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On the Existence of Scaling Multi-Centered Black Holes

Abstract

For suitable charges of the constituents, the phase space of multi-centered BPS black holes in \(\mathcal {N}=2\) four-dimensional supergravity famously exhibits scaling regions where the distances between the centers can be made arbitrarily small, so that the bound state becomes indistinguishable from a single-centered black hole. In this note, we establish necessary conditions on the Dirac product of charges for the existence of such regions for any number of centers, generalizing the standard triangular inequalities in the three-center case. Furthermore, we show the same conditions are necessary for the existence of multi-centered solutions at the attractor point. We prove that similar conditions are also necessary for the existence of self-stable Abelian representations of the corresponding quiver, as suggested by the duality between the Coulomb and Higgs branches of supersymmetric quantum mechanics.

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Notes

  1. In the appendix, we adapt the proof of the final inequalities when this condition is not satisfied, i.e. when there are more vanishing arrows than necessary: in this case, there are less than |I| independent potential constraints

  2. This minimization property was first observed in examples of quivers associated to non-compact Calabi–Yau threefolds in [21].

  3. The fact that the inequalities for the existence of self-stable representations are stronger than the one for the existence of scaling solutions is consistent with the fact that the contribution of scaling solutions may cancel against the contribution of regular collinear solutions when computing the equivariant Dirac index of the phase space of multi-centered configurations using localization [22].

  4. This complex is related to the cellular homology complex \(0{\rightarrow }{\mathbb {R}}^{Q_1}\overset{\partial _1}{\rightarrow }{\mathbb {R}}^{Q_0}\rightarrow 0\) of the graph considered as a cellular space. The cellular homology group \(H^0\) gives the number of connected components of the graph, see [23, Sec 2.2].

  5. The notions of biconnected graph (also known as ’non-separable’), biconnected components of a graph (also known as ‘blocks’), strongly connected directed graph and strongly connected components are standard in graph theory, see, for example, [24, Sec 3,6].

  6. Since i is the only non-Abelian node, we abuse notation and denote by (ak) and (bk) the lift of the arrows \(a:i\rightarrow j\) and \(b:j\rightarrow i\) on Q to the k-th copy of the node i on \(Q^d\).

References

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Acknowledgements

We are grateful to Guillaume Beaujard, Jan Manschot, Swapnamay Mondal and Olivier Schiffmann for useful discussions, and to the anonymous referees for their careful reading and valuable suggestions. The research of BP is supported by Agence Nationale de la Recherche under contract number ANR-21-CE31-0021.

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Communicated by Claude-Alain Pillet.

A Proofs

A Proofs

In this Appendix, we provide mathematical proofs for some of the technical results used in the body of the paper.

A.1 Conserved Currents and Graph Homology

A strictly positive conserved current is an element of \(\ker (\partial _1)\cap ({\mathbb {R}}^\star _+)^{Q_1}\). A quiver is strongly connected if and only if for any two nodes \(i,j\in Q_0\) , there is an oriented path from i to j and from j to i. The key result which allows us to derive constraints on the existence of scaling or attractor solutions is the following:

Lemma A.1

(i) If Q admits a strictly positive conserved current, then Q is strongly connected.

(ii) A strictly positive conserved current \(\lambda \) can be expressed as a sum of strictly positive conserved currents circulating on all simple oriented cycles of the quiver, i.e. \(\lambda =\sum _{w\in Q_2}\mu _w\partial _2(w)\), with \(\mu _w>0\).

Proof

  1. i)

    Suppose that Q is not strongly connected. The strongly connected components of Q form a connected tree with at least two nodes: one has then a nontrivial partition \(Q_0=Q_0'\sqcup Q_0''\) such that there is at least one arrow \(Q_0'\rightarrow Q_0''\), and no arrow \(Q_0''\rightarrow Q_0'\). Consider \(\lambda \in \ker (\partial _1)\cap {\mathbb {R}}^{Q_1}\): by summing the conservation of \(\lambda \) at each node of \(Q_0'\), one has \(\sum _{(a:Q_0'\rightarrow Q_0'')\in Q_1}\lambda _a=0\), and then \(\lambda \not \in ({\mathbb {R}}^\star _+)^{Q_1}\).

  2. ii)

