5d SCFTs from Decoupling and Gluing

We systematically analyse 5d superconformal field theories (SCFTs) obtained by dimensional reduction from 6d $\mathcal{N}=(1,0)$ SCFTs. Such theories have a realization as M-theory on a singular Calabi-Yau threefold, from which we determine the so-called combined fiber diagrams (CFD) introduced in arXiv:1906.11820, arXiv:1907.05404, arXiv:1909.09128. The CFDs are graphs that encode the superconformal flavor symmetry, BPS states, low energy descriptions, as well as descendants upon flavor matter decoupling. To obtain a 5d SCFT from 6d, there are two approaches: the first is to consider a circle-reduction combined with mass deformations. The second is to circle-reduce and decouple an entire gauge sector from the theory. The former is applicable e.g. for very Higgsable theories, whereas the latter is required to obtain a 5d SCFT from a non-very Higgsable 6d theory. In the M-theory realization the latter case corresponds to decompactification of a set of compact surfaces in the Calabi-Yau threefold. To exemplify this we consider the 5d SCFTs that descend from non-Higgsable clusters and non-minimal conformal matter theories. Finally, inspired by the quiver structure of 6d theories, we propose a gluing construction for 5d SCFTs from building blocks and their CFDs.

A natural question is whether 5d SCFTs are related to 6d SCFTs upon circle compactification, and even whether they can be classified by descending from 6d. In [1][2][3], we utilized the relation between 6d and 5d SCFTs by systematically studying the singularity resolutions of the singular Calabi-Yau threefolds that underlie the construction of 6d theories in F-theory.
In M-theory, these geometries model the Coulomb branch of 5d gauge theories. The superconformal flavor symmetry of the 5d SCFTs is a key datum to characterize these theories, and in [1][2][3]25] we initiated a classification, which keeps manifestly track of the strongly coupled flavor symmetries. These can be computed geometrically, and more strikingly can be summarized in graphs, the combined fiber diagrams (CFD). The CFDs are very powerful tools.
They contain as marked subgraphs, the Dynkin diagrams of the enhanced flavor symmetries as well as nontrivial information about the BPS states of the theory. Furthermore, they comprehensively encode all mass deformations that trigger RG-flows with new 5d UV fixed points.
The classification strategy in [1][2][3] can be succinctly summarized as follows: begin with a 6d SCFT, defined by a singular elliptic Calabi-Yau geometry in F-theory. These geometries have so-called non-minimal singularities. Depending on the resolution of the singularities, we obtain different M-theory compactifications, which model 5d gauge theories on the Coulomb branch. From this geometry, we extract the CFD, which encodes the flavor symmetry at the origin of the Coulomb branch, as well as the mass deformations, which trigger RG-flows. The systematic exploration of 5d SCFTs hinges then on obtaining the CFDs for the marginal or KK-theories, from which all descendant 5d SCFTs can be obtained by simple graph operations on the CFDs.
The CFDs defined in [1] not only encode key non-perturbative information of the theory but also the trees of descendant 5d SCFTs with the same dimension of the Coulomb branch (rank).
This approach is always applicable in the case of so-called very Higgsable 6d theories [46], i.e. geometries where the base of the elliptic fibration is smooth.
In cases when the 6d theory is not very Higgsable, the approach requires substantial generalization. Specifically, the base of the elliptic fibration for a 6d SCFT is in general an orbifold C 2 /Γ, where Γ is a finite subgroup of U (2). These are called non-very Higgsable theories. Examples are the non-Higgsable clusters (NHCs) and non-minimal conformal matter theories, corresponding to N > 1 M5-branes probing R × C 2 /Γ ADE .
Again the circle reduction of this class of 6d SCFTs lead to 5d KK-theories, which uplift back to 6d SCFTs in the UV, but differently from the very Higgsable case, they do have an IR description in terms of a marginal theory with flavor matter (as opposed to bifundamental matter). In [46], it was conjectured that the reduction to 5d of non-minimal conformal matter leads in the IR to a 5d quiver gauge theory coupled to an extra dynamical SU (N ) vector multiplet, and only the decoupling of the latter can result in a theory with a 5d UV fixed point. We prove this conjecture geometrically for the case of non-minimal conformal matter and some of the single node tensor branch theory with a gauge group, which contains the NHCs. For these theories the only possible way to get 5d SCFTs consist of the following two options: either the theory in the IR allows decoupling of a bifundamental hypermultiplet, whereby the resulting 5d theory factorizes into two SCFTs -this case is not of interest to us.
The second option -which is the main objective of this paper -is the possibility of decoupling an entire gauge sector, which then allows for the existence of a UV fixed point.
Geometrically, this means sending the volume of the compact divisors, that engineer this gauge sector, to infinity -i.e. they are decompactified.
We study the geometries for the single curve with gauge group as well as the non-minimal conformal matter theories. We also present the CFDs before and after decompactification, which match the expected flavor symmetry enhancements [46]. For the NHCs we always get the geometry and CFD corresponding in the IR to the 5d gauge theory analog to the 6d theory in the tensor branch. We also study the possible IR low-energy descriptions for nonminimal conformal matter theories as predicted by the CFDs. This can contain interesting strongly couple matter, leading to the construction of 5d generalized quiver, which happens when non-perturbative flavor symmetries are gauged.
Finally, we propose a gluing construction, motivated by the structure of the 6d tensor branch. We propose gluing condition in order to realize higher rank theories, such as nonminimal conformal matter starting with lower rank building blocks, where we use the 6d tensor branch as a guide for this gluing procedure. In particular we implement this from the graph theoretic perspective of the CFDs, and describe some rules how to glue them. We verify this gluing from geometry, and from the perspective of the IR gauge theory description. The existence of strongly coupled matter as well as these gluing procedure resemble the punctured sphere and their combination for 4d N = 2 theories of class S, [47][48][49].
The present paper is structured as follows: in section 2 we both revise some basics on 5d SCFTs, gauge theories and M-theory geometry. We furthermore give an in depth analysis and derivation of the concept of CFDs as flop-invariants. In section 3, we present the general strategy to get 5d SCFTs given a general 6d SCFTs, which include mass deformation and the decoupling of a subsector of the theory and complement it with a decompactification in the M-theory geometry. In section 4, we list the CFDs for non-Higgsable clusters, before and after decompactification. In section 5 we present the CFDs and geometries for non-minimal conformal matter, and discuss various, dual low-energy descriptions that are motivated by the CFDs. In section 6, based on the geometry we propose gluing rules for CFDs. We conclude in section 7. Appendix A has a summary of all building blocks, including the tensor branches in 6d, as well as the CFDs in 5d. The remaining appendices provide details of the geometric computations that underlie the main text, in particular the derivations of CFDs for NHCs and non-minimal conformal matter theories.
Note added: While we were completing this work, the paper [50] appeared which proposes the decoupling idea from 6d to 5d as well. Our findings are consistent with the criteria described therein.

5d SCFTs, CFDs and all that
In this section we review some basic and crucial aspects of 5d SCFTs and their construction from M-theory on non-compact Calabi-Yaus with a canonical singularity. We will then review how these geometries are captured into graphs called combined fiber diagrams (CFD), which encode the superconformal flavor symmetry, as well as information on the BPS states.

5d SCFTs and Gauge Theories
5d SCFTs are always strongly coupled and do not have a Lagrangian description at all energy scales. However, they allow for effective descriptions at low energies. For instance, mass deformations of SCFTs with lead to effective gauge theories at low energies. This implies that we can write an effective Lagrangian in the IR, [1,4,22,25,26,28]. Note that this allows for quiver gauge theories. Here r = i r i is the rank with r i the ranks of the simple factor. Every vector multiplet corresponding to G r I I contains a real scalars that can take vev in the Cartan of the gauge group. This defines the Coulomb branch of the G gauge I . The dynamics of the gauge theory on the Coulomb branch is parametrized by a real, one-loop exact prepotential (2.5) A 5d SCFT can have many IR gauge theory descriptions, which are dual in the UV, meaning that they have the same UV fixed point. For this reason, the gauge redundancies apart from providing information about the Coulomb branch of the theory, they are not enough to specify the UV fixed points completely. At the SCFT point one can describe the theory in terms of operators, correlation functions and states, which are specified by quantum numbers.
An important factor is, in fact, given by the flavor symmetry, which in the UV is different from the one observed in the IR. For instance, from a gauge theory perspective there can be a flavor symmetry rotating the hypermultiplets with an associated current J F cl µ , transforming in the adjoint representation of G F cl . In addition, there is a U (1) topological current in 5d defined by, From the gauge theory point of view there are massive non-perturbative states (like instanton particles), but they become massless at the UV fixed point. Moreover, when they are generically charged under the classical flavor current as well as the topological U (1) T , it happens that these symmetries mix quantum mechanically, leading to enhanced flavor symmetry at strong coupling [6,12].

5d SCFTs from M-theory
M-theory on a non-compact Calabi-Yau threefold provides a geometric framework to model the Coulomb branch and UV fixed points of 5d theories. For instance, it allows to track the theory from the IR effective description in the Coulomb branch up to the fixed point in the UV. The SCFT corresponds to the singular point (canonical singularity), and the Coulomb branches is given by its crepant resolutions [2,3,22,25,26,29], The resolution introduces compact surfaces which supply (1,1) forms that model the Cartans of the gauge group. In particular r is the rank of G gauge . These surfaces intersect along curves S i · S j = C ij , and they can also intersect with non-compact divisors D α · S i = C iα . Weakly coupled gauge theory descriptions exist, if the reducible surface S admits a ruling, i.e. admit a f i = P 1 fibration over a collection of genus g curves. In particular, if the surfaces intersect along section of these ruling such that they form a Dynkin diagram of G gauge , they will define the Coulomb branch of the gauge theory in the following way 1. The Cartan of the gauge symmetry is given by expanding the 3-form potential, C 3 = ω (1,1) ∧ A i where dual (1, 1)-form ω (1,1) are the Poincaré dual of S i , and A i are U (1) gauge potentials.
2. The W-bosons are given by M2-branes wrapping the generic fiber of the ruling f i , whose self intersection are f i · S i f i = 0 and S i · f i = −2.
In this way S can then collapse to a curve of singularities of the type of G gauge type, and we notice that in this limit all these gauge theory states become massless.
Another situation is given if the surfaces are still ruled, but some of them intersect at special fibers instead of sections. For example, we could have S i · S j ⊆ f i , f j . In this case the geometry collapses to multiple intersecting curves of singularities, and realizes a 5d quiver gauge theories, where the matter transforming in two connected gauge groups G I × G J comes from M2-branes (antibranes) wrapping fibral O(−1)⊕O(−1) curves, which intersect the curve S i · S j between the two surfaces. If there is no consistent assignment of ruling and section curves that apply to all S i ·S j , the theory does not have a gauge description. In such instances, there can however still exist a low energy effective theory specified by some generalization of quivers, we will discuss this situation for example in section 5.4.
The geometric prepotential is computed from the triple intersection numbers c ij = S i ·S j ·S in the Calabi-Yau threefold

