Coulomb branches of quiver gauge theories with symmetrizers

We generalize the mathematical definition of Coulomb branches of $3$-dimensional $\mathcal N=4$ SUSY quiver gauge theories in arXiv:1503.03676, arXiv:1601.03686, arXiv:1604.03625 to the cases with symmetrizers. We obtain generalized affine Grassmannian slices of type $BCFG$ as examples of the construction, and their deformation quantizations via truncated shifted Yangians. Finally, we study modules over these quantizations and relate them to the lower triangular part of the quantized enveloping algebra of type $ADE$.


Introduction
Let I be a finite set. Recall (c ij ) i,j∈I is a symmetrizable Cartan matrix if • c ii = 2 for all i ∈ I, and c ij ∈ Z ≤0 for all i = j, • there is (d i ) ∈ Z I >0 such that d i c ij = d j c ji for all i, j. When d i = 1 for any i ∈ I, a mathematical definition of the Coulomb branch of a 3d N = 4 quiver gauge theory associated with two I-graded vector spaces V = V i , W = W i was given in [Nak16,BFN18a], and its properties were studied in [BFN19]. In this note, we generalize the definition to more general symmetrizable cases. This new definition is motivated by works of Geiss, Leclerc and Schröer ( [GLS17] and the subsequent papers [GLS18a,GLS16,GLS18b,GLS18c,GLS18d]) which aim to generalize various results on relations between symmetric Kac-Moody Lie algebras and quivers to symmetrizable cases. They modify quiver representations by replacing vector spaces on vertices by free modules of truncated polynomial rings. They use different variables for polynomials, which are related to each other according to d i . This modification allows them to relate quiver representations to symmetrizable Kac-Moody algebras. Their work, and ours, is also partly motivated by the theory of modulated graphs [DR80,NT16], another approach to quivers in symmetrizable types.
In [BFN18a] we assign vector bundles over the formal disk D = Spec C [[z]]. Since we can take different variables z i for each vertex i ∈ I, the definition has the same modification.
A similar construction was considered in the context of 4d N = 2 quiver gauge theories by Kimura and Pestun [KP18] under the name of fractional quiver gauge theories.
Let us recall that we defined Coulomb branches of quiver gauge theories associated with a symmetrizable Cartan matrix in a different way in [BFN19, §4]. There, we realize a symmetrizable Cartan matrix by a folding of a graph. This folding gives a finite group action on the Coulomb branch of the quiver gauge theory of the (unfolded) graph. Then we may define the Coulomb branch of the symmetrizable theory as the corresponding fixed point subscheme. This construction recovers the twisted monopole formula by Cremonesi, Ferlito, Hanany and Mekareeya [CFHM14], as the Hilbert series of the coordinate ring. This gives supporting evidence that the folding construction is a reasonable candidate for a mathematical definition of the Coulomb branch.
Our new construction also gives the twisted monopole formula. It is natural to believe that the folding construction and the new one give isomorphic varieties. However various properties of the Coulomb branch are obvious in the new construction, while they are not in the old one. For example, the twisted monopole formula requires a proof in the old construction, while it is obvious in the new construction. We also do not know how to show the normality in the old construction, while the proof in [BFN18a] works for the new construction. Therefore we believe that the new construction has its own meaning. In addition, work in progress of de Campos Affonso will identify the new definition with the symmetric bow varieties introduced in [dCA18] for quiver gauge theories of non-symmetric affine Lie algebras of classical type. This identification is not clear for the old construction of the Coulomb branch as a fixed point subscheme.
In fact, we will give a second potential definition for the Coulomb branch of a quiver gauge theory with symmetrizers in §C. In many cases both definitions agree, and in particular this is true in finite BCFG types. However, in general type they are different. This alternative definition applies to more general data than quivers with symmetrizers, which may be of independent interest.
As a generalization of one of the main results in [BFN19], we show that our Coulomb branches are generalized slices in the affine Grassmannian when the Cartan matrix is of type BCF G (Theorem 4.1). Therefore the geometric Satake correspondence, as modified in [Kry18], says that the direct sum of hyperbolic stalks of the intersection cohomology complexes of our Coulomb branches has a structure of a finite dimensional irreducible representation of the Langlands dual Lie algebra. We expect that the same should be true for arbitrary symmetrizable Kac-Moody Lie algebras as a symmetrizable generalization of the conjecture in [BFN19, §3(viii)]. (See also [Nak18] for a refinement of the conjecture.) Also as a generalization of the main result in seven authors' (BFK 2 NW 2 ) appendices of [BFN19] and also of [Wee19], we show that the quantization of the Coulomb branch is a truncated shifted Yangian when the Cartan matrix is of type BCF G. (See Theorem 5.8.) Its modules can be analyzed by using techniques of the localization theorem in equivariant homology groups, even though we use infinite dimensional varieties [VV10,Web19,Nak19]. We study the fixed point subvariety with respect to a C × -action in an infinite dimensional variety used in the definition of the Coulomb branch in §B. It turns out that the fixed point subvariety is the same as one appears in the Coulomb branches of type ADE, which is a disjoint union of varieties appearing Lusztig's work on canonical bases of U − q of type ADE [Lus91]. This implies that a certain category of modules of the truncated shifted Yangian of type BCF G categorifies U − q of type ADE. (See Theorem B.6. We only explain a parametrization of simple modules for simplicity.) It is interesting to understand the relation between this analysis and the geometric Satake correspondence explained above, as we obtain different Lie algebras, type ADE and BCF G.