    Suppose now that Q is strongly connected. We must show that \(\ker (\partial _1)\cap ({\mathbb {R}}^\star _+)^{Q_1}\subset \partial _2(({\mathbb {R}}^\star _+)^{Q_2})\). We begin by showing \(\ker (\partial _1)\cap ({\mathbb {R}}_+)^{Q_1}\subset \partial _2(({\mathbb {R}}_+)^{Q_2})\): for this we reason inductively on the number of arrows which carry a non-vanishing current. For \(\lambda \in \ker (\partial _1)\cap ({\mathbb {R}}_+)^{Q_1}\), we consider \(Q^\lambda =(Q_0,\{a\in Q_1|\lambda _a>0\})\), the (possibly non connected) quiver where we keep only the arrows with a non-vanishing current. If \(Q^\lambda _1\ne \emptyset \), take \((a:i\rightarrow j)\in Q^\lambda _1\) such that \(\lambda _a>0\) is minimal. One has \(\lambda \in \ker (\partial _1)\cap ({\mathbb {R}}^\star _+)^{Q^\lambda _1}\), and then each connected component of \(Q^\lambda \) is strongly connected from (i): there is then a path \(v:j\rightarrow i\) in \(Q^\lambda \), and then a simple oriented cycle \(w:=av\in Q_2\) containing a, such that the arrows of w are in \(Q^\lambda \). One has \(\lambda ':=\lambda -\partial _2(\lambda _a w)\in \ker (\partial _1)\), and, because \(\lambda _a\) is minimal, \(\lambda '\in ({\mathbb {R}}_+)^{Q_1}\). Moreover, \(\lambda '_a=0\), and then \(Q^{\lambda '}_1\subsetneq Q^\lambda _1\): by induction on the number of arrows of \(Q^\lambda \), one has \(\lambda '\in \partial _2(({\mathbb {R}}_+)^{Q_2})\), and then \(\lambda \in \partial _2(({\mathbb {R}}_+)^{Q_2})\). We have then \(\ker (\partial _1)\cap ({\mathbb {R}}_+)^{Q_1}\subset \partial _2(({\mathbb {R}}_+)^{Q_2})\). Consider \(\lambda \in \ker (\partial _1)\cap ({\mathbb {R}}^\star _+)^{Q_1}\): one has \(0<\epsilon \ll 1\) such that \(\lambda -\partial _2(\epsilon \sum _{w\in Q_2}w)\in \ker (\partial _1)\cap ({\mathbb {R}}_+)^{Q_1}\), and then \(\lambda =\partial _2\sum _{w\in Q_2}(\mu _w+\epsilon )w\) with \(\mu _w\in {\mathbb {R}}_+\), i.e. \(\lambda \in \partial _2(({\mathbb {R}}^\star _+)^{Q_2})\).

\(\square \)

A.2 Biconnected Components of a Quiver

A quiver is biconnected if there is no node i of the quiver such that removing i (and then also the arrows with source or target i) disconnects the quiver. On a quiver Q, the biconnected components are defined as the maximal subquivers of Q being biconnected. We prove the following fact about biconnected components:

Lemma A.2

(i) The biconnected components give a partition of the arrows of the quiver Q, and two different biconnected components of a quiver can share at most one node.

(ii) Define K as the unoriented graph with one node for each biconnected component and one node for each node of the quiver shared between different biconnected components, and an edge between the node i and the biconnected component b if \(i\in b\). Then, K is a connected tree, i.e. has no loops.

Proof

This result can be deduced from [24, Prop 3.5], but we prove it here from clarity and completeness. Suppose that there is a sequence of distinct biconnected components \(b_1,...,b_p\) and a sequence of distinct nodes \(i_1,...,i_p\) such that \(i_k\in b_k\cup b_{k+1}\), \(i_p\in b_p\cup b_1\), with \(p\ge 2\). Consider the subquiver \(b_1\cup ...\cup b_p\) of Q, and remove a node i: one can consider up to a circular permutation that \(i\ne i_1\),..., \(i\ne i_{p-1}\). Consider two nodes \(j,j'\) such that \(j\in b_k\), \(j'\in b_{k'}\), and suppose up to exchanging j and \(j'\) that \(k\le k'\). Because the \(b_{k''}\) are biconnected, there is an unoriented path between j and \(i_k\) in \(b_k\) avoiding i, unoriented paths between \(i_{k''}\) and \(i_{k''+1}\) in \(b_{k''}\) avoiding i for \(k\le k''\le k'\) and an unoriented path between \(i_{k'}\) and \(j'\) in \(b_{k'}\) avoiding i. By concatenation, these give an unoriented path between j and \(j'\) in \(b_1\cup ...\cup b_p\) avoiding i, i.e. \(b_1\cup ...\cup b_p\) is still connected when one has removed i. One obtains then that \(b_1\cup ...\cup b_p\) is biconnected, but it is strictly bigger that the \(b_k\), which were assumed to be maximal biconnected subquivers of Q, giving a contradiction.

In particular, for \(p=2\), one obtains that two different biconnected components can share at most one node: because an arrow is adjacent to two nodes, two different biconnected components cannot share an arrow, proving (i). The argument above for general p gives exactly that K has no cycle, i.e. is a tree. Consider two biconnected components \(b,b'\), and two nodes \(i\in b,i'\in b'\): because Q is connected, there is an unoriented path in Q between i and \(i'\): by denoting \(b,b_1,...,b_n,b'\) the biconnected components crossed by these paths, b and \(b'\) are then connected by a sequence of biconnected components sharing a node, i.e. the graph K is connected. This concludes the proof of (ii). \(\square \)

