# GRT learning seminar winter 2020: Symplectic Duality

Where? HU1018 The seminar has now moved online! The link to join the zoom chat is here.

When? Wednesday 12:00-2:00

Topic: The general theme is 'symplectic duality'. The list of references below indicates a rough outline of the topics we'll try to cover, with emphasis depending on participants' interests. Other useful references include the Warthog 2017 webpage and the videos from the 21st Midrasha. I'd recommend Braverman's first talk at the school for a broad overview of the subject. The link to last semester's learning seminar is here.

## Schedule

Day Speaker Title Abstract
Feb 5 Dylan Butson Overview I'll give a short review of Poisson geometry and filtered quantization, introduce a few important running examples, and outline some of the main structures involved in symplectic duality.
Feb 7 Ben Webster (S)ymplectic duality (research seminar) Around 2007, Braden, Licata, Proudfoot and I came up with the slightly daft idea that certain singular symplectic varieties come in dual pairs whose definition we could not explain, with relations between them that we struggled to articulate, let alone prove. By 2014, we were able to produce at least a coherent list of properties we expected to match in dual pairs, the most striking being a Koszul duality between associated category O's. This was either excellent or terrible timing (depending on your perspective), since immediately afterwards Braverman-Finkelberg-Nakajima produced a uniform construction of a huge number of examples, including almost all of those known to us. I'll take this talk as an opportunity to organize where we stand on the relationship between our work and BFN's, as well as the copious follow-up work which has appeared since their paper.
Feb 12 Hyungseop Kim Poisson deformations and Weyl groups We review definitions and properties of Poisson deformations of symplectic resolutions, which involve the notion of (birational) Weyl groups and chamber structures.
Feb 19 Dylan Butson Quantization of symplectic resolutions and category O I I'll review quantization of Poisson varieties, quantum Hamiltonian reduction, classification of formal and graded quantizations via formal geometry, and the relation to Poisson deformations in the case of symplectic resolutions.
Mar 4 Dylan Butson Quantization of symplectic resolutions and category O II I'll give the definition of category O for a quantized symplectic resolution, review some aspects of the theory of highest weight categories, and explain the highest weight structure on category O geometrically in several examples. We'll also try to give a preview of Koszul duality between categories O for dual pairs of symplectic resolutions, which will be the topic of the next few talks.
Mar 11 Balazs Elek Koszul Duality I We will introduce Koszul duality and try to understand it through several concrete examples including categories O and the symmetric and alternating algebras.
Mar 18 Balazs Elek Koszul Duality II We will continue to investigate the principal block of category O. We will discuss how one can define a grading on the endomorphism ring of a projective generator, and on the category itself.
Mar 18 Daniel Le Koszul Duality III Riemann—Hilbert and Beilinson–Bernstein give an equivalence between Category O and the category of perverse sheaves on the corresponding flag variety. I’ll discuss how descent of such sheaves to a finite field gives a grading on these categories.
Mar 25 Dylan Butson Koszul Duality IV I'll review Koszulity, and Koszul duality, for graded algebras and mixed categories. Further, I'll explain how parabolic-singular Koszul duality of Beilinson-Ginzburg-Soergel can be understood from the perspective of symplectic duality, and compute everything in the case of SL_3.
Apr 1 Shotaro Makisumi Towards a triply-graded Koszul duality for the Hecke category The "Hecke category" associated to a complex connected reductive (or Kac--Moody) group G participates in a monoidal "Koszul duality," which exchanges it with the category of "free-monodromic complexes" associated to the Langlands dual group. This duality categorifies the bijection between twisting and shuffling functors for the symplectic duality of T*(G/B). After discussing this duality, I will explain joint work with Matthew Hogancamp in which we propose to enhance it with a third grading. If time permits, I will discuss how in type A this duality is related to a (conjectural) q-t symmetry in link homology.
