Fundamental monopole operators and embeddings of Kac-Moody affine Grassmannian slices

Braverman, Finkelberg, and Nakajima define Kac-Moody affine Grassmannian slices as Coulomb branches of $3d$ $\mathcal{N}=4$ quiver gauge theories and prove that their Coulomb branch construction agrees with the usual loop group definition in finite ADE types. The Coulomb branch construction has good algebraic properties, but its geometry is hard to understand in general. In finite types, an essential geometric feature is that slices embed into one another. We show that these embeddings are compatible with the fundamental monopole operators (FMOs), remarkable regular functions arising from the Coulomb branch construction. Beyond finite type these embeddings were not known, and our second result is to construct them for all symmetric Kac-Moody types. We show that these embeddings respect Poisson structures under a mild"goodness"hypothesis. These results give an affirmative answer to a question posed by Finkelberg in his 2018 ICM address and demonstrate the utility of FMOs in studying the geometry of Kac-Moody affine Grassmannian slices, even in finite types.

1. Introduction 1.1.Kac-Moody Affine Grassmannian slices.Braverman, Finkelberg, and Nakajima [BFN19] proved the spectacular result that affine Grassmannian slices of finite ADE type arise as Coulomb branches of 3d N " 4 quiver gauge theories.These slices W λ µ model the singularities between spherical Schubert varieties in the affine Grassmannian, and thereby encode essential aspects of the geometric Satake correspondence.Here λ and µ are coweights with λ dominant and λ ě µ.
Remarkably, the Braverman-Finkelberg-Nakajima construction works for arbitrary quivers.One can thus take the corresponding Coulomb branches as a definition of Kac-Moody affine Grassmannian slices W λ µ for general symmetric Kac-Moody types.Starting with the work of Braverman and Finkelberg [BF10], these slices are proposed as a fruitful setting for developing the geometric Satake correspondence for Kac-Moody groups [Nak09,BF12,BF13,BFN19,Fin18,Nak18].
The Coulomb branch is defined as the spectrum of a certain convolution algebra (see §3.1 and the references within).Despite the abstract definition, the construction is well suited for deducing algebraic properties of Kac-Moody affine Grassmannian slices (via general results for Coulomb branches from [BFN18]).However, the geometry of these varieties remains quite mysterious.For example, in Finkelberg's 2018 ICM address [Fin18, §5.6 and §6.4], he sought certain embeddings W λ 1 µ ãÑ W λ µ for general Kac-Moody types.These embeddings are well-known in finite types, but even in this case the relationship with Coulomb branches has been mysterious (see §1.2 below for more detail).
The goal of this paper is to remedy this situation.In finite types, we show that the above closed embeddings respect certain functions called fundamental monopole operators which arise naturally in the Coulomb branch picture (see Theorem 1.2).For general symmetric Kac-Moody type we exhibit closed embeddings W λ 1 µ ãÑ W λ µ (see Theorem 1.4), and study their interaction with Poisson structures (see Theorem 1.6).
1.2.Embeddings of slices in finite type.Let us restrict our attention to finite ADE types.One geometric phenomenon that is obvious from the loop group-theoretic picture of affine Grassmannian slices is the fact that slices embed into one another: for λ 1 another dominant coweight with λ ě λ 1 ě µ, there is a closed embedding: For readers familiar with affine Grassmannians, this is analogous to and indeed a direct consequence of the fact that the spherical Schubert variety Gr λ 1 embeds into Gr λ .This closed embedding, which is a basic feature of the geometry, is mysterious from the perspective of Coulomb branches.Moreover, any Coulomb branch construction of this embedding must necessarily be subtle: all of the basic geometric operations on the convolution algebra defining a Coulomb branch lift to its quantization, but algebraically one can easily see that the closed embedding (1.1) need not lift to the quantization (see Remark 4.60).
1.3.Fundamental monopole operators.Our first goal is to understand how the closed embedding (1.1) interacts with the Coulomb branch descriptions of both sides.To this end, we study its effect on fundamental monopole operators (FMOs).The FMOs are regular functions on W λ µ that are natural from the Coulomb branch perspective, corresponding to certain algebraic cycles in the convolution algebra defining the Coulomb branch.A priori, these functions have nothing to do with the closed embedding (1.1).Nonetheless, we prove the following result: Theorem 1.2.Under the closed embedding (1.1), FMOs map to FMOs or to zero.
See Theorem 2.28 in the main text for the precise formula.By work of the second author [Wee19], the FMOs generate the coordinate ring of W λ µ , so the closed embedding (1.1) is completely characterized by this theorem.
There are explicit rational expressions (see (2.18)) for the FMOs, which arise from the localization theorem.Unfortunately, the denominators in these rational expressions vanish on the image of the closed embedding (1.1), so it is not clear how they restrict.To prove the theorem, we move the problem to studying the "adding defect" map of zastava spaces (see §2.5) where the denominators can be controlled.1.4.Embeddings of slices in Kac-Moody type.Theorem 1.2 suggests a natural guess for how to define a closed embedding of Kac-Moody affine Grassmannian slices: we should map FMOs using the same formula.Unfortunately, we do not know the relations among the FMOs, so it is unclear that such a map extends to an algebra homomorphism.Indeed, this is an important open problem.
Problem 1.3.Describe a complete set of relations among the fundamental monopole operators.
If we had a solution to this problem, we could proceed by simply stating where FMOs map and checking relations.Instead, we construct the map another way, which is our second main result.
Theorem 1.4.Suppose further that λ 1 satisfies a certain "conicity condition" with respect to λ.Then there is a closed embedding of Kac-Moody affine Grassmannian slices (1.5) under which the FMOs map to FMOs or to zero.
See Theorem 4.9 and 4.42 for precise statements.We mention that the conicity condition (see Definition 4.1) is automatic in finite-type, and it holds in affine type for all non-zero levels, which are the cases of primary interest (see Theorem A.8).We also note that Theorem 1.4 does not subsume Theorem 1.2.We only know that the two maps agree in finite type because of our result showing that FMOs map the same way in both cases.
To construct the map in the theorem we make use of several natural Coulomb branch operations from [BFN18].However, we require an additional map.The coordinate rings of Coulomb branches carry gradings corresponding to the monopole formula, and the conicity condition guarantees that W λ λ 1 is a cone with respect to this grading (see Lemma A.4).We require the embedding of the cone point into this cone.We do not know of any natural Coulomb branch construction of the cone point (e.g. it does not lift to the quantization in general) but we are able to proceed by simply appealing to the grading.Finally, we verify that the above map has the correct behavior on FMOs.Also, our map is a priori only rationally defined, and only because of our FMO calculation do we conclude that the map is regular.
Theorems 1.2 and 1.4 show that the geometry of Kac-Moody affine Grassmannian slices is closely coupled with the FMOs, and we propose that they should be standard tools for future investigation.In particular, understanding Problem 1.3 is crucial.
1.5.Symplectic leaves and Poisson subvarieties.In [Wee22], the second author proved that Coulomb branches of quiver gauge theories have symplectic singularities for all quivers with no loops and no multiple edges.By work of Kaledin [Kal06], we conclude that these Coulomb branches have finitely many closed Poisson subvarieties, and their symplectic leaves are obtained precisely as the smooth loci of irreducible closed Poisson subvarieties.We prove the following.
Theorem 1.6 (Theorem 4.58 in the main text).Assume that λ 1 satisfies a certain "good condition" for λ.Then the closed embedding is compatible with Poisson structures.In particular, the inclusion of the smooth locus pW λ 1 µ q reg ãÑ W λ µ defines a symplectic leaf.
We mention that the good condition (see Definition 4.55) is automatic in finite-type, and it holds in affine type for all levels greater than or equal to two (see Theorem A.8).
In our previous work [MW19], we constructed all symplectic leaves (and therefore all closed Poisson subvarieties) for finite-type slices, and indeed they are exactly the closed embeddings we study here.However, in infinite Kac-Moody type it is known that there are further closed Poisson subvarieties.
For an arbitrary Coulomb branch, Nakajima has made precise predictions for the enumeration of symplectic leaves based on symplectic duality [Nak15,§2].For quiver gauge theories -and thus for Kac-Moody affine Grassmannian slices -this enumeration should be dual to the enumeration of symplectic leaves for Nakajima quiver varieties.A complete enumeration of the latter is given by Bellamy and Schedler [BS21, Theorem 1.9].It would be very interesting to generalize Theorems 1.4 and 1.6 to obtain all of the "dual" symplectic leaves on the affine Grassmannian side.1.6.Previous work in affine type A. In addition to finite type, there is another important case where the geometry of affine Grassmannian slices is well understood: affine type A. In this case Nakajima and Takayama [NT17] prove that the Kac-Moody affine Grassmannian slices are isomorphic to Cherkis bow varieties.In particular, they obtain an explicit geometric invariant theory description of these varieties and from that a full description of their symplectic leaves, transversal slices, and torus action.In later work [Nak18], Nakajima has proved further geometric facts about bow varieties and has succeeded in generalizing the geometric Satake correspondence in this setting by geometrically constructing the irreducible representations of affine type A.
From Nakajima and Takayama's work, we see that our construction recovers many, but not all, of the symplectic leaves [NT17, Theorem 7.26].Furthermore, in describing symplectic leaves, they discover a distinction between level one and levels greater than one.In particular, the good hypothesis in Theorem 1.6 is sharp in this setting (cf.Theorem A.8). 1.7.Relation to physics.Coulomb branches of 3d N " 4 gauge theories originate in theoretical physics, see for example [Nak16] and references therein.In addition to these physical origins, an important physically-motivated ingredient in the present work is the monopole formula [CHZ14], which encodes the Hilbert series of the coordinate ring of a Coulomb branch with respect to a certain natural grading.See Appendix A for an overview.In particular, the "good condition" from Theorem 1.6 corresponds to being a good theory in physics terminology, while the "conicity condition" from Theorem 1.4 corresponds to a good or ugly theory.
The study of symplectic leaves of Coulomb branches is also of physical significance, see for example the works of Hanany and collaborators [BCG `20, GH20, CH18].
1.8.Outline of the paper.In §2, we prove Theorem 2.28: closed embeddings of finite-type slices are compatible with FMOs.Aside from the formula for the FMOs, Coulomb branches do not enter the discussion in this section.In §3, we recall the fundamentals of Coulomb branches and the definition of Kac-Moody affine Grassmannian slices as Coulomb branches of quiver gauge theories.In §4, we construct the embeddings of Kac-Moody affine Grassmannian slices and prove Thoerem 4.9.We also explain in §4.3 about compatibility with Poisson structure.Finally, in Appendix A we overview the monopole formula and its consequences for quiver gauge theories.1.9.Notation.For a positive integer v, we write rvs " t1, ¨¨¨, vu.We will work with schemes and ind-schemes over a field k.By a variety we mean a integral scheme of finite-type over k.Let I be a finite set (later it will be the vertex set of a quiver), and let Z I denote the set of I-tuples of integers.We write 0 P Z I for the I-tuple consisting of all zeros.Given two elements m " pm i q iPI and v " pv i q iPI of Z I , we write m ď v to mean m i ď v i for all i P I. Given v P Z I , we denote by S v " ś iPI S v i the corresponding product of symmetric groups.
1.10.Acknowledgements.D.M. was supported by JSPS KAKENHI Grant Number JP19K14495.A.W. was supported by an NSERC Discovery Grant.We are grateful to the organizers of the conference "Bundles and Conformal Blocks with a Twist" which took place in June 2022 at the ICMS where some of this work was completed.