Denote by B the set of biconnected components of a quiver, and \(Q^b\) the quiver associated with a biconnected component. Two biconnected components have distinct arrows, and then, one has a decomposition \({\mathbb {R}}^{Q_1}=\bigoplus _{b\in B}{\mathbb {R}}^{Q^b_1}\). Consider a simple unoriented cycle of Q: it cannot pass through different biconnected components, since otherwise it would project to a cycle in K, which is forbidden by the above lemma, i.e. it is included in a single biconnected component. In particular this applies to simple oriented cycles, and then one has \({\mathbb {R}}^{Q_2}=\bigoplus _{b\in B}{\mathbb {R}}^{Q^b_2}\), giving a decomposition of the complex:

$$\begin{aligned} \bigoplus _{b\in B}{\mathbb {R}}^{Q^b_2}\overset{\sum _{b\in B}\partial ^b_2}{\rightarrow }\bigoplus _{b\in B}{\mathbb {R}}^{Q^b_1}\overset{\sum _{b\in B}\partial ^b_1}{\rightarrow }{\mathbb {R}}^{Q_0}\overset{\partial _0}{\rightarrow }{\mathbb {R}}\rightarrow 0 \end{aligned}$$
(A.1)

The cellular homology group \(H^1\) of the graph gives the loops of the quiver (see [23, Sec 2.2]); hence, \(\ker (\partial _1)\) is generated by the simple unoriented cycles of Q, each one lying in a single biconnected component. It gives then the decomposition \(\ker (\partial _1)=\bigoplus _{b\in B}\ker (\partial ^b_1)\):

Lemma A.3

A current on a quiver is conserved if and only if its restriction to each biconnected component is a conserved current.

Lemma A.4

(i) A quiver is strongly connected if and only if all its biconnected components are strongly connected.

(ii) Suppose that Q is strongly connected. Consider the equivalence class \(\sim \) on the arrows of Q generated by \(a\sim b\) if \(a,b\in w\) for \(w\in Q_2\). The equivalence classes of \(\sim \) correspond to the biconnected components of Q.

Proof

(i):

Suppose that Q is strongly connected. Consider \(i,i'\) in a biconnected component b. There is an oriented path \(v:i\rightarrow i'\) in Q, giving a cycle \(\bar{v}:b\rightarrow b\) on the graph K. If this cycle is trivial, then v stays in the biconnected component b. If it is not trivial, since K is a tree, the cycle \(\bar{v}\) must be of the form \(b\rightarrow j\rightarrow ...\rightarrow j\rightarrow b\), i.e. the path v can be decomposed as \(i\overset{v_1}{\rightarrow }j\overset{v_2}{\rightarrow }j\overset{v_3}{\rightarrow }i'\), with \(v_3\) and \(v_1\) oriented paths in b. One obtains then an oriented path \(v_3v_1:i\rightarrow i'\) in b, i.e. b is strongly connected.

Suppose that each biconnected component of Q is strongly connected. Consider \(i,i'\) two nodes of Q, with \(i\in b,i'\in b'\). The graph K is connected, consider a path \(b\ni i_1\in b_1...i_{n-1}\in b_{n-1}\ni i_n\in b' \). Because \(b,b_k,b'\) are strongly connected, consider oriented paths \(v:i\rightarrow i_1,v_k:i_k\rightarrow i_{k+1},v':i_n\rightarrow i'\), respectively, in \(b,b_k,b'\). This gives an oriented path \(v'v_{n-1}...v_1v:i\rightarrow i'\) in Q, i.e. Q is strongly connected.

(ii):

Because a simple oriented cycle \(w\in Q_2\) is contained in a single biconnected component of Q, an equivalence class of \(\sim \) is contained in a single biconnected component of Q. One has then to show that a biconnected quiver contains only one single class of \(\sim \). Two equivalence classes do not share any arrows by definition: one can construct an unoriented graph \({\tilde{K}}\) with a node corresponding to each equivalence class s of \(\sim \), and a node for each node \(i\in Q_0\) shared between different equivalence classes, and an edge between i and s when \(i\in s\).

Considering two nodes ij, there is a path between i and j in Q: the sequence of equivalence classes of \(\sim \) crossed by this path gives a path in \({\tilde{K}}\) between i and j, i.e. \({\tilde{K}}\) is connected. Consider a cycle \(i_1\in s_1\ni i_2...i_n\in s_n\ni i_1\) with no node or edge repeated in \({\tilde{K}}\). Because each equivalence class is strongly connected, one can choose an oriented path \(v_k:i_k\rightarrow i_{k+1}\) with no node repeated in each equivalence class \(s_k\), the concatenation giving an oriented cycle \(v_n...v_1\) in Q. Because each \(v_k\) contains no repeated node, there is a subcycle w of \(v_n...v_1\) which is simple and contains arrows of different equivalence classes of \(\sim \), giving a contradiction. The graph \({\tilde{K}}\) is then a connected tree. If there were more than one equivalence classes of \(\sim \), then there would be a node \(i\in {\tilde{K}}\) such that removing i disconnects \({\tilde{K}}\), and then disconnects the quiver Q, a contradiction because Q was assumed to be biconnected. There is therefore a single equivalence class of \(\sim \) in a biconnected quiver Q.

\(\square \)

A.3 Cuts and R-Charge

Lemma A.5

The following assertions are equivalent:

(i):

Q admits a cut.

(ii):

Q admits an R-charge.

(iii):

Each maximal weak cut of Q is a cut.

Proof

\((iii)\Rightarrow (i)\): \(\emptyset \) is a weak cut, and the poset of weak cuts is finite, then Q admits a maximal weak cut, which is a cut by assumption.

