Symplectic duality is a network of correspondences between pairs of conical symplectic resolutions. It exchanges structures that look unrelated on one side—symplectic leaves, torus-fixed points, category O, deformation parameters and twisting parameters—with complementary structures on the other.
The route extends Nakajima Quiver Varieties, Symplectic Reflection Algebras and Koszul Duality.
Conical symplectic resolutions
A symplectic resolution π:X→X_0 resolves a singular Poisson variety by a smooth symplectic variety. Conical examples carry a C* action contracting X_0 toward a distinguished point while scaling the symplectic form with positive weight.
Examples include cotangent bundles of flag varieties, many Nakajima quiver varieties, hypertoric varieties and resolutions related to slices in affine Grassmannians.
Quantization
A quantization replaces the commutative coordinate algebra by a filtered noncommutative algebra whose associated graded recovers functions on X_0. Different periods or deformation parameters produce families of quantizations.
Representation categories of these quantizations carry highest-weight-like structures analogous to category O.
Category O for symplectic resolutions
Choose a Hamiltonian torus action with isolated fixed points under suitable hypotheses. Category O consists roughly of modules whose behavior is bounded with respect to the attracting direction of that action.
Standard objects are associated with fixed points and attracting sets. Their partial order is geometric rather than imposed externally.
What duality exchanges
A symplectic-dual pair X and X^! is expected to exchange several structures: fixed points with fixed points, symplectic leaves with certain special subvarieties, deformation parameters with equivariant or twisting parameters, and categories O through Koszul-type duality.
The exact dictionary depends on the class of examples. Symplectic duality is a framework of related conjectures and theorems, not one universal elementary formula.
Higgs and Coulomb branches
Three-dimensional N=4 gauge theories naturally produce a Higgs branch and a Coulomb branch. Mirror symmetry exchanges them.
Mathematically, many proposed symplectic-dual pairs arise as Higgs/Coulomb branches of mirror theories. This gives the duality its most useful intuition: geometry built from matter and quotient data on one side becomes geometry built from monopole and gauge data on the other.
Hypertoric model
Hypertoric varieties are hyperkähler analogues of toric varieties. Their combinatorics is controlled by hyperplane arrangements.
Gale duality of arrangements produces natural dual hypertoric varieties. In this setting, many symplectic-duality correspondences can be seen explicitly through combinatorial linear algebra.
Quiver varieties
Nakajima quiver varieties provide richer examples. Changing dimension vectors, framings and quiver data can produce dual descriptions whose categories and leaf structures exhibit the predicted exchange.
The same quiver data also connects to Kac–Moody representations, Yangians and Coulomb-branch constructions.
Koszul duality
One of the strongest categorical predictions is that suitable graded categories O attached to symplectic-dual varieties are Koszul dual.
Standard, costandard, simple, projective and tilting structures are reorganized under this duality. Geometry therefore determines a homological exchange between representation categories.
Twisting and shuffling
Derived autoequivalences called twisting and shuffling functors arise from changing quantization parameters or torus chambers. Symplectic duality exchanges these two families.
This is a categorical version of exchanging two kinds of geometric motion: changing the algebra versus changing the attracting direction.
Leaves and supports
Symplectic singularities are stratified by symplectic leaves. Supports of modules and characteristic cycles interact with this stratification.
Under duality, leaf closure order and support data are expected to correspond to complementary structures on the dual side, often reversing natural orders.
Verification workflow
- Verify that the spaces in question are conical symplectic resolutions or the appropriate singular analogues.
- Specify the quantization and torus action.
- State the precise category O being used.
- Distinguish proven dual pairs from conjectural ones.
- Do not assume every hyperkähler pair is symplectic dual.
- Track which parameters are exchanged.
- Separate physical mirror-symmetry intuition from mathematical theorem statements.
Practice
1. What is a conical symplectic resolution? 2. What labels standards in category O? 3. What does mirror symmetry exchange? 4. Why does Koszul duality appear? 5. What are twisting and shuffling? 6. Why are symplectic leaves important?
Answers. A symplectic resolution with a contracting scaling action; torus-fixed points/attracting geometry; Higgs and Coulomb branches; dual categories O often have Koszul-dual graded algebras; two families of derived equivalences exchanged by duality; leaves stratify singular Poisson geometry and control supports.
Representation Mathematics — Batch 14
Read Soergel Bimodules and Character Sheaves, then continue to Coulomb Branches. Return to the BTT Mathematics Learning Hub.