Flavor Symmetry of 5d SCFTs, Part 1: General Setup

A large class of 5d superconformal field theories (SCFTs) can be constructed by integrating out BPS particles from 6d SCFTs compactified on a circle. We describe a general method for extracting the flavor symmetry of any 5d SCFT lying in this class. For this purpose, we utilize the geometric engineering of 5d N=1 theories in M-theory, where the flavor symmetry is encoded in a collection of non-compact surfaces.


Introduction
By now, the existence of a huge number of interacting superconformal field theories (SCFTs) in five dimensions has been proposed. These theories are typically defined as certain decoupling limits of string theory compactifications on singular geometries  or of intersecting brane configurations in string theory [24][25][26][27][28][29][30][31][32][33][34][35][36][37]. 1 A purely field theoretic handle on these theories can also be obtained in certain cases where the SCFT reduces in the IR, upon a mass deformation, to a 5d N = 1 supersymmetric gauge theory.

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We will approach this topic from the point of view of the construction of 5d SCFTs in terms of M-theory compactified on Calabi-Yau threefolds 2 (CY3). The advantage of using this approach is that it applies to all known 5d SCFTs. Indeed, at the time of writing this paper, there is no known 5d SCFT which (or some discretely gauged 3 version of it) does not admit such a construction. 4 Moreover, we will restrict ourselves to the sub-class of those 5d SCFTs which can be obtained (on their extended 5 Coulomb branch) by integrating out BPS particles from the extended Coulomb branch of a known 6d SCFT compactified on a circle (possibly with a twist by a discrete flavor symmetry of the 6d SCFT). This subclass is expected to capture "almost all" of the 5d SCFTs that admit a smooth M-theory construction. Evidence for this statement was provided in [17,19] where it was shown that among the class of 5d SCFTs which can be deformed to some 5d N = 1 gauge theory with a simple gauge algebra, the 5d SCFTs not arising by integrating out of BPS particles from circle compactified 6d SCFTs form an extremely tiny subset.
In Part II [69], we apply the method discussed in this paper to determine the flavor symmetry of all 5d SCFTs which can be deformed to some 5d N = 1 gauge theory with a simple gauge algebra and lying in the above class. For many of these theories, the flavor symmetry at the conformal point is larger than the flavor symmetry of the N = 1 gauge theory, a phenomenon referred to as the enhancement of the flavor symmetry at the conformal point. Thus, in this context, our method can be viewed as a tool to determine this enhancement. Some of our results in Part II [69] have been found using other methods discussed above, but many of the results are new.
Let us now provide an overview of our method, which is similar in spirit to the CFD based method [12][13][14] and bears many resemblances to it. The essential idea is that the flavor symmetry algebra of a 5d SCFT is captured in the spectrum of non-compact surfaces inside the resolved non-compact Calabi-Yau threefold describing the theory on the extended Coulomb branch of the 5d SCFT. 6 The non-abelian part of this flavor symmetry 2 It should be noted that we do not discuss algebraic models for the construction of the Calabi-Yau singularities that describe the 5d SCFTs at their conformal points. In fact, for many of the theories discussed in this paper, such an algebraic description is not known at the moment, and would be an interesting future direction. Instead, in this paper we only discuss resolved Calabi-Yaus that describe other points on the extended Coulomb branch of the 5d SCFTs, and it is conjectured that the limit to the singular Calabi-Yau can be taken without any obstructions, as long as certain conditions are satisfied by the resolved Calabi-Yau. See [7] for more details on these conditions. Another conjecture that this paper relies on is that if a resolved Calabi-Yau is obtained by blowing down a resolved Calabi-Yau associated to a 6d SCFT, then the former resolved Calabi-Yau can be flopped in such a way that after the flops, it would satisfy the conditions of [7]. 3 One can also gauge after stacking by an SPT phase. Here we include gaugings of higher-form symmetries and also more general gaugings, some of which were discussed in [70]. 4 See also the recent paper [21] where some examples of exotic 5d SCFTs have been proposed which, despite admitting a CY3 construction in M-theory, cannot be directly studied using methods of this paper, since the CY3 cannot be completely resolved in those cases. 5 Extended Coulomb branch of a 5d SCFT is defined to be the total space formed by fibering Coulomb branch of its mass deformations upon the space of supersymmetric mass deformations. 6 This idea is not new. It was used by [71] to determine flavor symmetries of Rank-1 Seiberg theories in 5d. Here we provide tools that lets one perform such computations for a huge class of 5d SCFTs of arbitrary ranks.

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admits an even nicer description in this context, which is captured in the spectrum of P 1 fibered non-compact surfaces having a non-compact base curve. It is easy to see that this should be the case from the well-known statement that non-abelian part of the gauge symmetry is captured by P 1 fibered compact surfaces (possibly with additional blowups). M2 branes wrapping the fibers of such compact surfaces give rise to W-bosons for the gauge algebra, and M2 branes wrapping the base curves of these compact surfaces give rise to instantonic BPS particles whose mass is controlled by the inverse gauge coupling of the gauge algebra. Since the mass of a particle arising from an M2 brane wrapping a compact curve is controlled by the volume of that curve, the volumes of the base curves of the Hirzebruch surfaces are then identified with the inverse gauge coupling of the gauge algebra. Turning the gauge coupling off turns the gauge algebra into a flavor algebra, and correspondingly decompactifies the base curves of the compact P 1 fibered surfaces. Thus, we expect that non-abelian part of the flavor symmetry algebra is captured by the P 1 fibered non-compact surfaces.
We start by providing a recipe to explicitly identify the non-compact surfaces capturing the non-abelian part of the flavor symmetry algebra of a 6d SCFT compactified on a circle 7 (possibly with a twist). We do not need to identify the non-compact surfaces corresponding to the abelian part of the flavor symmetry algebra. The reason for this is that the number of abelian factors in the flavor symmetry algebra can be easily tracked when we integrate out BPS particles, as we will explain in more detail below. Now, the process of integrating out a BPS particle is achieved by performing a sequence of flops on the set of compact surfaces until a compact curve is taken out of the compact surfaces into the non-compact surfaces. The process of integrating out is fully completed when we send this compact curve to infinite volume after it has been taken out of the set of compact surfaces. This last step of decompactifying the compact curve ends up modifying the non-compact surfaces so that in general some of the P 1 fibered non-compact surfaces do not remain P 1 fibered anymore. Thus, after integrating out the BPS particle we obtain a CY3 describing a 5d SCFT with a different collection of non-compact surfaces admitting P 1 fibration. We propose that this set of P 1 fibered non-compact surfaces is the maximal set of P 1 fibered non-compact surfaces for the CY3 obtained after the integrating out process is complete. In other words, the non-abelian part of the flavor symmetry of the corresponding 5d SCFT can be read from the data of the P 1 fibered non-compact surfaces that we have deduced from the data of the P 1 fibered non-compact surfaces of the parent 5d KK theory.
The abelian part of the flavor symmetry of the descendant 5d SCFT is obtained by noticing that the above integrating out procedure reduces the total rank of the flavor symmetry by one. Thus, the number of abelian factors in the flavor symmetry of the 5d SCFT can be deduced from the knowledge of the rank of the non-abelian part of the flavor symmetry of the 5d SCFT along with the number of abelian factors in flavor symmetry algebra of the parent 5d KK theory.

