No title
January 01, 1970
Abstract
2cm2cm Based on the Decay and Fission Conjecture, we provide a classification of unitary quivers whose 3d \(\mathcal{N}=4\) Coulomb branches exhibit isolated singularities. This yields the complete list of isolated conical symplectic singularities that can arise in this way. In the process, we identify three new families of stable quivers: two giving rise to previously unknown isolated symplectic singularities, and one offering a novel realization of a known family.
UWThPh 2025-13
Classifying Isolated Symplectic Singularities
via 3d \(\mathcal{N}=4\) Coulomb Branches
Antoine Bourget\({}^a\), Quentin Lamouret\({}^a\)
Sinan Moura Soysüren\({}^b\), Marcus Sperling\({}^b\)
Email: antoine.bourget@ipht.fr,
quentin.lamouret@ipht.fr
Email: sinan.moura.soysueren@univie.ac.at,
marcus.sperling@univie.ac.at
1 Introduction↩︎
Mathematics Motivations↩︎
Symplectic singularities were defined by Beauville in 1999 [1] to capture the notion of singular symplectic varieties. The simplest examples of symplectic singularities are isolated, meaning that the singular locus is a point, and conical, meaning there is a \(\mathbb{C}^{\ast}\) action compatible with the symplectic structure. We henceforth call them Isolated Conical Symplectic Singularities, ICSSs for short. Well-known ICSSs include the Kleinian/Du Val surface singularities \(\mathbb{C}^2\slash \Gamma_{ADE}\) and the closures of minimal nilpotent orbits \(\overline{\mathcal{O}_{\min}} (\mathfrak{g})\) of semi-simple Lie algebras \(\mathfrak{g}\). These encode all appearing minimal transverse slices for nilpotent orbits of semi-simple Lie algebras [2]–[4].
Beauville [1] raised the following question: What are more examples of ICSSs with trivial local fundamental group, beyond closures of minimal nilpotent orbits? This remained open for about 20 years, until recently. In [5] the authors identified a new such family \(\mathcal{Y}(\ell)\) as singularities in the blowup of the quotient of \(\mathbb{C}^{4}\) by the dihedral group of order 2\(\ell\). Shortly after, [6] provided the construction of what we call \(\overline{h}_{n,\sigma}\) singularities by using (toric) hyper-Kähler quotients.
Another possible source of examples are 3d \(\mathcal{N}=4\) Coulomb branches [7], [8], which under certain assumptions have symplectic singularities [9], [10]. Here, we focus on quiver gauge theories with unitary gauge groups, generalized by adding non-simply laced edges [11]. The isolated character of the singularity can be detected following the conjectural Decay and Fission algorithm [12], [13], which computes the stratification of such a Coulomb branch into partially ordered symplectic leaves (this poset is encoded in a “Hasse diagram”, see Figure 1). Based on this conjecture, we provide a classification of isolated conical symplectic singularities realized as 3d \(\mathcal{N}{=}4\) Coulomb branches of quiver gauge theories with unitary gauge groups. Remarkably, this encompasses all the previously known ICSSs — except for \(D\) and \(E\) type surface singularities — and adds two new infinite families to the list, see below. We also get a new Coulomb branch identification, see 18 .
Our classification also is of physical interest, as discussed in the next paragraph and illustrated in Figure 1. This, however, can be skipped by readers only interested in the mathematical content.
Physics Motivations↩︎
Superconformal quantum field theories (SCFTs) with 8 supercharges in spacetime dimensions \(d=3,4,5,6\) typically exhibit a moduli space \(\mathcal{M}\) of supersymmetric vacuum solutions. A distinguished branch \(\mathcal{M}_H \subset \mathcal{M}\), called the Higgs branch, can be defined as the locus left invariant by all of the superconformal algebra, except for the \(R\)-symmetry factor \(\mathfrak{su}(2)_R\). This is a conical symplectic singularity (CSS). Despite the difficulty of rigorously defining the quantum field theory framework (QFT), the Higgs branch is an object that is mathematically well-defined [14]–[16], and which can be used as a rich invariant for the SCFT — indeed SCFTs can be defined using very diverse languages. The geometric properties of \(\mathcal{M}_H\) are in one-to-one correspondence with features of the physical theory. For instance, isometries correspond to flavor symmetries and the finite stratification into partially ordered symplectic leaves corresponds to the Higgs mechanism [17]. From that perspective, an ICSS corresponds to an elementary Higgs mechanism. More generally, ICSSs are viewed as elementary building blocks of Higgs branches, which motivates our effort to gather as many examples as possible.
In \(d=3\) spacetime dimensions, another branch is also a CSS, the Coulomb branch \(\mathcal{M}_C \subset \mathcal{M}\) [7], [8], [18]. The coexistence of these two branches is the basis of 3d \(\mathcal{N}=4\) mirror symmetry [19], which exchanges the Higgs and Coulomb branch of two dual theories. Many insights into the Coulomb branch of Lagrangian theories have been developed in recent years (see [20], [21] and subsequent works), building on the realization that monopole operators [22], [23] are a suitable starting point for the quantum behavior of these spaces. Following this, the 3d \(\mathcal{N}=4\) Coulomb branch has been appreciated as a new construction methods for symplectic singularities. In physics, this is particularly prominent in the magnetic quiver program (see [24]–[26] and subsequent works), which uses this new construction to study quantum Higgs branches of higher dimensional theories with 8 supercharges, as discussed in the previous paragraph. This often relies on string theory which provides constructions of a vast class of SCFTs in dimension 3 to 6, using e.g. brane systems. These allow in many cases to derive magnetic quivers based on intersection numbers of branes in the magnetic phase, see for example [24], [25]. Such magnetic quivers are, in the simplest cases (in particular in the absence of certain orientifold planes), so-called unitary quivers. These are 3d \(\mathcal{N}=4\) quiver gauge theories with gauge group \(G=\prod_{i}\textrm{U}(n_{i})\), and generalizations thereof with so-called non-simply laced edges [27].
