Branes, Quivers, and the Affine Grassmannian
Antoine Bourget, Julius F. Grimminger, Amihay Hanany, Marcus Sperling, Zhenghao Zhong
IImperial/TP/21/AH/01
Branes, Quivers, and the Affine Grassmannian
Antoine Bourget, ל Julius F. Grimminger, ל Amihay Hanany, ל Marcus Sperling, ב andZhenghao Zhong ל ל Theoretical Physics Group, The Blackett Laboratory, Imperial College London, Prince ConsortRoad London, SW7 2AZ, UK ב Yau Mathematical Sciences Center, Tsinghua University, Haidian District, Beijing, 100084,China
E-mail: [email protected] , [email protected] , [email protected] , [email protected] , [email protected] Abstract:
Brane systems provide a large class of gauge theories that arise in string theory.This paper demonstrates how such brane systems fit with a somewhat exotic geometricobject, called the affine Grassmannian. This gives a strong motivation to study physicalaspects of the affine Grassmannian. Explicit quivers are presented throughout the paper,and a quiver addition algorithm to generate the affine Grassmannian is introduced. Animportant outcome of this study is a set of quivers for new elementary slices. a r X i v : . [ h e p - t h ] F e b ontents , C ) and PSL(2 , C ) affine Grassmannians 82.5 Orbits in the Affine Grassmannian 102.6 Slices in the affine Grassmannian 12 A -type affine Grassmannian 354.1.1 The A affine Grassmannian 384.1.2 The A affine Grassmannian 404.2 The B -type affine Grassmannian 434.2.1 The B affine Grassmannian 454.2.2 The B affine Grassmannian 464.3 The C -type affine Grassmannian 494.3.1 The C affine Grassmannian 504.3.2 The C affine Grassmannian 524.4 The D -type affine Grassmannian 554.4.1 The D affine Grassmannian 57 – i – Branes, ON Planes and Quivers 76
A.1 ON + − (cid:103) ON − Branes play a crucial role in our current understanding of string theory. Their world-volume low energy effective action has been occupying numerous papers and is still usedas an essential tool in studying a whole variety of string backgrounds. Furthermore, theexistence of brane systems opens up a window to the study of theories when an effectiveLagrangian either does not exist or is beyond reach. This again increases the level ofunderstanding in a significant way.This paper is devoted to yet another conceptual advancement in the study of branesystems in string theory. It has to do with a somewhat exotic object – and so far lessstudied within the string theory community – the affine Grassmannian [1–6]. This objectplays a prominent role in the so-called geometric Satake correspondence in the geometricLanglands program [7, 8], which is partly responsible for the considerable interest it hasattracted in mathematics in the past two decades. A close inspection reveals that theaffine Grassmannian is also precisely the type of object one should study when trying tounderstand brane systems, hence making it an important element in physical systems.As is argued below, brane systems of the type of Hanany-Witten setups [9] fit intothe affine Grassmannian, and they do so in a host of space-time dimensions ranging from6 to 3, and possibly even lower. A definition of the affine Grassmannian, suitable to aphysics audience is presented below, together with examples and explanations. Specialattention is paid to slices in the affine Grassmannian [10–12] which turn out to fit withthe physical setting, namely each slice is a moduli space vacua of an interacting theory,with or without a Lagrangian; or viewed from the brane perspective, a Kraft-Procesi tran-sition [13, 14] between two phases of the brane system [15, 16]. The connection betweenCoulomb branches viewed as moduli spaces of singular monopoles [9, 17–19] and slices inthe affine Grassmannians was made in [12, 20, 21], using previous developments [22–28].The mathematical properties of the slices in connection with symplectic geometry and therelation with quiver theories have been further studied in [11, 29–31].We note that so far we do not have a brane construction for the affine Grassmannian ofexceptional groups. Slices in these affine Grassmannians can nevertheless be studied usingquivers. This involves a new algorithm of quiver addition, which is explained in detailbelow.Let us turn to Hasse diagrams [32]. Hasse diagrams are a depiction of a partial order.They are a useful tool for characterizing the phase structure of a given theory. In suchdiagrams, consisting of nodes connected by edges, each node represents a fixed set of– 1 –assless states of the physical system. Tuning moduli without moving to a different phasecorresponds to changing the effective masses of massive particles, while keeping the setof massless states fixed. When the set of massless states of one phase, call it phase a ,is contained in the set of massless states of another phase, call it phase b , we assign theorder b < a . Suppose there exists no phase c such that b < c < a then a and b areconnected by an edge due to the partial order b < a . The edge connecting node a to node b represents the minimal set of moduli one needs to tune in order to move from phase a phase b . The tuned moduli form a conical singularity and supersymmetry dictates the typeof such a singularity. For 3 d N = 4 Coulomb branches, and Higgs branches of theorieswith 8 supercharges in dimensions 3 −
6, this singularity is a symplectic singularity [33], ora hyper K¨ahler cone – as more commonly referred to in the physics literature.As Hasse diagrams for moduli spaces are composed of nodes and edges, it is naturalto put some additional structure by embedding such a diagram into a lattice. Indeed, itis one of the features of the affine Grassmannian of a Lie group G , that the Hasse dia-gram for any slice originates from such a lattice, making the conceptual understanding ofsuch Hasse diagram much easier. The leaves of the affine Grassmannian (called Schubertcells) are in one-to-one correspondence with dominant coweights. Transverse slices be-tween two leaves are in one-to-one correspondence with lattice vectors that connect onenode to another. These lattice vectors, in turn, are spanned by positive coroots, and ifthe linear combination is non-negative then there exists a directed path between one pointand another; hence, inducing a natural partial order in the lattice [34]. Therefore, slicesin the affine Grassmannian are particularly nice, as their Hasse diagrams can be under-stood as all directed paths between two points in the principal Weyl chamber. The factthat this structure fits beautifully with brane systems is not surprising, as brane systemscan be viewed as algebraic objects. This just emphasizes the point that slices in the affineGrassmannian form very interesting physical systems. The stratification of the slices in theaffine Grassmannian into symplectic leaves can be related to monopole bubbling [35–38].The simplest affine Grassmannian is that of SL(2 , C ), where the Hasse diagram isa semi infinite linear diagram. Each slice in the affine Grassmannian of SL(2 , C ) is anSQCD type moduli space (specifically its Coulomb branch) with a unitary gauge group asopposed to special unitary. Hasse diagrams of SQCD theories were already shown to belinear in [32, Table 1]. One important point made in the present paper is that this line isactually connecting two points on the principal Weyl chamber of SL(2 , C ). It is reassuringto see that for the simplest possible, and most studied, gauge theories – SQCD – there is acorrespondence with the simplest possible affine Grassmannian – the affine Grassmannianof SL(2 , C ). If one identifies SQCD theories with the more mathematical name, framed A quiver theories, then it is easy to see the generalization. A slice in the affine Grassmannianof the group G is given by the Coulomb branch of a framed Dynkin diagram of type G .It is useful to introduce a simple characterisation of a moduli space by an integer num-ber, for example the dimension or the number of leaves. A new measure of the simplicity We note that these Schubert cells may be of odd complex dimension, and their closure hence not asymplectic singularity. The transverse slice between two Schubert cells, however, is a symplectic singularity. – 2 –f a moduli space is given by the following integer number: For a generic moduli space M with a Hasse diagram H one can define the disposition D ( n ) of a node n ∈ H as thenumber of minimal degenerations of the corresponding leaf, D ( n ) = number of minimal degenerations of n. (1.1)One can further define the disposition D ( M ) of a moduli space as the maximal dispositionover all nodes in its Hasse diagram, D ( M ) = max n ∈ H ( D ( n )) . (1.2)One would say that a moduli space with a smaller disposition is simpler than a modulispace with a bigger disposition. The disposition of the affine Grassmannian is the rank ofthe group. The disposition of a slice in the affine Grassmannian is hence bounded fromabove by the rank of G . The disposition of the Coulomb branch or Higgs branch of SQCDis 1; the lowest possible. The disposition of the entire moduli space of SQCD is 2. Ingeneral it is much easier to obtain and generalise Hasse diagrams with a low disposition.However, Hasse diagrams for slices in the affine Grassmannian can be well understood dueto their embedding into the coweight lattice, even when the disposition is very high.Another important integer number is the minimal number of generators of the chiralring, call it g . A striking feature of any slice in the affine Grassmannian of a group G isthe fact that g/ dim( G ) is again an integer number.One should further make a distinction between finite dimensional Lie groups and infi-nite dimensional generalizations. By analogy with the finite dimensional case, one wouldidentify (the Coulomb branch of) a framed affine quiver with a slice in the affine Grassman-nian of the affine Lie group. Naturally, the structure of the moduli space, and consequentlyits Hasse diagram, become significantly more complex. The disposition for a generic slicein the affine Grassmannian of an affine group appears not to be bounded from above.Consequently, if the group is infinite dimensional one expects a much more complicatedmoduli space than that of a finite dimensional group. Prominent examples of slices in theaffine Grassmannian of affine groups are moduli spaces of instantons. While they are wellstudied spaces in physics, we see that they are much more challenging objects than slicesin the affine Grassmannian of finite groups.The present paper focuses on slices in the affine Grassmannian of a finite dimensionalLie group, which in this sense are simpler than moduli spaces of instantons. This is reflectedin several ways. Many of the slices are complete intersections. Many of these are closuresof nilpotent orbits. The generators of the chiral ring are simple and, as argued above, theHasse diagram for slices of algebras of finite type is simpler than the Hasse diagram forthe moduli space of instantons. These features make these moduli spaces more tractableobjects and, hence, simpler to study.In the long run, the lessons learned from the affine Grassmannian of finite dimensionalgroups will hopefully help us to tackle more difficult moduli spaces. We propose the name disposition to mean the tendency of the theory to gain more massless states inseveral inequivalent ways. – 3 –lice Framed quiver Unframed quiver a n · · · · · · b n · · · · · · c n · · · · · · d n · · · · · · e e e
12 3 4 5 6 4 23 12 3 4 5 6 4 23 Slice Framed quiver Unframed quiver f g ac n
11 1 · · · · · · ag cg h n,k · · · k · · · kkh n,k
11 1 · · · k · · · kkA n n + 11 11 n + 1 Table 1 . Most up-to-date, but incomplete list of unitary quivers without loops for elementary slices usable in the quiver subtraction algorithm.In each case we provide two quivers, a framed version and an equivalent unframed version, where a U (1) should be ungauged on the long node.For a n , b n , c n , d n , ac n , h n,k and ¯ h n,k there are n gauge nodes in the framed quiver and n + 1 gauge nodes in the unframed quiver. Notice that h n, = H n , h n, = c n , h , = cg , h n, = a n , h n, = ac n , and h , = ag . –4– nice by-product of studying the affine Grassmannian is the encounter of the so-called quasi-minimal singularities introduced in [10], which are elementary slices that donot appear in the study of nilpotent orbits. This is an important update to our arsenalof quivers to be used in quiver subtraction. We display all unitary quivers without loopswhich we know to be elementary slices in Table 1. The classification of these slices is stillan open problem, and every new addition is hence very exciting. Plan of the paper
In Section 2, we provide a lightening review of the construction ofthe affine Grassmannian, and put forward the important concepts of orbits, stratificationand transverse slices. In Section 3, we construct quivers for the transverse slices andcompute them explicitly for rank 1 and 2 groups, along with the Hasse diagrams. Thisallows to identify new quivers for the quasi-minimal singularities. We then show how thisconstruction can be reproduced using brane setups with orientifolds for the classical groupsin Section 4, and for any group using the new algorithm of quiver addition in Section 5.Finally, we end with an analysis of the generators of the infinite dimensional transverseslices using Hilbert series in Section 6.
In this section, we give the definitions that apply to the discussions in this paper. Ournotations are summarized at the end of this section in Table 3.Our description tries to avoid excessive technicalities as it is aimed at physicists pri-marily. For instance, we describe the affine Grassmannian as an infinite dimensional varietyand not as an ind-scheme; we also identify schemes with their underlying topological space.For a more formal treatment, we refer to [1, 39–43]. A summary of the main properties ofthe affine Grassmannian is presented in [22, Section 2]. For a very explicit treatment withexamples, see [44, 45]. For an introduction in the context of gauge theory, see [46, 47].
