Loop Grassmannian of quivers and Compactified Coulomb branch of quiver gauge theory with no framing

Zhijie Dong

(February 2015)

Abstract

Mirković introduced the notion of loop Grassmannian for symmetric integer matrix $\kappa$ . It is a two-step limit of the local projective space $Z_{\kappa}^{\alpha}$ , which generalizes the usual Zastava for a simply laced group $G$ . The usual loop Grassmannian of $G$ is recovered when the matrix $\kappa$ is the Cartan matrix of $G$ . On the other hand, Braverman, Finkelberg, and Nakajima showed that the Compactified Coulomb branch $\mathbf{M}_{Q}^{\alpha}$ for the quiver gauge theory with no framing also generalizes the usual Zastava. We show that in the case when $\kappa$ is the associated matrix of the quiver $Q$ , these two generalizations of Zastava coincide, i.e $\mathbf{M}_{Q}^{\alpha}\cong Z_{\kappa(Q)}^{\alpha}$ .

1 Introduction

1.1 Generalization of loop Grassmannian

Let $G$ be a reductive group over a field $k=\overline{k}$ with char( $k$ ) $=0$ . Let $G_{\mathcal{K}}$ and $G_{\mathcal{O}}$ be its formal loop group and formal arc group, respectively. Define $\mathcal{G}(G)=G_{\mathcal{K}}/G_{\mathcal{O}}$ as the loop Grassmannian of $G$ , which is an ind-scheme over $k$ .

We consider extensions of the concept of the loop Grassmannian to an arbitrary Kac-Moody group $G_{KM}$ . It is not clear how to define the quotient $(G_{KM})_{\mathcal{K}}/(G_{KM})_{\mathcal{O}}$ as an ind-scheme. The standard approach is to construct a system of finite dimensional schemes. In [BF10, BFN19], normal slices to certain orbits in the (undefined) loop Grassmannian were considered. In two approaches considered here ([BFN19, Mir23]) these schemes are projective and one constructs the whole loop Grassmannian $\mathcal{G}(G_{KM})$ as a certain colimit [Mir23].

1.2 Zastava spaces [FM99]

We recall Zastava spaces $Z_{G}$ for a semisimple simply-connected group $G$ . In 1.3 and 1.4 we will consider two constructions of generalization of Zastava for a quiver.

1.2.1 Intersections of semiinfinite orbits in $\mathcal{G}(G)$

First, let’s get a feeling how the procedure in 1.1 can be done when $G$ is a simply-connected group.

We start by fixing a Cartan subgroup $T$ and a pair of opposite Borel subgroups $(B^{+},B^{-})$ such that $B^{+}\cap B^{-}=T$ . Let $N^{\pm}$ be the unipotent subgroups such that $B^{\pm}=TN^{\pm}.$ For any cocharacter $\alpha$ of $T$ , let $t^{\alpha}\in G_{\mathcal{K}}$ be the point corresponding to $\alpha$ and $\overline{t^{\alpha}}$ be the corresponding point in $\mathcal{G}(G)$ . For two cocharacter $\lambda,\mu$ of $T$ , let $S^{+}_{\lambda}$ and $S^{-}_{\mu}$ be the $N_{\mathcal{K}}^{+}$ -orbit of the point $\overline{t^{\lambda}}$ and the $N_{\mathcal{K}}^{-}$ -orbit of the point $\overline{t^{\mu}}$ , respectively. Furthermore, let $\overline{S^{+}_{\lambda}}$ and $\overline{S^{-}_{\mu}}$ be their closures in $\mathcal{G}(G).$

For each cocharacter $\alpha$ , we define $F^{\alpha}=\overline{S^{-}_{-\alpha}}\cap\overline{S^{+}_{0}}$ . Now for cocharacters $\alpha,\beta$ such that $\alpha\leq\beta$ , we have a closed embedding $F^{\alpha}\xhookrightarrow{}F^{\beta}$ , which we refer to as the growth structure.

Taking the inductive limit of $F^{\alpha}$ for $\alpha\geq 0$ , we recover $\overline{S_{0}}\stackrel{{\scriptstyle def}}{{=}}\overline{S^{+}_{0}}$ . Let $\overline{S_{0}}\xrightarrow[]{\times t^{\alpha}}\overline{S_{0}}$ be the multiplication map that maps $\overline{S_{-\alpha}}\subset\overline{S_{0}}$ to $\overline{S_{0}}.$ Subsequently, taking the direct limit of $\overline{S_{0}}\xrightarrow[]{\times t^{\alpha}}\overline{S_{0}}$ for $\alpha\geq 0$ (the direct system will be referred to as the shift structure), we obtain the entire $\mathcal{G}(G)$ when $G$ is simply connected.

This is a case of the prescription from 1.1 with finite-dimensional schemes $F^{\alpha}$ and the ind-system given by the growth structure between $F^{\alpha}$ and the shift structure between $\overline{S_{0}}.$

1.2.2 Ordered Zastava $Z^{\prime}_{G}$ from global loop Grassmannian

Consider the global version of the loop Grassmannian $\mathcal{G}(G).$ This is the Beilinson-Drinfeld Grassmannian $\mathcal{G}^{C}(G)$ defined for a reductive group $G$ and a smooth curve $C$ . It has a natural projection $\mathcal{G}^{C}(G)\xrightarrow[]{\pi}\sqcup_{n\in\mathbb{N}}C^{n}$ . This space has the so-called factorization property. E.g., $\mathcal{G}^{C}(G)_{a}\cong\mathcal{G}(G)$ and $\mathcal{G}^{C}(G)_{(a,b)}\cong\mathcal{G}^{C}(G)_{a}\times\mathcal{G}^{C}(G)_{b}$ for $a\neq b.$ For cocharacters $\lambda_{1},\cdots,\lambda_{n}$ , we can also define the global version of $N^{\pm}_{\mathcal{K}}$ -orbit¹¹1 First, one defines the section of map $\mathcal{G}^{C}(G)\xrightarrow[]{\pi}C^{n}$ corresponding to $\lambda_{1},\cdots,\lambda_{n}$ [Zhu09, 1.1.8]. Then one defines the global loop group of $N$ [AR, 2.2.3]. $S_{\lambda_{1},\lambda_{2},\cdots,\lambda_{n}}^{BD,\pm}\subset\mathcal{G}^{C}(G)$ , their orbit closures $\overline{S_{\lambda_{1},\lambda_{2},\cdots,\lambda_{n}}^{BD,\pm}}$ and intersections $F_{BD}^{\alpha_{1}\cdots,\alpha_{n}}=\overline{S_{-\alpha_{1},\cdots,-\alpha_{n}}^{BD,-}}\cap\overline{S_{0,\cdots,0}^{BD,+}}$ for cocharacters $\alpha_{1},\cdots,\alpha_{n}$ . E.g., the space $S_{\lambda_{1},\lambda_{2}}^{BD}$ is over $C^{2}$ and its fiber over $(a,b)$ is $S_{\lambda_{1}}\times S_{\lambda_{2}}$ for $a\neq b$ and its fiber over $(a,a)$ is $S_{\lambda_{1}+\lambda_{2}}$ ²²2The space $\overline{S_{\lambda_{1},\lambda_{2}}^{BD,+}}$ is over $C^{2}$ and its fiber over $(a,b)$ is $\overline{S_{\lambda_{1}}}\times\overline{S_{\lambda_{2}}}$ for $a\neq b$ . But here, we only know that the reduced scheme of its fiber over $(a,a)$ is $\overline{S_{\lambda_{1}+\lambda_{2}}}$ .. For $\alpha=\alpha_{k_{1}}+\cdots\alpha_{k_{n}}$ , where $\alpha_{k_{i}}$ is simple coroot, let $n_{i}$ be the number of $j^{\prime}s$ such that $\alpha_{k_{j}}=\alpha_{i}$ . Let $C^{\alpha}=\prod_{i\in I}C^{n_{i}}$ . Define the ordered Zastava $Z_{G}^{\prime\alpha}=F_{BD}^{\alpha_{k_{1}},\cdots,\alpha_{k_{n}}}$ over $C^{\alpha}$ . Here, the space $F_{BD}^{\alpha_{k_{1}},\cdots,\alpha_{k_{n}}}$ is canonically isomorphic to $F_{BD}^{\alpha^{\prime}_{k_{1}},\cdots,\alpha^{\prime}_{k_{n}}}$ for a different order of decomposition $\alpha=\alpha^{\prime}_{k_{1}}+\cdots+\alpha^{\prime}_{k_{n}}$ (they are different as subspaces of $\mathcal{G}^{C}(G)$ , through). Hence the ordered Zastava $Z_{G}^{\prime\alpha}$ is independent of the order, which justifies the notation.

1.2.3 Zastava $Z_{G}$ from quasimaps

It turns out that $Z_{G}^{\prime\alpha}$ and its limit constructions can be defined without using the Beilinson-Drinfeld Grassmannian $\mathcal{G}^{C}(G)$ .

First, the affine Zastava³³3We will fix a curve $C$ so we drop $C$ from this notation. The affine Zastava was called Zastava in [FM99] and Zastava was called compactified Zastava in [BFN19]. $Z^{\text{aff},\alpha}_{G}$ is the space of based quasimaps from the curve $C$ to the flag variety of $G$ of degree $\alpha\in\mathbb{N}[I]$ . It has a natural map $\pi$ to $\mathcal{H}^{\alpha}_{C\times I}$ , the Hilbert scheme of point of $C\times I$ of length $\alpha$ . Then Zastava $Z^{\alpha}_{G}$ is defined as certain compactification (fiberwise with respect to $\pi$ ) of $Z^{\text{aff},\alpha}_{G}$ ( see remark 3.7 in [BFN19] and the reference therein). Now $Z_{G}^{\alpha}$ is the partially symmetrized version⁴⁴4The map $Z_{G}^{\prime\alpha}\xrightarrow[]{\pi}\prod_{i\in I}C^{n_{i}}$ descends to $Z_{G}^{\alpha}=Z_{G}^{\prime\alpha}/\prod_{i\in I}S^{n_{i}}\xrightarrow[]{\pi}\prod_{i\in I}C^{n_{i}}/S^{n_{i}}\cong\mathcal{H}^{\alpha}_{C\times I}.$ Define the intersection $F_{\text{aff},BD}^{\alpha_{1}\cdots,\alpha_{n}}=S_{\alpha_{1},\cdots,\alpha_{n}}^{BD,-}\cap\overline{S_{0,\cdots,0}^{BD,+}}$ . We have $Z_{G}^{\prime\text{aff},\alpha}=F_{\text{aff},BD}^{\alpha_{k_{1}},\cdots,\alpha_{k_{n}}}$ . Then $Z_{G}^{\text{aff},\alpha}$ is the partially symmetrized version of $Z_{G}^{\prime\text{aff},\alpha}$ . of $Z_{G}^{\prime\alpha}.$ Construction of affine Zastava $Z_{G}^{\text{aff}}$ by quasimap was extended to $G$ being Kac-Moody group in [BFG06], and we expect that the construction of Zastava extends easily.

In this paper, we will consider two other approaches that recover $Z^{\alpha}_{G}$ when $G$ is simply-laced⁵⁵5It is an open question to identify these two approaches with the quasimap construction for any quiver besides ADE quivers. We expect that this identification can be reduced to the identification of three constructions of affine Zastava once the quasimap definition of Zastava has been made. For identification of the Coulomb branch affine Zastava (see 1.4) and quasimap affine Zastava, it suffices to show that the quasimap affine Zastava $\text{Qmap}^{\text{aff},\alpha}_{Q}$ is normal or flat over $\mathcal{H}_{C\times I}$ [BFN19]. This is known when $Q$ is ADE or affine type A [BF14] where they constructed a resolution of $\text{Qmap}^{\text{aff},\alpha}_{Q}$ and prove it is normal and Cohen-Macaulay, hence flat over $\mathcal{H}_{C\times I}$ .. These two do not directly deal with $G_{KM}$ , but with a quiver $Q$ . In the approach in 1.4 below, the growth and shift structure can also be constructed. Hence, we accomplished the prescription in 1.1 and have a definition of $\mathcal{G}(G_{KM})$ when $G_{KM}$ is associated with a quiver $Q$ .

1.3 Compactified Coulomb branch for quiver gauge theory with no framing

This is introduced in [BFN18, BFN19]. For any quiver $Q$ (possibly with loops) and dimension vector $\alpha\in\mathbb{Z}[I]$ , they defined a convolution algebra on certain equivariant homology space of a certain ind-scheme $\mathcal{R}$ , and showed it is commutative. They consider certain positive part $\mathcal{R}^{+}$ of $\mathcal{R}$ . Denote the Spec of the convolution subalgebra corresponding to $\mathcal{R}^{+}$ by $\textsf{M}^{\alpha}_{Q}.$ They also defined a filtration on this subalgebra and defined $\mathbf{M}^{\alpha}_{Q}$ as the Proj of its Rees algebra. When the underlying diagram of the quiver $Q$ is the Dynkin diagram of $G$ , they showed that $\textsf{M}^{\alpha}_{Q}$ ( or $\mathbf{M}^{\alpha}_{Q}$ ) coincides with $Z_{G}^{\text{aff},\alpha}$ ( or $Z_{G}^{\alpha}$ resp) for $C=\mathbb{A}^{1}$ (remark 3.7 in [BFN19]). However, defining the growth structure in this approach requires some work [MW24]. This approach, when accompanied by framing, extends the concept of generalized transversal slices to any quiver $Q$ [BFN19].

1.4 Local projective spaces [MYZ21, Mir23]

1.4.1 Motivations

We motivate this construction by explaining how to reconstruct $Z^{\alpha}_{G}$ directly from the Cartan matrix of $G$ . We abbreviate $\mathcal{H}_{C\times I}^{\alpha}$ by $\mathcal{H}^{\alpha}$ . Now we assume $G$ is simple and simply connected. The factorizable part of the Picard group of $\mathcal{G}^{C}(G)$ is $\mathbb{Z}.$ Let $\mathcal{O}(1)$ be the positive generator of it. Denote the descent of $\mathcal{O}(1)|_{Z^{\prime}_{G}}$ (under the partial symmetrization map $Z^{\prime}_{G}\xrightarrow[]{}Z_{G}$ ) to $Z_{G}$ also by $\mathcal{O}(1).$ For a sheaf $\mathcal{F}$ on $X$ and a map $X\xrightarrow[]{}Y$ , denote by $\mathcal{F}_{X/Y}$ the pushforward of $\mathcal{F}$ from $X$ to $Y$ (the notation is used when the map $X\xrightarrow[]{}Y$ is clear from the context).

We have the Kodaira embedding $Z^{\alpha}_{G}\xrightarrow[]{i}\mathbb{P}((\mathcal{O}(1)_{Z^{\alpha}_{G}/\mathcal{H}^{\alpha}})^{*})$ . Now we further assume that $G$ is simply laced. A crucial observation is that the restriction map $\mathcal{O}(1)_{Z^{\alpha}_{G}/\mathcal{H}^{\alpha}}\cong\mathcal{O}(1)_{(Z^{\alpha}_{G})^{T}/\mathcal{H}^{\alpha}}$ is an isomorphism. Moreover, $(Z_{G}^{\alpha})^{T}$ has a moduli description that solely depends on $\alpha$ . It is $Gr(\mathcal{T}^{\alpha})$ , the Hilbert scheme of the tautological bundle $\mathcal{T}^{\alpha}$ over $\mathcal{H}^{\alpha}$ . The generic fiber of the map $Z^{\alpha}_{G}\xrightarrow[]{\pi}\mathcal{H^{\alpha}}$ is an $n$ -th power of $\mathbb{P}^{1}$ ’s, where $n$ is the length of $\alpha$ . These $\mathbb{P}^{1}$ ’s can be regarded as the fibers of $Z^{\alpha_{i_{k}}}\xrightarrow[]{\pi}\mathcal{H}^{\alpha_{i_{k}}}$ , where $\alpha$ decomposes into the sum of simple roots $\alpha_{i_{k}},1\leq k\leq n,i_{k}\in I$ .

Over the regular part of $C^{\alpha}$ ⁶⁶6This is defined as the pullback under $C^{\alpha}\xrightarrow[]{}\mathcal{H}_{C\times I}^{\alpha}$ of the regular part $\mathcal{H}_{reg}^{\alpha}$ of $\mathcal{H}_{C\times I}^{\alpha}$ , the line bundle $\mathcal{O}(1)$ on $Z^{\prime\alpha}$ is canonically isomorphic to the outer tensor product (over $\mathcal{H}^{\alpha}$ ) of $\mathcal{O}(1)$ on these $Z^{\alpha_{i_{k}}},1\leq k\leq n$ , regardless of $G$ . Denote the isomorphism by $loc$ . Consider the ordered Zastava $Z^{\prime\alpha}_{G}=Z^{\alpha}_{G}\times_{\mathcal{H}^{\alpha}}C^{\alpha}$ (see footnote 4 on page 3). Denote the quotient map $C^{\alpha}\xrightarrow[]{q^{\alpha}}\mathcal{H}^{\alpha}$ . We have

Here $\mathbb{P}^{1}$ means $\mathbb{P}^{1}_{C_{reg}^{\alpha}}$ and $\mathbb{P}^{1}\times\cdots\times\mathbb{P}^{1}$ means the fiber product over $C_{reg}^{\alpha}$ . The space $(pr_{i})^{*}Gr(\mathcal{T}^{\alpha_{k_{i}}})$ is the pullback of $Gr(\mathcal{T}^{\alpha_{k_{i}}})$ under the projection $C_{reg}^{\alpha}\xrightarrow[]{pr_{i}}C^{\alpha_{k_{i}}}=\mathcal{H}^{\alpha_{k_{i}}}.$ In the left column, the first map is the Kodaira embedding. The second is induced by the isomorphism $\mathcal{O}(1)_{Z^{\prime\alpha}_{G}/C^{\alpha}}\cong\mathcal{O}(1)_{(Z^{\prime\alpha}_{G})^{T}/C^{\alpha}}\cong\mathcal{O}(1)_{(q^{\alpha})^{*}Gr(\mathcal{T}^{\alpha})/C^{\alpha}}$ . The third is embedding from the regular part to the whole. In the right column, the first map is the Kodaira embedding. The second map is induced by $\mathcal{O}(1)_{Z^{\prime\alpha}_{G}/C^{\alpha}}\cong\mathcal{O}(1)_{(Z^{\prime\alpha}_{G})^{T}/C^{\alpha}}\cong\mathcal{O}(1)_{(q^{\alpha})^{*}Gr(\mathcal{T}^{\alpha})/C^{\alpha}}$ for $\alpha=\alpha_{k_{i}}$ .