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One can go on in this fashion to integrate out another BPS particle from the above descendant 5d SCFT, which again in general destroys the P 1 fibrations of some of the P 1 fibered non-compact surfaces, resulting in a reduction of the non-abelian part of the flavor symmetry. The number of abelian factors can again be computed by using the fact that the integrating out of each BPS particle reduces the total rank of the flavor symmetry algebra by one. Similarly, we can continue to integrate out even more BPS particles to obtain even more 5d SCFTs and determine their corresponding flavor symmetry algebras.
The paper is organized as follows: in section 2, we discuss the flavor symmetry of 6d SCFTs, making a key distinction between localized and delocalized flavor symmetries. This prepares the ground for a discussion of flavor symmetry of 5d KK theories in section 3. In section 4, we discuss the coupling of non-compact surfaces corresponding to flavor symmetries of 5d KK theories to the compact surfaces describing the extended Coulomb branch of the 5d KK theories. In section 5, we illustrate using a simple example how flavor symmetry of 5d SCFTs obtained by integrating out BPS particles from 5d KK theories can be obtained by using the formalism set up in the previous sections of this paper. Further more complicated examples are discussed in Part II [69] of this series of papers. We finish this paper with a quick discussion in section 6 on how the method for computing flavor symmetry discussed in this paper compares with another method based on CFDs [12][13][14].
Throughout this paper, we use notation and background about geometric constructions and 5d KK theories that can be found in section 5 and appendix A of [11]. Background and notation about geometric construction of 5d N = 1 gauge theories can be found in section 2 of [54] and section 3.2 of [19]. Background on flops can be found in [15].

Flavor symmetry of 6d SCFTs
The starting point of our analysis of flavor symmetry of 5d SCFTs is the flavor symmetry of 6d SCFTs. 8 In this paper, we describe 6d SCFTs 9 in terms of their tensor branch data which can be captured in terms of a graph. Briefly, the structure of the graph captures the Green-Schwarz matrix Ω ij associated to the 6d theory, the algebras on the nodes describe the gauge algebras appearing on the tensor branch of the 6d theory, and the edges between the nodes encode the hypermultiplet matter content transforming under the gauge algebras. See section 2 of [11] for more details on this graphical notation which we will employ throughout this paper.
The flavor symmetry of 6d SCFTs can be classified into two types: localized and delocalized. Localized flavor symmetries are associated to a single node or edge in the graph associated to the 6d SCFT, and delocalized flavor symmetries are associated to multiple edges and nodes in the graph associated to 6d SCFTs. Moreover, as discussed in the introduction, we will only need to focus on the non-abelian part of the flavor symmetry for the purposes of this paper. In what follows, we will often use the term "flavor symmetry" even when we are talking about non-abelian part of the flavor symmetry. The precise meaning of the phrase "flavor symmetry" should be clear from the context of discussion.

Localized flavor symmetries
Let us consider a localized flavor symmetry associated to a node i in the graph associated to the 6d SCFT. If the gauge algebra g i associated to the node is non-trivial, then this flavor symmetry acts on some hypers that are not gauged by any other gauge algebra g j for j = i. We can represent localized flavor symmetries by extending the graph of the 6d SCFT by some extra nodes that we refer to as 'flavor nodes'. So, a localized flavor symmetry f i associated to node i is represented by a flavor nodeî carrying label [f i ] and joined to the node i by an edge. The edge can be of different kinds (solid, dashed etc.) depending on the representation formed by the hypers under g i that f i rotates. Moreover, the edge between i and i can carry a label which is specified by the positive integer −Ω iî appearing in the Green-Schwarz coupling where Fî denotes the field strength of the background gauge field associated to f i . When −Ω iî = 1, the edge between i andî carries no label. But when −Ω iî > 1, we insert a label in the middle of the edge specifying the value of −Ω iî . We can compute Ω iî concretely as and n i , nî are numbers associated to algebra g i , f i according to the table 1 of [77]. If R i ⊗Rî is a pseudo-real irrep of g i ⊕ f i , then we have All the possible localized flavor symmetries for non-trivial g i have been collected in tables 1 and 2. In these tables we show the data of the node i, the gauge algebra f g i and the attached localized flavor symmetries f i . Let us explain these tables a little:

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where the su(4) node with Ω ii = 1 contains a total of 12 hypers in fundamental and one hyper in 2-index antisymmetric. The hypers can be further gauged by other neighboring gauge algebras, but the antisymmetric cannot be gauged in a 6d SCFT. So, we obtain an su(m ≤ 12) flavor symmetry from the fundamentals and an su (2) flavor symmetry from the antisymmetric. The flavor symmetry arising from the fundamentals is joined by a solid edge and the flavor symmetry arising from the antisymmetric is joined by a dashed edge.
• Consider a node g i = su (2) and Ω ii = 2. There are a total number of 8 half-hypers transforming in fundamental representation of su (2). It is known that the maximal flavor symmetry algebra is only so (7) under which the hypers transform in the spinor representation of so (7). The spinor representations are always denoted by a dashed edge, as in the case of 2 su(2) [so (7)] (2.6) It is also possible for a 6d SCFT carrying such a node i to have only 6 or 7 out of the 8 half-hypers not transforming under some other neighboring gauge algebra. When we have 7 such half-hypers we have a g 2 flavor symmetry, and when we have 6 such half-hypers we have an su(3) flavor symmetry.
• For g i = so(n), we join flavor symmetries associated to hypers transforming in spinor representation of so(n) by a dashed edge, and the flavor symmetries associated to hypers transforming in vector representation of so(n) by a solid edge. A dotted edge is also used in the case of so(n) = so (8) to denote hypers transforming in cospinor representation of so(n) = so(8).
Now we turn to a discussion of localized flavor symmetries for nodes i with trivial g i . First consider the case of Ω ii = 1 for which we denote the trivial gauge algebra as g i = sp(0). In the case of 6d SCFTs, all the neighboring nodes j of such a node i have Ω ij = −1 and ⊕ j g j ⊆ e 8 . Then, the flavor symmetry algebra associated to node i is the commutant of ⊕ j g j inside e 8 . If one of the neighbors has g j = su(8), then there are two possible embeddings into e 8 . One of them has commutant su(2) and the other has a u(1) commutant. We differentiate these two cases by attaching a theta angle θ to the sp(0), with θ = 0 if the commutant is su(2) and θ = π if the commutant is u(1). It is possible for some factors of the non-abelian flavor symmetry algebra to have embedding index larger than 1. In such cases, we insert a label denoting the embedding index in the middle of the edge connecting the nodeî associated to this factor of flavor symmetry algebra and the node i. We do so because the value of Ω iî coincides with this embedding index when g i is trivial.
Second, consider the case of Ω ii = 2 for which we denote the trivial gauge algebra as g i = su (1). If this node has a neighbor with g j = su(1) or g j = su(2), then there is no non-trivial localized flavor symmetry. Only if the node has no neighbor or has a neighbor with g j = sp(0) do we obtain an su(2) localized flavor symmetry.  12 1 ≤ k ≤ 8 All the possible localized flavor symmetries for these two types of nodes are collected in tables 3, 4, 5 and 6, where we list all the neighbors of the node i. In these cases, we display all the abelian factors in the flavor symmetry explicitly (for which, unlike the non-abelian case, we do not need to define a notion of embedding index) since it is difficult to track the number of abelian factors for non-gauge-theoretic nodes as compared to the case of gauge-theoretic nodes discussed above.