Affine Grassmannian slices
Let G be a split semisimple simply-laced and simply-connected group with a fixed pinning.We make the simply-connectedness assumption so that G has all fundamental representations, but this is largely not necessary (see Remark 2.10 below).The pinning determines a pair B `and B óf opposite Borel subgroups.Let U `and U ´be the unipotent radicals of B `and B ´, and let T " B `X B ´, a maximal torus.
Let I denote the vertices of the Dynkin diagram.Let tα i u iPI denote the set of simple roots, and tΛ i u iPI denote the set of fundamental weights.Let P denote the weight lattice of T , and P ``the set of dominant weights.Let tα _ i u iPI denote the set of simple coroots, and let Q _ `" À iPI Z ě0 α _ i denote the positive coroot cone.Let P _ denote the coweight lattice of the torus T , and let P _ `denote the set of dominant coweights.Finally, recall that the dominance order on P _ is defined by λ ě µ iff λ ´µ P Q _ `.
For each dominant weight Λ P P ``, let Wp´Λq denote the Weyl module with lowest weight ´Λ, and let Sp´Λq denote the Schur module with lowest weight ´Λ.We interpret both of these as left representations of G, and we interpret their vector-space duals as right representations.Write Vp´Λq for a representation that is either Wp´Λq or Sp´Λq, but where we have not specified which.The ´Λ weight space Vp´Λq ´Λ is one dimensional, and we fix a generator |v ´Λy.Let xv ˚Λ| be the unique weight covector that pairs to 1 with |v ´Λy.We use the "bra" and "ket" notation to emphasize on which side G acts.The pinning on G determines, for each i P I, a fixed choice of |v ´si pΛq y P Vp´Λq ´si pΛq and unique weight covector xv ˚si pΛq | that pairs to 1 with |v ´si pΛq y.
2.0.1.Colored divisors.Let γ P Q _ `, which we may write as γ " ř iPI v i α _ i where v i " xγ, Λ i y.Write A pγq for the variety of I-colored divisors of degree γ on A 1 .Explicitly, points of A pγq are given by I-tuples of monic polynomials pL i pzqq iPI such that L i pzq has degree v i for each i P I.The non-leading coefficients of L i pzq are regular functions on A pγq , and the coordinate ring krA pγq s is a polynomial ring in these non-leading coefficients.2.0.2.Quiver orientation.The definition of fundamental monopole operators will require us to orient the edges of the Dynkin diagram.We choose a quiver pI, Eq whose underlying unoriented graph is our Dynkin diagram.Here I is the vertex set of the quiver, and E is the arrow set.On the arrow set E, we have source and target maps s, t : E Ñ I.
2.1.Affine Grassmannian slices.We will follow the notation in [MW19] closely.Let Gppz ´1qq denote the Laurent series loop group of G (in z ´1), and let Grzs denote the subgroup of positive loops.We have an embedding P _ ãÑ Gppz ´1qq of the coweight lattice, denoted by µ Þ Ñ z µ .For any affine algebraic group H we may also consider the subgroup Hrrz ´1ss Ă Hppz ´1qq of negative loops.Denote by H 1 rrz ´1ss the kernel of the "evaluation at z " 8" map to H, i.e. we have the short exact sequence of groups: For the rest of this section, fix coweights λ P P _ ``and µ P P _ such that λ ě µ.Define elements v " pv i q iPI and w " pw i q iPI of Z I by v i " xλ ´µ, Λ i y (2.2) for all i P I.Note that all w i , v i ě 0, and that λ ´µ " (2.4) Consider also the closed sub-ind-scheme X λ " Grzsz λ Grzs Ă Gppz ´1qq, defined as the preimage of the Schubert variety Grzsz λ Grzs{Grzs Ă Gppz ´1qq{Grzs in the thick affine Grassmannian under the natural map Gppz ´1qq Ñ Gppz ´1qq{Grzs.
The affine Grassmannian slice is defined by We will often write xaz µ y P W λ µ for elements where x P U 1 rrzss, a P T 1 rrzss, and y P U 1 rrz ´1ss.Finally, we write Apλ, µq " krW for the coordinate ring of W λ µ .
Remark 2.7.When µ is dominant, the variety W λ µ embeds into the affine Grassmannian of G, providing a transverse slice between spherical Schubert varieties [KWWY14].This motivates the name "affine Grassmannian slice".Note that when µ is not dominant W λ µ is called a generalized affine Grassmannian slice in [BFN19].In this paper, we have elected to drop the word "generalized".
It is easy to see that we have a closed embedding X λ ãÑ X λ that is a bijection on points, i.e.X λ is a possibly non-reduced thickening of X λ .Define: Then there is a closed embedding W λ µ ãÑ W λ µ that is a bijection on points.Because W λ µ is known to be reduced [BFN19, §2(ii)], we could alternatively define W λ µ as the reduced scheme of W λ µ .2.2.Fundamental monopole operators.
2.2.1.Birational coordinates.Recall from (2.2) that v i " xλ ´µ, Λ i y.For any g P Gppz ´1qq, we can consider z xλ,Λ i y xv ˚Λi |g|v ´Λi y, which is a Laurent series in z ´1.If we further restrict that g " xaz µ y P W λ µ , then z xλ,Λ i y xv ˚Λi |g|v ´Λi y is a monic polynomial in z of degree v i , which we denote Q i pzq and think of as a regular function on W λ µ (with values in monic polynomials of degree v i ).Observe that Q i pzq depends only on the a-factor of g " xaz µ y.We obtain a map W λ µ Ñ A pλ´µq given sending a point of W λ µ to the colored divisor pQ i pzqq iPI .Similarly we define P i pzq " z xλ,Λ i y xv ˚si pΛ i q |xaz µ y|v ´Λi y.This is a (not necessarily monic) polynomial of degree less than or equal to v i ´1, and we think of P i pzq as a regular function on W λ µ .Observe that P i pzq depends only on the x and a-factors of g " xaz µ y.Finally, it is known that the coefficients of the Q i pzq and P i pzq for i P I form a system of birational coordinates on W λ µ [FKMM99, BDF16], cf.§2.4 below.
Remark 2.10.This is the first time we make use of the simply-connectedness hypothesis, in order to have all the fundamental representations.However, this is largely not needed.Let G 1 be a group with universal cover equal to G. Then U `and U ´are canonically identified with the positive and negative unipotent subgroups of G 1 .If we assume that pchar k, |π 1 pG 1 q|q " 1, then T 1 rrz ´1ss is identified with the corresponding group for G 1 .In particular, because Q i pzq depends only on the a-factor and P i pzq depends only on the a and x-factors of a point xaz µ y P W λ µ , we see that the functions Q i pzq and P i pzq are well-defined in this case as well.
If |π 1 pG 1 q| is divisible by char k, then we suggest to take the Coulomb branch characterization of W λ µ (Theorem 3.20) as a definition, because this definition behaves well over general base rings.