Let us also remark that our construction can be applied to more general situations than considered here. For example, the first-named author originally introduced the Coulomb branch via cohomology with compact support of the moduli space of twisted maps from P 1 to the Higgs branch M H (viewed as a quotient stack) with coefficients in the sheaf of vanishing cycles [Nak16]. This definition can be generalized to our setting, just changing the domain P 1 for each vertex i ∈ I. This view point might shed a new light on the Higgs branch M H corresponding to our new construction: we cannot make sense of M H , but the space of maps to M H does make sense. In particular, enumerative problems for M H , such as discussed in [Oko17], are meaningful.
We also hope that our view point is useful to make advance in the program of Geiss, Leclerc, Schröer. We may hope to use the above space of maps to M H to realize representations of the Lie algebra, or its cousins the Yangian and the quantum loop algebra associated with the symmetrizable Cartan matrix (c ij ).
The paper is organized as follows. In §2 we give the definition of Coulomb branches for symmetrizable Cartan matrix (c ij ). Since it is a modification of the original one in [BFN18a], we only explain where we change the definition. In §3 we determine Coulomb branches in some cases when the Cartan matrix is 2 × 2, and there are no framed vector spaces W i . In §4 we show that Coulomb branches are generalized slices in the affine Grassmannian when the Cartan matrix is of type BCF G. The proof is the same as in [BFN19, §3], once examples in §3 are determined. In §5 we discuss quantized Coulomb branches. We show that they are isomorphic to truncated shifted Yangians in type BCF G. In §A we give an explicit presentation of the coordinate ring of the zastava space of degree α 1 + α 2 of type G 2 . This is used in §3. Contents in §B are already explained above. In §C we present a second possible definition for the Coulomb branch associated to a quiver with symmetrizers, as mentioned above.

Definition
2(i). A valued quiver. Let (c ij ) i,j∈I be a symmetrizable Cartan matrix. We assign a valued graph where it has vertices i ∈ I and unoriented edges between i, j for c ij < 0 with values (|c ij |, |c ji |). A valued quiver is a valued graph together with a choice of an orientation of each edge. Following [GLS17], we set g ij = gcd(|c ij |, |c ji |), f ij = |c ij |/g ij when c ij < 0. Note that these are independent of d i . We take the formal disk D i = Spec C[[z i ]] for each vertex i ∈ I. For a pair (i, j) with c ij < 0 we take the formal disk D = Spec C[[z]] and consider its branched coverings of coordinate rings. The disk D depends on (i, j), but we drop i, j from the notation. Let D * i , D * j , D * denote the punctured formal disk for D i , D j , D respectively.
is imposed, where ε i , ε j are edge loops at i and j respectively, and α (g) ij is the g-th arrow from j to i. It means that we have z Thus it differs from our convention by f ij ↔ f ji . This is probably compatible with geometric Satake correspondence: We will obtain generalized slices in the affine Grassmannian for G for (c ij ) below, and hence representations of G ∨ , by the work of Krylov [Kry18]. On the other hand, the space of constructible functions on modules over the quiver with the relation (H2) is the enveloping algebra of the upper triangular subalgebra n of the Lie algebra g for (c ij ). Since we hope to compare representations of the same Lie algebra in Coulomb branches and [GLS17], we need to take Langlands dual relation of (H2).
Note also that the relation imposed in [HL16] for a cluster algebra related to the quantum loop algebra U q (Lg) is the same as ours. See [GLS17, §1.7.1]. We believe that this is compatible with our results in §5, as the K-theoretic version of our construction in §5 should yield U q (Lg)-modules. However, the K-theoretic version of our construction does not immediately give a new approach to the results of [HL16]. Modules obtained in this way are infinite dimensional, while [HL16] discussed Kirillov-Reshetikhin modules, which are finite dimensional. Nevertheless we expect that it gives a first step towards in that direction.
We consider the moduli space R parametrizing the following objects: • a rank v i vector bundle E i over D i together with a trivialization ϕ i : (π ji * ϕ j ) −1 extends to D, where c ij < 0 and there is an arrow j → i in the quiver. The moduli space of pairs (E i , ϕ i ) as above is the affine Grassmannian Gr GL(V i ) for GL(V i ).
Dropping the extension conditions in the second and third, we have a larger moduli space T , which is an infinite rank vector bundle over i Gr GL(V i ) . Then R is a closed subvariety in T .
When c ij = c ji , R is nothing but the variety of triples introduced in [BFN18a, §2(i)].
We have a G O -action on R by change of trivializations ϕ i , and we consider the G O -equivariant Borel-Moore homology group H G O * (R) with complex coefficients. This is defined rigorously as a double limit as in [BFN18a, §2(ii)].
The spaces N O , N K appear during the construction of the convolution product in [BFN18a, §3(i)]. They were the space of sections (resp. rational sections) of the vector bundle associated with the trivial G-bundle. In our setting, N O is defined as the direct sum of For N K , we take homomorphisms over O D * i and O D * . We have maps Π : R → N O and T → N K .