Delocalized flavor symmetries
The reason for the delocalization of flavor symmetries has to do with nodes of type k su(n) (2.7) except for the n = 2, k = 1 case. For example for k = 2 and n ≥ 3, one would naively expect a u(2n) flavor symmetry associated to this node. However, the diagonal u(1) inside u(2n) is anomalous at the quantum level. Thus, the above node is associated to an su(2n) flavor symmetry only which has rank one less than the naive u(2n) flavor symmetry. For n = 2 and k = 2, something similar occurs. Since the fundamental representation is pseudo-real, naively one would expect an so(8) flavor symmetry. However, the true flavor symmetry is only so(7) [43,75] which again has rank one less as compared to the rank of the naively expected flavor symmetry. It should be noted that for n = 2 and k = 1, one does not find such a reduction in the rank of flavor symmetry. Now consider a chain of the form 2 su(n k ) 2 where all n i ≥ 3. Then there are various u(1) factors in the naive flavor symmetry. Each node i carries m i hypers in fundamental of su(n i ) which are not charged under any su(n j ) for j = i. If m i ≥ 1, then there is a diagonal u(1) i rotating all the m i hypers for each i. Moreover, the edge between any two consecutive nodes i and i + 1 denotes the presence of a hyper transforming in bifundamental of su(n i ) ⊕ su(n i+1 ). Then there is u(1) symmetry rotating each such bifundamental. k linear combinations of all these u(1)s are anomalous and thus the true flavor symmetry has rank k less than the naive flavor symmetry. Moreover, the remaining u(1)s are linear combinations of all the above mentioned localized u(1)s, thus the remaining u(1)s are "delocalized". We can say that each su(n i ) is responsible for reducing the rank of flavor symmetry by 1, thus leading to a difference of k in the ranks of the true and the naive flavor symmetries. The case of all n i equal is helpful to understand the delocalization: In this case, along with bifundamentals between adjacent nodes, we have n fundamentals transforming under the leftmost su(n) and n fundamentals transforming under the rightmost su(n). Thus, we have a total of k + 1 u(1) symmetry naively. Removing k of these JHEP09(2021)186 su (2) [su (2)] [su(2)]  Table 6. Possible localized flavor symmetries for nodes with a trivial gauge algebra continued from table 5. leaves only a single u(1) symmetry which is a linear combination of all the above k + 1 u(1)s, thus the remaining u(1) is not localized around any subset of nodes in (2.9). Now, consider a limit n = 2 of (2.9) 2 su(2) 2 su(2) · · · 2 su(2) (2.10) In this case, each bifundamental is a strictly-real representation and hence is rotated by an su(2) symmetry. The leftmost node carries an extra 2 fundamental hypers rotated by an so(4) = su(2) ⊕ su(2) symmetry. Similarly, the rightmost node also carries an extra 2 fundamental hypers rotated by an so(4) = su(2) ⊕ su(2) symmetry. In total the naive flavor symmetry is comprised of k + 3 su(2)s. We claim that just as in the n > 2 case, each su(2) gauge node is responsible for the reduction of rank by 1.

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We expect its flavor symmetry to be e 7 ⊕su(2) 3 where the e 7 factor is the localized symmetry associated to the sp(0) node. In particular we expect the flavor symmetry to have rank 10. If we would compactify this 6d SCFT on a circle of finite non-zero size, we would expect the flavor symmetry of the resulting 5d KK theory to have rank 11 = 10 + 1 where the extra rank is associated to the size of the circle. It is known that this 5d KK theory on special loci in its extended Coulomb branch is also described by the 5d gauge theory which has a rank 11 flavor symmetry (accounting for the instanton current). Thus, our proposal for the rank of the 6d SCFT passes this test. Later, we will solidify our proposal for the e 7 ⊕ su(2) 3 flavor symmetry by manifesting it in the geometry. Finally, consider the n = 1 case of (2.9) which is well-known to have an su(2) delocalized flavor symmetry. From the above considerations, we can derive the following rule for the determination of flavor symmetry f of a 6d SCFT T that has nodes of type (2.7): first, we define a naive flavor symmetry f N associated to the theory. First part of f N arises from the flavor symmetry of the 6d N = (1, 0) gauge theory associated to T. Second part of f N arises from the flavor symmetries localized at all the sp(0) nodes in the theory. And the final and third part of f N arises from chains of su(1) nodes. Let c be the number of connected subset of nodes of T such that the connected subset contains only su(1) nodes and none of the nodes in the connected subset has a neighboring node carrying su(2) gauge algebra. Then the third contribution to f N is su(2) c . Now, let m 1 be the number of nodes of type (2.7) in T that have n ≥ 3 and let m 2 be the number of nodes of type (2.7) in T that have k = n = 2.
Then, one would expect that More precisely, each node of type (2.7) for n ≥ 3 must remove a u(1) factor from f N due to ABJ anomaly. The removal of this u(1) causes some of the remaining u(1) factors to become delocalized. On the other hand, each node of type (2.7) for k = n = 2 should typically remove an su(2) factor, but can also remove a u(1) factor. 11 Then, it is possible for some of the remaining su(2) and u(1) factors to delocalize. Moreover, if a node of type (2.7) for k = n = 2 is not a part of chain of nodes of type (2.9), then it can also reduce the rank of naive flavor symmetry associated to it in more complicated ways, like so(8) → so (7) or so(7) → g 2 or so(6) → su(3) or so(5) → su(2) (see section 3.2 for more details). The remaining delocalized factors in f are su(2) factors associated to connected subsets comprising of more than one su(1) nodes without su(2) neighbors.