5d SCFTs from 6d and Flavor Symmetries
An alternative, though closely related approach to studying 5d SCFTs, is to compactify 6d (1, 0) theories on a circle with holonomies in the flavor symmetry turned on; these correspond to mass deformations in 5d. The 6d theories can be engineered from elliptic Calabi-Yau compactifications in F-theory, and the associated 5d theories are constructed using M-theory/Ftheory duality.
Usually a standard circle compactification of a 6d theory leads to a KK-theory, which UV completes back in 6d. These KK-theory can sometimes have marginal gauge theory description in the IR [28], where marginal means that the metric of the Coulomb branch is positive semi-definite, see also [51,52]. Mass deformations of these marginal theories lead to a tree of descedant theories, which in the UV complete to 5d SCFTs. The mass deformations can be of two types: 1. Decoupling a matter hypermultiplet: Field-theoretically this is giving mass to a hypermultiplet that is charged under the flavor symmetry and sending the mass to infinity. In the M-theory geometry this corresponds to a flop of an O(−1) ⊕ O(−1) curve C, which is flopped out of the reducible surface S.
In the singular limit, when vol(S) → 0, the state obtained by an M2-brane wrapping C decouples.
2. Decoupling of a gauge sector: One can also decouple an entire sector of the theory, such as a gauge vector multiplet. In the geometry, this corresponds to the decompactification, vol(S k ) → ∞, of some of the compact surface components S k ⊂ S, thereby sending the associated gauge couplings to zero.
Both of these are key in the construction of 5d SCFTs from 6d, and will be important in the comprehensive study of all 5d theories obtained in this way. We will give a detailed discussion of these in section 3.
In 6d the flavor symmetries are encoded in the Kodaira singular fiber type over noncompact curves in the base B 2 of the elliptic Calabi-Yau three-fold Y 3 . These flavor symmetry generators will be denoted by D i which are ruled surfaces, with fibers P 1 i , intersect in the affine Dynkin diagram of the flavor symmetry group. Considering M-theory on resolutions of the same Calabi-Yau threefold results in 5d gauge theories on the Coulomb branch. The compact surfaces that arise in this resolution, S = ∪S k correspond to the Cartans of the gauge group.
If a flavor curves that are contained in S (i.e. they are fibral curves D i ·S) determine the flavor symmetry at the UV fixed point [1,25]: in the limit vol(S) → 0 the states charged under the corresponding flavor symmetry become massless. In the next subsection we will review the geometric/graph theoretic tool, the CFD introduced in [1][2][3], to track these flavor symmetries in the process of decoupling hypermultiplets.

CFDs and BG-CFDs
We now summarize and extend the definition of CFDs and BG-CFDs introduced in [1][2][3]. We associated to each 5d SCFT a graph, the combined fiber diagram (CFD), which encodes the flavor symmetry of the UV fixed point as well as the BPS states. Each CFD is an undirected, marked graph, where each vertex C α has two integer labels (n, g): the self-intersection number C 2 α = n and genus g. Two vertices C α and C β are connected with C α · C β = m αβ edges. Depending on the values of n and g, the vertices can be divided into the following types: The vertex is marked (in green), and it corresponds to a Cartan node of the non-Abelian part of superconformal flavor symmetry G F .
This vertex can be thought as a combination of p disconnected (n, g) = (−2, 0) vertices, which also contributes to the non-Abelian part of G F . It is hence a marked (green) vertex as well. This type of vertex only appears as the short root of a non-simply laced This vertex is considered as the "extremal vertex" that generates a CFD transition, as we introduce it shortly after. It is unmarked and does not contribute to the non-Abelian This vertex is a combination of p disconnected (n, g) = (−1, 0) vertices, which is unmarked. In the CFD transition, these p vertices need to be removed together.
(V5) All the other cases: For other values of (n, g), these vertices are unmarked and they do not contribute to the non-Abelian part of G F . Nonetheless, they still could generate the Abelian part and G F and should be drawn in the figure. Note that certain combinations of (n, g) are forbidden, such as the cases with n < −2, g = 0.
In the M-theory geometry picture, an unmarked vertex C α with (n, g) = (−1, 0) can be thought as a complex curve with normal bundle O(−1) ⊕ O(−1). The BPS state from M2-branes wrapping C α is a 5d massive hypermultiplet in the Coulomb branch of IR gauge theory. Removing such a vertex will correspond to decoupling a hypermultiplet in the gauge theory. In the CFD language, this operation generates a "CFD transition" with the following modifications on the graph.
CFD transition after removing a vertex C α with (n, g) = (−1, 0) 1. ∀C β with (n β , g β ) and m αβ > 0, the (n , g ) of C β in the new CFD are: (2.9) 2. ∀C β , C γ (β = γ) with m αβ > 0, m αγ > 0, in the new CFD, the number of edges between C β and C γ is: On the other hand, a vertex C α with (n, g) = (−p, −(p − 1)) (p > 1) can be thought as p complex curves with normal bundle O(−1) ⊕ O(−1), which are homologous in the Calabi-Yau threefold and needs to be removed simultaneously. After the CFD transition generated by removing C α , the graph is modified as: CFD transition after removing a vertex C α with (n, g) = (−p, −(p − 1)) 1. ∀C β with (n β , g β ) and m αβ = pm > 0, where m ∈ Z + , the (n , g ) of C β in the new CFD are: (2.11) 2. ∀C β with (n β , g β ) and p m αβ , the (n , g ) of C β in the new CFD are: (2.12) 3. ∀C β , C γ (β = γ) with m αβ > 0, m αγ > 0, in the new CFD, the number of edges between C β and C γ is: For the 5d KK-theory, obtained by circle-reduction of a given 6d (1,0) SCFT, we define an associated marginal CFD, which often contains green vertices that form Dynkin diagrams of affine Lie algebras. For this marginal CFD, we can construct the CFD transitions in all possible ways, which generates all the 5d SCFT descendants from the KK theory.
More generally, the non-Abelian part of G F can be clearly read off from the sub-diagram of marked (green) vertices of a CFD. The intersection matrix between the vertices C α is exactly the symmetrized Cartan matrix where the αs are the roots of the Lie algebra. For non-simply laced Lie algebra factor H F , the short roots correspond to vertices with (n, g) = (−2p, −(p − 1)), where the integer p is the ratio between the length of the long roots and the length of the short roots.
For a given 5d SCFT, the CFD is not necessarily uniquely determined. There are two possible ways to get equivalent CFDs: 1. Adding vertices into the CFD that are linear combinations of the existing ones: In the CFD, one can always add more copies C * of the existing vertex C α , which are connected with m * ,α = n α edges. The resulting CFD is trivially equivalent to the original one, but this procedure is useful in the gluing construction that we discuss in section 6.
More generally, one can add linear combination of vertices C * = k α=1 a α C α to the graph, which do not change the flavor symmetry or transitions. The values of (n * , g * ) and number of edges with other vertices are The detailed criterion and derivation of these formula will be discussed in the next section. The main constraint is that the vertices of type (V1)-(V4) that can be added cannot change the flavor symmetry or transitions.
2. Two CFDs with different marked subgraphs, but same G F : In this case, the sub-diagram of marked vertices are different in the two CFDs, but after inclusing of BPS states from the (n, g) = (−1, 0) and (n, g) = (−p, −(p − 1)) vertices, one can combined them into non-trivial representations of a larger Lie algebra.
Detailed examples will be presented in the CFD building block section of appendix A.
This geometrically means that we have chosen two different complex structures of ∪S i , resulting in two different looking CFDs.
Finally, we introduce the way of reading off IR classical flavor symmetry G cl F from the CFD, which leads to constraints on the IR gauge theory descriptions. We define a set of graphs, the box graph combined fiber diagram (BG-CFD), in table 1. These were introduced in order to characterize the IR descriptions using box graphs [53], which succinctly characterize the Coulomb branch of 5d gauge theories. If k of these BG-CFDs with G gauge,i and G F cl ,i (i = 1, . . . , k) can be embedded in the CFD without connected to each other, then the IR gauge theory can have gauge factors k i=1 G gauge,i with classical flavor symmetry k i=1 G cl F,i . Note that the gauge groups are not fixed in this procedure, but they are constrained with the information of the total flavor rank and gauge rank, see [3].

