Flavor symmetry of 5d SCFTs. Part II. Applications

In Part I of this series of papers, we described a general method for determining the flavor symmetry of any 5d SCFT which can be constructed by integrating out BPS particles from some 6d SCFT compactified on a circle. In this part, we apply the method to explicitly determine the flavor symmetry of those 5d SCFTs which reduce, upon a mass deformation, to some 5dN\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{N} $$\end{document} = 1 gauge theory carrying a simple gauge algebra. In these cases, the flavor symmetry of the 5d gauge theory is often enhanced at the conformal point. We use our method to determine this enhancement.


Introduction
In this series of papers (Part 1 [1] and Part 2), we study the flavor symmetry algebras of 5d SCFTs 1 . In Part 1 [1], we provide a general recipe for computing the flavor symmetry of any 5d SCFT that can be obtained (on its extended Coulomb branch) by integrating out BPS particles from the extended Coulomb branch of a known 5d KK theory 2 . This is done by utilizing the construction of the extended Coulomb branch of a 5d KK theory in terms of M-theory compactified on Calabi-Yau threefolds (CY3) [10,10,12,24]. The flavor symmetry of this 5d KK theory is encoded in terms of P 1 fibered non-compact surfaces coupled to the compact surfaces inside the CY3. The RG flows associated to integrating out BPS particles lead to the decoupling of some of the non-compact surfaces, leading to a new set of non-compact surfaces which encodes the flavor symmetry of the resulting 5d SCFT.
In this part, we apply the method discussed in Part 1 [1] to explicitly determine the flavor symmetry of 5d SCFTs which reduce upon a mass deformation to a 5d N = 1 gauge theory with a simple gauge algebra, and can be obtained by integrating out matter from a 5d KK theory. See [20] for the list of all such 5d SCFTs which are known to exist at the time of writing of this paper.
So, consider a 5d SCFT T which admits a mass deformation to a 5d N = 1 gauge theory G. Let G carry a semi-simple gauge algebra g with matter content being organized as n i copies of hypermultiplets transforming in some irrep R i of g. Then, there is a classically visible flavor symmetry algebra f G that we can assign to G. If R i is a complex representation, then we obtain a factor of u(n i ) in f G . If R i is a strictly real representation, then we obtain a factor of sp(n i ) in f G . If R i is a pseudoreal representation, then n i is half-integral and we obtain a factor of so(2n i ) in f G . Moreover, for each simple gauge algebra g a appearing in the semi-simple gauge algebra g = ⊕ a g a , we obtain an additional u(1) a factor in the flavor symmetry algebra whose current is provided by the instanton number for g a . One might then wonder whether the full flavor symmetry algebra f T of T is the same as f G . It is well-known that this is not the case. In general, f G is only a subalgebra of f T , but an important point is that the rank of f G equals the rank of f T . This is usually stated by saying that the classical flavor symmetry f G of T is enhanced to f T at the superconformal point, and f T is then referred to as enhanced flavor symmetry. A classic example of enhanced flavor symmetry is provided by the Seiberg E n (where n ≤ 8) theories [2,3,25] which admit a mass deformation to su(2) gauge theory with n − 1 full hypers in fundamental representation. The classical flavor symmetry f G = so(2n − 2) ⊕ u(1) which is known to enhance for n ≥ 2 to f T = e n where e 5 := so(10), e 4 := su(5), e 3 := su(3) ⊕ su(2) and e 2 := su(2) ⊕ u (1).
We emphasize that the method for determining the flavor symmetry of a 5d SCFT described in Part 1 does not depend on the existence of a mass deformation reducing the 5d SCFT to a 5d gauge theory. That is, our method always captures the full enhanced flavor symmetry f T of the 5d SCFT T. In this part, we use our method to tabulate the 5d gauge theories with simple gauge algebra whose (associated classical) flavor symmetries are enhanced when they are UV completed into a 5d SCFT (where the precise meaning of the UV completion has been discussed above). See Section 2 for a quick reference list of such gauge theories, where we have arranged the gauge theories according to the rank of their gauge algebra. The detailed derivation of these results has been provided in the following Section 3.
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 [12]. We use some notation about P 1 fibered surfaces that can be found in Section 4.1 of Part 1. Background and notation about geometric construction of 5d N = 1 gauge theories can be found in Section 2 of [64] and Section 3.2 of [20]. Background on flops can be found in [16].

Flavor symmetry of 5d SCFTs: Summary of results
In this section, we collect our results for flavor symmetry of 5d SCFTs that admit a mass deformation to a 5d N = 1 gauge theory carrying a simple gauge algebra. These flavor symmetry of a subset of these theories has been studied from other points of view in [13-15, 19, 21, 23, 30-32, 34, 35, 38, 49-51, 54-56, 59] and our results agree with the analysis of those papers.
We will denote such theories as where g is the simple gauge algebra and n i R i denotes that the theory contains n i hypermultiplets in irreducible representation R i of g. To account for half-hypermultiplets, we allow n i to be half-integral for pseudo-real representations. We will further abbreviate the names of various irreducible representations as follows: • F denotes the fundamental representations for su(n) and sp(n), the vector representation for so(n), and irreducible representations of dimensions 7, 26, 27, 56 for g 2 , f 4 , e 6 , e 7 respectively.
• A denotes the adjoint representation.
• Λ n denote the irreducible n-index antisymmetric representations for su(n) and sp(n).
• S 2 denotes the 2-index symmetric representation for su(n).
• S denotes irreducible spinor representation for so(n).
Furthermore, for g = su(n) we have to specify a Chern-Simons level 3 k, which we include as a subscript of su(n), and describe such a theory as su(n) k + i n i R i . For sp(n) we sometimes have to specify a theta angle θ which can take values 0, π only, and we describe such a theory as sp(n) θ + i n i R i . The list of 5d gauge theories with simple gauge algebra that are known to UV complete to 5d SCFTs has been compiled in [20], to which we refer the reader. The only gauge theories in their list which cannot be obtained from 5d KK theories by integrating out BPS particles are as follows [18,20]: • f 4 + nF for 1 ≤ n ≤ 3.
In this section, we provide the flavor symmetry of all 5d SCFTs appearing in [20] except for the three kinds of theories listed above. We will use either T or T n to denote the theories and f(T) or f(T n ) to denote their flavor symmetries. Some 5d SCFTs can reduce to multiple 5d gauge theories (with a simple gauge algebra) if one deforms them by different mass parameters. In this case, one says that the different 5d gauge theory descriptions are related by 5d dualities. Below, we account for such dualities by placing an '=' sign between the different 5d gauge theory descriptions. For example, the 5d SCFTs appearing in (2.2) have two gauge theory descriptions; one of them being su(m + 2) n 2 + (2m + 8 − n)F, and the other being sp(m + 1) + (2m + 8 − n)F.
Below, we will only mention theories for which there is a non-trivial enhancement of flavor symmetry at the conformal point. The flavor symmetry for theories not being mentioned in this section, but appearing in [20], is simply the classical flavor symmetry associated to the gauge theory. As an example, for n = 2m + 7 and n = 2m + 8 in (2.2) there is no enhancement of classical flavor symmetry, and hence those cases are omitted. On the other hand, some of the gauge theories have an enhancement that is visible from the viewpoint of a dual gauge theory. Such cases are not omitted below. An example of such a case is (2.2) for 3 ≤ n ≤ 2m + 6.