We define three important structures that underlie all the construction of the affine Grass-mannian, reviewed below. We use throughout a formal variable t . • First, we have the ring of formal power series in t , denoted C [[ t ]]. Here formalmeans that we do not worry about convergence issues. For examples of elements of C [[ t ]], let us mention the polynomials in t , the rational functions like − t = (cid:80) ∞ i =0 t i ,transcendental functions like e t = (cid:80) ∞ i =0 t i /i !, or non-convergent series like (cid:80) ∞ i =0 t i i !.Note that the important condition is that an element of C [[ t ]] is a formal series in t with only non-negative powers of t . Geometrically, C [[ t ]] can be seen as the ring offunctions on the unit complex disk D = { z ∈ C | | z | < } . Equivalently, the unit disk D is the spectrum of C [[ t ]]. • The second object we consider is the ring of polynomials in t − , denoted C [ t − ]. Wedraw the attention of the reader on two differences with C [[ t ]]: here the powers of t are ≤
0, and the series expansion needs to terminate (a polynomial has a finitedegree). – 5 –
Finally, we need the field of fractions of C [[ t ]], denoted C (( t )). This can be defined asthe set of all fractions f ( t ) /g ( t ) for f ( t ) , g ( t ) ∈ C [[ t ]] with g ( t ) (cid:54) = 0. Equivalently, thisis the set of formal series that can be written in the form (cid:80) ∞ i = N a i t i for some N ∈ Z and a i ∈ C . The spectrum of C (( t )) is the punctured disk D ∗ = { z ∈ C | < | z | < } .The ring C [[ t ]] is a discrete valuation ring , where the valuation is given by the degreeof the lowest non-zero monomial. More generally for f = ∞ (cid:88) i = N f i t i ∈ C (( t )) (2.1)with N ∈ Z and f N (cid:54) = 0, the valuation is ν ( f ) = N . As the name indicates, C (( t )) is afield, so every non-zero element is invertible. We remark that C [[ t ]] is the ring of integersof C (( t )), and that the multiplicative group C [[ t ]] ∗ of invertible elements in C [[ t ]] is C [[ t ]] ∗ = (cid:40) ∞ (cid:88) i =0 a i t i | a i ∈ C and a (cid:54) = 0 (cid:41) . (2.2)Finally, the only invertible elements in C [ t − ] are the non-zero constants. G (( t ))For any characteristic 0 field K , GL( n, K ) is the group of invertible n × n matrices withentries in K . As mentioned in the previous subsection, C (( t )) is a field and we can use it toconsider the group GL( n, C (( t ))) of invertible n × n matrices with entries in C (( t )). Sucha matrix is invertible if and only if its determinant is not identically zero. More generally,let G be an algebraic subgroup of GL( n, C ), which means that G = { M ∈ GL( n, C ) | P j ( M ) = 0 } (2.3)for a certain collection of polynomials P j , j = 1 , ..., J . This includes the classical groupsSL( n, C ), SO( n, C ), Sp(2 n, C ), and also the exceptional groups. We then define G (( t )) := { M ∈ GL( n, C (( t ))) | P j ( M ) = 0 } . (2.4) Example
For G = SL(2 , C ), we have G (( t )) := (cid:40)(cid:32) a ( t ) b ( t ) c ( t ) d ( t ) (cid:33) | a ( t ) , b ( t ) , c ( t ) , d ( t ) ∈ C (( t )) and a ( t ) d ( t ) − b ( t ) c ( t ) = 1 (cid:41) . (2.5) Note that Sp(2 n, C ) denotes the symplectic group of rank n over the complex numbers. – 6 – he groups G [[ t ]] and G [ t − ]We can mimic this construction with C (( t )) replaced with C [[ t ]]. One can consider thegroup GL( n, C [[ t ]]) of matrices with coefficients in C [[ t ]] and which are invertible in C [[ t ]].Note that a necessary and sufficient condition for this to be the case is that the determinantshould belong to C [[ t ]] ∗ , i.e. have a series expansion with non-zero coefficient of degree 0.Up to this subtlety, the definition of G [[ t ]] goes exactly as before: G [[ t ]] := { M ∈ GL( n, C [[ t ]]) | P j ( M ) = 0 } . (2.6)Similarly, one can construct G [ t − ], bearing in mind that by the remark above GL( n, C [ t − ])is characterized by the property that the determinant is a non-vanishing constant. The groups G [ t − ]Finally, we define G [ t − ] := { M ∈ G [ t − ] | M | t →∞ = } . (2.7)In other words, G [ t − ] is the subgroup of G [ t − ] where the degree 0 term in the seriesexpansion of the elements are the identity matrix. In this section, we finally give the definition of the affine Grassmannian for the group G .This involves spaces of lattices, so we begin as a warm-up with a brief reminder aboutlattices.Consider a vector space V of dimension n . A lattice in V is a discrete subgroupof V which is isomorphic to Z n . One way to construct a lattice explicitly is as follows:pick a basis ( v , . . . , v n ) of V , and consider the set of all linear combinations with integercoefficients: L = (cid:40) n (cid:88) i =1 a i v i | a i ∈ Z (cid:41) . (2.8)So a basis of V defines a unique lattice L ⊂ V , but the converse is not true: many differentbases give the same lattice. Two bases correspond to the same lattice if and only if oneis obtained from the other by multiplication by a matrix in GL( n, Z ), the set of invertiblematrices with integer coefficients and whose inverse also has integer coefficients. On theother hand, the set of all bases of V can be identified with the set GL( n, R ), once areference basis is picked: one just writes the components of the basis vectors as columns of the matrix. We now turn to the crucial point of the argument: we have chosen to writethe basis vectors as columns of the matrix M ∈ GL( n, R ); then for any P ∈ GL( n, Z ), thematrix M P corresponds to the same lattice, as the columns of
M P are linear combinationswith integer coefficients of the columns of M . However P M does not a priori gives thesame lattice! Therefore the set of inequivalent lattices isSet of inequivalent lattices in V = GL( n, R ) / GL( n, Z ) , (2.9)– 7 –here the quotient is taken on the right . Multiplication on the left gives instead an actionof GL( n, Z ) on the set of lattices. In the context of the affine Grassmannian, it is thisaction which generates the orbits we are interested in.We now repeat the above discussion but replace our base field R by C (( t )) and theintegers Z by C [[ t ]]. A lattice L in V = C (( t )) n is a free C [[ t ]]-submodule of rank n . Alattice is fully specified by a basis, in other words an n × n matrix of elements of C (( t ))with non-zero determinant, or equivalently a matrix in GL( n, C (( t ))). We identify thebasis vectors with the columns of the matrix. Multiplying such a matrix on the right by a matrix in GL( n, C [[ t ]]) gives a matrix representing the same lattice. On the otherhand, multiplication on the left by a matrices in GL( n, C [[ t ]]) gives a family of possiblyinequivalent lattices, that we call the orbit of the initial lattice. We can identify the setof lattices as the set of equivalence classes of elements of GL( n, C (( t ))) with equivalencerelation given by multiplication on the right by GL( n, C [[ t ]]). Definition
The affine Grassmannian Gr G for the group G = GL( n, C ) is, as a set, theset of all lattices in C (( t )) n . By the remark above, we haveGr GL( n, C ) (cid:39) GL( n, C (( t ))) / GL( n, C [[ t ]]) . (2.10)More generally, for a group G we defineGr G (cid:39) G (( t )) /G [[ t ]] . (2.11)This can be seen as an infinite dimensional variety. The group G [[ t ]] acts on the lefton the points of the affine Grassmannian Gr G , thus generating orbits , which are Zariskiopen algebraic sets. The closure of the orbits are finite dimensional (generically singular)projective varieties. SL(2 , C ) and PSL(2 , C ) affine Grassmannians In this subsection, we leave our general discussion to illustrate it with explicit computationsat rank 1. Some of these computations can be found in [1, Proposition 2.6].
Points of Gr SL(2 , C ) In this paragraph, we set G = SL(2 , C ). The group G (( t )) has been explicitly written in(2.5). A matrix in this set represents a lattice as:Λ (cid:34)(cid:32) a bc d (cid:33)(cid:35) = (cid:40) u ( t ) (cid:32) a ( t ) c ( t ) (cid:33) + v ( t ) (cid:32) b ( t ) d ( t ) (cid:33) | u, v ∈ G [[ t ]] (cid:41) . (2.12)For any matrix R ∈ G [[ t ]], it is clear with this description thatΛ (cid:34)(cid:32) a bc d (cid:33)(cid:35) = Λ (cid:34)(cid:32) a bc d (cid:33) R (cid:35) . (2.13)The points in Gr G are equivalence classes of matrices M = (cid:32) a bc d (cid:33) ∈ G (( t )) under M ∼ M R , R = (cid:32) α βγ δ (cid:33) ∈ G [[ t ]] . (2.14)– 8 –his relation is non-trivial because R is restricted to lie in G [[ t ]], so α ( t ), β ( t ), γ ( t ) and δ ( t ) can not have negative powers of t , while a ( t ), b ( t ), c ( t ) and d ( t ) may have negativepowers. We can see this in action explicitly by trying to implement Gaussian elimination.Let us assume for definiteness that the most singular coefficient of M (the coefficient withlowest valuation) is d , and set λ = − ν ( d ) ≥ d = t − λ/ d where ν ( d ) = 0. The first step of Gaussianelimination uses R = (cid:32) d − ct λ/ d − (cid:33) ∈ G [[ t ]] giving M R = (cid:32) t λ/ bd − t − λ/ (cid:33) (2.15)Then for the second step one can write bd − = λ/ − (cid:88) i = ν ( b ) x i t i + t λ/ b (cid:48) (2.16)where ν ( b (cid:48) ) ≥
0, so that we obtain R = (cid:32) − b (cid:48) (cid:33) ∈ G [[ t ]] giving M R R = t λ/ λ/ − (cid:80) i = − λ/ x i t i t − λ/ (2.17)So the points in Gr G are parametrized by λ ∈ N , and then for a given λ by λ complexnumbers ( x − λ/ , x − λ/ , . . . , x λ/ − ). Orbits
We now describe the orbits under the right action of G [[ t ]]. From the point of view ofGaussian elimination, this means we can now combine rows of the matrix M together. It isclear from the explicit representative (2.17) that all the points of Gr G corresponding to thesame λ ∈ N form a single orbit, and conversely points corresponding to λ (cid:54) = λ (cid:48) belong todifferent orbits. Therefore, the orbits, which we denote by [Gr G ] λ , are labeled by λ ∈ N ,and can be defined by a distinguished element M λ :[Gr G ] λ = G [[ t ]] · M λ , M λ = (cid:32) t λ/ t − λ/ (cid:33) . (2.18)Elements of 2 N can be seen as the positive coweights of G .Consider an orbit [Gr G ] λ . For every u ∈ C ∗ , we have (cid:32) u − u − tu − (cid:33) M λ (cid:32) u − u − t λ − u (cid:33) = (cid:32) t λ/ − u t − λ/ t − λ/ (cid:33) (2.19)which when u → G ] λ − . This shows that the orbit [Gr G ] λ is openand contains [Gr G ] λ − in its closure. Therefore there is a partial order in the orbits definedby closure inclusions. The difference between the coweights labeling two adjacent orbits, λ − ( λ −
2) = 2, can be interpreted as the positive coroot in G .– 9 – = 0 λ = 2 λ = 4 λ = 6Component I = 0 ∈ Z λ = 1 λ = 3 λ = 5 λ = 7Component I = 1 ∈ Z Figure 1 . Hasse diagram for the orbits in the affine Grassmannian of PSL(2 , C ). Points and orbits of Gr PSL(2 , C ) In this paragraph we discuss PSL(2 , C ), seen as GL(2 , C ) quotiented by scalar matrices.In GL(2 , C ), the difference with the previous analysis is that the determinant conditionis lifted, so the orbits can be labeled by two integers a and b , with a ≥ b . In PSL(2 , C )though we have (cid:32) t a t b (cid:33) ∼ (cid:32) t a − b