CFDs from Geometry as Flop-Invariants
In this section, we present a systematic way of deriving the CFD of a 5d SCFT from the Calabi-Yau threefold geometry introduced in section 2.2.
g=-1 g=-1 g=-1 g=-1 g=0 Table 1: The list of BG-CFDs with the gauge theory descriptions and classical flavor symmetry G F cl . The grey vertices denote (n, g) = (−1, 0) vertices in the CFD that can be removed via a CFD transition. In the Sp(N R ) case, the two BG-CFDs are equivalent in the sense that the intersection matrices are rescaled by a factor of two. Note that the grey node in the BG-CFD with (n, g) = (−4, −1) vertices should be a type (V4) vertex with (n, g) = (−2, −1).
The complex curves on the surfaces include the intersection curves S i · S j (i = j) between different compact surfaces and the intersection curves C iα = S i · D α with a non-compact surface D α in Y 3 . In the CFD, each node C α is essentially a linear combination of C iα on each S i , and we want to read off the labels (n, g) of C α directly from the geometry. In [2,3], the correspondence between resolution geometry and CFDs is developed only for compactifications of some classes of very Higgsable (VH) 6d theories, focusing on minimal conformal matter, i.e.
collisions of codimension one singularities at a smooth point in the base. We will generalize this in the present paper to include non-very Higgsable (NVH) theories, in particular nonminimal conformal matter theories, for which we now define CFDs more generally. This will be consistent with the previous cases, but generalizes it substantially. operations. For all curves D α · S i , we need to introduce a multiplicity factor ξ i,α , such that the normal bundle of the curve is invariant under such flops. Then the label (n, g) of the corresponding vertex in the CFD is given by With this multiplicity factor, the number of edges between two vertices in the CFD is In this formula, it is assumed that the two curves D α ·S i and D β ·S i have the same multiplicity factor if they intersect each other on S i . Note however that in certain cases there can be subtleties as we will discuss later in the rank 2 E-string example.
Similarly, to properly define the linear combination of vertices C α (α = 1, . . . , k) in the CFD: one of the following two conditions need to be satisfied for a particular flopped geometry, and the formula (2.15) hold: 1. All the multiplicity factors ξ i,α are the same if a α S i · D α is non-zero in the second line of (2.20). If this is the case, then which is in the form of a complete intersection curve.
2. All the curve components D α · S i in (2.20) lie on the same surface S j . If this happens, 22) which is also in the form of a complete intersection curve.
To define the multiplicity factors, we first study the curve components D α · S i with normal bundle O(−1) ⊕ O(−1), which intersects other S j s at one or more points. As an example, consider the geometric configuration in the figure 1, where the curve D 1 · S 3 intersects both S 1 and S 2 . After the curve D 1 · S 3 is shrunk, both S 1 and S 2 are blown up at one point, and the curves D 1 · S 1 and D 1 · S 2 will have normal bundle O(−1) ⊕ O(−1). As we can see, if we define the multiplicity factors ξ 3,1 = 2, ξ 1,1 = ξ 2,1 = 1, where the notation with " " denotes the quantities after the flop, then the linear combination (2.17) is invariant.
For more general cases, in principle one needs to perform a sequence of flops r i=1 S i → r i=1 S i among the compact surface components, such that all the curve components D α · S i for a fixed α only intersect other S j at one or zero points. From the perspective of the noncompact surface D α , this corresponds to a blow down sequence of D α which terminates when D α cannot be blown down, or all the O(−1) ⊕ O(−1) curves D α · S i only intersect another S j at one point.
In this "terminated" geometry, all the multiplicity factors ξ i,α for S i · D α would be trivially one. Then the multiplicity factor ξ i,α of the original curve D α · S i can be counted as: (2.23) The flop operation on an example with D 2 1 ·S 3 = −1, D 1 ·S 1 ·S 3 = 1 and D 1 ·S 2 ·S 3 = 1. In the picture, each line segment denotes an intersection curve S · S , and the integer label on the side of S (or S ) is the triple intersection number S · (S ) 2 (or S · S 2 ). After shrinking the curve D 1 · S 3 , the surface geometry ofD 1 andS i has a conifold singularity. Then after blowing upS 1 andS 2 , the geometry will become the D 1 and S i , which is the flopped geometry of the original one.
An equivalent simple way to read off the multiplicity factors is to consider the collapse of the following sequence of curves: We put the self-intersection number of curves in the brackets and the multiplicity factors over them. In the terminated geometry on the bottom line of (2.24), we assign trivial multiplicity factor one to all of the curves. Then when we go up (blow up D α ), the multiplicity of the old curves remain the same, while the multiplicity of the new exceptional (−1)-curve is given the sum of the two multiplicity factors on each side. After repeating the procedure, we can get all the multiplicity factors in the original geometry. The procedure can be easily generalized to non-toric D α as well, where the multiplicity of the new exceptional (−1)-curve is given the sum of the multiplicity factors on all the old curves that intersect the new (−1)-curve.
In fact, it can be shown that the quantities (2.17) and (2.18) are invariant under the operation (2.24). For the example in figure 1, this sequence is We can see that the multiplicity of the middle curve D 1 · S 3 on S 3 is indeed two.
If initially we already have , if it intersects another curve D β · S i with a well defined multiplicity factor ξ i,β on S i , then we define ξ i,α = ξ i,β . This is called "neighbor principle" in the later references.
If this does not happen, then we simply take ξ i,α = 1.
Finally, for a curve D α · S i with normal bundle O(−1) ⊕ O(−1) that does not intersect another S j , one can directly flop it out of r i=1 S i and that corresponds to a CFD transition. Its multiplicity ξ i,α equals to a previously defined ξ i,β if S i · D α · D β > 0, unless there are two different ξ i,β = ξ i,γ , where S i · D α · D β , S i · D α · D γ > 0. If the latter situation happens, then ξ i,α is not uniquely defined. We will discuss this situation in the rank-two E-string example latter.
To illustrate this general framework, we consider two, somewhat more complicated examples: In figure 2, we show a non-compact surface D 1 with four compact surfaces S i , S 1 , S 2 and S 3 . In the first flop, we shrink the curve D 1 · S i and blow up the surface S 1 , S 2 . After this flop, D 1 · S 2 still intersects S 1 and S 3 , hence we need to further shrink the curve D 1 · S 2 and blow up the surface S 1 , S 3 . Finally, on the surface components S i , all the multiplicity factors associated to D 1 equal to one, and we can see that D 1 should correspond to a vertex with n(D 1 ) = −2. From (2.23) applied with S i , we can see that the correct multiplicity factor of D 1 · S i on the original surface S i is ξ i,1 = 3, and the multiplicity factor of D 1 · S 2 on S 2 is Figure 2: The necessary flop operations to determine the multiplicity factor ξ i,1 of the noncompact surface D 1 on S i . The procedure includes two conifold transitions, where we do not draw the singular geometry explicitly.
We can also apply the procedure (2.24), which explicitly generates the correct multiplicity factors: In this case, the shrinking of D α · S i will change the genus of the compact intersection curve S i ·S j . In [29], it was shown that such genus changing transition involves changing the complex structure moduli of the surfaces. For example, if one wants to transform a genus-one curve S i · S j into a genus-zero curve, then one needs to first take the singular limit of S i · S j where the torus is pinched at a point and then blow up that double point singularity. Nonetheless, it is still possible to define invariant quantities n α under this kind of geometric transition.
For example, see the resolution geometries of (E 8 , SU (2)) conformal matter (rank-two Estring) in figure 3, which was discussed in [2]. The non-compact divisors D · S 2 intersects S 1 at two points. After this curve is shrunk, the geometry is flopped such that the intersection curve S 1 · S 2 has genus one instead of zero. On the new geometry, we have S 1 · (D SU (2) 1 ) 2 = −2. Then from the formula (2.23), we can see that the multiplicity factor of D SU (2) 1 · S 2 on the original geometry equals to two.
In this case, we can also apply the procedure (2.24), keeping in mind that D Figure 3: The flop operation on the resolution geometry of (E 8 , SU (2)) conformal matter (rank-two E-string) rank-two E-string. On each surface component S i (labeled by the letter in the box), each node D α corresponds to the intersection curve S i · D α . The number besides the node is the intersection number (D α ) 2 · S i . The genus of such a curve is by default zero unless otherwise labeled. In this geometry, the intersection curve S 1 · S 2 has genus-zero. After the flop, the intersection curve S 1 · S 2 has genus-one.
Then we can determine the multiplicity factors of the other curves with normal bundle O(−1) ⊕ O(−1), that intersects another S j : and S 2 at a single point.
For the other curves, the multiplicities can be read off by the neighbor principle, which all equal to one except for D x · S 2 on S 2 . D x · S 2 is a O(−1) ⊕ O(−1) curve connected to a curve D SU (2) 1 · S 2 with multiplicity two and another curve D E 8 7 · S 2 with multiplicity one. Hence the multiplicity of such "interpolating curve" is not uniquely defined. Despite of this subtlety, there are two equivalent ways to present it in the CFD: as a (n, g) = (−2, 0)-node, and the node D x is drawn as a node with (n, g) = (−2, −2), or two (−1)-nodes in the same circle (as in the (E 7 , SO(7)) case in [3]). In the edge multiplicity formula (2.19), the multiplicity factor ξ i,α is always taken as that of D becomes a node with (n, g) = (0, 0), but the node D E 8 7 will become a node with (n, g) = (−1, 0).
The descendant CFDs are exactly the same, no matter which convention is used to compute them.

5d SCFTs from Decoupling in 6d
Our goal is to understand 5d SCFTs that descend from S 1 -compactifications of a general 6d SCFTs. A trivial compactification does not lead to a 5d SCFT, but rather to a KKtheory [28], which can have many IR descriptions in terms of a marginal gauge theories. In order to obtain a genuine 5d SCFT in the UV, one usually needs to mass deform the KKtheory, which corresponds to turning on Wilson lines for the flavor symmetry. For some 6d SCFT, different choices of mass deformation lead to different 5d SCFT. This, however, is not always the case and it highly depend on the 6d theory we start with.
A general 6d SCFT can be characterized by the tensor branch, which can be geometrically classified, and is comprised of smaller building blocks -in the geometry these are curves in the base of the elliptic Calabi-Yau three-fold, which intersect in a quiver, that obeys certain rules [18]. Field-theoretically, this is modeled by constructing higher rank tensor branches by consistently gauging and adding tensor multiplets.
The same logic can be implemented for the 5d SCFTs obtained by circle compactification and deformations. In particular, we start by defining some fundamental building blocks, which are reduction of 6d SCFTs on S 1 . We then develop rules how these building blocks are consistently glued together. We implement this both from the (gauge) effective field theory prospective as well as using the geometry and CFDs intoduced in [1,2]. We note that the theories discussed in [1,2] form one class of building blocks in 5d, which descend from 6d conformal matter type theories. In the present paper, we develop the methodology how to generalize this to an arbitrary 6d theory as a starting point.

Mass Deformations vs. Decoupling
It is relevant for our purpose to divide 6d SCFTs in two classes [46,54,55]. We give in each case the field theoretic description, the Calabi-Yau threefold geometry in F-theory, as well as the tensor branch structure. The latter is characterized by a collection of intersection rational curves, with self-intersection numbers Σ 2 = (−n). Blowing down (−1) curves allows moving to the origin of the tensor branch, which transforms their self-intersection numbers as follows We denote the endpoint of the tensor branch by B end . The two types of theories in 6d are distinguished as follows: • Very Higgsable Theories (VH Theories): These are 6d theories which can be higgsed completely to free hypermultiplets. Geometrically, this means that the non-minimal singularity of the F-theory model occurs at a smooth point in the base. In terms of the the resolved tensor branch geometry (which is a collection of rational curves in the base of the elliptic Calabi-Yau threefold) we get the endpoint configuration, which for very Higgsable thoeries is • non-very Higgsable Theories (NVH Theories): These are 6d theories which cannot be higgsed completely, but always have residual non-trivial 6d SCFTs in the Higgs branch. In F-theory geometry these correspond to singular elliptic fibrations over an orbifold base C 2 /Γ, with Γ ⊂ U (2) [18], where the endpoint configuration is

5d SCFTs from very Higgsable Theories
A large class of 5d SCFTs arise from the dimensional reduction of VH theories, and mass deformations. Rank one and two theories are of this type [1][2][3]29], and more generally min-imal conformal matter theories, whose descendants and flavor symmetry enhancements were systematically studied in [1][2][3].
This approach generates a tree of 5d SCFTs connected by RG-flows triggered by mass deformations, where the tree originates from the marginal theory, i.e. the 6d SCFT on S 1 without Wilson-lines. Most of these SCFTs have at least one IR effective gauge theory description and the mass deformation corresponds to decoupling an hypermultiplet at a time by sending their mass, m f → ±∞.
From the point of view of M-theory geometry, a 5d SCFT is defined by M-theory on a Calabi-Yau threefold with a canonical singularity. This implies that the resolution is given by a collection of intersecting compact surfaces, which collapse to a point at the UV fixed point.
Starting with the marginal theory, on an S 1 results in a 5d theory with an additional KK-U (1).
To get a theory that UV completes in 5d, we first need to mass deform the U (1)-KK. In the geometry this means we need to flop the (−1)-curve that corresponds to the states charged under the affine node of the 6d flavor symmetry.