Here, we focus on symplectic singularities realized as the Coulomb branch of such unitary quivers. Within this class of theories, new isolated symplectic singularities have recently been found:
The \(\mathcal{Y}(\ell)\) singularities [5] have been given a quiver realization in [28].
The \(\overline{h}_{n,\sigma}\) singularities [6] have been realized as quivers in [26], [29].
Further singularities, called \(\mathcal{J}_{2,3}\) and \(\mathcal{J}_{3,3}\), have been found via unitary quivers [28].
Hyper-Kähler quotient singularities \(h_{n,\delta,\sigma}\) by discrete cyclic groups have been realized as quivers in [29] (building on earlier special cases of [26]).
Another (quaternionic) 4-dimensional singularity \(gb_2\) has been conjectured in [12], [13], which is extended here to a whole family \(gb_n\) of new ICSSs.
The purpose of this work is to complete the classification of such isolated symplectic singularities, realized as Coulomb branches of 3d \(\mathcal{N}=4\) quiver theories, by using the Decay and Fission algorithm [12], [13].
Summary of Results↩︎
All ICSSs discussed above, arising from different constructions, remarkably show up in our classification, in a completely uniform language, based on the Decay and Fission algorithm (Conjecture [conj:Decay-Fission]). Our main result is:
Assuming Conjecture [conj:Decay-Fission] holds, Table 2 provides the complete list of unitary quivers (as defined in Definition 1) whose Coulomb branches are ICSSs.
Note in particular the addition of the new quiver families1 labeled \(gb_n\), \(gc_n\) and \(gd_n\), which we claim to complete the list of unitary quivers whose Coulomb branch is an ICSS. As a first characterization, Table 1 provides the isometry algebra as well as the Highest Weight Generating (HWG) function, see Definition 10. As with almost all known ICSSs2, the HWG has a polynomial plethystic logarithm (see Definition 11), which is indicative of the simplicity of the moduli space. The HWG of \(gc_n\) coincides with that of \(\overline{h}_{2n+1,(3,1,\dots,1)}\), hinting at a new realization of that geometry, see Section 3.4.
| ICSS | Symmetry | \(\PL(\mathrm{HWG})\) |
|---|---|---|
| \(gb_n\) | \(\mathfrak{so}_{2n+1}\) | \(\mu_2 t^2 + (1 + \mu_1^2 + \mu_1^3) t^4 + \mu_1^3 t^6 -\mu_1^6 t^{12}\) |
| \(gc_n\) | \(\mathfrak{u}_1 \oplus \mathfrak{su}_{2n+1}\) | \((1+ \mu_1 \mu_{2n}) t^2 + (q \mu_1^3 + q^{-1} \mu_{2n}^3) t^4 - \mu_1^3 \mu_{2n}^3 t^8\) |
| \(gd_n\) | \(\mathfrak{so}_{2n}\) | \(\mu_2 t^2 + (1 + \mu_1^2 + \mu_1^3) t^4 + \mu_1^3 t^6 -\mu_1^6 t^{12}\) |
1.0.0.1 Future Directions.
We conjecture that the local fundamental group of the new families is trivial. It would be of great interest to prove this, in order to answer Beauville’s question. More generally, one should aim at proving the isolated character of the symplectic singularities in our list directly from the Coulomb branch construction, and not using the Decay and Fission Conjecture. In turn, such a proof could constitute a first step in a proof of the conjecture itself.
1.0.0.2 Outline.
In Section 2, we introduce relevant definitions for quivers and other tools, like the Decay and Fission algorithm. Thereafter, we state and prove Theorem [thm:final95result] in intermediate steps in Section 3. Lastly, we relax the initial assumptions and conjecture the result to hold in a more general quiver setting, see Section 4.
1.0.0.3 Acknowledgments.
We thank Paul Levy for sharing his insights regarding the new quiver families. We thank Julius Grimminger, Amihay Hanany, Daniel Juteau and Ben Webster for stimulating discussions. The work of SMS and MS is supported by the Austrian Science Fund (FWF), START project “Phases of quantum field theories: symmetries and vacua” STA 73-N [grant DOI: 10.55776/STA73]. SMS and MS also acknowledge support from the Faculty of Physics, University of Vienna. SMS acknowledges the financial support by the Vienna Doctoral School in Physics (VDSP). The work of QL is supported by École Normale Supérieure - PSL through a CDSN doctoral grant. MS gratefully acknowledges support from the Simons Center for Geometry and Physics, Stony Brook University, during the Workshop “Symplectic Singularities, Supersymmetric QFT, and Geometric Representation Theory,” at which the final stages of this work were performed.
2 Definitions and Tools↩︎
2.1 Good Quivers↩︎
Definition 1. A quiver \(Q\) is a triple \((V,A,K)\) where \(V\) is a finite set, \(A\) a function \(V\times V\to \mathbb{Z}\) and \(K\) a function \(V\to \mathbb{Z}_{>0}\), such that
for all \(x \in V\), \(A(x,x) = -2+2g_x\), for some \(g_x \in \mathbb{Z}_{\geq 0}\). If \(K(x) = 1\), then \(A(x,x)=-2\).
for all \(x\neq y \in V\), \(A(x,y)=0\) if and only if \(A(y,x)=0\). If they are non-zero, then both are positive and one is a divisor of the other.
there exists a function \(L: V\to\mathbb{Z}_{>0}\) such that for every \(x,y\in V\), \(A(x,y) L(y)= A(y,x)L(x)\).
Given a quiver \(Q = (V,A,K)\) and an integer \(N \geq 1\), we denote by \(N \cdot Q\) the quiver \((V,A, N K)\).
Elements of \(V\) are called vertices of \(Q\), \(A\) the adjacency matrix, and \(K\) the weight function. We say that \(K(x)\) is the weight of \(x \in V\). When \(V = \{ 1,\ldots,n\}\), we omit it and write \(Q = (A,K)\) where \(A\) is a matrix and \(K\) a column vector. Two vertices are neighbors if they are distinct and have a non-zero adjacency matrix coefficient.