00 1 (cid:33) , (2.20)so the orbits can be labeled by λ = a − b ∈ N . A computation analog to (6.4) shows thatagain the orbit (cid:2) Gr PSL(2 , C ) (cid:3) λ − lies in the closure of (cid:2) Gr PSL(2 , C ) (cid:3) λ . There are two connectedcomponents in Gr PSL(2 , C ) : one is the union of all the orbits (cid:2) Gr PSL(2 , C ) (cid:3) λ for λ ∈ N , whichgives Gr SL(2 , C ) , and the other is the union of all the orbits (cid:2) Gr PSL(2 , C ) (cid:3) λ for λ odd. Thecomplex dimension of (cid:2) Gr PSL(2 , C ) (cid:3) λ is λ in all cases, so it is even in one component andodd in the other. This is summarized in Figure 1. We now come back to the general discussion and spell out how the observations of theprevious subsections generalize to an arbitrary (semisimple complex algebraic) group G .An important role is played by the coweights and the coroots of G , so we begin with areminder of a few definitions. – 10 –et us pick a maximal torus and a Borel subgroup T ⊂ B ⊂ G , with Lie algebras h ⊂ b ⊂ g . A character of G is a homomorphism T → C ∗ while a cocharacter is ahomomorphism C ∗ → T . The differentials of characters are weights, while the differentialsof cocharacters are coweights. We call Λ the coweight lattice of G and Λ the corootlattice of G . The goal of this subsection is to define a collection of elements M λ ∈ G (( t ))generalizing (2.18) from which we will build orbits in the affine Grassmannian, so it isnatural to require that λ be a coweight , and not a weight. The coweights of G can be usedto label the (highest-weight) irreducible representations of the Langlands dual group G ∨ .We use the Killing form to identify the space of weights and the space of coweights. TheBorel subalgebra b fixes a set of simple roots, which in turn defines a set of fundamentalweights in the usual fashion. We call λ + (respectively Λ +0 ) the set of linear combinationswith nonnegative integer coefficients of the fundamental coweights (resp. simple coroots).Any weight or coweight can then be expressed in the basis of the fundamental weights( (cid:36) i ) i =1 ,...,r , and we denote a (co)weight by its coordinates in this basis λ = r (cid:88) i =1 λ i (cid:36) i . (2.21)The coweights are partially ordered as follows. For λ and µ two coweights, λ ≤ µ ⇐⇒ µ − λ is a linear combination ofsimple coroots with coefficients in N ⇐⇒ µ − λ ∈ Λ +0 . (2.22)This partial order defines a Hasse diagram for the dominant coweights. This Hasse diagramhas connected components labeled by classes Λ / Λ . Definition
Given a coweight λ we have a corresponding homomorphism C ∗ → T . Denot-ing by t ∈ C ∗ the formal variable, the image of this homomorphism is called M λ ( t ) := M λ .We have identified the affine Grassmannian of G with the spaceGr G = G (( t )) /G [[ t ]] , (2.23)where a matrix in G (( t )) specifies a lattice by giving an explicit basis, and the quotient by G [[ t ]] eliminating the arbitrariness in the choice of such a basis. We also recall that given amatrix M ∈ G (( t )), multiplying on the right by an element of G [[ t ]] gives the same lattice,as we just said, but multiplying on the left gives a different lattice. We can then denotepoints and orbits in the affine Grassmannian as follows: • M · G [[ t ]] is the point in Gr G corresponding to M • G [[ t ]] · M · G [[ t ]] is the orbit of that point in Gr G .Given a coweight λ of G , we can build a full orbit in the affine Grassmannian, whichis denoted: [Gr G ] λ = G [[ t ]] · M λ · G [[ t ]] . (2.24)– 11 –wo coweight λ and λ (cid:48) which belong to the same Weyl group orbit give the sameorbit in the affine Grassmannian, [Gr G ] λ = [Gr G ] λ (cid:48) , so we can restrict our attention to theorbits [Gr G ] λ for λ dominant. It turns out these orbits provide a partition of the affineGrassmannian, which can be written as a disjoint unionGr G = (cid:91) λ ∈ Λ + [Gr G ] λ . (2.25)The (Zariski) closure of [Gr G ] λ is[Gr G ] λ = [Gr G ] ≤ λ = (cid:91) µ ∈ Λ + ,µ ≤ λ [Gr G ] µ . (2.26)Therefore, the partial order on the dominant coweights Λ + provides the Hasse diagramfor the G [[ t ]] orbits in the affine Grassmannian Gr G . There is one Hasse diagram for eachconnected component of Gr G , and by construction we have π (Gr G ) = Λ / Λ = π ( G ) . (2.27)Elements of this group will be denoted by an index I . We now introduce the notion oflowest dominant coweights. A coweight Ω is a lowest dominant coweight if it is dominant,and if in addition there is no other dominant coweight ω such that ω < Ω. There are asmany components in the affine Grassmannian of G as there are lowest dominant coweightsof G ; we call them Ω I , where the index I ∈ π ( G ). We list the lowest dominant coweightsfor simple groups in Table 2.The [Gr G ] λ are called Schubert cells and the [Gr G ] ≤ λ are called Schubert varieties . TheSchubert cell [Gr G ] λ is a smooth variety of dimensiondim C [Gr G ] λ = (cid:104) ρ, λ (cid:105) , (2.28)where ρ is the Weyl vector [41, Prop 2.1.5]. The Schubert varieties can be singular spaces,but are well behaved: the singularities are always normal, Gorenstein and Cohen-Macaulay[41, Theorem 2.1.21]. Their complex dimension can be even or odd, but the parity of thecomplex dimension remains the same in a given connected component of Gr G . One can use other groups than G [[ t ]] to generate orbits. In particular, we can use G [ t − ]to generate the orbit [Gr G ] λ := G [ t − ] · M λ · G [[ t ]] , (2.29)which is called an opposite Schubert cell . From this one can build an (infinite dimensionalbut finite codimensional) opposite Schubert variety [Gr G ] ≥ λ := (cid:91) µ ∈ Λ + ,µ ≥ λ [Gr G ] µ . (2.30)– 12 – π ( G ) = Z ( G ∨ ) Ω I PSL( n + 1 , C ) Z n +1 Ω = [0 , . . . , l = [0 , . . . , (cid:124) (cid:123)(cid:122) (cid:125) l − , , . . . , n + 1 , C ) Z Ω = [0 , , . . . , = [1 , , . . . , n, C ) Z Ω = [0 , . . . , , = [0 , . . . , , n, C ) Z × Z Ω (0 , = [0 , , . . . , , , (1 , = [1 , , . . . , , , (1 , = [0 , , . . . , , , (0 , = [0 , , . . . , , , n + 2 , C ) Z Ω = [0 , , . . . , , , = [0 , , . . . , , , = [1 , , . . . , , , = [0 , , . . . , , , E / Z Z Ω = [0 , , , , , = [1 , , , , , = [0 , , , , , E / Z Z Ω = [0 , , , , , , = [1 , , , , , , E trivial Ω = [0 , , , , , , , F trivial Ω = [0 , , , G trivial Ω = [0 , Table 2 . Lowest dominant coweights Ω I (given in the basis of fundamental coweights) for centerlesssimple groups G . We denote elements of π ( G ) = Z ( G ∨ ) I . To each Ω I there is a component labelled I in the affine Grassmannian of G . Note that [Gr G ] λ ∩ [Gr G ] λ = G · M λ · G [[ t ]] is not reduced to a point. This motivates theintroduction of a last type of orbits, in which G does not act at degree 0, namely[ W G ] λ := G [ t − ] · M λ · G [[ t ]] . (2.31)The intersection of these orbits with Schubert cells and Schubert varieties give the thespaces which are the main focus of this paper:[ W G ] µλ = [ W G ] λ ∩ [Gr G ] µ and [ W G ] µλ = [ W G ] λ ∩ [Gr G ] µ (2.32)The spaces [ W G ] µλ intersect [Gr G ] ν transversely for any λ ≤ ν ≤ µ , and are called the slices in the affine Grassmannian. They are varieties with symplectic singularities [11, 33], with– 13 –imension dim C [ W G ] µλ = (cid:104) ρ, µ − λ (cid:105) . (2.33)In particular, when µ − λ is a simple coroot α ∨ , the two orbits [Gr G ] µ and [Gr G ] λ areadjacent in the Hasse diagram, the elementary transverse slice has complex dimension is2, and one can show that [11, Example 2.2][ W G ] λ + α ∨ λ = C / Z (cid:104) λ,α (cid:105) = A (cid:104) λ,α (cid:105) . (2.34)This shows that the “generic” elementary slice in the affine Grassmannian is a Kleiniansingularity of type A . More generally, elementary slices correspond to pairs of adjacentcoweights according to the partial order (2.22). This problem has been studied in detailin [10, 34, 48], and the conclusion is that the elementary transverse slices fall into threecategories [10]: • Kleinian singularities of type A , (2.34), when µ − λ is a simple coroot. These aredenoted A n . • Closure of minimal nilpotent orbits, when µ − λ is the short dominant coroot of thesubalgebra defined by the nonvanishing of µ − λ . These are denoted a n , b n , c n , d n , e , e , e , f and g . • Quasi-minimal singularities, which can be of three types:1. An infinite family called ac n , for n ≥ ag singularity;3. The cg singularity.The existence of the elementary slices other than Kleinian of type A is caused by geometricconstraints in the Weyl chambers when approaching walls. The disposition D of the affineGrassmannian Gr G is r = rank( G ). Close to a codimension d wall of the Weyl chamber,the disposition of a generic coweight decreases to r − d .In the next section we will see this at work in the affine Grassmannian at rank 2. Thequasi-minimal singularities will be studied in detail in Section 3.3. In this section we explore the affine Grassmannian, or rather the slices in the affine Grass-mannian, of simple groups. The two coweights which define a slice can be used to producea quiver, whose Coulomb branch is the slice. The terminology comes from [10]. The Disposition D of a leaf and a moduli space is defined in (1.1) and (1.2) respectively in the Intro-duction. – 14 –otation Explanation C [ t ] Ring of polynomials in t C [[ t ]] Ring of formal power series in t C (( t )) Field of formal Laurent series in t ,also field of fractions of C [[ t ]] G Simple algebraic subgroup of GL( n, C ) g Lie algebra of GC g Cartan matrix of g Λ Coweight lattice of G Λ Coroot lattice of G ( α i ) i =1 ,...,r , ( α ∨ i ) i =1 ,...,r Simple roots and simple coroots of g ( (cid:36) i ) i =1 ,...,r , ( (cid:36) ∨ i ) i =1 ,...,r Fundamental weights and fundamental coweights of Gρ Weyl vector, half-sum of the positive roots G [ t ] Group G “with coefficients in C [ t ]” G [[ t ]] Group G “with coefficients in C [[ t ]]” G (( t )) Group G “with coefficients in C (( t ))” G [ t − ] Subgroup of G [ t − ] where the t part is the identity of G Gr G = G (( t )) /G [[ t ]] Affine Grassmannian of GM λ Image in G (( t )) of the coweight λ ∈ Λ[Gr G ] λ = G [[ t ]] · M λ · G [[ t ]] Schubert cell associated to dominant coweight λ ,a smooth variety of complex dimension (cid:104) ρ, λ (cid:105) [Gr G ] λ = (cid:70) σ ≤ λ [Gr G ] λ Schubert variety, the Zariski closure of theSchubert cell [Gr G ] λ , of dimension (cid:104) ρ, λ (cid:105) [Gr G ] λ = G [ t − ] · M λ · G [[ t ]] Opposite Schubert Cell (infinite-dimensional)[ W G ] λ = G [ t − ] · M λ · G [[ t ]] Transverse slice to Schubert cellin Gr G (infinite dimensional)[ W G ] µλ = [ W G ] λ ∩ [Gr G ] µ Transverse slice from µ to λ , a symplecticsingularity of complex dimension (cid:104) ρ, µ − λ (cid:105) Table 3 . Summary of the notations. – 15 – .1 General formula for slices
From now on we assume that G is simple with rank r , so that its Lie algebra can becharacterized by a connected Dynkin diagram. In [12, 20], following the series of works[22, 24, 27, 28] the slices in the affine Grassmannian Gr G have been identified with Coulombbranches of certain 3d N = 4 good quiver gauge theories. In this subsection, we summarizethis connection and give explicit formulas. From slice to quiver.
Let λ and µ be two coweights. We assume that • µ is dominant. • µ − λ is in the positive coroot lattice.Then we can construct a quiver Q µλ as follows: • There are r gauge nodes, connected as in the Dynkin diagram of g , with node ranks k i = (cid:104) (cid:36) i , µ − λ (cid:105) . (3.1)When λ and µ are identified with their components in the basis of fundamentalweights, the vector of gauge nodes ranks is k = C − g · ( µ − λ ). • There are r flavor nodes, connected to the r gauge nodes, with ranks given by N i = (cid:104) α ∨ i , µ (cid:105) . (3.2)When µ is identified with its components in the basis of fundamental weights, N = µ .Note that the condition that µ − λ be in the positive coroot lattice guarantees preciselythat k is an element of N r . From good quiver to slice.
Consider a good quiver such that the gauge nodes formthe Dynkin diagram of a simple Lie algebra g . We denote by k = ( k i ) i =1 ,...,r the ranks ofthe gauge nodes and N = ( N i ) i =1 ,...,r the ranks of the flavor nodes, with r the rank of g .To this quiver we associate two coweights: • The coweight µ is given by the flavor nodes, µ = r (cid:80) i =1 n i (cid:36) i ; • The coweight λ is given by λ = N − C g · k . Note that the components of the coweight λ gives the balance of the gauge nodes of the quiver. Theassumption that the quiver is good implies that λ is dominant. – 16 – xample. For instance if one considers the algebra C and coweights expressed in thebasis of fundamental weights λ = [2 ,
0] and µ = [2 , C g = (cid:32) − − (cid:33) one gets µ − λ = [0 , C − g · ( µ − λ ) = [1 ,
2] sothe quiver is Q µλ = . (3.3)Note that the gauge node on the right is balanced, corresponding to the vanishing compo-nent of λ .If instead one considers λ = [0 ,
1] and µ = [2 , µ − λ = [2 ,
1] is not in thepositive coroot lattice. Accordingly, C − g · ( µ − λ ) = [ ,
3] is not a vector of integers andno quiver can be defined.Note that it is not necessary that the coweight λ be dominant. For instance considerthe following pair: λ = [ − , µ = [4 , Q µλ = . (3.4)Note that the node on the right is under-balanced. Generalized Slices.