The map in the second row is denoted by $loc$ which comes from a certain isomorphism (still denoted by) $loc$ between line bundles. Same for the map $loc$ on the next row. The isomorphism in the first row is induced by the isomorphism $loc$ between line bundles that appear in the second row⁷⁷7This isomorphism does not descend to an isomorphism $Z_{G,reg}^{\alpha}\cong\mathbb{P}^{1}_{\mathcal{H}_{reg}^{\alpha}}\times_{\mathcal{H}_{reg}^{\alpha}}\cdots\times_{\mathcal{H}_{reg}^{\alpha}}\mathbb{P}^{1}_{\mathcal{H}_{reg}^{\alpha}}$ .

Under the isomorphism $loc$ , we get certain sections $S_{can}$ of $\mathcal{O}(1)_{(q^{\alpha})^{*}Gr(\mathcal{T}_{reg}^{\alpha})/C_{reg}^{\alpha}}$ corresponding to the canonical sections of the outer tensor of $\mathcal{O}(1)$ on $\mathbb{P}^{1}.$

Now it suffices to know how $Z^{\prime\alpha}_{reg}$ embeds into $\mathbb{P}((\mathcal{O}(1)_{(q^{\alpha})^{*}Gr(\mathcal{T}^{\alpha})/C^{\alpha}})^{*})$ , since then we can reconstruct $Z^{\prime\alpha}$ as the closure of $Z^{\prime\alpha}_{reg}$ in $\mathbb{P}((\mathcal{O}(1)_{(q^{\alpha})^{*}Gr(\mathcal{T}^{\alpha})/C^{\alpha}})^{*})$ by the fact that the map $Z^{\prime\alpha}\xrightarrow[]{\pi}C^{\alpha}$ is flat. This will be determined by the singularities of these sections $S_{can}$ of $\mathcal{O}(1)_{(q^{\alpha})^{*}Gr(\mathcal{T}_{reg}^{\alpha})/C_{reg}^{\alpha}}$ along the diagonal of $C^{\alpha}$ . The singularities can be classified by a symmetric integer matrix $\kappa$ , which is the Cartan matrix of $G$ .

1.4.2 Construction of Zastava $Z_{\kappa}$ for a matrix $\kappa$

On the other hand, for any symmetric matrix $\kappa\in M_{I}(\mathbb{Z})$ , there is a line bundle $L^{\kappa}$ (2.6) on $\mathcal{H}_{C\times I}$ with certain locality property. Let $pr_{1},pr_{2}$ be the maps $\mathcal{H}_{C\times I}\xleftarrow{pr_{1}}Gr(\mathcal{T}^{\alpha})\xrightarrow[]{pr_{2}}\mathcal{H}_{C\times I}$ such that $pr_{1}(D^{\prime}\subset D)=D^{\prime}$ and $pr_{2}(D^{\prime}\subset D)=D.$ This defines a vector bundle $\mathcal{I}^{\alpha}=(pr_{2})_{*}pr_{1}^{*}(L^{\kappa})$ . Denote the dual of $\mathcal{I}^{\alpha}$ by $V^{\alpha}$ . Its locality structure will determine an embedding $\Pi^{n}_{i=1}\mathbb{P}_{\mathcal{H}_{reg}^{\alpha}}^{1}\xrightarrow[]{i}\mathbb{P}(V^{\alpha})$ . Then one defines the space $Z^{\alpha}_{\kappa}$ as the closure of the image of $i$ in $\mathbb{P}(V^{\alpha})$ .

As we briefly discussed above, when the matrix $\kappa$ is the Cartan matrix of $G$ , Mirkovic showed that $Z^{\alpha}_{\kappa}$ coincides with $Z_{G}^{\alpha}$ ⁸⁸8 This is done by showing $(q^{\alpha})^{*}Z^{\alpha}_{\kappa}\cong(q^{\alpha})^{*}Z_{G}^{\alpha}=Z_{G}^{\prime\alpha}$ . It seems natural to use the notation $Z_{\kappa}^{\prime\alpha}$ for $(q^{\alpha})^{*}Z^{\alpha}_{\kappa}$ as we do for $Z_{G}^{\prime\alpha}$ but we will not use that.. In principle, the ” $\kappa$ -approach” is more general than the Coulomb branch construction since we can consider any symmetric matrix that is not necessarily the (modified) incidence matrix of a quiver (possibly with loops). In the upcoming paper[Don], we will provide some examples where the projection $Z^{\alpha}_{\kappa}\xrightarrow[]{\pi}\mathcal{H}^{\alpha}_{C\times I}$ is not flat when the matrix $\kappa$ is not the incidence matrix of a quiver. Therefore, the merit of zastava for such $\kappa$ is subject to skepticism.

The drawback of this approach is that $Z_{\kappa}^{\alpha}$ is defined as a closure and we lack a direct description. Additionally, it is unclear how to define the generalized transverse slices of $\mathcal{G}(G_{KM})$ . However, the advantage is that the growth structure [Mir23] is much easier to define.

1.5 Identification of two generalizations of Zastava

Now Let $C=\mathbb{A}^{1}$ . The main result of this paper is that the local (projective) space construction coincides with the Coulomb branch construction when $\kappa$ is the incidence matrix $\kappa(Q)$ of a quiver $Q$ . We will clarify the following parallel structures for Coulomb branch $\mathbf{M}^{\alpha}_{Q}$ and local projective space $Z^{\alpha}_{\kappa(Q)}$ . In the table below, we omit the subscripts $Q$ and $\kappa(Q)$ . Also, we fix $\alpha=(n_{i})_{i\in I}$ . Let $G=\prod_{i\in I}GL(k^{n_{i}})$ and $T$ be the diagonal subgroup of $G$ ⁹⁹9These $G,T$ depend on $\alpha$ and are different from the $G,T$ that appeared in the Zastava before. It is clear which $G,T$ we mean from the context.. Here, we omit $\alpha$ for the notations involving $\mathcal{R}$ .

	$\mathbf{M}^{\alpha}$	$Z^{\alpha}$
projective embedding	$\mathbf{M}^{\alpha}\xhookrightarrow{j_{\alpha}}\mathbb{P}_{\mathcal{H}^{\alpha}}(H_{}^{{G}}(\mathcal{R}^{+}_{\leq 1})^{})$	$Z^{\alpha}\xhookrightarrow{i_{\alpha}}\mathbb{P}_{\mathcal{H}^{\alpha}}(V^{\alpha})$
regular fibers over $\mathcal{H}_{reg}$	product of $\mathbb{P}^{1}$ ’s	product of $\mathbb{P}^{1}$ ’s
global basis	basis of $H_{*}^{G}(\mathcal{R}^{+}_{\leq 1})$	basis of $\Gamma(\mathcal{H}^{\alpha},\mathcal{I}^{\alpha})$

Here $\mathcal{R}^{+}_{\leq 1}$ is certain subspace of the positive part $\mathcal{R}^{+}$ of $\mathcal{R}.$ The embedding $j_{\alpha}$ follows from Lemma 6, while the embedding $i_{\alpha}$ is based on the definition of the local space. The space $\mathbf{M}^{\alpha}$ is over $Spec(H_{G}^{*}(pt))\cong\mathcal{H}^{\alpha}.$ Let $C^{\alpha}=\prod_{i\in I}C^{n_{i}}.$

Let us see how to identify $\mathbf{M}^{\alpha}$ and $Z^{\alpha}$ after pulling back under $C^{\alpha}\xrightarrow[]{q^{\alpha}}\mathcal{H}^{\alpha}$ .

Under the pull back $q^{\alpha}$ , the space $H_{*}^{G}(\mathcal{R}^{+}_{\leq 1})$ becomes $H_{*}^{T}(\mathcal{R}^{+}_{\leq 1})$ and $\Gamma(\mathcal{H}^{\alpha},\mathcal{I}^{\alpha})$ becomes $\Gamma(C^{\alpha},(q^{\alpha})^{*}\mathcal{I}^{\alpha}).$ We will identify $H_{*}^{T}(\mathcal{R}^{+}_{\leq 1})\xrightarrow[\cong]{rel}\Gamma(C^{\alpha},(q^{\alpha})^{*}\mathcal{I}^{\alpha})$ in section 4.1, as free rank 1 modules¹⁰¹⁰10In each component. over $H^{*}_{T}(\mathcal{R}^{+}_{\leq 1})\cong\mathcal{O}((q^{\alpha})^{*}Gr(\mathcal{T}^{\alpha})).$

In component, the isomorphism $H_{*}^{T}(\mathcal{R}_{\varpi_{\beta}})\xrightarrow[\cong]{rel}\Gamma(C^{\alpha},(q^{\alpha})^{*}\mathcal{I}^{\alpha,\beta})$ is given by the ring isomorphism $H_{*}^{T}(\mathcal{R}_{\varpi_{\beta}})\cong\mathcal{O}((q^{\alpha})^{*}Gr^{\beta}(\mathcal{T}^{\alpha}))$ and choosing basis $[\mathcal{R}_{\varpi_{\beta}}]$ of $H_{*}^{T}(\mathcal{R}_{\varpi_{\beta}})$ and basis $u^{\beta}\otimes 1$ of $\Gamma(C^{\alpha},(q^{\alpha})^{*}\mathcal{I}^{\alpha,\beta})$ (see 4.1 for the notations).

By a simple lemma (Lemma 4.3), it suffices to prove that the dotted arrow below exists as an isomorphism, i.e., the isomorphism $rel$ restricts to an isomorphism of two embeddings over the regular part $\mathcal{H}_{reg}$ (lemma 18).

To produce the dotted arrow, we will show their homogeneous coordinate rings are isomorphic. Any basis of $H_{*}^{T}(\mathcal{R}^{+}_{\leq 1})$ (resp. $\Gamma(C^{\alpha},(q^{\alpha})^{*}\mathcal{I}^{\alpha})$ ) over $H^{*}_{T}(\mathcal{R}^{+}_{\leq 1})$ (resp. $\mathcal{O}((q^{\alpha})^{*}Gr(\mathcal{T}^{\alpha}))$ generate the homogeneous coordinate rings of $\mathbb{P}_{C^{\alpha}}(H_{*}^{T}(\mathcal{R}^{+}_{\leq 1})^{*})$ (resp. $\mathbb{P}_{C^{\alpha}}((q^{\alpha})^{*}\mathcal{I}^{\alpha})$ ) over $\mathcal{O}(C^{\alpha}).$ There are canonical choices of basis for the localization over $C_{reg}$ of both $H_{*}^{T}(\mathcal{R}^{+}_{\leq 1})$ and $\Gamma(C^{\alpha},(q^{\alpha})^{*}\mathcal{I}^{\alpha})$ , which we will call the local bases¹¹¹¹11There are no canonical choices of basis before localization. We call any of them a global basis. Here we use parallel terminologies for global and local basis but they have different natures. The local basis of $H_{*}^{T}(\mathcal{R}^{+}_{\leq 1})$ arises naturally from the localization theory of equivariant homology, and the local basis of $\Gamma(C^{\alpha},(q^{\alpha})^{*}\mathcal{I}^{\alpha})$ is inherent in the definition of local projective spaces. We then show that these are identified via $rel$ . Finally, we verify that they satisfy the same relations through direct computation, using our explicit understanding of multiplication formulas on both sides.

1.6 Background and further directions

The loop Grassmannians associated to quivers $Q$ contain a class of loop Grassmannians for affine groups, so it is a part of a current effort by mathematicians and physicists to lift features of Langlands program to dimension 2. This paper unifies approaches to $\mathcal{G}_{Q}$ via Coulomb branches and local spaces. For any quiver $Q$ the latter constructs Zastavas and Beilinson-Drinfeld Grassmannian [Mir23] (more generally, for symmetric integral matrices), and the former [BFN19] constructs Zastava and generalized slices.

This seems to make some features of Satake equivalence for general quiver $Q$ more accessible at the moment. This includes conjectural geometric constructions of the positive part $U(\mathfrak{g}^{+}_{Q})$ of the enveloping algebra for a Kac-Moody algebra associated with $Q$ (using Zastava) [FKM20, 5.4], and of its irreducible highest weight module $V(\lambda)$ (using generalized slices) in [BFN19, Conjecture 3.25]. Also, one hopes that the Geometric Casselman-Shalika theorem, i.e., the Whittaker version of Satake equivalence [Ras21, 6.36.1] can be extended to loop Grassmannian $\mathcal{G}_{Q}$ associated to quivers.

2 Mirković local projective space (Zastava)

We recall the notion of local space and local projective space introduced by Mirkovic[MYZ21, Mir23].

2.1 Hilbert scheme $\mathcal{H}_{C\times I}$

Let $I$ be a finite set. Let $C$ be a smooth algebraic curve over an algebraic closed field $k$ . We consider the Hilbert scheme $\mathcal{H}_{C\times I}$ , the moduli of finite subschemes of $C\times I$ . Let $C_{i}=C\times i\subset C\times I$ and $\mathcal{H}^{n_{i}}_{C_{i}}$ be the Hilbert scheme of finite subschemes $D$ of length $n_{i}$ of $C_{i}$ . For $\alpha=(n_{i})_{i\in I}\in\mathbb{N}[I],$ we define $\mathcal{H}^{\alpha}_{C\times I}=\prod_{i\in I}\mathcal{H}^{n_{i}}_{C_{i}}.$ We understand $\alpha$ as a dimension vector.

We have $\mathcal{H}_{C\times I}=\sqcup_{\alpha\in\mathbb{N}[I]}\mathcal{H}^{\alpha}_{C\times I}.$ We also call $D=(D_{i})_{i\in I}\in\mathcal{H}_{C\times I}$ an $I$ -colored divisor where $D_{i}\in H_{C_{i}}$ is an effective divisor of $C_{i}$ for $i\in I$ . Since $C$ is a smooth algebraic curve, the Hilbert scheme $\mathcal{H}^{n}_{C}$ can also be viewed as the $n$ -th symmetric power $S^{n}C$ of $C$ , which is also the categorical quotient $C^{n}//S_{n}.$

2.2 The regular part of $\mathcal{H}_{C\times I}$

For $i,j\in I$ we define the $(i,j)$ -diagonal divisor $\Delta_{ij}\subset\mathcal{H}_{C\times I}$ . When $i\neq j$ the condition for $D\in\Delta_{ij}$ is that the component divisors $D_{i}$ and $D_{j}$ meet. Also, $\Delta_{ii}$ is a discriminant divisor, a subscheme $D\in\mathcal{H}_{C\times I}$ lies in $\Delta_{ii}$ if $D_{i}$ is not discrete, i.e., some point has multiplicity $>1$ . We call the complement of the union $\cup\Delta_{ij}\subset\mathcal{H}_{C\times I}$ the regular part of $\mathcal{H}_{C\times I}$ and denote it by $\mathcal{H}_{reg}$ . We call a divisor $D\in\mathcal{H}_{reg}$ a regular divisor. We say two divisors $D$ and $D^{\prime}$ are disjoint if they are disjoint after forgetting the colors.

2.3 Local line bundle $L$ on $\mathcal{H}_{C\times I}$ .

Definition 1.

A locality structure of a vector bundle $\mathcal{V}$ over $\mathcal{H}_{C\times I}$ is a system of isomorphisms : For two disjoint divisors¹²¹²12Here, divisor means closed subschemes of $C\times I\times S$ that are flat over $S$ $D,D^{\prime}\in\mathcal{H}_{C\times I}$ ,

\mathcal{V}_{D}\otimes\mathcal{V}_{D^{\prime}}\xrightarrow[]{i_{D,D^{\prime}}}\mathcal{V}_{D\cup D^{\prime}}

that satisfy the associative, commutative and unital properties.
Similarly, a local structure on a space $Y$ over $\mathcal{H}_{C\times I}$ is a system of isomorphisms

Y_{D}\times Y_{D^{\prime}}\xrightarrow[]{i_{D,D^{\prime}}}Y_{D\cup D^{\prime}}

that satisfy the same properties.

Remark 1.

The above structure (map between functors) is represented by the following. For any $\alpha=\alpha_{1}+\alpha_{2}$ , denote the vector bundle $\mathcal{V}$ on the component $\mathcal{H}^{\alpha}$ by $\mathcal{V}^{\alpha}.$ ¹³¹³13Here in the notation $\mathcal{V}^{\alpha}$ , the superscript $\alpha$ indexes the connected component, not power. We have the addition of divisors

\mathcal{H}^{\alpha_{1}}\times\mathcal{H}^{\alpha_{2}}\xrightarrow[]{a^{\alpha_{1},\alpha_{2}}}\mathcal{H}^{\alpha}.

Let $pr_{i},i=1,2$ and $\mathcal{H}^{\alpha_{1}}\times\mathcal{H}^{\alpha_{2}}\xrightarrow[]{pr_{i}}\mathcal{H}^{\alpha_{i}}$ be the $i$ -th projection. Denote $(\mathcal{H}^{\alpha_{1}}\times\mathcal{H}^{\alpha_{2}})_{disj}$ by the open part of $\mathcal{H}^{\alpha_{1}}\times\mathcal{H}^{\alpha_{2}}$ that consists $(D,D^{\prime})$ such that $D\in\mathcal{H}^{\alpha_{1}},D^{\prime}\in\mathcal{H}^{\alpha_{2}}$ are disjoint. A locality structure on a vector bundle $\mathcal{V}$ over $\mathcal{H}_{C\times I}$ is a system of isomorphisms over $(\mathcal{H}^{\alpha_{1}}\times\mathcal{H}^{\alpha_{2}})_{disj}$

\mathcal{V}^{\alpha_{1}}\boxtimes\mathcal{V}^{\alpha_{2}}:=pr_{1}^{*}\mathcal{V}^{\alpha_{1}}\otimes pr_{2}^{*}\mathcal{V}^{\alpha_{2}}\xrightarrow[]{i^{\alpha_{1},\alpha_{2}}}(a^{\alpha_{1},\alpha_{2}})^{*}\mathcal{V}^{\alpha},

for any $\alpha_{1},\alpha_{2}$ that satisfy the associative, commutative and unital properties. Later, we will have vector bundles with the notation $L,\mathcal{I}$ and $V$ , where $L$ means it is of rank 1, i.e. a line bundle.