5d SCFTs from non-very Higgsable Theories
If the starting point is a NVH 6d SCFT, one needs to do something more drastic in order to actually get a 5d SCFT. In fact, the circle reduction in the Higgs branch gives a 5d SCFT coupled to an extra sector, which is usually an extra gauge vector multiplet [46]. Since, the Higgs branch moduli space does not mix with the Coulomb branch in 5d, one can turn off the Higgs branch vevs, without decoupling the gauge theory. At the origin of the the Higgs and Coulomb branch the resultant KK-theory will be a 5d SCFT non-trivially coupled to a gauge theory [46]. In this cases we will encounter the following situation where the 5d SCFT S 5d whose flavor symmetry is (or contains) G, is modded out by gauge group G redundancies, or in other words, part of its flavor symmetry is gauged. Let us assume that the effective gauge theory of S 5d has a quiver gauge theory effective description at low energies. S 5d couples to the extra gauge theory with gauge group G, and we can explicitly illustrate this coupling in terms of an effective Lagrangian where (φ i = 2πR, A i = 2πRa i ) are the Coulomb branch parameters and U (1) Cartan gauge vector fields for the gauge theory with gauge group G, and R is the radius of S 1 . We can notice that this extra gauge theory couples to the kinetic terms of the quiver theory, as well as non-perturbatively to their U (1) T currents. Moreover, the gauge coupling is and because the couplings of the quiver theory and the extra gauge theory have different dependence in terms of the S 1 radius, g 2 G ∼ R and g 2 i ∼ R −1 , we can conclude that they cannot have a common strongly coupled limit. This implies that, in order to obtain a 5d SCFT we need to isolate S 5d and decouple the extra gauge theory with gauge group G.
NVH 6d SCFTs are geometrically constructed from F-theory on a non-compact singular elliptically fibered Calabi-Yau threefold, where the base is itself an orbifold singularity. In order to get 5d SCFTs from the circle compactification of these 6d theories we have two possible geometric transitions: • The only situation we encounter in where we can flop out a curve is when two compact In this case the resulting geometric transition and decompactification of that curve leads to two disconnected, reducible surface components, and thus a reducible SCFT. Field theoretically this procedure corresponds to a mass deformation, which from 5d SCFT or KK-theory leads to multiple factorized 5d SCFTs. In terms of effective gauge theory, it correspond to a bifundamental hypermultiplet getting decoupled. This case is in fact excluded on purpose in the description of CFDs, since we do not allow the factorization into lower rank 5d SCFTs after a CFD transition as these are expected not to result in new lower rank SCFTs.
• The second possibility corresponds to decoupling the extra gauge theory, and this is achieved by a decompactification limit of the compact surfaces which are dual to the Cartans of the extra gauge group G that needs to be decoupled.
In particular this decompactification retains a compact part of the theory, in particular taking vol(S i∈s G ) → ∞ whilst keeping the volume of all other compact surfaces in S finite, a necessary condition is that the curves S i · S j for i ∈ s G and any j / ∈ s G remain at finite volume. This in particular requires potentially flopping curves before decoupling the surfaces. Table 2: The 5d affine quiver gauge theory descriptions of the KK reductions of the nonminimal conformal matter theories of N M5-branes probing C 2 /Γ G .

G Quiver
An example of these theories are Non-Higgsable Clusters (NHCs), and these two possible geometric operations in order to get 5d SCFTs were discussed in [23]. More generally, we also discuss non-minimal conformal matter theories in detail in this paper.
This decoupling/decompactification process will be one of the main foci of this paper, and before describing these geometric operations in many examples, we briefly illustrate what happens in terms of the effective IR field theories for cases, where the theory is very Higgsable to 6d (2, 0) SCFTs.

Non-Minimal Conformal Matter
An illustrative class of theories Higgsable to 6d (2, 0) SCFTs is provided by non-minimal 6d conformal matter theories. They are defined as N M5 branes probing an ADE singularity C 2 /Γ G . Their circle compactification leads to KK-theories which UV complete into the 6d SCFT they originate from. At low-energy they admit an effective gauge theory in terms of the quivers in table 2.
Note that for A n−1 , the first and last SU (N ) are identified obtaining the a circular quiver.
The rank of the classical flavor symmetries, which is obtained by counting the baryonic and topological U (1)s, matches the dimension of the following 6d flavor groups where the dimension of the flavor groups is given by the total number of nodes of the Dynkin diagrams respectively, plus an extra node which can be interpreted as a shared affine extension of the flavor symmetry algebras. This is consistent with the fact that they uplift to 6d in the UV.
We can first notice that these theories have only bifundamental matter charged under the gauge groups, and they do not have any flavor matter. As already anticipated, by giving mass to these the quiver will factorize into subquivers, which might lead to fixed points.
The second, more interesting prospect is to decouple the extra gauge theory, as first proposed in [46]. These KK-theories consist of 5d SCFTs coupled to a 5d N = 1 SU (N ) gauge theory. From the point of view of the classical gauge theories mentioned above, the difference now is that the SU (N ) gauge node corresponding to the affine becomes a flavor group. The theories are summarized in table 3.
The dimension of the flavor symmetries is given by the number of nodes of the Dynkin diagrams, consistently with the fact that these theory leads to 5d SCFTs in the UV.
Once we have decoupled the extra SU (N ) gauge theory, a low energy alternative description of these theories is the given by the 5d analog of the partial tensor branch quivers in 6d, where G is of ADE type, and cm stands for the circle compactification of the conformal matter, where a G or a G × G subgroup of the superconformal flavor symmetry has been gauged. In particular, for the G × G gauging, if we assume that the matter has a weakly coupled gauge theory description, we have a contradiction. That is, if there exists a gauge theory, the compact surfaces describing this generalized matter are ruled, but not all the G × G generator curves are fibers of this ruling. In particular, one of them corresponds to the topological U (1), and it is a section. Field theoretically, it means that the non-perturbative symmetry U (1) T is gauged, which implies that there is no weakly coupled matter charged under the hypothetical gauge groups. However, this is very analogous to what happens between gluing by tubes of sphere with punctures of 4d N = 2 Gaiotto theories [48].
We have seen that the decoupling of the SU (N ) vector leads to a 5d SCFT with at least two effective descriptions, which might not be always weakly coupled. Geometrically, this is the affine SU (N ) node. However, we also need to make sure that the U (1) T SU (N ) associated to the SU (N ) gauge theory is decoupled from the 5d SCFT. Geometrically, this can require is related to the affine node in the elliptic fibration, and the additional flop makes sure that U (1) T decouples in the decompactification process. This breaks the affine structure of the flavor symmetry, and the resulting theory no longer UV-completes to a 6d SCFT.
Keeping track of these operations is very important in order to define a good geometry where all the surfaces are shrinkable to a point. Moreover it will allow us to write a CFD for the decompactified geometry, and study its descendant automatically.

CFDs for NHCs
Non-Higgsable clusters (NHCs) are an example of NVH theories and they are key building blocks for 6d SCFTs. We now discuss their counterpart in the reduction to 5d and determine the associated CFDs, implementing the decoupling philosophy.
NHCs are characterized by a single (−n) self-intersection curve with the following gauge To determine the CFDs we first need to compute the resolution geometries. The surface components in the marginal resolution geometry was presented in [23,31]. In table 4, we summarize the CFDs read off from the geometry. In this section, we will only discuss the    Table 4: CFDs for NHCs: the first column denotes the self-intersection number of the rational curve in the base of the elliptic fibration, g is the non-Higgsable gauge group. The last two columns show the CFDs before and after the decoupling of the gauge sector.
We plot the curve configurations on the surfaces as follows: The letter in the box labels the compact surface component, and each node on each surface component denotes a complete intersection curve between two surfaces. In this case, the letter V , z, x and y correspond to non-compact surfaces. The number next to a node is the selfintersection number of such complete intersection curve. By default, these curves are rational (with genus-0). If this is not the case, we will label it out with g = the genus. When g < 0, it describes a reducible curve with multiple (rational) components. For example, here on the surface component U , the complete intersection curve V · U is a rational 0-curve on U , and the curve z · U is a rational 1-curve on U . The complete intersection curve U · u 1 , U · u 2 and u 1 ·u 2 coincides, which are all (−1)-curves on U , u 1 and u 2 . One can check that the adjunction formula is always satisfied, where g(D 1 · D 2 ) is the genus of the complete intersection curve D 1 · D 2 .
In the pre-decoupled CFD, there are four nodes that correspond to the non-compact surfaces V, x, y, z. The self-intersection number n and genus g of each node are read off by: and the adjunction formula (4.3). Hence V is a node with (n, g) = (0, 1), and x, y, z all corresponds to nodes with (n, g) = (1, 0). The number of edges between each nodes are read off by We hence get the CFD before decoupling To get a 5d SCFT, we decompactify the surface component U , see also [23]. Then the remaining compact surfaces are two F 1 intersecting along an O(−1)⊕O(−1) curve, which corresponds to the rank-2 5d SCFT with gauge theory description SU (3) 0 [29]. In [2], it was shown to have the following CFD: However, in the geometry (4.2), the intersection curve u 1 · u 2 can also be interpreted as complete intersection curve between the non-compact surface U after the decoupling. Indeed, if this curve is shrunk, the two remaining compact surface components become two disconnected P 2 , which leads to two decoupled copies of rank-one 5d SCFTs. In [2], such flop transition is not allowed as the CFD tree only includes irreducible rank-two theories. However, for the purpose of gluing, it is convenient to attach an additional (−1)-node corresponding to U , which leads to the final CFD . (4.8) As the 0-node is actually a combination of two 0-curves, after flopping this (−1)-node, the CFD will become two disconnected CFD of the rank-one with (+1)-nodes.
Note that for all the single curve NHC geometries, the decoupled surface U is always a Hirzebruch surface F m , and the intersection curve U · u 1 with another compact surface is the section curve Σ with self-intersection Σ 2 = −m. Then we can always decompactify the P 1 fiber of F m , which is consistent with the decoupling criterion, [50] and section 3.3. The remaining cases are discussed in the appendix and are summarized in table 4.

Non-Minimal Conformal Matter
As we already discussed in section 3.4 the non-minimal conformal matter theories in 6d are examples of NVH theories, and they can be higgsed to 6d N = (2, 0) theories. This implies that upon circle reduction, the KK-theory is described by a 5d SCFT coupled to an SU (N ) vector multiplet. The 5d SCFT are isolated by decoupling the extra sector via decompactification of the M-theory geometry. In this section we derive the 5d CFDs before and after decoupling for these models of type (G, G) from which all descendants can be obtained by the usual CFD transition rules [1][2][3]. The geometric derivation of the CFDs starts with the tensor branch geometries in 6d.