When λ is not dominant, the construction of the quiver Q µλ stillmakes sense. The quiver is not good in the sense of [49], and its Coulomb branch does notcorrespond to a slice in Gr G . However the Coulomb branch of Q µλ can be identified withthe generalized slices of [12]. Explicit quivers
The quivers Q µλ for algebras of classical types are gathered in Table 4.We use the following explicit form for the inverse of the Cartan matrix for g = A n : C − g = (cid:18) min( i, j ) − ijn + 1 (cid:19) ≤ i,j ≤ n . (3.5) In this subsection, we use the description reviewed above to construct the bottom part ofthe Hasse diagrams for complex simple groups of ranks 1 and 2. This can be achieved byfocusing only on the centerless groups associated to the Lie algebra (the affine Grassman-nian for groups with non-trivial center are obtained by taking only the relevant connectedcomponents). This is presented in Figures 2 to 13. For conventions regarding the C algebra, see Figure 9. – 17 – aption for Figures 2-13. For each Lie algebra, we first draw a diagram with ourLie algebra conventions. This diagram represents both the Cartan subalgebra and itsdual, identified via the Killing form. The red arrows represent the roots, the black arrowrepresent the fundamental weights. The simple roots and coroots, and the fundamentalweights and coweight are explicitly labeled. The dots denote the coweight lattice, wheredifferent colors are used for different elements of the group (2.27), and therefore correspondto different connected components in (the Hasse diagram of) the affine Grassmannian. TheWeyl chamber, defined by the fundamental coweights, is shaded in gray.Then for each element of the group (2.27) we draw a Hasse diagram for the few lowestorbits in the corresponding component of the affine Grassmannian. The dots in the Hassediagram are the coweights and correspond to orbits (2.24), while the lines connect adja-cent coweights according to the partial order (2.22). Dotted lines indicate that the Hassediagram is infinite.A connected component can be labeled by a lowest coweight Ω I . In that component,next to each dot corresponding to a coweight µ , we draw the quiver Q µ Ω I . The Coulombbranch of this quiver is the transverse slice to the orbit at the origin of the Hasse diagram,[ W G ] µ Ω I . Finally, next to each line we indicate the nature of the elementary transverse slice,using the notations specified at the end of Section 2.6. Disposition and quasi-minimal slices.
From the Hasse diagrams presented in thissection, one can observe that the appearance of slices beyond Kleinian singularities of type A is related to space constraints close to the walls of the Weyl chamber. In the A case, anon-zero coweight λ always has at least one coweight µ such that µ − λ is a simple coroot.This is related to the basic geometric fact that the Weyl chamber has an angle of π at theorigin, which is the angle for equilateral triangles.In the B case, the Weyl chamber has an angle π at the origin. Adding any simplecoroot to the coweight (cid:36) ∨ in Figure 6 gives a coweight outside the Weyl chamber. In orderto stay in the Weyl chamber, one has to add the non simple coroot α ∨ + α ∨ . This leads toan elementary slice with quasi-minimal singularity in Figure 8. The same obviously appliesto the C case.Finally in the G case, there are two non-zero coweights which are such that addingany simple coroot gives a coweight outside the Weyl chamber, namely (cid:36) ∨ and 2 (cid:36) ∨ . Thisgives rise to two elementary slices with quasi-minimal singularities in Figure 13.– 18 – = α ∨ (cid:36) = (cid:36) ∨ A A A Component 0 ∈ Z A A A Component 1 ∈ Z Figure 2 . Bottom of the Hasse diagram for PSL(2 , C ). See detailed caption on page 18. – 19 – α (cid:36) (cid:36) Figure 3 . Summary of A data. See detailed caption on page 18. The red dots correspond to thecoroot lattice Λ , while the blue dots are the lattice Λ + (cid:36) and the green dots are the latticeΛ + (cid:36) . a A A A A A A A A A A A A A Figure 4 . Bottom of the Hasse diagram for PSL(3 , C ), component 0 ∈ Z . See detailed captionon page 18. – 20 – A A A A A A A A A A Figure 5 . Bottom of the Hasse diagram for PSL(3 , C ), component 1 ∈ Z . See detailed captionon page 18. The third component − ∈ Z is obtained by symmetry. – 21 – = α ∨ α ∨ (cid:36) ∨ α (cid:36) = (cid:36) ∨ (cid:36) Figure 6 . Summary of B data. See detailed caption on page 18.The black dots correspond to thecoroot lattice Λ , while the blue dots are the lattice Λ + (cid:36) ∨ . – 22 – A A A A A A A A A A A A A A A A
000 0110 1 122 0220 2 232 1 244 0330 3 342 2 354 1 366 0440 4 452 3 464 2 476 1 488 0
Figure 7 . Bottom of the Hasse diagram for PSO(5 , C ), component 0 ∈ Z . See detailed captionon page 18. – 23 – c A A A A A A A A A A A A A A A A
111 1 123 0221 2 233 1 245 0331 3 343 2 355 1 367 0441 4 453 3 465 2 477 1 489 0
Figure 8 . Bottom of the Hasse diagram for PSO(5 , C ), component 1 ∈ Z . See detailed captionon page 18. – 24 – α ∨ α = α ∨ (cid:36) = (cid:36) ∨ (cid:36) (cid:36) ∨ Figure 9 . Summary of C data. See detailed caption on page 18. The black dots correspond tothe coroot lattice Λ , while the blue dots are the lattice Λ + (cid:36) ∨ . – 25 – A A A A A A A A A A A A Figure 10 . Bottom of the Hasse diagram for Sp(4 , C ), component 0 ∈ Z . See detailed caption onpage 18. – 26 – c A A A A A A A A Figure 11 . Bottom of the Hasse diagram for Sp(4 , C ), component 1 ∈ Z . See detailed caption onpage 18. – 27 – = α ∨ α α ∨ (cid:36) = (cid:36) ∨ (cid:36) (cid:36) ∨ Figure 12 . Summary of G data. See detailed caption on page 18. – 28 – cg A ag A A A A A A A A A A A A A A A A A A A
121 1 32242 6126 41 116 32 127 4 12823 13851 137 13 11711 53363 2 6421 74484 12 8531 955105 3 9622 106
Figure 13 . Bottom of the Hasse diagram for G . See detailed caption on page 18. – 29 –lgebra Quiver A n n (cid:80) j =1 jν n +1 − j n +1 · · · n (cid:80) j =1 (cid:16) min( i, j ) − ijn +1 (cid:17) ν j · · · n (cid:80) j =1 (cid:16) n ( n +1) − njn +1 (cid:17) ν j µ µ i µ n B n n (cid:80) j =1 ν j · · · i (cid:80) k =1 n (cid:80) j = k ν j · · · n − (cid:80) k =1 n (cid:80) j = k ν j n (cid:80) k =1 n (cid:80) j = k ν j µ µ i µ n − µ n C n n − (cid:80) j =1 ν j + ν n · · · i (cid:80) k =1 n − (cid:80) j = k ν j + iν n · · · n − (cid:80) k =1 n − (cid:80) j = k ν j + ( n − ν n n − (cid:80) k =1 n − (cid:80) j = k ν j + nν n µ µ i µ n − µ n D n n − (cid:80) j =1 ν j + ν n − + ν n · · · i (cid:80) k =1 n − (cid:80) j = k ν j + ν n − + ν n /i · · · n − (cid:80) k =1 n − (cid:80) j = k ν j + ν n − + ν n / ( n −
2) 12 n − (cid:80) k =1 n − (cid:80) j = k ν j + nν n − +( n − ν n n − (cid:80) k =1 n − (cid:80) j = k ν j + ( n − ν n − + nν n µ µ i µ n − µ n − µ n Table 4 . For algebras of classical types, we depict the quivers Q µλ where µ = n (cid:80) j =1 µ i (cid:36) i and λ = n (cid:80) j =1 λ i (cid:36) i are coweights with µ dominant and ν := µ − λ in the positive coroot lattice. If λ is dominant, their 3d N = 4 Coulomb branch is the transverse slice [ W G ] µλ . –30– .3 Quivers for Quasi-Minimal Singularities In this section we focus on the elementary slices in the affine Grassmannian that did notmake an appearance in [32], corresponding to the quasi-minimal singularities reviewed inSection 2.6: • The ac n slices are the lowest slices in the component of Gr Sp( n ) not connected to theidentity. • The slices cg and ag , which appear in Gr G , and were identified in Figure 13.In this section, we study magnetic quivers for these elementary slices, which are crucial inthe algorithm of quiver subtraction and quiver addition, see Section 5. They are deducedfrom the coweights that are used to define them, using the formulas of Section 3.1. Thequivers are gathered in Table 1.We start with the ac n singularity. It is the Coulomb branch of the quivers ac n : 1 1 . . . n (3.6)The global symmetry is su ( n ) × u (1), which is consistent with the rightmost node beingoverbalanced, while the other nodes are balanced and form the A n − Dynkin diagram. TheCoulomb branch Hilbert series can be neatly encapsulated in the form of a highest weightgenerating function (HWG) [50]:HWG = 1 − µ µ n − t (1 − t )(1 − µ µ n − t )(1 − µ t q )(1 − qµ n − t ) (3.7)= PE (cid:104) t + µ µ n − t + ( qµ k + q − µ kn − ) t k +1 − µ k µ kn − t k +2 (cid:105) where µ i and q are Dynkin fugacities of su ( n ) and u (1) respectively. From the HWG wecan also obtain the unrefined Hilbert series. For ac , the unrefined Hilbert series is:HS = 1 + 2 t + 4 t + 2 t + t (1 − t ) (1 − t ) (1 + t + t ) (3.8)In addition, we also provide the refined plethystic logarithm (PL) of the Hilbert series whichencodes representation content of the generators and relations in the Coulomb branch inTable 5.For G affine Grassmannian, we have two new elementary slices which are the ag and cg . For ag , the quiver takes the form: ag : 1 11 1 (3.9)– 31 –ith global symmetry su (2) × u (1). The HWG takes the form:HWG = (1 − µ t )(1 − t )(1 − µ t )(1 − µ t q )(1 − µ qt ) (3.10)= PE[ t + µ t + ( q + q − ) µ t − µ t ]where µ and q are the Dynkin fugacities of su (2) and u (1) respectively. The unrefinedHilbert series is HS = 1 + 2 t + 8 t + 2 t + t (1 − t ) (1 − t ) (3.11)and the refined PL is given in Table 6. For cg the quiver takes the form: cg : 1 1 1 (3.12)where the global symmetry is su (2) × u (1). The HWG is:HWG = 1 + µ t + µ t (1 − t ) (cid:16) − µ t q (cid:17) (1 − qµ t ) (3.13)= PE[ t + µ t + ( q + q − ) µ t − µ t ]where µ and q are the Dynkin fugacities of su (2) and u (1) respectively. The unrefinedHilbert series takes the form: HS = 1 − t + 5 t − t + t (1 − t ) (1 + t + t ) (3.14)with the refined PL given in Table 5.These three singularities, along with Kleinian singularities and closures of minimalnilpotent orbits are all the singularities that occur as elementary slices in affine Grassman-nians. However, this does not mean we have a complete list of all possible elementary slicesfor symplectic singularities. In [51], we introduced a two parameter family h n,k ∼ = H n / Z k of singularities which generalizes the c n singularities ( c n = h n, ). They were used to char-acterize certain elementary slices in Higgs branches of 4d N = 2 rank 1 SCFTs. The quiverfor h n,k is . . .
11 1 1 1 1 n k (3.15)with u ( n ) global symmetry. – 32 – Beginning of the expansion of the PL2 t : [2] + [0] t : ( q + q )[2] t : − [0] t : − ( q + q )([0] + [2])3 t : [11] + [00] t : q [20] + q [02] t : − [00] − [11] t : − q ([01] + [12] + [20]) − q ([02] + 10] + [21])4 t : [101] + [000] t : q [200] + q [002] t : − [000] − [020] − [101] t : − q ([010] + [111] + [200]) − q ([002] + [010] + [111]) Table 5 . ac n refined PL. [ . . . ] are the Dynkin labels for SU ( n ). Beginning of the expansion of the PL ag t : [2] + [0] t : − [0] + q [3] + q [3] t − (1 + q + q )[0] − [2] − [4] cg t : [2] + [0] t q [3] + q [3] t : − [0] + q [3] + q [3] t : − ( q + q )([1] + [3]) Table 6 . ag and cg refined PL. [ . . . ] are the Dynkin labels for SU (2). Along a similar line of logic, we expect the generalization of ac n and ag a two param-eter family of singularities that we call ¯ h n,k , with a = ¯ h n, , ac n = ¯ h n, and ag = ¯ h , .The quiver for ¯ h n,k is . . .
11 1 1 1 1 n k (3.16)where the global symmetry is also u ( n ). The Coulomb branch is no longer an orbifold.– 33 –he HWGs for the singularities h n,k and ¯ h n,k areHWG[ h n,k ] = PE (cid:104) t + µ µ n − t + ( qµ k + q − µ kn − ) t k +1 − µ k µ kn − t k (cid:105) (3.17)HWG[¯ h n,k ] = PE (cid:104) t + µ µ n − t + ( qµ k + q − µ kn − ) t k +1 − µ k µ kn − t k +2 (cid:105) (3.18)where µ i and q are the Dynkin fugacities of u ( n ) ∼ = su ( n ) × u (1). A highest weight varietyfor h n,k is C × C / Z k , with equation xy = z k . A highest weight variety for ¯ h n,k is thethreefold defined by the equation xy = z k w for ( x, y, z, w ) ∈ C . Unframed quivers.