2.4 Induced vector bundle $\mathcal{I}$

To a local line bundle $L$ over $\mathcal{H}_{C\times I}$ , we associate an induced vector bundle $\mathcal{I}$ over $\mathcal{H}_{C\times I}$ .

Let $\mathcal{T}\subset\mathcal{H}_{C\times I}\times(C\times I)$ be the tautological scheme over $\mathcal{H}_{C\times I}$ , where the fiber $\mathcal{T}_{D}$ at $D\in\mathcal{H}_{C\times I}$ is $D\subset C\times I.$ Let $Gr(\mathcal{T})$ be the relative Hilbert scheme $Gr(\mathcal{T}/\mathcal{H}_{C\times I})$ over $\mathcal{H}_{C\times I}$ such that the fiber at $D\in\mathcal{H}_{C\times I}$ is $Gr(D):=\mathcal{H}_{D}$ , where $\mathcal{H}_{D}$ is the Hilbert scheme of all subschemes $D^{\prime}\subset D$ .

Let $\mathcal{T}^{\alpha}$ be the component of $\mathcal{T}$ over $\mathcal{H}^{\alpha}_{C\times I}$ and $Gr^{\beta}(\mathcal{T}^{\alpha})$ be the component of $Gr(\mathcal{T}/\mathcal{H}_{C\times I})$ whose fiber at $D$ is $Gr^{\beta}(D):=\mathcal{H}^{\beta}(D),$ the Hilbert scheme of subschemes $D^{\prime}\subset D$ of length $\beta$ .

The scheme $Gr(\mathcal{T})$ is a self correspondence of $\mathcal{H}_{C\times I}$ ,

\mathcal{H}_{C\times I}\xleftarrow{pr_{1}}Gr(\mathcal{T})\xrightarrow[]{pr_{2}}\mathcal{H}_{C\times I},

where $pr_{1}(D^{\prime}\subset D)=D^{\prime}$ and $pr_{2}(D^{\prime}\subset D)=D.$

Lemma 1.

The map $pr_{2}$ is finite flat so $\mathcal{I}$ is a vector bundle.

Proof.

For brevity, assume $I=\{1\}$ . For $p=(p_{1}<\cdots<p_{k})$ , let $Gr_{p}(\mathcal{T})$ be the partial flag space of all filtrations $D_{1}\subset\cdots\subset D_{k}\subset D$ with $|D_{i}|=p_{i}.$ The maps $Gr_{q}(\mathcal{T})\xrightarrow[]{}Gr_{p}(\mathcal{T})$ are clearly finite flat when $q$ is obtained from $p$ by adding $p_{s+1}$ after some $p_{s}$ . Build a tower of maps from $p=\{0\}$ to $p=(1,\cdots,n)$ . The claim follows by the property that if $X\xrightarrow[]{f}Y$ is flat, $Y\xrightarrow[]{g}Z$ is flat if and only if $x\xrightarrow[]{g\circ f}Z$ is flat. ∎

Now define the induced vector bundle $\mathcal{I}=(pr_{2})_{*}pr_{1}^{*}(L)$ and its dual $V=\mathcal{I}^{*}$ .

Lemma 2.

A local structure of $L$ canonically induces a local structure on $V$

Proof.

The locality structure is naturally induced from its dual vector bundle so we check the locality structure for $\mathcal{I}$ . Here we only check the case where $D,D^{\prime}$ are regular for later use. The general case is similar. We compute the fiber $\mathcal{I}_{D}$ for $D=\sqcup^{n}_{k=1}p_{k}$

\mathcal{I}_{D}=((pr_{2})_{*}(pr_{1})^{*}L)_{D}=\oplus_{E\subset D}L_{E}.

Now for disjoint divisors $D,D^{\prime}$ ,

\mathcal{I}_{D}\otimes\mathcal{I}_{D^{\prime}}\cong(\oplus_{E\subset D}L_{E})\otimes(\oplus_{E^{\prime}\subset D^{\prime}}L_{E^{\prime}})\cong\oplus_{E\subset D,E^{\prime}\subset D^{\prime}}L_{E}\otimes L_{E^{\prime}}\xrightarrow[]{i^{\mathcal{I}}_{D,D^{\prime}}}\oplus_{E\cup E^{\prime}\in D\cup D^{\prime}}L_{E\cup E^{\prime}}\cong\mathcal{I}_{D\cup D^{\prime}},

where $i^{\mathcal{I}}_{D,D^{\prime}}$ is the direct sum of the locality isomorphisms of $L$ . We have the last isomorphism since $D,D^{\prime}$ are disjoint, any subscheme $F\subset D\cup D^{\prime}$ can be uniquely written as the union of $E\subset D$ and $E^{\prime}\subset D^{\prime}$ . ∎

2.5 Local projective space of a local vector bundle

To a local vector bundle $V$ over $\mathcal{H}_{C\times I}$ we will associated its local projective space $\mathbb{P}^{loc}(V)\subset\mathbb{P}(V)$ .

First, at a point $D=p\in C\times I\subset\mathcal{H}_{C\times I}$ the locality condition is empty, so we define $\mathbb{P}^{loc}(V)_{p}=\mathbb{P}(V_{p})$ . Now over a regular divisor $D=\sum p_{k}$ , as required by locality condition we define

\mathbb{P}^{loc}(V)_{D}=\prod^{n}_{k=1}\mathbb{P}(V_{p_{k}}).

Then the locality structure of $\otimes^{n}_{k=1}V_{p_{k}}\cong V_{D}$ gives the Segre embedding

\mathbb{P}^{loc}(V)_{D}\hookrightarrow\mathbb{P}(V)_{D}.

Now we have a subspace ${P}^{loc}_{reg}(V)$ in $\mathbb{P}(V)$ defined over $\mathcal{H}_{reg}.$ We define the local projective space $\mathbb{P}^{loc}(V)$ as the closure of $\mathbb{P}^{loc}_{reg}(V)$ in $\mathbb{P}(V).$ By definition $\mathbb{P}^{loc}_{reg}(V)$ is a local space over $\mathcal{H}.$

Remark 2.

Now we represent the Segre embedding. Denote by $e_{i}\in\mathbb{N}[I]$ which is $1$ in the $i$ -th coordinate and $0$ in the others. Denote $C^{\alpha}=\prod_{i\in I}C_{i}^{n_{i}}$ Let $q^{\alpha}$ be the quotient map $C^{\alpha}\xrightarrow[]{q^{\alpha}}\mathcal{H}^{\alpha}$ . For $i\in I,1\leq j\leq n_{i}$ , let $pr_{ij}$ be the compositions of $C^{\alpha}\xrightarrow[]{\text{i-th projection}}C^{n_{i}}_{i}\xrightarrow[]{\text{j-th projection}}C_{i}$ .

Let $C^{\alpha}_{reg}=(q^{\alpha})^{-1}(\mathcal{H}^{\alpha}_{reg})$ and $C_{reg}=\sqcup_{\alpha\in\mathbb{N}[I]}C^{\alpha}_{reg}$ . Recall we denote the restriction of $V,\mathcal{I}$ to $C_{i}$ by $V^{e_{i}},\mathcal{I}^{e_{i}}$ . For $\alpha=(n_{i})_{i\in I}$ , the local structure of $\mathcal{I}$ defines the isomorphism over $C_{reg}$

\bigotimes_{i\in I}\bigotimes_{j\in\{1,\cdots a_{i}\}}pr_{ij}^{*}\mathcal{I}^{e_{i}}\xleftarrow[]{i_{\alpha}}(q^{\alpha})^{*}\mathcal{I}^{\alpha}.

(1)

and the Segre embedding over $C_{reg}$ is the corresponding embedding

\prod_{i\in I}\prod_{j\in\{1,\cdots a_{i}\}}pr_{ij}^{*}\mathbb{P}(V^{e_{i}})\xhookrightarrow{\mathbf{i}^{*}_{\alpha}}(q^{\alpha})^{*}\mathbb{P}(V^{\alpha}).

(2)

It is more convenient to work over $C^{\alpha}$ other than $\mathcal{H}^{\alpha}$ when doing calculations. We will always pullback objects over $\mathcal{H}^{\alpha}$ by $q^{\alpha}$ . In practice, we will consider the image of $\mathbf{i}^{*}_{\alpha}$ and its closure over $C^{\alpha}$ which descents to $\mathbb{P}^{loc}_{reg}(V)$ and $\mathbb{P}^{loc}(V)$ respectively under $q^{\alpha}$ .

Remark 3.

We compute the fiber $V_{p}$ for $p$ a point. Since $Gr(p)=\{\emptyset,p\}$ , we have $V_{p}=L^{*}_{\emptyset}\oplus L^{*}_{p}\cong k\oplus L^{*}_{p}$ and $\mathbb{P}(V_{p})\cong\mathbb{P}^{1}.$ By locality, the fiber $V_{D}$ at a regular divisor $D$ of length $\alpha$ is¹⁴¹⁴14Since we just compute the fiber we can ignore the index $I$ and only count the total length $|\alpha|$ of $D$ .

V_{D}=(\oplus_{D^{\prime}\subset D}L_{D^{\prime}})^{*}\cong k^{2^{|\alpha|}}.

Hence over regular divisors, the fiber of $\mathbb{P}^{loc}(V^{\alpha})$ is a product of $\mathbb{P}^{1}$ ’s and the local space $\mathbb{P}^{loc}(V^{\alpha})$ can be viewed as a degeneration of a product of $\mathbb{P}^{1}$ ’s in $\mathbb{P}^{2^{|\alpha|}-1}.$

2.6 Zastava space $Z_{\kappa}$ of a symmetric matrix $\kappa$

We define a local line bundle over $\mathcal{H}_{C\times I}$ associated with a symmetric integral matrix $\kappa\in M_{I}(\mathbb{Z})$ as

L_{\kappa}=\mathcal{O}_{\mathcal{H}}(-\kappa\Delta),

where $\kappa\Delta$ is the divisor $\sum_{i\leq j}\kappa_{ij}\Delta_{ij}$ in $\mathcal{H}_{C\times I}.$ On each connected component, we can embed $L_{\kappa}$ into the sheaf of the fraction field of $C$ . The locality structure of $L$ is then the multiplication of rational functions.

For the local line bundle $L_{\kappa}$ , denote the induced vector bundle by $\mathcal{I}_{\kappa}$ and its dual by $V_{\kappa}$ . We define $Z^{\alpha}_{\kappa}=\mathbb{P}^{loc}(V_{\kappa}^{\alpha}).$ Now we fix a matrix $\kappa$ and drop $\kappa$ from the notation. Denote the projection $Z^{\alpha}\xrightarrow[]{\pi}\mathcal{H}^{\alpha}.$ Abusing notation, we will also denote the pullback of $\pi$ under $q^{\alpha}$ by $\pi$ .

2.7 $Z^{\alpha}$ for $C=\mathbb{A}^{1}$

From now on, we set $C=\mathbb{A}^{1}$ to be the affine line. We fix a dimension vector $\alpha$ . Since the matrix $\kappa$ is also fixed we write $L=L_{\kappa},\mathcal{I}=\mathcal{I}_{\kappa},V=V_{\kappa}.$ Denote by $\mathcal{S}_{\mathcal{H}^{\alpha}}(\mathcal{I}^{\alpha})$ the symmetric algebra generated by $\mathcal{I}^{\alpha}$ over $\mathcal{H}^{\alpha}$ . Let $A_{0}$ be¹⁵¹⁵15The notation $A_{0}$ is only used in section 2.7. It does not reflect its dependence on $\alpha$ . Since we fix $\alpha$ in 2.7, this will not cause confusion. The same happens for notation $\mathbf{I}$ . its pullback under $q^{\alpha},$ hence $Proj(A_{0})\cong(q^{\alpha})^{*}\mathbb{P}(V^{\alpha}).$ Let $A_{0,reg}$ be its localization over $C^{\alpha}_{reg}=(q^{\alpha})^{*}\mathcal{H}_{reg}$ .
Let $\mathbf{I}_{reg}$ be an ideal in $A_{0,reg}$ such that $Proj(A_{0,reg}/\mathbf{I}_{reg})\cong(q^{\alpha})^{*}Z^{\alpha}|_{reg}.$
Since $\mathcal{I}^{\alpha}$ is a free sheaf over $\mathcal{H}^{\alpha}$ , once we choose a trivialization of it, we can write the algebra $A_{0}$ as a polynomial ring.

2.7.1 Explicit description of locality structure on $\mathcal{I}$

In this subsection we will study the ideal $\mathbf{I}_{reg}$ that defines the Segre embedding on $C_{reg}.$

Since we work over $C^{\alpha}$ rather than $\mathcal{H}^{\alpha}$ , it is necessary to understand the pullback of the line bundle $L$ under $q^{\alpha}$ . We fix a coordinate $a$ on $C=A^{1}$ so get a coordinate system $(a^{i}_{j})_{i\in I,j\in\{1,\cdots,n_{i}\}}$ for $C^{\alpha}$ .

Lemma 3.

The pullback of the divisors are

(q^{\alpha})^{*}\Delta_{ii}=div(\prod_{l\neq j}(a^{i}_{l}-a^{i}_{j}),\text{ }(q^{\alpha})^{*}\Delta_{ii^{\prime}}=div(\prod_{l,j}(a^{i}_{l}-a^{i^{\prime}}_{j})).

Proof.

The second formula is clear. This first follows from the case when $I=\{1\}$ and $\alpha=2$ (recall $\alpha\in\mathbb{Z}^{I}$ is a dimension vector and when $I$ is a one-point set, $\alpha\in\mathbb{Z})$ . ∎

Denote the map replacing $\mathcal{I}$ by $L$ in (1) by $i^{L}_{\alpha}$ . We will describe the map $i_{\alpha}$ in (1) from the corresponding maps $i^{L}_{\alpha}$ of $L$ . Note in lemma 2, we described the locality structure of $\mathcal{I}$ by morphisms between functor of points. Recall the map

\mathcal{H}\xleftarrow[]{pr_{1}}Gr(\mathcal{T})\xrightarrow[]{pr_{2}}\mathcal{H}

On the component $\mathcal{H}^{\alpha},$ it is

\sqcup_{0\leq\beta\leq\alpha}\mathcal{H^{\beta}}\xleftarrow[]{pr^{\alpha}_{1}}\sqcup_{0\leq\beta\leq\alpha}Gr^{\beta}(\mathcal{T}^{\alpha})\xrightarrow[]{pr^{\alpha}_{2}}\mathcal{H}^{\alpha},

which is the disjoint union of

\mathcal{H^{\beta}}\xleftarrow[]{pr^{\alpha,\beta}_{1}}Gr^{\beta}(\mathcal{T}^{\alpha})\xrightarrow[]{pr^{\alpha,\beta}_{2}}\mathcal{H}^{\alpha},

(3)

Let $\mathcal{I}^{\alpha,\beta}=(pr^{\alpha,\beta}_{2})_{*}(pr^{\alpha,\beta}_{1})^{*}L.$ For $\alpha$ we define an $I$ -colored index set $S^{\alpha}=\sqcup_{i\in I}\{1,\cdots,n_{i}\}.$ For any subset $S\subset S^{\alpha},$ let $S_{i}$ be the $i$ -th component of $S$ and let $|S|=(|S_{i}|)_{i\in I}.$ Define $C^{S}=\prod_{i\in I}\prod_{j\in S_{i}}C_{ij}$ , where $C_{ij}=C$ so that $C^{\alpha}=C^{S^{\alpha}}.$ Now we will see what $(q^{\alpha})^{*}\mathcal{I}^{\alpha,\beta}$ is. We introduce the following

\mathcal{H}^{\beta}\xleftarrow[]{pr_{3}^{\alpha,\beta}}(q^{\alpha})^{*}Gr^{\beta}(\mathcal{T}^{\alpha})\xrightarrow[]{pr^{\alpha,\beta}_{4}:=(q^{\alpha})^{*}(pr_{2}^{\alpha,\beta})}C^{\alpha},

(4)

where $pr_{3}^{\alpha,\beta}$ is the composition of maps from $(q^{\alpha})^{*}Gr^{\beta}(\mathcal{T}^{\alpha})$ to $Gr^{\beta}(\mathcal{T}^{\alpha})$ and $Gr^{\beta}(\mathcal{T}^{\alpha})\xrightarrow[]{pr^{\alpha,\beta}_{1}}\mathcal{H}^{\beta}$ . We have

(q^{\alpha})^{*}\mathcal{I}^{\alpha,\beta}\cong(pr^{\alpha,\beta}_{4})_{*}(pr^{\alpha,\beta}_{3})^{*}(L).

(5)

Over $C^{\alpha}_{reg},$ $(q^{\alpha})^{*}Gr^{\beta}(\mathcal{T}^{\alpha}_{reg})\cong\sqcup_{S,|S|=|\beta|}C^{\alpha}_{reg}$ and the map $pr_{1}^{\alpha,\beta}$ factors through $\sqcup_{S,|S|=|\beta|}C^{\beta}_{reg}$ so (4) becomes

\mathcal{H}_{reg}^{\beta}\xleftarrow[]{\sqcup q^{\beta}}\sqcup_{S,|S|=|\beta|}C^{\beta}_{reg}\xleftarrow[]{\sqcup pr^{\alpha,S}}\sqcup_{S,|S|=|\beta|}C^{\alpha}_{reg}\xrightarrow[]{\sqcup id}C^{\alpha}_{reg},

(6)

where $pr^{\alpha,S}$ is the projection of the $S$ -components of $C^{\alpha}$ to $C^{S}$ composed with the identification map $C^{S}\cong C^{\beta}$ (and we still denote by the same notation $pr^{\alpha,S}$ when restricting on the regular part). On the $S$ -component, this is

\mathcal{H}_{reg}^{\beta}\xleftarrow[]{q^{\beta}}C^{\beta}_{reg}\xleftarrow[]{pr^{\alpha,S}}C^{\alpha}_{reg}=C^{\alpha}_{reg}.