Tensor Branch Geometries
The non-minimal (G, G) conformal matter theories of rank N correspond to the 6d theory of N M5-branes probing a C 2 /Γ G singularity, which have flavor symmetry is G × G. We will first summarize the tensor branch geometries in 6d.
The tensor branch for the (A n−1 , A n−1 ) non-minimal conformal matter theory is a quiver with nodes su(n) on (−2), i.e. denote by N − 1 the number of (−2)-curves. In the standard 6d notation 1 , the tensor branch is The non-minimal (D n , D n ) conformal matter is contructed by the following base geometry in 6d F-theory: where there are N − 1 (−2) curves in the middle. In the full tensor branch, the base geometry becomes There are N (−1) curves and N − 1 (−4) curves in the middle.
The non-minimal (E 6 , E 6 ) conformal matter is contructed by the following base geometry in 6d F-theory: where there are N − 1 (−2) curves in the middle.
The non-minimal (E 7 , E 7 ) conformal matter is contructed by the following base geometry in 6d F-theory: where there are N − 1 (−2) curves in the middle.
In the full tensor branch, the base geometry becomes Similarly, the non-minimal (E 8 , E 8 ) conformal matter is given by: where there are N − 1 (−2) curves in the middle.
In the full tensor branch, the base geometry is

Example Geometry: (SO(8), SO(8)) non-minimal CM
We now determine the CFDs for the decoupled theories from the tensor branch geometry.
We exemplify this for one Calabi-Yau threefold geometry, the non-minimal (SO (8), SO (8)) conformal matter. We will discuss the decoupling procedure, CFD and the IR gauge theory descriptions from the geometric perspective. The remaining cases are discussed in the appendix E and are summarized in tables 5 and 6.
The minimal (SO(8), SO (8)) conformal matter theory is equivalent to the rank-one Estring theory with E 8 flavor symmetry. In the resolution geometry, there is a generalized dP 9 (rational elliptic surface) over the (−1)-curve on the base, which has the following set of genus-zero curves 2 : This figure is exactly the (SO(8), SO(8)) marginal CFD in table 8.
With this rational elliptic surface as building blocks, we study the non-minimal (SO(8), SO (8)) conformal matter with N = 2, which has the following tensor branch: Figure 4: The configuration of curves on the seven compact surfaces (U, u 1 , u 2 , u 3 , u 4 , S 1 , S 2 ) in the resolution geometry of (SO (8), SO (8)) N = 2 non-minimal conformal matter theory. Here u i denotes u 1 , u 2 and u 3 , which has the topology of Hirzebruch surface F 2 . . are denoted by S 1 and S 2 , from left to the right.
We plot the configuration of curves on the seven compact surfaces (U, u 1 , u 2 , u 3 , u 4 , S 1 , S 2 ) in figure 4. The two surfaces S 1 and S 2 have exactly the same curve configurations as (5.10).
The surface u 1 , u 2 , u 3 , u 4 are all Hirzebruch surface F 2 and the surface U is the Hirzebruch surface F 0 , as expected in [23].
From this geometry, the CFD vertex corresponding to non-compact surface z is given by the following combination of curves where all the multiplicity factors equal to one. From (2.17), (2.18), it corresponds to a node C z with (n, g) = (0, 0). Similarly, the nodes C y i corresponding to y i (i = 1, . . . To get a 5d SCFT with descendants, we need to decouple the extra SU (2) gauge theory by decompactifying the surface u 4 in figure 4. The surface u 4 will give rise to a new vertex in the CFD after this operation. However, from this geometry we will naively get and g(u 4 ) = 0, which is not allowed in a valid CFD. Moreover, if we want to keep the curves U · u 4 , S 1 · u 4 and S 2 · u 4 compact, since they are parts of the remaining compact surfaces, then all the curves on u 4 are compact. This is because the 0-curves and the (-2)-curve on u 4 generate the Mori cone of u 4 , and the decompactification of u 4 will not be allowed [50]. To resolve this issue, we need to flop the curves z · S 1 and z · S 2 on S 1 and S 2 into u 4 , which results in the following geometry Now the surface u 4 has two more Mori cone generators u 4 · v 4 and u 4 · w 4 , which can be made non-compact. Then there is no issue in decompactifying u 4 . In the general case of non-minimal conformal matter, this flop should always happen before the decompactification, as expected in the field theory analysis in section 3.
Since all the multiplicity factors are trivially one, and we can read off the corresponding CFD (the letters label the corresponding non-compact surfaces) In the geometry (5.15), we can assign the following P 1 rulings on each surface component: With this assignment, the surfaces U, u 1 , u 2 , u 3 will form the Cartans of an SO(8) gauge group after the above ruling curves are shrunk to zero size, while S 1 and S 2 gives rise to two SU (2)s.
We hence have the following quiver gauge theory description: Although the geometry (5.16) does not apparently have an SU (4) × SU (2) 3 quiver gauge theory description, we can do a few flops to get it. We shrink the curves y i · S 1 (i = 1, 2, 3) on S 1 and y i · S 2 (i = 1, 2, 3) on S 2 , and consequently blow up the compact surfaces u i (i = 1, 2, 3) two times for each. After the six flops, the curve configurations are The assignment of section/rulings is shown in the figure explicitly. We can hence read off the following quiver description, where each letter in the bracket denotes the Cartan node of the Finally, we can generalize this story to higher N , with more (−1) and (−4) curves in the tensor branch: The resolution geometry then would become ....
where S 1 , . . . , S N denotes the N rational elliptic surfaces over the (−1)-curves, and U (k) , u 4 (k = 1, . . . , N − 1) are the compact surfaces corresponding to the k-th affine SO (8). In this case, the curves z ·S i and y j ·S i for i = 2, . . . , N −1, j = 1, . . . , 3 have non-trivial multiplicity factors ξ S i ,z = ξ S i ,y j = 2, because they are O(−1) ⊕ O(−1) curves which intersect two other compact surfaces. Hence the correct n(z) and n(y j ) are computed as: The resulting marginal CFD is exact the same as the N = 2 case (5.13), which has no descendant. In the flop and decoupling process to get a 5d SCFT, we we make all the surfaces u (k) geometry: (1) ....
The CFD is exactly given by the (D 4 , D 4 ) row in table 5. The superconformal flavor symmetry , which is consistent with [46].
However, in this geometry, there is no consistent assignment of rulings that give rise to a weakly coupled quiver gauge theory We will also explain why this cannot happen from a field theoretic point of view in section 5.4. The reason is that on the middle surfaces S k (k = 2, . . . , N − 1), we need to assign the following linear combination of curves as the ruling Although they are both curves with self-intersection number zero and genus zero, they mutually intersect at two points. Hence they cannot both be the ruling curve of a P 1 fibration structure. This point was already discussed in section 3.4. Nonetheless, the theory will have a strongly coupled quiver description with SO(6) × SO(6) classical flavor symmetry, which will be discussed in section 5.4.    Table 5: CFDs for non-minimal N (G, G) conformal matter. The left hand picture shows the CFD before decoupling, the right hand one after.

(G, G)
CFD before decoupling CFD after decoupling Table 6: CFDs for non-minimal N (E 7 , E 7 ) and (E 8 , E 8 ) conformal matter. The left hand picture shows the CFD before decoupling, the right hand one after. The CFDs of the (D 4 , D 4 ) case have already been derived in the previous section, and we will present the geometric derivation of (D n , D n ) and (E 6 , E 6 ) cases in appendix E. The (A n−1 , A n−1 ) type case follows from the geometry of the single node SU (n) on a (−2) gauge theory that we derive in appendix A. The CFD after decoupling will be derived in [57] using toric methods. For the cases (E 7 , E 7 ) and (E 8 , E 8 ), we also derive the (n, g) = (−2, 0) and (−1, 0) vertices from the geometry in appendix E.
Given these non-minimal conformal matter CFDs, as well as the quiver structure of the 6d parent theory, it is natural to wonder, whether there is a gluing construction for CFDs. We will return to this in section 6, where we propose building blocks for CFDs and gluing rules. In this context we will re-derive the CFDs for the (D n , D n ) and (E 6 , E 6 ) non-minimal conformal matter theories lower rank theories in section 6.4. In this context we also give evidence for the CFDs of (E 7 , E 7 ) and (E 8 , E 8 ).

Low-Energy Descriptions and Dualities
We will now describe the possible low-energy effective descriptions of the 5d SCFTs and corresponding geometries discussed in this section. We also remind that not all the effective theories will be weakly coupled. For instance, it will sometimes be necessary to introduce strongly coupled matter, e.g. the 5d analog of conformal matter. In fact, it can happen that the non-perturbative part of the flavor symmetry is gauged. For example a subgroup, H, of the superconformal flavor symmetry, G 5d F , has to be gauged and, in particular, it contains the U (1) T associated to the gauge vector, Geometrically this corresponds to two surface components S 1 , S 2 intersecting along C 12 , which is a section for the ruling of S 1 and a fiber for the ruling of S 2 .
A straightforward set of examples is given by the 5d theories originating from the circle reduction of 6d theories, which are single curve with a gauge groups in the tensor branch.
Upon decompactification, or decoupling, we get exactly the 5d analog of the 6d gauge theory in the tensor branch. For instance, the geometry corresponding to the NHC su 3

(5.28)
consists of three F 1 intersecting along (−1) curves. Decompactifying one of the surfaces leads to two F 1 intersecting along the (−1) curve, and this geometry exactly corresponds to the SU (3) 0 theory, as we already seen from the CFD prospective in section 4. This procedure applies also to the rest of the circle reductions of theories, which in the tensor branch are single curve with self intersection (−n) and n > 1.
We now list some of the possible low-energy descriptions of 5d SCFTs coming from decompactification of the geometries corresponding to the 6d non-minimal conformal matter, which are determined by embedding the BG-CFDs into the CFDs. In order to construct these dual IR theories of the same UV SCFT, we will sometimes need to locally dualize gauge nodes of known quiver theory description, and in particular we will use the following duality, which descend from higher rank (D n , D n ) conformal matter theories [1].
In addition, we will obtain some description with maximum amount of flavor matter. The descendant 5d SCFTs are obtained from matter mass deformation, which consists of decoupling the flavor hypermultiplets in the IR gauge theory descriptions. Their superconformal flavor symmetries can be straightforwardly read off from the CFD transition, i.e shrinking (−1) vertices.
A-Type non-minimal conformal matter As already explained a 5d SCFT can be obtained from (SU (n), SU (n)) non-minimal ((N > 1) conformal matter upon decoupling of the extra gauge theory. We can deform the SCFT and study the theory in the IR, which can be a quiver gauge theory. The embedding of the classical flavor symmetries are shown in figure 5a. Two weakly coupled descriptions are pretty manifest, and the 5d SCFT in the UV after mass deformation leads in the IR to [46]: ...
..  where the first one has been already discussed in section 3 as decoupling of the affine SU (N ) quiver node, and matches the first BG-CFD embedding in figure 5b, whereas the second one corresponds to the 5d copy of the partial tensor branch quiver. The links cm − are the first descendant of (SO(2n), SO(2n)) conformal matter KK-theory [1]. We notice that this 5d low-energy effective description has already some strongly coupled sectors. At the interior of the quiver the matter cannot have a direct weakly coupled description. This is due to the fact that the full superconformal flavor is gauged by SO(2n) × SO(2n). More precisely also the non-perturbative topological symmetry of a putative gauge theory description is also gauged.
On the other hand, at the two tails there is still an global [SO(2n)], and in fact the IR theory is also described by the quiver:  The 5d SCFT from non-minimal (E 6 , E 6 ) conformal matter has the two dual low-energy descriptions: The first one is again given by the decoupling of the affine gauge node vector multiplet, and matches the second BG-CFD embedding in figure 5c. The second description is the the 5d analog of the 6d tensor branch after decompactification, where the links are given by the first mass deformation of the KK-theory coming from straight circle compactification of N = 1 (E 6 , E 6 ) conformal matter [2]. In the interior the link do not have a direct weakly coupled description in terms of gauge theory, since the full superconformal flavor symmetry is gauged.
In this case, also for the tails of the quiver we cannot have a complete description in terms  Non-minimal (E 7 , E 7 ) conformal matter on a circle and after decoupling of the extra gauge theory and mass deformation has the following dual low-energy descriptions: In the spirit of [3], we propose a description of the 5d conformal matter at the two tails, which has a weakly coupled part compatible with the classical flavor symmetry embedding into the CFD in figure 5d. That is given by a gauge theory with some matter hypermultiplets, which is also coupled to a residual strongly coupled theory, * − transforming in Sp(n 1 ) × E 7 or  The 5d SCFT descending from non-minimal (E 8 , E 8 ) conformal matter can be described in the IR by the following dual theories: The first theory comes from decoupling the gauge vector of the affine node of the affine quiver, and the classical flavor symmetry U (N ) shows that is corresponds to the first BG-CFD embedding in 5e. The second one is the 5d analog of the partial tensor branch quiver where we have strongly coupled conformal matter transforming under E 8 × E 8 . This matter is the first descendant of the KK-theory coming from circle reduction of N = 1 (E 8 , E 8 ) 6d conformal matter.
Similarly to [3], since only a single E 8 has been gauged, we propose a partial weakly coupled description which is compatible with the CFD in figure 5e. That is 6 Gluing CFDs from Building Blocks

