breakfast ↓ Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |

09:00 - 10:00 |
Jennifer Brown: Defects and Quantum Character Stacks ↓ This talk is motivated by a small difference in two actions which leads to the need for defects in quantized character stacks.
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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.
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Familiarity with skeins and stacks will not be assumed. (TCPL 201) |

10:00 - 10:30 | Coffee Break (TCPL Foyer) |

10:30 - 11:30 |
Richard Rimanyi: 3d mirror symmetry for stable envelopes via geometric fusion ↓ 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. (TCPL 201) |

11:30 - 13:00 |
Lunch ↓ Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |

14:00 - 15:00 |
Takuro Mochizuki: Asymptotics of the Hitchin Metric on the Moduli of Higgs Bundles ↓ 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. (TCPL 201) |

15:00 - 15:30 | Coffee Break (TCPL Foyer) |

15:25 - 16:25 | Marina Logares: Hitchin Systems (TCPL 201) |

16:30 - 17:30 |
Andrey Smirnov: Frobenius structures for quantum differential equations and mirror symmetry ↓ 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. (TCPL 201) |

17:30 - 19:30 |
Dinner ↓ A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building. (Vistas Dining Room) |

20:00 - 21:00 | Vincent Bouchard: Cluster Algebras, tau-Functions, and Topological Recursion. An Introduction (TCPL 201) |

Wednesday, October 18 | |
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07:00 - 08:45 |
Breakfast ↓ Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |

09:00 - 10:00 |
Kenji Fukaya: Exact Immersed Lagrangian Floer Theory of the Cotangent Bundle ↓ 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. (TCPL 201) |

10:00 - 10:30 | Coffee Break (TCPL Foyer) |

10:30 - 11:30 |
Vladimir Fock: Cluster Integrable Systems and the Tame Symbol ↓ 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. (TCPL 201) |

11:30 - 13:00 |
Lunch ↓ Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |

13:30 - 17:30 | Free Afternoon (Banff National Park) |

17:30 - 19:30 |
Dinner ↓ |

20:00 - 21:00 |
Thematic Group Discussions ↓ We meet in TCPL 201, TCPL 202, and TCPL 102/106 (Other (See Description)) |

Thursday, October 19 | |
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07:00 - 08:45 |
Breakfast ↓ |

09:00 - 10:00 | Joerg Teschner: Tau-Functions and Cluster Algebra Structures on the Space of Stability Conditions (TCPL 201) |

10:00 - 10:30 | Coffee Break (TCPL Foyer) |

10:30 - 11:30 | Masahito Yamazaki: Hitchin Moduli Space and 4d Chern-Simons Theory (TCPL 201) |

11:30 - 13:00 |
Lunch ↓ |

14:00 - 15:00 |
Hiroshi Iritani: Fourier Analysis of Equivariant Quantum Cohomology ↓ 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. (TCPL 201) |

15:00 - 15:30 | Coffee Break (TCPL Foyer) |

15:25 - 16:25 |
Kohei Iwaki: Nonlinear Stokes Phenomenon for Painlev ́e tau-Function and Topological Recursion ↓ 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]. (TCPL 201) |

16:30 - 17:30 |
Pierrick Bousseau: Holomorphic Floer Theory and Donaldson-Thomas Invariants ↓ 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. (TCPL 201) |

17:30 - 19:30 |
Dinner ↓ |

20:00 - 21:00 |
Thematic Group Discussions ↓ We meet in TCPL 201, TCPL 202, and TCPL 102/106 (Other (See Description)) |

Friday, October 20 | |
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07:00 - 08:45 |
Breakfast ↓ |

09:00 - 10:00 |
Szilárd Szabó: Separation of Variables for Rank 2 Irregular Connections over Curves ↓ 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. (TCPL 201) |

10:00 - 10:30 | Coffee Break (TCPL Foyer) |

10:30 - 11:00 |
Checkout by 11AM ↓ 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. (Front Desk - Professional Development Centre) |

11:00 - 12:00 | Motohico Mulase: Final Discussions (TCPL 201) |

12:00 - 13:30 | Lunch from 11:30 to 13:30 (Vistas Dining Room) |