We say that \(Q\) is simply-laced if its adjacency matrix is symmetric. Note that this allows for multiple edges between two vertices.
We call any function \(L\) in (iii) a length function. A Cartan matrix with such an \(L\) is known as symmetrizable Cartan matrix. Note here, that property (ii) is more restrictive.
The integer \(g_{x}\) is by definition the number of loops for the vertex \(x\in V\), and by definition a node of weight \(1\) has no loops.
Let \(Q = (V,A,K)\) be a quiver. Its underlying graph is the graph whose set of vertices is \(V\) and with a edge between \(x,y\in V,x\neq y,\) when \(A(x,y)\neq 0\). We say that a quiver is connected (resp.cyclic, a tree, linear, etc.) if its underlying graph is. The degree of a vertex is its number of neighbors.
We say that two quivers are isomorphic if there is a bijection between their sets of vertices preserving the adjacency matrices and the weight vectors.
Definition 2. A connected quiver \(Q\) is good if it is non-empty and one of the following holds true:
Its Hilbert Series \(\HS_Q(t)\) (see Definition 9) converges, is non-constant, and has in its series expansion no coefficient at order \(t\).
\(Q\) is \(N \cdot A_0^{(1)}\) for \(N \geq 2\) or \(N \cdot X_n^{(r)}\) for \(N \geq 1\) and \(X_n^{(r)}\) is one of the affine Dynkin quivers shown in Figure 3.
A necessary condition [30] for the “good” property reads \[\begin{align} \forall\, x\in V\, :\quad \sum_{y \in V} A(x,y)K(y) \geq 0 \; , \label{eq:good} \end{align}\tag{1}\] which is sufficient if and only if \(Q\) is simply-laced and not of type \(N \cdot X_n^{(r)}\).
The \(N \cdot X_n^{(r)}\) quivers with \(N>1\) are not stable, see Definition 7. Therefore, their Coulomb branches are not ICSSs. For \(N \cdot X_n^{(r)}\) with \(N=1\) both (i) and (ii) hold.
If \(\PL(\HS_Q(t)) = 2h\, t\) for some \(h\in \mathbb{Z}_{>0}\), then \(Q\) is said to be free. If \(\PL(\HS_Q(t)) = 2h \, t + \ldots\), then \(Q\) is said to contain free parts. (See Definition 11, for the Plethystic Logarithm (PL).)
Definition 3. Let \(Q= (V,A,K)\) be a quiver. A subquiver of \(Q\) is a quiver \(Q' =(V',A', K')\) where \(V'\subseteq V\), \(K'\leq K|_{V'}\) and \(A'\) is defined by \(A'(x,y) = -2\) if \(x=y\) and \(K(x) = 1\) and \(A'(x,y) = A(x,y)\) otherwise.
If \(V'\) is a subset of \(V\), then the subquiver on \(V'\) is \(Q' =(V',A', K|_{V'})\) with \(A'\) defined as before.
A subquiver of a good quiver is not necessarily good.
2.2 Moduli Space of Vacua, Coulomb Branch, and Symplectic Leaves↩︎
A 3d \(\Ncal=4\) gauge theory admits two maximal branches of the space of supersymmetric vacua: the Higgs and Coulomb branch. Here, the emphasis is placed on the Coulomb branch. Its mathematical definition was established in [7], [8], [11]. It was later proven in [9], [10] that Coulomb branches of all quiver gauge theories have symplectic singularities in the sense of [1]. It then follows that 3d \(\Ncal=4\) Coulomb branches admit a finite stratification into symplectic leaves [31]. As this is a finite partially ordered set, the stratification is naturally encoded in a Hasse diagram.
Definition 4. Let \(Q\) be a good quiver. We call \(\mathcal{M}_C (Q)\) its Coulomb branch, as defined in [11]. This is a conical symplectic singularity.
2.3 The Decay and Fission Algorithm↩︎
The Decay and Fission algorithm is reviewed here. It conjecturally provides a combinatorial construction of the Hasse diagram of \(\mathcal{M}_C (Q)\).
Definition 5. Let \(Q=(V,A,K)\) be a quiver. A fission product of \(Q\) is a multiset \(\multiset{Q_1,\ldots,Q_n}\) with \(n\geq 0\), where \(Q_i = (V_i,A_i,K_i)\) are quivers such that :
for each \(1\leq i\leq n\), \(Q_i\) is a good connected subquiver of \(Q\);
\(\sum_{i=1}^n K_i\leq K\), where we define each \(K_i\) to vanish on \(V\backslash V_i\)
Let us call \(\mathcal{L}(Q)\) the set of fission products of \(Q\).
A decay product of \(Q\) is a quiver \(Q'\) such that \(\multiset{ Q'} \in \mathcal{L}(Q)\). If \(Q,Q'\) are two quivers such that \(Q\) has a decay product isomorphic to \(Q'\), we write \(Q\rightsquigarrow Q'\).
The empty fission product \(\multiset{}\) is always in \(\mathcal{L}(Q)\). \(Q\) is a decay product of itself if and only if it is good and connected.
The set of fission products can be equipped with a natural partial order:
Definition 6. Let \(Q\) be a quiver and \(\multiset{Q_1,\ldots,Q_n},\multiset{Q'_1,\ldots,Q'_m}\) two fission products of \(Q\). We write \(\multiset{Q_1,\ldots,Q_n} \preccurlyeq \multiset{Q'_1,\ldots,Q'_m}\) if there exists a partition \(\{ 1,\ldots,n\} = \bigsqcup_{j=1}^m I_j\), with the \(I_j\) possibly empty, such that for every \(1\leq j \leq m\), \(\multiset{Q_i:i\in I_j}\in\mathcal{L}(Q'_j)\).
This leads to the following conjecture:
There exists a 1-to-1 correspondence between the poset of symplectic leaves of the Coulomb branch of a good 3d \(\mathcal{N}=4\) quiver theory \(Q\) and the poset \((\mathcal{L} , \succcurlyeq)\) of decay and fission products of the good quiver \(Q\)
The Decay and Fission algorithm also allows us to determine the minimal transitions, or, transverse slices in form of a quiver. Using this, the underlying geometry can be determined.