Building Blocks and Gluing
Any 6d SCFT, in particular NVH theories, in its partial tensor branch can be seen as a generalized quiver [20], where the nodes are given by where n = Σ 2 ≤ −1 is the self-intersection number of a compact rational curve Σ. Over Σ, the elliptic fiber can be singular, which is associated to the gauge group G in 6d. There can be matter hypermultiplet transforming under the flavor symmetry G 6d F . The matter can be either given by standard (half) hypermultiplet, or by VH 6d SCFTs with where * − the notation means that the link is non-conventional matter and it has a G 6d F i × G 6d F j manifest flavor symmetry. An important class of examples of this type is the minimal is connected to a node T 6d (G, G 6d F ) by gauging the flavor symmetry G 6d F i , which should be exactly identical to G of the T 6d (G, G 6d F ). Repeating this procedure leads to the generalized quivers of [20]. In this way we can construct general 6d tensor branches, whose origin corresponds to a 6d SCFT.
We implement a similar strategy in 5d based on the M-theory geometry. The building blocks are defined by the resolution geometries associated to the tensor branch building blocks in 6d: F ), which in fact corresponds to matter. If n < −1, then we need to decompactify one compact surface in the M-theory geometry, in order to decouple the extra SU (2) gauge theory.
where a decoupling occurs, we first need to mass deform H 5d ij (G i , G j ) before the gluing. In the corresponding M-theory geometry, we flop a curve out of the reducible surface. This geometric transition is usually necessary to decouple the U (1) T of the extra gauge theory when we start from the 6d tensor branch, The bottom-up construction of 6d SCFTs is guided by the definition of a consistent tensor branch with a superconformal fixed point at its origin. Inspired by the tensor branch geometries, we propose a set of rules which allow us to glue the geometries associated to S 5d (G, G 6d F ) and H 5d ij (G 6d F i , G 6d F j ). Furthermore, we propose a gluing rule for CFDs, which then allows determining the 5d superconformal flavor symmetries through a gluing. The input for this construction are the geometries/theories/CFDs for building blocks that are descendants of simple constituent of the 6d tensor branch. A class of these building blocks are the circle-reduction of single curve tensor branches, as listed in [30,33]. In appendix A, determine and summarize these constituents and their CFDs, including a single (−1)-curve, gauge group on single curves (including NHCs) and minimal conformal matter. This is not a comprehensive list of building blocks, e.g. we do not consider those 6d tensor branch geometries where the flavor symmetry is not manifest. The building blocks that we computed in appendix A are single gauge node components, where the flavor symmetry is manifest as well as minimal conformal matter. These will then be used to propose a gluing construction.

CFDs from Gluing
We now propose a gluing rule on CFDs, which proceeds in two steps: first we gauge a common flavor symmetry, and then define how to combine the CFDs. Suppose that we have already constructed the CFDs for S 5d (G, G 6d F ) and H 5d ij (G 6d F i , G 6d F j ) S 1 , which are CFD (1) and CFD (2) . Denote their vertices by C and then remove all the vertices C (i) α from both CFDs.
The (n, g) of such linear combinations needs to satisfy the following "gauge conditions": The reasoning is that each C (i) α can be considered as a linear combination of curves/vertices Then v i can be written as the following curve in the Calabi-Yau threefold Now assume that all the weight factors ξ j,α = 1 identically, then this is a well-defined Thus we can identify the two curves v 1 and v 2 with the following gluing condition In general, if the building block S 5d (G, G 6d F ) has non-Abelian flavor symmetry G 6d F , then we need to choose two sets of v 1 and v 2 to gauge both G 6d F and G 6d

Combine:
After we have removed the vertices C (i) α , we need to connect the remaining parts of the two CFDs. We define the set of vertices connected to C where µ (i) β is a "weight factor" or multiplicity of vertices in the CFD, which appears in certain gluing processes. This is an analogy of the weight factor ξ i,α of curves in each surface component. We will give concrete examples of this in the following. Finally, after the vertices in S 1 and S 2 are combined pair-wise, the other parts of the two CFDs connecting to S 1 and S 2 remain the same. This gluing is motivated by the geometric structure that we observe in higher rank theories.
We will now exemplify it with gluing of NHCs and E-strings, as well as higher rank conformal matter theories, and show that it provides a consistent framework.

Example: (−1)-NHC Quivers
In this section, we present the simplest example of the gluing philosophy, which is a single 6d Quiver 6d Tensor Branch CFD  Table 7: CFDs for 5d theories obtained by reduction from 6d quivers with two nodes (−1) − (−n), where (−n) corresponds to an NHC. The tensor branch geometry is shown in the middle and the CFDs in the right-most column.
In these cases, after blowing down the (−1)-curve on the tensor branch, we always end up with a curve with self-intersection −n + 1 ≤ −2. Hence the 5d KK theory has an SU (2) vector multiplet associated to it, which we need to be decoupled. Geometrically, we need to decompactify the surface associated to this, and the CFDs can be derived from directly in appendix D. We summarize the results in table 7.
From the perspective of CFD gluing, it is useful to pick a convenient representation of the rank-one E-string marginal CFD in table 9, such that the apparent flavor symmetry is G × H.
In other words, the rank-one E-string theory can be thought as a rank-one (G, H) conformal matter, which acts as the link theory H 5d ij (G, H). After the decoupling process, the marginal CFD needs to be flopped once, as we have discussed before. The actual building blocks should be this "sub-marginal" CFD with G F = G × H 5d flavor symmetry and the CFD of the NHC after decoupling in table 4.
We summarize the gluing process of these two building blocks for n = 3, 4, 6, 8, 12 in figure   6. In order to satisfy the condition for decoupling in section 3.3, we first need to flop one of the (−1) curves in the E-string CFD. This is the first step figure 6. Then we identify the curves that we use to gauge a flavor symmetry. The combinations of curves involved in the gauging part are encircled in yellow, and the vertices that get combined are colored orange in the gluing process. As one can see, the orange vertices are matched pair-wise, and they are never flavor vertices in the building block CFDs. The details of matching the orange vertices should be read off from the geometry in appendix D. However, in many cases we observe that the discrete symmetries of the CFDs select which curves (orange) need to be combined.
It would be interesting to understand better the role of these discrete symmetry and their interplay with the 6d and 5d flavors. Note that the multiplicity factors µ in (6.9) are always trivially one in these cases.

Non-Minimal Conformal Matter from Gluing
Another class of theories that can be studied also from the gluing, are the higher rank conformal matter theories. The gluing of the (A n−1 , A n−1 ) will be discussed in [57], where using a toric description it will be even simpler. The first interesting non-trivial case to consider is (D 4 , D 4 ) non-minimal conformal matter. We already discussed the geometry of the tensor branches as well as the CFDs from the geometry of the tensor branch plus decoupling in section 5.
We can also get (D 4 , D 4 ) N = 2 non-minimal conformal matter by gluing two rank-one E-string the Σ 2 = (−4) NHC (see table 4). The CFD for rank-one E-string is taken to be the one with explicit G i × G j = SO(8) × SO (8). We again first take the descendants of the Below the graphs we shown the 6d tensor branch quiver building blocks. The resulting CFD is in agreement with the one derived directly from the geometry in section 5.
To obtain higher N , we iterate this process as follows . (6.12) Note that the (−1) vertices in the middle building block have µ = 2 multiplicity. The gluing for the general (D k , D k ) non-minimal conformal matter theory works along the same logic.
We should now comment on the matter of the multiplicities that are key in the gluing: one might think that determining this requires considering the full resolved geometry as in section 2.5. However, we will be able to extract a relatively simple rule, from considerations of the tensor branch structure. The main point is that the multiplicity µ of the curves (6.9) used to combine the CFDs, the orange-colored curves, has contributions from both the multiplicities ξ in the building block as well as from the surfaces that are getting compactified in the gauging.
For the building blocks in this paper, only the minimal conformal matter building blocks in appendix A.3 have non-zero intrinsic ξs. For the other simpler building blocks, we have ξ = 0.
Let us consider a set of CFD vertices {C β , β ∈ Φ gauge }, where Φ gauge is the set of roots that we gauge. They correspond to the set of non-compact surfaces that get compactified in the gluing process, i.e. these are associated with the Cartans of the flavor symmetry that is getting gauged. The multiplicity of the orange curves gets modified, as they intersect these surfaces, according to the (2.23). In general, the new multiplicity µ is computed with the following formula: where ξ(C β ) is the multiplicity factor intrinsic to a (n, g) = (−1, 0) vertex C β of the building block CFD (e.g. for (E 6 , E 6 ) there are multiplicity ξ = 1 curves, see appendix A.3). Note that m βγ is the number of edges between the vertices C β and C γ in the CFD.
For instance in (6.12), we observe that the gluing procedure requires to identify the middle orange (−1) vertices, as well as the yellow SO(8) curves on the left, on the right and the (0) in the middle, which are then removed by gauging. In the case of (D k , D k ) conformal matter, there is no intrinsic multiplicity factor ξ. Then the multiplicity (6.13) of a (−1) in the gluing procedure is simply given by the number of its adjacent vertices that are gauged (marked yellow).
Let us apply this to the N = 2 (E 6 , E 6 ) non-minimal conformal matter theory. This is glued from two minimal conformal matter theories along an NHC with Σ 2 = (−6), which results in su (3) . (6.14) Here the (−1) gluing nodes have multiplicity µ = 2, following the general rule stated above.
Again the CFD is the one we obtained from a direct computation in the geometry in section 5.
We can iterate this and obtain the N = 3 (E 6 , E 6 ) from gluing as follows -note the additive nature of the multiplicity µ: su (3) [ Finally, we consider the case of the (E 7 , E 7 ) and (E 8 , E 8 ) for which we earlier conjectured the CFDs. Although we will require a detailed knowledge of the geometry to compute the multiplicity factors, which are key to deriving the labels of the unmarked vertices with n > −1, we can determine the part of the CFD, that encodes the superconformal flavor symmetry (marked vertices) as well as the mass deformations (i.e. the (−1) vertices). For (E 7 , E 7 ) theories with N = 2 the tensor branch suggests the following gluing of two minimal conformal matter theories of type (E 7 , E 7 ) with the NHC with Σ 2 = −8 in table 4 The multiplicity factor is µ = 3. Likewise for the (E 8 , E 8 ) theory with N = 2, we glue two minimal conformal matter theories with the NHC Σ 2 = −12 to obtain The multiplicity factor here is µ = 5. Note that these multiplicity factors can be exactly seen from the resolution geometry after the flop and decompactification, see Appendix E.3 and Appendix E.4.