So far we have considered quivers with an explicit framing (flavornodes). Magnetic quivers for higher dimensional supersymmetric gauge theories (4 d N = 2,5 d N = 1 and 6 d N = (1 ,
0) theories) usually arise from brane configurations, and are moreconveniently written without framing. For unframed quivers, it is understood that a U(1)node has to be ungauged on the long side [32, 52]. The quivers discussed in this sectionare summarized in both framed and unframed form in Table 1.
In this section we construct brane systems for the affine Grassmannian of classical groupsconsisting of NS5, D5, and D3 branes in Type IIB String Theory as first developed in[9], possibly in presence of an ON plane. The method of reading quivers from such branesystems, including quivers for transverse slices, is reviewed in Appendix A.
Affine Grassmannian.
As discussed in Section 2, the affine Grassmannian [Gr G ] of agroup G consists of several connected components, one for each lowest dominant coweightΩ I , see Table 2. Each connected component of the affine Grassmannian is infinite dimen-sional, as are the transverse slices [ W G ] λ . The transverse slices [ W G ] µλ however are finitedimensional and can be constructed as a Coulomb branch of a quiver theory as discussedin Section 3. Every such quiver describes the low energy theory of a brane system, whichcan be obtained using simple rules (see Appendix A). The goal of this section however, isto turn this around and to generate the affine Grassmannian, or rather any [ W G ] λ , froma brane system. For every component I in the affine Grassmannian a brane system isproposed whose phases correspond to the symplectic leaves of [ W G ] Ω I . From this branesystem all of [ W G ] Ω I can be constructed. Since the objective is to construct an infinitedimensional space, an infinite number of moduli (D3 branes) must be present in the branesystem. To study the symplectic leaves of [ W G ] Ω I from bottom-up, we restrict the move-ments of almost every D3 brane and only turn on few moduli at a time, moving from onesymplectic leaf of [ W G ] Ω I to another. Starting at the lowest leaf, i.e. all D3 branes restat the origin, this process generates the transverse slices to the lowest leaf in the compo-nent at hand. This bottom-up construction solely uses Kraft-Procesi transitions [13–15] inbrane systems, without the need to rely on the results of Section 3. As a result, the affineGrassmannian arises naturally in brane systems. Higgs branch constructions are possible but not discussed in this paper. – 34 –ll other finite slices [ W G ] µλ in the affine Grassmannian are contained in the branesystems as (generically non-elementary) Kraft-Procesi transitions. Furthermore any [ W G ] λ may be constructed bottom-up from the proposed brane systems by first moving to thephase in the brane system which corresponds to [ W g ] λ Ω I , ignoring the already turned onD3 moduli, and proceeding with Kraft-Procesi transitions. A -type affine Grassmannian The affine Grassmannian of PSL( n, C ) consists of n connected components. We proposea brane system for each such component. More precisely, we propose a brane system for (cid:2) W P SL ( n, C ) (cid:3) Ω I for every lowest dominant coweight Ω I . Component ∈ Z n The component connected to the origin (component 0) is in factthe affine Grassmannian of SL( n, C ). The proposed brane system for (cid:2) W P SL ( n, C ) (cid:3) [0 ,..., consists of n NS5 branes with an infinite number of D5 branes on both sides, and aninfinite number of D3 branes extended between every neighbouring pair of D5 branes. · · ·· · · · · · ∞ D ∞ D3 ∞ D n NS5 ∞ D5 ∞ D5 (4.1)When drawing the brane system, we generally do not depict branes which are not essentialto study the symplectic leaf at hand, i.e. we suppress those branes resting at the origin.Depicting the origin of (cid:2) W P SL ( n, C ) (cid:3) [0 ,..., and suppressing all irrelevant branes leads to thefollowing: · · · n NS5 (4.2)– 35 –or this brane system we read the empty quiver:00 00 00 · · · (4.3)i.e. the origin of (cid:2) W P SL ( n, C ) (cid:3) [0 ,..., corresponding to the lowest dominant coweight Ω =[0 , , . . . , Other components.
The transverse slices to the lowest leaf in the remaining connectedcomponents of the affine Grassmannian of PSL( n, C ) can be constructed from a branesystem where a single D5 is in one of the intervals between two neighbouring NS5 branes.For example, placing a D5 brane in the first interval we obtain the following brane system: · · · n NS5 . (4.4)For this brane system we read the quiver:01 00 00 · · · (4.5)which represents the origin of [ W PSL n ] [1 , ,..., , corresponding to the lowest dominant coweightΩ = [1 , , . . . , thanany of the D5 branes in (4.2). Quotients of
SL( n, C ) . For quotients SL( n, C ) / Z k , where k divides n , k connected com-ponents make up its affine Grassmannian. The starting points for the respective compo-nents are those with a single D5 placed in the I -th interval satisfying the condition I mod nk = 0 , (4.6) All prescriptions for linking numbers are equivalent and can be used. – 36 –hich has k solutions for I ∈ { , . . . , n − } .Example: The possible starting intervals for a D5 (depicted by an × ) to be placed fora respective component of the affine Grassmannian of SL(6 , C ) / Z k are as follows: k = 6 × × × × × k = 3 × × k = 2 × k = 1 (4.7)Note that a brane system without a D5 in any interval corresponds to the componentconnected to the origin, the 0-th component, which is always present for any SL( n, C ) / Z k . Identifying the component.
Given a brane system for any slice in the affine Grass-mannian of PSL( n, C ), with n l D5 branes in the l -th interval, there is a simple formula tocompute which component the corresponding slice belongs to: n − (cid:88) l =1 ln l mod n = I , (4.8)where I labels the component. Now that we addressed the various components of theA-type affine Grassmannian, we can start exploring the stratification of each component. Symplectic leaves, minimal transitions.
In order to study the different leaves of agiven transverse slice in the affine Grassmannian one can start turning on Coulomb branchmoduli of the brane system. However, this has to be done in a systematic way in orderto identify every leaf. One has to carefully turn on a ‘minimal’ amount of moduli in thefollowing two ways.In order to open up Coulomb branch directions, one needs to consider an interval ofD5 branes that contains at least 2 NS5 branes. In order to open up a minimal direction,the D5 branes need to be the closest two D5 branes separated by at least two NS5 branes.This leaves two options:1. Activating a D3 brane between two neighbouring D5 branes (i.e. there is no D5 branebetween them) separated by p NS5 branes, where p ≥
2. This corresponds to the– 37 –ransverse slice a p − . · · · p a p − . (4.9)2. Activating a D3 brane between two D5 branes which have exactly 2 NS5 branesbetween them, with possibly q ≥ A q +1 . q · · · A q +1 . (4.10)Clearly (4.9) for p = 2 and (4.10) for q = 0 are the same transition, as a = A .This allows a transition from one leaf to another. The quiver can be read from thebrane system. Its Coulomb branch contains the moduli which are turned on, and themoduli space is the closure of the corresponding leaf. This is analysed in detail for rank 1and 2 in the following two sections. A affine Grassmannian The affine Grassmannian for PSL(2 , C ) has two connected components, corresponding tothe two lowest dominant coweights, [0] and [1]. Component ∈ Z (cid:2) W PSL(2 , C ) (cid:3) [0] , for fte 0-th component of the affine Grassmannian ofPSL(2 , C ), which is the affine Grassmannian of SL(2 , C ), is given by a brane system withinfinitely many D3 branes which are free to move between two NS5 branes in the presenceof D5 branes. The origin of (cid:2) W PSL(2 , C ) (cid:3) [0] is where all these infinitely many branes arecoincident. Let us draw the two NS5 branes and suppress the infinitely many D3 branes– 38 –t the origin, as well as infinitely many D5 branes to the left and right. (4.11)We move to the lowest non-trivial leaf by pulling a D3 brane from the stack of infinitelymany branes at the origin and placing it between two D5 branes and letting it slide alongthe NS5 branes, corresponding to an A transition: . (4.12)The stuck D3 branes between the two pairs of D5 and NS5 can be annihilated by a Hanany-Witten transition and we can read the quiver whose Coulomb branch is the closure of theleaf: 12 . (4.13)The only way to keep moving onto higher leaves in the affine Grassmannian is by repeatingthe same process. However, the D5 branes already placed between the NS5 branes makethe transverse slices A k − for the k -th transition of this type. A general leaf and thecorresponding quiver are: 2 k · · · ... k k k . (4.14)– 39 –hich is indeed what we find in Section 3. Component ∈ Z For the first connected component of PSL(2 , C ) we start with thefollowing brane system: . (4.15)Which depicts the origin of (cid:2) W PSL(2 , C ) (cid:3) [1] . The transverse slice for the k -th transition is A k while the brane system and quiver for the leaf are:2 k + 1 · · · ... k k k + 1 . (4.16)The quivers in (4.14) correspond to slices to the lowest leaf (origin) in the 0-th componentof the affine Grassmannian, while the quivers in (4.16) correspond to slices to the lowest leafin the 1-st component. The other slices, between any two leaves in a connected component,are realised as (generically non-minimal) Kraft-Procesi transitions [15] in the brane system.The form of the quiver is a framed A quiver, or an SQCD theory with U ( k ) gauge groupand N flavors, satisfying N ≥ k . k is the number of minimal slices between the twoleaves, and N is even for slices in the 0-th component, while N is odd for slices in the1-st component. In this way every good quiver of A type is realised in one of the twobrane systems corresponding to the two connected components of the affine Grassmannianof SL(2 , C ). A affine Grassmannian The affine Grassmannian of PSL(3 , C ) has 3 disconnected components. Component ∈ Z The brane systems for the lower symplectic leaves and elementaryslices between them for the component connected to the origin are depicted in Figure 14.
Component , ∈ Z The brane systems for the lower symplectic leaves and elementaryslices between them for one of the components not connected to the origin are depictedin Figure 15. The brane system for the other component is related to the one depicted inFigure 15 by a Z action x (cid:55)→ − x , and we do not draw it.– 40 – · · · · · a A A A A A ··· A Figure 14 . Brane depiction of the low dimensional leaves [and elementary slices between them] of (cid:2) W PSL(3 , C ) (cid:3) [0 , . –41– · · · · · · · · · · · A A A A A A ··· A ··· A A ··· A Figure 15 . Brane depiction of the low dimensional leaves [and elementary slices between them] of (cid:2) W PSL(3 , C ) (cid:3) [0 , . –42– .2 The B -type affine Grassmannian For this setup, we require a (cid:103) ON − plane. In the main text we always draw the full coveringspace and refer to half branes simply as branes. Component ∈ Z The the affine Grassmannian of Spin(2 n +1 , C ) is the 0-th componentof the affine Grassmannian of SO(2 n + 1 , C ). The brane system for (cid:2) W SO(2 n +1 , C ) (cid:3) [0 , ..., isproposed to be · · · n · · · n (cid:103) ON − (4.17)Depicted at the origin, with all D3 branes and D5 branes suppressed. The quiver repre-senting the origin is: 0 0 · · · Component ∈ Z The brane system for the 1-st component of the affine Grassmannianof SO(2 n + 1 , C ), is proposed to be · · · n · · · n (cid:103) ON − (4.19)Depicted at the origin, with all D3 branes and D5 branes suppressed. The quiver repre-senting the origin is: 0 0 · · · (cid:2) W SO(2 n +1 , C ) (cid:3) [1 , ,..., . Minimal transitions.
When activating D3 branes between D5 branes, one can followthe same procedure as for the A-type affine Grassmannian, when considering branes awayfrom the ON (keeping in mind that in the covering space one has to move mirrors imagesalike). If however one wants to activate a D3 brane between a D5 brane and its image, inorder to move the D3 brane along an NS5 brane and its image, one has to keep in mind theboundary conditions reviewed in Appendix A. The D3 brane ending on a D5 brane and itsimage needs to be accompanied by its own mirror image. The associated transition is b n ifthe D5 branes are separated by n NS5 branes (and their mirror images). · · · n · · · n b n , n > n = 1 one has to be slightly more careful. If we attempt to move the D3brane and its mirror along the NS5 brane we would obtain a set up breaking the S-rule:Breaks S-rule! (4.22)We therefore need to include another D5 brane and its mirror image and can now performa b = A transition: b = A . (4.23)Drawing the D5 branes at different vertical positions is purely for convenience. In (4.23)more D5 branes could be present between the NS5 and the (cid:103) ON − plane leading to A k transitions. – 44 –hen dealing with the 1-st component for SO(5 , C ), i.e. n = 2, one also has to becareful with the first transition. Activating a minimal number of moduli without breakingsupersymmetry has to be done like this: ac . (4.24)This transition is called ac as it appears also in the type C affine Grassmannians, and weuse the isomorphism C = B . For n > B affine GrassmannianComponent ∈ Z The affine Grassmannian of Spin(3 , C ), i.e. component 0 of theaffine Grassmannian of SO(3 , C ), is given by the brane system: , (4.25)depicted at the origin of (cid:2) W SO(3 , C ) (cid:3) [0] . We can now move onto the lowest non-trivial leafthrough a b = A transition as in (4.23). After a Hanany-Witten transition we can readoff the quiver: 12 (4.26)The only way to keep moving onto higher leaves in (cid:2) W SO(3 , C ) (cid:3) [0] is by repeating the samething, however the D5 branes already placed between the NS5 branes make the transverseslices A k − for the k -th transition of this type. A general leaf and the corresponding quiver– 45 –re: 2 k · · · k · · · ... k ... k k k (4.27)Which completely agrees with the construction in Section 4.1.1. Component ∈ Z For the second connected component of SO(3 , C ) we start with: , (4.28)depicting the origin of (cid:2) W SO(3 , C ) (cid:3) [1] The transverse slice for the k -th transition is A k andthe brane system and quiver for the leaf are:2 k + 1 · · · k + 1 · · · ... k ... k k k + 1 (4.29)Which is again in agreement with Section 4.1.1. B affine GrassmannianComponent ∈ Z Is in Figure 16.