(7)

Denote the pullback of $L_{reg}^{\beta}$ under $pr^{\alpha,S}\circ q^{\beta}$ by $L_{reg}^{\alpha,S}$ . Hence $(q^{\alpha})^{*}\mathcal{I}^{\alpha}|_{reg}$ is the direct sum of line bundles $L_{reg}^{\alpha,S}$ , i.e.

(q^{\alpha})^{*}\mathcal{I}^{\alpha}_{reg}\cong\oplus_{0\leq\beta\leq\alpha}(q^{\alpha})^{*}\mathcal{I}^{\alpha,\beta}_{reg}\cong\oplus_{0\leq\beta\leq\alpha}\oplus_{S,|S|=|\beta|}L^{\alpha,S}_{reg}=\oplus_{S,S\subset S^{\alpha}}L^{\alpha,S}_{reg}.

(8)

Here when $\beta=0$ , $S=\emptyset$ and $L^{\alpha,\emptyset}$ is the pullback from $C^{0}$ to $C^{\alpha}$ , which is the trivial line bundle (structure sheaf). Apply (8) to the case where $\alpha=e_{i}$ , since $q^{e_{i}}=id$ and $C^{e_{i}}_{reg}=C^{e_{i}}$ , we have

\mathcal{I}^{e_{i}}\cong L^{e_{i},\emptyset}\oplus L^{e_{i}}.

Pulling back under $C^{\alpha}\xrightarrow[]{pr_{ij}}C^{e_{i}}$ , we have

pr_{ij}^{*}\mathcal{I}^{e_{i}}\cong L^{\alpha,\emptyset}\oplus pr_{ij}^{*}L^{e_{i}}.

Now the locality isomorphism $i_{\alpha}$ of $\mathcal{I}$ in (1) (to be described) becomes

(\otimes_{i\in I}\otimes_{j\in(S_{\alpha})_{i}}(L^{\alpha,\emptyset}\oplus pr_{ij}^{*}L^{e_{i}}))|_{C^{\alpha}_{reg}}\xleftarrow[]{i_{\alpha}}\oplus_{S,S\subset S_{\alpha}}L^{\alpha,S}_{reg}.

Expanding the tensor product, since $L^{\alpha,\emptyset}\otimes\mathcal{L}$ is the trivial line bundle, for any $S\subset S^{\alpha}$ the $S$ -component of the map $i_{\alpha}$ is denoted by $i^{S}_{\alpha}$

\otimes_{i\in I}\otimes_{j\in S_{i}}pr_{ij}^{*}L^{e_{i}}|_{C^{\alpha}_{reg}}\xleftarrow[]{i_{\alpha}^{S}}L_{reg}^{\alpha,S}.

(9)

Finally, we can describe the isomorphism $i^{S}_{\alpha}$ . It is the pullback under $C^{\beta}_{reg}\xleftarrow[]{pr^{S}}C^{\alpha}_{reg}$ for $\beta=|S|$ of an open part¹⁶¹⁶16In remark (1), we decompose $\beta$ completely. of the locality structure of $L^{\beta}$

\otimes_{i\in I}\otimes_{j\in(S_{\beta})_{i}}pr^{*}_{ij}(L^{e_{i}})|_{C^{\beta}_{reg}}\xleftarrow[\cong]{i^{L}_{\beta}}(q^{\beta})^{*}L^{\beta}|_{C^{\beta}_{reg}}.

(10)

It is not hard to see that $i_{\alpha}$ as described above are the isomorphisms that represent the locality isomorphisms in lemma 2.

2.7.2 Bases of global section $\Gamma(C^{\alpha},(q^{\alpha})^{*}\mathcal{I})$ as $\mathcal{O}(C^{\alpha})$ -module (global bases)

For each $\beta$ , we fix a basis $s^{\beta}\in\Gamma(C^{\beta},(q^{\beta})^{*}L^{\beta})$ adapted to the above local structure as follows. First we fix a basis $s_{i}$ of global section of $L^{e_{i}}$ over $\mathcal{H}^{e_{i}}.$ For any $\beta=(k_{i})_{i\in I}\in\mathbb{N}[I],$ let

s^{\beta}=l(\beta)i_{\beta}(\otimes_{i\in I}\otimes_{j\in(S_{\beta})_{i}}pr^{*}_{ij}s_{i})\in\Gamma(C^{\beta}_{reg},(q^{\beta})^{*}L^{\beta}|_{C_{reg}}),

(11)

where ¹⁷¹⁷17Here, we choose a basis according to $\kappa$ , later, we will define $l_{Q}(\beta)$ depending on the quiver $Q$ .

l(\beta)=\prod_{i<i^{\prime},l\in(S_{\beta})_{i},j\in(S_{\beta})_{i^{\prime}}}(a^{i}_{l}-a^{i^{\prime}}_{j})^{\kappa_{ii^{\prime}}}\prod_{i\in I,l\neq j,l,j\in(S_{\beta})_{i}}(a^{i}_{l}-a^{i}_{j})^{\kappa_{ii}}

(12)

is in the fraction field $\mathcal{K}(\mathcal{H}^{\beta})$ of $\mathcal{H}^{\beta}.$ Here we still denote by $i_{\beta}$ the induced map between global sections of line bundles in (10). By lemma 3, the section $s^{\beta}$ extend across the diagonal to a section of $C^{\beta}.$ Let $W^{\beta}$ be the group such that $C^{\beta}/W^{\beta}\cong\mathcal{H}^{\beta}$ . It is easy to check that $s^{\beta}$ is $W^{\beta}$ -invariant. Let $u^{\beta}\in\Gamma(\mathcal{H}^{\beta},L^{\beta})$ be the descent of $s^{\beta}$ under $q^{\beta}.$

Let $M^{\beta}$ be the global sections of $L^{\beta}$ over $\mathcal{H}^{\beta}$ . It is a rank 1 free $\mathcal{O}(\mathcal{H}^{\beta})$ -module generated by $u^{\beta}$ . By (5),

\Gamma(C^{\alpha},(q^{\alpha})^{*}\mathcal{I}^{\alpha,\beta})=M^{\beta}\otimes_{\mathcal{O}(\mathcal{H}^{\beta})}\mathcal{O}((q^{\alpha})^{*}Gr^{\beta}(\mathcal{T}^{\alpha}))

as $O(C^{\alpha})$ -module. By choosing any basis $\{y^{\alpha,\beta}_{p}\}$ of $\mathcal{O}((q^{\alpha})^{*}Gr^{\beta}(\mathcal{T}^{\alpha}))$ as $\mathcal{O}(C^{\alpha})$ -module, which is chosen as a basis of $\mathcal{O}(Gr^{\beta}(\mathcal{T}^{\alpha}))$ as $\mathcal{O}(\mathcal{H}^{\alpha})$ -module, we have the basis $\{z^{\alpha,\beta}_{p}=u^{\beta}\otimes y_{p}\}$ of $\Gamma(C^{\alpha},(q^{\alpha})^{*}\mathcal{I}^{\alpha,\beta})$ . When $\alpha$ is fixed, we abbreviate $z^{\alpha,\beta}_{p}$ as $z^{\beta}_{p}$ . Such a basis is called a global basis.

2.7.3 The basis of $\Gamma(C^{\alpha}_{reg},(q^{\alpha})^{*}\mathcal{I}_{reg})$ (local basis)

From $1\in\mathcal{O}((q^{\alpha})^{*}Gr^{\beta}(\mathcal{H}^{\alpha}))$ , we get an element $u^{\beta}\otimes 1$ in

\Gamma(C^{\alpha},(q^{\alpha})^{*}\mathcal{I}^{\alpha,\beta})=M^{\beta}\otimes_{\mathcal{O}(\mathcal{H}^{\beta})}\mathcal{O}((q^{\alpha})^{*}Gr^{\beta}(\mathcal{T}^{\alpha})).

Restricting $u^{\beta}\otimes 1$ to $C^{\alpha}_{reg}$ , by (6), we have $s^{\beta}|_{reg}\otimes 1$ in

\Gamma(C_{reg}^{\alpha},(q^{\alpha})^{*}\mathcal{I}_{reg}^{\alpha,\beta})=\bigoplus_{S,|S|=\beta}\Gamma(C_{reg}^{\alpha},L^{\alpha,S}_{reg})=\bigoplus_{S,|S|=\beta}\Gamma(C_{reg}^{\beta},(q^{\beta})^{*}L^{\beta}_{reg})\otimes_{\mathcal{O}(C^{\beta}_{reg})}{\mathcal{O}(C^{\alpha}_{reg})}.

In the decomposition in (8), denote the $S$ -component of $s^{\beta}|_{reg}\otimes 1$ by $s^{\alpha,S}\in\Gamma(C^{\alpha}_{reg},L^{\alpha,S}_{reg})$ . We will refer to $s^{\alpha,S}$ as the local basis. Now we fix an $\alpha$ and abbreviate $s^{\alpha,S}$ as $s^{S}.$ By our choices of bases $s^{\beta}$ in (11) and the relation between (9) and (10), we have

s^{S}=l(S)i^{S}_{\alpha}(\otimes_{i\in I}\otimes_{j\in S_{i}}pr^{*}_{ij}s_{i}),

(13)

where

l(S)=\prod_{i<i^{\prime},l\in S_{i},j\in S_{i^{\prime}}}(a^{i}_{l}-a^{i^{\prime}}_{j})^{\kappa_{ii^{\prime}}}\prod_{i\in I,l\neq j,l,j\in S_{i}}(a^{i}_{l}-a^{i}_{j})^{\kappa_{ii}}

(14)

is the pullback of $l(\beta)$ under $C^{\beta}_{reg}\xleftarrow[]{pr^{\alpha,S}}C^{\alpha}_{reg}$ (see (7)) and we still denote by $i^{S}_{\alpha}$ the induced map between global sections of line bundles in (9). By definition $\{s^{S},S\subset S^{\alpha}\}$ generate the algebra $A_{0,reg}$ over $\mathcal{O}(C_{reg}^{\alpha})$ . Now we can describe the ideal $\mathbf{I}_{reg}$ in terms of local basis $s^{S}$ .

Lemma 4.

The ideal $\mathbf{I}_{reg}$ is generated by

\frac{s^{A}}{l(A)}\frac{s^{B}}{l(B)}=\frac{s^{A\cup B}}{l(A\cup B)}\frac{s^{A\cap B}}{l(A\cap B)}

for all $A,B\subset S_{\alpha}.$

Proof.

This is a standard result of Segre embedding. Since we cannot find a reference, we give a proof in the appendix. ∎

2.7.4 Transition between a global basis $z^{\beta}_{p}$ and the local basis $s^{S}$ over the regular part

First, we state a simple lemma about the module structure under localization.

Lemma 5.

The following diagram commutes

where the horizontal maps are the ring actions on the modules.

Proof.

Clear from the definitions. ∎

Now we will study the following map in more details.

\mathcal{O}(Gr^{\beta}(\mathcal{T}^{\alpha}))\xrightarrow[]{|_{reg}}\mathcal{O}(Gr^{\beta}(\mathcal{T}_{reg}^{\alpha}))\cong\bigoplus_{S,S\subset S_{\alpha},|S|=|\beta|}\mathcal{O}(\mathcal{H}^{\alpha}_{reg}).

(15)

Let $\beta=\sum_{i\in I}k_{i}i$ . Denote $f^{s}()$ the $s$ -th elementary symmetric functions on the variables inside the parenthesis.

Theorem 1.

\mathcal{O}(Gr^{\beta}(\mathcal{T}^{\alpha}))\cong\bigotimes_{i\in I}\mathcal{O}(Gr^{k_{i}}(\mathcal{T}^{n_{i}})).

b) The map $(pr_{1},m)$

Gr^{k_{i}}(\mathcal{T}^{n_{i}})\xrightarrow[]{(pr_{1},m)}\mathcal{H}^{k_{i}}\times\mathcal{H}^{n_{i}-k_{i}},

where $m((D^{\prime},D)=D-D^{\prime}$ is an isomorphism.
c) The projection $Gr^{k_{i}}(\mathcal{T}^{n_{i}})\xrightarrow[]{pr_{2}}\mathcal{H}^{n_{i}}$ is finite flat and $\mathcal{O}(Gr^{k_{i}}(\mathcal{T}^{n_{i}}))$ is an algebra extension over $\mathcal{O}(\mathcal{H}^{n_{i}})$ generated by formal variables $c^{i}_{l},d^{i}_{j},1\leq l\leq k_{i},1\leq j\leq n_{i}-k_{i}$ subject to the relations

<\sum_{1\leq l\leq k_{i},l+j=s_{i}}c^{i}_{l}d^{i}_{j}-f^{s_{i}}(a^{i}_{1},\cdots,a^{i}_{n_{i}})>

for all $s_{i}$ such that ${1\leq s_{i}\leq n_{i}}.$
d) Under the above isomorphism and (15), the image of $c^{i}_{l}$ in the $S_{i}$ component is $f^{l}(a^{i}_{r})_{r\in S_{i}}$ and the image of $d^{i}_{j}$ in the $S_{i}$ component is $f^{j}(a^{i}_{r})_{r\in\{1,\cdots,n_{i}\}\setminus S_{i}}$ .

Proof.

a) Since

(Gr^{\beta}(\mathcal{T}^{\alpha}))\cong\prod_{i\in I}(Gr^{k_{i}}(\mathcal{T}^{n_{i}})).

b) Clear.
c) We omit the index $i$ . The map $pr_{2}$ being finite flat is already proved. Since we have the map $Gr^{k}(\mathcal{T})\xrightarrow[]{pr_{1}}\mathcal{H}^{k}$ , where $pr_{1}(D^{\prime},D)=D$ , we can pull back functions on $\mathcal{H}^{k}$ to $Gr^{k}(\mathcal{T})$ . Let $\mathcal{S}_{h}$ be the symmetric group of order $h$ . We have $f^{l}(a_{1},\cdots,a_{k})\in k[a_{1},\cdots,a_{k}]^{\mathcal{S}_{k}}\cong\mathcal{O}(\mathcal{H}^{k}).$ Let their pullback under $pr_{1}$ be $c_{1},\cdots c_{k}$ . We also have a map $Gr^{k}(\mathcal{T})\xrightarrow[]{m}\mathcal{H}^{n-k}$ where $m(D^{\prime},D)=D-D^{\prime}$ . Similarly, we have $f^{j}(a_{1},\cdots,a_{n-k})\in k[a_{1},\cdots,a_{n-k}]^{\mathcal{S}_{n-k}}\cong\mathcal{O}(\mathcal{H}^{n-k}).$ and let their pullback under $m$ be $d_{1},\cdots d_{n-k}$ . For $(D^{\prime},D)\in Gr(\mathcal{T})$ , where $D=\sum^{n}_{t=1}p_{t},D^{\prime}=\sum_{t\in S}p_{t}$ , We have

\sum_{l+j=s}c_{l}d_{j}((D^{\prime},D))=\sum_{l+j=s}c_{l}(D)d_{j}(D-D^{\prime})\\

=\sum_{l+j=s}f^{l}((a_{r})_{r\in S})f^{j}((a_{t})_{t\in\{1,\cdots,a\}\setminus S})=f^{s}(a_{1},\cdots,a_{n}).

Since $f^{s}(a_{1},\cdots,a_{n})$ is $\mathcal{S}^{n}$ invariant, it can be viewed as functions on $\mathcal{H}_{n}$ of the same form pulling back by $pr_{2}$ .
By part(b), the algebra $\mathcal{O}(Gr^{k}(\mathcal{T}^{n}))$ is already generated by $c_{l},d_{j}$ . Over $\mathcal{H}_{reg}$ since $pr_{2}$ is finite flat, the degree can be seen over $\mathcal{H}_{reg}$ and from (15) it is $\tbinom{n}{k}.$ Now, these relations exhaust all from the well-known algebraic fact that the degree of the algebra extension is $\tbinom{n}{k}.$
d) Clear from the definition. ∎

Restricting to the regular part (as $\mathcal{O}(C_{reg})$ -module), we still denote by $\{z^{\beta}_{p}\}$ as a basis of the $\mathcal{O}(C_{reg})$ -module $\Gamma(C_{reg}^{\alpha},(q^{\alpha})^{*}\mathcal{I}_{reg}^{\alpha,\beta}).$

3 Coulomb branch

We briefly review the mathematical definition of the Coulomb branch[BFN18].

Let $G$ be a reductive group and $N$ be a representation of $G$ . Let $\mathcal{K}=\mathbb{C}((t))$ and $\mathcal{O}=\mathbb{C}[[t]]$ . We denote by $\mathcal{G}(G)=G_{\mathcal{K}}/G_{\mathcal{O}}$ the loop Grassmannian of $G$ . Since $G$ acts on $N$ , $G_{\mathcal{O}}$ acts on $N_{\mathcal{O}}$ and we get an associated bundle $G_{\mathcal{K}}\times_{G_{\mathcal{O}}}N_{\mathcal{O}}$ . It has an embedding $i$ into $\mathcal{G}(G)\times N_{\mathcal{K}}$ given by $i(\overline{(g,v)})=(\overline{g},g\cdot v)$ . Let $\mathcal{R}_{(G,N)}$ be the preimage of $\mathcal{G}(G)\times N_{\mathcal{O}}$ under $i$ . In this paper, we will call Borel-Moore homology just homology. In [BFN18], they define an algebra structure on the $G_{\mathcal{O}}$ -equivariant homology of $\mathcal{R}$ and define the Coulomb branch of the pair $(G,N)$ to be the Spec of this algebra.