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3 Flavor symmetry of 5d KK theories 5d KK theories are produced by compactifying 6d SCFTs on a circle of finite, non-zero radius. One then has the option of introducing, around the circle, holonomies for discrete flavor symmetries of the 6d SCFT. If no such holonomy is present, we say that the compactification is untwisted. If such a holonomy is present, we say that the compactification is twisted. From the point of view of the tensor branch of 6d SCFTs, the possible twists involve discrete symmetries related to outer automorphisms of the gauge algebras appearing on the tensor branch, permutation of the tensor multiplets, and parity transformation in O(2n) flavor symmetry groups. See [9][10][11]23] for a discussion of all the known 5d KK theories. Upon an untwisted compactification, it is known that the KK mode affinizes each gauge algebra g i appearing on the tensor branch to its untwisted affine version g (1) i . The same is true for flavor symmetries. Each localized or delocalized flavor symmetry is affinized to its affine version. This does not mean that the 5d KK theory admits a flavor symmetry given by an affine Lie algebra. It simply means that the mass parameters of the 5d KK theory are valued in the Cartan subalgebra of the affine Lie algebra. A linear combination of these mass parameters can be identified with the inverse radius of the compactification circle. Thus, as long as the radius of the compactification circle remains finite, the flavor symmetry of the 5d KK theory is always provided by a finite Lie algebra whose Dynkin diagram is a proper subset of the Dynkin diagram of the affine Lie algebra. Different such subsets arise at different loci in the extended Coulomb branch of the 5d KK theory. We represent a 5d KK theory arising via untwisted compactification of a 6d SCFT by a graph which is exactly the same as the graph for the 6d SCFT but replace each gauge algebra g i by g (1) i and each localized flavor algebra fî by f (1) i . For a 5d KK theory arising via a twisted compactification of a 6d SCFT, the graph for the 5d KK theory is obtained in general by folding the graph of the 6d SCFT and replacing the gauge algebras g i by affine Lie algebras g i . The possible building blocks for graphs of 5d KK theories arising by twisted compactifications were discussed in [11,23] where the effect of the twist on flavor symmetry was ignored. In this paper, we extend their analysis to account for flavor symmetries in these building blocks.

Localized flavor symmetries
For nodes carrying a non-trivial gauge algebra, the localized flavor symmetries are collected 12 in table 7. Let us explain some parts of this table: (2) su(m) (2) 1 when we have in a 6d SCFT an su(n) node with an associated su(m) localized flavor symmetry for m ≥ 3. The edge between su(n) and su(m) carries hypers transforming in bifundamental F ⊗ F of su(n) ⊕ su(m). The fields inside this hypermultiplet are left invariant by an action of order two outer-automorphism on su(n) only if we simultaneously act by an order two outer-automorphism on su(n). This can be understood by noting that the two outer automorphisms send a hyper in F ⊗ F of su(n) ⊕ su(m) to a hyper inF ⊗F which is simply a hyper in F ⊗ F sinceF ⊗F is a complex conjugate of F⊗F. Thus, the 5d KK theory produced after the above outer automorphism twist carries k su(n) (2) su(m) (2) (3.2) If we consider the case m = 2 of (3.1), then the outer automorphism of su(n) leaves the bifundamental invariant since the F of su(2) is real.

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• Consider now a 6d SCFT carrying for m ≥ 3. The edge between su(4) and su(m) carries a hyper in bifundamental, and the edge between su(2) and su(4) carries a half-hyper in F ⊗ Λ 2 of su(2) ⊕ su (4). The outer automorphism of su(4) sends F →F but leaves Λ 2 invariant. Thus, to convert this outer automorphism into a symmetry, we need to only combine it with the outer automorphism of su(m). The resulting KK theory then carries • Consider a 6d SCFT carrying Consider a Z 2 outer automorphism which exchanges the fundamental and the spinor reps of so (8). This can be a symmetry of the theory if m = n. The resulting 5d KK theory then contains k so(8) (2) sp(n) (1) where we have folded the above graph to identify the flavor symmetries associated to F and S of so (8). The partially dashed and partially solid edge between so(8) (1) and sp(n) (1) edge is supposed to denote that this edge comes by combining a solid and a dashed edge of the 6d theory. The direction of the edge denotes the direction of folding and the label 2 in the middle of the edge can be thought of as denoting the fact that this directed edge arises from combining 2 undirected edges via a folding process. More precisely, the label 2 captures an off-diagonal entry in a matrix of Chern-Simons couplings arising in the low-energy abelian gauge theory on the extended Coulomb branch of the 5d KK theory [11]. • The 5d KK theory configuration where the e 7 carries 6 half-hypers in 56 and hence the localized flavor symmetry of the 6d SCFT configuration is so (6). In fact, the flavor symmetry group is O (6). The compactification resulting in the above 5d KK theory configuration contains a twist by a Z 2 element lying in the disconnected component of O (6). Thus one of the halfhypers is projected out, and we expect the 5d KK theory to have an so(5) (1) flavor symmetry.
For nodes carrying a trivial gauge algebra, the flavor symmetries are collected in tables 8-12. Let us explain some parts of these tables below: The BPS string arising from the sp(0) node transforms as adjoint under the total e 8 symmetry of the sp(0). Under e 8 → su(5) ⊕ su(5), the adjoint of e 8 decomposes as The above decomposition is not preserved if we perform outer-automorphism on only one of the su(5) factors, but it is preserved if we perform outer-automorphism on both su(5) factors. Thus the configuration Table 10. Continued from table 9: possible localized flavor symmetries in 5d KK theories for nodes with a trivial gauge algebra having no incoming directed edges. has an su(5) (2) flavor symmetry localized at the sp(0) (1) node, and we denote the full configuration as • Consider e 8 → so(12) ⊕ su(2) ⊕ su (2) under which the adjoint of e 8 decomposes as From this we see that outer-automorphism of so (12) is a symmetry only if we exchange the two su(2). Thus, compactifying the configuration

