Differences

This shows you the differences between two versions of the page.

--- blog:2023-07-30 [2023/07/31 06:05] – pzhou
+++ blog:2023-07-30 [2023/07/31 08:24] (current) – [Coulomb branches] pzhou
@@ Line 3: / Line 3: @@
 ===== 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. BFN did a combination of both.
+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.
+===== Steve Jobs =====
+"... Apple at the core -- its core value-- is that we believe that people with passion can change the world for the better. "
+"... And that those people that are crazy enough to think that they can change the world are the ones that actually do. ..."
-Why is it hard? Is it hard to write down the result, or is it hard to prove that the answer is right?
-In the end, it turns out, we need an affine variety that maps over the base $T$ with some conic singular fiber and with some affine blow-up. Then, we just need to take Weyl cover. If you don't do the affine blow-up just do the abelian gauge theory with matter, then you almost get it right, except you need to blow-up near the root hyperplane. The question is maybe, how do you do the affine blow-up?
-It is always like: first make the ring bigger, then make the ring smaller. Take affine blow-up makes the ring bigger, then passing to