Peng Zhou

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====== 2023-07-30 ======
  * abelian with matter is the trouble 

===== Coulomb branches =====
Compare with pure abelian gauge theory, the Coulomb branch of general cases needs two direction modifications, with matter representations, and with non-abelian gauge group. 

In the abelian case, how do we deal with matter? We specify the multiplication table for the 'monopole operators', a vector space basis over the base coefficient ring. It is a bit brutal force. 

How do we do the affine blow-up? Well, you pick two hypersurfaces in the total space, you blow-up their intersections, then remove the proper transform of one divisor. 

For example, just $(n)-[m]$ type quiver. We need to first do the abelian case, we have
$$ x_i^+ x_i^- = (y_i-a_1) \cdots (y_i - a_m), \quad i=1,\cdots, n $$
OK, not too bad. Then, we do the blow-up. For each $i \neq j$, we consider $\{x_i^+ = x_j^+\}$ and $\{y_i = y_j\}$. You would complain, why not use $\{x_i^- = x_j^-\}$ and $\{y_i = y_j\}$ ? Then I would say, they cut-out the same loci. If you ask, 'why not choose $x_i^+ = x_j^-$?' then I would say, they are isomorphic. Hold on, so this is it? We have this matter induced relation, and we introduce these $u_{ij}$ for the blow-up coordinates? Yes, I guess so. Affine space, check; dimension is right, check; desired behavior over the smooth part of the descriminant loci, check. Then, that is it. 

This gives me hope that things might not be so hard. Consider $(1)-(2)-[3]$. 
$$ x_1^+ x_1^- = (y_1-y_{2,1})(y_1-y_{2,2}). $$
$$ x_{2,i}^+ x_{2,i}^- = (y_{2,i}-y_1)(y_{2,i}-a_1)\cdots(y_{2,i}-a_3), \quad i=1,2$$
OK, now we need to abelianize, so we introduce 
$$ \frac{x_{2,i}^+ - x_{2,j}^+}{y_{2,i}-y_{2,j}}. $$

Well, here is the trouble, remember the case for $T^*\P^2$? Or the simpler one $(1)-(1)$? If you go by the rule of $x_1^+x_1^- = y_1-y_2, \quad x_2^+x_2^-=y_1-y_2$, then you lose. You know what this is suppose to be. You need more generators and relations. You need the $r_{1,1}$ and $r_{-1,-1}$. Then you are in good shape. In general, you need to take a good size of blobs of weights, and do those generators, and get those relations. 

Then, when you do the blow-up, you can do the simple things. Just pick some variables and blow them up. 

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