GKLO embedding.
For each i P I, we define variables w i,1 , . . ., w i,v i and u i,1 , . . ., u i,v i and form the ring: We further define r Apλ ´µq loc to be the ring obtained by inverting all polynomials of the form w i,r ´wi,s for i P I and 1 ď r ‰ s ď v i .The group S v acts on this ring by permuting variables.
Recall from (2.6) that we denote the coordinate ring of W pz ´wi,r q (2.14) and for all i P I.
Remark 2.16.The above embedding differs from that in [BFN19] by a sign.More precisely, define an involution on r Apλ ´µq by Composing this involution with the map (2.13), we obtain precisely the h " 0 limit of [BFN19, Theorem B.15], cf.[BFN19, Lemma B.27].We make this change of signs to simplify the statement of certain other results, such as Theorem 2.28 below.The above involution may also be interpreted as the action of a certain element of the torus T Ă G, acting via the adjoint action on W λ µ .
The map from the theorem is a slight variation of one constructed for (quantized) zastava spaces by Gerasimov, Kharchev, Lebedev, and Oblezin [GKLO05].For that reason we will call this the GKLO embedding of Apλ, µq.This map was also studied in [KWWY14], in the case where µ is dominant.Note that because the P i pzq and Q i pzq are birational coordinates on W λ µ , the map is uniquely determined by formulas (2.14) and (2.15).
Consider the subalgebra of Apλ, µq generated by the coefficients of the Q i pzq.Under the GKLO embedding, this subalgebra is identified with the subalgebra krw i,r s S v iPI,rPrv i s Ă r Apλ ´µq of symmetric polynomials.For each i P I, the coefficients of Q i pzq are identified (up to a sign) with the elementary symmetric functions in the variables w i,r for r P rv i s, by (2.14).

Fundamental monopole operators.
For each tuple m " pm i q iPI P Z I with 0 ď m ď v, we write for the corresponding ring of partially symmetric polynomials.Below we will consider sums over tuples Γ " pΓ i q iPI such that Γ i Ď rv i s and #Γ i " m i .For each such tuple Γ , let σ " pσ i q iPI P S v be an element such that σ i prm i sq " Γ i .Given f P Λ v m , define f| Γ " σpfq, which is an element of Â iPI krw i,r : 1 ď r ď v i s.Because f is partially symmetric, f| Γ does not depend on which σ was chosen.Finally, we define u Γ " ś iPI ś rPΓ i u i,r .
Definition 2.18.Let m " pm i q iPI P Z I with 0 ď m ď v, and let f P Λ v m .We define the (positive) dressed fundamental monopole operator M mpfq P r Apλ ´µq loc by the formula: We define the (negative) dressed fundamental monopole operator M ḿpfq P r Apλ ´µq loc by the formula: Remark 2.22.When m " 0, meaning that m i " 0 for all i P I, we interpret Furthermore, for general m we have Λ 0 Ď Λ m , and a linearity property over this subring Λ 0 : For brevity, we will abbreviate "dressed fundamental monopole operator" to FMO.Observe that the FMOs lie in r Apλ ´µq S v loc , so via Theorem 2.12 the FMOs are rational functions on W  The above results come from the mathematical theory of Coulomb branches, which we review in §3, and in particular rely on Braverman, Finkelberg and Nakajima's groundbreaking Theorem 3.20.The FMOs arise very naturally in this setting, see §3.2.3.
Remark 2.25.Up to a sign, the elements M mpfq do not depend on the choice of orientation of the Dynkin diagram.This is explained most naturally in the context of Coulomb branches, see §3.2.3 and equation (3.17) below.

Compatibility of FMOs with inclusions of affine Grassmannian slices. Fix another λ 1 P P _
``such that λ ě λ 1 ě µ.Then there is a closed embedding X λ 1 ãÑ X λ of (preimages of) spherical Schubert varieties; this is easy to see for the spaces X λ from (2.8).Intersecting with the space W µ from (2.4), there is thus a closed embedding: (2.26) Define the tuple v 1 " pv 1 i q iPI by v 1 i " xλ ´λ1 , Λ i y for each i P I. Let m " pm i q iPI P Z I with 0 ď m ď v. Suppose further that m ď v 1 .Then we can define a map by setting the variables w i,r with r ą v 1 i equal to zero for all i P I. We can now state the main theorem of this section.
Theorem 2.28.Let m " pm i q iPI P Z I with 0 ď m ď v, and let f P Λ v m .Under the restriction map of functions krW λ µ s Ñ krW λ 1 µ s corresponding to the closed embedding (2.26), we have In the theorem and in the proof below, we assume that the FMOs on both sides are defined using the same orientation of the underlying Dynkin diagram.But this assumption is ultimately of little importance, as the FMOs are independent of orientation up to a sign by Remark 2.25.

