Decay and Fission of Magnetic Quivers

In exploring supersymmetric theories with 8 supercharges, the Higgs branches present an intriguing window into strong coupling dynamics. Magnetic quivers serve as crucial tools for understanding these branches. Here, we introduce the decay and fission algorithm for unitary magnetic quivers. It efficiently derives complete phase diagrams (Hasse diagrams) through convex linear algebra. It allows magnetic quivers to undergo decay or fission, reflecting Higgs branch RG-flows in the theory. Importantly, the algorithm generates magnetic quivers for the RG fixed points and simplifies the understanding of transverse slice geometry with no need for a list of minimal transitions. In contrast, the algorithm hints to the existence of a new minimal transition, whose geometry and physics needs to be explored.


I. INTRODUCTION
The study of phases of quantum field theory (QFT), and transitions between these phases, is fundamental to our understanding of nature.One example is the Higgs mechanism, where a scalar field acquires a vacuum expectation value (VEV) that subsequently breaks the gauge symmetry [1][2][3][4].Beyond its central role in the electroweak sector of the Standard Model, this mechanism can be observed in any gauge theory with charged scalar fields.It can further be generalized to intrinsically strongly coupled theories, which lack a Lagrangian description, but in this case a systematic understanding is still lacking, in part due to the extraordinary diversity of the landscape of QFTs.
This can be remedied by adding simplifying assumptions.A particularly relevant playground in that respect is supersymmetry.Supersymmetric theories (SQFTs) encompass a very rich set of phenomena, from Standard Model-like theories to strongly coupled systems, while maintaining computational control.SQFTs with 8 supercharges in space-time dimension 3, 4, 5 and 6 generically possess a continuous space of vacua known as the Higgs branch, denoted H, parameterized by scalar fields when the field approach is available.The Higgs mechanism is then geometrized, and corresponds to hitting singularities in H.The phase diagram can then be identified with the structure of nested singularities [5].It also encodes how SQFTs are related through deformations, tuning of gauge couplings, and RG flows.
The description of H beyond the perturbative regime, which includes most conformal SQFTs, is challenging.Fortunately, a powerful technique has recently been introduced: one defines H by means of an auxiliary combinatorial object called a magnetic quiver (MQ) .In the simplest cases, a MQ defines an auxiliary 3d N = 4 superconformal field theory (SCFT) whose Coulomb branch coincides by definition with H.It is known that a MQ encodes the Higgs phase diagram, and a precise algorithm, dubbed quiver subtraction [5,17,29,35,36], has been developed over the last five years.However, this algorithm has shortcomings: • It requires as an external input the knowledge of a list of magnetic quivers for isolated symplectic singularities, which is still incomplete.• It requires the introduction of so-called decorated quivers, whose definition is still unclear; • It is computationally complex, it being an algorithm acting on the graphs underlying the quivers.
• It provides rather limited information on the SQFTs at the end of the Higgs branch RG-flows.This note introduces a powerful operation on quivers which addresses all the above problems: it provides the phase diagram of nested singularities of H without relying on any input, all the intermediate steps are well-defined, and the computation involves vectors, and not graphs.Moreover, the algorithm generates the magnetic quivers of SQFTs at the end of Higgs branch RG-flows; therefore, allowing to identify those SQFTs.In a companion paper [37], we apply it to SQFTs in various dimensions and show perfect agreement with the literature, along with several new results.We also give there the details of the geometry of transverse slices, which are omitted here.