Definition 7. A quiver \(Q\) is stable if it is good, non-empty, and its only decay product is itself.
This is equivalent to \(\mathcal{L}(Q) = \{\multiset{},\multiset{Q}\}\). Geometrically, it means that the Coulomb branch of \(Q\) only has two leaves: the singular point and the regular part of the Coulomb branch.
3 Results and Proofs↩︎
In this section, Theorem [thm:final95result] is proven first for simply-laced and then for non-simply-laced quivers assuming Conjecture [conj:Decay-Fission]. Before proceeding, we recall two facts:
The stable Abelian quivers are given in Table 2.
Moreover, any quiver that contains either \((i)\) two nodes connected with a simply-laced edge of multiplicity \(N\geq 2\) or \((ii)\) a vertex \(x\in V\) with \(g_{x}\geq1\) cannot be a stable quiver, as it admits a decay product isomorphic to the quivers with geometry \(A_{N-1}\) or \(D_{g_x +1}\) (see Table 2). Therefore, any such quiver can be omitted in the following discussion.
3.1 Simply-laced Quivers↩︎
To begin with, we first consider quivers with simply-laced edges.
The quivers in Figure 3 are all good and stable.
![Figure 3: All simply-laced quivers that are stable. There are the affine Dynkin quivers A_n^{(1)} (n \geq 2), D_n^{(1)} (n \geq 4), and E_{6,7,8}^{(1)}; conventions follow [32]. The subscript in the name of each quiver is one less than the number of nodes. In addition, there is the 2-vertex quiver A_{N-1} with an edge of multiplicity N.](/article/2504.05373/_pdflatex/zhqvduyj.png "fig:")
Proof. The two-vertex quiver has the \(A\)-type Kleinian surface singularity \(\mathbb{C}^2\slash \mathbb{Z}_{N}\) as Coulomb branch. The untwisted affine Dynkin quivers \(A_n^{(1)}\), \(D_{n}^{(1)}\), \(E_{6,7,8}^{(1)}\) (see Figure 3) have Coulomb branches given by the closure of the minimal nilpotent \(\overline{\mathcal{O}}_{\min}^{\mathfrak{g}}\). All of which are isolated symplectic singularities.
Alternatively, it can be checked that these quivers contain no non trivial subquiver satisfying the necessary condition 1 . ◻
The main result in this section is the following theorem:
Let \(Q\) be a non-empty good simply-laced quiver. Then one of the quivers listed in Figure 3 is a decay product of \(Q\).
Proof. Let \(Q\) be a non-empty good simply-laced quiver. Each connected component of \(Q\) is a decay product, therefore we only have to prove the theorem for connected quivers. If \(Q\) contains a cycle, then this cycle, with all weights set to \(1\), is a decay product isomorphic to \(A^{(1)}_{n}\) for some \(n\). So we only have to prove the theorem for acyclic quivers.
As \(Q\) is finite and acyclic, it must have a node with exactly one neighbor. To ensure that \(Q\) is good at this node, the neighbor must have weight greater or equal to \(2\). Therefore, \(Q\) is non-Abelian.
We can construct a decay product of \(Q\) in the following way: choose a connected component in the subgraph of non-Abelian nodes. Add all the nodes with weight \(1\) that are direct neighbors of these. If needed, we can remove a weight one vertex to ensure that the subquiver does not correspond to an over-extended \(N \cdot X_n^{(r)}\) quiver3. The resulting subquiver is good and, therefore, a decay product. It has the property that all \(U(1)\) nodes have exactly \(1\) neighbor. Hence proving the theorem for this decay product, implies the result for the original quiver. Therefore, we can assume that all nodes of weight \(1\) have only one neighbor.
If \(Q\) has a node with four neighbors or more, then \(Q\) decays to \(D^{(1)}_4\). If \(Q\) has two nodes with three neighbors, then it decays to \(D^{(1)}_n\) for some \(n \geq 5\). If \(Q\) has no vertices with three or more neighbors, then it is linear. Let \(a_1,\ldots,a_n\) be the weights of its nodes. Then, setting \(a_0 = a_{n+1} = 0\), the “good” constraint 1 reads: \[\forall k \in\{1,\ldots,n\}: \quad 2a_k \leq a_{k-1} + a_{k+1} \,,\] which is a convexity inequality. It implies : \[\label{eq:convexity951} \forall k \in\{1,\ldots,n\}: \quad a_k \leq \frac{n+1-k}{n+1} a_0 + \frac{k}{n+1}a_{n+1} = 0 \,.\tag{2}\] Therefore, we can assume that \(Q\) has exactly one node with three neighbors, with all other nodes having one or two: \(Q\) is of the form \[\begin{align} \raisebox{-.5\height}{ \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/eotucfni.png}\tag{3}\end{figure} } \tag{4} \end{align}\] Let \(a_0=b_0=c_0 =m\) be the weight of the trivalent node, and \((a_1,a_2,\ldots)\), \((b_1,b_2,\ldots)\), \((c_1,c_2,\ldots)\) the sequences of weights along the three legs. We have \(m\geq 2\) and \(a_1,b_1,c_1 \geq 1\). As before, these sequences are convex and vanish eventually. A convexity inequality like 2 implies that \(a_1,b_1\) and \(c_1\) are strictly smaller than \(m\), and more generally that the sequences are strictly decreasing until they reach zero. The “good” constraint 1 at the trivalent node reads: \[\label{eq:proof95E95balance1} 2m\leq a_1 + b_1+c_1 \, .\tag{5}\] Together with \(a_1,b_1,c_1\leq m-1\), it implies \(m\geq 3\).
If \(a_2,b_2\) and \(c_2\) are greater or equal to \(1\), then, as the sequences are strictly decreasing, \(a_1,b_1\) and \(c_1\) are greater or equal to \(2\) and \(m \geq 3\). In this case, \(Q\) decays to \(E^{(1)}_6\). Therefore we can assume, without loss of generality, that \(c_2= 0\).