Conclusions and Outlook
In this paper we investigate the possible ways of getting 5d superconformal field theories (SCFTs) coming from 6d on a circle. In general, the circle reduction of a 6d SCFTs leads to a KK-theory, which in the UV completes back into the original 6d theory. To obtain a genuine 5d SCFT we need to consider mass deformations of the KK-theory, or equivalently, holonomies in the flavor symmetry. There are two type of possible mass deformation which we studied, which lead to 5d SCFTs: 1. The first corresponds to in the gauge theory description in the IR to the decoupling of matter hypermultiplets. In the M-theory Calabi-Yau geometry this corresponds to flopping the associated (−1)-curves out of the compact surfaces of the geometry.
2. The second one is more drastic and require the decoupling an entire sector of the theory, like a gauge vector multiplet. In geometry this corresponds to the decompactification of some surfaces.
In [1][2][3], we mainly focused on the so-called very Higgsable theories, where the natural mass deformations are those of the first type: giving masses to the hypermultiplets. We determined the starting points for such 5d RG-flows, and encoded these in the CFDs, which enabled tracking the complete tree of descendants and their superconformal flavor symmetries.
In this work we focused on the exploration of the second possibility, in particular it turns out that for many not very Higgsable theories the first possibility is not an option if we want to get a single unfactorized 5d SCFT, and the second approach is unavoidable. We prove that the decoupling of an entire sector can be necessary for instance if the starting point is a 6d SCFT single curve with gauge group theory in the tensor branch, as is the case for non-Higgsable clusters (NHCs). In addition, we studied the circle-reduction of non-minimal conformal matter theories, i.e. the 6d theory of N > 1 M5-branes probing an ADE singularity. We show that by decoupling an SU (N ) gauge theory, which geometrically correspond to decompactifying resolution surfaces, we obtain 5d SCFTs of arbitrary rank. In particular, the geometries describing the Coulomb branched of these theories present very interesting features. We characterize these theories in terms of CFDs, which again encode flavor symmetries, mass deformations, descendant structure, and BPS states. We did not study in detail the descendants for these theories, but they are easily accessible by applying the descendant rules.
Finally, inspired by the 6d classification, we propose a gluing procedure in order to get higher rank 5d SCFTs from lower rank building blocks. We first define the building blocks, which are the single node tensor branch theories (which are not very Higgsable), and the minimal conformal matter theories. We then use the tensor branch resolution of the 6d SCFT, to motivate the gluing rules, and cross-check these against direct computations for simple quivers and non-minimal conformal matter theories.
It would be interesting to generalize this gluing procedure further, in order to capture the vast landscape of 5d SCFTs which originate from 6d on a circle. In particular we did not consider the building blocks in 6d [58], which do not have the manifest 6d superconformal flavor symmetry realized geometrically. It would be interesting to compute the CFDs for such models, and consider the BPS states to determine the 5d CFDs and flavor symmetry enhancements in those cases as well.

A.1 Rank 1 E-string Building Blocks
In the 6d tensor branch descriptions, a single (−1)-curve by itself corresponds to the rank-1 E-string theory with flavor symmetry G F = E 8 . In the tensor branch resolution geometry, the compact surface S will be a rational elliptic surface (generalized dP 9 ) over the base (−1)curve. Nonetheless, if there are two curves with simple gauge groups g 1 and g 2 connected to it: then the surface S serves as a connection surface between the Cartan divisors of g 1 and g 2 . On the rational elliptic surface S, there are degenerate elliptic fibers corresponding to affine Lie algebraĝ 1 andĝ 2 . Namely, the (−2)-curves on S form the affine Dynkin diagram ofĝ 1 ×ĝ 2 , and there are (−1)-curves connected between them.
In principle, for any g 1 ⊕ g 2 ⊂ E 8 , such a rational elliptic surface exists. However, only for a subset of (g 1 , g 2 ), the number of (−1)-curve on the rational elliptic surface is finite. These surfaces are called "extremal" and has been classified in [59,60]. In this paper, we only list the ones that will be used in the later gluing discussions, which happens to satisfy the extremal Table 8: CFDs for the marginal rank 1 E-string obtained by different collisions of G 1 and G 2 singularities. The maximal manifest flavor symmetry is realized only in the (E 8 , ∅) model. For all other the manifest flavor symmetry is G 1 × G 2 , and enhances toÊ 8 by including the additional BPS states. However these different realizations are useful for the gluing process.
criterion. They can be generated by blowing up the generalized del Pezzo surfaces in [61] or putting together set of (−2)-curves. These are summarized in table 8 In some cases where decoupling happens, the rational elliptic surface needs to be blown down to gdP 8 . The set of curves are transformed according to the usual rule of shrinking (−1)-curves.

A.2 Single Curves with Gauge Group
In this section, we discuss the building block of a single curve with a (tuned) non-Abelian gauge group on it, which is not an NHC. The flavor symmetry and F-theory realization of such building blocks are discussed in [58]. We focus on the cases where the flavor symmetry is identical in Table 2 and Table 3  The 6d global symmetry is SU (2n), and the tensor branch can be chosen as: where 0 ≤ m ≤ n. After the decompling process, the 5d gauge theory description is SU (n) 0 + 2nF , which is a descendant of (D n+2 , D n+2 ) conformal matter. For m = 0, the CFD is a descendant of the (D 2n+4 , D 2n+4 ) CFD tree in [1]. The CFDs before and after decoupling are ...
The 6d global symmetry is Sp(2n − 8), and the tensor branch can be chosen as: where 0 ≤ m ≤ n − 4. After the decoupling process, the 5d gauge theory description is When m = 0, the CFD can be read off from the geometry in appendix C.3. The CFDs are summarized in table 11.

A.3 Minimal Conformal Matter
Another useful class of building blocks is the minimal conformal matter where the 6d tensor branch have rank higher than one. For many of these theories, their marginal CFDs have been constructed in [1][2][3]. Here we summarize their marginal CFDs in table 12. We are going to shortly discuss the subtleties associated to non-trivial intrinsic multiplicities involved in the gluing section 6.4 and non-simply laced Lie algebra.

(E 8 , E 8 ):
In this case, there is a non-trivial intrinsic multiplicity factor ξ = 4 for the (n, g) = (−1, 0) vertex that connects the two affine node ofÊ 8 . In the sub-marginal CFD generated after one CFD transition, the two new (−1, 0) vertices both have ξ = 4 as well.
4. (E 8 , SU (2k)): Similar to the rank-two E-string case in section 2.5, one can choose to draw a "green (−1)-node" that still contributes to the non-Abelian flavor symmetry. Alternatively, one can draw a (n, g) = (−2, 0) node instead, but the (n, g) = (−1, 0) node will become (7)) an "interpolating node" that connects to the E 8 node at one edge but connects to the SU (2) node at two edges.
It connects to the E 7 node with one edge but connects to the SO(7) node with two edges. After the CFD transition where it is removed, the E 7 node above it will become an (n, g) = (−1, 0) node, while the SO(7) node below it becomes an (n, g) = (−2, −1) node.

B Geometry for NHCs
For a single (−4) curve, the 6d non-Higgsable gauge group is SO (8). In the marginal geometry, the five surface components are arranged as: Here u 4 denotes the affine node.
We plot the curve configurations on the surfaces here: V , z and y i (i = 1, . . . , 3) are non-compact surfaces. Before the decoupling, the CFD is read off to be: where the middle node with genus g = 1 corresponds to the non-compact surface V and the four (+2)-nodes correspond to z and y i .
To get a 5d SCFT, we decompactify the surface component u 4 . Then the remaining compact surfaces are three F 2 with one F 1 , with the following triple intersection numbers: The compact surface components are connected via the sections of the P 1 fibration on the Hirzebruch surface components, and there are no O(−1) ⊕ O(−1) curves in the geometry.
From this, we conclude that the 5d gauge theory description is a pure SO(8) gauge theory. We can read off the following CFD from the geometry, where the three (+2)-curves correspond to y 1 , y 2 , y 3 and the 0-curve corresponds to V : For a single (−5) curve, the 6d non-Higgsable gauge group is F 4 . In the marginal geometry, the five surface components are arranged as: U is the affine Cartan divisor that will be decompactified.
The curve configuration is Note that on u 3 and u 4 , the intersection curve with non-compact divisor V is a reducible 0-curve with two identical fiber components. V , z and y are non-compact surfaces. Before the decoupling, the CFD is given by: where the (+3)-node corresponds to z, the middle (n, g) = (0, 1) node corresponds to V and the (+8)-node corresponds to y. From the geometry of compact surface u 4 in (B.7), we can clearly see that the y-node should connect to the V node via a double line.
After we decompactify U , the remaining compact surfaces have the following triple intersection numbers: After the ruling curves on u i (i = 1, . . . , 4) are shrunk to zero size, the intersection matrix between the ruling curves and surface components on each surface component are: which is the (−C) ij of Lie algebra F 4 . Hence the 5d gauge theory description of this geometry is a pure F 4 gauge theory. The CFD can be read off as follows, where the (+1)-curve corresponds to U , the 0-curve corresponds to V and the 8-curve corresponds to y: For a single (−6) curve, the 6d non-Higgsable gauge group is E 6 . In the marginal geometry, the seven surface components are arranged as: U is the affine Cartan divisor that will be decompactified.
The curve configuration is (B.13) Before the decoupling, the CFD is read off as: (B.14) where the middle node corresponds to the non-compact surface V and the (+4)-nodes correspond to non-compact surfaces x, y, z.
After we decompactify U , the remaining compact surfaces have the following triple intersection numbers: As one can see, U , u 1 , u 2 , u 3 and u 5 are F 5 , F 3 , F 1 , F 1 and F 4 respectively, while u 4 , u 6 and u 7 are blow ups of Hirzebruch surfaces. They are arranged into the affine E 7 Dynkin diagram as: where U is the affine node. Before the decoupling, the CFD is read off as: where the middle node corresponds to V , the (+5)-node corresponds z, the (+7)-node corresponds to y 1 and the (+2)-node corresponses to y 2 .
After we decompactify U , the remaining compact surfaces have the following triple intersection numbers: The ruling structures of the surface components are The 5d gauge group is E 7 after the ruling curves are shrunk to zero size. In this case, there are three O(−1) ⊕ O(−1) curves in the ruling: Their charge C i · u j under the Cartans of E 7 are weight vectors in the 56 representation of E 6 . However, we cannot flop them out of the compact surface, since the surface components u 4 , u 6 and u 7 cannot be blown down twice. From these information, we conclude that the 5d gauge theory description should be E 7 + 1 2 56. The CFD can be read off as: (B.23) For a single (−8) curve, the 6d non-Higgsable gauge group is E 7 and there is no matter field.
In the marginal geometry, the eight surface components are arranged as: The configuration of curves are: (B.25) Before the decoupling, the CFD is read off as: where the middle node corresponds to V , the (+2)-node corresponds to y 2 and the (+6)-nodes correspond to z and y 1 .
After we decompactify U , the remaining compact surfaces have the following triple intersection numbers: . (B.28) For a single (−12) curve, the 6d non-Higgsable gauge group is E 8 and there is no matter field.
In the marginal geometry, the nine surface components are arranged as: The configuration of curves are: Before the decoupling, the CFD is read off as: where the middle node corresponds to V , the (+10)-node corresponds to z, the (+2)-node corresponds to y and the (+4)-node corresponds to x.
After we decompactify U , the remaining compact surfaces have the following triple intersection numbers: Note that the intersection curve U · v 3 is reducible, with two (−1)-curve components. Note that this collection of surfaces does not satisfy the shrinkability condition [29], because there exists a genus-one fibration structure where the singular fiber is a ring of three P 1 s and the sections are the intersection curves U · U 1 , U · U 2 , U 1 · U 2 .
To get a 5d SCFT geometry, we need to either flop curves out of these compact surfaces or decompactify a surface component. The former choice is only possible if we shrink the curve U 1 · U 2 and results in the same surface geometry U with two P 2 s. However, this geometry has no gauge theory description either, since the surface P 2 does not have a ruling structure.
For the latter choice, we can flop curves out of U and then decompactify U , and we consequently get a theory with more descendants. We first shrink the two (−1)-curves that consist of U · v 3 , and then shrink U · v 2 , U · v 4 , U · v 1 and U · v 5 consequently.
In this process, U is blown down six times, while U 1 and U 2 are blown up three times for each. The final surface geometry after the process is: In this geometry, the surface U is F 0 and U 1 , U 2 are two identical gdP 4 s. After U is decompactified, the (−2) curves U · u 1 and U · u 2 are actually unrelated. Similarly, the (−1)curves V · u 1 and V · u 2 become independent, since we can shrink one of these (−1)-curves without changing the geometry of the other surface component. Hence in this case, these curves should not be combined, in contrary to the usual rule of extracting CFD from the geometry. The only combined curves are v 3 · u 1 and v 3 · u 2 , and the CFD is read off as: which is exactly the CFD of the 5d rank-2 gauge theory SU (3) 0 + 6F [2].
Similarly, we can study the resolution geometry for the equivalent tensor branch The intersection U · v N is reducible. We can flop curves out of the affine Cartan divisor U