Apr 3 Vasily Krylov On isomorphisms between quiver varieties of type A and slices in the affine Grassmannian In my talk, I will discuss isomorphisms between quiver varieties of type A and transversal slices in the affine Grassmannian for $GL_d$. Such an isomorphism was first constructed by Mirkovi\’c and Vybornov. We will explain a geometric construction of isomorphisms between quiver varieties and transversal slices that follows by combining ideas of Braverman-Finkelberg and Nakajima. This approach will allow us to obtain explicit formulas for these isomorphisms. Time permitting we will explain why they coincide with Mirkovi ́c-Vybornov’s.
Apr 8 Anne Dranowski Another Mirkovic-Vybornov isomorphism Let O be a nilpotent orbit. Mirkovic and Vybornov defined a family of slices T transverse to O (as well as its closure) and showed that any given intersection \bar O \cap T is isomorphic to a certain affine quiver variety. The last talk relied on the high tech Bun(_G)-y description of the affine Grassmannian, Gr, to construct an isomorphism of a slice, W, in Gr and an affine quiver variety. In this talk we present a direct construction of the implicit isomorphism of O \cap T and W, relying on the elementary lattice description of Gr. We then discuss how all of these isomorphisms can be seen as a geometric version of the classical skew Howe duality.
Apr 15 Surya Raghavendran Slodowy varieties, Parabolic W-algebras, and an Introduction to Shifted Yangians Slodowy varieties are certain symplectic varieties related to Slodowy slices in reductive lie algebras. I'll introduce these gadgets and their quantizations which are certain generalizations of finite W-algebras. I'll end with an introduction to shifted Yangians via a result of Brundan-Kleschev that uses them to give presentations of finite W-algebras in gl_n.
Apr 22 Yehao Zhou Slices to orbits and shifted Yangians In this talk, we will discuss the thick affine Grassmannian, orbits and their transverse slices. We will explain the symplectic structure on Gr_G and deduce some properties of slices. Then we move on to a conjectural generator set of the ideal defining those slices and use these generators to show that there is a natural surjective map from Y^{\lambda}_{\mu} modulo \hbar to O(Gr^{\lambda}_{\mu}).
Apr 29 Joel Kamnitzer Symplectic duality between affine Grassmannian slices and quiver varieties Using the geometric Satake correspondence and the theory of quiver varieties, we will see how we can get two geometric realizations of tensor product representations of semisimple Lie algebras. Then we will see that the framework of symplectic duality can be used to understand this relationship. We will also examine skew Howe duality in this context.
May 13 Dylan Butson Introduction to the Coulomb branch construction I'll give some motivation from symplectic duality and topological field theory for the Braverman-Finkelberg-Nakajima (BFN) Coulomb branch construction. I'll review some aspects of sheaf theory on pro-finite type and/or ind schemes and define the Borel-Moore homology of the variety of triples, roughly following section 2 of their paper. I'll also review some formalism around factorization and convolution, define the Coulomb branch algebra, discuss some of its elementary properties, and explain its calculation in easy examples, roughly following sections 3 and 4 of their paper.
May 20 Dylan Butson Introduction to the Coulomb branch construction II
May 27 Stefan Dawydiak Coulomb branches and symplectic duality (at least for tori) Having seen the construction of the Coulomb branch associated to the pair (G,N) last week, we will pick up from there and follow approximately sections 3.5–4.5 of BFN II. Some main features from the first half will be the property 3(vii) (d) and Proposition 3.18, which we will use to illustrate the exchange of equivariant and Kahler parameters when passing from the Higgs branch to the Coulomb branch, providing the promised example of symplectic duality. Then in sections 4.1 to 4.5 we will see a hands-on description of the product structure on Borel-Moore homology of the BFN space for a torus, building on the vector space description we saw last week.
June 17 Justin Hilburn BFN Springer Theory Given a representation N of a reductive group G, Braverman-Finkelberg-Nakajima have defined a remarkable Poisson variety called the Coulomb branch. Their construction of this space was motivated by considerations from 3d gauge theories and symplectic duality. The coordinate ring of this Coulomb branch is defined as a convolution algebra, using a vector bundle over the affine Grassmannian of G. This vector bundle over the affine Grassmannian maps to the space of loops in the representation N. We study the fibres of this maps, which live in the affine Grassmannian. We used these BFN Springer fibres to construct modules for (quantized) Coulomb branch algebras. These constructions have a natural motivation coming from the category of line operators in A-twisted 3d N=4 theories. Which we will discuss.