2(iii). Twisted monopole formula. Recall that the monopole formula for the Hilbert series of the Coulomb branch of a gauge theory [CHZ14] is interpreted as the Poincaré polynomial of H G O * (R) with a suitable modification in the ordinary untwisted case [Nak16,BFN18a]. The twisted monopole formula is given in [CFHM14] to cover Coulomb branches of quiver gauge theories for certain symmetrizable Cartan matrices. It is of the same form λ t 2∆(λ) P G (t; λ) as the untwisted monopole formula, where the summation runs over the set of dominant coweights λ of the gauge group G, and P G (t; λ) is the Poincaré polynomial of the equivariant cohomology ring H * Stab G (λ) (pt). Only ∆(λ) is changed from the untwisted monopole formula: if i, j ∈ I, the ordinary 2∆(λ) contains contribution |λ a ..,v j are components of λ for vertices i, j respectively. In the twisted monopole formula, this contribution is simply replaced by |f ji λ a i − f ij λ b j |. Let us check that our new R gives the twisted monopole formula as the Poincaré polynomial. The argument is a simple modification of [BFN18a, §2(iii)]. We do so under an additional assumption: In particular all finite types satisfy this assumption. Let Gr λ G denote the G O -orbit in Gr G corresponding to a dominant coweight λ of G. Let R λ , T λ denote the inverse image of Gr λ G under the projection π : R → Gr G , π : T → Gr G respectively. As in [BFN18a, Lemma 2.2], T λ /R λ is a vector bundle over Gr λ G . The fiber of T λ at λ is Here z = z Following the notation of [BFN18a, Section 2(iii)], we may formally write the Poincaré polynomial of R. Let R ≤µ denote the inverse image of the closure Gr µ G = λ≤µ Gr λ G in R. As in [BFN18a, Proposition 2.7]: Proposition 2.3. The Poincaré polynomial for R ≤µ is given by where the sum is over dominant coweights λ with λ ≤ µ.
In particular, taking the limit over µ we formally obtain: However, we note that this expression may not converge even as a Laurent series.
The monopole formula is closely related to this Poincaré polynomial: the contribution ∆(λ) mentioned above is given by depends only on the sums a λ a i , b λ b j . In particular, it is possible to view the twisted monopole formula as the Poincaré polynomial of R, but with respect to a different grading (i.e. different from the homological grading). See [BFN18a, Remark 2.8].
Remark 2.5. If the assumption (2.2) does not hold, then the ranks of T λ /R λ are generally not given by such a simple formula. The corresponding Poincaré polynomial (and monopole formula) is thus more complicated.
2(iv). Convolution product. The definition of the convolution product on H G O * (R) goes exactly as in [BFN18a,3(iii)]. Moreover, we have an algebra embedding z * : to that which appears in the ordinary construction (i.e. all z i 's are replaced by z). In particular, is commutative as well. We define the Coulomb branch as is filtered in the same was as [BFN18a, Section 6(i)], and as in [BFN18a, Proposition 6.8] we can prove that A is finitely generated.
The proof of the normality in [BFN18a, Proposition 6.12] was given by the reduction to the cases when the gauge group is C × , SL(2) or PGL(2). That argument is applicable in our situation, and we are reduced to the case a quiver with a single vertex. Then our modification of the definition of the Coulomb branch is unnecessary and returns back to the original situation. Therefore we see that M C is normal.
Finally, M C has a natural deformation quantization A defined below in §5. This endows M C with a Poisson structure, which is generically symplectic as in [BFN18a, Proposition 6.15]. The subalgebra H * G (pt) is Poisson commutative, and defines an integrable systems on M C , see [BFN18a, Section 1(iii)].
Theorem 2.6. M C is an irreducible normal variety of finite type. It carries a Poisson structure which is generically symplectic, with an integrable system M C → Spec H * G (pt). Remark 2.7. As in [BFN19, Remark 3.9(3)], one can also consider the K-theoretic Coulomb branch.

Examples
3(i). We consider the case I = {1, 2}, c 12 = −1, c 21 = −m (where m ∈ Z >0 ), w 1 = w 2 = 0, and v 1 = v 2 = 1. We choose the orientation 1 ← 2. Note that G is the two dimensional torus. We consider the embedding z * : Let w 1 , w 2 be generators of the equivariant cohomology ring of a point for the first and second factors of G = (C × ) 2 . Let u a,b denote the fundamental class of the point (a, b) ∈ Gr G = Z 2 . We have with u a,b u a ,b = u a+a ,b+b . Let y a,b denote the fundametal class of the fiber of R → Gr G over (a, b). Then we have Note that y 1,m = u 1,m is invertible, with inverse y −1,−m . Therefore On the other hand, let us consider the folding of the Coulomb branch of the quiver gauge theory I = {1, 2 1 , 2 2 , . . . , 2 m } with edges 1-2 j for all j = 1, . . . , m with w i = 0, v i = 1 for all i ∈ I. We consider the Z/m-action on the quiver given by 2 1 → 2 2 → · · · → 2 m → 2 1 . See Figure 1 for m = 3. In order to distinguish the two groups for this theory and the former gauge theory, let us writeĜ = GL(V i ). Note that the diagonal scalar C × inĜ acts trivially on N, and so we haveĜ ∼ = C × × (C × ) m , N = C m where the first C × acts trivially on N, and (C × ) m acts on N in the standard way. Therefore the (usual) Coulomb branch for (Ĝ, N) is M C = A × A × × (A 2 ) m and Z/m acts by cyclically permutating the factors of (A 2 ) m . Therefore the fixed point locus is also A × A × × A 2 . Thus the former Coulomb branch is isomorphic to the fixed point locus of the latter Coulomb branch: Proposition 3.1. For the above data, there is an isomorphism More concretely w 1 , w 2 are identified with equivariant variables for GL(V 1 ) and GL(V 2 j ), where the latter is independent of j on the Z/m-fixed point locus. The function y a,b is identified with the restriction of the functionŷ a,b 1 ,...,bm on M C given by the fundamental class over (a, b 1 , .