Rank 3
su(4) n 2 + Λ 2 + (10 − n)F = T n = f(T 1 ) su (12) = f(T n ) u(12 − n) n ≥ 2 ; 1 ≤ n ≤ 9 ; (2.31) su(4) n−1 (2.32) (7) = f(T 2 ) so(12) ⊕ so (7) = f(T 3 ) su(6) ⊕ so (7) = f(T 4 ) su(2) ⊕ su(4) ⊕ so (7) = f(T n ) u(8 − n) ⊕ so(7) n ≥ 5 ; (2. 33) su(4) n−1 = so(7) + 5S + F = so(7) + 6S (2.52) (2.54) su(2) 3 n = 1, 2 ; (2.58) sp(4) ⊕ so (7) 1 ≤ n ≤ 3 ; (2.69) (2.74) (2.76) (2.83) (2.96) (2.98) Let us start with the derivation of (2.2). The theories sp(m + 1) + (2m + 8 − n)F can be obtained from sp(m + 1) + (2m + 8)F (3.1) by integrating out fundamental hypers. It is known that the 5d N = 1 gauge theory (3.1) is a 5d KK theory and can be obtained by an untwisted circle compactification of the 6d SCFT whose tensor branch description is provided by the 6d N = (1, 0) gauge theory sp(m) + (2m + 8)F. We denote this fact by an equation of the following form where the notation for 5d KK theories is borrowed from [12]. According to [20,64], the above equality can be seen geometrically as follows. Consider the resolved CY3 geometry described by Now applying the isomorphism S (which exchanges e and f in a surface F b 0 ) on the left-most surface of (3.3) leads to the geometry which describes the 5d gauge theory sp(m+1)+(2m+8)F. This isomorphism establishes (3.2). Since the 6d SCFT 1 sp(m) has a so(4m + 16) flavor symmetry, we expect to be able to couple the geometry (3.3) to a collection of non-compact P 1 fibered surfaces N i such that their associated intersection matrix 4 gives rise to the Cartan matrix for the affine Lie algebra so(4m + 16) (1) . According to the gluing rules, this coupling takes the following form where N i denote 5 the non-compact surfaces corresponding to so(4m + 16) (1) . The e curves living in N i are non-compact sections whose crucial property is that We emphasize that any section in N i satisfying (3.8) is being denoted by e in our notation. Correspondingly different appearances of e for a single non-compact surfaces should be regarded as two different sections which may not even be in the same homology class inside the surface. For example, there are three such sections of N 2 appearing in (3.7), namely the curves gluing N 2 to N 0 , N 1 and N 3 . Despite all these three sections being denoted by e, these three sections should be understood as three different sections without any apriori relationship between their homology classes inside N 2 . Now to integrate out an F of sp(m + 1), we have to first flop the curve f − x 1 living in F 2m+8 0 of (3.9) to obtain the following geometry where we have relabeled the blowups living in the resulting surface F 2m+7 1 . The flopped curve can be identified with the blowup x living in N 1 . To complete the process of integrating out of the flavor, we have to expand this blowup x to infinite volume while keeping all the curves living in the compact surfaces at finite volume. In particular, we need to keep the curve f − x 1 living in F 2m+7 1 , which, since it is identified with the curve f − x living in N 1 , implies that the curve f living in N 1 must go to infinite volume as well. Thus, the P 1 fibration of the non-compact surface N 1 is destroyed once we integrate out the flavor. After this process, we obtain the following geometry comprised of compact surfaces and P 1 fibered non-compact surfaces f e e e e e e e e (3.11) which implies that sp(m+1)+(2m+7)F carries an so(2m+16) flavor symmetry, as can be seen by computing the intersection matrix of the remaining P 1 fibered non-compact surfaces in the above geometry. Now, removing another flavor corresponds to flopping f − x 1 living inside F 2m+7 1 of (3.11). This leads to the following geometry where we have again relabeled the blowups on the resulting surface F 2m+6 2 . By similar argument as above, sending the volume of the blowup x living in N 2 to infinity decouples the surface N 2 , and we are left with the geometry f e e e e e e (3.13) implying that the flavor symmetry for sp(m + 1) + (2m + 6)F is so(2m + 12) ⊕ su (2).
At the next step, an interesting phenomenon occurs. Integrating out another flavor corresponds to flopping f − x 1 living in F 2m+6 2 of (3.13) and leads to Thus, integrating out this flavor decouples two non-compact surfaces namely N 0 and N 3 , thus reducing the rank of the non-abelian part of the flavor symmetry by two. However, since we have only integrated out a single flavor, the rank of the full flavor symmetry algebra should only reduce by one. This implies that a u(1) factor should arise in the full flavor symmetry algebra of the resulting theory. That is, the flavor symmetry for sp(m + 1) + (2m + 5)F should be so(2m + 10) ⊕ u(1). In this paper, we are not going to track u(1) factors in the geometry, but instead track them by matching the rank of the non-abelian part of the flavor symmetry (as deduced from geometry) with the rank of the full flavor symmetry, in order to obtain the number of missing u(1) factors.
Continuing in this fashion we observe that the geometry for sp(m + 1) + F contains no non-compact P 1 fibered surfaces. Consequently, the geometry for pure sp(m + 1) θ won't contain any non-compact P 1 fibered surfaces, irrespective of the value of θ. Thus, the flavor symmetry for sp(m + 1) θ with m ≥ 1 is u(1) for θ = 0, π.