II. DECAY AND FISSION
The algorithm can be described using analogies with nuclear reactions.It applies to good quivers, meaning that at each node with rank k, the sum of the ranks of adjacent nodes is ≥ 2k.A more general and rigorous definition is given in Section III.Given a good magnetic quiver Q, it can decay, i.e. become lighter, or fission into exactly two smaller quivers.These processes are then repeated in all possible ways until one reaches stable quivers, which can neither decay nor fission into non-trivial quivers.The reaction diagram then coincides with the Higgs phase diagram of the theory of which Q was a magnetic quiver.Up to a few restrictions spelled out in Section III, the general rule is to perform every possible decay and fission that keeps all involved quivers good.During this transition, a given quiver (specified by shape and ranks) "decays" into a quiver of the same shape, but with reduced ranks.For example, consider which reflects the Higgs branch RG-flow of the 5d SCFT with low-energy effective description SU(5) 1 gauge theory with N AS = 2, N f = 4 to the SCFT fixed point of SU(4) 1 with with N AS = 2, N f = 4.During decay, the change in ranks (or equivalently, the dimension of the transition) can be any integer between one and the entire sum of the ranks.The latter is a terminal decay; i.e. the entire quiver decays to nothing.
A given quiver can also "fission" into two quivers of the same shape, but with smaller ranks.For instance, the same quiver as above fissions as follows: which translates into the Higgs branch RG-flow of the SU( 5) theory into a product of fixed points defined by SU(3) 1 with N f = 6 and SU(2) with N f = 4.In contrast to decay, the change in (sum of all) ranks during fission is always one; thus, these are 1-dimensional transitions.
The decay transition finds its origin in 3d mirror symmetry [38][39][40][41], in which the correspondence with Higgsing is manifest.The fission transition, on the other hand, can be understood as a generalization of adjoint Higgsing in theories with higher supersymmetry, like N = 4 SYM in 4d with gauge group U(k), which can be Higgsed to U(k ′ ) × U(k − k ′ ).In that case, the phases are labelled by partitions [42], which fission obviously implement: Our main result is that these two fundamental processes, when appropriately combined, cover all 8 supercharges theories in any dimension.Further confirmation comes from agreement with both physical and mathematical literature on dozens of examples, as well as direct insights from string theory, see Section IV.

III. PRECISE ALGORITHM
In this section, we give the precise rules that govern the decay and fission processes introduced above.The underlying graph of a quiver with n unitary gauge nodes can be encoded into an adjacency matrix A ∈ Mat(n × n, Z), with A i,j ≥ 0 the number of links from node i to node j.By convention A i,i = 2g − 2 where g ≥ 0 is the number of loops on node i.If A i,j < A j,i , we assume further that Aj,i Ai,j = ℓ ∈ Z >0 ; the edge between nodes i and j is then ℓ-non-simply laced.The quiver itself is specified by its rank vector K ∈ N n .Given two rank vectors K 1,2 ∈ N n , we say The quiver is assumed to be good, i.e. it satisfies AK ≥ 0.

A. Partial Order
Given a quiver (A, K), the finite set of possible decay and fission products is defined by where property P 1 detects the subquivers of the form U(1) with one or more adjoints.Property P 2 detects the subquivers that correspond to instanton moduli spaces.
To implement fission, we then assemble the products in all possibles ways: with and Elements of L m are the multisets of m vectors of V 0 whose sum is ≤ K (repetitions are allowed).Finally, we write This is the set of vertices of our Hasse diagram, which correspond to the symplectic leaves of the 3d N = 4 Coulomb branch of the initial quiver (A, K).For an element ℓ ∈ L, there is a unique m ∈ N such that ℓ ∈ L m , we call it the length of ℓ, denoted length(ℓ).We also denote by Σℓ ∈ N n the sum of the elements of ℓ.
Next, we define a partial order on L. Let ℓ 1 , ℓ 2 ∈ L. We write The relation ⇝ is reflexive and antisymmetric, but not transitive in general.Let us denote by ≽ its transitive closure, i.e.
This is a partial order relation.We claim that (L, ≽) coincides with the poset of symplectic leaves in the 3d N = 4 Coulomb branch of the quiver.

B. Elementary transitions
The previous paragraph has shown how to associate to any quiver (A, K) a poset of (multisets of) quivers.The partial order can be depicted using a Hasse diagram, in which elements of the poset are represented as points, and lines are drawn between adjacent elements: there is a line between ℓ 1 ≽ ℓ 2 if there is no leaf ℓ 3 ̸ = ℓ 1 , ℓ 2 satisfying ℓ 1 ≽ ℓ 3 ≽ ℓ 2 .This has a physical and a geometric incarnation.Physically, this plays the role of a Higgs branch phase diagram, in which lines represent elementary phase transitions.Geometrically, we are in presence of a conical symplectic singularity, and lines correspond to minimal degenerations [43,44].
It is of both physical and mathematical relevance to classify these elementary transitions.Interestingly, our algorithm, originally designed to compute the phase diagram for a particular theory, can be twisted to achieve this goal.Namely, search systematically through the set of all magnetic quivers, and identify those which have a phase diagram that consists of exactly two phases; the quiver then serves as a signature for that phase transition.A proof of principle is provided in Section V.