\((i)\Rightarrow (ii)\) Consider a cut I of Q. Recall that we have defined the homomorphism \(R_I:{\mathbb {R}}^{Q_1}\mapsto {\mathbb {R}}\) by \(R_I(a)=2\) if \(a\in I\) and \(R_I(a)=0\) otherwise. Each cycle \(w\in Q_2\) contains exactly one arrow \(a\in I\), and \(R_I \circ \partial _2(w)=R_I(a)=2\), i.e. \(R_I\) is a R-charge.

\((ii) \Rightarrow (iii)\). Consider a maximal weak cut I, and \(R:{\mathbb {R}}^{Q_1}\rightarrow {\mathbb {R}}\) such that \(R\circ \partial _2\ge R_I\circ \partial _2\), and \(R_I\circ \partial _2(w)=2\Rightarrow R\circ \partial _2(w)=2\). We must show that \(R\circ \partial _2= R_I\circ \partial _2\). Consider \((w=a_n...a_1)\in Q_2\) such that \(R_I\circ \partial _2(w)=0\) i.e. \(a_k\not \in I\) for each k (see Fig. 10): in particular one has \(R\circ \partial _2(w)\ge 0\). Because I is maximal, each arrow \(a_k\in w\) is contained in a simple cycle \(w_k=v_ka_k\in Q_2\) such that \(v_k\) contains exactly one arrow of I (since otherwise \(I\cup \{a_k\}\) would be a larger weak cut). The oriented cycle \(\bar{w}=v_1...v_n\) is a product of simple oriented cycles, and satisfies:

$$\begin{aligned}&R\circ \partial _2(\bar{w})\ge R_I\circ \partial _2(\bar{w})=2n\nonumber \\&\partial _2(w)=\sum _k\partial _2(w_k)-\partial _2(\bar{w})\nonumber \\ \Rightarrow&R\circ \partial _2(w)=2n-R\circ \partial _2(\bar{w})\le 0\nonumber \\ \Rightarrow&R\circ \partial _2(w)=0=R_I\circ \partial _2(w) \end{aligned}$$
(A.2)

where the third line holds because \(R_I\circ \partial _2(w_k)=2\). By disjunction of cases, one has then \(R\circ \partial _2=R_I\circ \partial _2\). When R is an R-charge, one obtains then that \(R_I\) is a R-charge, i.e. I is a cut. \(\square \)

Fig. 10
figure 10

If Q admits an R-charge, then each maximal weak cut is a cut

Lemma A.6

For a quiver with an R-charge and a cycle \(w_0\) passing through all the nodes, the cuts are given by \(I=\{a:i\rightarrow j|i>j\}\) for each labeling of the nodes such that \(w_0:1\rightarrow 2\rightarrow ...\rightarrow n\rightarrow 1\).

Consider a cut I of Q: it contains exactly one arrow \(a\in w_0\): we choose the labeling of \(Q_0\) such that \(a:n\rightarrow 1\) (see Fig. 11). Consider an arrow \((b:i\rightarrow j)\in Q_1\), and denote by \(v_{ji}\) the minimal path going from j to i on the cycle \(w_0\): \(v_{ji}\) contains an arrow of I if and only if \(i<j\), and the simple oriented cycle \(v_{ji}b\in Q_2\) contains exactly one arrow of the cut I, i.e. \(b\in I\) if and only if \(j<i\). One concludes that each cut of Q is of the form \(I=\{b:i\in j|j<i\). for a specific cyclic ordering of \(w_0\).

Fig. 11
figure 11

Under assumptions of Lemma A.6, all cuts are of the form \(I=\{a:i\rightarrow j| i>j\}\)

Conversely, consider the set \(I=\{b:i\rightarrow j|j<i\)}, and a cycle \(w: i_1\overset{a_1}{\rightarrow } i_2...i_r\overset{a_r}{\rightarrow } i_1\in Q_2\), such that \(R\circ \partial _2(w)=2\) (see Fig. 12). We will show that w contains exactly one arrow of I. The cycle \(v_{i_1 i_r}...v_{i_2 i_1}\) is equal to the m-th iteration \(w_0^m\), for some \(m\in {\mathbb {N}}\). Since \(2=R\circ \partial _2(w)=\sum _{k=1}^rR\circ \partial _2( v_{i_{k+1}i_{k}}a_k)-mR\circ \partial _2(w_0)=2r-2m\), the number of iterations is \(m=r-1\). But \(w_0\in Q_2\) contains exactly one arrow of I, and each cycle \(v_{i_{k+1}i_{k}}a_k\in Q_2\) contains exactly one arrow of I. Indeed, there are two options: a) either \(i_k>i_{k+1}\), and then \(a_k\in I\) but \(v_{i_{k+1}i_{k}}\) does not contain any arrow of I, or b) \(i_k<i_{k+1}\), and then \(a_k\not \in I\) but \(v_{i_{k+1}i_{k}}\) contains the arrow \(n\rightarrow 1\in I\). Evaluating the R-charge, we get \(R_I\circ \partial _2(w)=\sum _{k=1}^rR_I\circ \partial _2( v_{i_{k+1}i_{k}}a_k)-(r-1)R_I\circ \partial _2(w_0)=2\). Therefore, \(w\in Q_2\) contains exactly one arrow of I, i.e. I is a cut. \(\square \)