If \(a_3,b_3 \geq 1\), then the sequences being strictly decreasing implies \(m \geq 4\), and \(Q\) decays to \(E^{(1)}_7\).
Therefore we can assume that \(b_3 =c_2= 0\). We then have \(a_1\leq m-1\), \(b_1 \leq \frac{2m}{3}\) and \(c_1 \leq \frac{m}{2}\). Inserting two or three of these into 5 , we find \(2m \leq \frac{13}{6}m -1\), hence \(m\geq 6\), and \[a_1 \geq \frac{5m}{6} \, , \qquad b_ 1 \geq \frac{m}{2} + 1 \, , \qquad c_1 \geq \frac{m}{3}+1 \geq 3 \, . \label{eq:proof95E95balance2}\tag{6}\]
Convexity also implies that \(\forall k\geq 0, a_k\geq (1-k)m +ka_1\) and similar equations for the other two sequences. Using these in combination with 6 , we get, for all \(k\geq 0\) \[a_k \geq \frac{m}{6}(6-k) \, , \qquad b_k \geq \frac{m}{2}(2-k) +k \, .\] Finally, we find that for \(k\in \{0,1,\ldots,6\}\), \(a_k\geq 6-k\) and for \(k\in\{0,1,2\}\), \(b_k \geq 6-2k\). Therefore \(Q\) decays to \(E^{(1)}_{8}\). This concludes the proof. ◻
The quivers of Proposition [prop:list95stable95simply95laced95quivers] are exactly all the stable simply-laced quivers.
Proof. None of those quivers are decay products of the others. ◻
![Figure 4: All affine Dynkin-type quivers that allow for a modification by a non-simply laced edge connected to a vertex of weight 1, see Section 3.2. In any quiver, all \ell_i should be pairwise coprime. For \overline{\mathcal{O}_{\text{min}}}\left(\mathfrak{a}_{2n-1}\right) and \overline{\mathcal{O}_{\text{min}}}\left(\mathfrak{e}_{6}\right), the \ell_i are odd. For \overline{\mathcal{O}_{\text{min}}}\left(\mathfrak{f}_{4}\right), \ell_2 is odd. For \overline{\mathcal{O}_{\text{min}}}\left(\mathfrak{d}_{4}\right), \mathrm{gcd}(\ell_1,3)=1. The naming of the quivers follows the conventions of affine Dynkin diagram of [32]. All the quivers in the first five rows have n+1 vertices.](/article/2504.05373/_pdflatex/gyjflero.png "fig:")
3.2 From Quiver Classification to Geometry Classification↩︎
It is well-known that two non-isomorphic quivers \(Q\) and \(Q'\) can have the same Coulomb branch, \(\mathcal{M}_C (Q) = \mathcal{M}_C (Q')\). So even though \(Q\) and \(Q'\) are distinct as combinatorial objects, they represent the same geometry. In this section, we provide a class of instances of this phenomenon which is crucial for our classification, see Table 4. This amounts to enlarging slightly what we call a Dynkin graph, so as to encompass all quivers whose Coulomb branch is a minimal orbit closure.
Let \(Q = (V,A,K)\) be a good, stable, non-Abelian quiver. Assume \(Q\) has an edge \(E\) between a vertex \(x\) of weight \(K(x)=1\) and a vertex \(y\) of weight \(K(y) \geq 2\), with \(A(x,y)=A(y,x)=1\). Let \(Q' = (V,A',K)\) where \(A=A'\) except for \(A'(x,y) = \ell \geq 1\). If \((V,A',1)\) is free then \(\mathcal{M}_C (Q) = \mathcal{M}_C (Q')\).
Proof. The quiver \(Q\) (and likewise for \(Q'\)) can be non-simply-laced, and therefore may not define a morphism \(\prod_i U(n_i)\to \prod_{\langle i;j\rangle} U(n_in_j)\). However, it does define a representation of the maximal torus \(R:\prod_i U(1)^{n_i}\to \prod_{\langle i;j\rangle} U(n_in_j)\). The monopole formula involves the quotient of \(T = \prod_i U(1)^{n_i}\) by \(\operatorname{Ker}(R)_0\) the connected part of the kernel of \(R\).
The kernel \(\operatorname{Ker}(R)\) is included in the diagonal subgroup \(D= \prod_i U(1)_{\rm{diag}}\) and the restriction \(R|_D\) of \(R\) to this diagonal subgroup is the morphism associated with the Abelian quiver obtained from \(Q\) by setting all weights to \(1\). As \(Q\) is stable and non-Abelian, this quiver must be free. This implies, in particular, that the kernel \(\operatorname{Ker}(R)\) is connected. The monopole formula for \(Q\) therefore only depends on the image of \(R\).
Likewise, because of the assumption on \(\ell\), we see that \(\operatorname{Ker}(R')\) is connected and the monopole formula for \(Q'\) therefore only depends on the image of \(R'\). To conclude, notice that we have a commutative diagram:
The horizontal arrow is the morphism obtained by applying \(z \in U(1)\mapsto z^\ell \in U(1)\) to the node \(x\). In particular, this is a surjective morphism. Therefore, \(\operatorname{Im}(R) = \operatorname{Im}(R')\) and both quivers have the same Coulomb branch. In particular, \(Q'\) is a stable quiver. ◻
Note that in the physics literature, this construction can be understood via framing/unframing operations of the gauge group encoded in the quiver. This is also known as Crawley-Boevey moves [33] in the mathematics literature.
3.3 Non-simply-laced Quivers↩︎
We now extend the classification to include quivers with non-simply-laced edges. Our main result, Theorem [thm:final95result], is a direct consequence of the two following propositions.
Let \(Q\) be a good quiver. Then \(Q\) admits a decay product isomorphic to one of the quivers listed Proposition [prop:list95stable95non95abelian95non95simply95laced95quivers].