2.3.1.
Examples.Although the definition of a general FMO M mpfq is given by a complicated rational expression, we recover some familiar functions as special cases: pz ´wi,r q ¯, P i pzq " M èi ´vi ź r"2 pz ´wi,r q ¯(2.30) Here e i P Z I denotes the i-th standard basis vector, and we have extended the definition of M mpfq to allow for f P Λ v m rzs.Equation (2.30) follows easily from Theorem 2.12 and Definition 2.18.Similarly, in §2.6 below we define functions P í pzq, which are the images of P i pzq under the Chevalley involution.These functions are also related to FMOs: Remark 2.32.Note that Q i pzq, P i pzq and P í pzq are all expressed in terms of representation-theoretic data: they are matrix coefficients.The same is therefore true of the corresponding FMOs in (2.30) and (2.31), which in particular confirms Theorem 2.23 in these cases.It would interesting to find a representation-theoretic or group-theoretic interpretation for more general FMOs M mpfq.We are not aware of such an interpretation even for G " SL 2 .
2.4.Zastava spaces.Let γ P Q _ `.Define Z γ `to be the following closed subscheme of Gppz ´1qq Z γ `" !xaz ´γ P U 1 rrz ´1ssT 1 rrz ´1ssz ´γ : val `xaz ´γ|v ´Λy ˘ě 0 for all Λ ) (2.33) where Λ P P ``varies over all dominant weights and |v ´Λy P Vp´Λq is the lowest weight vector, and Vp´Λq varies over Wp´Λq and Sp´Λq.It is not clear to us whether Z γ `is a reduced scheme, so we define the zastava space Z γ `to be the reduced scheme of Z γ `.
We have a map Z γ `Ñ A pγq sending the point xaz ´γ to the colored divisor `Qi pzq ˘iPI where each Q i pzq " z xγ,Λ i y xv ´Λi |a|v ´Λi y.
Remark 2.34.The zastava space is often defined as a space of quasimaps from a rational curve into a flag variety.Our definition of the zastava rather is most naturally isomorphic to the equivalent Beilinson-Drinfeld picture of the zastava (see e.g.[FM99,§6]).Given a point xaz ´γ P Z γ `, we extract a colored divisor as above; let us call it D. It is easy to see that xaz ´γ P Z γ `implies that x is in fact a rational function of z regular away from the support of D. So using x ´1, we obtain a rational trivialization on the trivial U-bundle on P 1 , and our valuation condition exactly translates into the required pole conditions along D.
We have a natural map where ˚is the diagram automorphism given by "minus the longest element of the Weyl group".Because G is equipped with a pinning, this diagram automorphism induces an automorphism of G, which induces an isomorphism `is exactly the composed map.
2.4.1.GKLO embedding for the zastava space.Let r A `pλ ´µq " krw i,r , u i,r s iPI,rPrv i s , and define r A `pλ ´µq loc to be the localization at all w i,r ´wi,s for all 1 ď r ‰ s ď v i .Note that in these rings we do not invert the elements u i,r .Then analogously to Theorem 2.12, we have a birational embedding krZ λ´µ `s ãÑ r A `pλ ´µq loc (2.36) sending Q i pzq and P i pzq to the elements given by formulas (2.14) and (2.15).Similarly to (2.6), we will denote A `pλ ´µq " krZ λ´µ `s.
In particular, we have the following commutative square of birational algebra embeddings: Observe that the positive FMOs define rational functions on Z λ´µ `.In fact they are regular functions, a consequence of the construction of the zastava space in Coulomb branch terms [BFN19, §3(ii)], see also §3.2.1 below.An analogue of Theorem 2.24 also holds: the positive FMOs M mpfq generate the ring of regular functions on Z λ´µ 2.4.2.Compatibility with closed embeddings.If γ 1 ď γ are elements of Q _ `, then we have a closed embedding Z γ 1 `ãÑ Z γ `given by xaz ´γ1 Þ Ñ xaz ´γ, which is compatible with the closed embedding of slices.We state this as the following proposition.
We see then that for positive FMOs, Theorem 2.28 reduces to the following analogous theorem for zastava spaces.
`ãÑ Z γ `and the corresponding restriction map krZ γ `s Ñ krZ γ 1 `s of functions.Let m " pm i q iPI P Z I with 0 ď m ď v, and let f P Λ m .Then under the restriction map we have The difficulty with proving Theorem 2.28 is that the FMOs are expressed as rational expressions, and the image of the closed embedding W λ 1 µ ãÑ W λ µ lies in the vanishing locus of the denominators of those rational expressions.Therefore, it is not clear how to restrict these functions to the closed subset.Unfortunately, Theorem 2.40 also suffers from this difficulty.
Instead, our strategy is to prove Theorem 2.40 by proving a stronger statement (Theorem 2.57) for the "adding defect" map of zastava spaces that we will recall below.The image of the adding defect map is no longer contained in the vanishing locus of the denominators of the rational expressions defining the FMOs, so we can restrict the FMOs.
2.5.Adding defect.Let γ 2 P Q _ `.Observe that A pγ 2 q is isomorphic to the variety of a 2 P T 1 rrz ´1ss such that z xγ 2 ,Λ i y xv ˚Λi |a 2 |v ´Λi y is a polynomial in z for all i P I.The isomorphism sends a 2 to the colored divisor pL i pzqq iPI , where L i pzq " z xγ 2 ,Λ i y xv ˚Λi |a 2 |v ´Λi y.
Definition 2.42.Let γ 1 , γ 2 P Q _ `and let γ " γ 1 `γ2 .The adding-defect map is defined by Fix γ 1 , γ 2 , and γ as in the definition.Recall the regular functions Q i pzq and P i pzq on Z γ `.We will write Q i pzq and P i pzq for the corresponding functions on Z γ 1 `.As above, we have regular functions L i pzq on A pγ 2 q .It follows directly from the formula defining the adding defect map that (2.45) and: Furthermore, because Q i pzq and P i pzq are birational coordinates, we see that these formulas uniquely determine the map.
2.5.1.GKLO and adding defect.Define v 1 " pv 1 i q i , v 2 " pv 2 i q i , and v " pv i q i by γ ‚ " where ‚ P t 1 , 2 , ∅u.Define Λ pv 1 ,vs " krw i,r s iPI,rPtv 1 i `1,...,v i u .In addition to the GKLO embeddings for the zastava space, we have an embedding krA pγ 2 q s ãÑ Λ pv 1 ,vs (2.47) given by: to be the localization given by inverting w i,r ´wj,s for all pairs pi, rq ‰ pj, sq.The GKLO embedding and (2.47) induce an embedding: Finally, we define a map by sending w i,r Þ Ñ w i,r and by sending: for all i P I and 1 ď r ď v i .
Theorem 2.53.Let γ 1 , γ 2 , and γ be as in Definition 2.42, and let v 1 , v 2 , v be as above.The following diagram commutes: 2.5.2.FMOs and adding defect.Fix m " pm i q iPI P Z I such that 0 ď m ď v.We have a embedding: Given f P Λ v m , we use Sweedler notation and write for the image of f under (2.55).
Theorem 2.57.Let γ 1 , γ 2 , and γ be as in Definition 2.42, and let v 1 , v 2 , v be as above.Let m " pm i q iPI P Z I with 0 ď m ď v, and let f P Λ v m .Write f " ř f p1q b f p2q under the embedding (2.55).Under the restriction map on coordinate rings corresponding to the adding defect map (2.43) we have: Observe that the closed embedding Z γ 1 `ãÑ Z γ `factors through the adding defect map as Z γ 1 `ãÑ Z γ 1 `ˆA pγ 2 q Ñ Z γ `where the closed embedding Z γ 1 `ãÑ Z γ 1 `ˆA pγ 2 q is given by the closed point of A pγ 2 q corresponding to the unique colored divisor supported at 0 P A 1 .Observe also that given f P Λ v m expressed as ř f p1q b f p2q under (2.55), we have r f " ř f p1q f p2q p0q, where f p2q p0q is obtained by setting all variables w i,r with r ą v 1 i equal to zero.We therefore obtain Theorem 2.40 as a corollary of Theorem 2.57.As explained above, we therefore obtain Theorem 2.28 for positive FMOs.
Proof of Theorem 2.57.Recall the rational expression (2.19) for M mpfq.By Theorem 2.53, we are reduced to applying the map ϕ to this rational expression.Because all u i,r with r ą v 1 i map to zero under ϕ in the sum (2.19), we need only consider Γ " pΓ i q iPI such that Γ i Ď rv 1 i s.That is, we are computing: If m i ą v 1 i for any i, we immediately see that the sum is empty, and we get 0.So we may assume m i ď v 1 i for all i.Consider a single term in the sum (2.59), i.e. fix a Γ " pΓ i q iPI such that Γ i Ď rv 1 i s Ď rv i s and #Γ i " m i for all i.First, observe that: In the numerator of our single term, we write ź aPE ź rPΓ spaq ,sRΓ tpaq pw tpaq,s ´wspaq,r q " (2.61) and in the denominator, we similarly write: Finally, observing that we see that (2.59) is equal to which is exactly the rational expression for ř M mpf p1q qf p2q .
2.6.Chevalley anti-involution on slices.In this section, we will recall an involution on the slice W λ µ that swaps positive and negative FMOs and allows us to finish the proof of Theorem 2.28 (by proving it for the negative FMOs).This involution and its non-commutative version for Yangians is studied in detailed in [BFN19, §2(vii) and Appendix B], and the statements of this section can be obtained there.We present the key facts necessary for the present commutative situation.
As the group G is equipped with a pinning, we have the Chevalley anti-involution ι : G Ñ G obtained by swapping positive and negative root subgroups.We get a corresponding anti-involution on the group Gppz ´1qq that preserves the slice W λ µ .We denote this involution by the same symbol ι : W λ µ Ñ W λ µ .We also write ι : krW λ µ s Ñ krW λ µ s for the corresponding involution of coordinate rings.Write P ì pzq " P i pzq and define the regular function P í pzq on W λ µ by P í pzq " z xλ,Λ i y xv ˚Λi |xaz µ y|v ´si pΛ i q y (2.66) for xaz µ y P W λ µ .Then we have ιpQ i pzqq " Q i pzq and ιpP ì pzqq " P í pzq.Because the Q i pzq and P ì pzq are birational coordinates, this uniquely determines ι.In order to study how this involution interacts with the GKLO embeddings, we need to compute the image of P í pzq under the GKLO embedding.
Proposition 2.67.Under the GKLO embedding (2.13), s"1 s‰r z ´wi,s w i,r ´wi,s ‹ ‚w As in Remark 2.16, this result agrees with the h " 0 limit of [BFN19, Theorem B.15], up to the corresponding sign.
Proof.Observe that (2.68) is in the form of the Lagrange interpolation formula, so it suffices to compute P í pw i,r q for all r P rv i s.Define D i pzq to be the regular function on W λ µ given by D i pzq " z xλ,Λ i y xv ˚si pΛ i q |xaz µ y|v ´si pΛ i q y (2.69) for xaz µ y P W λ µ .Then we have the following identity (see e.g.[FZ99, Theorem 1.17], [GKLO05, (2.19)]): (2.70)For each r P rv i s, we substitute z " w i,r and solve for P í pw i,r q, noting that Q i pw i,r q " 0. As the involution ι is compatible with closed embeddings (2.26), we obtain Theorem 2.28 for negative FMOs and therefore complete its proof.2.7.The other zastava space.Let γ P Q _ `.Define Z γ ´to be the following closed subscheme of Gppz ´1qq:

Z γ
´" az ´γy P T 1 rrz ´1ssz ´γU 1 rrz ´1ss : az ´γy|v ´Λy ě 0 for all Λ ( (2.79) where Λ P P ``varies over all dominant weights and |v ´Λy P Vp´Λq is the lowest weight vector defined above, and Vp´Λq varies over Wp´Λq and Sp´Λq.As before, define Z γ ´to be the reduced scheme of Z γ ´.There is a map W λ µ Ñ Z λ´µ ´given by xaz µ y Þ Ñ az µ´λ y.The involution ι induces an isomorphism: Applying ι, all of the discussion above about Z γ `holds for Z γ ´as well.Furthermore, we obtain a closed embedding: which is given explicitly on points by: We can interpret this embedding in terms of FMOs: the positive FMOs M mpfq generate the coordinate ring of Z γ `, the negative FMOs M ḿpfq generate the coordinate ring of Z γ ´, and the map on coordinate rings corresponding to (2.81) is surjective because the coordinate ring of W λ µ is generated by all the FMOs M mpfq together.

Kac-Moody affine Grassmannian slices via Coulomb branches
In this section we explain how the Braverman-Finkelberg-Nakajima theory of Coulomb branches gives a definition of affine Grassmannian slices for all symmetric Kac-Moody types.First we will discuss Coulomb branches in general.We then focus on the Coulomb branches of quiver gauge theories, which are those which produce Kac-Moody affine Grassmannian slices.

Coulomb branches in general.
We first recall the general definition of Coulomb branches due to Braverman, Finkelberg and Nakajima [BFN18].Let G be a split reductive group, and let N be a finite dimensional representation of G, both defined over C. (Note that G will not be the same as the group G from the previous section.For us G will always be a product of general linear groups.) We will refer to [BFN18, §3, Definition 3.13] for the precise definitions and here only recall the basic ingredients in the definition.Let K " Cpptqq and O " Crrtss, and define the ind-scheme R G,N (see [BFN18, §2(i)]) by: R G,N " !pg, nq P GpKq{GpOq ˆNpOq : gn P NpOq where H GpOq ‚ denotes equivariant Borel-Moore homology (with coefficients in k).Note that R G,N is not an inductive limit of finite type schemes unless N " 0, so the definition of its equivariant Borel-Moore homology requires some care, see [BFN18, §2(ii)].
Observe that there is a natural map R G,N Ñ GpKq{GpOq to the affine Grassmannian for G. Using a variation of the convolution structure on this affine Grassmannian, Braverman, Finkelberg, and Nakajima define a multiplication on H GpOq ‚ pR G,N q.By formal properties of convolution algebras, this multiplication is associative [BFN18, Theorem 3.10].However, a special feature of this setting is that the multiplication is in fact commutative ([BFN18, Proposition 5.15]).One can thus define the Coulomb branch M C pG, Nq as where we view the roots β P ∆ G as elements of H ‚ T pptq.In our main case of interest, the quiver gauge theories to be recalled below, we will give a precise formula for this embedding in terms of the GKLO embedding (see §3.2.2 below).
The algebra ApT , Nq is relatively simple to describe: it is free as a module over its subalgebra H ‚ T pptq, with a basis tr γ u γ indexed by the set of all coweights γ : G m Ñ T .Moreover there is an explicit multiplication formula for these elements [BFN18, Theorem 4.1].
3.1.3.Forgetting matter.Suppose that N 1 and N 2 are two G-representations.Then we have the forgetting matter homomorphism [BFN18, Remark 5.14]: To define this map, first consider the case where G " T is a torus.In this case, the map z ˚: ApT , N 1 ' N 2 q ãÑ ApT , N 1 q is H ‚ T pptq-linear and for all cocharacters γ : G m Ñ T we have Here the product runs over weights ξ of N, thought of as elements of H ‚ T pptq [BFN18, §4(vi)].For general G, let T Ă G be a maximal torus.Then the map (3.6) is determined by the following commutative square involving the respective localization maps:

3.8)
In particular, the map (3.6) is linear over H ‚ G pptq.
3.1.4.Fourier transform.The Fourier transform is an isomorphism [BFN18, §6(viii)]: This map is again compatible with localization: The bottom arrow is determined by The latter map is H ‚ T pptq-linear, and on monopole operators is given by a sign [BFN18, §4(v)]: In particular, the map (3.9) is H ‚ G pptq-linear.
3.1.5.Minuscule monopole operators.Let γ : G m Ñ T be a cocharacter.Consider the orbit GpOqt λ GpOq{GpOq ãÑ GpKq{GpOq.We say γ is minuscule precisely when this orbit is closed; this is equivalant to asking that the dominant translate of γ under the Weyl group W G be minuscule in the usual sense.Let R γ G,N denote the preimage of the orbit under the map R G,N Ñ GpKq{GpOq.Because of our assumption that γ is minuscule, is a vector bundle (of infinite rank), and we have a closed embedding which induces an embedding: where W G is the Weyl group of G, and W G,γ is the stabilizer of γ.
Finally, observe that the sign appearing in (3.11) is constant on W G -orbits.So we see that the Fourier transform acts on MMOs by a sign: with notation as in §3.1.4, (3.17) 3.2.Coulomb branches of quiver gauge theories and Kac-Moody affine Grassmannian slices.Fix a quiver pI, Eq with no edge loops, which we think of as an orientation of a Dynkin diagram of symmetric Kac-Moody type.Note that multiple edges are allowed, for example as in the Dynkin diagram of the affine Lie algebra p sl 2 .We keep the same notation from §2 for coweights, dominant coweights, simple roots, etc., in this general Kac-Moody setting.
Fix λ P P _ ``and µ P P _ with λ ě µ.Define v " pv i q iPI and w " pw i q iPI by the same formulas (2.2) and (2.3).Fix finite dimensional I-graded C-vector spaces V " À iPI V i and W " Finally, we pick bases of each V i , which determines a pinning of G.In particular, we get a maximal torus T of G.
In physics terminology, G and N determine a "quiver gauge theory", and associated to this data, we can form a Coulomb branch.Braverman, Finkelberg, and Nakajima prove the following astonishing theorem, which is the reason Coulomb branches enter our story.We mention that this result can be interpreted physically in terms of moduli spaces of singular monopoles [BFN19,BDG17].We state a version twisted by the diagram automorphism ˚from Remark 2.35.Theorem 3.20 ([BFN19, Theorem 3.10]).Suppose the quiver pI, Eq is finite ADE type.Let λ and µ be as above, and let G and N be defined by (3.18).Then there is an explicit isomorphism: Motivated by this theorem, we have the following definition.Definition 3.22.Let λ and µ be as above, and let G and N be defined by (3.18).Then the (Kac-Moody) affine Grassmannian slice is defined by: To match the notation from the previous section, we also write: When the quiver pI, Eq is affine type, W λ µ is often called a double affine Grassmannian slice (short for "affine affine Grassmannian slice").Finally, observe that W λ µ does not depend on the orientation of the quiver because of the Fourier transform from §3.1.4.3.2.1.Zastava spaces.In the special case where G is a product of general linear groups, one can define a positive part of the ind-scheme R G,N , which we denote R G,N .The Borel-Moore homology forms a subalgebra of ApG, Nq and Braverman, Finkelberg, and Nakajima prove the following theorem.
Theorem 3.26 ([BFN19, Corollary 3.4]).Suppose the quiver pI, Eq is finite ADE type.Let λ and µ be as above, and let G and N be defined by (3.18).Then there is an explicit isomorphism: Analogous to Definition 3.22, in Kac-Moody type, we define the zastava space Z λ´µ `by formula (3.27).To match the notation from the previous section, we also write: Similarly, there is a negative part R Ǵ,N , we analogously define A ´pλ, µq " A ´pG, Nq, and Theorem 3.26 holds analogously for Z λ´µ ´.

GKLO embedding.
At this point we can explain where Theorem 2.12 comes from.The ring r Apλ ´µq is identified with ApT , 0q: the u i,r are particular generators for this torus Coulomb branch, and the w i,r are the equivariant parameters generating H ‚ T pOq pptq.The GKLO embedding is exactly the composed map: Note that the positive roots β " w i,r ´wi,s for i P I and 1 ď r ă s ď v i .Thus Theorem 2.12 holds exactly the same in the Kac-Moody case.The only difference is that the regular functions Q i pzq and P i pzq are not defined via matrix coefficients, but rather are characterized uniquely by formulas (2.14) and (2.15).Similarly, the GKLO embedding (2.36) for Z λ´µ `also holds exactly as before.
3.2.3.Fundamental monopole operators.The definition of FMOs (Definition 2.18), the fact that they are regular functions (Theorem 2.23), and the fact that they generate the coordinate ring (Theorem 2.24) continue to hold without change.We can now say where FMOs come from: they are particular examples of minuscule monopole operators.Let m " pm i q iPI P Z I with 0 ď m ď v.We define ̟ m to be the following coweight of G: it is the sum over i of the m i -th fundamental coweight p1, . . ., 1, 0, . . ., 0q of the factor GLpV i q of G.For this coweight, we have an identification between Λ v m and H pptq.The coweight ̟ m is minuscule, and for f P Λ v m , and similarly to ([BFN19, Proposition A.2]) we have: The coweight ´̟m is also minuscule, and we have an identification between Λ v m and H where "sign" is given by (2.21).

Embeddings of Kac-Moody affine Grassmannian slices
Our main goal in this section is to generalize Theorem 2.28 to Kac-Moody affine Grassmannian slices.The main difficulty is that we do not a priori have a definition of the closed embedding (2.26), because we only have the Coulomb branch definition of Kac-Moody affine Grassmannian slices.Because we know generation by FMOs, we could try to use Theorem 2.28 to define the closed embedding.Unfortunately, this does not work because we do not know a priori that the map on generators extends to an algebra homomorphism.
Instead, we will proceed another way and construct the closed embedding (2.26) using the Coulomb branch definition.Theorem 2.28 appears during the proof of the construction.4.1.Conicity condition and closed embeddings.Let λ 1 P P _ ``with λ ě λ 1 ě µ.In finite type, we automatically have a closed embedding of W λ 1 µ into W λ µ , but in general Kac-Moody type, we need a further condition on λ 1 to obtain a closed embedding.
Let C denote the Cartan matrix of our Kac-Moody type, which is a symmetric matrix by assumption.
Definition 4.1.Let λ, λ 1 P P _ ``with λ ě λ 1 .Define w as above and v 2 " pv 2 i q iPI by: We say that λ 1 satisfies the conicity condition for λ if u ¨pw ´Cv 2 q `u ¨pCuq ě 1 (4.3) for all non-zero u with 0 ď u ď v 2 .Here ¨denotes the usual dot product on Z I .
Remark 4.4.As we will recall in Appendix A, the conicity condition corresponds exactly to quiver gauge theories that are either "good" or "ugly" in the physics terminology.
Remark 4.5.The conicity condition is automatic in finite type, and for most affine types.See Theorem A.8 for details.
Proposition 4.6.If λ 1 satisfies the conicity condition for λ, then the Kac-Moody slice W λ λ 1 is a conical variety.That is, Apλ, λ 1 q " krW λ λ 1 s admits a grading by Z ě0 with the 0-graded piece consisting just of k.
This result is a special case of Corollary A.6.In particular, we obtain an algebra homomorphism: which sends all elements of non-zero degree to zero.More precisely, let m " pm i q iPI P Z I with 0 ď m i ď v 2 i for all i P I, and let f P Λ v 2 m .Then the map (4.7) is defined by: The above notation fp0q means setting all equivariant parameters in f to zero.
We can now state the main theorem of this section.
Theorem 4.9.Let λ, λ 1 P P _ ``and µ P P _ with λ ě λ 1 ě µ.Further assume that λ 1 satisfies the conicity condition for λ.Then there is a closed embedding under which the FMOs restrict exactly as in Theorem 2.28.4.2.Proving Theorem 4.9.Let λ, λ 1 , and µ be as in the Theorem statement.Let w and v be as above, v 2 as in Definition 4.1, and v 1 as in §2.3.uniquely determined by the commutativity of the following diagram: Because the MMOs generate ApG, Nq, it suffices to show that their images in ApT , Nq " β

´1‰ βP∆
G are equal to the images of elements of ApL, Nq In fact, we will show that the image of each MMO is a linear combination of MMOs from ApL, Nq with coefficients only involving the allowed denominators.
Let γ be a minuscule dominant weight for G, and let f P H ‚ T pptq W G,γ be a dressing for γ.Then under localization to the torus, the corresponding MMO is: Let tγ 1 , . . ., γ N u be the elements of W G ¨γ that are dominant for L. For each 1 ď j ď N, choose σ j P W G such that γ j " σ j pγq.Then (4.13) is equal to: where σ j P W G is any element such that γ j " σ j pγq.