Derivation of (2.3):
To produce theories listed in (2.3), we start with which is implemented by doing an S transformation on both F 2m+8 0 and F 0 in (3.7), which gives to the bottom-most compact surface F 0 to rewrite the above geometry as The theory su(m + 2) 0 + (2m + 6)F is produced by flopping f − x living in F 1 0 and f − x 1 living in F 2m+7 0 out of the geometry. This leads to the geometry The reader can verify in the same way as above that demanding all curves inside compact surfaces to have finite volume implies that f of N 2m+7 goes to infinite size. According to the above geometry, we find that the flavor symmetry for su(m + 2) 0 + (2m + 6)F is su(2m + 8). Subsequent theories in (2.3) are produced by flopping and integrating out the curves f − x i living in the top-most compact surface as discussed above for the case of (2.2).

Derivation of (2.4):
Let us flop x 2m+6 from the top-most compact surface to the bottom-most compact surface in (3.18). This leads to the geometry The theory su(m + 2) 0 + (2m + 4)F is produced by integrating out f − x in F 1 1 and f − x 1 in F 2m+5 1 . Other theories in (2.4) are produced by successively integrating out f − x i from the top-most compact surface. The reader can easily check that integrating out these curves leads precisely to the results mentioned in (2.4). The reader can also check that the theories su(m + 2)n−1 for m, n, p ≥ 1 that can also be produced by integrating out matter from (3.15) have no enhancement of flavor symmetry.

Derivation of (2.5) and (2.6):
We can produce these theories by integrating out fundamental matter from the KK theory The flavor symmetry for the 6d SCFT  is produced by applying S on the top-most and bottom-most compact surfaces (3.26) The theories in (2.5) are produced by integrating out curves f − x i from the top-most compact surface of the above geometry. The first step corresponds to integrating out f − x m+6 and we can see that it destroys the P 1 fibration of the surface N 3 thus leading to an su(m + 8) non-abelian part of the flavor symmetry. Combining it with the extra u(1) flavor symmetry descending from the 6d SCFT we find that the flavor symmetry for su(m + 1)1 2 + Λ 2 + (m + 6)F is u(m + 8), as claimed in (2.5). The reader can similarly check the remaining claims in (2.5).
The theories in (2.6) can be produced by first integrating out f − x from the bottom-most compact surface followed by integrating out the curves f − x i from the top-most compact surface in (3.26).
Finally, note that we have only derived (2.6) for m ≥ 5. For m = 4, we will derive it in Section 3.5.

Derivation of (2.7-2.10):
This class of theories can be produced by integrating out matter from the KK theory The corresponding 6d SCFT has an e 8 ⊕ su(2) flavor symmetry. The e 8 factor arises from the sp(0) node and the su(2) factor is a delocalized flavor symmetry associated to the su(1) nodes. Correspondingly we expect that the compact part of the geometry for the above KK theory can be coupled to non-compact P 1 fibered surfaces whose intersection matrix comprises the Cartan matrix for e (1) 8 ⊕ su(2) (1) . We will denote the non-compact surfaces comprising e (1) 8 as N i and the non-compact surfaces comprising su(2) (1) as M i . The geometry can be written as x, y which manifests the sp(m + 1) + Λ 2 + 8F 5d gauge theory description of the KK theory. The theories in (2.7) can be produced by successively integrating out x i living in the top-most compact surface. It is easy to read how these flops affect the non-compact surfaces. At the first step, integrating out x 8 integrates out N 0 and M 1 , thus leading to an e 8 ⊕ su(2) flavor symmetry. Subsequent flops only affect the surfaces N i and so an su(2) factor is present in the flavor symmetry for all 5d SCFTs in this class. The geometry for can be written as Now, integrating out x living in the top-most compact surface leads to the theory sp(m + 1) 0 + Λ 2 , while integrating out f − x living in the top-msot compact surface leads to the theory sp(m + 1) π + Λ 2 . The former RG flow preserves both N 7 and M 0 while the latter RG flow only preserves M 0 , thus implying that the flavor symmetry is su(2) 2 when θ = 0 but only u(2) when θ = π. Combining this with the duality we derive the results (2.8-2.10).

