5d SCFTs from $(E_n,E_m)$ Conformal Matter

We determine 5d $\mathcal{N}=1$ SCFTs originating from 6d $(E_n,E_m)$ conformal matter theories with $n\neq m$ by circle reduction and mass deformations. The marginal geometries are constructed and we derive their combined fiber diagrams (CFDs). The CFDs allow for an enumeration of descendant SCFTs obtained by decoupling matter hypermultiplets and a description of candidate weakly coupled quivers.

1 Introduction 5d N = 1 superconformal field theories (SCFTs) are non-perturbative in nature. The first such theories were understood as the UV completion of non-renormalizable 5d supersymmetric gauge theories realized by an embedding into string theory [1]. Geometrizations of these constructions have proven to be an efficient method for systemizing large classes of 5d SCFTs  and have greatly expanded the list of known examples and their properties.
The classification of 6d N = (1, 0) SCFTs is organized by an enumeration of admissible F-theory geometries [24,25]. These geometries are elliptically fibered Calabi-Yau 3-folds with the classification amounting to a characterization of the possible base geometries and permitted singular fibers above these. The base geometries are built from non-Higgsable clusters connected by conformal matter theories and other links. These conformal matter theories [26] are SCFTs themselves and essential building blocks of the classification result. Sharpening this classification, it was argued in [27] that under certain Higgs and tensor branch flows all classified 6d N = (1, 0) SCFTs are generated starting from a small set of UV-progenitor theories, the rank k (E 8 , G ADE ) orbi-instanton theories.
Circle compactifications of 6d N = (1, 0) SCFTs yield 5d KK-theories for which suitable mass deformations trigger an RG flow to 5d N = 1 SCFTs. Geometrically the mass deformations are realized as partial resolutions of the fiber singularities of the Calabi-Yau 3-fold used to engineer the 6d SCFT. By M-/F-theory duality the 5d SCFT is then realized by M-theory on this partially resolved Calabi-Yau 3-fold. This observation has been systematized to a classification programme in [7-9, 13-15, 17] and suggests an avenue to utilize the classification of 6d N = (1, 0) SCFTs for 5d N = 1 SCFTs. As a first step in this programmme 5d SCFTs originating from 6d conformal matter theories of types (E n , A m ), (D n , D n ), (E n , E n ) and non-Higgsable clusters were analysed [13][14][15]. We report on 5d SCFTs originating from the 6d (E n , E m ) with n = m conformal matter theories. This serves as an initial step in analyzing the associated circle reductions of the 6d UV-progenitor theories with the ultimate goal of systematically determining all 5d theories that descend through 6d SCFTs. Extending this approach to include 5d SCFTs reached through RG flows triggered by Higgs branch vacuum expectation values would further connect to recent results in [28][29][30][31][32][33][34][35].
We study the 5d N = 1 SCFTs at generic points of their Couloumb branches via the associated resolved Calabi-Yau 3-folds. These exhibit a non-flat fiber given by a reducible surface S = ∪ k S k which collapses in the singular limit and characterizes the SCFT. Much of the resolution independent data of the surface S can be subsumed into combined fiber diagrams (CFDs), introduced in [12,13], which then manifestly encode many properties of the SCFT such as the superconformal flavor symmetry, BPS states, mass deformations and possible quiver descriptions. This constitutes a uniform geometric formulation of many known results from field theoretic and brane web considerations [36][37][38][39][40][41][42][43][44][45][46]. This paper is organized as follows. In section 2 we discuss the singular geometries realizing (E n , E m ) conformal matter and their resolutions. We compute the marginal fiber diagrams and the underlying geometries of the irreducible components of the non-flat surface S. We conclude the section with a derivation of the marginal CFDs for (E n , E m ) conformal matter.
In section 3 we utilize the CFDs to enumerate 5d SCFTs descending from the marginal theories via mass deformations. Furthermore we derive possible weakly coupled quiver descriptions of the marginal theories and its descendants, which are not excluded by consistency constraints imposed by their CFDs and those derived in [47] .