Component ∈ Z Is in Figure 17. – 46 – A A A Figure 16 . Brane depiction of the low dimensional leaves [and elementary slices between them] of (cid:2) W SO(5 , C ) (cid:3) [0 , . – 47 – c A A A Figure 17 . Brane depiction of the low dimensional leaves [and elementary slices between them] of (cid:2) W SO(5 , C ) (cid:3) [1 , . – 48 – .3 The C -type affine Grassmannian For this setup, we require a ON + plane. In the main text we draw the full covering spaceand refer to half branes simply as branes. Component ∈ Z The brane system for the affine Grassmannian of Sp(2 n, C ), i.e. the0-th component of the affine Grassmannian of PSp(2 n, C ), is proposed to be · · · n · · · n ON + (4.30)Depicted is the origin of (cid:2) W PSp(2 n, C ) (cid:3) [0 ,..., , , with all D3 branes and D5 branes suppressed.The quiver representing the origin is:0 0 · · · Component ∈ (cid:50) The brane system for the 1-st component of the affine Grassmannianof PSp(2 n, C ), is proposed to be · · · n · · · n ON + (4.32)Depicted is the origin of (cid:2) W PSp(2 n, C ) (cid:3) [0 ,..., , , with all D3 branes and D5 branes suppressed.The quiver representing the origin is:0 0 · · · inimal transitions. In the C -type brane systems, we may perform the same transi-tions as in the A -type case. Additionally there are two more minimal transitions: · · · n · · · n c n (4.34)and · · · n · · · n ac n (4.35)For n = 1 there are transitions corresponding to Kleinian singularities studied in detail inthe next section. C affine GrassmannianComponent ∈ Z The affine Grassmannian of PSp(2 , C ) is given by the brane system: , (4.36)depicted at the origin of (cid:2) W PSp(2 , C ) (cid:3) [0] . We can now move onto the lowest non-trivial leafby activating a D3 brane between two D5 branes, which are mirror images, and movingthe D3 brane along the NS5 branes, corresponding to a A transition: . (4.37)– 50 –fter a Hanany-Witten transition we can read off the quiver:12 . (4.38)The only way to keep moving onto higher leaves in the affine Grassmannian is by repeatingthe same thing, however the D5 branes already placed between the NS5 branes makethe transverse slices A k − for the k -th transition of this type. A general leaf and thecorresponding quiver are: k · · · k · · · ... k ... k k k . (4.39)This is in agreement with the construction in both Section 4.1.1 and 4.2.1. Component ∈ Z For the second connected component for PSp(2 , C ) we start with: . (4.40)Depicting the origin of (cid:2) W PSp(2 , C ) (cid:3) [1] . The transverse slice for the k -th transition is A k and the brane system and quiver for the leaf are: k · · · k · · · ... k ... k k k + 1 . (4.41)This is in agreement with the construction in both Section 4.1.1 and 4.2.1.– 51 – .3.2 The C affine Grassmannian The two components are represented in Figures 18 and 19.– 52 – A A A A A A A Figure 18 . Brane depiction of the low dimensional leaves [and elementary slices between them] of (cid:2) W PSp(4 , C ) (cid:3) [0 , . – 53 – c A A A A A A A Figure 19 . Brane depiction of the low dimensional leaves [and elementary slices between them] of (cid:2) W PSp(4 , C ) (cid:3) [0 , . – 54 – .4 The D -type affine Grassmannian For this setup, we require a ON − plane. In the main text we draw the full covering spaceand refer to half branes simply as branes. Component (0 , ∈ Z × Z or ∈ Z The brane system for the affine Grassmannianof Spin(2 n, C ), i.e. the (0 , n even) or 0-th component (for n odd) of the affineGrassmannian of PSO(2 n, C ), is proposed to be · · · n · · · n ON − (4.42)Depicted is at the origin of (cid:2) W Spin(2 n, C ) (cid:3) [0 , ,..., , , , with all D3 branes and D5 branessuppressed. The quiver representing the origin is:0 0 · · · Component (1 , ∈ Z × Z or ∈ Z · · · n · · · n ON − (4.44)– 55 –epicted at the origin of (cid:2) W Spin(2 n, C ) (cid:3) [1 , ,..., , , , with all D3 branes and D5 branes sup-pressed. The quiver representing the origin is:0 0 · · · Component (1 , ∈ Z × Z or ∈ Z · · · n · · · n ON − (4.46)Depicted at the origin of (cid:2) W Spin(2 n, C ) (cid:3) [0 , ,..., , , , with all D3 branes and D5 branes sup-pressed. The quiver representing the origin is:0 0 · · · Component (0 , ∈ Z × Z or ∈ Z · · · n · · · n ON − (4.48)– 56 –epicted at the origin of (cid:2) W Spin(2 n, C ) (cid:3) [0 , ,..., , , , with all D3 branes and D5 branes sup-pressed. The quiver representing the origin is:0 0 · · · Minimal transitions.
In the D -type brane systems, we may perform the same transi-tions as in the A -type case. Additionally there is one more minimal transition: · · · n · · · n d n (4.50) D affine Grassmannian There are four components:
Component (0 , ∈ Z This component is studied in Figure 20. The origin is (4.51)
Component (0 , ∈ Z This component is studied in Figure 21. The origin is (4.52)– 57 – omponent (1 , ∈ Z This component is studied in Figure 22. The origin is (4.53)
Component (1 , ∈ Z This component is studied in Figure 23. The origin is HW ←→ (4.54) Transition.
While not strictly a Hanany-Witten transition, the following two brane websare equivalent in that they depict the same leaf in the Coulomb branch: ∼
11 22(4.55)In Figure 20 we use the representation on the left. Furthermore the following two transverseslices are equivalent and correspond to an A transition: ∼ (4.56)The only difference between the two brane systems in (4.55) and (4.56) lies in their Higgsbranch. – 58 – A A A A A Figure 20 . Brane depiction of the low dimensional leaves [and elementary slices between them] of (cid:2) W PSO(4 , C ) (cid:3) [0 , . –59– A A A A A Figure 21 . Brane depiction of the low dimensional leaves [and elementary slices between them] of (cid:2) W PSO(4 , C ) (cid:3) [0 , . –60– A A A A A Figure 22 . Brane depiction of the low dimensional leaves [and elementary slices between them] of (cid:2) W PSO(4 , C ) (cid:3) [1 , . –61– A A A A A Figure 23 . Brane depiction of the low dimensional leaves [and elementary slices between them] of (cid:2) W PSO(4 , C ) (cid:3) [1 , . –62– Quiver addition
In recent papers [15, 16, 32, 51–58], the notion of quiver subtraction was developed to obtainthe Hasse diagram for symplectic singularities, and has been used in various scenarios [59–65]. In this section, we consider the reverse of this process: quiver addition . Aspects of thisnotion have been discussed in [60] for D -type Dynkin quivers. In this work, we show thatthe quiver addition algorithm presented below precisely reproduces the Hasse diagrams andquivers for the slices in the affine Grassmannian of any finite dimensional simple complexLie group. For classical groups the quiver addition algorithm can be viewed as the quiverversion of the brane constructions of Section 4, expressed in a way more suitable e.g. toteach a computer. For exceptional groups however no brane systems are known, and thequiver addition algorithm provides a simple but powerful way to obtain the Hasse diagramof any transverse slice in the affine Grassmannian. Before proceeding to quiver addition,let us review the procedure of quiver subtraction.One way to obtain the Coulomb branch Hasse diagram of a quiver Q is to use quiversubtraction. In quiver subtraction one identifies an elementary slice with quiver D fromTable 1 and subtracts it from Q following the algorithm given in [32, Appendix A] toproduce a new quiver Q (cid:48) = Q − D . We restrict to the case where a given elementaryslice cannot be subtracted more than once consecutively. In this case, the Coulombbranch of Q (cid:48) is the closure of a minimal degeneration of the highest leaf in the Coulombbranch of Q . One key point in the algorithm is that the balance of all the gauge nodesremains invariant during subtraction. If more than one quiver can be subtracted, the Hassediagram bifurcates. As introduced above, we call the number of elementary slices whichcan be subtracted at any given step the disposition D of the leaf.Conversely, to a given quiver one can generically add an infinite number of elementaryquivers. However, if one imposes the restriction that the Dynkin diagram formed by thegauge nodes remains invariant then the number of possible slices allowed to be addedbecomes finite. We now discuss the algorithm. Quiver addition algorithm.
Let Q be a quiver of unitary gauge nodes forming a rank r finite Dynkin diagram with Cartan matrix ( C g ) i,j =1 ,...,r . Let the ranks of the gauge nodesbe k i ≥
0, and the ranks of the flavors nodes be N i ≥ i = 1 , . . . , r . Identification of the elementary quivers S to add . Three kinds of elementaryquivers can be added to Q . • Define the subset of gauge nodes without any flavors J = { j ∈ { , . . . , r } | N j = 0 } . (5.1) If the same slice can be subtracted more than once, we are in the realm of affine Grassmannians ofaffine type, rather than finite type. We emphasize that k i = 0 is allowed, even when N i > – 63 –o this section of Q One can add the following S · · · · · · ac n cg ag Table 7 . Identification of the situations where the quivers for ac n , cg and ag can be added.We insist on the fact that the flavor nodes must be exactly as indicated in the first column, andthat there should be no flavorless gauge node connected to the portions of quiver drawn in the firstcolumn, so that quiver of the second column can be added. The nodes labeled by J form a union of connected Dynkin diagrams. For eachconnected Dynkin diagram, one can add the quiver S for the closure of theminimal nilpotent orbit (see Table 1). • Consider the set of gauge nodes J = (cid:8) j ∈ { , . . . , r } | N j ≥ ∀ j (cid:48) ∈ { , . . . , r } , N j (cid:48) ≥ − ( C g ) j (cid:48) j (cid:9) (5.2)such that gauge node itself has flavours and also its connected neighbours haveenough flavour. For each j ∈ J , one can add on that node the quiver for aKleinian singularity A N j +1 . • If there is a subset of the gauge nodes which realize the quivers on the firstcolumn of Table 7 with the flavors exactly as indicated , and with no flavorlessgauge node connected to that subset , then one can add the S in the secondcolumn.2. Addition of the elementary quivers . The result of the addition of the elementaryquiver S to Q is a quiver Q (cid:48) with gauge nodes given by the Dynkin diagram specifiedby ( C g ) i,j =1 ,...,r , where • The ranks of the gauge groups k (cid:48) i are the sums of the ranks in Q and S . • The flavors N (cid:48) i are given by the corresponding flavor in S if k i (cid:54) = k (cid:48) i and byrebalancing otherwise. – 64 –ote that the total number of flavors either stays the same or increases when a sliceis added. As we grow the Hasse diagram further, we reach a stage where all gaugegroups have non-zero flavor nodes. Once this happens, the only slice that can be addedare Kleinian A k singularities. Hence, as the Hasse diagram grows, more and more Kleiniansingularities populate the Hasse diagram. When all the nodes have flavors, the dispositionis equal to the rank of the group, and can not decrease upon quiver addition. Let us exemplify how the algorithm works in the case of good linear quivers. The set J of unflavored nodes decomposes into smaller A -type diagrams, and for each such diagram,an a n -type slice can be added. For instance, consider the following scenario: . . . . . .. . .m m m Can add a m . . .m Can add a m . . .m Can add a m . . .m k a k b k c k d k f k g k h k i k j N N N (5.3)There are three possible slices one can add a m , a m , and a m . For example, the result ofadding the a m slice produces: . . . . . .. . .m m m k a k b k c k d +1 k f +1 k g k h k i k j N − N − N Just like the number of D5 branes between NS5 branes stays the same or increases when moving upwardsin the Hasse diagrams in Section 4 The transition in the corresponding brane system is (4.9). – 65 –n the other hand the A -type Kleinian singularities can be added in the following situa-tion: N N N . . . . . . Can add A N +1 N − N +2 N − . . . . . .k a k b k c k d k e N +2 k a k b k c + 1 k d k e (5.5)which corresponds to nodes in the set J . Single unbalanced gauge group.