Fig. 12
figure 12

Under assumptions of Lemma A.6, the set \(I=\{a:i\rightarrow j| i>j\}\) is a cut

When Q admits no R-charge (see Figs. 5, 4 for example), the set \(I=\{b:i\in j|j<i\}\) is a minimal strong cut of Q which is not a cut: each simple oriented cycle contains at least one arrow of I, but there must be a cycle \(w\in Q_2\) containing more than one arrow of Q. In particular, if Q has another cycle \(w_1\) passing through all the nodes in a different order than \(w_0\), then it has no R-charge. Indeed, for a labeling of the nodes such that \(w_0:1\rightarrow 2\rightarrow ...\rightarrow n\rightarrow 1\), the cycle \(w_1\) is of the form \(1=\sigma (1)\rightarrow \sigma (2)\rightarrow ...\rightarrow \sigma (n)\rightarrow \sigma (1)\), with \(\sigma \) a permutation of \(\{1,...,n\}\) different from the identity. There must then be at least two arrows of \(w_1\) such that \(\sigma (i)>\sigma (i+1)\), and then \(w_1\) contains at least two arrows of \(I=\{a:i\rightarrow j|i>j\}\), i.e. I is not a cut, and then Q has no R-charge. More generally, the criterion of the above lemma gives a simple algorithm to check if a quiver with a cycle passing through all the nodes has an R-charge.

A.4 Current Decomposition for Abelianized Quivers

Lemma A.7

For Q a biconnected quiver and d a dimension vector with \(d_i\ge 1\) for \(i\in Q_0\), a strictly positive conserved current on \(Q^d\) can be expressed as a sum of (not necessarily positive) currents circulating on the cycles in \(p^{-1}(Q_2)\).

Proof

Consider a strictly positive conserved current \(\lambda \) on \(Q^d\): in particular \(Q^d\), and therefore Q, is strongly connected from (i) of Lemma A.1, and \(\lambda \) can be expressed as a sum of positive currents circulating on the cycles in \(Q^d_2\). One must then show that for Q a biconnected strongly connected quiver each cycle in \(Q^d_2\) can be expressed as a linear combination of cycles in \(p^{-1}(Q_2)\), i.e. \(\partial _2({\mathbb {R}}^{Q^d_2})\subset \partial _2({\mathbb {R}}^{p^{-1}(Q_2)})\). One can construct the Abelianized quiver by using a finite sequence of elementary steps, where one node is split into two nodes at each step. Each of these steps preserves the fact that the quiver is biconnected and strongly connected; thus, it suffices to prove the statement for an elementary step.

Consider a biconnected strongly connected quiver Q, a node \(i\in Q_0\) and the dimension vector d such that \(d_i=2\) and \(d_j=1\) for \(j\ne i\). Consider a cycle \(w\in Q^d_2-p^{-1}(Q_2)\): it necessarily passes through the two nodes (i, 1), (i, 2), i.e. it is of the formFootnote 6 (d, 1)v(c, 2)(b, 2)u(a, 1), where ac are arrows of Q with source i, bd arrows of Q with target i, and u and v paths of Q avoiding i, such that \(dvc,bua\in Q_2\) (see Fig. 13). We claim that there is a sequence of arrows \(b_0=b,b_1,...,b_n\) with target i, a sequence of arrows \(c_0,...,c_n=c\) with source i, and sequences of paths \(u_1,...,u_n\) and \(v_1,...,v_{n-1}\) in Q, such that \(b_ku_kc_k\) and \(b_{k+1}v_kc_k\) are simple oriented cycles in \(Q_2\) for \(1\le k\le n-1\). Then,

$$\begin{aligned}&\partial _2\left( (d,1)v(c,2)(b,2)u(a,1)\right) =\partial _2\left( (d,1)v(c,1)+(b,1)u(a,1) \right. \nonumber \\&\quad +\sum _{k=0}^{n-1} \left( (b_k,2)u_k(c_k,2)-(b_k,1)u_k(c_k,1)+(b_{k+1},1) v_k(c_k,1) -(b_{k+1},2)v_k(c_k,2) \right) \end{aligned}$$
(A.3)

where each of the cycles appearing on the right belongs to \(p^{-1}(Q_2)\), i.e., project to a simple cycle of Q. Thus, \(\partial _2((d,1)v(c,2)(b,2)u(a,1))\in \partial _2({\mathbb {R}}^{p^{-1}(Q_2)})\), and then \(\partial _2({\mathbb {R}}^{Q^d_2})\subset \partial _2({\mathbb {R}}^{p^{-1}(Q_2)})\) which concludes the proof.

Fig. 13
figure 13

Proof of the fact that a simple cycle of \(Q^d\) can be expressed as a signed sum of cycles in \(p^{-1}(Q_2)\)

Proof of the claim: Consider a strongly connected biconnected quiver Q, and a node \(i\in Q_0\). Consider the equivalence \(\sim \) on the arrows of Q with source or target i, generated by \(a\sim b\), with a (resp. b) with source (resp. target) i if there is a path v in Q such that bva is a simple oriented cycle in \(Q_2\). We need to show that there is a single equivalence class under \(\sim \). Let S be the set of equivalence classes. To each \(s\in S\) of \(\sim \) one can associate the subquiver \(Q^s\subset Q-\{i\}\) whose nodes are the nodes j such that there is a path \(v:j\rightarrow i\) passing only once through i and ending by an arrow in the given equivalence class.