Proof. Let \(Q\) be a good quiver. Any connected component of \(Q\) is a decay product, so we only have to deal with the case where \(Q\) is connected.
If \(Q\) is Abelian or admits a good Abelian decay product (or even a non-free Abelian subquiver), then we can apply Theorem [thm:list95stable95abelian95quivers]. Therefore, we can assume that \(Q\) has only free Abelian subquivers. In particular, \(Q\) is non-Abelian, has no cycles4 and if two non-simply-laced edges are directed towards each other, then their lacedness are coprime.
Just as in the simply-laced case, we can construct a decay product of \(Q\) where all nodes of weight \(1\) have exactly one neighbor. Since the result for this decay product implies the result for \(Q\), we can assume that \(Q\) has this property.
Let us call trivial a non-simply-laced edge whose source is a node of rank \(1\). As discussed in Section 3.2, the field theory associated with \(Q\), and in particular its Coulomb branch, is the same as the one of \(Q'\), obtained by replacing all trivial non-simply-laced edges by simply-laced ones. Explicitly, if we prove the theorem for \(Q'\) and find a decay \(Q' \rightsquigarrow Q''\), then \(Q''\), after adding back trivial edges if needed, is a decay product of \(Q\). We can therefore restrict to the case where \(Q\) has no trivial non-simply-laced edges.
If \(Q\) has an edge of lacedness greater or equal to \(4\), then it decays to \(\mathcal{Y}(\ell)\) (cf.Table 2). Therefore we can assume that all non-simply-laced edges have lacedness \(2\) or \(3\).
If \(Q\) has an edge of lacedness \(3\) whose source has two or more neighbors, then \(Q\) decays to \(G_2^{(1)}\) (cf. Table 4), \(\mathcal{J}_{2,3}\) or \(\mathcal{J}_{3,3}\) (cf. Table 2). Therefore, we can assume that any triple edge is sourced at a node of \(Q\) with exactly one neighbor.
If \(Q\) has two or more non-simply-laced edges, then, by considering two such edges such that all edges in between are simply-laced, we see that \(Q\) decays to a \(\tilde{B}_n^{(2)}\) (or \(A_{2n}^{(2)}\)) (cf.Table 4), \(B_n^{(2)}\) (or \(D_{n+1}^{(2)}\)) (cf.Table 4), \(gc_n\) (cf.Table 2), or \(gb_n\) (cf.Table 2).
If it has only simply-laced edges, Theorem [thm:simply95laced95decay] applies. Therefore, we can assume that \(Q\) has exactly one non-simply-laced edge.
If \(Q\) has a vertex of degree greater or equal to \(3\), then \(Q\) decays to \(C_n^{(2)}\) (or \(A_{2n-1}^{(2)}\)) (Table 4), \(B_n^{(1)}\) (Table 4), or \(gd_n\) (Table 2). Therefore, \(Q\) is a linear quiver with one non-simply-laced edge of lacedness \(\ell \in \{2,3\}\). If \(\ell =3\), then \(Q\) is shaped as in \[\begin{align} \raisebox{-.5\height}{ \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/rlgvnyzu.png}\tag{7}\end{figure} } \tag{8} \end{align}\] Let \(a\) be the weight of the long node and \(b_0,b_1,\ldots\) the weight of the short nodes. The balance conditions read: \[\begin{align} 2a &\leq 3 b_0 \, ,\tag{9}\\ 2b_0 &\leq a +b_1 \,, \tag{10}\\ 2b_n &\leq b_{n-1} + b_{n+1} \quad (\forall n \geq 1) \,. \end{align}\] The sequence \((b_n)_{n\geq 0}\) is convex and vanish for \(n\) large enough. Therefore, it is positive and strictly decreasing before it reaches \(0\). In particular, \(b_1 \leq b_0 - 1\). Together with equations (9 ) and (10 ), this implies \(a\geq 3\), \(b_0\geq 2\) and \(b_1 \geq 1\). Hence \(Q\) decays to \(D_4^{(3)}\) (\(G_2^{(3)}\)) (cf. Table 4).
We are only left with the case \(\ell = 2\). If a long node has three neighbors, then \(Q\) decays to a \(BD\)-shaped quiver. Therefore, we can assume that \(Q\) is a linear quiver \[\begin{align} \raisebox{-.5\height}{ \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/apufxqvc.png}\tag{11}\end{figure} } \tag{12} \end{align}\] Let us write \(a_0,a_1,\ldots\) for the weights of the long nodes and \(b_0,b_1,\ldots\) for the weights of the short nodes. We already know that \(a_0 \geq 2,b_0\geq 1\). The balance conditions read, with \(n\in \mathbb{Z}_{>0}\): \[\begin{align} 2a_0 &\leq 2 b_0 + a_1 \,, \tag{13}\\ 2b_0 &\leq a_0+b_1 \,,\tag{14} \\ 2a_n &\leq a_{n-1} + a_{n+1}\,, \tag{15}\\ 2b_n &\leq b_{n-1} + b_{n+1} \,.\tag{16} \end{align}\] As before, the sequences \((a_n)_{n\in\mathbb{N}}\) and \((b_n)_{n\in\mathbb{N}}\) are convex, positive and strictly decreasing before they reach \(0\). In particular \(b_n \leq b_0 - n\) and \(a_n\leq a_0 - n\). This inequalities for \(n=1\), together with 13 and 14 imply that: \[\begin{align} a_0 +1 \leq 2b_0 \,,\\ 1+b_0 \leq a_0 \,.\label{eq:2long956} \end{align}\tag{17}\] All the solutions of this system of inequalities satisfy \(a_0\geq 3\) and \(b_0\geq 2\). Then, from (13 ) and (14 ), we also see that \(a_1\geq 2\) and \(b_1\geq 1\).
If \(a_2 \geq 1\), we see that \(Q\) decays to \(F_4^{(1)}\) as \(a_1\geq2\) (cf. Table 4), so we can now assume that \(a_2=0\). Equation 15 reduces to \(2a_1 \leq a_0\).