IV. A COMPLETE EXAMPLE
To illustrate the decay and fission algorithm, we consider the magnetic quiver for the SCFT-fixed point of 5d N = 1 SU(5) 1 gauge theory with N AS = 2 antisymmetric and N f = 4 fundamental hypermultiplets, and CS-level 1 [18].Despite the low-energy effective description, the emergence of massless (gauge) instantons at the SCFT-fixed point means that the phase diagrams at the fixed point and away from it differ drastically.This is because the Higgs branch moduli at the fixed point include these instanton operators as well.The lack of a Lagrangian description at the fixed-point implies that semi-classical approaches are insufficient.In contrast, the decay and fission algorithm outputs the phase diagram for the full "quantum" Higgs branch, as shown in Figure 1.The starting point (8) can undergo two distinct processes: the decay (1) and the fission (2).The resulting MQs readily allow the identification of the SCFTs which reside at the end of the Higgs branch RG-flows, due to the comprehensive catalogues of MQs for 5d theories [6,10,14,15,18].As a consequence, the decay and fission algorithm allows to construct the Higgs branch RG-flow diagram, as indicated in blue in Figure 1.
Physically, the 5d N = 1 theory can be realized in Type IIB superstring theory by a 5-brane web [45,46] with two O7 − orientifolds.Crucially, the 5-brane web of the SU(5) theory with N AS = 2 requires two fractional NS5 branes on the orientifolds [47]; the N f = 4 fundamental hypermultiplets are constructed by adding D7 branes.Here, we use the classical brane system to gain insights; the quantum-mechanical brane web is obtained after resolving O7 − planes.The latter gives rise to the MQs at the conformal fixed point.
The 5-brane web allows for two distinct Higgs branch transitions: firstly, a "decay" of the SU(5) theory to the SU(4) theory.This is realized by moving a single D5 brane (a gauge 5-brane) off to infinity (along the flavor 7-branes), see Figure 2 (Top).Secondly, the fission is enabled by the presence of two O7 − orientifolds (and the fractional NS5s).To see this, suppose that the gauge 5branes all terminate on both fractional NS5 branes.One possible brane motion along the Higgs branch is to move one of the factional NS5 along the O7 − .However, this is only allowed if the total number of gauge branes is even, as required by the orientifold projection.Thus, the fractional NS5s cannot be moved for an odd number of gauge 5-branes.Instead, for any number N of gauge 5branes, one can "fission" the stack into 2ℓ and (N −2ℓ), for ℓ ∈ {1, 2, . . ., ⌊ N 2 ⌋}.The stack of 2ℓ branes can be moved away from the fractional NS5s, along the O7 − ; while the stack of (N −2ℓ) branes remains suspended between the fractional NS5s, see Figure 2 (Center).The latter subsystem then realizes an SU(N − 2ℓ) world-volume SU(4) brane web Sp(1) brane web SU(3) brane web gauge theory with N AS = 2 and some fundamental hypermultiplets.The former 5-brane subweb is 2ℓ 5-branes intersecting two O7 − planes (without any fractional NS5s) and some N f flavor 7-branes.This is the T-dual of 2l D4 branes inside a stack of N f coincident D8s in the presence of a single O8 − -the brane system realising the moduli space of ℓ SO(2N f )-instantons [48,49].Therefore, the stack of 2ℓ 5-branes between the two O7 − planes together with the flavor 7-branes yields an Sp(ℓ) world-volume theory with N AS = 1 and some fundamental matter.In the case of Figure 1 with N = 5, ℓ = 1, the Higgs branch RG-flow takes the SU(5) theory to a product of an SU(3) and an SU(2) ∼ = Sp(1) theory.
Similarly, the Higgs branch flows of the SU(4) theory are understood via the 5-brane web.The decay to an SU(3) theory is realized by moving a single gauge 5-brane off to infinity (along the flavor 7-branes).The decay to an Sp(2) theory, on the other hand, is realized by moving one of the fractional NS5 branes along one of the O7 − branes off to infinity.This follows, because the difference between an SU(4) brane web with N AS = 2 and an Sp(2) brane web with N AS = 1 is whether there are two or one fractional NS5 branes [47], respectively, see Figure 2 (Bottom).
As we have seen, the considered 5d theory is intimately related to instantons and we comment on the arising MQs.Specifically, the moduli space of two SO(10)instanton on C 2 appears, indicated by the affine D 5 Dynkin quiver wherein all ranks equal twice the dual Coxeter labels.While this quiver is "bad" in the sense of [50], we argue that it is a meaningful combinatorial object that allows us to find all the symplectic leaves via the decay and fission algorithm.To see this, view instantons as realized as k Dp branes within a stack of coincident D(p+4) branes [48,49], in the presence of a suitable O(p+4) orientifold.Then an affine Dynkin quiver, wherein all ranks equal the k-fold of the dual Coxeter labels, symbolizes k coincident Dp branes.The only possible Higgs branch motions are to split the branes into two stacks of (k−ℓ) and ℓ coincident branes, for ℓ ∈ {1, 2, . . ., ⌊ k 2 ⌋}.The magnetic quiver consequently fissions into two Dynkin quivers, one with ranks (k−ℓ)fold of the dual Coxeter labels and the other the ℓ-fold.Only if one of the stacks is a single Dp brane another Higgs branch motion arises: the Dp dissolves in the D(p+4) stack and realizes the instanton moduli.From the MQ view point, an affine Dynkin quiver (ranks equal the dual Coxeter labels) decays completely.
Returning to Figure 1 and the fission of the Sp(2) theory into two copies of the Sp(1) theory, this is seen on the 5-brane web as follows: the brane system has two gauge branes, two O7 − plane one of which supports a fractional NS5 brane.The system can fission into two self-consistent 5-brane webs: one with a gauge 5-brane between two O7 planes, ending in one fractional NS5.The other subsystem is a gauge 5-brane between the two O7 planes.Both subsystem still perceive the presence of all flavor 7-branes.Crucially, both subwebs realize an Sptype world-volume theory; the only difference is that in one system the anti-symmetric matter is manifest while it is not in the other [47].The distance between these subwebs is setting the required vacuum expectation value for this RG-flow.The remaining transitions in Figure 1 can be analysed with similar arguments.