Since Q is strongly connected, for each \(j\in Q_0-\{i\}\) there is a path from j to i, which up to truncating the path, can be considered as passing only once through i, i.e. \(j\in Q_0^s\) for at least one \(s\in S\). Consider \(j\in Q_0^s\), \(j'\in Q_0^{s'}\) and \(u:j\rightarrow j'\) an oriented path of Q avoiding i. Consider paths \(v:j\rightarrow i\), \(v':j'\rightarrow i\) passing only once through i and ending respectively by \(b\in s\), \(b'\in s'\). Because Q is strongly connected, there is a path \(u':i\rightarrow j\) beginning by an arrow a with source i: the two cycles \(vu'\) (resp. \(v'uu'\)) contain a simple cycle containing ba (resp. \(b'a\)), and then \(b\sim a\sim b'\), i.e. \(s=s'\). For \(j=j'\) and u the trivial path at j, one obtains that the \(Q_0^s\) form a partition of \(Q_0^s-\{i\}\), and for \(u=a:j\rightarrow j'\) an arrow of Q one obtains that \(Q^s\) and \(Q^{s'}\) are disconnected in \(Q-\{i\}\). If there were different equivalence classes under \(\sim \), \(Q-\{i\}\) would be disconnected, contradicting the assumption that Q is biconnected. Thus, there is a single equivalence class under \(\sim \), which proves the claim. \(\square \)

A.5 Higgs Branch

Lemma A.8

Let Q be a quiver, \(C\subset Q_2\) a set of cycles of Q such that there exists a subset \(I\subset Q_1\) of arrows such that each cycle of C contains exactly one arrow of I. For any dimension vector \(d\in {\mathbb {N}}^Q_0\), there exists a dense open subset \(U_{C,d}\subset {\mathbb {C}}^C\) such that for any \((\nu _w)_{w\in C}\in U_{c,d}\) and any d-dimensional representation \(\phi \) of the quiver Q with potential \(W=\sum _{w\in C}\nu _w w\), the trace \(\mathrm{Tr}(w)\) vanishes on the representation \(\phi \) for all cycles \(w\in C\) appearing in the potential.

Proof

We follow an argument of Kontsevich quoted in the proof of [29, Prop. 3.1]. For \(d\in {\mathbb {N}}^{Q_0}\), denote by \(M_d\) the smooth quasi-projective connected space of d-dimensional representations of Q. Now, consider a potential cut I of W, such that \(\mathrm{Tr}(W)=\sum _{a\in I}\mathrm{Tr}(a\partial _a W)\): by homogeneity, the critical points of \(\mathrm{Tr}(W)\) on \(M_d\) necessarily have \(\mathrm{Tr}(W)=0\). Let \(f: M_d \rightarrow {\mathbb {C}}^C\) be the map which associates to any d-dimensional representation of Q the vector \((\mathrm{Tr}(w)_{w\in C}\). Away from the locus \(f^{-1}(0)\), the image of this map descends to the projective space \({\mathbb {P}}^{|C|-1}\). The trace of the potential \(W=\sum _{w\in C}\nu _w w\) is obtained by composing f with a linear form, corresponding to a hyperplane section of \({\mathbb {P}}^{|C|-1}\). By applying Bertini’s theorem to the complete linear system \(\mathrm{Tr}(W)^{-1}(0)\), one finds that, for \((\nu _w)_{w\in C}\) in a dense open subset \(U_{C,d}\subset {\mathbb {C}}^C\), \(\mathrm{Tr}(W)^{-1}(0)\) is a smooth connected strict sub-variety of \(M_d\) away from the zero locus \(f^{-1}(0)\). In particular, its tangent space at any point \(x\in \mathrm{Tr}(W)^{-1}(0)-f^{-1}(0))\) is strictly included in the tangent space of \(M_d\),

$$\begin{aligned} T_x(\mathrm{Tr}(W)^{-1}(0))\subsetneq T_x(M_d) \end{aligned}$$
(A.4)

hence \(\delta (\mathrm{Tr}(W))|_x\ne 0\). It follows that the critical points of \((\mathrm{Tr}(W)\) lie in \(f^{-1}(0)\), hence \(\mathrm{Tr}(w)=0\) for \(w\in C\) on a d-dimensional representation of (QW). Specializing to the dimension vector \(d=(1,...,1)\), one obtains that for \((\nu _w)_{w\in C}\in U_C\) a dense open subset of \({\mathbb {C}}^C\), and for any Abelian representation of (QW), the cycles w vanish for all \(w\in C\). \(\square \)

Now, for any weak cut \(I\subset Q_1\), we denote by \(Q_2^I\) the set of cycles containing exactly one arrow of I, and for \(W=\sum _{w\in Q_2}\nu _w w\), we define \(W_I:=\sum _{w\in Q_2^I}\nu _w w\). Let \(p_I:{\mathbb {C}}^{Q_2}\rightarrow {\mathbb {C}}^{Q_2^I}\) be the natural projection.