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along with an outer-automorphism twist of so(12) induces a permutation twist on the two flavor su(2) nodes, and the resulting 5d KK theory configuration is which carries a single su(2) (1) flavor symmetry.
• Consider the 6d SCFT configuration which we want to compactify with an outer-automorphism twist of su (7). The adjoint of e 8 decomposes under where the subscripts denote u(1) charges. Thus, the outer-automorphism of su (7) is a symmetry only if we also perform outer-automorphism of u(1) which takes a charge q representation of u(1) to charge −q representation. Due to the presence of this outer-automorphism, the 5d KK theory does not obtain any continuous flavor symmetry from the u(1) factor present in the 6d SCFT, and hence we omit it and write the resulting 5d KK theory configuration simply as There are multiple ways to twist this configuration, all of which lead to different flavor symmetries. The adjoint of e 8 decomposes under e 8 → su(3) 4 as

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Performing outer-automorphism on two of the su(3)s forces us to perform outerautomorphisms on the other two su(3)s as well, thus leading to the 5d KK theory configuration Performing outer-automorphism on only one of the su(3)s while performing no outerautomorphism on another su(3) forces us to exchange the remaining two su(3)s leading to the 5d KK theory configurations Similarly, exchanging two su(3)s forces us to perform outer-automorphism on one of the remaining two su(3)s leading to the 5d KK theory configuration Finally, we want to twist the gauge su(3)s to obtain a 5d KK theory configuration of the form This can be done by sending the left gauge su(3) to the right gauge su(3) along with an outer-automorphism, and the right gauge su(3) to the left gauge su (3) without an outer-automorphism, thus exchanging the two su(3)s with a relative outerautomorphism involved in the exchange. This operation forces us to also exchange the two flavor su(3)s in the same fashion implying that the full 5d KK theory configuration involving the flavor algebras is The adjoint of e 8 decomposes as From this, we see that if we perform an outerautomorphism twist on so (8) and no outer-automorphism twist, then one of the u(1)s is projected out. Similarly, if we perform outer-automorphism twist on both so (8) and su (3), then again one of the u(1)s is projected out. However, if we perform outer-automorphims twist on su(3) but no outer-automorphism twist on so (8), then both the u(1)s are projected out, and the resulting 5d KK theory configuration has no continuous flavor symmetry, thus this configuration does not appear in the tables.
• A 6d SCFT configuration of the form can simply be folded to result in a 5d KK theory configuration of the form

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which is e = 2 version of the first entry in table 11. Similarly, the e = 3 version is produced by folding the 6d SCFT configuration of the form 1 sp(0) (1) k Similarly, one can also obtain e > 3 versions of the entry. In a similar fashion, one can construct the first entry of table 12.
• The 5d KK theory configuration where we now have two BPS strings to account for. Let us choose an outer-automorphism frame for the left su(3) so that the left BPS string transforms as under su(3) left ⊕ e 6 and an outer-automorphism frame for the right su(3) so that the right BPS string transforms as Then, it is clear that exchanging the two strings and the two su(3) while simultaneously performing outer-automorphism on e 6 is a symmetry and no outer-automorphism of su(3) is needed. Thus, the full 5d KK theory configuration displaying the flavor symmetry is

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Now, the 5d KK theory configuration Now there is no way to choose outer-automorphism frame for the three su(3)s so that a cyclic permutation of the three su(3) combined with an outer-automorphism of e 6 is a symmetry of the spectrum of BPS strings. to convert the above operation into a symmetry, one is forced to insert a relative outer-automorphism between the three su(3)s while cyclically permuting them. Thus, the corresponding 5d KK theory configuration with flavor symmetry included is

Ω matrix associated to a 5d KK theory
As we saw above, there is a graph that can be associated to any 5d KK theory which also incorporates the flavor nodes. In the case of 6d SCFTs, we associated matrix Ω to such a graph where the diagonal entry of Ω for flavor nodes was not specified since it is not well-defined. In the same fashion, we can associate a matrix Ω to the graph of a 5d KK theory, which was done for non-flavor nodes in [11]. Their analysis can be extended to incorporate flavor nodes. If we have a 5d KK theory configuration of the form This rule extends over to flavor nodes, and for the 5d KK theory configuration of the form we associate Ω ii = 1 rather than Ω ii = 2.

Delocalized flavor symmetries
Now we turn to a discussion of delocalized flavor symmetries in 5d KK theories that descend from delocalized flavor symmetries in 6d SCFTs (see section 2.2). The nodes responsible for the delocalization are of the form except for the n = 2, k = 1 case. These nodes remove u(1) (1) and su (2) (1) factors from the naive flavor symmetry associated to a 5d KK theory. The counting of u(1) (1) factors in the naive flavor symmetry algebra deserves some comments since the outer-automorphism twists may project out some of the u(1) factors in the flavor symmetry of the parent 6d SCFT. For example, in the 5d KK theory configuration k su(m) (2) l there is no u(1) (1) flavor factor associated to the edge since the u(1) flavor factor associated to the edge in the parent 6d SCFT configuration is complex conjugated by the outer-automorphism implementing the twist, and hence the u(1) factor is projected out upon carrying out the twist. Similarly, there is not u(1) (1) factor associated to the edge in the following 5d KK theory configuration On the other hand, there is a single u(1) (1) factor associated to the edge in For example for e = 2, the above configuration is produced by applying a permutation twist on the following 6d SCFT configuration which has a u(1) factor associated to each edge, but the two u(1) factors are combined into a single u(1) (1) factor by the permutation twist.
Once the naive flavor symmetry algebra for the 5d KK theory has been figured out, the rule is that each node of the form (3.43) for n ≥ 3 removes a u(1) (1) factor, thus delocalizing other u(1) (1) factors. The effect of (3.43) for n = k = 2 is same as for the case of 6d SCFTs. If such a node appears without a gauge theoretic neighbor then it modifies the naive flavor symmetry localized at the node as so (8) (1) . If such a node appears such that naive flavor symmetry localized at the node is so(4) (1) or so(3) (1) , then it removes an su(2) (1) . If such a node appears such that naive flavor symmetry localized at the node is so(2) (1) , then it removes a u(1) (1) . If such a node appears such that naive flavor symmetry localized at the node is trivial but the naive flavor symmetry localized at a neighboring edge is u(1) (1) , then it removes a u(1) (1) . If such a node appears such that naive flavor symmetry localized at the node is trivial but the naive flavor symmetry localized at a neighboring edge is su(2) (1) , then it removes an su(2) (1) . If such a node appears such that naive flavor symmetry localized at the node is trivial and the naive flavor symmetry localized at all the neighboring edges is trivial, then it removes nothing. Below we present several examples demonstrating the rules discussed in this paragraph.
So, for example, the configuration 2 flavor symmetry. The configuration also has a g 2 flavor symmetry which is viewed as a non-standard affinization of the su(3) flavor symmetry of the 6d SCFT. This phenomenon is discussed in the detailed analysis for (2.77) in Part II of this series of papers. The configuration has an su(2) (1) flavor symmetry. The configuration