4.2.2.
Constructing the closed embedding.For each i P I, we decompose where V 1 i is the span of the first v 1 i basis vectors of V i , and V 2 i is the span of the remaining v 2 i basis vectors.Taking the product over i P I of the corresponding subgroups GLpV 1 i qˆGLpV 2 i q Ă GLpV i q, we obtain a standard Levi subgroup G 1 ˆG2 Ă G.As a representation of G 1 ˆG2 , we have where: By definition, we have We have assumed that λ 1 satisfies the conicity condition for λ, so there is an algebra homomorphism as in (4.7): By the Künneth formula we also have [BFN18, §3(vii)(a)]: Combining this with (4.24) gives us a surjective map: Proposition 4.27.There is a surjective map π : ApG 1 ˆG2 , Nq ։ ApG 1 , N 1 w ' N mix q which is uniquely determined by the commutativity of the following square: where the vertical arrows are obtained by forgetting N mix , and the lower horizontal arrow is (4.26).Furthermore, let γ " pγ 1 , γ 2 q be a minuscule coweight for G 1 ˆG2 , and let where Clearly π is a well-defined rational map.For the rational map π, the formulas (4.29) follow by direct calculation using (3.7) and (4.8).
By [Wee19, Remark 3.8], ApG 1 ˆG2 , N 1 w ' N 2 w q and ApG 1 , N 1 w q are each generated by MMOs.Formula (4.29) tells us that all the MMOs map to MMOs, and that each MMO is the image of an MMO.So we conclude that π is a regular map, and it is surjective.We can write N mix " N mix 1 ' N mix 2 where: The Fourier transform from §3.1.4provides us with a isomorphism For each i P I, we fix a vector subspace with dim W 1 i " w 1 i .This is possible because one computes the right hand side of (4.33) to have dimension w 1 i `2v 2 i .Let X i be any complementary vector subspace.Then we define and: In particular, we have: By construction, we can G 1 -equivariantly identify: N 1 w ' pN mix 1 q ˚' N mix 2 q " N p3q ' N p4q (4.37) So we view (4.32) as an isomorphism: For each i P I, let us further choose a vector-space decomposition X i " X ì ' X í with dim X ì " dim X í " v 2 i , and write and Then we further have a Fourier transform isomorphism

4.41)
To prove Theorem 4.9, we prove the following more elaborate statement.
Theorem 4.42 (Elaborated version of Theorem 4.9).There is a surjective homomorphism Apλ, µq " ApG, Nq ։ ApG 1 , N p3q q " Apλ 1 , µq that is uniquely determined by the commutativity of the following diagram: Additionally, under this map, the FMOs restrict exactly as in Theorem 2.28.
Proof.Let m " pm i q iPI P Z I with 0 ď m ď v, and let f P Λ v m .Recall from (3.30) that we have: By Propositions 4.11 and 4.27, we have: If not, we get zero.All the weights of N mix 1 pair negatively with ̟ m , so the first Fourier transform acts by the identity, see (3.17).However, the second Fourier transform introduces a sign, sending (4.45) to: Finally, z ˚sends this to: Observe that the signs cancels, and this is exactly equal to Now we turn to the negative FMOs.Analogously to before we have: if m i ď v 1 i for all i P I, and zero otherwise.This time the first Fourier transform introduces a sign sending (4.49) to and the second Fourier transform acts by the identity.Finally, z ˚sends this to: The signs do not cancel, and we get: Recall (2.20) that the negative FMO M is defined as a sign times M G 1 ,N p3q ´̟m pfq.In particular, the sign appearing (4.52) exactly compensates for this sign, and we conclude that As both source and target of the left vertical arrow in (4.43) are generated by FMOs, we conclude that the map is regular and surjective.
Remark 4.54.We do not know a priori that this closed embedding is the same as the closed embedding (2.26) defined group-theoretically for finite-type affine Grassmannian slices.Rather, we only know this a posteriori because both maps have the same effect on FMOs.In particular, Theorem 4.42 does not subsume Theorem 2.28.4.3.Poisson structure.Coulomb branch coordinate rings each come equipped with a one-parameter non-commutative deformation quantization, see [BFN18,§3(iv)].Geometrically, this extra parameter h corresponds to considering equivariance under an additional C ˆcorresponding to "loop rotation", and we obtain an algebra over kr hs " H ‚ C ˆpptq.This deformation quantization endows the commutative Coulomb branch with a Poisson structure.So we might ask when the closed embedding constructed in Theorem 4.9 respects the Poisson structure.To answer this question, we need the following strengthening of the conicity condition.Definition 4.55.Let λ, λ 1 P P _ ``with λ ě λ 1 .Define w as above and v 2 " pv 2 i q iPI by: v 2 i " xλ ´λ1 , Λ i y (4.56) We say that λ 1 satisfies the good condition for λ if u ¨pw ´Cv 2 q `u ¨pCuq ě 2 (4.57) for all non-zero u with 0 ď u ď v 2 .Here ¨denotes the usual dot product on Z I .
Remark 4.60.One might ask if Theorem 4.58 holds for the quantized Coulomb branches.Remarkably, the answer is no!The simplest example is the case of the A 1 quiver, λ " α _ , and λ 1 " 0. In this case, W λ µ is the nilpotent cone for SL 2 , and W λ 1 µ is a point.The map (4.59) quantizes precisely if the Coulomb branch quantization of W λ µ has a non-trivial one-dimensional module, and this property fails in this example.
More precisely, let us work over k " C. The Coulomb branch quantization A h pλ, µq is an algebra over Cr hs, and we seek a surjection A h pλ, µq ։ A h pλ 1 , µq " Cr hs over Cr hs.Specializing at h " 1, we would obtain a non-trivial one-dimensional module for the algebra A h"1 pλ, µq.In our example, one can check that the Coulomb branch quantization A h"1 pλ, µq is a particular central quotient of Upsl 2 q (corresponding to ´ρ), which does not have a one-dimensional module.(Of course, there does exist a quantization possessing a one-dimensional module, but this corresponds to a different central quotient of Upsl 2 q.)This is a fundamental reason why the construction of Theorem 4.9 is necessarily complicated: any simple geometric construction would also make sense for the quantization.It is all the more remarkable that we obtain such a nice formula for the restriction of the geometrically defined FMOs.
Proof of Theorem 4.58.It suffices to check that the top horizontal map and all the right vertical maps in (4.43) are Poisson.All the maps except π are Poisson by the results of [BFN18].As for π, using Corollary A.6 the good condition implies that π is Poisson.4.4.Adding defect map and closed embeddings for zastava spaces.We do not have an a priori definition of adding defect like Definition 2.42.Instead, we first define the map (2.51) on localized rings by the exact same formulas.Now we claim that there is a (necessarily uniquely determined) algebra homomorphism krZ γ `s Ñ krZ γ 1 `ˆA pγ 2 q s making the square (2.54) commute.We prove this by checking that the FMOs, which generate krZ γ `s, map to elements of krZ γ 1 `ˆA pγ 2 q s under the map (2.51) of localized rings: this is exactly what we computed in the proof of Theorem 2.57.
A.1.The case of quivers.Consider now the case of a quiver gauge theory defined as in §3.2.Given λ P P _ ``and µ P P _ with λ ě µ, recall that we associate w, v by equations (2.2) and (2.3), and thus associate pG, Nq as in (3.18) and (3.19).
Recall that the algebra Apλ, µq " ApG, Nq is generated by the FMOs M mpfq where 0 ď m ď v, see §3.2.3.Since these FMOs are precisely the dressed minuscule monopole operators for the minuscule coweights ˘̟m of G, their degrees are given by the formula (A.3).Since these elements generate the algebra, to verify whether pG, Nq is good or ugly, it suffices to the check that 2∆p˘̟ m q ě 2 for all non-zero 0 ď m ď v (resp.2∆p˘̟ m q ě 1 and this bound is saturated).
Lemma A.5.For any 0 ď m ď v, we have: 2∆p˘̟ m q " m ¨pw ´Cvq `m ¨pCmq This lemma is based on calculations from [Nak15,§5].We include the brief proof.
Let us consider the formula (A.1) for ∆p̟ m q.Recall that ̟ m is the sum over i of the m i -th fundamental coweights p1, . . ., 1, 0, . . ., 0q.Working one GLpV i q at a time, it is easy to see that the sum over positive roots α of ś i GLpV i q is given by ÿ α |xα, ̟ m y| " ÿ i m i pv i ´mi q Next, the weights of N are of two types: for each i P I there are the weights of HompW i , V i q, and for each edge a P E there are the weights of HompV spaq , V tpaq q.It is not hard to see that: ÿ ξ |xξ, ̟ m y| dim Npξq " ÿ iPI m i w i `ÿ aPE `mtpaq pv spaq ´mspaq q `mspaq pv tpaq ´mtpaq q Substituting into the formula (A.1), we therefore have 2∆p̟ m q " ´2 ÿ i m i pv i ´mi q `ÿ iPI m i w i `ÿ aPE `mtpaq pv spaq ´mspaq q `mspaq pv tpaq ´mtpaq q which in turn can be easily rearranged to give: ÿ i m i w i ´2 ÿ i m i v i `ÿ aPE pm spaq v tpaq `mtpaq v spaq q `2 ÿ i m i m i ´ÿ aPE pm spaq m tpaq `mtpaq m spaq q Observe that this sum does not depend on the orientation of our quiver, and is exactly m ¨pw Ćvq `m ¨pCmq.
Summarizing the above discussion, we can finally explain our previous definitions of conicity (Definition 4.1) and goodness (Definition 4.55): Corollary A.6.If µ satisfies the conicity condition for λ (Definition 4.1), then pG, Nq is good or ugly, and in particular Apλ, µq is conical.Moreover, if µ satisfies the good condition for λ (Definition 4.55), then pG, Nq is good and the map Apλ, µq ։ k is Poisson.
Remark A.7.Using the above lemma, it is easy to see that the surjection Apλ, µq ։ Apλ 1 , µq from Theorem 4.9 respects the gradings discussed above.Indeed, recall that M mpfq Þ Ñ M mp r fq or zero.Assume the former case.For any 0 ď m, it follows by construction that m ¨pw ´Cvq " m ¨pw 1 ´Cv 1 q, and therefore deg M mpfq " deg M mp r fq by (A.3) and the lemma.
Finally, we conclude by studying the finite and affine type cases.Recall that in affine type, the level of a coweight µ is its pairing with the imaginary root δ.Following the notation of [Kac90, Theorem 4.8], recall that we can write δ " ř iPI a i α i for some positive integers a i .
(2) If the quiver is affine type, then: (a) If the level of µ is ě 2, then µ is good for λ.
(b) If the level of µ is 1, then µ satisfies the conicity condition for λ, but is not good for λ.(c) If the level of µ is 0, then µ does not satisfy the conicity condition for λ.
In particular, part (2c) above demonstrates that for general symmetric Kac-Moody types, µ being dominant does not necessarily imply the conicity condition.
Proof.The fact that µ is dominant means precisely that m ¨pw ´Cvq ě 0 for any m ě 0. Note also that we can identify m ¨pCmq with the standard bilinear form on the root lattice.In finite type we therefore have m ¨pCmq ě 2 for any non-zero m.This implies that µ is good for λ, since for any 0 ă m ď v we see that m ¨pw ´Cvq `m ¨pCmq ě 0 `2 " 2 The same argument applies in affine type, except in those cases where m corresponds to a multiple of the imaginary root δ, in which case m ¨pCmq " 0 [Kac90, Theorem 4.3].Note that we still require 0 ă m ď v, and these m corresponding to multiples of δ are thus only allowed if a i ď v i for all i P I. To complete the proof, we will study the latter case more carefully.
First observe that it suffices to study the case where m corresponds exactly to δ, meaning that m i " a i for all i P I.In this case m ¨pw ´Cvq is exactly the level of µ.So on the one hand, if the level of µ is ě 2, then we have m ¨pw ´Cvq `m ¨pCmq ě 2 `0 " 2, This shows that µ is good for λ.On the other hand, suppose that the level of µ is 0 or 1.Then λ also has level 0 (resp.1).Since λ ą µ are both dominant and we are in symmetric affine type, the only possibility is that µ " λ ´kδ for some integer k ą 0 (this follows from [Kac90, §12.4-12.6]).This automatically implies that the vector m " pa i q iPI corresponding to δ satisfies 0 ă m ď v, and we have m ¨pw ´Cvq `m ¨pCmq " 0 (resp.1) Since this is equal to 2∆p̟ m q, this shows that µ does not satisfy the conicity condition for λ (resp.satisfies the conicity condition, but not the good condition).