3.1 Localization

Let $T\subset G$ be a Cartan subgroup and $\mathfrak{t}$ be its Lie algebra. Let $\mathcal{R}_{T,N_{T}}$ be the variety of triples for the pair $(T,N_{T})$ , where $N_{T}$ is the $N$ considered as a representation of $T$ . Denote the localization of $\mathfrak{t}$ at all roots of $G$ and weights of $N_{T}$ by $\mathfrak{t}_{reg}.$ Let $\mathcal{R}_{T,N_{T}}\xrightarrow[]{\iota}\mathcal{R}$ be the embedding, the pushforward homomorphism $\iota_{*}$

H^{T_{\mathcal{O}}}_{*}(\mathcal{R}_{T,N_{T}})\xrightarrow[]{\iota_{*}}H^{T_{\mathcal{O}}}_{*}(\mathcal{R})

(16)

gives an algebra homomorphism, which becomes an isomorphism over $\mathfrak{t}_{reg}$ [BFN18, Lemma (5.17)].¹⁸¹⁸18The notation $\mathfrak{t}_{reg}$ ignores its dependence on $N$ .

3.2 Formula for $H^{T_{\mathcal{O}}}_{*}(\mathcal{R}_{T})$

Now we fix a representation $N$ and abbreviate $\mathcal{R}_{N_{T},T}$ by $\mathcal{R}_{T}$ . We have $\mathcal{R}_{T}\xrightarrow[]{p}\mathcal{G}(T)$ . Since we consider the homology, we only count the reduced part of $\mathcal{G}(T)$ so we denote the reduced part of $\mathcal{G}(T)$ (still) by $\mathcal{G}(T)$ . Then $\mathcal{G}(T)$ is disjoint union of points $t^{\chi}$ , where $\chi$ are cocharacters of $T$ . The fiber of $p$ over $t^{\chi}$ is a vector space, which we denote by $V_{\chi}$ . So we have $\mathcal{R}_{T}=\sqcup_{\chi}V_{\chi}.$ Denote the $T_{\mathcal{O}}$ -equivariant fundamental class of $V_{\chi}$ by $r^{\chi}$ . Let $\xi_{i}$ be the characters of $T$ that appear in the representation $N$ . Denote by $\xi_{i}(\chi)$ the pairing of $\xi_{i}$ and $\chi$ . For two integers $k,l$ , let us set

d(k,l)=\left\{\begin{aligned} &0&\text{if }k\text{ and }l\text{ have the same signs.}\\ &min(|k|,|l|)&\text{if }k\text{ and }l\text{ have different signs.}\\ \end{aligned}\right.

Theorem 2.

[BFN18, Theorem 4.1] The algebra $H^{T_{\mathcal{O}}}_{*}(\mathcal{R}_{T})$ is generated by $r^{\chi}$ for all $\chi\in X_{*}(T)$ over $H^{T_{\mathcal{O}}}_{*}(pt)$ .
For two cocharacters $\lambda$ and $\mu$ , the multiplication of $r^{\lambda}$ and $r^{\mu}$ is

r^{\lambda}r^{\mu}=\prod^{n}_{i=1}\xi_{i}^{d(\xi_{i}(\lambda),\xi_{i}(\mu))}r^{\lambda+\mu}.

(17)

Remark 4.

The coefficient before $r^{\lambda+\mu}$ depends on $\lambda,\mu$ and the representation $N$ . Denote the Grothendieck group of $T$ by $K^{0}(Rep(T))$ and the monomials of $X_{*}(T)$ by $Mon(X_{*}(T))$ . For fixed $\lambda,\mu$ , the map $K^{0}(Rep(T))\xrightarrow[]{co^{\lambda,\mu}}Mon(X_{*}(T))$ given by $co^{\lambda,\mu}(N)=\prod^{n}_{i=1}\xi_{i}^{d(\xi_{i}(\lambda),\xi_{i}(\mu))}$ is a homomorphism of monoids, i.e. for $N=N_{1}\oplus N_{2}$ , we have

co^{\lambda,\mu}(N)=co^{\lambda,\mu}(N_{1})co^{\lambda,\mu}(N_{2}).

3.3 Compactified Coulomb branch $\mathbf{M}^{\alpha}_{Q}$ of quiver gauge theory

Here we consider a special case where the pair $(G,N)$ is given from a quiver $Q$ . Let $Q=(I,E)$ be the quiver where $I$ is the set of vertices and $E$ is the set of arrows. Given $\alpha=\sum_{i\in I}n_{i}i\in\mathbb{N}[I]$ , we view it as a dimension vector $\alpha=(n_{i})_{i\in I}$ . Let $V_{i}=k^{n_{i}}$ and $V=(V_{i})_{i\in I}$ is an $I$ - graded vector space. Let $N=\oplus_{i\xrightarrow[]{}j}Hom(V_{i},V_{j})$ and $G=\prod_{i}GL(V_{i})$ act on $N$ by conjugation.

Following [BFN18] 3(ii), for any vector space $U$ , we define $\mathcal{G}^{+}(GL(U))\subset\mathcal{G}((GL(U))$ as the moduli space of vector bundles $\mathcal{U}$ on the formal disc $D$ with a trivialization $\sigma:\mathcal{U}_{D^{*}}\cong U\otimes\mathcal{O}_{D^{*}}$ on the punctured disc that extends through the puncture as an embedding $\sigma:\mathcal{U}_{D}\hookrightarrow U\otimes\mathcal{O}_{D^{*}}$ . Let $\mathcal{G}^{+}_{GL(V)}=\prod_{i\in I}\mathcal{G}^{+}_{GL(V_{i})}$ . Define $\mathcal{R}^{+}$ as the preimage of $\mathcal{G}^{+}_{GL(V)}\subset\mathcal{G}_{GL(V)}$ under $\mathcal{R}\xrightarrow[]{}\mathcal{G}_{GL(V)}$ . The homology group $H^{GL(V)_{\mathcal{O}}}_{*}(\mathcal{R}^{+})$ forms a convolution subalgebra of $H^{GL(V)_{\mathcal{O}}}_{*}(\mathcal{R})$ .

In general, given a $\mathbb{Z}^{\geq 0}$ filtration of an algebra $R$ , $0\subset F_{1}\subset F_{2}\cdots\subset R$ , we can get a $\mathbb{Z}^{\geq 0}$ -graded algebra, the Rees algebra $Rees_{F}(R):=\oplus_{i}F_{i}t^{i}$ , where $t$ is a formal variable. Then $Proj(Rees_{F}(R))$ is a compactification of $Spec(R)$ . When the filtration $F$ is fixed, we often omit $F$ from the notation $Rees_{F}(R)$ .

Now we give the algebra¹⁹¹⁹19Since we fix the dimension vector $\alpha$ , we omit $\alpha$ for the notation $\mathcal{A}$ . $\mathcal{A}\stackrel{{\scriptstyle\mathrm{def}}}{{=}}H_{*}^{GL(V)_{\mathcal{O}}}(\mathcal{R}^{+})$ a $\mathbb{Z}_{\geq 0}$ filtration, which is the pullback of the filtration in remark 3.7 [BFN18] under the diagonal embedding of $\mathbb{Z}_{\geq 0}\xrightarrow[]{}\mathbb{Z}_{\geq 0}^{I}$ , as follows. Recall $n_{i}=dimV_{i}$ . The $GL(V)_{\mathcal{O}}$ -orbits in $Gr^{+}_{GL(V)}$ are numbered by $I$ -colored partitions $(\lambda^{(i)})_{i\in I}$ , $\lambda^{(i)}=(\lambda^{(i)}_{1}\geq\lambda^{(i)}_{2}\geq\cdots),$ such that the number of parts $l(\lambda^{(i)})\leq n_{i}$ . Given a dimension vector $(m_{i})_{i\in I}$ , we define a closed $GL(V)_{\mathcal{O}}$ -invariant subvariety $\overline{Gr_{GL(V)}}^{+,(m_{i})}\subset Gr^{+}_{GL(V)}$ as the union of orbits $Gr^{(\lambda^{(i)})_{i\in I}}_{GL(V)}$ such that $\lambda^{(i)}_{1}\leq m_{i}$ for any $i\in I$ . For $m\in\mathbb{N}$ , we define $\mathcal{R}^{+}_{\leq m}\subset\mathcal{R}^{+}$ as the preimage of $\overline{Gr}_{GL(V)}^{+,(m,\cdots,m)}$ under $\mathcal{R}^{+}\xrightarrow[]{}Gr^{+}_{GL(V)}$ . This gives a $\mathbb{Z}^{\geq 0}$ filtration $F_{m}=H_{*}^{GL(V)_{\mathcal{O}}}(\mathcal{R}^{+}_{\leq m})$ of the algebra $\mathcal{A}.$

Define $\mathbf{M}_{Q}^{\alpha}\stackrel{{\scriptstyle\text{def}}}{{=}}Proj(Rees(\mathcal{A}))$ and call $\mathbf{M}_{Q}^{\alpha}$ the compactified Coulomb branch.

3.4 Embedding of $\mathbf{M}_{Q}^{\alpha}$ into $\mathbb{P}_{\mathcal{H}}(H^{G_{\mathcal{O}}}(\mathcal{R}^{+}_{\leq 1})^{*})$

We fix an ordered basis of $V.$ Let $T_{i}$ be corresponding diagonal subgroup of $GL(V_{i})$ and $\mathfrak{t_{i}}$ be its Lie algebra. Let $W_{i}$ be the Weyl group of $GL(V_{i}).$ Denote by $(a^{i}_{j})_{1\leq j\leq n_{i}}$ the corresponding standard basis of $\mathfrak{t_{i}}$ . Notice that in the last section, we already denote certain generators of $\mathcal{O}(C^{\alpha})$ by $a^{i}_{j}$ . Here we identify $H^{*}_{G_{\mathcal{O}}}(pt)\hookrightarrow H^{*}_{T_{\mathcal{O}}}(pt)$ with $\mathcal{O}(\mathcal{H}^{\alpha})\hookrightarrow\mathcal{O}(C^{\alpha})$ via $Sym((\mathfrak{t}/W)^{*})\hookrightarrow Sym(\mathfrak{t}^{*})$ . The algebra $\oplus_{m}H^{G_{\mathcal{O}}}(R^{+}_{\leq m})t^{m}$ is a module over $H^{*}_{G_{\mathcal{O}}}(pt)\cong H^{*}_{G}(pt)$ so we have a projection

\mathbf{M}_{Q}^{\alpha}\xrightarrow[]{}\mathcal{H}^{\alpha}.

(18)

Lemma 6.

As a $H^{*}_{G_{\mathcal{O}}}(pt)$ -algebra, the Rees algebra $\oplus_{m}H_{*}^{G_{\mathcal{O}}}(\mathcal{R}^{+}_{\leq m})t^{m}$ is generated by $H_{*}^{G_{\mathcal{O}}}(\mathcal{R}^{+}_{\leq 1})t$ .

Proof.

We recall some notations in [BFN19]. Let $\mathcal{R}_{\lambda}=\mathcal{R}\cap\pi^{-1}(Gr^{\lambda})$ ( $\mathcal{R}_{\leq\lambda}=\mathcal{R}\cap\pi^{-1}(\overline{Gr^{\lambda})}$ ) be the restriction of $\mathcal{R}$ on the $G_{\mathcal{O}}$ -orbit $Gr^{\lambda}$ ( $G_{\mathcal{O}}$ -orbit closure $\overline{Gr^{\lambda}}$ , resp). Let $\mathcal{R}_{<\lambda}$ be the complement $\mathcal{R}_{\leq\lambda}\setminus\mathcal{R}_{\lambda}$ . It is a closed subvariety. Lemma 6.2 in [BFN18] says that the Mayer-Vietoris sequence splits into short exact sequences

0\xrightarrow[]{}H^{G_{\mathcal{O}}}_{*}(\mathcal{R}_{<\lambda})\xrightarrow[]{}H^{G_{\mathcal{O}}}_{*}(\mathcal{R}_{\leq\lambda})\xrightarrow[]{}H^{G_{\mathcal{O}}}_{*}(\mathcal{R}_{\lambda})\xrightarrow[]{}0.

Moreover, as $H^{*}_{G_{\mathcal{O}}}(pt)$ -module, this exact sequence splits canonically so we have

H^{G_{\mathcal{O}}}_{*}(\mathcal{R}_{\leq\lambda})=H^{G_{\mathcal{O}}}_{*}(\mathcal{R}_{<\lambda})\oplus H^{G_{\mathcal{O}}}_{*}(\mathcal{R}_{\lambda}).

(19)

It suffices to show that any $a\in H^{G_{\mathcal{O}}}(\mathcal{R}^{+}_{\leq m})t^{m}$ is generated by $H^{G_{\mathcal{O}}}(\mathcal{R}^{+}_{\leq 1})t$ . Let $\Lambda(m)$ be the set of all maximal $\lambda$ such that $\lambda^{(i)}_{1}\leq m$ . By definition of $\mathcal{R}^{+}_{\leq m}$ and $\overline{Gr_{\lambda}}=\cup_{\mu:\mu\leq\lambda}Gr_{\mu}$ , we have $\mathcal{R}^{+}_{\leq m}=\sqcup_{\lambda\in\Lambda(m)}\mathcal{R}^{+}_{\leq\lambda}$ . It suffices to prove the claim for any $a\in H^{G_{\mathcal{O}}}(\mathcal{R}^{+}_{\leq\lambda})t^{m}$ for $\lambda\in\Lambda(m)$ . We prove this by induction on $\lambda$ . Suppose for any $\mu<\lambda$ , the theorem holds. By formula (19) and induction hypothesis, it suffices to show when $a\in H^{G_{\mathcal{O}}}_{*}(\mathcal{R}_{\lambda})$ for some $\lambda$ . The second paragraph after proposition 6.1 [BFN19] says

H_{*}^{G_{\mathcal{O}}}(R_{\lambda})\cong H^{*}_{Stab_{G}(\lambda)}(pt)\cap[\mathcal{R}_{\lambda}]\cong\mathbb{C}[\mathbf{t}]^{W_{\lambda}}[\mathcal{R}_{\lambda}],

where $\cap$ is the cap product, $W_{\lambda}$ is the stabilizer of $\lambda$ in the Weyl group $W$ and $[\mathcal{R}_{\lambda}]$ is the $G_{\mathcal{O}}$ -equivariant fundamental class of $\mathcal{R}_{\lambda}$ . So it suffices to prove the case $a=[\mathcal{R}_{\lambda}]t^{m}.$ Recall we denote $H^{GL(V)_{\mathcal{O}}}_{*}(\mathcal{R}^{+})$ by $\mathcal{A}$ . In the proof of [BFN19] proposition 6.8, regarding $[\mathcal{R}_{\lambda}]$ as an element in $gr\mathcal{A}$ , we have

[\mathcal{R}_{\lambda}][\mathcal{R}_{\mu}]=[\mathcal{R}_{\lambda+\mu}]

in $gr\mathcal{A}$ when $\lambda,\mu$ are in the same ”generalized Weyl chamber”. In this case, any weight of $N$ is a root of $G$ , so their ”generalized Weyl chamber” is the same as the usual Weyl chamber. Let $\varpi_{\beta}$ be the fundamental weights of $GL(V)$ . Here, the fundamental weights of $GL_{n}$ consist of all $\varpi_{k}=(1,1\cdots,1,0,0\cdots,0)$ for $1\leq k\leq n$ where $k$ is the number of $1$ ’s. The fundamental weights of the product of $GL(V_{i})$ ’s consist of the disjoint union of fundamental weights of $GL(V_{i})$ ’s. Since $\mathcal{R}_{\beta}\subset\mathcal{R}_{\leq 1}$ , we have $H^{G_{\mathcal{O}}}(\mathcal{R}_{\beta})t\subset Rees\mathcal{A}.$ We can assume $max_{i}\lambda^{(i)}_{1}=m$ . It is easy to see that we can choose $m$ fundamental weights $\beta_{i}$ such that $\sum_{i}\beta_{i}=\lambda$ . So in $\mathcal{A}$ , we have $[\mathcal{R}_{\beta_{1}}][\mathcal{R}_{\beta_{2}}]\cdots[\mathcal{R}_{\beta_{m}}]-[\mathcal{R}_{\lambda}]\in H^{G_{\mathcal{O}}}(\mathcal{R}_{<\lambda})$ . In $Rees\mathcal{A}$ , we have $[\mathcal{R}_{\beta_{1}}]t[\mathcal{R}_{\beta_{2}}]t\cdots[\mathcal{R}_{\beta_{m}}]t-[\mathcal{R}_{\lambda}]t^{m}\in H_{*}^{G_{\mathcal{O}}}(\mathcal{R}_{<\lambda})t^{m}$ , which is generated by $H_{*}^{G_{\mathcal{O}}}(\mathcal{R}_{\leq 1})$ by induction. Hence we claim $[\mathcal{R}_{\lambda}]t^{m}$ is generated by $H_{*}^{G_{\mathcal{O}}}(\mathcal{R}_{\leq 1})$ .

∎

Corollary 1.

We have an embedding

\mathbf{M}^{\alpha}\xhookrightarrow{j_{\alpha}}\mathbb{P}_{\mathcal{H}^{\alpha}}(H_{*}^{G}(\mathcal{R}^{+}_{\leq 1})^{*}),

where $H_{*}^{G}(\mathcal{R}^{+}_{\leq 1})^{*}$ is understood as a vector bundle over $Spec(H^{*}_{G}(pt))\cong\mathcal{H}^{\alpha}.$

Proof.

By lemma 6, we have the surjection of graded algebras

\mathcal{S}(H_{*}^{G_{\mathcal{O}}}(\mathcal{R}^{+}_{\leq 1})t)\twoheadrightarrow\oplus_{m}H_{*}^{G_{\mathcal{O}}}(\mathcal{R}^{+}_{\leq m})t^{m}.

Taking Proj, we get the embedding. ∎

3.5 Bases of $H^{G_{\mathcal{O}}}_{}(\mathcal{R}^{+}_{\leq 1})$ as $H_{G_{\mathcal{O}}}^{}(pt)\cong\mathcal{O}(\mathcal{H}^{\alpha})$ -module

Lemma 7.