Derivation of (2.11-2.15):
These theories can be produced by using the KK theory The corresponding 6d SCFT has an e 7 ⊕ su(2) 3 flavor symmetry. The e 7 arises from the sp(0) node, one su(2) arises from the two fundamental hypers situated at the left end of the chain of su(2) nodes, one su(2) arises from the two fundamental hypers situated at the right end of the chain of su(2) nodes, and one su(2) is a delocalized symmetry rotating all the bifundamentals between the su(2) nodes. For m even, we write the geometry for the KK theory as (3.33) where we have have labeled the compact surfaces as i b n which denotes F b n and i is simply a label allowing us to refer to this surface as S i , which we shall do in what follows. We have also displayed all the P 1 fibered non-compact surfaces. However, we have omitted all the "mutual" edges, that is edges between compact and non-compact surfaces, and edges between non-compact surfaces comprising different simple factors of the flavor symmetry algebra (or its affinized version). The data of these omitted edges is displayed in the following gluing rules: For m odd, we write the geometry for the KK theory as along with the following gluing rules The theories in (2.11) are produced by successively integrating out x i living in S m+1 . This integrates out P 0 , Q 1 and M 0 for m even, and P 0 , Q 0 and M 0 for m odd. The affect on surfaces N i is same in both cases. Thus the flavor symmetry takes the form f⊕su(2) 3 (where the subfactor f originates from the surfaces N i ) irrespective of whether m is even or odd.
To produce theories in (2.12), we first integrate out f − x 1 living in S m+1 , which integrates out N 1 , P 1 , Q 0 , M 0 for m even, and N 1 , P 1 , Q 1 , M 0 for m odd. Then, we successively integrate out other x i living in S m+1 . The combined effect is that only M 1 survives out of the surfaces M i , P i and Q i , irrespective of whether m is even or odd. The effect on N i is same for both cases. Thus non-abelian part of the global symmetry takes the form f ⊕ su(2) for all these theories.
To produce theories in (2.13), we first integrate out In all these cases only M 1 survives out of the surfaces M i , P i and Q i . Thus the non-abelian part of the flavor symmetry takes the f ⊕ su(2) where f is read from the surviving N i .

Derivation of (2.16-2.18):
These theories can be produced by using the KK theory where the corresponding 6d SCFT has an so(16) ⊕ su(2) 2 flavor symmetry. The so (16) factor arises from eight fundamental hypers charged only under sp(1), one su(2) factor arises from the two fundamental hypers charged under the right-most su(2) node only, and the other su(2) factor corresponds to a delocalized symmetry rotating all the bifundamentals. For m odd, the geometry can be written as (3.36) along with the following gluing rules: For m even, the geometry can be written as (3.37) along with the following gluing rules: The theories in (2.16) are produced by integrating out x i living in S m+1 , the theories in (2.17) are produced by integrating out f − x 1 before integrating out remaining x i living in S m+1 , and the theories in (2.18) are produced by integrating out

Derivation of (2.19):
These theories can be produced by using the KK theory for which the corresponding 6d SCFT has an e 7 ⊕ su(2) 2 flavor symmetry. For m odd, we write the geometry as where one of the su(2) factors in the flavor symmetry is represented in a non-affine form (via surface M 1 ) in order to simplify the presentation. This lack of information does not influence the computation of flavor symmetry for 5d SCFTs appearing in (2.19) because of the following reason: In the affinized form, this su(2) flavor symmetry of the 6d SCFT appears as two noncompact surfaces M 0 , M 1 with intersection matrix being the Cartan matrix for su(2) (1) . After an RG flow to a 5d SCFT, either one of these surfaces or both of these must be integrated out since the flavor symmetry of a 5d SCFT can not have an affine Lie algebra as a factor. We will see that this RG flow does not integrate out M 1 , thus M 0 must have been integrated out.
The gluing rules for the above geometry are: • y 1 , y 2 in S m+2 are glued to x 1 , x 2 in P 0 .
For even m, we write the geometry as After doing some flops, we can write the geometry for m odd as (3.41) along with the following gluing rules Similarly, performing some flops, we can write the geometry for m even as (3.42) along with the following gluing rules After performing an isomorphism on S 2m+1 , we can write the geometry for odd m as (3.43) which manifests the 5d gauge theory description of the 5d KK theory. The following gluing rules are Similarly, the geometry for m even takes the form (3.44) along with the following gluing rules The theories in (2.19) are produced by successively integrating out x i living in S m+1 . As can be seen from the gluing rules, any such RG flow integrates out P 0 for m odd and P 1 for m even, while preserving M 1 in both cases. The effect on N i is same for both cases.

Derivation of (2.20):
These theories can be produced using the KK theory for which the corresponding 6d SCFT has an so(16) ⊕ su(2) flavor symmetry. Again, it turns out to be enough to know the coupling of the compact part of the geometry only to non-affinized su (2), as in the previous case 6 . For m even, we write the geometry as For m odd, we write the geometry as By performing similar manipulations as for the case of (2.19), we can rewrite the above geometry for m even as (3.48) along with the following gluing rules: and the geometry for odd m as (3.49) along with the following gluing rules: The theories in (2.20) are obtained by successively integrating out x i living in S m+2 from the above two geometries.

Derivation of (2.21) and (2.22):
These theories can be constructed from the KK theory The corresponding 6d SCFT is The matter content is a bifundamental along with m fundamentals charged under each su(m). For m ≥ 3, there is a u(1) symmetry rotating the bifundamental and an su(m) 2 symmetry rotating the two sets of fundamentals. After twisting, the u(1) survives, while the two flavor su(m)s are identified with each other. Thus, for m ≥ 3, we expect to be able to couple the compact part of the geometry to non-compact P 1 fibered surfaces whose intersection matrix is the Cartan matrix of su(m) (1) .
For m ≥ 3 and m = 2n, the geometry can be written as (3.53) In both the cases, the gluing rules are as follows: The theories in (2.21) for m ≥ 3 are produced by successively integrating out y i living in S m . To produced the theories in (2.22), we have to first integrate out f − y 1 living in S m , which integrates out N m−2 . Then, we successively integrate out other y i living in S m .

Derivation of (2.23):
These can be produced using the 5d KK theory so(m + 2) + mF = 2 su(m) (2) (3.54) We expect to be able to couple the compact part of the geometry to P 1 fibered noncompact surfaces with intersection matrix being the Cartan matrix for su(2m) (2) . For m = 2n, the geometry is and, for m = 2n + 1, the geometry is The theories in (2.23) are obtained by successively integrating out x i living in the top-most compact surface.