In the cases m = 2, 3, we can identify our modified Coulomb branch with an open zastava spaceZ α . Recall thatZ α is the moduli space of based maps from P 1 to the flag variety, of degree α, see [BDF16,§2] Lemma 3.2. For m = 2 (resp. m = 3), M C is isomorphic to the open zastava space Z α 1 +α 2 of type B 2 (resp. type G 2 ).
Proof. Explicitly, we identify our description for m = 2 with the B 2 type open zastava space [BDF16, §5.7] 1 by w 1 = −A 2 , w 2 = −A 1 , y 1,2 = b 03 , y 0,1 = b 01 , and y 0, Similarly, in the m = 3 case we appeal to the description of the G 2 type open zastava in terms of coordiantes, from §A.
Another perpsective on this result is via folding. Recall that there is anétale rational coordinate system (y i,r , w i,r ) i∈I,1≤r≤v i on the open zastava spaceZ α for finite type [FKMM99,BDF16]. We claim that it is compatible with folding of the same type described above, namely the coordinate system for B 2 , G 2 is the restriction of the coordinate system for A 3 , D 4 to the Z/m-fixed point (m = 2, 3), respectively. For B 2 with the above choice of v, this can be checked directly from [BDF16, §5.7] as y i = b 01 = y 0,1 , In general, it is enough to check the assertion when v i is 1 dimensional for a single vertex i and 0 otherwise by the compatibility of the coordinate system and the factorization in [BDF16, Th. 1.6(3)], as the factorization and folding are compatible. In that case, a based map factors through P 1 via the embedding of P 1 into the flag variety corresponding to the vertex i. Then the assertion is clear. Alternatively we use the description from §A for G 2 to argue as in the B 2 case.
Recall that the isomorphism betweenZ α and the corresponding Coulomb branch was defined so that the coordinate system (y i,r , w i,r ) is mapped to (y i,r , w i,r ), where the latter w i,r is an equivariant variable as above, and y i,r is the fundamental class of the fiber over the point corresponding to w i,r [BFN19, §3]. Since the coordinate system is compatible with the above folding, and y 1,0 , y 0,1 for B 2 , G 2 are restriction of appropriateŷ 1,0,...,0 , y 0,1,0,...,0 for A 3 , D 4 , the coordinate system (y 1,0 , y 0,1 , w 1 , w 2 ) for B 2 , G 2 is identified with (y 1,0 , y 0,1 , w 1 , w 2 ). 3(ii). As in [BFN18a, §3(ii)], we may consider the positive part Gr + G of the affine Grassmannian Gr G : the subvariety consisting of (E i , ϕ i ) such that ϕ i extends through the puncture as an embedding Let us consider H G O * (R + ) for the example in §3(i). It is the subalgebra generated by w 1 , w 2 , y a,b with a, b ≥ 0. It is easy to check that it is, in fact, generated by w 1 , w 2 , y 0,1 , y 1,0 , y 1,1 , . . . , y 1,m . We have y 1,b y 0,1 = (w 1 − w 2 )y 1,b+1 for 0 ≤ b ≤ m − 1, and If m = 2, the only nontrivial relation of the latter type is y 1,0 y 1,2 = y 2 1,1 . This coincides with the presentation of the B 2 type zastava space [BDF16,§5.7 If m = 3, we have two more relations y 1,0 y 1,3 = y 1,1 y 1,2 , y 1,1 y 1,3 = y 2 1,2 . We cannot find this presentation of the zastava space for G 2 for degree α 1 + α 2 in the literature. Therefore we include the proof in the appendix A.
Together, we obtain: Remark 3.4. For general m, a complete set of relations of the latter type are as follows:

Slices
Consider an adjoint group G of BCF G type, with fundamental coweights {Λ i } and simple coroots {α i }. Given a dominant coweight λ for G, and a coweight µ such that λ ≥ µ, we define the corresponding generalized affine Grassmannian slice W λ µ as in [BFN19, §2(ii)]. Recall that in the case when µ is itself dominant, W λ µ is isomorphic to an ordinary affine Grassmannian slice in Gr G as defined in [BF14,§2], [KWWY14,§2B].
The proofs of properties of W λ µ , given in [BFN19, §2], work for non simply-laced types. In particular, W λ µ is Cohen-Macaulay, normal, and affine. It has an integrable system Thanks to the analysis in the previous section, we can apply the argument in [BFN19, §3] to symmetrizable cases: Theorem 4.1. Suppose that the valued quiver is of type BCF G, with adjoint group G as above. Then Proof. Since the proofs are essentially the same as in [BFN19, §3], we simply indicate the differences. In both parts we wish to appeal to [BFN18a, Thm. 5.26]: in a certain precise sense, it suffices to identify the varieties in codimension 1. This result generalizes to our present setting, with the same proof.