If \(b_2 = 0\), we have \(2b_1 \leq b_0\). Together with 13 and 14 , we get \(\frac{3}{4} a_0 \leq b_0 \leq \frac{2}{3} a_0\) which is a contradiction as \(a_0 >0\). Therefore \(b_2\geq 1\). This implies \(b_1\geq 2\), \(b_0\geq 3\), and, using 17 , \(a_0\geq 4\). We conclude that \(Q\) decays to \(F_4^{(2)}\) (or \(E_6^{(2)}\)) (cf. Table 4). ◻
3.4 HWG Computations↩︎
Part of the insight gained in this work are the three new families of unitary quivers (cf.Table 2) — \(gb_n\), \(gc_n\), \(gd_n\). As a first analysis of the geometry, the (unrefined) Hilbert Series (see Definition 9) for the lowest dimensional cases are calculated to be \[\begin{align} \HS_{gb_{2}}(t)=&\frac{1+6t^{2}+42t^{4}+71t^{6}+122t^{8}+71t^{10}+42t^{12}+6t^{14}+t^{16}}{\left(1-t^{2}\right)^{4}\left(1-t^{4}\right)^{4}} \\ \HS_{gc_{2}}(t)=&\frac{1+20t^{2}+175t^{4}+590t^{6}+1290t^{8}+1550t^{10}+1290t^{12}+590t^{14}+175t^{16}+20t^{18}+t^{20}}{\left(1-t^{2}\right)^{5}\left(1-t^{4}\right)^{5}} \\ \HS_{gd_{3}}(t)=&\frac{1+10t^{2}+85t^{4}+239t^{6}+545t^{8}+602t^{10}+545t^{12}+239t^{14}+85t^{16}+10t^{18}+t^{20}}{\left(1-t^{2}\right)^{5}\left(1-t^{4}\right)^{5}} \end{align}\] In addition, for the lowest \(n\) cases we computed the refined Hilbert series, which allows to derive the Highest Weight Generating function (recall Definition 10). The obtained HWGs are then expected to generalize into the full \(gb_n\), \(gc_n\), \(gd_n\) families as shown in Table 1. For higher \(n\) cases, this proposed HWG has been tested against unrefined Hilbert series computations. Note the remarkable fact that the HWG for \(gb_n\) and \(gd_n\) are exactly the same (even though the symmetry algebra is not the same), and they do not depend on \(n\).
The HWG for \(gc_n\) coincides with that of \(\overline{h}_{2n+1,\sigma = (3,1,\dots,1)}\) [26].5 This leads to the conjecture that the two geometries are the same. If this is true, the \(gc_n\) still should appear in our result as a stable quiver, but the associated Coulomb branch is a particular case of the \(\overline{h}_{n,\sigma}\) family: \[\tag{18} \mathcal{M}_C \left( \raisebox{-.5\height}{\begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/ivfycrht.png}\tag{19}\end{figure}} \right) = \mathcal{M}_C \left( \raisebox{-.5\height}{ \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/ysmpordw.png}\tag{20}\end{figure}} \right) \, .\]
4 Generalization to \((p,q)\)-edges↩︎
Let us relax assumption (ii) in Definition 1 to allow for quivers with \((p,q)\)-edges.
Definition 8. A generalized quiver \(Q\) is a triple \((V,A,K)\) where \(V\) is a finite set, \(A\) a function \(V\times V\to \mathbb{Z}\) and \(K\) a function \(V\to \mathbb{Z}_{>0}\), such that
for all \(x \in V\), \(A(x,x) = -2+2g_x\), with \(g_x \in \mathbb{Z}_{\geq 0}\). If \(K(x) = 1\), then \(A(x,x)=-2\).
there exists a function \(L: V\to\mathbb{Z}_{>0}\) such that for every \(x,y\in V\), \(A(x,y) L(y)= A(y,x)L(x)\).
A \((p,q)\)-edge is the extension to the case that if \(A(x,y)>0\) for some \(x,y\in V\), \(A(x,y)\) does not need to be a divisor of \(A(y,x)\), and vice versa. Such an edge is called a \((p,q)\)-edge with \(p=A(x,y)\) and \(q=A(y,x)\) [34]. The Cartan matrix \(A\) is symmetrizable, due to the length function.
Based on analysis of the abelian generalized quivers in [29], it is natural to propose the following:
The Decay and Fission algorithm holds true for good quivers of Definition 8.
Let \(Q\) be a generalized quiver, then it admits a decay product isomorphic to a quiver listed in Tables 2 and 4, if one allows for \(gc_1\).
Proof. The proof requires an extension of the proof of Theorem [thm:non-simply95laced], assuming Conjecture [conj:Decay95Fission95p-q] holds.
First, we realize that the \(\mathcal{Y}(\ell)\) quiver is naturally extended to the \((p=\ell^\prime,q=\ell)\)-edges cases, \[\begin{align} \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/fecqtmps.png}\tag{21}\end{figure} \qquad \leftrightarrow\qquad \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/ctiuvldb.png}\tag{22}\end{figure} \end{align}\] analogous to Section 3.2. Therefore, we can restrict to \((p,q)\)-edges between non-Abelian vertices with \(p,q<4\).
Next, by the linear Abelian quivers \(h_{n=1,\delta,\sigma}\) (see Table 2), we can restrict to co-prime \(p,q\), as otherwise a decay to an Abelian quiver exists. Thus, without loss of generality, we are led to a \((p,q)\)-edge with \(p=3\) and \(q=2\). This is precisely \(gc_1\), for which one computes \[\begin{align} gc_1: \begin{figure}\includegraphics[width=0.8\textwidth]{_pdflatex/zbekgnyi.png}\label{iwojrzeh}\end{figure} \;,\qquad \HS_{gc_1}(t)= \frac{1+6t^2+29t^4+30t^6+29t^8+6t^{10}+t^{12}}{(1-t^2)^3(1-t^4)^3} \,, \end{align}\tag{23}\] which is compatible with the \(n=1\) limit of \(\HWG_{gc_n}\) of Table 1. This concludes the proof. ◻
5 Hilbert Series and Highest Weight Generating Functions↩︎
In general, the \(\mathbb{C}^*\)-action on a CSS induces a grading on the coordinate ring. When the dimensions of the subspaces of fixed degree are finite, the Hilbert series is defined as their generating function.