As an illustration, we consider linear quivers withone unbalanced node, with imbalance +1. These quivers cover all cases of transverse sliceto the lowest leaf in the components of Gr
PSL( N, C ) not connected to the trivial lattice.Quiver addition then reproduces the Hasse diagram and quivers for the transverse slicesin these components of Gr PSL( N, C ) . Specifically, Figure 24 shows the first few leaves inthe affine Grassmannian Hasse diagram of Gr PSL( m + m +2 , C ) for m , m ≥ m = [0 . . . (cid:124) (cid:123)(cid:122) (cid:125) m . . . (cid:124) (cid:123)(cid:122) (cid:125) m ] . (5.6)It is clear that for quivers where all gauge groups have non-zero ranks, the global symmetryis A m × A m × u (1). As a feature of Hasse diagrams, the non-abelian part of the globalsymmetry is given by the two slices directly connected to the origin. Note, since m and m can be different, the Hasse diagram becomes less symmetric as one moves away fromthe bottom.We observe that in the Hasse diagram of Gr PSL( m + m +2 , C ) , if we keep the leaves whereonly the ranks of the first m gauge nodes are non zero, then the resulting diagram is finiteand is the Hasse diagram of the nilpotent orbits of sl ( m + 1 , C ). Non-linear D N , E , , quivers. The treatment is almost identical to the A -type affineGrassmannians except for the novelty that one may add d n , e , e , e elementary slices.For A -type affine Grassmannian, we mentioned how n unflavored gauge groups betweentwo flavored gauge groups allow the addition of an a n slice. We can now generalize thisby considering the subset of n flavorless gauge nodes that are bounded by flavored gaugenodes. Then, one is allowed to add the elementary slice that is the closure of the minimalnilpotent orbit of g n , where g n is the finite Dynkin diagram in the shape of the unflavored The transition in the corresponding brane system is (4.10). – 66 – . .m m . . . . . . . . . . . . . . . . . . . . . . . . . . .
01 33 112 . . . . . .
11 110 . . .
11 1 1 . . .
00 11 . . .
11 2 2 . . .
00 11 . . .
11 2 2 . . . . . .
21 2 2 . . . . . . . . . . . . . . .
11 2 211 1 . . .
221 221 . . . a m a m − A a m − a m +2 a m +1 a m +1 a m a m − a m +2 A a m − a m − a m − a m a m ... ... ... ... ... ... ...00 0 000 00 Figure 24 . Hasse diagram for the first orbits in the component of Gr
PSL( m + m +2 , C ) corresponding to the lowest coweight Ω m = [0 . . . (cid:124) (cid:123)(cid:122) (cid:125) m . . . (cid:124) (cid:123)(cid:122) (cid:125) m ]for m , m ≥
6. Blue nodes represent unbalanced gauge groups. –67– auge nodes. Let us demonstrate this with some slices of E affine Grassmannian:1 11 1 1 1 1add a k a k b k c k d k e k f k a +1 k b +1 k c +1 k d +1 k e +1 k f N N −
11 1 (5.7)Here, we can add a a slice to the nodes that make up a a finite Dynkin diagram in thered box. 1 2 2 1add d k a k b k c k d k e k f N k a k b +1 k c +2 k d +2 k e +1 k f +1 N − d slice to the nodes that make up a d finite Dynkin diagram in thered box. – 68 – 2 1add d k a k b k c k d k e k f N N k a k b +1 k c +2 k d +1 k e k f +1 N − N −
11 (5.9)Here, we can add a d slice to the nodes that make up a d finite Dynkin diagram in thered box. We now turn to non simply laced quivers. We consider the case of F , which displays allthe subtleties of the algorithm. This is summarized in Figure 25, where we show the 12types of quiver additions that can be made as a function of the ranks of the flavor nodes. In this section, the Hilbert series of infinite dimensional slices, arising from the affineGrassmannian, are explored.To every orbit [Gr G ] λ , labelled by a coweight λ , in the affine Grassmannian of a certaingroup G , there is an infinite dimensional transverse slice [ W G ] λ . To this transverse slicewe can associate a Hilbert series HS G,λ , which can be obtained from an infinite rank limitof a quiver [66], and which can be expressed as an infinite product:HS
G,λ ( t ) = (cid:89) j =1 PE (cid:88) α ∈ ∆(adj) t |(cid:104) λ,α (cid:105)| +2 j = PE (cid:88) α ∈ ∆(adj) t |(cid:104) λ,α (cid:105)| − t , (6.1)where PE denotes the plethystic exponential. This can be further refined using fundamentalweight fugacities x i as follows:HS G,λ ( t ; x ) = PE (cid:88) α ∈ ∆(adj) x α t |(cid:104) λ,α (cid:105)| − t (6.2)– 69 – N N N ≥ A N +2 N ≥ N N ≥ N A N +2 N N ≥ N N ≥ A N +2 N N N ≥ N A N +2 N N ≥ a a N N ≥ c N ≥ N ≥ ac N ≥ N ≥ c N ≥ ac N ≥ b f N ≥ Figure 25 . Quiver addition rules regarding slices in the affine Grassmannian of F . We just illus-trate step 1 of the algorithm, as once the slices to be added are identified, step 2 is straightforward.The red boxes indicate where the quivers for the slices, which can be found in Table 1, have to beadded. When no condition is written on a flavor node rank N i , it means that it can take any value N i ≥ where x λ = r (cid:89) i =1 x (cid:104) λ,α ∨ i (cid:105) i (6.3)and r is the rank of G .The Hilbert series HS µG,λ of a (finite dimensional) transverse slice [ W G ] µλ = [ W G ] λ ∩ [Gr G ] µ approximates the Hilbert series for the infinite dimensional transverse slice (6.2),– 70 –nd this approximation gets better when µ gets large. More precisely,lim (cid:104) ρ,µ (cid:105)→∞ HS µG,λ ( t ; x ) = HS G,λ ( t ; x ) , (6.4)where ρ is the Weyl vector. In this equation, HS µG,λ can be computed using the monopoleformula on the quiver Q µλ . When µ gets larger, the ranks of the gauge nodes in Q µλ increasewhile the imbalance stays fixed. In the remainder of this section, we elaborate on theappearance of new generators in the chiral ring when these ranks increase, focusing on twoexamples with G = Sp(4 , C ) and λ = 0 and λ = (cid:36) respectively. The case λ = 0 . By means of the Hilbert series, we are studying the moduli space gener-ators and the relations between them. Let us consider slices in the affine Grassmannian of G where all gauge groups are balanced (i.e. slices to the origin, λ = 0). If all the generatorsappear at order t in the Hilbert series, the Coulomb branch is the closure of a nilpotentorbit of G [67]. However, by increasing the ranks of the gauge groups, whilst maintainingthe balance, new generators show up at higher orders.For example, let us consider slices of the G = Sp(4 , C ) affine Grassmannian for λ =[0 , , C t + . . . PL[HS]Slices of (cid:2) Gr Sp(4 , C ) (cid:3) λ =0 Slices of (cid:2) Gr Sp(4 , C ) (cid:3) λ =0 [2 , C t + . . . [2 , C t + [2 , C t + . . . [2 , C t + [2 , C t + . . . [2 , C t + [2 , C t + . . . [2 , C t + [2 , C t + [2 , C t + . . . (6.5)where the first few terms of the PL of the Hilbert series are written, giving the quantumnumbers of the generators of the Coulomb branch. In general, one starts with the affineDynkin quiver of C which contributes [2 , t for the generators. Once all gauge nodesincrease their rank by 1, a generator transforming in [2 , t is added. This pattern persists– 71 –or any addition of 1 to all gauge nodes, and we see the quivers in the blue boxes areprecisely those where new generators in the adjoint representation appear. In general, forthe quiver 1 + x x
01 + x (6.6)one findsPL (cid:104) HS µ =[1+ x, , C ) ,λ =[0 , (cid:105) = x (cid:88) i =0 [2 , C t i ) + O ( t x ) ) = [2 , C t − t x − t + O ( t x ) ) . (6.7)As x → ∞ , we see the Coulomb branch Hilbert series takes the following form:HS Sp(4 , C ) , [0 , PE (cid:20) [2 , C t − t (cid:21) . (6.8)In this equation, it is understood that [2 , C stands for a character whose variables par-ticipate in the plethystic exponential. Explicitly, using fundamental weight fugacities x i of C , this isHS Sp(4 , C ) , [0 , ( t ; x i ) = PE (cid:20)(cid:18) x + x + x x + x x + 2 + x x + x x + 1 x + 1 x (cid:19) t − t (cid:21) (6.9)This reproduces the limit (6.4).The reasoning behind this is as follows: if one increases the rank of all the gaugegroups, then new generators appear. This is most easily seen in the cases of algebras oftype A and C where the comarks are all equal to 1. If we increase all the ranks by 1, thenthe number of new generators is the dimension of the adjoint representation. For type A and C and all nodes balanced, we are able to simultaneously increase the rank of all gaugegroups by 1 and at the same time maintaining a fully balanced quiver. Since the balanceis maintained, the new generators transform always in the adjoint representation. Eachtime this is done, a new set of generators in the adjoint appears at higher and higher order.One can consider repeating this counting process for other Lie algebras, but since somecomarks are >
1, the situations is more involved and is left for future work.
The cases λ (cid:54) = 0 . If λ (cid:54) = 0, the quiver is unbalanced. However, the total number ofgenerators is still an integer multiple of the dimension of the adjoint representation of G ,– 72 –ut with generators spread over different orders of t . Consider λ = [0 ,
1] for G = Sp(4 , C ):1 2301 1112 212 2 3312 4503 313([2] A + [0] A ) t ([2] A + [0] A ) t +([2] A + [0] A ) t +( q + q − )[2] A t + . . . +( q + q − )[2] A t +( q + q − )[2] A t + . . . +([2] A + [0] A ) t +( q + q − )[2] A t + . . . PL[HS] PL[HS]Slices of (cid:2) Gr Sp(4 , C ) (cid:3) λ =[0 , Slices of (cid:2) Gr Sp(4 , C ) (cid:3) λ =[0 , ([2] A + [0] A ) t +( q + q − )[2] A t + . . . ([2] A + [0] A ) t +([2] A + [0] A ) t +( q + q − )[2] A t +( q + q − )[2] A t + . . . ([2] A + [0] A ) t +([2] A + [0] A ) t +( q + q − )[2] A t +( q + q − )[2] A t + . . . ([2] A + [0] A ) t +([2] A + [0] A ) t +( q + q − )[2] A t +( q + q − )[2] A t (6.10)Due to the overbalanced gauge node, the global symmetry is now su (2) × u (1) with [ ] A and q Dynkin labels respectively. Compared with (6.5), we see that the total number ofgenerators (at all orders of t combined) remains the same. However, due to the unbalancedgauge node, some generators appear at higher orders and acquire non-trivial u (1) charge.The precise irreducible representations for these generators depend on which node of thebalanced quiver it is connected to [68]. This easily generalizes to λ = [0 , λ ] such that inthe large rank limit, we can once again derive the freely generated Hilbert series:HS Sp(2) , [0 ,λ ] ( t ) = PE (cid:34) ∞ (cid:88) i =1 (dim([2] SU (2) ) + 1) t i + 2dim([2] SU (2) )) t i + λ (cid:35) = PE (cid:34) ∞ (cid:88) i =1 t i + 6 t i + λ (cid:35) (6.11)and refining with x i and q , the fundamental weight fugacities of su (2) × u (1), yields:HS Sp(2) , [0 ,λ ] ( t ; x i , q ) = PE (cid:34) (2 + x + x ) t + ( q + q )(1 + x + x ) t λ − t (cid:35) . (6.12)– 73 –his agrees with (6.2). A similar analysis can be made for any coweight λ and any group G . Observation about number of generators.