Marginal Geometries for Conformal Matter
Marginal theories in 5d are circle reductions of 6d N = (1, 0) gauge theories which UV complete to a 6d N = (1, 0) SCFT. Consider a marginal theory given by a 5d N = 1 gauge theory with gauge group G and gauge algebra g of rank r = rank g coupled to massive matter. At generic points of its Coulomb branch the field content is given by r massless U (1) vector multiplets and massive hypermultiplets associated with the W-bosons of the broken gauge symmetry and the original matter multiplets. Integrating out the W-bosons the dynamics of the low energy effective theory is governed by a prepotential F cubic in the U (1) vector multiplets. The terms involving the scalars φ i of the r vector multiplets parametrizing the Coulomb branch reads Here g 5d is the 5d Yang-Mills coupling constant, h ij = tr T i T j is the metric on the moduli space involving the Lie algebra generators T i , the integer k is the Chern-Simons level, ) is a symmetric group theoretic quantity, R f are the representations of the massive hypermultiplets with masses m f and W R f is the weight system of these representations [1,2].
Let the 6d N = (1, 0) SCFT associated to this marginal theory be realized by F-theory on a singular Calabi-Yau 3-fold X 3 . Then the marginal theory is realized by M-theory at low energies on a crepant resolution Z 3 of this Calabi-Yau 3-fold. Given a basis of Kähler classes J i ∈ H 1,1 (Z 3 ) of unit volume and defining coordinates J = φ i J i the triple intersections c ijk = S i · S j · S k = Z 3 J i ∧ J j ∧ J k set the cubic term of the prepotential 1 6 c ijk φ i φ j φ k ⊂ F describing the marginal theory [2,48]. Here S i are complex surfaces dual to the basis J i of Kähler classes. Masses are set by volumes of two-cycles within Z 3 and their intersection structure determines the representation theoretic details of (2.1).
Mass deformations of the marginal theory and subsequent RG flows of the 5d theories correspond in geometry to partial singular limits Z 3 → X 3 parametrized by the RG flow. For suitably chosen mass deformations this procedure results in a family of 5d SCFTs enumerated by partial resolutions of X 3 . These SCFTs are thereby derived from the smooth geometry Z 3 of the 5d marginal theory, they are referred to as descendants of the marginal theory.
Consequently the starting point to the analysis of this tree of descendants and their properties is the marginal geometry Z 3 .

Singular Elliptically Fibered Calabi-Yau 3-folds
The geometries we use to engineer 5d N = 1 conformal matter are non-compact singular elliptically fibered Calabi-Yau 3-folds with a holomorphic section and non-isolated, non-minimal singularities. We begin the construction of these geometries with an elliptically fibered 3-fold Singular Fibers Figure 1: Sketch of the singular Calabi-Yau 3-fold geometry X 3 (2.5). Minimal singularities of Kodaira-type are supported along base divisors u, v = 0 and enhance to a non-minimal singularity upon collision at the origin.
where E is the elliptic fiber and B 2 is the 2-fold Kähler base. The 3-fold X 3 is realized as a hypersurface within the weighted projective bundle over B 2 given by with the sections a n ∈ Γ(B 2 , K −n B 2 ) encoding base dependence. The holomorphic section of the fibration is given by s : thus intersects X 3 precisely along the image of s and once in each fiber.
We introduce singularities of Kodaira type above the base divisors S u = {u = 0} ⊂ B 2 and S v = {v = 0} ⊂ B 2 . This amounts to prescribing the vanishing orders of the sections a n ∈ Γ(B 2 , K −n B 2 ) along S u , S v when expanded in the coordinates (u, v). Denoting the two sets of exponents for u and v by (i 1 , i 2 , i 3 , i 4 , i 6 ) and (j 1 , j 2 , j 3 , j 4 , j 6 ) respectively the Tate form (2.4) now reads with singularities along [0 : 0 : 1] ∈ E over S u , S v and a generically non-minimal singularity at the point u = v = 0 in the base. We sketch the setup in figure 1. The classes of various section are x : 2σ + 2c 1 , y : 3σ + 3c 1 , where c 1 = c 1 (T B 2 ) is the first chern class of the base B 2 .

