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--- blog:2023-06-21 [2023/06/21 17:53] – [The Geometry of Coulomb branch] pzhou
+++ blog:2023-06-21 [2023/06/25 15:53] (current) – external edit 127.0.0.1
@@ Line 8: / Line 8: @@
 ===== The need to talk =====
+  * ZS, LhS, SyB, CyM, BG, JH, PY, CE, MM, BhF, YfS, MH, PhL, XJ, EZaz, DNad
+  * RB, BW, BE, YL, MKap, MKon
@@ Line 14: / Line 16: @@
   * 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. We did not use the integrable system structure, except it is a holomorphic map. (this feels wrong. what is a natural Lagrangian in an algebraic integrable system? the fiber, the section. ) Now, is our T-brane
+  * 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.
+==== abelian story ====
+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