This is the webpage for a working seminar on Coulomb Branch Varieties. The first edition happened in autumn/winter 2019/20. As of July 2020, the seminar is now **asynchronous**, so it can be attended virtually at any time. The live talks have been replaced with a set of notes on Overleaf – each ‘speaker’ now adds a section to the notes for their ‘talk’. The plan is for new ‘talks’ to be added about once a fortnight. The notes should also double as a discussion board – all participants are encouraged to add frequent questions and comments to the notes (which have some special LaTeX commands to make this easier).

A **read-only** version of the notes can be found here. **To obtain a read/write link (which is essential to ask questions), please email Dougal Davis dougal.davis@ed.ac.uk.** Also email Dougal to sign up for email updates.

Suggested talks: (Please volunteer!)

(Online!) **Overview of Coulomb branch varieties** (Dougal)

(Online!) **Definition, toy examples and basic properties of Coulomb branch varieties, I** (Dougal)

(Online!) **Definition, toy examples and basic properties of Coulomb branch varieties, II** (Dougal)

Braverman-Finkelberg, Coulomb branches of 3-dimensional gauge theories and related structures.

Braverman-Finkelberg-Nakajima, Towards a mathematical definition of Coulomb branches of 3-dimensional N = 4 gauge theories, II

Nakajima, Introduction to a provisional mathematical definition of Coulomb branches of 3-dimensional N = 4 gauge theories

Nakajima, Towards a mathematical definition of Coulomb branches of 3-dimensional N =4 gauge theories, I

Nakajima, Questions on provisional Coulomb branches of 3-dimensional N = 4 gauge theories.

**More sophisticated examples** (???)

Braverman-Finkelberg-Nakajima, Coulomb branches of 3d N = 4 quiver gauge theories and slices in the affine Grassmannian

Bezrukavnikov-Finkelberg-Mirkovic, Equivariant homology and K-theory of affine Grassmannians and Toda lattices

**BFN resolutions and deformations** (???)

Braverman-Finkelberg-Nakajima, Towards a mathematical definition of Coulomb branches of 3-dimensional N = 4 gauge theories, II, sections 3(vii)-(ix)

Weekes, Quiver gauge theories and symplectic singularities

**Symplectic duality and Koszul duality** (???)

Bernstein-Ginzburg-Soergel, Koszul Duality Patterns in Representation Theory.

Braden-Licata-Proudfoot-Webster, Quantizations of conical symplectic resolutions II: category O and symplectic duality.

**Can someone do any physics about what underlies this** (???)

Seiberg and Witten, Gauge Dynamics And Compactification To Three Dimensions

Bullimore, Dimofte, and Gaiotto, The Coulomb Branch of 3d N = 4 Theories

Bullimore, Dimofte, Gaiotto, and Hilburn, Boundaries, Mirror Symmetry, and Symplectic Duality in 3d N=4 Gauge Theory

Bullimore, Ferrari, and Kim, Twisted Indices of 3d N = 4 Gauge Theories and Enumerative Geometry of Quasi-Maps

**Representation theory of Coulomb branch algebras** (???)

Hilburn, Kamnitzer, and Weekes, BFN Springer theory