Resolution of Singularities
We resolve the singularities of the Calabi-Yau 3-fold (2.5) by blowing up in the base once to remove the non-minimal singularity located at u = v = 0 and subsequently resolving the codimension 1 and 2 singularities in the fiber. We adhere to the resolution procedure and notation presented in [52,53], which we reintroduce where necessary.
The non-minimal singularity is removed by the blowup together with the rescaling x → 2 x and y → 3 y which introduces the exceptional divisor The rescaling of the two sections x, y previously belonging to Γ(B 2 , K −k B 2 ) with k = 2, 3 respectively is due to the canonical bundle shifting to K B 2 = K B 2 + E. Physically this resolution amounts to moving onto the tensor branch of the SCFT.
The chern class c 1 and base divisor classes S u , S v are all shifted by a copy of the exceptional divisor E to The partially resolved geometry Y 3 is explicitly given by substituting (2.7) into (2.5) with an overall power 6 removed by a proper transform (2.8) This gives a hypersurface in where B 2 is the blowup of the base B 2 . The coordinates u, v can no longer vanish simultanesouly, the non-minimal singularity is removed.
The Calabi-Yau 3-fold Y 3 still exhibits singularities in codimension 1 and 2 which can be removed with additional blowups in the ambient space [54][55][56][57]. Singularities at Codim 1 Singularities : 0 = s a = s b = s c , where s a , s i are place holders for generic sections of the Calabi-Yau, are resolved by the replacements Codim 1 Blowup : together with a proper transform which removes a factor of s 2 d , s k from the transformed Tate form. We abbreviated these replacements together with their proper transforms by This fibration is non-flat, i.e. it contains fibers S of complex dimension 2 which encode the SCFT data. Non-flat fibrations of this kind have most recently been studied in [5,6,10].

Geometries for (E n , E m ) Conformal Matter
We select two distinct sets of vanishing orders for the ordered set of coefficients appearing in (2.8) corresponding to E-type singularities and substitute these into the Tate-model (2.5). The vanishing orders (2.13) can be found in the lists of possible elliptic fiber degeneracies presented e.g. in [50,52]. We blowup in the base B 2 as in (2.7) to find the Tate models for the partially resolved geometries (2.14) We resolve each E-type singularity individually using two of the blowup sequences  We have listed these blowups in the notation introduced in (2.11) here with the generic sections s a , s i now explicitly given by x, y, u i , v i . The Cartan divisors intersecting according to the affine E n -Dynkin diagrams among the exceptional divisors of (2.15) are Finally the remaining singularities are resolved by the cross term blowups, i.e. blowups involving a mix of sections and introducing δ k , {x, y, ; δ 8 }, {y, δ 8 ; δ 9 }, {δ 8 , δ 9 ; δ 10 }, {x, δ 9 ; δ 11 }, {δ 9 , δ 11 ; δ 12 }, The projective relations introduced by the blowups prohibit the sections δ 2 and δ 2 , δ 9 , δ 14 and δ 2 , δ 10 , δ 15 , δ 18 from vanishing for (E 6 , E 7 ) and (E 6 , E 8 ) and (E 7 , E 8 ) respectively. For all other sections , δ i restricting the associated divisors to the smooth Calabi-Yau 3-fold yields an irreducible complex surface. The number of these surface components is the rank r of the associated SCFT, we have

Intersection Ring
The intersection ring of the divisors of the fully resolved Calabi-Yau 3-fold Z 3 ⊂ Z 4 determines the 5d physics. There are vertical and horizontal divisors in Z 3 . The vertical divisors are pull introduced in the blowups of (2.11) restricted to Z 3 together with the divisor associated to the holomorphic section σ.
The intersection rings of the 3-fold Z 3 and its base B 2 are related as After the base blowup (2.7) the base coordinates u, v can no longer vanish simultaneously implying that their associated divisors do not intersect The exceptional divisor E is a curve of self-intersection −1 while S u , S v intersect exactly once in B 2 and thus we derive For horizontal divisors in the 3-fold Z 3 we have the relations as the centers of the blowups introducing the exceptional divisors D u i , D v i , D δ i are located in the w = 1 patch of Z 3 . Finally note that the intersection of any three divisors D i in Z 3 can be lifted to an intersection in the ambient space Z 4 by while the base blowup gives rise to (2.20). These relations can be used to evaluate mixed intersections involving distinct horizontal divisors [52].
A subset of the exceptional divisors D u i , D v i are Cartan divisors D En α i labelled by the root α i they correspond to within the affine E n root system. On the 3-fold Z 3 these are fibered as where ν = u, v depending on which coordinate is associated with the E n singularity. We The position of the fibers P 1 α i within the exceptional locus π −1 (W ν ) are determined by the pull back properties of the projection π : Z 3 → B 2 , which are derived from the replacement relations (2.7) and (2.10) and condensed in the transformations (2.26) Here m are the Dynkin labels of the respective Lie algebra E n , E m and we take the relations (2.26) as a definition for the multiplicity integers ξ . This implies that the pull backs of the vanishing loci W ν and the exceptional base divisor E are given by where the divisors {ν i = 0} = D ν i and surfaces S k = {δ k = 0} for k = 2, . . . , r and S 1 = The fiber P 1 α i of a Cartan divisor D En α i , D Em α i is called a flavor curve if it is fully contained in the reducible surface S = ∪ r k=1 S k . Consider a divisor D En α i associated to the complex base coordinate u and therefore giving W u when projected to the base. Then the intersection α i associated to simply laced algebras one therefore has where we have indicated the coordinates (u, v) the algebra E n is associated with by superscripts rather than the algebra. Intersecting the expansion of Identical arguments apply with u, v interchanged.