V. PERSPECTIVES
The decay and fission algorithm places particular emphasis on quivers that cannot decay or fission into other non-trivial quivers.These stable quivers lead to 3d Coulomb branches [53,54] characterized by isolated conical symplectic singularities (ICSS).These can be seen as the geometric incarnation of elementary phase transitions on Higgs branches, in addition to offering interest to mathematics in their own right.This simple observation offers a well-defined perspective to list all ICSS that can be realized as Coulomb branches of unitary quivers, reducing the problem to convex linear algebra, as stated in Section III B. As a proof of concept, we implemented FIG. 3. List of all 2-and 3-nodes good unitary quivers that give rise to elementary Higgsing transitions.We use notations from [29,43,51].It includes the recently introduced slices Y(ℓ) [52], and a new transition, in red.
this principle on all quivers with up to three nodes.The resulting list is presented in Figure 3. Physically, this covers a surprisingly vast array of theories: these quivers appear within Higgs branches of theories ranging from 4d [11] to 6d SCFTs [29].Remarkably, our exploration adds one new element (highlighted in red), corresponding to a new isolated singularity of complex dimension 8 with isometry so 5 .We predict that the associated phase transition should show up in some yet to be discovered SCFTs.
The above exploration will be carried out more generally to arbitrary size quivers in a future work, thereby charting a significant corner of the landscape of possible Higgs transitions.Combined with an extension to decorated quivers, we expect to be able to describe every single minimal degeneration using the Coulomb branch construction.Finally, we can further adapt our algorithm towards quiver growth.This adaptation, combined with our deep understanding of elementary transitions, paves the way for a bootstrap method; it opens possibilities for classifying theories based on their Higgs branch geometry, thereby complementing the Coulomb branch approach of [55,56].

2 a 4 a 6 A 1 2d 5 d 5 FIG. 1 .
FIG.1.Decay and fission algorithm for the 5d N = 1 SCFT with SU(5)1 gauge theory description plus NAS = 2 antisymmetric and N f = 4 fundamental matter fields.The algorithm generates the entire Hasse diagram by constructing for each leaf one or more MQs.The slices can be obtained in a subsequent step, detailed in[37].The corresponding 5d N = 1 theories are indicated in blue.