2N times by shrinking
As a result, the surface components U 1 and U N −1 are blown up N times for each, and the resulting surface geometry is After U is decompactified, the curves V · U 1 and V · U 2 can be independently shrinked.
Similarly, U · U 1 and U · U 2 becomes independent curves. The CFD is then

C.3 SO(2n) gauge group on (−4)-curve
We study the resolution geometry of the tensor branch: We use the resolution sequence of SO(2n) in [62], and the resolution sequence for Sp(2n − 8) is given by: Denote the Cartan divisors of SO(2n) by U, U 1 , U 2 , . . . , U n , the configuration of curves on each surface component are: Note that the curves with g = −1 are a double copy of a rational curve on the surface.
Moreover, the intersection curve U n−1 · U n consists of (2n − 8) copies of rational curve. The intersection relations among surface components are: which is consistent with the geometric picture in [31].
To get a 5D SCFT, we need to decompactify the surface U , and the CFD can be read off as the one in table 11. Note that the vertices V, V 1 , . . . , V 2n−8 form the BG-CFD of Sp(2n−8), see table 1. The nodes with n > 0 are given by U , y 1 , y 2 and y 3 .
We can similarly work out the resolution geometry for as well.

D (−1)-NHC Gluing From Geometry
In this appendix we derive the CFDs from the tensor branch geometry. by U, u 1 , u 2 . The curve configurations on the compact surfaces are: In this case, to circumvent the multiplicity factor subtlety, we do a flop on (D.1) by shrinking the single intersection curve among the surfaces U, u 1 and u 2 . Consequently, the surface S is blown up at the point where three curves U · S, u 1 · S and u 2 · S intersect. The curve configurations after this flop are: To get a valid 5d theory, we need to flop the curve z ·S on S into U , and then decompactify U . This leads to the following curve configurations on the remaining compact surfaces: Hence we can read off the following CFD: The rank-3 5d theory has G F = E 6 × SU (2) superconformal flavor symmetry and an IR quiver gauge theory description of: The BG-CFD of SU (2) + 4F can be embedded in (D.4).

(−1)(−4): G = SO(8)
The curve configurations in the tensor branch resolution are S is the vertical divisor over the (−1)-curve, which is a gdP 9 . We flop the curve z · S on S into U and then decompactify U , which result in the following curve (D.8) The CFD can be read off as (D.9) The resolution geometry is similar to the G = SU (3)  (D.10) Similar to the previous cases, we shrink the curve z ·S on S and decompactify U . The resulting CFD is going to be: In the resolution geometry, we denote the compact Cartan divisors of E 7 by U, u 1 , . . . , u 7 , the compact vertical divisor over (−1)-curve by S and the non-compact Cartan divisors of SU (2) by V, v 1 . The curve configurations on the compact surfaces are: After we shrink the curve z · S on S and decompactify U , the resulting CFD is: (D.13) In this case, there will not be any non-Abelian flavor symmetry on the non-compact curve.
We just have a resolution geometry of a gdP 9 glued with the exceptional divisors of a II * Kodaira singularity. The curve configurations are: . (D.14) After the shrinking of curve z · S on S and decompactification of U , we just get a CFD with no (−1) or lower node: The tensor branch of non-minimal (D n , D n ) (n > 4) conformal matter is given by: When N = 1, the theory is the minimal (D n , D n ) conformal matter, with the equivalent descriptions of marginal CFD in table 9.
In the tensor branch (KK) resolution geometry, we label the Cartan divisors of each SO(2n) by U (j) and U   curves V (j) · z, V (j) · y 1 and V (j) · y 2 has non-trivial multiplicity two, and the curves V n−4 has multiplicity two as well.
Hence in the CFD from this geometry, the vertices z, y 1 , y 2 and y 3 have: The CFD, see tables 5 and 6, then does not have any possible transitions to a 5d SCFT descendant. To get a 5d SCFT, we shrink the curves z · V (j) for j = 1, . . . , N and then decompactify the surfaces U (j) for j = 1, . . . , N − 1. The configuration of curves on the new compact surfaces is plotted in figure 8.
The curves with multiplicity two are still V (j) ·y 1 , V (j) ·y 2 and V (j) n−4 ·y 3 with j = 2, . . . , N −1, and we can read off: n(U (j) ) = (U (j) ) 2 · (V (j) + V (j+1) ) = −2 (j = 1, . . . , N − 1) ,   Figure 8: The configuration of curves in the decoupled and flopped geometry of figure 7. Note that the decompactified surfaces U (j) has been removed. Then we plot the curve configurations on each surface components in figure 9, where x, y and z corresponds to non-compact divisors. For the surfaces Q (k) , q (k) 1 , q (k) 2 label k goes from 1 to N , and we effectively have As we can see, the surfaces S i are gdP 9 s withÊ 6 and SU (3) singular fibers, and the configuration of Mori cone generators is exactly given in the table 8. The divisors Q (i) , q are Hirzebruch surface F 1 sharing a common O(−1) ⊕ O(−1) curve. In this geometry, the multiplicity factor of the intersection curves between the compact surfaces with x, y, z can be computed from the procedure (2.24). For z, obviously the intersection curves on the noncompact surface z form the following chain, which is exactly the same as the tensor branch base geometry: Then we try to shrink this chain of curves by blow down the (−1)-curves in the middle of the chain, and finally we get: . (E.5) Now we assign weight factor one to all the curves in the chain above, and blow up back to the original chain (E.4). In the process, the weight factor of a new (−1)-curve is given by the sum of its two neighbors. Finally, we get all the weight factors for the chain (E.4), which is labeled above each curve: Then for the vertex z in the CFD, we can compute: for any N .
Since there is a permutation symmetry among x, y and z, we can carry over the same analysis to x and y, and compute n(x) = n(y) = 0 as well. The CFD is then given in tables 5 and 6, with no descendants.
In the decoupling process, we first flop all the (−1)-curves z · S i on each S i into the surface components Q (k) , and then shrink the (−1)-curves z · Q (k) . After these flops, we decompactify the divisors U (i) , (i = 1, . . . , N − 1). Finally, we get the following configuration of curves on the compact surfaces in figure 10.
The CFD will contain vertices V , v i , W , w i , x, y and U (k) for k = 1, . . . , N − 1. Note that the O(−1) ⊕ O(−1) curves U (k) · S 2k and U (k) · S 2k+1 have multiplicity two as well, which can be derived from the chain of curves on U (k) : Note that the decompactified surfaces U (j) has been removed.
(E.9) and we can read off the CFD in table 5 and 6, with G F = E 6 × E 6 × SU (N ).

E.3 Non-Minimal (E 7 , E 7 ) Conformal Matter
In the resolution geometry of the tensor branch, we label the exceptional divisors as follows (E.10) The divisors U (0) , u To compute the n(U (k) ) for k = 1, . . . , N − 1, we need to compute the correct multiplicity factors for the curves U (k) · S 2k and U (k) · S 2k+1 in figure 12. The chain of curves on U (k) and the multiplicity factors are: . (E.11) The number in the bracket denotes the self-intersection number of that complete intersection curve inside U (k) . Hence the multiplicity factors for U (k) ·S 2k and U (k) ·S 2k+1 (k = 1, . . . , N −1) are 3, and we can compute that the vertices U (k) (k = 1, . . . , N − 1) in the CFD has n(U (k) ) = −2, g(U (k) ) = 0. Then we can read off the CFD in table 5 and 6, with G F = E 7 ×E 7 ×SU (N ).

E.4 Non-Minimal (E 8 , E 8 ) Conformal Matter
In the resolution geometry of the tensor branch, we label the exceptional divisors as follows (0 ≤ k ≤ N ): The divisors U (0) , u . (E.13) The number in the bracket denotes the self-intersection number of that complete intersection curve inside U (k) . Thus the curves U (k) · S 2k and U (k) · S 2k+1 (k = 1, . . . , N − 1) have multiplicity factors 5, and we can compute that the vertices U (k) (k = 1, . . . , N − 1) in the CFD has n(U (k) ) = −2, g(U (k) ) = 0. Then we can read off the CFD in table 5 and 6, with