## References

### Quantizations of symplectic resolutions and category O

Quantizations of conical symplectic resolutions I: local and global structure - Tom Braden, Nicholas Proudfoot, Ben Webster
Quantizations of conical symplectic resolutions II: category O and symplectic duality - Tom Braden, Anthony Licata, Nicholas Proudfoot, Ben Webster
On categories O for quantized symplectic resolutions - Ivan Losev
Symplectic singularities - A. Beauville
Symplectic singularities from the Poisson point of view - D. Kaledin
Poisson deformations of affine symplectic varieties - Yoshinori Namikawa
Fedosov quantization in algebraic context - R. Bezrukavnikov, D. Kaledin

### Koszul duality for category O

Gale duality and Koszul duality - Tom Braden, Anthony Licata, Nicholas Proudfoot, Ben Webster
Koszul duality between Higgs and Coulomb categories O - Ben Webster
Koszul duality patterns in representation theory - Beilinson, Ginzburg, Soergel
Langlands' philosophy and Koszul duality - Soergel
Quadratic duals, Koszul dual functors, and applications - Volodymyr Mazorchuk, Serge Ovsienko, Catharina Stroppel
Koszul duality and mixed Hodge modules - Pramod N. Achar, S. Kitchen

### The BFN 'Coulomb Branch' construction

Towards a mathematical definition of Coulomb branches of 3-dimensional N=4 gauge theories, II - Alexander Braverman, Michael Finkelberg, Hiraku Nakajima
Equivariant (K-)homology of affine Grassmannian and Toda lattice - Roman Bezrukavnikov, Michael Finkelberg, Ivan Mirković
Ring objects in the equivariant derived Satake category arising from Coulomb branches - Alexander Braverman, Michael Finkelberg, Hiraku Nakajima

### Nakajima varieties, slices to orbits in the affine Grassmannian, shifted Yangians, and finite W algebras

Coulomb branches of 3d N=4 quiver gauge theories and slices in the affine Grassmannian - Alexander Braverman, Michael Finkelberg, Joel Kamnitzer, Ryosuke Kodera, Hiraku Nakajima, Ben Webster, and Alex Weekes
Yangians and quantizations of slices in the affine Grassmannian - Joel Kamnitzer, Ben Webster, Alex Weekes, Oded Yacobi
On category O for affine Grassmannian slices and categorified tensor products - Joel Kamnitzer, Peter Tingley, Ben Webster, Alex Weekes, Oded Yacobi
A quantum Mirković-Vybornov isomorphism - Ben Webster, Alex Weekes, Oded Yacobi
Shifted Yangians and finite W-algebras - Jonathan Brundan, Alexander Kleshchev

### Connections to Enumerative Geometry

Instanton counting via affine Lie algebras I: Equivariant J-functions of (affine) flag manifolds and Whittaker vectors - Alexander Braverman
A finite analog of the AGT relation I: finite W-algebras and quasimaps' spaces - Alexander Braverman, Boris Feigin, Leonid Rybnikov, Michael Finkelberg
Gauge theory and mirror symmetry - Constantin Teleman
The role of Coulomb branches in 2D gauge theory - Constantin Teleman
Quantum cohomology of the Springer resolution - Alexander Braverman, Davesh Maulik, Andrei Okounkov
On the crossroads of enumerative geometry and geometric representation theory - Andrei Okounkov
The quantum Hikita conjecture - Joel Kamnitzer, Michael McBreen, Nicholas Proudfoot

### Connections to 'Physics'

Towards a mathematical definition of Coulomb branches of 3-dimensional N=4 gauge theories, I - Hiraku Nakajima
The Coulomb Branch of 3d N=4 Theories - Mathew Bullimore, Tudor Dimofte, Davide Gaiotto
Boundaries, Mirror Symmetry, and Symplectic Duality in 3d N=4 Gauge Theory - Mathew Bullimore, Tudor Dimofte, Davide Gaiotto, Justin Hilburn
Vortices and Vermas - Mathew Bullimore, Tudor Dimofte, Davide Gaiotto, Justin Hilburn, Hee-Cheol Kim

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