Reduced Intersection Matrices
The reduced intersection matrices encode how the flavor curves P 1 α i (2.25) of the Cartan divisors D En α i are contained within the surface components S k . It is motivated by (2.29) and defined by the triple intersections within Z 3 . The criterium (2.29) for a Cartan divisor to contribute a flavor curve now becomes   Taking these multiplicities into account we find the three tables of triple intersections

Fiber Diagrams and Surface Geometries
In the singular limit Z 3 → X 3 the SCFT originates from the collapse of the reducible surface S = ∪ k S k in the non-flat fibration. M5 branes wrapping irreducible components of S give rise to tensionless magnetically charged strings and M2 branes wrapping curves in S generate a tower of electrically charged states. Both enhance the spectrum in the singular limit. The nature of these enhancements depends crucially on the geometry of S = ∪ k S k which we study here in detail for the geometries of (E n , E m ) conformal matter given in (   within Z 3 where D p,q run over all Cartan divisors D En α i , D Em α j together with the divisors D x , D y , D w , D δ k associated to the sections x, y, w, δ k . Each surface S k has its own fiber diagram and depicts the intersection matrix (2.34) for fixed k. Nodes represent divisors and are labelled by the associated sections. The self-intersection of each curve S k · D p is recorded in the center of the representing node. The genus of a curve C ⊂ S k is fixed from the intersection matrix by the relation with the double intersections taken in the surface S k and the triple intersections taken in the Calabi-Yau 3-fold. Here K S k is the canonical bundle of the surface.
We now explicitly discuss the rank 10 (E 6 , E 7 ) marginal geometry, giving the results for the other cases of (E n , E m ) without derivation as the analysis extends unaltered to these cases. Here flopping a curve from S k into S l involves a blow down in S k and a blowup in S l . Further blowups and blow downs decrease and increase the degrees of the surfaces by 1 such that the degrees of S 1 , S 5 , S 6 are now 6, 1, 1 respectively. The reduced intersection matrix for the flopped geometry reads where we have denoted the degree of the surfaces as given in figures by a superscript and  Figure 4: The picture shows an (E 6 , E 7 ) marginal geometry related to the blowup sequence in (2.17). Each diagram depicts the full set of generators for the Cox ring of the surface S k . The curves C i in individual surfaces are distinct and enumerate excess generators not directly associated to section of the Calabi-Yau geometry. Their relation to the divisors of the Calabi-Yau restricted to S k is listed in (2.41). Flavor curves of self-intersection (−2) are colored green, manifest gluing curves are colored yellow and the remaining curves are colored white. Homologous curves are listed by '/ '.  Figure 5: (E 6 , E 7 ) marginal geometry. Figure 4 continued.

Combined Fiber diagrams (CFDs)
A combined fiber diagram (CFD) is generated from a collection of fiber geometries associated to the surface components S k by jointly representing the Mori-cone generators of the surfaces   Figure 7: Marginal Combined fiber diagram (CFD) for (E 6 , E 7 ) conformal matter. The genus of all depicted curves vanishes. curves Σ kl = S k ∩ S l are excluded. In addition curves of vanishing self-intersection are shown.
The CFDs are independent of the flop transitions moving curves between the S k and describe all Coulomb branch phases of a gauge theory equally [14].
In general the (−1) curves will be a linear combination of the curves depicted in figure   4 and 5, however manipulating the geometry using the flop transitions we can make these manifest. For the (E 6 , E 7 ) geometry these flops are depicted in figure 6 with the resulting Alternatively we can compute the reduced intersection matrices with z = x, y, w and sum over all surface components S k . In all cases (n, m) = (6, 7), (6,8), (7,8)

Descendants and Weakly Coupled Quivers
The combined fiber diagrams (CFDs) derived in section 2 distill key features of the resolutions (2.15) and (2.17). Taking the CFDs in figures 7, 8 and 9 as starting points we turn to discuss descendant 5d SCFTs and weakly coupled quiver descriptions of (E n , E m ) conformal matter.

