\[\newcommand{\C}{\mathbb{C}} \newcommand{\Gr}{\operatorname{Gr}} \newcommand{\Kc}{\mathcal{K}} \newcommand{\Oc}{\mathcal{O}} \newcommand{\Rc}{\mathcal{R}} \newcommand{\Tc}{\mathcal{T}} \renewcommand{\Kc}{\mathcal{K}} \newcommand{\sym}{\operatorname{Sym}} \newcommand{\tf}{\mathfrak{t}} \newcommand{\coker}{\operatorname{coker}}\]

In arXiv:1601.03586, coulomb branches are defined when the matter representation $M=N\oplus N^*$ is of cotangent type. Also computed is the coulomb branch when the gauge group is abelian.

The computation there is however extremely confusing to me, largely due to the rather involved definition of convolution product and their approach of exposition (and my incompetence of course!)

I will try to dot the i's and cross the t's.

For simplicity, assume our group is $G=T=\C^\times$, and the representation $N = \C_\xi$ is given by a single character $\xi$.

A quick recap on the definition of $\Rc$. Define $\Tc = G_\Kc\times_{G_\Oc}N_\Oc$ and consider the embedding $\Tc \hookrightarrow \Gr_G \times N_\Kc$ given by quotient on the first factor and multiplying on the second. The variety of triples are then given by $\Rc := \Tc \cap \Gr_G \times N_\Oc$.

Recall that when $G$ is the torus $\C^\times$, the affine grassmannian is just the coweight lattice $Y = \mathbb{Z}$.

We then have,

\[\Rc = \coprod_\lambda \{\lambda\} \times (z^\lambda N_\Oc \cap N_\Oc).\]

It follows that as a vector space,

\[H_*^{G_\Oc}(\Rc) = \oplus_\lambda \C[\tf] r^\lambda.\]

Here $r^\lambda$ is the fundamental class of the component corresponding to $\lambda$.

So in order to get the ring structure, it suffices to compute $r^\lambda r^\mu$.

Now we look at the following convolution diagram.

\[\begin{CD} \Rc \times \Rc @<<< p^{-1}(\Rc\times\Rc) @>>> q(p^{-1}(\Rc\times\Rc)) @>>> \Rc \\ @VVV @VVV @VVV @VVV \\ \Tc \times \Rc @<p = (p_\Tc, p_\Rc)<< G_\Kc \times \Rc @>q>> G_\Kc \times_{G_\Oc} \Rc @>m>> \Tc \\ \end{CD}\]

The difficulty of analyzing this convolution product lies in the fact that it is defined via the rather mysterious restriction with support morphism giving a map $H_\bullet^{G_\Oc}(\Rc) \otimes H_\bullet^{G_\Oc}(\Rc) \rightarrow H_\bullet^{G_\Oc \times G_\Oc}(p^{-1}(\Rc\times\Rc))$ , which is the usual restriction unless everything is smooth. Good news is everything is indeeed smooth in the abelian case.

Now identify $T_\Kc \times_{T_\Oc} \Rc$ with

\[\coprod_{\lambda, \nu} \{\lambda\} \times \{\nu\} \times z^\lambda N_\Oc \cap z^\nu N_\Oc\]

with the map given by $[g_1, [g_2, s]] \mapsto ([g_1], [g_1g_2], [g_1g_2s])$.

To calculate $r^\lambda r^\mu$ it is then enough to look at the component given by $(\lambda, \nu = \lambda + \mu)$.

Let's first have a look at the map $m$. Under the identification $\Tc = \coprod {\lambda} \times z^\lambda N_\Oc$, it is simply given by the projection.

\[\lambda \times \nu \times z^\lambda N_\Oc \cap z^\nu N_\Oc \rightarrow \nu \times z^\nu N_\Oc\]

The trick is to define a map $p':T_\Kc \times_{T_\Oc} \Rc \rightarrow \Tc \times \Rc$ by sending $(\lambda,\nu,s)$ to $(\lambda, s, \nu - \lambda, z^{-\lambda}s)$.

It enjoys the property that $ p^\prime_{\Tc} q = p_\Tc, p^\prime_\Rc q a = p_\Rc $,

where $ a $ is an automorphism of $ T_\Kc\times \Rc $ inducing identity map on the quotient $ T_\Kc\times_{T_\Oc} \Rc $.

Therefore, the convolution can be computed as

\[r^\lambda r^\mu = m_*p'^*(r^\lambda \boxtimes r^\mu) = m_* p'^*([z^\lambda N_\Oc \times (N_\Oc \cap z^\mu N_\Oc)]\]

Recall that if $\iota: V \hookrightarrow X$ is a $T$-invariant smooth closed subvariety, we have $\iota^\ast\iota_\ast([V]) = e(N_{V/X})[V]$ in equivariant (Borel-Moore/co-) homology.

Thus we have

\[p'^*_\Tc([N_\Oc \cap z^\lambda N_\Oc]) = e(z^\lambda N_\Oc/N_\Oc\cap z^\lambda N_\Oc) [z^\lambda N_\Oc \cap z^\nu N_\Oc],\] \[p'^*_\Rc([N_\Oc \cap z^\mu N_\Oc]) = [z^\lambda N_\Oc \cap z^\nu N_\Oc]\]

So

\[p'^*([(N_\Oc \cap z^\lambda N_\Oc) \times (N_\Oc \cap z^\mu N_\Oc)]) = e(z^\lambda N_\Oc/N_\Oc\cap z^\lambda N_\Oc)[z^\lambda N_\Oc \cap z^\nu N_\Oc]\] \[m_*p'^*([(N_\Oc \cap z^\lambda N_\Oc) \times (N_\Oc \cap z^\mu N_\Oc)]) = e(z^\nu N_\Oc/z^\lambda N_\Oc\cap z^\nu N_\Oc) e(z^\lambda N_\Oc/N_\Oc\cap z^\lambda N_\Oc) [z^\nu N_\Oc]\]

We know $r^\lambda r^\mu$ is supported on $R_{\nu}$, so it suffices to divide by the normal bundle of $\Rc_\nu$ in $\Tc_\nu$ to get the coefficient.

Therefore,

\[r^\lambda r^\mu = m_*p'^*([(N_\Oc \cap z^\lambda N_\Oc) \times (N_\Oc \cap z^\mu N_\Oc)]) = \frac { e(z^\lambda N_\Oc/N_\Oc\cap z^\lambda N_\Oc) e(z^{\nu}N_\Oc/z^{\lambda}N_\Oc\cap z^\nu N_\Oc) } { e(z^\nu N_\Oc/z^\nu N_\Oc \cap N_\Oc) } r^{\lambda + \mu}.\]

Counting dimensions case-by-case shows that the above is equal to

\[r^\lambda r^\mu = \xi^{d(\langle\lambda,\xi\rangle, \langle\mu, \xi\rangle)} r^{\lambda + \mu}.\]

Here $d(k, l):=[kl < 0]\min(\lvert k \rvert,\lvert l \rvert)$.

This computation generalizes trivially to higher-dimensional tori and representations, giving

\[r^\lambda r^\mu = \prod_i \xi_i^{d(\langle\lambda,\xi_i\rangle, \langle\mu, \xi_i\rangle)} r^{\lambda + \mu}\]

if the representation $N$ is given by characters $\xi_i$.