2023-06-21

I don't like the feeling of 'having to do something', such as, writing this paper. I need to persuade myself again and again to write it. The only reason that I have is, I need to write this paper, so I am done with it. That's not a good feeling.

Andrei is saying, writing paper is not a waste of time. The process of telling a coherent story forces one to think better and express better. So, here I am.

I think the obstacle for me to write anything is a lack of 'actionable' plan. I don't like the word of 'actionable', maybe call it 'executable'. Let me think about what I need to do here.

• ZS, LhS, SyB, CyM, BG, JH, PY, CE, MM, BhF, YfS, MH, PhL, XJ, EZaz, DNad
• RB, BW, BE, YL, MKap, MKon

We have a clear understanding of what this space is and what this superpotential is. I just need to explain it.

• We have the definition of Coulomb branch from BFN, which is about the spec of a convolution algebra. We also have the superpotential.
• We want to understand the fiber of the algebraic integrable system. There are two tricks
  • abelianization, we write the Coulomb branch as cotangent bundle of the dual Cartan torus mod Weyl
  • add matter, we glue two copies, and do affinization.

This is supposed to be trivial, but I still don't understand it.

Consider $G=GL_1(\C)$, $V=\C$, standard representation.

• what physicis say about $M_C(G,V)$? some principal $G$-bundle and connection, some associated bundle. minimization of some field configuration, that suppose to be the vacuum, no? Start again. Consider the path integral, over this gigantic space of field configuration, all possible kind. Trouble is, we don't know the integration measure. (we don't? even for the Euclidean signature? don't we have Gaussian integral after the Wick rotation?) What's the physical story? (STILL DON'T KNOW....)
• What BFN says, again some principal $G$-bundle