Gluing rules
To every 5d KK theory, one can associated a CY3 such that M-theory compactified on it manufactures the 5d KK theory on its Coulomb branch. The CY3 surface geometry can be constructed using the data of the graph (restricted to non-flavor nodes) associated to the 5d KK theory. To each non-flavor node one associates a collection of compact surfaces intersecting each other. To an edge between two non-flavor nodes, one associates a set of "gluing rules" which describe how the two collections of surfaces associated to the two nodes intersect each other. See [9][10][11]23] for more details.
In this section, we expand their procedure to incorporate flavor nodes (carrying nonabelian flavor symmetries) into the geometry. For localized flavor symmetries, the story JHEP09(2021)186 is the same as above. One associates a collection of non-compact P 1 fibered surfaces (intersecting each other) to each such flavor node. For each edge between a flavor and a non-flavor node, one associates a set of gluing rules which describe how the collection of non-compact P 1 fibered surfaces associated to the flavor node intersect the collection of compact surfaces associated to the non-flavor node. For delocalized flavor symmetries, the gluing rules describe how the collection of non-compact P 1 fibered surfaces associated to the flavor symmetry intersect the collection of compact surfaces associated to the non-flavor nodes over which the flavor symmetry is delocalized.
From this point on, we use a lot of notation and geometric background which can be found in section 5 and appendix A of [11]. We also explain a lot of this notation example by example as we move through this section.

Description of P 1 fibered surfaces
In this subsection, we describe a notation for P 1 fibered surfaces that we use to describe gluing rules in this paper and Part II [69]. We will denote such surfaces as N i or M i etc. where i is simply a label to distinguish different surfaces. We also add a superscript sometimes and write N p i which means that the P 1 fibered surface N i carries p blowups. We denote the P 1 fibers as f i and various blowups as x a , y a etc.
We label an arbitrary section of N which intersects f at p points as pe + α a x a (where x a are some blowups and α a are arbitrary coefficients) instead of displaying it as pe + α a x a + βf . That is, we do not display the number of f involved in the section. The rationale for this is that and Since only the above two types of intersection numbers are relevant while discussing gluing rules, we see that we can omit the data about the number of f involved in the section, which can be easily restored using the general consistency conditions discussed in section 5 and appendix A of [11]. For a collection N i of non-compact P 1 fibered surfaces, we also do not display individual gluing curves between two non-compact surfaces N i and N j if those gluing curves are noncompact. So, if N 0 and N 1 are non-compact, the geometry is represented as the geometry where we have replaced e + 2f by e and combined the two gluing curves together. This is because the splitting of a non-compact gluing curve between two non-compact surfaces

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into multiple non-compact gluing curves is not relevant for any physical information about the resulting 5d theory. The above notation also allows us to present the gluing rules between two compact P 1 fibered surfaces, or a compact P 1 fibered and a non-compact P 1 fibered surface, or two non-compact P 1 fibered surfaces on an equal footing in what follows.

Condition on the gluing rules
In this subsection, we recall an important condition that gluing rules are supposed to satisfy [19,58]. Let us label the surfaces associated to node i (which could be flavor or non-flavor) as S i,a where different values of a label different surfaces associated to this node i. Then to this node we assign a linear combination of surfaces where d ∨ i,a are dual Coxeter numbers 13 associated to the affine algebra g (q i ) i associated to this node. If the node does not have a non-trivial algebra associated to it, then there is only a single surface S i,0 and we define S i = S i,0 .
Moreover, a curveẽ i for each collection of surfaces S i,a was defined in section 3.3.3 of [58]. Then, we must have − S i ·ẽ j = Ω ij as homology classes in the full CY3, where x α are arbitrary blowups living in surfaces S i,a , and m ab , n ab , p abc , q abα are some coefficients determined by the gluing rules, and we impose the condition that gcd(m ab , n ab , p abc , q abα ) = 1. Nowẽ i := e i,b for that particular b such that m ab = 1 for all a. There are be multiple b which satisfies this condition, and in such a case we can choose any such b to defineẽ i without any change in the consequences. Another fact to note is that if the algebra (flavor or non-flavor) g (q i ) i associated to the node i has is an untwisted affine algebra, then we can always chooseẽ i = e i,0 where S i,0 corresponds to the affine node in the Dynkin diagram of g i . Another consistency condition for the introduction of a blowup x living in any surface S j,b is that, we must have x · S i = 0

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and we must also have f i,a · S j,b = 0 (4.10) for i = j.

Gluing rules for localized flavor symmetries: without sp(0) (1) nodes
In this subsection, we will describe gluing rules of the form (4.11) or of the form Actually, the gluing rules do not depend on the data of diagonal entries of Ω ii or on whether the node i or j is a flavor or a non-flavor node, we will denote the gluing rules for (4.11) as (4.13) and the gluing rules for (4.12) as Some of the gluing rules have been previously discussed in [9][10][11]23] as the gluing rules for two gauge nodes. We thus don't need to discuss them again, and hence only display the gluing rules which do not appear in those works. Unfortunately, we will not be able to provide the gluing rules for cases where the non-flavor node is e (1) 7 since these cases are computationally quite intensive. However, we do not expect these cases to be special in any way: it should be possible to figure out the gluing rules for these cases with enough effort.
Gluing rules for su(4) (1) su(2) (1) . Here, as discussed before, the dashed edge means that su(2) gauges an antisymmetric of su(4) rather than gauging two fundamentals of su(4) (which would be represented by a solid edge).
The P 1 fibered surfaces corresponding to su(4) (1) and su(2) (1) can be represented as where M i parametrize su(4) (1) and N i parametrize su(2) (1) . We are using the notation for P 1 fibered (compact or non-compact) surfaces described in section 4.1. Moreover, we only display blowups which are used in the process of gluing. Other blowups are omitted. We represent a surface N i as when we want to label the i b i number of blowups of N i by different alphabets. We label the first b 1 blowups as x i where i = 1, · · · , b 1 , the next b 2 blowups as y i where i = 1, · · · , b 2 , the next b 3 blowups as z i where i = 1, · · · , b 3 , the next b 4 blowups as w i where i = 1, · · · , b 4 , and so on. In keeping line with this notation, we label M 1 in the above set of surfaces as M 1+1 1 and label the two blowups as x and y. We do not put a subscript on the blowups if there is only a single blowup of that type.
Each edge describes a gluing between the surfaces that the edge is subtended between. The labels at the end of each edge describe the gluing curve in the corresponding participating in that gluing. Whenever we write x i or y i , we mean a sum over all the blowups of that type. So, the gluing curve 2e − x i living in N 0 for its gluing with N 1 should be read as 2e − The gluings between su(4) (1) and su (2) (1) are referred to as the gluing rules, and are listed below: Whenever we say that C i , D i , E i , · · · in M i are glued to C j , D j , E j , · · · in M j , the gluings are supposed to be read in order. That is, Gluing rules for so (9)