Rank 1 Derivation of (2.24) and (2.25):
These can be produced by integrating out fundamentals from the KK theory su(2) + 8F = 1 where the corresponding 6d SCFT has an e 8 flavor symmetry. The geometry for the KK theory is Removing fundamentals corresponds to successively integrating out x i from the compact surface. This leads to the enhanced flavor symmetries shown in (2.24). The geometry from su(2) + F is found to be The integrating out process corresponds to sending the volume of x to infinity in the above geometry, which implies that the P 1 fibration for the non-compact surface is destroyed, thus implying that the flavor symmetry of pure 5d su(2) π gauge theory is u(1).

Rank 2
Derivation of (2.26-2.28): These theories can be produced by integrating out matter from the following KK theory (3.62) The 6d SCFT is produced by compactifying the 6d SCFT with a twist by charge conjugation symmetry which acts on the flavor su(12) by an outer automorphism, and thus we expect to couple the geometry corresponding to (3.64) to a collection of non-compact surfaces with intersection matrix su(12) (2) . According to the gluing rules presented in previous sections, the combined geometry (after some flops) can be written as The g 2 description is obtained after applying S on the top-most compact surface To remove the first fundamental, we have to first flop f − x 6 living in the top-most compact surface to obtain and now we integrate out f − x living in the bottom-most surface which destroys the P 1 fibration of N 1 . As a result, we find that g 2 + 5F has an enhanced sp(6) flavor symmetry. In a similar fashion, one can successively integrate out the curves f − x i living in the top-most compact surface to obtain the flavor symmetry for other theories mentioned in (2.26) and (2.27).
To derive (2.28), we write the geometry for g 2 + F in the su(3) frame as follows Pure su(3) 6 is produced by integrating out f − x living in the top compact surface which preserves N 6 , thus leaving an su(2) flavor symmetry as claimed in (2.28).

Derivation of (2.29):
Let us start with the geometry for g 2 + 3F as derived from the above analysis We can express this as the geometry for sp(2) + 2Λ 2 + F by first applying I 1 on the top-most compact surface using the blowup x 3 , and then applying S on the top-most compact surface. After performing these isomorphisms, the geometry is written as

Derivation of (2.30):
This can be produced using the KK theory The corresponding 6d SCFT is The matter content is a bifundamental along with 2 fundamentals charged under each su (2). There is an su(2) flavor symmetry rotating the bifundamental and an su(2) 2 flavor symmetry rotating the two sets of fundamentals. After twisting, the su(2) associated to bifundamental survives, while the other two flavor su(2)s are identified with each other. Thus, we expect to be able to couple the compact part of the geometry to non-compact P 1 fibered surfaces whose intersection matrix is the Cartan matrix of su(2) (1) ⊕ su(2) (1) . Indeed, the geometry can be written as The theory in (2.30) is produced by integrating out f − x living in the top compact surface. This integrates out N 0 and M 1 leaving an su(2) 2 flavor symmetry.

Rank 3 Derivation of (2.31) and (2.32):
These theories can be obtained by integrating out matter from the case m = 3 of (3.21), but the flavor symmetry of the corresponding 6d SCFT is su(12) instead of u(m + 8) = u (11). The geometry for su(4) 0 + Λ 2 + 10F coupled to P 1 fibered non-compact surfaces corresponding to su(12) (1) is These can be produced from the KK theory The 6d SCFT has an e 7 ⊕so(7) flavor symmetry where the e 7 part is the flavor symmetry associated to the sp(0) node and so (7) is the flavor symmetry associated to the su(2) node 7 . The geometry for the KK theory is then found to be 7 and M i parametrize non-compact surfaces corresponding to so(7) (1) . su(4) 0 + 2Λ 2 + 8F is obtained by ap-7 Naively one might think that there is an so(8) flavor symmetry rotating the 4 fundamental hypers charged under su(2). However, it is known that there is a reduction in the rank of the flavor symmetry and the flavor symmetry is infact so(7) with the 4 hypers transforming in the strictly-real spinor representation of so (7).  x-y e (3.77) We can then produce su(4) 1 2 +2Λ 2 +7F by integrating out y 4 living in the middle compact surface, which removes N 6 and M 1 implying that the flavor symmetry is e 7 ⊕ so (7). Removing other y i living in the middle compact surface we reach su(4) 2 + 2Λ 2 + 4F. To go beyond this point and obtain other theories in (2.33), we need to successively integrate out the curves f − x i living in the middle compact surface. The reader can verify that these processes lead to the flavor symmetry claimed in (2.33).

plying S on the top-most and bottom-most compact surfaces to obtain
To obtain theories in (2.34), we first integrate out x 4 , which decouples N 0 and M 0 , and then successively integrate out y i living in the middle compact surface until we reach su(2) 3 2 + 2Λ 2 + 3F, from which point onward we integrate out the remaining f − x i living in the middle compact surface. Similarly, to obtain theories in (2.35), we first integrate out x 4 , x 3 , y 4 (in that order), which decouples N 0 , N 1 , N 6 , M 0 and M 1 , and then successively integrate out y i living in the middle compact surface until we reach su(2) 1 + 2Λ 2 + 2F, from which point onward we integrate out the remaining f − x i living in the middle compact surface. In a similar fashion, we can also obtain theories in (2.36) and (2.37) by integrating out more x i before we start integrating out

Derivation of (2.38):
These theories can be produced from the KK theory The su(1) node as an sp(1) flavor symmetry which is fully gauged by the sp(1) node. The sp(1) node carries 10F of sp(1) but a 1 2 F is trapped in coupling to su(1), leaving an so (19) flavor symmetry for the corresponding 6d SCFT. The geometry for the 5d KK theory turns out to be To obtain the su(4) 3 2 + 2Λ 2 + 7F of the above geometry, we have to first apply S on the top-most and bottom-most compact surfaces, and flop x, y living in the middle compact surface, to obtain

.80)
Applying some isomorphisms upon the top-most compact surface, the above geometry can now be written as When all the fundamentals are integrated out, we obtain an so(5) ⊕ u(1) flavor symmetry, which is the classical flavor symmetry, not only for su(4) 5 + 2Λ 2 , but also for the dual gauge theory sp(3) + 1 2 Λ 3 + 5 2 F. Thus, the theories obtained by integrating out more fundamentals from sp(3) + 1 2 Λ 3 + 5 2 F would have no enhancement of flavor symmetry either.