Definition A.9

A potential \(W=\sum _{w\in Q_2}\nu _w w\) is said to be generic if \((\nu _w)_{w\in Q_2}\) is in the dense open subset \(\bigcap _{I}p_I^{-1}(U_{Q_2^I})\).

Proposition A.10

Consider a quiver with generic potential (QW). If there exists a self-stable Abelian representation of (QW), then for each weak cut I:

$$\begin{aligned} |I|\le |Q_1-I|-|Q_0|+1 \end{aligned}$$
(A.5)

Proof

Let \(\phi \) be a \(\zeta \)-stable representation of (QW), with \(\zeta ,W\) generic, and I a weak cut of Q. Denote by J the set of arrows of Q vanishing in \(\phi \). The representation \(\phi \) gives a \(\zeta \)-stable representation of \(Q_J\), the quiver where the arrows of J have been removed: in particular, the stability of \(\phi \) implies that \(Q_J\) is connected (otherwise, \(\phi \) would be a direct sum of representations supported on the connected components). Consider the set \(K\subset I-J\) of arrows \((a:i\rightarrow j)\in I-J\) such that i and j are connected in \(Q_{J\cup I}\). The quiver \(Q_{J\cup K}\) obtained by removing all arrows in K is then still connected. Sending the arrows of K to 0 in \(\phi \), one obtains a representations \(\psi \) of \(Q_{J\cup K}\) without vanishing arrows. Because \(Q_{J\cup K}\) is connected, the gauge group \(({\mathbb {C}}^*)^{Q_0}\) scaling the nodes acts on the space \(({\mathbb {C}}^*)^{Q_1-J-K}\) of representations of \(Q_{J\cup K}\) without vanishing arrows with stabilizer \({\mathbb {C}}^\star \), giving a smooth moduli space of dimension \(|Q_1-J-K|-|Q_0|+1\).

Consider the potential \(W_I\) obtained from the generic potential W by keeping only the cycles \(w\in Q_2\) which contain an arrow of I. Consider \(L\subset J\cup K\) the set of arrows contained in a cycle \(w\in Q_2\) such that \(R_{J\cup K}\circ \partial _2(w)= R_I\circ \partial _2(w)=2\). Consider \(a\in L\):

  • If \(a\in I\), because I is a weak cut, any cycle \(w\in Q_2\) which contains a is a cycle of \(W_I\), and contains no other arrow of I, and in particular no other arrows of K, i.e. :

    $$\begin{aligned} \partial _a W_I|_{\psi }=\partial _a W|_{\psi }=\partial _a W|_\phi =0 \end{aligned}$$
    (A.6)
  • Suppose \(a\in J-I\). There is a cycle \(w\in Q_2\) containing a such that \(R_{J\cup K}\circ \partial _2(w)= R_I\circ \partial _2(w)=2\). Hence, w contains an arrow \(b\in I\) different from a, which cannot be in K, hence \(b\in I-K\). Denote by \(i\sim j\) the equivalence relation identifying vertices i and j which are connected in \(Q_{J\cup I}\): one has \(t(b)\sim s(a)\) and \(t(a)\sim s(b)\) because w contains no other arrows of \(J\cup I\). Consider an other cycle \(w'\) containing a.

    • If \(w'\) contains an other arrow of J, then \(0=\partial _a w'|_\psi =\partial _a w'|_\phi \).

    • If \(w'\) contains no other arrow of J and an arrow \(c\in K\subset I-J\), by definition, \(s(c)\sim t(c)\): using \(t(a)\sim s(c)\) and \(t(c)\sim s(a)\) because \(w'\) contains no other arrows of \(J\cup I\), one obtains \(t(b)\sim s(b)\), a contradiction because \(b\in I-K\).

    • If \(w'\) contains no other arrow of J and no arrow of I, then \(t(a)\sim s(a)\), and then \(t(b)\sim s(b)\), a contradiction because \(b\in I-K\).

    • In the remaining case \(w'\) contains one arrow of J and one arrow of \(I-K\), then \(w'\) is a cycle of \(W_I\) and \(\partial _a w'|_\psi =\partial _a w'|_\phi \)

    By disjunction of cases one has \(\partial _a W|_\psi =\partial _a W|_\phi =0\), and because the only cycles contributing to \(\partial _a W|_\psi \) are cycles of \(W_I\), one has \(\partial _a W_I|_\psi =\partial _a W|_\psi =0\).

By disjunction on cases, \(\partial _a W_I|_\psi =0\) for \(a\in L\), hence \(\psi \) is a representation of the quiver with relations \((Q_{J\cup K},\partial _L W_I)\) without vanishing arrows. The tangent space of the moduli space of representations of \((Q_{J\cup K},\partial _{L}W_I)\) at \(\psi \) is given by the intersection of the kernel of the |L| differential forms \(\delta (\partial _a W_I)\) for \(a\in L\).