Definition 9. For a quiver \(Q=\left(V,A,K\right)\) the conformal dimension is defined as the function \[\begin{align} & \Delta_{Q}:\,\Lambda\equiv\prod_{x\in V}\mathbb{Z}^{K(x)}\rightarrow\mathbb{R} \\ & \Delta_{Q} (\vec{m}_1 , \dots , \vec{m}_n ) = \frac{1}{2} \sum\limits_{i,j=1}^n \mathrm{sgn} (A_{i,j}) \delta ( \vec{m}_i A_{i,j} , \vec{m}_j A_{j,i} ) \notag \\ & \textrm{with} \quad \delta (\vec{u} , \vec{v}) = \sum\limits \notag|u_i - v_j| \end{align}\] For the length function \(L\) we define \(\boldsymbol{L} \in \prod_{x\in V}\mathbb{Z}^{K(x)}\) as \(\boldsymbol{L}|_{\mathbb{Z}^{K(x)}} = L(x) \cdot (1,\ldots,1)\) where \((1,\ldots,1) \in \mathbb{Z}^{K(x)}\), such that \(\Delta_{(A.K)}(\boldsymbol{m}+\boldsymbol{L})=\Delta_{(A,K)}(\boldsymbol{m})\).
The Hilbert series of the Coulomb branch is given [11], if the series converges, by the Monopole Formula [20], [35] \[\HS_{Q} (t) = \frac{1-t^2}{|W|} \sum\limits_{\mathbf{m} \in \Lambda / \mathbf{L} \mathbb{Z}} \; \sum\limits_{w \in W (\mathbf{m})} \frac{t^{\Delta (\mathbf{m})}}{\det (1 - w t^2)} \, .\] where \(w\) is seen as a permutation matrix.
There is a list of quivers for which the monopole formula does not converge, but where the Coulomb branch is a CSS with a well-defined Hilbert series. They are given in (ii) of Definition 2.
When the Coulomb branch has a symmetry algebra \(\mathfrak{g}\), one can refine the Hilbert series, which becomes a power series in \(t\) with coefficients which are characters of \(\mathfrak{g}\), written as Laurent polynomials in \(\mathrm{rank}(\mathfrak{g})\) variables which we denote as \(z_1 , \dots , z_{\mathrm{rank}(\mathfrak{g})}\). Then the refined Hilbert series can be written in a unique way as \[\label{eq:refinedHS} \HS_{Q}(t ; z_1 , \dots , z_r ) = \sum\limits_{n \in \mathbb{N}} \sum\limits_{n_1 , \dots , n_r \in \mathbb{N}} a_{n,n_1 , \dots n_r} \chi_{[n_1 , \dots , n_r]} (z_1 , \dots , z_r ) t^n \, ,\tag{24}\] where \(\chi_{[n_1 , \dots , n_r]} (z_1 , \dots , z_r )\) is the character for the representation of \(\mathfrak{g}\) with highest weight specified by Dynkin labels \([n_1 , \dots , n_r]\). For the ordering of the labels, we choose conventions such that for \(\mathfrak{g}= \mathfrak{su}_n\), \(n \geq 3\), \([1,0,\dots,0]\) corresponds to the fundamental representation and \([1,0,\dots,0,1]\) to the adjoint representation. For \(\mathfrak{g}=\mathfrak{so}_n\), \([1,0,\dots,0]\) corresponds to the fundamental representation and \([0,1,\dots,0]\) to the adjoint representation for \(n \geq 7\). For \(\mathfrak{so}_6\), the adjoint is \([0,1,1]\), for \(\mathfrak{so}_5\), the adjoint is \([0,2]\).
Definition 10. The Highest Weight Generating (HWG) function [36] associated to 24 is defined to be \(\HWG_{Q} \in \mathbb{C}[[t , \mu_1 , \dots , \mu_r]]\) with \[\HWG_{Q} = \sum\limits_{n \in \mathbb{N}} \sum\limits_{n_1 , \dots , n_r \in \mathbb{N}} a_{n,n_1 , \dots n_r} t^n \mu_1^{n_1} \cdots \mu_r^{n_r} \, .\]
The plethystic logarithm, and its inverse the plethystic exponential, are natural operations when studying generating functions.
Definition 11. For \(f \in \mathbb{C}[[t_1 , \dots , t_k]]\) such that \(f(0,\dots,0)=1\), the Plethystic Logarithm (PL) is defined to be \[\PL[f](t_1 , \dots , t_k) = \sum\limits_{j=1}^{\infty} \frac{\mu (j)}{j} \log (f(t_1^j \dots , t_k^j)) \, .\] Here, \(\mu\) is the Möbius multiplicative function.
For some ICSSs, such as the minimal nilpotent orbits or the Coulomb branches of the \(gb_n,gc_n\) and \(gd_n\) quivers, the plethystic logarithm of the HWG turns out to be a simple polynomial.
References↩︎
Note that the quiver \(gb_2\) was already discussed in the initial papers on the Decay and Fission algorithm [12], [13], using the same method of derivation as in this work.↩︎
Recall, for \(\overline{\mathcal{O}_{\text{min}}}\left(\mathfrak{g}\right)\), \(\PL [ \mathrm{HWG} ]= \chi_{\mathrm{adj}} t^2\). In contrast, the HWG for \(\mathcal{J}_{2,3}\) and \(\mathcal{J}_{3,3}\) do not have polynomial PL.↩︎
An over-extended \(N\cdot X_n^{(r)}\) quiver includes an extra weight one vertex at the affine vertex of the Dynkin diagram.↩︎
The existence of a length function is required here, as it plays a crucial role in the proof [29] of Theorem [thm:list95stable95abelian95quivers].↩︎
We thank Paul Levy for pointing this out to us.↩︎