H^{G_{\mathcal{O}}}_{*}(\mathcal{R}^{+}_{\leq 1})\cong\bigoplus_{(k_{i})_{i\in I}}\bigotimes_{i\in I}H^{GL(V_{i})}_{*}(Gr(n_{i},k_{i})).

(20)

Proof.

Let $\{\varpi^{i}_{k_{i}},1\leq k_{i}\leq n_{i}\}$ be all fundamental weights of $GL(V_{i})$ , where $\varpi^{i}_{k_{i}}=(1,1\cdots,1,0,0\cdots,0)$ , $k_{i}$ =number of $1$ ’s. Now we allow $k_{i}=0$ , denote $\varpi^{i}_{0}=(0,\cdots,0)$ and let $\varpi_{(k_{i})_{i\in I}}\stackrel{{\scriptstyle def}}{{=}}(\varpi^{i}_{k_{i}})_{i\in I}$ . Recall $V=\oplus_{i\in I}V_{i}$ with dimension vector $\alpha=(n_{i})_{i\in I}.$ For the set $\{\varpi_{(k_{i})_{i\in I}},0\leq k_{i}\leq n_{i},i\in I\}$ , we have

\mathcal{R}^{+}_{\leq 1}=\bigsqcup_{0\leq k_{i}\leq n_{i},i\in I}\mathcal{R}_{\varpi_{(k_{i})_{i\in I}}}.

For each component of $\mathcal{R}^{+}_{\leq 1}$ , $\mathcal{R}_{\varpi_{(k_{i})_{i\in I}}}$ is a vector bundle over the $GL(V)_{\mathcal{O}}$ -orbit $\mathcal{G}_{\varpi_{(k_{i})_{i\in I}}}=\prod_{i}Gr(n_{i},k_{i})$ so their homologies are isomorphic (without grading). ∎

Now we fix a dimension vector $\beta=(k_{i})_{i\in I}$ . By Poincare duality, $H_{G}^{*}(\mathcal{R}_{\varpi_{\beta}})\xrightarrow[\sim]{p}H^{G}_{*}(\mathcal{R}_{\varpi_{\beta}})$ , where²⁰²⁰20again we ignore the grading $p(c)=c\cap[\mathcal{R}_{\varpi_{\beta}}]^{G}$ and $[\mathcal{R}_{\varpi_{\beta}}]^{G}$ is the $G$ -equivariant fundamental class of $\mathcal{R}_{\varpi_{\beta}}$ . The homology group $H^{G}_{*}(\mathcal{R}_{\varpi_{\beta}})$ is a free rank 1 module over the ring $H_{G}^{*}(\prod_{i\in I}Gr(n_{i},k_{i}))\cong\otimes_{i\in I}H_{GL(V_{i})}^{*}Gr(n_{i},k_{i}).$ For the cohomology of Grassmannian, we have a well-known result. Since the cohomology of the product of Grassmannians is just the tensor product of cohomologies of Grassmannians, for brevity, in lemma 8, we set $I=\{1\}$ so $GL(V)=GL_{n}$ and we abbreviate $a^{1}_{j}$ by $a_{j}$ for $1\leq j\leq n.$ Let $S$ be the tautological bundle over $Gr(n,k)$ and $Q$ be the quotient bundle. Denote $c_{i}$ the $i$ -th $G$ -equivariant Chern class and $c$ the total Chern class. Recall that the notation $f^{s}$ is the elementary symmetric functions, introduced before theorem 1.

Lemma 8.

H_{GL(V)}^{*}Gr(n,k)\cong H_{GL(V)}^{*}(pt)[c_{l}(S),c_{j}(Q)]/\mathrm{I},

where the ideal $\mathrm{I}$ is generated by

\sum_{1\leq l\leq k,l+j=s}c_{l}(S)c_{j}(Q)-f^{s}(a_{1},\cdots,a_{n})

for all $s$ such that ${1\leq s\leq n}.$

Proof.

We have an exact sequence $0\xrightarrow[]{}S\xrightarrow[]{}V\xrightarrow[]{}Q\xrightarrow[]{}0$ . It is well-known that as an $H_{GL(V)}^{*}(pt)$ -algebra, $H^{*}_{T}(Gr(n,k)$ has generators $c_{1}(S),\cdots,c_{k}(S),c_{1}(Q),\cdots c_{n-k}(Q)$ . From the exact sequence, we have $c(S)c(Q)=c(V)$ . $c(V)=\prod_{\chi}c(V_{\chi})=\prod_{\chi}(1+\chi)=\prod_{i}(1+a_{i})$ , where $\chi$ are characters of $T$ and $V_{\chi}$ is the $\chi$ -weight space of $V$ . Plug in $c=\sum_{i}c^{i}$ into $S,Q$ and expand $c(S)c(Q)=c(V)$ . Comparing the degree $s$ part, we get the relation in the lemma.

∎

Remark 5.

By the same argument, we get the isomorphism for $T$ -equivariant cohomology.

H_{T}^{*}(Gr(n,k))\cong H_{T}^{*}(pt)[c_{l}^{T}(S),c_{j}^{T}(Q)]/I,

for the ideal $I$ with the same generators from the lemma.

3.6 Pulling back $\mathbf{M}^{\alpha}$ under $C^{\alpha}\xrightarrow[]{q^{\alpha}}\mathcal{H}^{\alpha}$

To apply localization theory in (3.1), we consider $H^{T_{\mathcal{O}}}_{*}(\mathcal{R})$ which is an algebra over $H_{*}^{T_{\mathcal{O}}}(pt)$ . We identify $Spec(H_{*}^{T_{\mathcal{O}}}(pt))$ with $C^{\alpha}$ .

Lemma 9.

The pullback of $Spec(H^{G_{\mathcal{O}}}_{*}(\mathcal{R}))$ , $\textsf{M}^{\alpha}$ and $\mathbf{M}^{\alpha}$ under the quotient map $C^{\alpha}\xrightarrow[]{q^{\alpha}}\mathcal{H}^{\alpha}$ is $Spec(H^{T_{\mathcal{O}}}_{*}(\mathcal{R}))$ , $Spec(H^{T_{\mathcal{O}}}_{*}(\mathcal{R}^{+}))$ and $Proj(\oplus_{m}H_{*}^{T_{\mathcal{O}}}(\mathcal{R}^{+}_{\leq m})t^{m})$

Proof.

The following is a Cartesian diagram where $?$ is $\mathcal{R}$ , $\mathcal{R}^{+}$ and $\mathcal{R}_{\leq m}^{+}$ . ∎

Lemma 10.

For the quiver gauge theory case, under the identification of $\mathfrak{t}$ and $C^{\alpha}$ , the pushforward homomorphism $\iota_{*}$ for the inclusion $\mathcal{R}_{T,N_{T}}\xrightarrow[]{\iota}\mathcal{R}$

H^{T_{\mathcal{O}}}_{*}(\mathcal{R}_{T,N_{T}})\xrightarrow[]{\iota_{*}}H^{T_{\mathcal{O}}}_{*}(\mathcal{R})

becomes an isomorphism over $C^{\alpha}_{reg}$ .

Proof.

In (16), it says the pushforward homomorphism $\iota_{*}$ becomes an isomorphism over $\mathfrak{t}_{reg}$ . For each direct summand $Hom(V_{i},V_{i^{\prime}})$ in $N$ , the localization inverts all $a^{i}_{l}-a^{i^{\prime}}_{j}$ for $1\leq l\leq dimV_{i},1\leq j\leq dimV_{i^{\prime}}$ . It is clear that the localization of $C^{\alpha}$ to $C^{\alpha}_{reg}$ includes all roots of $G$ and weights of $N_{T}$ . ∎

We define $\mathcal{R}^{+}_{T,N_{T}}$ and its filtration $(\mathcal{R}^{+}_{T,N_{T}})_{\leq m}$ as the pullback under $\iota$ of $\mathcal{R}^{+}$ and its filtration $\mathcal{R}_{\leq m}^{+}$ . Then we get the Rees algebra $\oplus_{m}H_{*}^{T_{\mathcal{O}}}((\mathcal{R}^{+}_{T,N_{T}})_{\leq m})t^{m}$ of $H^{T_{\mathcal{O}}}_{*}(\mathcal{R}_{T,N_{T}}).$ For any space $X$ , we denote

\mathcal{H}^{T}(X)\stackrel{{\scriptstyle def}}{{=}}H^{T}(X)\otimes_{\mathcal{O}(C^{\alpha})}\mathcal{O}(C^{\alpha})_{reg}.

(21)

For example, by this convention, the localized Rees algebra of $\mathcal{R}^{+}_{T,N_{T}}$ is denoted by $\oplus_{m}\mathcal{H}^{T_{\mathcal{O}}}((\mathcal{R}_{T,N_{T}})^{+}_{\leq m})t^{m},$

We fix some notations. In our case $\mathcal{G}(T)$ is the disjoint union of points $t^{\chi}=(t_{i})^{\chi^{i}}_{i\in I}$ where $\chi=(\chi^{i})_{i\in I}$ is a cocharacter of $T=\prod_{i\in I}T_{i}$ and $\chi^{i}$ is a cocharactor of $T_{i}$ . By the standard isomorphism $T_{i}\cong(G_{m})^{n_{i}}$ we can write $\chi^{i}=(\chi_{j}^{i})_{1\leq j\leq n_{i}}$ .

3.6.1 Generators of the localized Rees algebra $\oplus_{m}\mathcal{H}^{T_{\mathcal{O}}}((\mathcal{R}_{T,N_{T}})^{+}_{\leq m})t^{m}$

From the definition of the filtration of $\mathcal{R}_{T,N_{T}}$ , it is easy to see that

(\mathcal{R}_{T,N_{T}})^{+}_{\leq m}=\bigsqcup_{0\leq\chi^{i}_{j}\leq m\text{ for any }i,j}V_{\chi}.

Lemma 11.

The localized Rees algebra $\oplus_{m}\mathcal{H}^{T_{\mathcal{O}}}((\mathcal{R}_{T,N_{T}})^{+}_{\leq m})t^{m}$ is generated by $\mathcal{H}^{T_{\mathcal{O}}}((\mathcal{R}_{T,N_{T}})^{+}_{\leq 1})t.$

Proof.

This follows from the formula (17) in theorem 2. ∎

3.7 $H^{T_{\mathcal{O}}}_{}((\mathcal{R}_{T,N_{T}}^{+})_{\leq 1})\xrightarrow[]{\iota_{}}H^{T_{\mathcal{O}}}_{*}(\mathcal{R}^{+}_{\leq 1})$ over $C_{reg}$

In the next lemma, we will study the map

H^{T_{\mathcal{O}}}_{*}(\mathcal{R}^{+}_{\leq 1})\xhookrightarrow{}\mathcal{H}^{T_{\mathcal{O}}}_{*}(\mathcal{R}^{+}_{\leq 1})\xrightarrow[]{(\iota_{*})^{-1}}\mathcal{H}^{T_{\mathcal{O}}}_{*}(\mathcal{R}_{T,N_{T}})^{+}_{\leq 1})\cong\bigoplus_{\chi|0\leq\chi^{i}_{j}\leq 1\text{ for any }i,j}\mathcal{H}^{T_{\mathcal{O}}}(V_{\chi}).

Since the preimage of $\mathcal{R}_{\varpi_{\beta}}$ under $\iota$ is $\sqcup_{\chi\in W\varpi_{\beta}}V_{\chi}$ , restricting the above map to $H_{*}^{T_{\mathcal{O}}}(\mathcal{R}_{\varpi_{\beta}})$ we get

H^{T_{\mathcal{O}}}_{*}(\mathcal{R}_{\varpi_{\beta}})\xhookrightarrow{}\bigoplus_{\chi\in W\varpi_{\beta}}H_{*}^{T_{\mathcal{O}}}(V_{\chi}).

(22)

For each $\chi\in W\varpi_{\beta}$ , let $x^{\chi}$ be the $\chi$ component of the image under (22) of the fundamental class of $[\mathcal{R}_{\varpi_{\beta}}]$ . For any $\chi$ such that $0\leq\chi^{i}_{j}\leq 1$ for all $i,j$ , we define a set $S(\chi)=\sqcup_{i\in I}S(\chi_{i})$ where $S(\chi_{i})=\{j|\chi^{i}_{j}=1\}.$ It is clear such $\chi$ is in bijection with $S$ such that $S\subset S^{\alpha},|S|=\beta$ . So given $S$ , we also use the notation $\chi(S)$ , meaning the $\chi$ such that $S(\chi)$ is the given set $S$ . We denote the set $\{1,\cdots,n_{i}\}$ by $[n_{i}]$ . Recall we denote the fundamental class $[V_{\chi}]$ by $r^{\chi}\in H_{*}^{T_{\mathcal{O}}}(V_{\chi})$ .

Lemma 12.

x^{\chi}=(Eu_{\chi})^{-1}r^{\chi},

where $Eu_{\chi}$ is the $T$ -equivariant Euler class of $N_{\mathcal{R}_{\varpi_{\beta}}/V_{\chi}}$ at the point $t^{\chi}$ . Here $N_{\mathcal{R}_{\varpi_{\beta}}/V_{\chi}}$ is the normal bundle of $V_{\chi}$ in $\mathcal{R}_{\varpi_{\beta}}.$
b)

Eu_{\chi}=\prod_{i\in I}\prod_{j\in S(\chi_{i}),l\in[n_{i}]\setminus S(\chi_{i})}(a^{i}_{j}-a^{i}_{l}).

Proof.

a) This is just localization theory. b) Let $(e^{i}_{j})_{i\in I,j\in[n_{i}]}$ be the $T$ -eigenvectors of $V=\oplus_{i\in I}V_{i}.$ Since the fiber of the projection $\mathcal{R}_{\varpi_{\beta}}\xrightarrow[]{}\mathcal{G}(GL(V))_{\varpi_{\beta}}=\prod_{i\in I}Gr(n_{i},k_{i}))$ at $t^{\chi}$ is $V_{\chi}$ , we have

(N_{\mathcal{R}_{\varpi_{\beta}}/V_{\chi}})_{t^{\chi}}=T_{[\oplus_{i\in I}\oplus_{j\in S(\chi_{i})}ke^{i}_{j}]}(\prod_{i\in I}Gr(n_{i},k_{i})),

where $[\oplus_{i\in I}\oplus_{j\in S(\chi_{i})}ke^{i}_{j}]$ is the point $t^{\chi}$ viewed as a point in $\prod_{i\in I}Gr(n_{i},k_{i})$ (i.e., a vector subspace in $V$ ). Now the formula follows from a standard computation about weight spaces of tangent spaces of a partial flag variety. ∎

Corollary 2.

The localized Rees algebra $\oplus_{m}\mathcal{H}^{T_{\mathcal{O}}}((\mathcal{R}_{T,N_{T}})^{+}_{\leq m})t^{m}$ is generated by $x^{\chi(S)}t,S\subset S^{\alpha}$ .

Proof.

Follows from lemma 11 and lemma 12 part (a). ∎

4 Identification of $\mathbf{M}^{\alpha}$ and $Z^{\alpha}$

For a quiver $Q$ , we forget the direction of arrows and define an integer matrix $\kappa(Q)$ as $\kappa_{ii}=1-$ number of self-loop of $i$ and $\kappa_{ij}$ =-number of arrow between $i$ and $j$ . For a symmetric integral matrix $\kappa$ , if $\kappa_{ii}\leq 1$ for all $i\in I$ and $\kappa_{ij}\leq 0$ for all $i,j\in I,i\neq j$ , we say $\kappa$ is of quiver-type, since in this case there exists a quiver $Q$ such that $\kappa=\kappa(Q).$

Remark 6 (Different conventions about the matrix $\kappa(Q)$ ).

For a diagram $Q$ (possibly with self-loops), define the Cartan matrix $C(Q)$ associated with $Q$ as $C_{ii}=2-2$ number of self-loop of $i$ and $C_{ij}$ =-number of arrow between $i$ and $j$ . So the relation is $C_{ii}=2\kappa_{ii},C_{ij}=\kappa_{ij}.$ For $Q$ of finite ADE type, let $G_{Q}$ be the simply-connected simple algebraic group corresponding to $Q$ . As mentioned in the introduction, in [MYZ21, Mir23], Mirkovic proved that $Z^{\alpha}_{\kappa(Q)}$ ²¹²¹21In our notation is isomorphic to the Zastava space $Z^{\alpha}_{G_{Q}}$ . The notation $\kappa$ in [MYZ21, Mir23] is $C(Q)$ here.

Theorem 3.

For a quiver $Q$ , the Compactified Coulomb branch $\mathbf{M}_{Q}^{\alpha}$ is canonically isomorphic to the local projective space $Z_{\kappa(Q)}^{\alpha}.$

Again, we will prove it after pullback under $C^{\alpha}\xrightarrow[]{a}\mathcal{H}^{\alpha}$ . We list what we have.

	$(q^{\alpha})^{*}\mathbf{M}^{\alpha}$	$(q^{\alpha})^{*}Z^{\alpha}$
embedding	$(q^{\alpha})^{}\mathbf{M}^{\alpha}\xhookrightarrow{j_{\alpha}}\mathbb{P}_{C^{\alpha}}(H_{}^{T}(\mathcal{R}^{+}_{\leq 1})^{*})$ (a)	$(q^{\alpha})^{}Z^{\alpha}\xhookrightarrow{i_{\alpha}}\mathbb{P}_{C^{\alpha}}((q^{\alpha})^{}V^{\alpha})$ (b)
over regular part $C^{\alpha}_{reg}$	product of $\mathbb{P}^{1}$ (c)	product of $\mathbb{P}^{1}$ (d)
global basis	basis of $H_{*}^{T}(\mathcal{R}^{+}_{\leq 1})$ (e)	basis of $\Gamma(C^{\alpha},(q^{\alpha})^{*}\mathcal{I}^{\alpha})$ (f)
local basis	$s^{S}$ (section 2.7.3)	$x^{\chi}$ (section 3.7, after (22))

(a) is by Corollary 1. (b) is by section 2.5. (c) is not proven yet but is easy to see. It is listed as a heuristic for the proof of theorem 3. (d) is by definition. (e) is by section 3.5. (f) is by section 2.7.2.