As in (3.8), a linear relation \(\sum _{a\in L}{\tilde{\psi }}_a \delta (\partial _aW_I)=0\) between these differential forms yields a representation \({\bar{\psi }}=(\psi _a,\epsilon {\tilde{\psi }}_b)_{a\in Q_1-L,b\in L}\) over \({\mathbb {C}}\epsilon /\epsilon ^2\) such that \(\delta _b W|_{{\bar{\psi }}}=0\) for \(b\in Q_1-J\cup K\). For an arrow \(b\in J\cup K\), because \(\partial _b|_\psi =0\), \(\partial _b|_{{\bar{\psi }}}\) is at most of order \(\epsilon \), and \({\bar{\psi }}_b\) is at most of order \(\epsilon \), hence \({\bar{\psi }}_b\partial _b|_{{\bar{\psi }}}=0\). Then, \({\bar{\psi }}_b\partial _b|_{{\bar{\psi }}}=0\) for \(b\in Q_1\). Since the each cycle of the potential \(W_I\) contains exactly one arrow of I, Lemma A.8 implies that the cycles of \(W_I\) vanish in \({\bar{\psi }}\). By definition, each arrow \(a\in L\subset J\cup K\) is contained in a cycle w of \(W_I\) in which it is the only arrow of \(J\cup K\). In particular all the other arrows are in \(Q_1-J-K\) and are non-vanishing: then \(\epsilon {\tilde{\psi }}_a\prod _{a\ne b\in w}\psi _b=0\), hence \({\tilde{\psi }}_a=0\). The |L| differential forms \(\delta (\partial _a W_I)\) for \(a\in L\) are then independent, hence:

$$\begin{aligned} |Q_1-J-K|-|Q_0|+1-|L|\ge 0 \end{aligned}$$
(A.7)

Defining

$$\begin{aligned} R(a)=\left\{ \begin{array}{ll} 0 &{} \text{ if } a\in Q_1-J \\ 1 &{} \text{ if } a\in J-L\\ 2 &{} \text{ if } a\in L\cap J \end{array}\right. \quad \quad R'(a)=\left\{ \begin{array}{ll} 0 &{} \text{ if } a\in Q_1-K \\ 1 &{} \text{ if } a\in K-L\\ 2 &{} \text{ if } a\in K\cap L \end{array} \right. \end{aligned}$$
(A.8)

one finds

$$\begin{aligned} \sum _{a\in Q_1}(R+R')(a)\le |Q_1|-|Q_0|+1 \end{aligned}$$
(A.9)

We will now show that the following inequality holds for all \(w\in Q_2\):

$$\begin{aligned} R_I\circ \partial _2(w)\le (R+R')\circ \partial _2(w) \end{aligned}$$
(A.10)

By disjunction of cases,

  • If w contains no arrow of I, one has directly \(R_I\circ \partial _2(w)=0\le (R+R')\circ \partial _2(w)\).

  • Suppose that w contains an arrow \(a\in I\). If w contains no arrow of J, then t(a) and s(a) are connected in \(Q_{J\cup I}\), hence \(a\in K\). Then, in any cases, w contains an arrow of \(J\cup K\). If it contains a single arrow of \(J\cup K\), then by definition it is an arrow of L, contributing to \(R+R'\) by 2. If it contains at least two arrows of \(J\cup K\), then each of them contributes to \(R+R'\) by 1. By disjunction of cases \(R_I\circ \partial _2(w)=2\le (R+R')\circ \partial _2(w)\).

The inequality (A.10) follows and can be rewritten:

$$\begin{aligned} (R_I-R')\circ \partial _2(w)\le R\circ \partial _2(w)\;\quad \forall \;w\in Q_2 \end{aligned}$$
(A.11)

We now assume that \(\zeta \) is a self-stability condition. Because the positive conserved current \(\lambda =(1+\delta _a+|\psi _a|^2)_{a\in Q_1}\) (with \(\delta _a\ll 1\)) corresponding with the self-stable representation \(\psi \) is a sum of positive currents supported of cycles in \(Q_2\) from Lemma A.1, it follows that:

$$\begin{aligned} (R_I-R')(\lambda )\le R(\lambda ) \end{aligned}$$
(A.12)

We deduce then:

$$\begin{aligned}&2|I|+\sum _{a\in Q_1}(R_I-R')(a)\delta _a \le \sum _{a\in Q_1}(R_I-R')(a)(1+\delta _a+|\phi _a|^2) +\sum _{a\in Q_1}R'(a)\nonumber \\ {}&\quad =(R_I-R')(\lambda )+\sum _{a\in Q_1}R'(a)\nonumber \\ {}&\quad \le R(\lambda )+\sum _{a\in Q_1}R'(a)=\sum _{a\in Q_1}R(a)(1+\delta _a+|\phi _a|^2)+\sum _{a\in Q_1}R'(a)\nonumber \\ {}&\quad =\sum _{a\in Q_1}(R+R')(a)+\sum _{a\in Q_1}\delta _aR(a)\nonumber \\ {}&\quad \le |Q_1|-|Q_0|+1+\sum _{a\in Q_1}\delta _aR(a)\nonumber \\&\quad \implies 2|I|\le |Q_1|-|Q_0|+1 \end{aligned}$$
(A.13)

Here we have used the facts that \((R_I-R')(a)\ge 0\) in the first line, \(\phi _a=0\) when \(R(a)\ne 0\) in the fourth line, (A.9) in the fifth line and \(\delta \ll 1\) in the last line. \(\square \)

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Descombes, P., Pioline, B. On the Existence of Scaling Multi-Centered Black Holes. Ann. Henri Poincaré (2022). https://doi.org/10.1007/s00023-022-01185-x

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