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A subscript in the middle of an edge between two surfaces N i and N j denotes the number of gluings between the two surfaces. In the above case, there are m gluings between M 1 and M 3 . The gluing is represented as x i − y i being glued to f , which means that x i − y i for each i is glued to a copy of f , leading to a total of m gluings since i = 1, · · · , m as is visible from the notation M m+m

3
. The gluing rules between M i and N j are: Gluing rules for so (10)  •

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Gluing rules for so (11) An edge starting from a surface N i and ending at the same surface N i represents a selfgluing of the surface N i . Above, we have 16 self-gluings of N 0 and 16 self-gluings of N 1 .
Here we have started abbreviating n copies of the same curve C in the gluing rules as nC.
Thus, for example, the second item in the above list should be read as: 12 , y 4 − y 5 , y 7 − y 8 , y 9 − y 10 , y 11 − y 12 in N i for i = 0, 1.

Gluing rules for localized flavor symmetries: with sp(0) (1) nodes
In this subsection, we discuss gluing rules when sp(0) (1) non-flavor nodes are involved except for the case of

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for which we have been unable to find a consistent geometry. 14 It would be interesting to investigate if this simply reflects the lack of ingenuity of the author or if something is pathological about this 5d KK theory (or its M-theory construction).
The gluing rules are again independent of whether the nodes are flavor or non-flavor, and hence we do not make this distinction in what follows.
where each node represents a curve in the compact surface dP 9 associated to sp(0) (1) . The P 1 hyperplane class is denoted as l and blowups are denoted as x i with i = 1, · · · 9. Number of edges between two nodes displays the intersection number of the two curves inside dP 9 . Each displayed curve is glued to a fiber of a P 1 fibered surface among the collection of P 1 fibered surfaces parametrizing su(2) (1) ⊕ su(8) (1) . Since these fibers intersect in the form of Dynkin diagram of su(2) (1) ⊕ su(8) (1) , the above displayed curves also intersect in the form of this Dynkin diagram. For example, we can see from the above that the curves 8 . See [10,11] for more details. Now we can add two of the curves 2l − 14 It is possible that the absence of a consistent geometry might be a hint that the discrete global symmetry used to twist the 6d theory has a 't Hooft anomaly. Thus, it would be interesting to study 't Hooft anomalies of the discrete global symmetries of 6d SCFTs appearing in [11,23]. The author thanks an anonymous referee for making this interesting remark.

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Moving on, we have the gluing rule

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From the sp(0) (1) gluing rules for e (1) 8 , we can obtain the gluing rule Another gluing rule is A node with multiple curves C, D, · · · indicates that the curves C, D, · · · inside dP 9 glue to different copies of P 1 fiber of the P 1 fibered surface corresponding to that node. Thus, in the above gluing rule, x 1 − x 4 and l − x 1 − x 3 − x 4 glue to two copies of P 1 fiber of the surface associated to that node of Dynkin diagram of so(13) (1) which corresponds to spinor representation of so (13). In such a case, the number of edges between two nodes denotes the total intersection number. Thus, we have spanned two edges between node corresponding to x 4 − x 5 and the node corresponding to

Gluing rules for delocalized flavor symmetries
In this section we will be using N i , M i etc. to denote only non-compact P 1 fibered surfaces. The compact surfaces have a P 1 base and thus are Hirzebruch surfaces. They are denoted as where i is simply a label, n is the degree of the P 1 fibration and b is the number of blowups. Such a surface is also denoted as F b n , referred to as a Hirzebruch surface of degree n and carrying b blowups.
where we refer to the Hirzebruch surface i b n as S i .
• f − x i , x i in N 0 are glued to x, y in S 2i−1 for i = 1, · · · , m/2 .
• f − x i , x i in N 0 are glued to y, x in S 2i for i = 1, · · · , m/2 .
The gluing rules for the su(2) (1) delocalized flavor symmetry of

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are described by The gluing rules for the su(2) (1) delocalized flavor symmetry of 2 su(1) (1) 2 are described by The gluing rules for the su(2) (1) delocalized flavor symmetry of are described by • f − x 5 , x 5 in N 0 are glued to x, y in S 8 .
• f − y 1 − y 2 in S m+2 is glued to f in P 1 .
Let us note, in order to avoid confusion, that there is no surface S m+1 in the above geometry. For m odd, the gluing rules are described by • y 1 , y 2 in S m+2 are glued to x 1 , x 2 in P 0 .
Let us note, in order to avoid confusion, that there is no surface S m+1 in the above geometry. The gluing rules for the (su(2) (1) ) ⊕2 delocalized flavor symmetry of for m odd are described by For m even, the gluing rules are described by The gluing rules for the (su(2) (1) ) ⊕2 delocalized flavor symmetry of  for m odd are described by
• f − y 1 − y 2 in S m+2 is glued to f in P 1 .
Let us note, in order to avoid confusion, that there is no surface S m+1 in the above geometry. For m even, the gluing rules are
• f − y 1 − y 2 in S m is glued to f in P 1 .
Let us note, in order to avoid confusion, that there is no surface S m+1 in the above geometry.
The (partial) gluing rules for the su(2) (1) delocalized flavor symmetry of • f − y 2 , y 1 , x, x in S 2m+1 are glued to x m , y m , z, w in M 0 .

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For m odd, the gluing rules are The gluing rules for the su(2) (1) delocalized flavor symmetry of The gluing rules for the su(2) (1) delocalized flavor symmetry of are described by 2ex i 2 (4.120) The gluing rules for the su(2) (1) delocalized flavor symmetry of 2e-x (4.124) • f in N 1 is glued to f − x − y in S i for i = 1, 2.
• f − x 1 , x 1 in N 0 are glued to x, y in S 1 .