Derivation of (2.39) and (2.40):
For the theories in (2.39), we use the KK theory which allows the coupling of non-compact surfaces comprising an e (1) 6 as shown in the geometry below Here we are displaying the compact surfaces in the geometry such that they manifest the 5d sp(3) gauge theory description. To manifest the KK theory description, one should apply S on the top-most compact surface. The 5F of sp(3) originate from the blowups x i living in the compact surface denoted by F 5+1+1+1 1 . Integrating out these blowups leads to the results claimed in (2.39). Now, let us consider the geometry when all 5F of sp(3) have been integrated out where we manifest the su(2) non-abelian flavor symmetry of the theory. After perform-ing some flops detailed in [20], we can write the above geometry as Applying S on the bottom-most compact surface of the above geometry, we obtain a geometry for su(4) 13 Now, we see that the curve f − x living in the bottom-most compact surface of the above geometry does not intersect the non-compact surface N 5 . Thus, the process of integrating it out preserves the non-abelian flavor symmetry su(2), and we obtain the result (2.40).

Derivation of (2.41-2.46):
We start with the 6d SCFT which has sp(7) flavor symmetry. The untwisted compactification of the above 6d SCFT is known to give rise to the 5d gauge theory so(7) + 5S + 2F [20]. The geometry for this 5d gauge theory can then be figured out to be x , y f 2e-x The two fundamentals of so(7) are encoded differently. One of the fundamentals corresponds to the blowups x, y in the bottom-most compact surface, and the other fundamental corresponds to the blowup y in the middle compact surface. We can integrate out one of the fundamentals by integrating out the curve f − y living in the middle surface. This process integrates out N 6 , thus leading to the sp(6) ⊕ su(2) flavor symmetry claimed in (2.41). Another fundamental can be integrated out by integrating out x, y living in the bottom-most compact surface, which integrates out N 7 and N 5 thus verifying the n = 1 case of (2.42).
The 5S are encoded in the 5 blowups x i living in the middle compact surface. They are integrated out by successively integrating out f −x i . Integrating out f −x 5 integrates out N 4 thus verifying the n = 2 case of (2.42). The reader can check that integrating out further S leads to a flavor symmetry pattern which shows no enhancement.
Similarly, the reader can also recover the results claimed in (2.43) and (2.44) by using the above geometry.
To obtain (2.45), let us have a look at the geometry for so(7) + 3S + 2F Flopping x, y living in the bottom-most compact surface, we get which can be written, after applying some isomorphisms on the middle compact surface, which identifies the above geometry as describing su(4) 2 + 3Λ 2 + 2F. We can obtain the n = 1 case of (2.45) by integrating out f − y 2 living in the middle compact surface, and the n = 2 case of (2.45) by further integrating out y 1 living in the middle compact surface. The reader can check that this leads to the results claimed in (2.45).
On the other hand, if we would integrate out f − y 1 after integrating out f − y 2 from the above geometry, we would obtain su(4) 3 + 3Λ 2 and read the flavor symmetry from the geometry to be sp(3) ⊕ u(1) which is indeed the classical flavor symmetry.
To obtain (2.46), we start from the geometry for so(7) + 4S + 2F obtained from (3.88) and convert it into a geometry for su(4) 3 2 + 3Λ 2 + 3F in a similar way as explained from which su(4) 0 + 3Λ 2 is produced by integrating out f − z and y i living in the middle compact surface. As is clear from the above geometry, these processes integrate out N 6 and N 7 leaving an sp(4) flavor symmetry as claimed in (2.46).

Derivation of (2.47-2.49):
These theories can be manufactured starting from the KK theory This theory allows the coupling of non-compact surfaces describing e 6 ⊕ su(6) (2) . This is because the corresponding 6d SCFT has an e 6 flavor symmetry coming from the sp(0) node and an su(6) flavor symmetry coming from the su(3) node as it carries 6 hypers transforming in fundamental of su(3). A discrete symmetry of the theory can be constructed if a Z 2 outer automorphism acts simultaneously on all these algebras.
According to [20], this KK theory can be described as so(7) + 4S + 3F and the geometry can be figured out to be  For (2.49), we can represent the geometry for so(7) + 3S + 3F, after some flops as Applying S on the bottom-most compact surface, we obtain an su(4) 5 2 + 3Λ 2 + 3F description of the geometry The theories in (2.49) are then obtained by integrating out f − x i living in the bottommost compact surface.

Derivation of (2.50):
These theories can be produced from the KK theory The corresponding 6d SCFT is to obtain the correct geometry, which can be worked out to be After applying some isomorphisms on the top-most compact surface, we can rewrite the above geometry as which manifests the sp(3) + 1 2 Λ 3 + Λ 2 + 5 2 F description of the KK theory. The fundamentals can now be integrated out by integrating out the blowups x 1 , y 1 living in the top-most compact surface. Integrating out x 1 integrates out N 3 leaving an sp(3) flavor symmetry, and further integrating out y 1 integrates out N 2 leaving an sp(2) flavor symmetry.

Derivation of (2.51):
This theory can be produced by using the KK theory so(7) + 2S + 4F = 1 su(5) (2) (3.103) The non-compact surfaces coupled to the above KK theory comprise an su(13) (2) and the geometry can be written in the following fashion (after some flops) 2e-x which describes so(7) + 2S + 4F gauge theory. We can remove a spinor by integrating out the blowup x living in the bottom-most compact surface which integrates out N 6 leaving an sp(6) flavor symmetry.

