3d mirror symmetry learning seminar

A hyperkahler manifold X is naturally the target for a supersymmetric N=4 3d theory, and possesses two distinct topological twists, which give rise to two 3d TQFTs: the 3d A-model and the 3d B-model associated to X. By the cobordism hypothesis, these TQFTs are determined by a 2-category of boundary conditions; while the 3d B-model 2-category appears to be taking the form of quasi-coherent sheaves of categories, the 3d A-model remains more elusive. The goal of this learning seminar is to better understand these 2-categories, and the phenomenon of 3d mirror symmetry: the exchange of the 3d A-model of a hyperkahler manifold with the 3d B-model of its dual.

The plan at least for the first half of the seminar is to understand some of what is known and expected of these 2-categories, and the 3d mirror symmetry equivalence in some more understood examples, such as in the "abelian" case, when the target manifold is the cotangent bundle of a torus. In the second half of the seminar, it would be nice to discuss the non-abelian case, which is related to questions in modern geometric representation theory. To this end, one of the goals of this seminar is to better understand the contents of Teleman's 2014 ICM address [Tel], and the current progress on the predictions made therein.

We will typically meet on Wednesdays at 11am in BA6183, except for weeks where this time is unavailable, in which case we will meet instead on Thursday, or in BA6180 (or both). Below I've provided a tentative schedule we could follow. Please let me know if you have suggestions for topics you would like to discuss.

Tentative schedule:

Here are some potential topics for talks that have yet to be claimed:

• Abelian 3d mirror symmetry and perverse schobers

   [Tel §4], [KaSch], [GamHM]
• Hypertoric varieties

   [Pr], [GamH1], [Bie[Da]
• Coulomb branches

   [BraFin], [Kam]
• Betti/de Rham Tate's thesis

   [HRa], [GamH2]

References:

• [Kap] Kapustin (2010): Topological Field Theory, Higher Categories, and Their Applications
• [DoRe] Doan—Rezchikov (2023): Holomorphic Floer theory and the Fueter equation
• [Tel] Teleman (2014): Gauge theory and mirror symmetry
• [BN] Ben-Zvi—Nadler (2025): Potent categorical representations
• [KaSch] Kapranov—Schechtman (2015): Perverse Schobers
• [Bou] Bousseau (2025): Holomorphic Floer theory and Donaldson—Thomas invariants
• [RoW] Rozansky—Witten (1997): Hyper-Kähler Geometry and Invariants of Three-Manifolds
• [Ga] Gaitsgory (2014): Sheaves of categories and the notion of 1-affineness
• [Ste1] Stefanich (2021): Higher Quasicoherent Sheaves (thesis)
• [Ste2] Stefanich (2025): Categorification of sheaf theory
• [KapRoSa] Kapustin—Rozansky—Saulina (2009): Three-dimensional topological field theory and symplectic algebraic geometry
• [KapRo] Kapustin—Rozansky (2009): Three-dimensional topological field theory and symplectic algebraic geometry II
• [PS] Pascaleff—Sibilla (2025): Speculations on higher Fukaya categories
• [BraFin] Braverman—Finkelberg (2018): Coulomb branches of 3-dimensional gauge theories and related structures
• [HRa] Hilburn—Raskin (2021): Tate's thesis in the de Rham Setting
• [GamHM] Gammage—Hilburn—Mazel-Gee (2023): Perverse schobers and 3d mirror symmetry
• [WeY] Webster—Yoo (2023): 3-dimensional mirror symmetry
• [ChL] Chan—Leung (2024): 3d mirror symmetry is mirror symmetry
• [Pr] Proudfoot (2007): A survey of hypertoric geometry and topology
• [GamH1] Gammage—Hilburn (2025): Hypertoric 2-categories O and symplectic duality
• [Gin] Ginzburg (2009): Lectures on Nakajima's Quiver Varieties
• [BieDa] Bielawski—Dancer (2000): The geometry and topology of toric hyperkahler manifolds
• [Kam] Kamnitzer (2022): Symplectic resolutions, symplectic duality, and Coulomb branches
• [GamH2] Gammage—Hilburn (2025): Betti Tate's thesis and the trace of perverse schobers