For (1),(2) we follow the proof of [BFN19, Thm. 3.1, 3.10]. Using the same notation, the only difference comes when comparing the varieties in the case when t lies on a diagonal divisor (w i,r − w j,s )(t) = 0 where i = j. In our present BCFG setting, we may meet factors of open zastava'sZ α 1 +α 2 of type B 2 , G 2 in addition to the usual A 1 × A 1 and A 2 types already discussed in [BFN19, Rem. 2.2]. In these new cases we apply Lemma 3.2 to complete the proof.

Quantization
In this section, we connect the deformed algebra Then the equation z Therefore we have an induced C × -action on R.

5(ii).
Embedding into the ring of difference operators. Consider a valued quiver along with vector spaces V = V i and W = W i as above. Consider the deformed algebra where we define an algebra For the homology classes of R associated to preimages R λ of closed G O -orbits, we can explicitly write down the image under the map z −1 (ι * ) −1 , following [BFN19, Proposition A.2]. Let λ be a miniscule dominant coweight, W λ ⊂ W its stabilizer, and f ∈ C[t] W λ . Then where e λ denotes the Euler class of the fiber of T over λ modulo the fiber of R over λ . Following [BFN19, Section A(ii)], we will compute these classes for the cocharacters i,n and * i,n of GL(V i ). We find Here α ∨ i denotes the simple root for i ∈ I. The properties of Y µ established in [FKP + 18] have straightforward extensions to all finite types. In particular, Y µ has a PBW basis, and for any coweights µ 1 , µ 2 with µ = µ 1 + µ 2 there is a filtration F • µ 1 ,µ 2 Y µ of Y µ . The associated graded gr Fµ 1 ,µ 2 Y µ is commutative, and the Rees algebras Rees Fµ 1 ,µ 2 Y µ are all canonically isomorphic as algebras (although not as graded algebras). For the purposes of this paper, we will choose µ 1 , µ 2 as follows: We write Y µ := Rees Fµ 1 ,µ 2 Y µ for the corresponding Rees algebra, which we view as a graded algebra over C[ ] with deg = 1.
Below, we work with the larger algebra Y µ [t 1 , . . . , by placing all t i in degree 1. We denote the corresponding Rees algebra by Y µ [t 1 , . . . , t N ]. Denote and define elements A i t −p 5(iv). A representation using difference operators. Recall the C[ , t 1 , . . . , t N ]-algebra A defined in §5(ii). This algebra has a grading, defined by deg = deg t k = deg w i,r = 1 and deg u ±1 i,r = 0. Denote The following result is a common generalization of [BFN19, Corollary B.17] and [KWWY14, Theorem 4.5], which were in turn generalizations of work of Gerasimov-Kharchev-Lebedev-Oblezin [GKLO05].
Theorem 5.4. There is a homomorphism of graded C[ , t 1 , . . . , t N ]-algebras In simply-laced type, a proof of this theorem was given in [BFN19, §B(iii)-B(vii)]. In all finite types, a geneneralization of this theorem for shifted quantum affine algebras was proven in [FT19]. We thus omit the proof.

5(v).
Relation to the quantized Coulomb branch. Consider the setup of §5(ii), restricted to BCFG type. Recall that in this case g ij = 1 and thus f ij = |c ij |, whenever c ij < 0. With this in mind, we see that the right-hand sides of equations (5.1), (5.2) for n = 1 are nearly identical to the images Φ λ µ (F i ) from the previous theorem, modulo shifts by in their respective numerators.
Choose σ i ∈ Z for each i ∈ Q 0 , which solve the following system of equations: for each h ∈ Q 1 , we require that Since (Q 0 , Q 1 ) is an orientation of a tree, a solution exists and is unique up to an overall additive shift. However, in general these equations depend upon the choice of orientation of the Dynkin diagram.
Theorem 5.6. Fix integers σ i satisfying (5.5). Then there is a unique graded C[ , t 1 , . . . , Remark 5.7. The integers σ i play the role of a "shift" in the action of the loop rotation from [BFN18a, Section 2(i)], where the loop C × also acts on N by weight 1/2. Indeed, in our present setting we could modify the loop action of C × from §5(i), so that it also scales V i , W i with weight σ i . (Thus when acting on R, in addition to rotating the discs D i , τ ∈ C × scales the morphism s ij by τ σ i −σ j , and scales s i by 1). With this modified action, no shifts by σ i would be needed in the statement of the theorem. Note that since this modified C × -action factors through the usual action of G × C × , the modified algebra is isomorphic to the original (c.f. [BFN18a, Remark 3.24(2)]).
Proof of Theorem 5.6. We may argue using the previous remark, and modify the loop C × -action while preserving the algebra A up to isomorphism. We give an equivalent elementary argument: Consider the automorphism σ of A defined by w i,r → w i,r + σ i and t k → t k + σ i k , while fixing the generators , u ±1 i,r . We claim that in A we have equalities i }, and where y ∈ A is the claimed image Φ The elements A as a Poisson algebra, under the Poisson bracket {a, b} = 1 (ab − ba). Since A is almost commutative, it is closed under Poisson brackets. It follows that there is a containment of graded C[ , t 1 , . . . , t N ]-algebras Since σz * (ι * ) −1 : A → A is an embedding, the homomorphism Φ λ µ exists as claimed.