λµ
by Apλ, µq.As a consequence of the Coulomb branch construction of the slice W λ µ , we have the following theorem: Theorem 2.12 ([BFN19]).There is a birational embedding Apλ, µq ãÑ r Apλ ´µq S v loc (2.13) of algebras under which following non-trivial theorem is a consequence of the Coulomb branch construction of W λ µ .Theorem 2.23 ([BFN19]).The FMOs are regular functions on W λ µ Additionally we have the following theorem of Weekes.

4.2. 1 .
Levi restriction.We need the following Levi restriction statement, generalizing the localization map from §3.1.2.Since we are in the setting of quiver gauge theories ApG, Nq is generated by FMOs, and in particular by the minuscule monopole operators M G,N γ pfq.Proposition 4.11.Let L be a standard Levi of G, and let ∆ L Ă ∆ G denote set of positive roots for L.There is an embedding ι ´1 ˚: ApG, Nq ãÑ ApL, Nq " β [FKMM99,BDF16]he functions P i pzq and Q i pzq on W λ µ are pulled back from regular functions on Z λ´µ `given by the same formulas.It is known[FKMM99,BDF16]that these functions are birational coordinates on Z λ´µ `.In particular, the map W λ µ Ñ Z λ´µ `is birational.Remark 2.35.Braverman, Finkelberg, and Nakajima [BFN19] make use of a map W λ μ˚Ñ Z λ´µ `, It suffices to check that Q i pzq and P i pzq map according to (2.45) and (2.46) under the top and right arrows.This is immediate for Q i pzq, and (2.52) is exactly what makes it hold for P i pzq.
Apλ ´µq loc,loc denote the further localization of r Apλ ´µq loc where we invert w i,r for all i P I, r P rv i s.Then we can define an automorphism Let r 2.6.1.Negative FMOs.With Proposition 2.75 in hand, we can directly calculate the following Proposition 2.77.Fix m " pm i q iPI with 0 ď m ď v and f P Λ v m .Under the involution ι : krW λ µ s Ñ krW λ µ s, we have ι `Mm pfq ˘" M ḿpfq (2.78) Braverman, Finkelberg and Nakajima prove that ApG, Nq is a finitely-generated k-domain, which tells us that M C pG, Nq is an irreducible k-variety [BFN18, Corollary 5.22 and Localization.Fix a pair of opposite Borel subgroups of G, which determines a maximal torus T and a set ∆ G of positive roots.Then using the localization theorem in equivariant Borel-Moore homology, we have an embedding [BFN18, Lemma 5.10, Remark 5.23] ι ´1 ˚: ApG, Nq ãÑ ApT , Nq " β G pptq ã−Ñ ApG, Nq (3.4)Moreover, multiplication by this subalgebra agrees with the natural geometric action of H ‚ G pptq " H ‚ GpOq pptq on the equivariant Borel-Moore homology ApG, Nq " H denote the stabilizer of t λ GpOq{GpOq.There is an action of H ‚ Stab t γ pptq on H q which is a rank-one free module with generator given by the fundamental class rR γ G,N s, see [BFN18, §6(ii)].For f P H ‚ Stab t γ pptq, denote the corresponding element of H called the (dressed) minuscule monopole operator (MMO) with dressing f.Applying the localization theorem, we obtain the following formula ([BFN18, Proposition 6.6]) γ j σσ j pfq ¨rσpγ j q σ ´śβP∆ L :xβ,γ j yą0 β ¨śηP∆ Gz∆ L :xη,γ j yą0 ηIn formula (3.16) for MMOs, we can abuse notation and formally consider the dressing to be an element of the fraction field of H ‚ Stab t γ pptq.With this generalization, we can write (4.16) more compactly as