Descendant SCFTs
The structure of the collapsing surface S = ∪ k S k determines the SCFT data in the singular limit and can be manipulated in various ways. In section 2.6 we flopped (−1) curves between surface components S k to access different gauge theory phases of the weakly coupled description of the marginal gauge theory. Alternatively (−1) curves can be flopped out of S resulting in a complex surface S which is not phase equivalent to the one it originates from. The SCFT generated by M-theory when collapsing the surface S is referred to as a descendant into unique manipulations on its associated CFD referred to as CFD transitions which were laid out in [12,13] and which we now reproduce.
Denote the nodes of a CFD by C i and label these with their self-intersection and genus (n i , g i ). The intersection matrix is denoted by m ij = C i · C j . Given this data a CFD transition generates a new CFD given by the labels (n i , g i ) and the intersection matrix m ij with the two rules: 1. Remove a curve C i of self-intersection (-1) and vanishing genus from the CFD, delete the corresponding row and column of the intersection matrix m ij and update the reduced matrix m jk according to with i = j, k. If C i intersects multiple curves apply the rule (3.1) pairwise.
2. Update the labels of the remnant curves C j according to We give an example of the CFD transition generating the first descendant of the marginal (E 6 , E 7 ) geometry in figure 10. Figure 11: List of box graph CFDs derived from the extended Coulomb branch of weakly coupled 5d quiver gauge theories. The integer N enumerate the number of blue curves of the box graph CFD. These necessarily embed into the CFD of any 5d SCFT to which the quiver theory completes. Only U (N ) and SU (N ) box graph CFDs embed into the marginal CFDs of (E n , E m ) conformal matter. The embeddings are shown in figure 12. This list is a partial recreation of a table found in [14].
The full tree of descendants is obtained by applying the CFD transition rules

Constraints on Weakly Coupled Quiver Descriptions
Given a marginal 5d theory its possible weakly coupled quiver gauge theory descriptions are heavily constrained: 1. Box Graph CFDs: The flavor symmetry of a quiver gauge theory is a subgroup of the flavor symmetry of the SCFT it completes to. Further, descendants of quiver gauge theories are weakly coupled descriptions of the descendants of the associated SCFT and the structure of the extended Coulomb branch must embed within the marginal geometry. As a consequence box graph CFDs derived from the extended Coulomb branch must form subgraphs of the CFD [14]. Conversely, the possible subgraphs of the marginal CFD correspond to partial quivers embedded within any consistent quiver gauge theory completing to the SCFT. The list of box graph CFDs is given in figure 11.
When multiple subgraphs are embedded they must not intersect. This ensures that the descendant structure of the quiver gauge theory is reproduced within that of the SCFTs.
2. Gauge and Flavor Rank: The gauge rank r G of a quiver gauge theory is given by the sum of the ranks of the gauge nodes and must coincide with the rank for the SCFT, i.e. the number of irreducible surface components of S = ∪ k S k as they were counted in (2.18).
The flavor rank r F is the rank of the total global symmetries. For quivers the total global symmetry receives a factor of the topological U (1) I abelian symmetry for every gauge node and a factor U (1) B for a single full hypermultiplet in the bifundamental of two gauge groups. Finally the classical flavor symmetries contribute. The flavor rank coincides with the rank of the enhanced flavor symmetry of the SCFT and for (E n , E m ) conformal matter is simply r F = n + m + 1.

Number of Hypermultiplets:
The number of hypermultiplets connecting to any single gauge node is constrained by positivity conditions on the Coulomb branch metric and monopole string tensions [47]. We list the implied restrictions on the matter content relevant for weakly coupled quiver descriptions of (E n , E m ) conformal matter in table 2.
We now repeatedly apply these rules to determine quiver candidates for (E n , E m ) conformal matter theories.