The image of Φ λ µ is called the truncated shifted Yangian, and is denoted by Y λ µ . We now give a generalization of [BFN19, Corollary B.28] and [Wee19, Theorem A] to BCFG types: Theorem 5.8. For any λ ≥ µ we have an isomorphism Y λ µ = A , and in particular is injective by definition, so we must prove surjectivity. When µ is dominant, this follows exactly as in the proof of [BFN19, Corollary B.28]. To extend to case of general µ, we follow the same strategy as the proof of [Wee19, Theorem 3.13]. First, we note that one can define shift homomorphisms for Y µ [t 1 , . . . , t N ] and A , which are compatible as in [Wee19, Lemma 3.14]. Second, we claim that A is generated by its subalgebras A ± corresponding to the loci R ± lying over the positive and negative parts of the affine Grassmannian (c.f. §3(ii)). Assuming this claim for the moment, the proof of [Wee19, Theorem 3.13] now goes through.
To prove the claim about generators, consider the semigroups of integral points in chambers of the generalized root hyperplane arrangement for A (see [BFN18a,Definition 5.2]). The hyperplanes in our situation are of three types: (i) w i,r − w i,s = 0 for all i ∈ I and 1 ≤ r, s ≤ v i , (ii) f ji w i,r − f ij w j,s = 0 for any c ij = 0 and 1 ≤ r ≤ v i , 1 ≤ s ≤ v j , and (iii) w i,r = 0 for any W i = 0 and 1 ≤ r ≤ v i . Even if W i = 0, we can always refine our arrangement by adding all hyperplanes w i,r . In this refined arrangement, any chamber is the product of its subcones of positive and negative elements. Thus we can choose generators for its semigroup of integral points which are each either positive or negative. Since the spherical Schubert variety through a positive (resp. negative) coweight lies inside Gr + (resp. Gr − ), we can lift the above semigroup generators to algebra generators for A which each lie in one of A ± . This proves the claim.
We give an explicit presentation of the coordinate ring of the zastava Z α 1 +α 2 of type G 2 , thought of as a variety over a field of characteristic zero (for simplicity, we will simply work over C). This presentation is similar to those for other rank 2 types given in [BDF16,].
Denote by g the Lie algebra of type G 2 , and write V (λ) for its irreducible representation of highest weight λ. Following the notation [FH91, Table 22.1], we pick a basis for the adjoint representation: Here X i , H i , Y i with i = 1, 2 are the Chevalley generators with respect to the Cartan matrix 2 −3 −1 2 . Note this is the transpose of the convention taken in §3(ii). We define X 4 ], X 6 = −[X 2 , X 5 ] and similarly for the Y i (but with opposite signs). In particular, X 6 is a highest weight vector. Following [FH91,pg. 354], we also pick a basis for the first fundamental representation: where V 4 is a highest weight vector and Using the above notation, recall that Z α 1 +α 2 has a description as Plücker sections [FM99, Section 5]: it is the space of pairs v i ∈ V ( i )[z] for i = 1, 2 such that (a) the coefficient of V 4 in v 1 (resp. X 6 in v 2 ) is monic of degree one, (b) the coefficients of all other basis vectors have degree zero, and (c) certain Plücker-type relations must hold (see the proof below for certain cases).
Consider an arbitrary pair v i ∈ V ( i )[z] for i = 1, 2 satisfying the degree requirements a), b) above. Using Sage, we compute the ideal defined by the above eigenvalue conditions for Ω 2 applied to v i ⊗ v j where 1 ≤ i ≤ j ≤ 2. We find that this ideal has two primary components, which have dimensions 4 and 1, respectively. Since Z α 1 +α 2 is a 4dimensional irreducible closed subscheme living inside the vanishing locus of this ideal, it must correspond to the 4-dimensional primary component. This yields the description claimed above.
Remark A.2. Comparing with §3(ii) in the case m = 3, we can identify the above coordinates with the generators of the Coulomb branch as follows: Remark A.3. To match the proposition with the conventions of [BDF16, Section 5.8], we take w i = −A 1 , w j = −A 2 , y i = b 0 and y j = −b 1 (we add overlines to avoid confusion with our notation for Coulomb branches). The equation of the boundary of Z α 1 +α 2 is then This is consistent with our comparison with the open zastava from §3(i): by the previous remark b 4 = y 1,3 , which is invertible in H G O * (R). It is also easy to see that H G O * (R) is generated by the inverse element y −1,−3 together with H G O * (R + ), as expected.

