Henry Liu

Learning Seminar on Symplectic Duality (Spring 2019)

The goal of this seminar is to understand various aspects of symplectic duality from a mathematical perspective.

• Organizer: Henry Liu
• Time/date: Thursdays 2:40-3:55pm
• Location: Math 622

I am live-TeXing notes for this seminar, available here.

Please email me at hliu at math dot columbia dot edu if you would like to be on the mailing list.

Possible topics and references

In physics, symplectic duality is a conjectural correspondence between 3d N=4 supersymmetric gauge theories \(X\) and \(X^\vee\). There are many aspects of this correspondence which have been formalized mathematically in varying levels of generality. The earliest (physics) papers describing the duality and providing examples seem to be:

• [IS96] Mirror Symmetry in Three Dimensional Gauge Theories
• [dBHOO96] Mirror Symmetry in Three-Dimensional Gauge Theories, Quivers and D-branes
• [dBHOOY96] Mirror Symmetry in Three-Dimensional Gauge Theories, SL(2,Z) and D-Brane Moduli Spaces.

Mathematically, a very explicit and combinatorial case is that of hypertoric varieties. Here symplectic duality is Gale duality. The following papers explain how to associate a category \(\mathcal{O}\) to a hypertoric variety, and later conical symplectic resolutions in general, and how Gale dual pairs give Koszul dual categories:

• [BLPW08] Gale duality and Koszul duality
• [BLPW10] Hypertoric category O
• [BLPW14] Quantizations of conical symplectic resolutions II: category \(\mathcal{O}\) and symplectic duality.

The Koszul duality part of the story is explained in the classical paper:

• [BGS96] Koszul Duality Patterns in Representation Theory.

In general, symplectic duality exchanges the Higgs and Coulomb branches of \(X\) and \(X^\vee\). These are moduli spaces associated to the gauge theories. For example, in the hypertoric case, Gale dual hypertoric varieties form a Higgs/Coulomb pair. More generally, for gauge theories associated to a quiver, the Higgs branch is the associated Nakajima quiver variety. When the quiver is of ADE type, quiver varieties have interpretations as moduli of instantons on ADE surfaces via a generalized ADHM construction:

• [KN90] Yang-Mills instantons on ALE gravitational instantons
• [Nak94] Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras
• [Nak18] Instantons on ALE spaces for classical groups.

For type A quivers, the Higgs branch of \(X^\vee\) (i.e. the Coulomb branch of \(X\)) is also a quiver variety of type A. Classically, symplectic duality in this case reduces to level-rank duality. The enumerative geometries, in the form of quantum K-theory, of the Higgs branches of \(X\) and \(X^\vee\) are expected to match. This can actually be explicitly checked for hypertoric varieties, and indirectly checked in general by matching monodromies of q-difference equations (up to gauge equivalence):

• [Onotes1] [Onotes2] (generously shared by Andrei Okounkov)
  Notes on quasimaps to hypertoric varieties, and matching their q-difference equations
• [AO16] Elliptic stable envelopes.

The Coulomb branch is more complicated than the Higgs branch. Physically, the moduli space is \((\mathbb{R}^3 \times S^1)^r\) classically but its metric receives quantum corrections, making a mathematical construction difficult. Only recently was such a construction proposed:

• [Nak16a] Towards a mathematical definition of Coulomb branches of 3-dimensional \(\mathcal{N}=4\) gauge theories, I
• [BFN16] Towards a mathematical definition of Coulomb branches of 3-dimensional \(\mathcal{N}=4\) gauge theories, II
• [Nak15] Questions on provisional Coulomb branches of 3-dimensional \(\mathcal{N}=4\) gauge theories.

There are also excellent surveys of the construction, with emphasis on different aspects of the story:

• [Nak16b] Introduction to a provisional mathematical definition of Coulomb branches of 3-dimensional \(\mathcal{N}=4\) gauge theories
• [BF18] Coulomb branches of 3-dimensional gauge theories and related structures.

The monopole formula plays a big role in formulating such a construction. Originating from physics, it is a combinatorial formula for what the Hilbert series of the Coulomb branch should be. Many examples are given in the original (physics) paper:

• [CHZ14] Monopole operators and Hilbert series of Coulomb branches of 3d \(\mathcal{N}=4\) gauge theories.

Schedule