4.1 Identification of $H_{}^{T}(\mathcal{R}^{+}_{\leq 1})$ and $\Gamma(C^{\alpha},(q^{\alpha})^{}\mathcal{I})$

We have the following decompositions into connected components.

	$\mathcal{R}_{\leq 1}$	$(q^{\alpha})^{*}Gr(\mathcal{T}^{\alpha})$
connected components	$\mathcal{R}_{\varpi_{\beta}}$	$(q^{\alpha})^{*}Gr^{\beta}(\mathcal{T}^{\alpha})$

The connected components on both sides are indexed by the same data $\beta=(k_{i})_{i\in I}$ and $0\leq k_{i}\leq n_{i}$ . On each component, we have

rank 1 free module	$H_{*}^{T}(\mathcal{R}_{\varpi_{\beta}})$	$\Gamma(C^{\alpha},(q^{\alpha})^{*}\mathcal{I}^{\alpha,\beta})$
over the ring	$H^{*}_{T}(\mathcal{R}_{\varpi_{\beta}})$	$\mathcal{O}((q^{\alpha})^{*}Gr^{\beta}(\mathcal{T}^{\alpha}))$
with basis	the fundamental class $[\mathcal{R}_{\varpi_{\beta}}]$	$u^{\beta}\otimes 1$

In the third line, the fundamental class $[\mathcal{R}_{\varpi_{\beta}}]$ is a basis of $H_{*}^{T}(\mathcal{R}_{\varpi_{\beta}})$ over $H^{*}_{T}(\mathcal{R}_{\varpi_{\beta}})$ by Poincare duality (See the paragraph after lemma 7, note there we used $G$ -equivariance and here $T$ -equivariance). The basis of $\Gamma(C^{\alpha},(q^{\alpha})^{*}\mathcal{I}^{\alpha,\beta})$ over $\mathcal{O}((q^{\alpha})^{*}Gr^{\beta}(\mathcal{T}^{\alpha}))$ is chosen as $u^{\beta}\otimes 1$ (defined at the beginning of 2.7.3²²²²22There, it is considered as a basis element for $\Gamma(C^{\alpha},(q^{\alpha})^{*}\mathcal{I}^{\alpha,\beta})$ as an $\mathcal{O}(C^{\alpha})$ -module. Here it is a basis as an $\mathcal{O}((q^{\alpha})^{*}Gr^{\beta}(\mathcal{T}^{\alpha,\beta}))$ -module.).

We will identify the rings in lemma 13 and therefore also the trivialized modules from line 1 of the above table.

Lemma 13.

There are canonical ring isomorphisms of the horizontal arrows of the following diagram

which makes the diagram commute. Here $loc$ is the localization map.

Proof.

The isomorphism for the upper horizontal map is clear from theorem 1 and remark 5, where in the case $I=\{1\}$ , it sends $c^{i}_{l}$ to $C_{i}(S_{l})$ and $d^{i}_{k}$ to $C_{i}(Q_{k})$ . The definition of the lower horizontal map and the commutativity of the diagram follow from the localization theory of equivariant homology. ∎

Denote the above isomorphism between these two rank 1 modules (by the above map between rings and trivialization maps from the bases $[\mathcal{R}_{\varpi_{\beta}}]$ and $u^{\beta}\otimes 1$ ) by $rel$ ,

\Gamma(C^{\alpha},(q^{\alpha})^{*}\mathcal{I}^{\alpha,\beta})\xrightarrow[\cong]{rel}H_{*}^{T}(\mathcal{R}_{\varpi_{\beta}}),\text{ }rel(u^{\beta}\otimes 1)=[\mathcal{R}_{\varpi_{\beta}}].

(23)

4.2 Matching the local basis

We recall a lemma in the localization theory of equivariant homology theory.

Lemma 14.

Let a torus $T$ act on a complex algebraic variety $X$ . The action of $H^{*}_{T}(X)$ on $H_{*}^{T}(X)$ is compatible with localizations, i.e.

Now we state a lemma about the decomposition of the two modules in 23 after localizations.

Lemma 15.

The following diagram commutes

where each term is considered as a module over the ring corresponding to the same place in the diagram in lemma 13. Moreover, when we decompose the terms on the bottom as $\mathcal{O}(C^{\alpha}_{reg})$ (resp. $\mathcal{H}^{T}_{*}(pt)$ )-modules, the decomposition is compatible with $rel$ , i.e. it maps the $S$ -summand to the $\chi(S)$ -summand.

Proof.

Follows from lemmas 5 and 14. ∎

Lemma 16.

Restricting $rel$ to $C_{reg}^{\alpha}$ , the basis $s^{S}$ (section 2.7.3) of $\Gamma(C_{reg}^{\alpha},L^{\alpha,S}_{reg})$ goes to the basis $x^{\chi(S)}$ (section 3.7, after (22)) of $\mathcal{H}_{*}^{T}(t^{\chi})$ .

Proof.

From

rel(u^{\beta}\otimes 1)=[\mathcal{R}_{\varpi_{\beta}}],

decompose $u^{\beta}\otimes 1$ and $[\mathcal{R}_{\varpi_{\beta}}]$ into $S$ (or $\chi(S)$ resp.)-component. Since $s^{S}$ and $x^{\chi(S)}$ are the $S$ -component of $u^{\beta}\otimes 1$ and $[\mathcal{R}_{\varpi_{\beta}}]$ respectively, by lemma 15,

rel(s^{S})=x^{\chi(S)}.

∎

4.3 A flatness lemma

Lemma 17.

Let $S$ be a base space and $U\subset S$ an open dense subspace. Let $Y=\mathbb{P}^{n}_{S}$ over $S$ . Let $X$ over $U$ be a closed subscheme in $Y_{U}=\mathbb{P}^{n}_{U}$ and $\overline{X}$ be its closure in $Y.$ Let $\mathcal{X}$ be a closed subscheme flat over $S$ . If $\mathcal{X}|_{U}=X$ , we have $\mathcal{X}=\overline{X}.$

Proof.

Since $\mathcal{X}$ is closed in $Y$ , $\overline{X}\subset\mathcal{X}$ . So for $n$ big enough, we have

\mathcal{O}(n)_{\mathcal{X}/S}\twoheadrightarrow\mathcal{O}(n)_{\overline{X}/S}.

Since $\mathcal{X}$ is flat over $S$ , the sheaf $\mathcal{O}(n)_{\mathcal{X}/S}$ is locally free for $n$ big enough. Now we claim the composition

\mathcal{O}(n)_{\mathcal{X}/S}\xrightarrow[]{}\mathcal{O}(n)_{\overline{X}/S}\xrightarrow[]{}\mathcal{O}(n)_{X/S}

is injective. Take a section $s$ in the kernel. By the assumption $\mathcal{X}|_{U}=X$ , $s|U=0$ but $s$ is a section in the locally free sheaf so it must be $0$ . Hence

\mathcal{O}(n)_{\mathcal{X}/S}\xrightarrow[]{}\mathcal{O}(n)_{\overline{X}/S}

is also injective so it is a isomorphism for $n>>0$ . This implies $\mathcal{X}=\overline{X}.$ ∎

4.4 Theorem 3 reduces to comparison on the regular part

Lemma 18.

The dotted arrow exists as an isomorphism, i.e. the isomorphism $rel$ restricts to an isomorphism for the two embeddings.

Given this lemma, we can prove theorem 3.

Proof of theorem 3.

By [BFN18, lemma 5.3], $(q^{\alpha})^{*}\mathbf{M}^{\alpha}$ is flat over $C^{\alpha}.$ Now we apply lemma 17. Let $S=C^{\alpha}$ and $U=C^{\alpha}_{reg}.$ Let $X=rel_{reg}((q^{\alpha})^{*}Z^{\alpha}_{reg})$ , $\mathcal{X}=(q^{\alpha})^{*}\mathbf{M}^{\alpha}$ and $Y=\mathbb{P}_{C^{\alpha}}(H_{*}^{T}(\mathcal{R}^{+}_{\leq 1})^{*}).$ By lemma 18, we have $\mathcal{X}|_{U}=X$ so applying lemma 17 gives $\mathcal{X}=X$ , i.e., $rel((q^{\alpha})^{*}Z)=(q^{\alpha})^{*}\mathbf{M}^{\alpha}.$

∎

Corollary 3.

The local space $Z_{\kappa(Q)}^{\alpha}$ is flat over $\mathcal{H}^{\alpha}$ .

Proof.

The space $(q^{\alpha})^{*}\mathbf{M}^{\alpha}$ is flat over $C^{\alpha}$ (see the proof of theorem 3), so $\mathbf{M}^{\alpha}$ is flat over $\mathcal{H}^{\alpha}$ . By theorem 3, $Z_{\kappa(Q)}^{\alpha}\cong\mathbf{M}^{\alpha}$ is flat over $\mathcal{H}^{\alpha}$ . ∎

4.5 Proof of the comparison lemma on the regular part (lemma 18)

Proof.

Recall that $s^{S},S\subset S^{\alpha}$ generate the algebra $A_{0,reg}$ (section 2.7.3) and $x^{\chi(S)}t,S\subset S^{\alpha}$ generate the localized Rees algebra $\oplus_{m}\mathcal{H}^{T_{\mathcal{O}}}((\mathcal{R}_{T,N_{T}})^{+}_{\leq m})t^{m}$ (Corollary 2). It suffices to prove that these satisfy the same relations when we identify $s^{S}$ with $x^{\chi(S)}$ by lemma 16.

By the multiplication formula (17) in theorem 2, we get the multiplication formula in the Rees algebra, (we need to add $t$ on the right hand side)

r^{\lambda}r^{\mu}=\prod^{n}_{i=1}\xi_{i}^{d(\xi_{i}(\lambda),\xi_{i}(\mu))}r^{\lambda+\mu}t.

Replace the indices $\lambda,\mu$ by $A,B$ and denote the coefficient before $r^{\lambda+\mu}$ by $\text{fc}_{Q}(A,B)$ , we rewrite the formula as

r^{A}r^{B}=\text{fc}_{Q}(A,B)r^{\lambda+\mu}t.

In particular,

r^{A\cup B}r^{A\cap B}=\text{fc}_{Q}(A\cup B,A\cap B)r^{\lambda+\mu}t.

hence the relations in the localized Rees algebra $\oplus_{m}\mathcal{H}^{T_{\mathcal{O}}}((\mathcal{R}_{T,N_{T}})^{+}_{\leq m})t^{m}$ can be written as

r^{A}r^{B}=\frac{\text{fc}_{Q}(A,B)}{\text{fc}_{Q}(A\cup B,A\cap B)}r^{A\cup B}r^{A\cap B}.

Recall by lemma 12 that

x^{S}=(Eu_{S})^{-1}r^{S}

so the relations in terms of $x^{S}$ are

x^{A}x^{B}=\frac{\text{fc}_{Q}(A,B)}{\text{fc}_{Q}(A\cup B,A\cap B)}\frac{Eu(A\cup B)Eu(A\cap B)}{Eu(A)Eu(B)}x^{A\cup B}x^{A\cap B}.

Now we consider the relations in the algebra $A_{0,reg}.$ Recall in (12), we defined $l(\beta)$ depending on $\kappa.$ Now we set $s^{\beta}$ by the same formula as in (11) replacing $l(\beta)$ by

l_{Q}(\beta)=\prod_{a\in E\text{ such that }s(a)=i,t(a)=i^{\prime},i\neq i^{\prime}l\in(S_{\beta})_{i},j\in(S_{\beta})_{i^{\prime}}}(x^{i}_{l}-x^{i^{\prime}}_{j})^{\kappa_{ii^{\prime}}}\prod_{i\in I,l\neq j,l,j\in(S_{\beta})_{i}}(x^{i}_{l}-x^{i}_{j})^{\kappa_{ii}},

where for an arrow $a\in E$ , $s(a)$ is the source and $t(a)$ is the target. The difference between $l(\beta)$ and $l_{Q}(\beta)$ is possibly a sign which depends on the directions of the arrows in $Q$ . So accordingly, in the formula (13) of $s^{S}$ , we replace $l(S)$ by

l_{Q}(S)=\prod_{a\in E\text{ such that }s(a)=i,t(a)=i^{\prime},i\neq i^{\prime},l\in S_{i},j\in S_{i^{\prime}}}(x^{i}_{l}-x^{i^{\prime}}_{j})^{\kappa_{ii^{\prime}}}\prod_{i\in I,l\neq j,l,j\in S_{i}}(x^{i}_{l}-x^{i}_{j})^{\kappa_{ii}}.

The relations in the algebra $A_{0,reg}$ for $s^{S},S\subset S^{\alpha}$ are

s^{A}s^{B}=\frac{l_{Q}(A)l_{Q}(B)}{l_{Q}(A\cup B)l_{Q}(A\cap B)}s^{A\cup B}s^{A\cap B}.

Now to check $s^{S},S\subset S^{\alpha}$ and $x^{\chi(S)}t,S\subset S^{\alpha}$ satisfy the same relations, it suffices to show that

\frac{\text{fc}_{Q}(A,B)}{\text{fc}_{Q}(A\cup B,A\cap B)}\frac{Eu(A\cup B)Eu(A\cap B)}{Eu(A)Eu(B)}=\frac{l_{Q}(A)l_{Q}(B)}{l_{Q}(A\cup B)l_{Q}(A\cap B)}.

(24)

This will be an elementary combinatorial calculation. For a general quiver $Q,$ denote by $Q_{s}$ the quiver removing all arrows between different vertices and $Q_{b}$ the removing all arrows.

•

We first prove the case where $Q$ has no edges, i.e. $Q=Q_{b}.$ In this case $N=0$ , so

\frac{\text{fc}_{Q_{b}}(A,B)}{\text{fc}_{Q_{b}}(A\cup B,A\cap B)}=1.

It suffices to show that

\frac{Eu(A\cup B)Eu(A\cap B)}{Eu(A)Eu(B)}=\frac{l_{Q_{b}}(A)l_{Q_{b}}(B)}{l_{Q_{b}}(A\cup B)l_{Q_{b}}(A\cap B)}.

(25)

From the description of $Eu(S)$ and $l_{Q_{b}}(S)$ , we can assume $I$ is one point. Let $W:=\{1,2,\cdots,n\}$ . Let $A\cap B=C,A\setminus B=D,B\setminus A=E,W\setminus(A\cup B)=F$ so $A=C\cup D,B=D\cup E,A\cup B=C\cup D\cup E,A\cap B=D,W=C\cup D\cup E\cup F.$ By lemma 12(b), we have

Eu^{-1}(A)=\prod_{l\in A,j\in W\setminus A}(a_{l}-a_{j})=\prod_{(l,j)\in A\times(W\setminus A)}(a_{l}-a_{j}).

Here, we rewrite the index set in the second equality as a product $A\times(W\setminus A)$ . It is clear to show 25, it suffices to compare the index set of the products. We write

X\overset{\text{supp}}{=}S

to mean

X=\prod_{(l,j)\in S}(a_{l}-a_{j}).

By this notation,

Eu^{-1}(A)\overset{\text{supp}}{=}A\times(W\setminus A)=(C\cup D)\times(E\cup F)=(C\times E)\cup(C\times F)\cup(D\times E)\cup(D\times F).

Eu^{-1}(B)=\prod_{l\in B,j\in W\setminus B}(a_{l}-a_{j})\overset{\text{supp}}{=}(D\times C)\cup(D\times F)\cup(E\times C)\cup(E\times F).

Eu^{-1}(A\cup B)=\prod_{l\in A\cup B,j\in W\setminus(A\cup B)}(a_{l}-a_{j})\overset{\text{supp}}{=}(C\times F)\cup(D\times F)\cup(E\times F)

Eu^{-1}(A\cap B)=\prod_{l\in A\cap B,j\in W\setminus(A\cap B)}(a_{l}-a_{j})\overset{\text{supp}}{=}(D\times C)\cup(D\times E)\cup(D\times F).

\frac{Eu^{-1}(A)Eu^{-1}(B)}{Eu^{-1}(A\cup B)Eu^{-1}(a\cap B)}\overset{\text{supp}}{=}(C\times E)\cup(E\times C).

For $l_{Q_{b}}$ , we have

l_{Q_{b}}(A)=\prod_{l\in A,j\in A}(a_{l}-a_{j})\overset{\text{supp}}{=}\{(C\times C)\cup(C\times D)\cup(D\times C)\cup(D\times D)\}.

l_{Q_{b}}(B)=\prod_{l\in B,j\in B}(a_{l}-a_{j})\overset{\text{supp}}{=}\{(D\times D)\cup(D\times E)\cup(E\times D)\cup(E\times E)\}.

l_{Q_{b}}(A\cup B)=\prod_{l\in A\cup B,j\in A\cup B}(a_{l}-a_{j})\overset{\text{supp}}{=}\\

\{(C\times C)\cup(C\times D)\cup(C\times E)\cup(D\times C)\cup(D\times D)\cup(D\times E)\cup(E\times C)\cup(E\times D)\cup(E\times E)\}.

l_{Q_{b}}(A\cap B)=\prod_{l\in A\cap B,j\in A\cap B}(a_{l}-a_{j})\overset{\text{supp}}{=}\{(D\times D)\}.

\frac{l_{Q_{b}}(A)l_{Q_{b}}(B)}{l_{Q_{b}}(A\cup B)l_{Q_{b}}(A\cap B)}\overset{\text{supp}}{=}(C\times E)\cup(E\times C)

and therefore (25) holds.

•

Now we prove the case where $Q=Q_{s}$ has only self-edges. Again it is clear we can assume $I$ is one point. Denote

l^{d}_{Q_{s}}=\frac{l_{Q_{s}}}{l_{Q_{b}}}.

Now it suffices to show that

\frac{\text{fc}_{Q_{s}}(A,B)}{\text{fc}_{Q_{s}}(A\cup B,A\cap B)}=\frac{l^{d}_{Q_{s}}(A)l^{d}_{Q_{s}}(B)}{l^{d}_{Q_{s}}(A\cup B)l^{d}_{Q_{s}}(A\cap B)}.

