• Discussion with all

If you point a gun at me, and ask me the gluing formula, I would say:

• introduce two identical copies of the pure Coulomb branch algebra (I don't know why two?)
• ok, you have two sets monopole operators, and the equivariant coefficient ring.
• then what?

Let $G$ be a compact Lie group. all nice stuff. $U(n), SU(n), SO(n)$, whatever you want.

ok, the basic space, pure gauge theory Coulomb branch $M_C(G,0)$ is spec of $H^G_*(\Omega G)$. The geometry is the algebra. It is like, taking the group algebra of the loop group. So, the resulting space is the dual group, like, the dual torus.

Here is another thing: if you consider $O(G /_{ad} G)$, that gives a subring of $O(G)$, which has a nice basis with the irreducible representations of $G$, we go like $$ O(\frac{G}{G}) = \oplus_{R \in irrep(G)} \chi_R $$ We have indeed orthogonality condition.

And even better, we have $$ O(G) \cong \oplus_{R \in irr(G)} End(R) $$ as $G$ representation. (? is this also as Hilbert space isom?)

Now, what are the eigenspace of $O(G)$? For a matrix element in $End(R)$, it would have eigenvalue proportional to $c_2(R)$.

Why do we like the $O(\frac{G}{G})$? It is the K-theory of $Rep(G)$. That's why it is called the character theory. direct sum goes to sum, product goes to product, very beautiful.

Back to that spherical BFM convolution algebra. It is roughly the coordinate ring of $T^* T^\vee_\C / W$. Some symplectic affine resolution.