Appendix B. Fixed point sets
Consider the category C of finitely generated right modules of the quantized Coulomb branch H G O C × * (R) such that (1) ∈ H * C × (pt) acts by a nonzero complex number, say 1, and (2) it is locally finite over H * G (pt), hence it is a direct sum of generalized simultaneous eigenspaces of H * G (pt). (When we include an additional flavor symmetry, we assume that the corresponding equivariant parameter acts by a complex number.) One can apply techniques of the localization theorem in equivariant (K)-homology groups of affine Steinberg varieties in [VV10] to study the category C. This theory, for the ordinary Coulomb branch, will be explained elsewhere [Nak19]. (See also [Web16,Web19] for another algebraic approach different from one in [VV10].) It also works in our current setting. As a consequence, we have for example Theorem B.1. Let λ ∈ t. There is a natural bijection between • simple modules in C such that one of eigenvalues above is given by evaluation Here t is the Lie algebra of a maximal torus of G, W is the Weyl group of G, and (λ, 1) is the element of the Lie algebra of T × C × , which acts on T and N K , as a subgroup of G O C × . Fixed point subsets are written as T (λ,1) , N (λ,1) K , and the projection is the restriction of Π : T → N K .
We study the fixed point set T (λ,1) , N (λ,1) K in this section. For simplicity, we assume λ is a differential of a cocharacter, denoted by the same symbol λ. (See Remark B.3 for general case.) Therefore we study the fixed point set with respect to a one parameter subgroup τ → (λ(τ ), τ ). B(i). Consider the affine Grassmannian Gr G . We have an action of G O C × on Gr G given by (h(z), τ ) · [g(z)] = [h(z)g(zτ )]. Take a cocharacter λ : C × → T and consider a homomorphism To see this let us identify Gr G with ΩG c the space of polynomial based maps (S 1 , 1) → (G c , 1), where G c is a maximal compact subgroup of G. Then g ∈ ΩG c is fixed if and only if λ(τ )g(zτ )g(τ ) −1 λ(τ ) −1 = g(z). It means that z → λ(z)g(z) is a group homomorphism.
Then g(z = 1) is well-defined and g(z) = z −λ g(z = 1)z λ , hence Z G K (λ(τ ), τ ) ∼ = G via g(z) → g(1). (We switch the notation from λ(z) to z λ .) Then the fixed point set is where µ is a dominant coweight of G, and z −λ+µ is regarded as a point in Gr G . The Z G K (λ(τ ), τ )-orbit through z −λ+µ is a partial flag variety G/P µ , where P µ is the parabolic subgroup corresponding to µ.
B(ii). More generally consider a homomorphism τ → (λ(τ ), τ m ) for m ∈ Z >0 . We suppose G = GL(V ) and decompose V = V (k) so that λ(τ ) acts on V (k) by τ k id V (k) . We consider k modulo m and decompose V as Here λ k (z) is defined so that it acts by z (l−k)/m on V (l). It is proved as follows. Take a based loop model g ∈ ΩG c . It is fixed if and only if λ(τ )g(zτ m )g(τ m ) −1 λ(τ ) −1 = g(z). Taking τ = ω, a primitive m-th root of unity, we see that g(z) preserves the decomposition Hence λ k (z)g k (z) is a group homomorphism, which we denoted by ϕ k (z).
= ϕ 1 ⊕ · · · ⊕ ϕ m . The connected component of Gr Note that the decomposition V = V {1} ⊕ · · · ⊕ V {m} and the group G depends on the choice of λ. If we take λ = 1 for example, we have V = V {m} and G = G.
Alternative description is as follows: Let It is isomorphic to G by g(z) → g(z = 1) ∈ G . Then the fixed point set is where µ is a dominant cocharacter of G , and the orbit Z G K (λ(τ ), τ m )·[z −λ +µ ] is isomorphic to the partial flag variety G /P µ .
Remark B.3. More generally we could study the fixed point set with respect to a cocharacter τ → (λ 1 (τ ), λ 2 (τ ), τ d ) for d ∈ Z >0 . But the fixed point set will be just the union of d copies of the fixed point set below, hence it does not yield a new space. On the other hand, this modification yields a new space when a quiver has a loop. See [VV10].

Then we have a natural bijection between
• simple objects in the category C such that their eigenvalues are evaluations at cocharacters of T , • canonical base elements of weight − dim(V i {k})α i k in the lower triangular part U − q of the quantized enveloping algebra of type ADE. Here i runs over the set of vertices of the original quiver, and k runs from 1 to d i . Concretely the correspondence between types is B n → A 2n−1 , C n → D n+1 , F 4 → E 6 , G 2 → D 4 .
Remark B.7. Note that the canonical base elements in the above theorem are in bijection also to simple objects in the category C (with the same constraint) of the quantized Coulomb branch of type ADE by the same analysis of the fixed point set as above. Recall that the quantized Coulomb branch A is a quotient of the shifted Yangian of type BCF G or ADE, the same type as quiver. Therefore we have a bijective correspondence between simple modules in quotients of shifted Yangian of type BCF G and of ADE. This result reminds us the result of Kashiwara, Kim and Oh [KKO19], where a similar bijection was found between simple finite dimensional modules of quantum affine algebras of types B n and A 2n−1 .

Appendix C. A second definition
In this section we present a second possible definition for a Coulomb branch associated to a quiver gauge theory with symmetrizers. In the case when the Cartan matrix satisfies assumption (2.2), this second definition agrees with that given in §2. But in general this is not the case. We note that this second definition applies to theories which are not of quiver type.
Similarly there are maps between the corresponding formal punctured discs, which we also denote ρ k| : D * k → D * by abuse of notation. These maps are equivariant for the C × -action by loop rotation, τ : N • ), consisting of G • = d k=1 G k a product of complex connected reductive groups, and N • = d k=1 N k a direct sum of complex finitedimensional representations of G • . In addition, we assume that G k acts trivially on N j , unless j|k.