Flops and blowdowns
In this section, we illustrate using a simple example how the flops and blowdowns associated to an RG flow from a 5d KK theory to a 5d SCFT transform the non-compact surfaces in the geometry associated to the 5d KK theory, which allows us not only to read the flavor symmetry of the 5d SCFT, but also describe how the non-abelian part of the flavor symmetry of the 5d SCFT is captured by non-compact surfaces in the geometry associated to the 5d SCFT. That is, we also obtain the data of how the non-compact surfaces associated to the flavor symmetry of the 5d SCFT are coupled to the compact part of the geometry.
Many more examples of diverse kinds of RG flows are discussed in Part II [69] of this series of papers.
The example we study is of the 5d KK theory The 6d SCFT has a tensor branch description as a 6d N = (1, 0) gauge theory sp(1) + 10F which has an so(20) flavor symmetry, and hence we expect to be able to couple the compact JHEP09(2021)186 part of the geometry (which was described in [9][10][11]) to non-compact P 1 surfaces comprising so(20) (1) , since there is no twist involved in the circle compactification. The gluing rules for sp(1) (1) so (20) (1) (5.2) were discussed in [10,11] from which we find that the full geometry containing both compact and non-compact surfaces can be written as where we display a Hirzebruch surface of degree n and carrying b blowups as F b n . Applying the isomorphism S (which exchanges e and f in F b 0 ) on the top compact surface, we can write the above geometry as Now from the compact surfaces we can read that the geometry also describes the 5d gauge theory sp(2) + 10F [54]. Integrating out F from this theory leads to 5d SCFTs sp(2) + nF JHEP09(2021)186 with n ≤ 9. We would like to determine the flavor symmetry for these 5d SCFTs. To integrate out one F, we can flop the curve f − x 1 living in the top compact surface, which leads to the following geometry f -x where let us note, in order to avoid confusion, that we don't have any blowup labeled as x 1 living in the top compact surface anymore. The process of integrating out F is completed when we send the size of the blowup x living in N 1 to infinity. However, during this process, we would like to keep the size of the compact curve f − x 2 living in the top compact surface finite, since sending it to infinity will decompactify the top compact surface and hence change the theory away from the one we want to study. Since f − x 2 living in top compact surface F 9 1 is glued to f − x living in N 1 , we must send the size of fiber f of N 1 to infinity. Thus, after integrating out F, the surface N 1 won't remain P 1 fibered anymore, thus leaving us with the geometry (where f i labels the P 1 fiber of N i ) which gives rise to the Cartan matrix for so (20) finite Lie algebra. However, the classical flavor symmetry, that is the flavor symmetry of the 5d N = 1 gauge theory sp(2) + 9F is only so(18) ⊕ u(1). Thus, we find that the flavor symmetry of the theory is enhanced from so(18) ⊕ u(1) to so (20) at the conformal point.
In a similar way, one can proceed to integrate out more F and determine the flavor symmetry for 5d SCFTs sp(2) + nF for n < 9. Let us instead apply S on the bottom compact surface, which leads to the following geometry Now the compact part describes the gauge theory su(3) 1 2 + 9F which only has a u(9) ⊕ u(1) classical flavor symmetry, but as we have seen above, the enhanced flavor symmetry for this theory is so (20).
Let us integrate out an F to obtain the theory su(3) 0 +8F. This is done by first flopping x 10 living in the top compact surface, which leads to the geometry To completely integrate out the F, we need to send the size of x 2 in N 9 to infinity which forces the size of x 1 in N 9 to go to infinity (in order to keep the size of f in bottom compact surface finite), which in turn forces f of N 9 to go to infinite size (to keep the size of x 9 in top compact surface finite). Thus, we see that N 9 decouples and we can write the resulting geometry as for which the geometry tells us that the classical flavor symmetry u(4) ⊕ u(1) is not enhanced. Actually the above geometry only manifests the non-abelian su(4) part of the flavor symmetry but the u(1) factors can be deduced by comparing with the rank of the flavor symmetry of the gauge theory.
Now we can integrate out the e curves living in both the compact surfaces to obtain a non-Lagrangian phase with geometry where we denote a P 2 with n blowups as dP n and a P 2 without any blowups as dP. According to the geometry, the flavor symmetry of the 5d SCFT corresponding to the above geometry is su(4). In fact, this 5d SCFT admits a 5d gauge theory description as su(2) ⊕ su(2) gauge theory with a bifundamental hyper and both theta angles being zero.

Comparison with CFDs
In a series of recent papers [12][13][14] a neat way of encapsulating the flavor symmetry of 5d SCFTs was proposed. This was done in terms of a graph referred to as the Combined Fiber Diagram (CFD) of the 5d theory. The approach in those papers is similar to the one pursued here. That is, one starts from the CFD of a 5d KK theory parent of a 5d SCFT whose flavor symmetry is desired. Then, the RG flows to that 5d SCFT descend to operations on the CFD of the KK theory. In the end, one obtains the CFD for the desired 5d SCFT, from which the flavor symmetry of the 5d SCFT can be readily read.

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However, the approach pursued there suffers from some limitations. The first limitation is that there is no known way of obtaining the CFD associated to an arbitrary 5d KK theory. The second limitation, which we discuss in this section, is that CFDs do not see all RG flows corresponding to integrating out BPS particles. If the BPS particle being integrated out descends from an M2 brane wrapping a gluing curve between two surfaces, then this RG flow is invisible in the CFD machinery. This invisibility is fine if integrating out this BPS particle decomposes the 5d theory into a product of two 5d theories of lower rank, since then the individual lower rank theories could be obtained inside the decoupling tree of a 5d KK theory of that lower rank. However, it is possible to preserve the rank of the 5d theory in such a flow. 15 An example of a 5d theory where such an RG flow is possible is provided by 2 su(n) (1) (6.1) which is obtained via an untwisted compactification of the 6d SCFT with tensor branch description su(n) + 2nF.
Let us discuss the case of n = 2 in the rest of this section. The 6d SCFT in this case has an so(7) flavor symmetry. Thus, we expect to be able to couple the compact part of the geometry (which was described in [9][10][11]) to a collection of non-compact P 1 fibered surfaces parametrizing so(7) (1) , the gluing rules for which were described in [10,11]. The combined full geometry can be represented as  15 The possibility of this scenario was pointed out in section 5 of [9] where the n = 1 version of the following example also appeared.

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Flopping x 4 living in the top compact surface, we can write the above geometry as The above geometry is isomorphic to the following geometry f Now we notice that we can integrate out the −1 curve e living in the bottom compact surface, which also integrates out e living in top compact surface, since the two e curves are glued to each other. Flopping this curve, we obtain

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Completing the integrating out process, i.e. sending x 2 living in N 3 to infinite volume, gives rise to the geometry which we recognize as the geometry (5.13) that we discussed at the end of the last section. There we had constructed this same theory by performing RG flows from a different 5d KK theory (5.1) and all the RG flows were of the kind that are visible at the level of CFDs. Using both methods, we have arrived at the same prediction su(4) for the flavor symmetry of the 5d SCFT corresponding to the above geometry.
In conclusion, we see that the methods of this paper can be used to perform RG flows of the kind which are invisible to CFDs. Moreover, we have arrived at the same theory (5.13) from two different 5d KK theories and both ways of arriving at this theory lead to the same prediction for the flavor symmetry of the corresponding 5d SCFT (5.13). This demonstrates the robustness of our proposal for determining flavor symmetries of 5d SCFTs.