So, I've been very interested recently with the developements of the (now mathematically precise) theory of Coulomb branches - in particular because of its recent applications on representation theory and symplectic geometry.

They have been considered by mathematical physicists for a time, but without a rigorous definition. The first attempt to give such a definition was made by H. Nakajima, Introduction to a provisional mathematical definition of Coulomb branches of 3-dimensional N=4 gauge theories.

This project was completed by A. Braverma, N. Finkelberg, H. Nakajima in

Towards a mathematical definition of Coulomb branches of 3-dimensional N=4 gauge theories, II, and their companion papers

Coulomb branches of 3d N=4 quiver gauge theories and slices in the affine Grassmannian. With two appendices by Braverman, Finkelberg, Joel Kamnitzer, Ryosuke Kodera, Nakajima, Ben Webster and Alex Weekes.

Ring objects in the equivariant derived Satake category arising from Coulomb branches. Appendix by Gus Lonergan.

Now, I've become rather convinced that to proceed with my research I need to have a rather firm grasp on the tree papers above by Braverman, Finkelberg and Nakajima.

Now, I unfortunately have no good physics intuition, and I've found that some parts of the above papers are very technical and are motived by a lot of different constructions in geometric representation theory in the range of the past 20 years. The final form of this theory is a remarkable achievement of novel ideas and technical mastery, and I'm feeling a little bit lost on what should be the relevant things to focus on (important previous work and mathematical techiniques and machinery).

Hence, the question: what is the royal road to Coloumb branches?

PS: the two papers I'm trying to fully appreciate are:

J. Kamnitzer, P. Tingley, B. Webster, A. Weekes and O. Yacobi, On category O for affine Grassmannian slices and categorified tensor products.


A. Weekes, Generators of Coulomb branches of quiver gauge theories, arXiv:1903.07734.


2 Answers 2


As for prerequisite for three papers, I recommend

Chriss-Ginzburg, Representation Theory and Complex Geometry, and Victor Ginzburg, Geometric Methods in Representation Theory of Hecke Algebras and Quantum Groups.

One also needs to know basics on affine Grassmannians and geometric Satake. There are lots of good survey articles on them, as well as the original paper, Mirkovic-Vilonen, Geometric Langlands duality and representations of algebraic groups over commutative rings.

If three papers are too technical, there are a few survey articles:

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

Michael Finkelberg, Double affine Grassmannians and Coulomb branches of 3d N=4 quiver gauge theories

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

For a physical intuition, besides my first paper, I recommend to look at

Stefano Cremonesi, Amihay Hanany, Alberto Zaffaroni, Monopole operators and Hilbert series of Coulomb branches of 3d N = 4 gauge theories

This paper is accessible by mathematicians.

  • $\begingroup$ Thank you very much! $\endgroup$
    – jg1896
    Jul 12, 2020 at 12:59

I've written a survey paper on this topic: Symplectic resolutions, symplectic duality, and Coulomb branches.

I begin with symplectic resolutions in general, then discuss symplectic duality, and then the Coulomb branch construction of Braverman-Finkelberg-Nakajima. Throughout, I discuss quiver varieties and (generalized) affine Grassmannian slices. I hope this is helpful!


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