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.*


and


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



A royal road to Coulomb branches of 3D $\mathcal{N}=4$ gauge theories
As for prerequisite for three papers, I recommend


Chriss-Ginzburg, [Representation Theory and Complex Geometry](https://www.springer.com/gp/book/9780817637927), and Victor Ginzburg, [Geometric Methods in Representation Theory of Hecke Algebras and Quantum Groups](https://arxiv.org/abs/math/9802004).


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](https://arxiv.org/abs/math/0401222).


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](https://arxiv.org/abs/1706.05154)


Michael Finkelberg, [Double affine Grassmannians and Coulomb branches of 3d N=4 quiver gauge theories](https://arxiv.org/abs/1712.03039)


Alexander Braverman, Michael Finkelberg, [Coulomb branches of 3-dimensional gauge theories and related structures](https://arxiv.org/abs/1807.09038)


For a physical intuition, besides [my first paper](https://arxiv.org/abs/1503.03676), 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](https://arxiv.org/abs/1309.2657)


This paper is accessible by mathematicians.


