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.

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:

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:

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

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:

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):

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:

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

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:

Schedule

Thurs Feb 07 Sam DeHority
Moduli spaces of instantons, quiver varieties and ALE spaces

Instantons are solutions to physical equations which are confined to a particular region of spacetime. Moduli spaces of instantons are extremely important in physics and mathematics, and can be described exactly in a number of cases. This talk will discuss examples of these moduli spaces, focusing on instantons on ALE spaces, which are ADE surfaces with particular hyperkahler metrics. These moduli spaces are constructed as Nakajima quiver varieites, and also come with hyperkahler metrics.

References: [KN90], [Nak94], Donaldson-Kronheimer The Geometry of Four-Manifolds.
Thurs Feb 14 Semon Rezchikov
Hypertoric Varieties and Hyperplane Arrangements

To a (suitably decorated) arrangement of hyperplanes in R^n and a choice of covector, one can associate a pair of combinatorially defined algebras A and B. When the hyperplanes are defined over Q, these algebras have geometric interpretations in terms of certain hypertoric varieties associated to the hyperplane arrangement. Gale duality, a certain operation on hyperplane-arrangements-with-covectors that is related to linear progamming, gives rise to a certain duality between the algebras A and B. This is supposed to be an instance of symplectic duality. We will give a leisurely and example-focused tour through some part of this construction.

References: [BLPW08], Proudfoot A survey of hypertoric geometry and topology
Thurs Feb 21 Yakov Kononov
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Thurs Feb 28 Speaker: TBA
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Thurs Mar 07 Speaker: TBA
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Thurs Mar 14 Speaker: TBA
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Thurs Mar 21 No meeting (spring break)
Thurs Mar 28 Speaker: TBA
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Thurs Apr 04 Speaker: TBA
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Thurs Apr 11 Speaker: TBA
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