Rank 4
Derivation of (2.52) and m = 4 case of (2.6): These theories can be obtained by integrating out matter from the case m = 4 of (3.21), but the flavor symmetry of the corresponding 6d SCFT is su(12) ⊕ su(2) instead of u(m + 8) = u (12). The geometry is The theories (2.52) are produced by successively integrating out f − x i living in the top-most compact surface. m = 4 case of (2.6) is produced by first integrating out x 1 living in the top-most compact surface and then successively integrating out f − x i living in the top-most compact surface.

Derivation of (2.53):
These theories can be produced using the KK theory The sp(0) node int he corresponding 6d SCFT gives rise to an e 7 flavor symmetry, and as we have discussed above, if the su(1) node was absent, then the flavor symmetry corresponding to the su(2) node would be so (7), so that the 8 half-hypers in the fundamental of su(2) transform as a spinor of so (7). We know that the su(1) node traps a half-hyper in the fundamental. Correspondingly, so (7) is broken to g 2 with the 7 remaining half-hypers transforming in F of g 2 . The associated geometry can be worked out to be f -x (3.107) The theories in (2.53) are produced by integrating out x i followed by f − y i living in the compact surface denoted by F 4+3 2 in the above geometry.

Derivation of (2.54) and (2.55):
These theories can be produced by starting from the 6d SCFT (3.98), but this time compactifying it with a Z 2 outer automorphism twist instead of the Z 3 twist employed there. Thus, we can represent the KK theory as The geometry can be worked out to be where we have manifested the so(9) + 3S + 3F description. The KK theory description can be manifested by applying S on the compact surface F 0 placed at the second position from the bottom of the diagram. The fundamentals can be integrated out by integrating out the curves x i living in the surface labeled F 3+3+3 0 . The reader can check that this leads to (2.54).
To obtain (2.55), we have to first apply S on F 3+3+3 0 in the above geometry to to obtain the geometry for su (5) 2e-x i e y 3 , z 3 The theories in (2.55) are then obtained by integrating out x i living in F 2+2+3 1 .

Derivation of (2.56) and (2.57):
We can manufacture these theories by using the KK theory (3.112) whose geometry is The fundamental of so(9) can be integrated out by integrating out x, y living in F 1+1 2 . This integrates out M 0 and N 4 leaving an su(2) ⊕ sp(4) flavor symmetry as claimed in (2.56). After doing some flops and isomorphisms (whose details can be found in [20]) on the geometry associated to so(9) + 4S, we can rewrite it as follows which describes su(5) 2 + 3Λ 2 + F. The so(9) description can be recovered by applying S onto F 3+3+1 0 . Now to obtain (2.57), we have to integrate out f − z living in F 3+3+1 0 which integrates out N 3 and leads to the claimed flavor symmetry.

Derivation of (2.58):
These can be produced by integrating out matter from the KK theory -90 -whose associated geometry is e e e f f (3.116) The KK theory description is recovered by applying S on the surface labeled as F 0 (without any blowups). The theories in (2.58) are obtained by successively integrating out z i living in the top-most compact surface.

Derivation of (2.59):
These theories can be obtained from the KK theory so(9) + 2S + 5F = 1 -91 -whose associated geometry is where we have made the so(9) description manifest. To make the KK description manifest, the reader can apply S on the surface labeled F 0 . (2.59) are obtained by integrating out x i living in the compact surface F 5 1 . (2.60) can be obtained by integrating out a spinor, which corresponds to integrating out half of the blowups blowups living in F 2+2+2+2 2 . This process involves too many flops, hence we now turn to the discussion of another KK theory which allows an easier derivation of (2.60).

Derivation of (2.60):
To produce the theories so(9) + S + (6 − n)F, we proceed using the KK theory = 1 su(7) (2) so(9) + S + 6F (3.119) whose associated geometry takes the form The fundamentals are integrated out by integrating out x i living in F 6 1 . Integrating out x 1 , we see that N 5 is integrated out, leading to an sp(2) ⊕ sp(5) flavor symmetry, as claimed in (2.60). Subsequently integrating out x 2 leads to integrating out N 4 and N 6 leading to an sp(4) ⊕ sp(1) ⊕ u(1) flavor symmetry, which is just the classical flavor symmetry associated to so(9) + S + 4F.

Derivation of (2.62):
We can produce it by integrating out C from the KK theory = 1 su(6) (2) so(8) + S + C + 5F (3.123) whose associated geometry is C can be integrated out by integrating out x 2 living in F 2+2 0 which integrates out N 0 leaving an sp(7) flavor symmetry.

Derivation of (2.63) and (2.64):
These can be constructed using the KK theory -95 -whose associated geometry can be written as . At the first step, we integrate out x 3 , which integrates out M 0 and N 4 leaving an sp(4) ⊕ so(9) flavor symmetry. At the next step, we further integrate out x 2 , which further integrates out M 1 leaving an sp(4) ⊕ su(4) flavor symmetry. This verifies (2.64).

Derivation of (2.65):
These theories can be produced using the KK theory = 1 so(7) (1) so(8) + 3S + 2C + 3F (3.127) -96 -whose associated geometry can be written as An S can be removed by integrating out f − y 3 living in F 2+3+3 0 . This destroys the P 1 fibrations of N 2 and M 2 leaving an sp(2) 2 ⊕ sp(4) flavor symmetry, thus verifying n = 1 case of (2.65). The geometry for so (8) F can be integrated out by successively integrating out f − z i living in F 2+2+3
To obtain the n = 2 case of (2.66), we need to look at the geometry for = 1 in more detail as in following e-x 1 , x 13 The above geometry manifests the su(6) 0 + 1 2 Λ 3 +13F description of the theory. The KK theory description can be manifested by applying S on the top-most and the bottommost compact surfaces. To obtain the n = 2 case of (2.66), we need to integrate out x 13 and f − x 1 living in the bottom-most compact surface. x 13 integrates N 12 and f − x 1 integrates out N 1 , thus leaving a u(11) ⊕ su(2) flavor symmetry as claimed in (2.66).