By remark 4, we can reduce to the case where $Q_{s}$ has one self-loop and it suffices to prove

\frac{\text{fc}_{Q_{s}}(A,B)}{\text{fc}_{Q_{s}}(A\cup B,A\cap B)}\overset{\text{supp}}{=}(C\times E)\cup(E\times C).

Now $N=Hom(V,V)$ is the adjoint representation of $GL(V).$ The characters $\xi_{i}$ is the set of all roots of $GL(V),$ which are all $(a_{l}-a_{j})$ for $l\neq j,l,j\in W.$ Compute the pairing between $A,B,A\cup B,A\cap B$ and $(a_{l}-a_{j})$ for $l\neq j,l,j\in W$ and $A,B$ we get the formula.

•

Now we prove the general case. Denote $l^{d}_{Q}=\frac{l_{Q}}{l_{Q_{s}}}$ and $\text{fc}^{d}_{Q}=\frac{\text{fc}_{Q}}{\text{fc}_{Q_{s}}}.$ It suffices to show that

\frac{\text{fc}^{d}_{Q}(A,B)}{\text{fc}^{d}_{Q}(A\cup B,A\cap B)}=\frac{l^{d}_{Q}(A)l^{d}_{Q}(B)}{l^{d}_{Q}(A\cup B)l^{d}_{Q}(A\cap B)}.

Again by remark 4, we can reduce to the case where $Q$ has two vertices and only one arrow $1\xrightarrow[]{}2.$ In this case $N=Hom(V_{1},V_{2})$ . The characters $\xi_{i}$ consist of $(a^{1}_{l}-a^{2}_{j})$ for all $l\leq l\leq n_{1},1\leq j\leq n_{2}.$ Compute the pairing between $A,B,A\cup B,A\cap B$ and $(a^{1}_{l}-a^{2}_{j})$ for all $l\leq l\leq n_{1},1\leq j\leq n_{2}$ and $A,B$ we get the formula.

∎

5 Appendix: Proof of lemma 4 (by Ivan Mirković)

5.1 Equations of Segre embeddings

The locality equations in the discrete range are just the Segre embedding equations of a special type $(\mathbb{P}^{1})^{n}\hookrightarrow\mathbb{P}^{2^{n}-1}$ . The standard list of Segre equations of general type $\prod_{p\in D}\mathbb{P}(V_{p})\hookrightarrow\mathbb{P}(V_{D})$ , is checked based on the case $\mathbb{P}(A)\times\mathbb{P}(B)\hookrightarrow\mathbb{P}(A\otimes B)$ in 5.1.1. In our case $(\mathbb{P}^{1})^{n}\hookrightarrow\mathbb{P}^{2^{n}-1}$ the combinatorics is stated in terms of $Gr(D)\subset\mathbb{N}[D]$ in 5.1.2.

5.1.1 Segre embeddings.

(i) Case of two factors. A choice of coordinates on $A$ and $B$ , $x_{a},a\in\mathcal{A}$ and $y_{b},b\in\mathcal{B},$ gives coordinates $z_{ab}=x_{a}\otimes y_{b}$ on $A\otimes B$ . A vector $v\in A\otimes B$ can be thought of as an operator $v:A^{*}\xrightarrow[]{}B$ and $v$ is a pure tensor iff the rank of $v$ is $\leq 1.$ In terms of the matrix $(z_{ab})_{\mathcal{A}\times\mathcal{B}}$ this condition is the vanishing of all of its $2\times 2$ minors $z_{ab}z_{a^{\prime}b^{\prime}}-z_{ab^{\prime}}z_{a^{\prime}b}.$
(ii) Segre embedding map $\prod_{p\in D}\mathbb{P}(V_{p})\xrightarrow[]{}\mathbb{P}(V_{D}).$ Let $x^{p}_{i},i\in\mathcal{B}_{p},$ be the coordinates on vector spaces $V_{p},p\in D,$ so that for $X\subset D$ on $V_{X}\stackrel{{\scriptstyle\text{def}}}{{=}}\otimes_{p\in X}V_{p}.$ We have coordinates $z_{\beta}^{X}\stackrel{{\scriptstyle\text{def}}}{{=}}\otimes_{p\in D}x_{\beta_{p}}^{p}$ indexed by $\beta\in\mathcal{B}_{X}\stackrel{{\scriptstyle\text{def}}}{{=}}\prod_{p\in X}\mathcal{B}_{p}.$ Then the Segre embedding map $\prod_{p\in D}\mathbb{P}(V_{p})\hookrightarrow\mathbb{P}(V_{D})$ is given by $z_{\beta}^{D}\mapsto\prod_{p\in D}x_{\beta_{p}}^{p}.$ We view $\prod_{p\in D}\mathbb{P}(V_{p})$ as the intersection of all $\mathbb{P}(V_{X})\times\mathbb{P}(V_{Y})\subset\mathbb{P}(V_{D})$ over all decomposition $D=X\sqcup Y.$ ²³²³23The claim $\mathbb{P}(A)\times\mathbb{P}(B\otimes C)\cap_{\mathbb{P}(A\times B\times C)}\mathbb{P}(A\times B)\times\mathbb{P}(C)=\mathbb{P}(A)\times\mathbb{P}(B)\times\mathbb{P}(C)$ means that if a vector $v$ in $V=A\otimes B\otimes C$ is a pure tensor for $A\otimes(B\otimes C)$ and $(A\otimes B)\otimes C$ , then it is also a pure tensor for $A\otimes B\otimes C.$ Proof. The assumption is $v=a\otimes\alpha=\gamma\otimes c$ with $a\in A,c\in C,\alpha\in B\otimes C,\gamma\in A\otimes B.$ Choose a basis $c_{i}$ of $C$ with a dual basis $c^{i}$ and $c=c_{0}$ . For $i\neq 0$ we have $c^{i}\perp c,$ hence $0=\langle v,c^{i}\rangle=a\otimes b_{i}$ hence $b_{i}=0$ . So, $v=a\otimes b_{0}\otimes c.$ ∎ So, by (i) equations are the minors $\begin{vmatrix}z_{\alpha^{\prime}\beta^{\prime}}&z_{\alpha^{\prime}\beta^{\prime\prime}}\\ z_{\alpha^{\prime\prime}\beta^{\prime}}&z_{\alpha^{\prime\prime}\beta^{\prime\prime}}\end{vmatrix}$ indexed by data $D=X\cup Y$ and $\alpha^{\prime},\alpha^{\prime\prime}\in\mathcal{B}_{X},\beta^{\prime},\beta^{\prime\prime}\in\mathcal{B}_{Y}$ (so that $\alpha^{\prime}\beta^{\prime}\in\mathcal{B}_{D}$ etc.).

Remark 7.

We can write these equations as $z_{\phi}z_{\psi}-z_{\phi^{C}}z_{\psi^{C}}$ for $(\phi,\psi,C)\in\mathcal{B}_{D}^{2}\times Gr(D);$ here $C\in Gr(D)$ acts on $\mathcal{B}_{D}^{2}$ by the unique involution $\sigma_{C}(\phi,\psi)=(\phi^{C},\psi^{C})$ that exchanges the values on $C$ , i.e., $\phi^{C}_{p}=\psi_{p}$ for $p\in C$ and $\psi^{C}_{p}=\psi_{p}$ for $p\notin C$ (and the same for $\psi^{C}$ ). So, the equation require invariance of products under $Gr(D)$ as a $D$ -degeneration of commutativity which is the case $C=D$ as $(\phi^{D},\psi^{D})=(\phi,\psi).$ (However, this $Gr(D)$ action is only defined on a chosen basis.)

5.1.2 The case of $(\mathbb{P}^{1})^{n}\hookrightarrow\mathbb{P}^{2^{n}-1}.$

Now we have identifications $\mathcal{B}_{p}\xleftarrow[\cong]{}\{\emptyset,p\}=Gr(p)$ and therefore $\mathcal{B}_{D}\xleftarrow[\cong]{}Gr(D)$ (by $\phi\mapsfrom X$ for $\phi_{p}=\delta_{p\in X}).$ We also embed $Gr(D)$ into a monoid $\mathbb{N}[D].$ (We often denote $z_{X}$ by $1_{X}$ .)

Corollary 4.

The Segre equations are now $z_{X}z_{Y}=z_{U}z_{V}$ , indexed by all $X,Y,U,V\subset D$ with $X+Y=U+V\in\mathbb{N}[D].$

Proof.

Involution $\sigma_{C},C\subset D$ preserve for each $p\in D$ the multiset $\phi_{p},\psi_{p}$ hence also the sum $\phi_{p}+\psi_{p}.$ Conversely, if $X+Y=U+V$ then $(U,V)=\sigma_{C}(X,Y)$ for $C=\{p\in D;X_{p}\neq U_{p}\}.$ ∎

Remark 8.

(0) One has $z_{i,j}z_{\emptyset}=z_{i}z_{j}$ and inductively $z_{X}z_{\emptyset}^{|X|-1}=\prod_{p\in D}z_{p}.$ (1) By setting $1_{\emptyset}=1$ one obtains an open affine subspace of $\mathbb{P}(V_{D})$ with functions $\mathbb{k}[1_{B},\emptyset\neq B\subset D]$ . Here, locality equations reduce to $z_{X}=\prod_{p\in D}z_{p}$ with solutions $\mathbb{A}^{D}.$

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[MYZ21] Ivan Mirković, Yaping Yang, and Gufang Zhao. Loop Grassmannians of quivers and affine quantum groups. In Representation Theory and Algebraic Geometry: A Conference Celebrating the Birthdays of Sasha Beilinson and Victor Ginzburg, pages 347–392. Springer, 2021.
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Loop Grassmannian of quivers and Compactified Coulomb branch of quiver gauge theory with no framing

Abstract

1 Introduction

1.1 Generalization of loop Grassmannian

1.2 Zastava spaces [FM99]

1.2.1 Intersections of semiinfinite orbits in 𝒢​(G)\mathcal{G}(G)

1.2.2 Ordered Zastava ZG′Z^{\prime}_{G} from global loop Grassmannian

1.2.3 Zastava ZGZ_{G} from quasimaps

1.3 Compactified Coulomb branch for quiver gauge theory with no framing

1.4 Local projective spaces [MYZ21, Mir23]

1.4.1 Motivations

1.4.2 Construction of Zastava ZκZ_{\kappa} for a matrix κ\kappa

1.5 Identification of two generalizations of Zastava

1.6 Background and further directions

2 Mirković local projective space (Zastava)

2.1 Hilbert scheme ℋC×I\mathcal{H}_{C\times I}

2.2 The regular part of ℋC×I\mathcal{H}_{C\times I}

2.3 Local line bundle LL on ℋC×I\mathcal{H}_{C\times I}.

Definition 1.

Remark 1.

2.4 Induced vector bundle ℐ\mathcal{I}

Lemma 1.

Proof.

Lemma 2.

Proof.

2.5 Local projective space of a local vector bundle

Remark 2.

Remark 3.

2.6 Zastava space ZκZ_{\kappa} of a symmetric matrix κ\kappa

2.7 ZαZ^{\alpha} for C=𝔸1C=\mathbb{A}^{1}

2.7.1 Explicit description of locality structure on ℐ\mathcal{I}

Lemma 3.

Proof.

2.7.2 Bases of global section Γ​(Cα,(qα)∗​ℐ)\Gamma(C^{\alpha},(q^{\alpha})^{*}\mathcal{I}) as 𝒪​(Cα)\mathcal{O}(C^{\alpha})-module (global bases)

2.7.3 The basis of Γ​(Cr​e​gα,(qα)∗​ℐr​e​g)\Gamma(C^{\alpha}_{reg},(q^{\alpha})^{*}\mathcal{I}_{reg}) (local basis)

Lemma 4.

Proof.

2.7.4 Transition between a global basis zpβz^{\beta}_{p} and the local basis sSs^{S} over the regular part

Lemma 5.

Proof.

Theorem 1.

Proof.

3 Coulomb branch

3.1 Localization

3.2 Formula for H∗T𝒪​(ℛT)H^{T_{\mathcal{O}}}_{*}(\mathcal{R}_{T})

Theorem 2.

Remark 4.

3.3 Compactified Coulomb branch 𝐌Qα\mathbf{M}^{\alpha}_{Q} of quiver gauge theory

3.4 Embedding of 𝐌Qα\mathbf{M}_{Q}^{\alpha} into ℙℋ​(HG𝒪​(ℛ≤1+)∗)\mathbb{P}_{\mathcal{H}}(H^{G_{\mathcal{O}}}(\mathcal{R}^{+}_{\leq 1})^{*})

Lemma 6.

Proof.

Corollary 1.

Proof.

3.5 Bases of H∗G𝒪​(ℛ≤1+)H^{G_{\mathcal{O}}}_{*}(\mathcal{R}^{+}_{\leq 1}) as HG𝒪∗​(p​t)≅𝒪​(ℋα)H_{G_{\mathcal{O}}}^{*}(pt)\cong\mathcal{O}(\mathcal{H}^{\alpha})-module

Lemma 7.

Proof.

Lemma 8.

Proof.

Remark 5.

3.6 Pulling back 𝐌α\mathbf{M}^{\alpha} under Cα→qαℋαC^{\alpha}\xrightarrow[]{q^{\alpha}}\mathcal{H}^{\alpha}

Lemma 9.

Proof.

Lemma 10.

Proof.

3.6.1 Generators of the localized Rees algebra ⊕mℋT𝒪​((ℛT,NT)≤m+)​tm\oplus_{m}\mathcal{H}^{T_{\mathcal{O}}}((\mathcal{R}_{T,N_{T}})^{+}_{\leq m})t^{m}

Lemma 11.

Proof.

3.7 H∗T𝒪​((ℛT,NT+)≤1)→ι∗H∗T𝒪​(ℛ≤1+)H^{T_{\mathcal{O}}}_{*}((\mathcal{R}_{T,N_{T}}^{+})_{\leq 1})\xrightarrow[]{\iota_{*}}H^{T_{\mathcal{O}}}_{*}(\mathcal{R}^{+}_{\leq 1}) over Cr​e​gC_{reg}

Lemma 12.

Proof.

Corollary 2.

Proof.

4 Identification of 𝐌α\mathbf{M}^{\alpha} and ZαZ^{\alpha}

Remark 6 (Different conventions about the matrix κ​(Q)\kappa(Q)).

Theorem 3.

4.1 Identification of H∗T​(ℛ≤1+)H_{*}^{T}(\mathcal{R}^{+}_{\leq 1}) and Γ​(Cα,(qα)∗​ℐ)\Gamma(C^{\alpha},(q^{\alpha})^{*}\mathcal{I})

Lemma 13.

Proof.

4.2 Matching the local basis

Lemma 14.

1.2.1 Intersections of semiinfinite orbits in $\mathcal{G}(G)$

1.2.2 Ordered Zastava $Z^{\prime}_{G}$ from global loop Grassmannian

1.2.3 Zastava $Z_{G}$ from quasimaps

1.4.2 Construction of Zastava $Z_{\kappa}$ for a matrix $\kappa$

2.1 Hilbert scheme $\mathcal{H}_{C\times I}$

2.2 The regular part of $\mathcal{H}_{C\times I}$

2.3 Local line bundle $L$ on $\mathcal{H}_{C\times I}$ .

2.4 Induced vector bundle $\mathcal{I}$

2.6 Zastava space $Z_{\kappa}$ of a symmetric matrix $\kappa$

2.7 $Z^{\alpha}$ for $C=\mathbb{A}^{1}$

2.7.1 Explicit description of locality structure on $\mathcal{I}$

2.7.2 Bases of global section $\Gamma(C^{\alpha},(q^{\alpha})^{*}\mathcal{I})$ as $\mathcal{O}(C^{\alpha})$ -module (global bases)

2.7.3 The basis of $\Gamma(C^{\alpha}_{reg},(q^{\alpha})^{*}\mathcal{I}_{reg})$ (local basis)

2.7.4 Transition between a global basis $z^{\beta}_{p}$ and the local basis $s^{S}$ over the regular part

3.2 Formula for $H^{T_{\mathcal{O}}}_{*}(\mathcal{R}_{T})$

3.3 Compactified Coulomb branch $\mathbf{M}^{\alpha}_{Q}$ of quiver gauge theory

3.4 Embedding of $\mathbf{M}_{Q}^{\alpha}$ into $\mathbb{P}_{\mathcal{H}}(H^{G_{\mathcal{O}}}(\mathcal{R}^{+}_{\leq 1})^{*})$

3.5 Bases of $H^{G_{\mathcal{O}}}_{}(\mathcal{R}^{+}_{\leq 1})$ as $H_{G_{\mathcal{O}}}^{}(pt)\cong\mathcal{O}(\mathcal{H}^{\alpha})$ -module

3.6 Pulling back $\mathbf{M}^{\alpha}$ under $C^{\alpha}\xrightarrow[]{q^{\alpha}}\mathcal{H}^{\alpha}$

3.6.1 Generators of the localized Rees algebra $\oplus_{m}\mathcal{H}^{T_{\mathcal{O}}}((\mathcal{R}_{T,N_{T}})^{+}_{\leq m})t^{m}$

3.7 $H^{T_{\mathcal{O}}}_{}((\mathcal{R}_{T,N_{T}}^{+})_{\leq 1})\xrightarrow[]{\iota_{}}H^{T_{\mathcal{O}}}_{*}(\mathcal{R}^{+}_{\leq 1})$ over $C_{reg}$

4 Identification of $\mathbf{M}^{\alpha}$ and $Z^{\alpha}$

Remark 6 (Different conventions about the matrix $\kappa(Q)$ ).

4.1 Identification of $H_{}^{T}(\mathcal{R}^{+}_{\leq 1})$ and $\Gamma(C^{\alpha},(q^{\alpha})^{}\mathcal{I})$

5.1.2 The case of $(\mathbb{P}^{1})^{n}\hookrightarrow\mathbb{P}^{2^{n}-1}.$