Quiver Descriptions of Maximal and Submaximal Depth
We now derive quiver descriptions consistent with the conditions above for (E n , E m ) conformal matter theories. We discuss each theory in turn.
From the box graphs in figure 11  CFD are shown in figure 12 together with the flavor symmetry they make manifest. Each of these embeddings gives rise to a subquiver which realizes this flavor symmetry as rotations on its hypermultiplets, the pairs are listed in table 1.
Next we connect the consistent subquivers by introducing additional gauge nodes and bifundamental hypermultiplets. The resulting quiver must have the gauge and flavor rank (E 6 , E 7 ) : (r G , r F ) = (10,14) , (3.4) be anomaly free and respect the consistency constraints on the number of attached hypermultiplets at each gauge node as listed in table 2. There are many such quivers and we restrict (4) The global symmetry rank must be 14 whereby the interior is either empty and the Sp(n) gauge nodes are connected by a full bifundamental hypermultiplet or it consists of a single gauge node G linked to the Sp(n) gauge nodes by half-hypermultiplets. Consequently such a node G must have a real fundamental representation. This in turn poses the additional constraint that this representation should be even dimensional as the theory is otherwise anomalous, due to an Sp(n) gauge node connecting to a total odd number of half-hypermultiplets.
We consider the case of an empty interior first and connect the two symplectic gauge groups by a single full bifundamental hypermultiplet. The global symmetry group of this quiver is  Table 1: We list the flavor subgroups and their corresponding subquiver for (E 6 , E 7 ) conformal matter derived from the possible embeddings of the box graph CFDs in figure 12 into the marginal (E 6 , E 7 ) CFD.
where the SU (2) B rotates the two bifundamental half-hyper multiplets. The global symmetry rank is as required. The remaining constraints then read Sp(n 1 ) : 2n 2 + 5 ≤ 2n 1 + 6 , Sp(n 2 ) : 2n 1 + 6 ≤ 2n 2 + 6 , r G : n 1 + n 2 = 10 , where the first two inequalities are derived from the constraint on the number of fundamental hypermultiplets as listed in table 2 for the gauge nodes Sp(n 1 ), Sp(n 2 ) respectively. The only admissible quiver of this type is thus The last consistency condition we apply is that the classical global symmetries of the descendants of quiver (3.8) must include into the flavor group of the corresponding CFD descendants.
The first descendant of the marginal (E 6 , E 7 ) CFD is shown in figure 10 and displays the global    The configuration of hypermultiplets connecting to a gauge node of a 5d quiver gauge theory are constrained if it is to complete to an SCFT. The table summarizes the constraints for SU (n), Sp(n) gauge nodes including their low rank outliers. The second column abbreviates the notation used in the third. Here Sym, AS, F, V, k denote the symmetric, antisymmetric, fundamental and vector representations as well as the Chern-Simons level respectively. For Sp(n), SO(n), F 4 the final column is an upper bound on the possible number hypermultiplets while for SU (n) the interpretation is more subtle, we refer to [47]. The above table is a partial recreation of tables found in [14].
It follows with the same reasoning as above that the last class of maximal depth quivers, which are of the structure   We move on to consider quivers of submaximal depth for (E 6 , E 8 ) and (E 7 , E 8 ) conformal matter theories. These are of the structure There are five choices of a connected interior G if we restrict to quivers without loops.
The gauge node G 1 of the quiver (3.22) necessarily has a real, even dimensional fundamental representation and thus G 1 = F 4 , SO(2r). The case G 1 = F 4 is found to be inconsistent.

Conclusion and Outlook
In this paper we reported on Calabi-Yau manifolds (2.15), (2.17) realizing marginal 5d gauge theories in M-theory that originate from 6d (E n , E m ) conformal matter theories and derived their associated combined fiber diagrams which are given in figures 7, 8, 9. These we used to constrain the list of quiver gauge theories of maximal and submaximal depth (3.14), (3.25), (3.26) which in the strong coupling limit potentially complete to the respective SCFTs.
The presented resolution of the (E n , E m ) singularities has no associated weakly coupled quiver gauge theory description as the geometries can not be consistently ruled. It would be interesting to study which quiver gauge theories can be realized by altering the resolution sequence such that the geometry allows for rulings. A systematic study of this requires understanding the structure with which the surface components S k glue to form the reducible surface S = ∪ k S k . Describing which geometric transitions mediate between surfaces with different ruling would facilitate an enumeration of all quiver gauge theories associated to a marginal geometry and is one possible avenue for further research. For rank 2 theories this is achieved in [9].
It is clear that the three constraints presented in section 3.2 which restrict the potential quiver gauge theories are not sufficient. The list of quivers (3.14) for (E 6 , E 7 ) contain in part quivers which at the same level of reasoning are candidates to UV complete to descendants of (E 7 , E 7 ) conformal matter, cf. [13]. Further constraints generalising the results of [47] such as recently explored in [20] are needed to decide the UV behaviour of the proposed quivers.
The results of this paper complete the list of 5d theories originating from 6d conformal matter theories, as initiated in [12,13], which are relevant to the discussion of circle reductions of the 6d UV-progenitor theories. The natural next step is the analysis of the higher rank progenitor theories with the final goal of systematizing all 5d theories that descend through Higgsable 6d SCFTs by RG flows induced through mass deformations and Higgs branch vacuum expectation values.