Given such a pair (G • , N • ), we define R G•,N• to be the moduli space of triples (P • , ϕ • , s • ), where P • = (P 1 , . . . , P d ), ϕ • = (ϕ 1 , . . . , ϕ d ), and s • = (s 1 , . . . , s d ) satisfy (a) P k is a principal G k -bundle over D k , (b) ϕ k is a trivialization of P k over D * k , (c) s k is a section of the associated bundle such that it is sent to a regular section of the trivial bundle under the trivialization k| ρ * k| ϕ over D * k . As usual we also define a larger moduli space T G•,N• by dropping the extension conditions in (c).
The group acts on R G•,N• by changing ϕ • . There is also an action of C × , acting by loop rotation of the discs D k as in the previous section. We can define a convolution product on H G •,O * (R G•,N• ) just as in [BFN18a]. By the argument in 2(iv), it is a commutative ring, and we define the Coulomb branch It has a deformation quantization defined by H G •,O C × (R G•,N• ), and in particular a Poisson structure.
The arguments from cite [BFN18a] apply with small modifications to M C (G • , N • ). In particular it is finite type, integral, normal, and generically symplectic. One useful observation in modifying the proofs is the following: Remark C.2. Suppose that G • = G consists of a single factor, and define its representation N = k| N ⊕( /k) k . Then M C (G • , N • ) is isomorphic to the usual Coulomb branch M C (G , N ) as defined in [BFN18a]. This comes from the fact that there is an isomorphism C(iii). The quiver case. As in §2(i), consider a valued quiver associated to a symmetrizable Cartan matrix (c ij ) i,j∈I . Also choose symmetrizers (d i ) ∈ Z d >0 . Recall that we denote g ij = gcd(|c ij |, |c ji |), f ij = |c ij |/g ij when c ij < 0. It is not hard to see that d i must be a multiple of f ji for any c ij < 0, so we may define integers d ij by the rule d i = d ji f ji . They satisfy d ij = d ji .
Remark C.3. In fact, lcm(d i , d j ) = d i f ij = d j f ji and gcd(d i , d j ) = d ij = d ji .
Choose vector spaces V i and W i for each i ∈ I. Given these choices, we define a pair (G • , N • ) according to the following rules: Then N • is a representation of G • in the natural way, and satisfies our assumption from the beginning of the previous section. By tracing through the definition one can see that the moduli space R G•,N• parametrizes: • a rank v i vector bundle E i over D d i together with a trivialization ϕ i : where c ij < 0 and there is an arrow j → i in the quiver. C(iv). Comparison. We now compare with the construction from §2(ii). For this it suffices to understand the case of a single edge j → i. As explained in Section 5(i), we can C × -equivariantly identify D i ∼ = D d i via z i → x d i , D j ∼ = D d j via z j → x d j , and D ∼ = D d i f ij = D d j f ji via z → x d i f ij . We also denote D = D d ij = D d ji . Then there are commutative diagrams of discs and their corresponding rings, as in [GLS18d, §4.2] 3 : Both squares are Cartesian, while the inclusion ] is of finite codimension over C. We also note that C For brevity, let us denote the covering maps ρ ij = ρ d ij |d j : D → D j and ρ ji = ρ d ji |d i : D → D i . Then the difference between the two constructions from §2(ii) and §C(iii) is simply in the definition the section s ij : whether it lies in (C.6) C g ij ⊗ C Hom O D (π ij * E j , π ji * E i ) or C g ij ⊗ C Hom O D (ρ * ij E j , ρ * ji E i ) We now reformulate both sides in terms of the above power series rings, ignoring the tensor product with C g ij in each case. Denote by E i the C[[x d i ]]-module corresponding to E i , and by E j the C[[x d j ]]-module corresponding to E j . Then on the one hand, the left side of (C.6) corresponds to  Thus we see that the difference between the two sides of (C.6), and thus between our two constructions, is captured by the finite codimension inclusion of rings Note that this map is an isomorphism if and only if f ij = 1 or f ji = 1.
Theorem C.7. For a general valued quiver, if the assumption (2.2) holds then our constructions from §2(ii) and §C(iii) are isomorphic. In particular, this is the case in all finite types. C(v). Twisted monopole formula. The previous section shows that the twisted monopole formula applies to R G•,N• in the case when assumption (2.2) holds. But in fact it is not hard to see that the twisted monopole formula is valid for R G•,N• even when this assumption does not hold. More precisely, Proposition 2.3 is valid for R G•,N• in all types, with the same expression for d λ from §2(iii).
The twisted monopole formula is related to the following generalization of the calculations from §3: for an arbitrary rank 2 Cartan matrix we find that z * (w 1 ) = w 1 , z * (w 2 ) = w 2 , z * (y a,b ) = (w 1 − w 2 ) g 12 ·max(f 12 b−f 21 a,0) u a,b .
Indeed, the fiber of T G•,N• over (a, b) ∈ Gr G = Z 2 is while the fiber of R G•,N• is its intersection with C g 12 ⊗ C Hom C (V 2 , V 1 )[[x d 12 ]]. The contribution to z * (y a,b ) above is the Euler class of the quotient, recalling that d 1 = d 12 f 21 and d 2 = d 12 f 12 .