We have the following generalization ofthe observations above. Take a slice [ W G ] λµ in the affine Grassmannian of a group G , withglobal symmetry G (cid:48) ⊂ G . The adjoint representation, identified with the Lie algebra, isdecomposed as g (cid:55)→ g (cid:48) (cid:124)(cid:123)(cid:122)(cid:125) appears at t + R (cid:48) (cid:124)(cid:123)(cid:122)(cid:125) appears at higher orders (6.13)where R (cid:48) is a (possibly reducible) representation of G (cid:48) . In the Hilbert series of [ W G ] λµ , g (cid:48) appears at t while irreducible representations in R (cid:48) appear at higher orders of t . The onlyfurther sets of generators (if there are any) are in the same representations g (cid:48) + R (cid:48) butappear at higher orders in t , shifted by a fixed integer. The total number of generators ishence an integer multiple of the dimension of G . One could say that the adjoint of G is“stretched” over higher orders. The affine Grassmannian is infinite dimensional, and this aspect pervades through allparts of this paper, in which we met infinite Hasse diagrams, configurations with infinitelymany branes, Hilbert series with infinitely many generators. However in all those cases,these infinite quantities do not represent a problem, because they can be approached in acontrolled way by finite quantities. In our language, this translates into equalities of thetype lim N →∞ C d (cid:32) NN (cid:33) = Gr SL(2 C ) . (6.14)The affine Grassmannian appears here as a controlled limit of finite dimensional Coulombbranches. The reason why the limit (6.14) makes sense is because each term in the sequenceis a subset of the next term: C d (cid:32) NN (cid:33) ⊂ C d (cid:32) N + 1) N + 1 (cid:33) . (6.15)This property is the reason why we can build the bottom part of the Hasse diagrams ofthe infinite dimensional affine Grassmannian using only data from the finite dimensionalCoulomb branches corresponding to transverse slices. In algebraic geometry, this is themain idea behind the construction of so-called ind-schemes , of which the affine Grassman-nian is a prime example. – 74 –ll the infinite quantities that appear in this paper can be regularized that way. Forinstance, the brane setup (4.1) should be seen as the limitlim N →∞ · · ·· · · · · · N − D N D3 N − D n NS5 N D5 N D5 (6.16)where the fundamental property that the system for a given N is included into the systemfor N + 1 is again satisfied. On that diagram, we can now compute the linking numberof the D5 branes at finite N to be N − ( N − − N + ( N −
1) = 0, independent of N .Therefore in the infinite limit, the D5 branes also have linking number 0. On the otherhand, if one adds a D5 brane between the two NS5 branes in (6.16), its linking number isequal to N − N − ( N −
1) + N = 1 for all N , and therefore is 1 in the limit, as claimedbelow the brane system (4.4).Finally, the same logic is behind the formulas for the Hilbert series as infinite products(6.2) obtained as a limit (6.4), and underlies the observations about the generators madein the previous subsection. In this paper we explore the Hasse diagrams and transverse slices of various affine Grass-mannians, using both quivers and branes. Every slice in the affine Grassmannian, specifiedby two dominant coweights λ ≤ µ , of a finite dimensional simple group G is the Coulombbranch of a good framed quiver in the shape of the Dynkin diagram of G . The coweight µ (incoweight basis) is the framing, while λ is the imbalance. For classical groups Kraft-Procesitransitions in brane systems naturally reproduce Hasse diagram of the affine Grassman-nian and all transverse slices. Quiver subtraction proves effective in the realm of the affineGrassmannian. A new notion called quiver addition is introduced. Asking for the reverseof quiver subtraction in the sense that for a quiver Q we take an ‘addable’ quiver S andproduce a new quiver Q (cid:48) , such that Q (cid:48) = Q − S leads to an infinite number of possibilities.When we limit to slices in the affine Grassmannian however, we have a finite number ofpossibilities. Starting with a coweight, quiver addition produces the entire Hasse diagramof slices to this coweight. This agrees perfectly with the brane construction. ‘Ugly’ and ‘bad’ quivers. Of course quivers, as well as brane systems, are not limited to‘good’ theories. Theories which have a negative imbalance have Coulomb branches dubbed generalized affine Grassmannian slices [12], which are also specified by two coweights, butthe lower coweight, λ , is not required to be dominant, i.e. the imbalance is allowed to benegative. Quivers and brane systems for these generalized slices are easily produced, and– 75 –he Hasse diagram may be obtained using quiver subtraction or Brane moves. It would benice to explore these theories in the future. Other brane systems.
We expect that for many brane systems consisting of D p , NS5,and D( p + 2) branes (with p ≤ Relation to nilpotent orbits and T σρ [ G ] theories. Quivers for slices in the affineGrassmannian of A -type and T σρ [ SU ] theories coincide. They are the good framed linearquivers. For other groups this is not the case. While the construction for T σρ theoriesfor other classical groups uses O3 orientifold planes and linear orthosymplectic quivers, theconstruction for affine Grassmannian slices uses ON planes and Dynkin shape unitary quiv-ers. The moduli spaces only coincide in few cases, and are related through quotients in fewmore. Exploring the relationship between T σρ theories and slices in the affine Grassmannianare left for future work. Symplectic duality.
The quivers associated to slices in the affine Grassmannian forsimply laced groups also possess a Higgs branch. Their Hasse diagrams are simply obtainedfrom inversion [57]. It could be interesting to ask for some notion of the symplectic dualof the entire affine Grassmannian.
Relation to other work.
We expect there to be many other interesting lines of researchinvolving the affine Grassmannian in physics, such as in the context of little strings [69, 70],or domain walls [71].
Acknowledgements
AH would like to extend special thanks to Jacob Matherne for explaining basic conceptsabout the affine Grassmannian, and for email exchange together with Santiago Cabrera,where early versions of the main topic of this paper were conceived. We are grateful to AlexWeekes for helping us understand the mathematical literature and for vital insights. Fur-thermore we would like to thank Travis Schedler for many helpful discussions throughoutthe years. AB would like to thank Daniel Juteau for many profitable exchanges. The workof AB, JFG, AH and ZZ is supported by STFC grant ST/P000762/1 and ST/T000791/1.The work of MS is supported by the National Thousand-Young-Talents Program of China,the National Natural Science Foundation of China (grant no. 11950410497), and the ChinaPostdoctoral Science Foundation (grant no. 2019M650616).
A Branes, ON Planes and Quivers
In this appendix we review allowed configurations of branes in the presence of ON planes.We construct brane systems of NS5 branes, D5 branes and D3 branes in Type IIB String– 76 – x x x x x x x x x NS5 x x x x x xD3 x x x xD5 x x x x x xON x x x x x x
Table 8 . Type IIB set up. The ’x’ mark the spacetime directions spanned by the various branesand, if present, the ON plane. All NS5 branes are localized at the same value of ( x , x , x ). AllD5 branes are localized at the same value of ( x , x , x ). Theory as first developed in [9], with the addition of ON planes. The space-time extensionof the branes are collected in Table 8.The low energy theory on the world-volume of D3 branes suspended between NS5branes is a 3 d N = 4 quiver gauge theory, where D5 branes provide flavour nodes. Forexample, the following set up:( x )( x , x , x )( x , x , x ) NS5 NS5origin k D3 N D5. . . kN (A.1)We will supress the ( x , x , x ) direction from now on and draw D5 branes as . Theposition of D3 branes along the NS5 branes, as depicted in (A.2), correspond to Coulombbranch moduli: · · · ... . (A.2) Including ON planes.
The gauge group on the world-volume of a stack of NS5 braneson top of an ON plane is given by the allowed D1 branes (and their mirror images) endingon NS5 branes and their mirror images. This is analogous to stacks of D3 branes on top ofO3 planes with fundamental strings stretched between them, studied in [72]. The allowedD1 states and the corresponding gauge/electric algebra on the world-volume of the NS5branes are: – 77 –lectric algebra
CDBD + ON − (cid:103) ON − , (A.3)where dashed lines correspond to mirror images (in the main text we use solid lines).The endpoints of D3 branes play the role of magnetic monopoles on the world-volumeof the NS5 branes, and the D3 branes play the role of roots of the GNO dual/magneticalgebra [73]. The allowed D3 states and corresponding magnetic algebras are:– 78 –agnetic algebra BDCD + ON − (cid:103) ON − quiver type CDB (A.4)However, the quiver representation of the low energy theory on the world-volume of D3branes suspended between NS5 branes in presence of an ON plane is not in the form ofthe Dynkin diagram associated to the magnetic algebra of the ON plane given in (A.4).Rather, D3 branes corresponding to a short root of the magnetic algebra will produce agauge node which is a long node in the quiver and vice versa, as pointed out in [74]. Thequiver type is also indicated in (A.4).
Reading the quiver.
In the following, and in the main text, we denote both a braneand its mirror image with a solid line. A stack of k D3 branes between two NS5 branesaway from an ON leads to a U ( k ) gauge node in the corresponding quiver, as depicted in(A.1). The presence of an ON plane does not have an effect on the type of gauge groups inthe quiver (they remain unitary), but one has to use the following rules to read the shapeof the quiver: – 79 – .1 ON + In the presence of an ON + we can study the following brane system. (D3 branes are drawnaway from the origin for clarity) · · · N · · · N · · · N · · · N ... k ... k ... k ... k (A.5)We can restrict to the physical system, move onto the Coulomb branch, and read thecorresponding quiver: · · · N · · · N ... k ... k k k N N (A.6) A.2 ON − In the presence of an ON − we can study the following brane system. · · · N · · · N · · · N · · · N · · · N · · · N ... k ... k + k ... 2 k ... 2 k ... k + k ... k (A.7)In order to move onto the Coulomb branch and read a quiver from the brane system, onehas to be careful. For a general system, where N >
0, there will be several quivers onecan read, depending on boundary conditions. This is investigated in detail in [75] and weonly state the possibilities here. The first option is the most natural one, identified in theliterature before [76]: · · · N · · · N · · · N ... k ... k ... k k k k N N N + 2 N (A.8)The second option is obtained, when one of the 2 k D3 branes crossing the ON − ends onone of the N D5 branes instead of NS5 branes. This leads to a different possibility of– 80 –reaking the D3 branes along the NS5 branes, and provides a different electric quiver (Thehorizontal separation of D5 branes is only for ease of reading): · · · N · · · N · · · N − k ... k + 1 ... k − k k + 1 k − N N + 2 N + 2( N − { N + 1 , k + 1 } possibilities, labelledby l ∈ { , . . . , min { N , k }} : · · · N · · · N · · · N − l ... k ... k + l ... k − l ... l k k + lk − lN N + 2 lN + 2( N − l )(A.10)Note that in order to transition between two leaves, whose closure is described by twoquivers with different l in (A.10), one needs to move onto a lower leaf which appears in thestratification of both spaces. A.3 (cid:103) ON − In the presence of an (cid:103) ON − we can study the following brane system. · · · N · · · N · · · N · · · N ... k ... 2 k ... 2 k ... k (A.11)We can restrict to the physical system and read the corresponding quiver: · · · N · · · N ... k ... k k k N N (A.12)– 81 – .4 Leaf Closures and Transverse Slices in Brane Systems In this section we investigate how to not only to identify the low energy theory living onD3 branes in a brane system, but how to analyse the stratification of its moduli spaceusing branes. For a given brane system there are various distinct phases, which correspondto the symplectic leaves that make up its moduli space. In the following we will onlyconsider Coulomb branch phases. For an analysis of the full moduli space, including abrane perspective, see [57]; we repeat what is needed for this paper in the following. Let C be the Coulomb branch of the low energy theory living on the brane system. For everysymplectic leaf L ⊂ C there is a Coulomb phase P in the brane system. From the phase P we can read a quiver Q , whose Coulomb branch C ( Q ) = L is the closure of the symplecticleaf L corresponding to the phase P . The phases of the brane system are distinguished,by how many branes coincide at the origin, i.e. how many massless states are present inthe field theory when on the leaf L on its Coulomb branch. Let us look at the exampleof U (2) with 4 fundamental hypers. The Hasse diagram of its Coulomb branch is straightforwardly computed, e.g. through quiver subtraction: L a L b L c A A for C (A.13)Where we labelled the leaves by L i , i = a, b, c . The three distinct Coulomb phases of thebrane system are: P a P b P c (A.14)The quiver for the most general Coulomb phase, P c , is easily read off. It is the theoryitself: Q c = 24 (A.15)The quiver not only describes the phase P c , but also the possibility of the D3 branesmoving to the origin. It is a quantum field theory in its own right with a moduli space– 82 –onsisting of several leaves. Hence its Coulomb branch is not the leaf L c itself, but the clo-sure of the leaf L c , or in other words, the transverse slice from the origin, L a , to the leaf L c .The quiver for the phase P b is more tricky to work out. There is one D3 brane resting atthe origin. We first have to make this D3 brane end on D5 branes rather than NS5 branes.This can be done through a Hanany-Witten transition. The resulting brane system is:(A.16)Now we have to ignore the possibility of the D3 brane, which is supposed to rest at theorigin, to move along the NS5 branes. We do this by simply not drawing it: (A.17)Now we can read a quiver from this brane system: Q b = 12 (A.18)The Coulomb branch of Q b is the closure L b .The quiver associated to P a , the origin, is trivial: Q a = 00 (A.19)– 83 –he quivers we have computed represent closures of the leaves. This corresponds to trans-verse slices from the origin L a to the leaf L i in question. We can also obtain a quiver forany other transverse slice. The remaining non-trivial slice for our theory is the one from L b to L c , we refer to the corresponding quiver as Q b − >c . We can obtain the quiver from thebrane system in the following way: Comparing the two phases P b and P c we have to ignorethe modulus which is turned on in both phases. We achieve this by simply not drawing theD3 brane in question; one can think of it as sending the D3 brane off to infinity . Theresulting brane system is: (A.20)From which a quiver is easily read off: Q b − >c = 14 (A.21)The Coulomb branch of this quiver is the transverse slice from L b to L c .Of course none of these quivers are new, as they are exactly the quivers involved inquiver subtraction. The point of this exercise was to show, how all transverse slices of theCoulomb branch show up as transitions in the brane system.In Section 4 we propose brane systems whose Coulomb branch is the transverse slice tothe lowest leaf in a connected component of the affine Grassmannian of a classical group.This space is infinite dimensional reflected in the fact that we have an infinite amount ofD3 branes present in the brane system. We fix all but a finite amount of D3 branes to beat the origin, in order to study the symplectic leaves the space is made up of from bottomup. For every such leaf we look at the corresponding phase in the brane system and obtainthe quiver whose Coulomb branch is the closure of the leaf. Furthermore one can associatea quiver to a transition between any two phases in the brane system. Every good quiverin shape of a classical Dynkin diagram appears this way in one of the brane systems. 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