Derivation of (2.67):
For 1 ≤ n ≤ 8, these theories are dual to which were already studied in (2.7). To study the n = 9 case, we consider the geometry for sp(5) + Λ 2 h h+f e e+f -y thus implying that the geometry also constructs su(6) 7 + 1 2 Λ 3 + F. Integrating out y in top-most compact surface leads to su(6) 15 2 + 1 2 Λ 3 , and since y does not intersect N, we verify the result quoted in (2.67).

Derivation of (2.68) and (2.69):
These theories can be constructed by using the KK theory = 1 whose associated geometry can be found to be

Derivation of (2.70):
These theories are dual to su(6) 3+n for 1 ≤ n ≤ 7, for which the flavor symmetry was derived in the detailed discussion for (2.20) (notice that the answer quoted there holds true for n = 7 as well, but it is not displayed since it matches the classical flavor symmetry for su(6) 5 + 2Λ 2 ). For n = 7, the geometry can be written as which manifests su(6) 5 + 1 2 Λ 3 + Λ 2 + F description. Integrating out z living in top-most compact surface leads to su(6) 11

Derivation of (2.71) and (2.72):
These theories can be produced using the KK theory whose associated geometry can be written as h h+f e e+ f -z2 -w1 -x1 -y1 , e-z2 -w2 The theories in (2.71) are produced by successively integrating out f − z i living in F 2+2+2+2

0
. Integrating out f − z 2 removes N 0 and N 3 . Subsequently integrating out f − z 1 removes N 1 as well. This verifies the result claimed in (2.71).
The theories in (2.72) are produced by successively integrating out z i living in F 2+2+2+2 0 . Integrating out z 1 removes N 0 and N 4 . Subsequently integrating out z 2 removes N 1 as well. This verifies the result claimed in (2.72).

Derivation of (2.73):
These theories can be obtained from the KK theory = 1 Including the flavor symmetries, we can represent the 5d KK theory as Using the gluing rules, we can figure out the geometry to be e-x, e-y z1 , z2 Integrating out f −z 2 removes P 1 , N 1 and M 0 , leading to an su(2) 2 ⊕g 2 flavor symmetry.
Further integrating out f −z 1 removes P 0 , thus leading to an su(2)⊕g 2 flavor symmetry. This verifies the results claimed in (2.73).

Derivation of (2.74-2.76):
These can be produced using the KK theory = 1 for which the geometry is

Derivation of (2.77):
These theories can be derived from the KK theory 2 su(1) (1) 1 The corresponding 6d SCFT has an e 8 flavor symmetry arising from the sp(0) node. The su(2) node carries 4 full hypers, out of which two half-hypers are trapped by the two su(1) nodes. As we know from before, if only one half-hyper is trapped, then the remaining 7 half-hypers transform as F of g 2 . Now, since another half-hyper is trapped, we expect the remaining 6 half-hypers to transform as F ⊕F of su (3). This would suggest that the KK theory admits a collection of non-compact surfaces comprising su(3) (1) . However, this leads to a wrong prediction for the flavor symmetry for su(6) 1 + Λ 3 + 8F. That is, it predicts an so(16) ⊕ u(2) flavor symmetry, while we know from the analysis for (2.74) that the correct flavor symmetry is so(16) ⊕ su(2) 2 .
We claim that the su(3) flavor symmetry actually affinizes to g (1) 2 rather than su(3) (1) , with the coupling of the corresponding non-compact surfaces shown below. This leads to the correct so(16) ⊕ so(4) flavor symmetry for su(6) 1 + Λ 3 + 8F as we verify below. The geometry manifesting the KK theory description is the following 148) The su(6) 3 2 + Λ 3 + 9F description is obtained by applying some flops and isomorphisms that can be found in [20]. This description is manifested by the following geometry e e x 1 -x 2 (3.149) Now, the theory su(6) 1 + Λ 3 + 8F can be obtained by integrating out f − x 1 living in F 9 1 , which can be seen to lead to the removal of N 7 and M 2 , implying that this theory has flavor symmetry so(16) ⊕ su(2) 2 . Thus, we see that if the affinization is g (1) 2 instead of su(3) (1) , then we obtain the correct flavor symmetry.
The theories (2.77) can be obtained by successively integrating out x i living in F 9 1 . The reader can check the results claimed in (2.77). For example, integrating out x 9 integrates out N 0 and M 0 , indeed leaving an e 8 ⊕ g 2 flavor symmetry.

Derivation of (2.78-2.80):
These theories can be produced using the KK theory The associated geometry can be written as 2e- 4f 4f To obtain (2.81), we have to successively integrate out x i living in F 3+3 6 .

Derivation of (2.82) and (2.83):
These can be produced using the KK theory = 1 so(10) (2) so(11) + 3 2 S + 5F (3.154) Drawing the geometry in a graphical form is quite tedious for this case. Thus, we only depict only partial data about the geometry graphically as follows where we have have labeled the compact surfaces as i b n which denotes F b n and i is simply a label allowing us to refer to this surface as S i , which we shall do in what follows. We have also displayed all the P 1 fibered non-compact surfaces. However, we have omitted all the "mutual" edges, that is edges between compact and non-compact surfaces, and edges between non-compact surfaces comprising different simple factors of the flavor symmetry algebra (or its affinized version). The data of these omitted edges is displayed in the following gluing rules: • e, e in S 4 are glued to Note that the gluings above should be read in the order they are presented. For instance, "C 1 , C 2 is glued to D 1 , D 2 " means that C 1 is glued to D 1 and C 2 is glued to D 2 . The theories in (2.82) are produced by successively integrating out x i living in S 1 . It is easy to see that during these processes M 1 is integrated out but M 0 always survives. Thus there is always an su(2) factor present in the flavor symmetry. The other non-abelian factors arise from the collection of surfaces N i and can be easily figured out from the above gluing rules.
To construct the theories in (2.83), we need to perform a few flops which lead to the following representation of the above geometry

Derivation of (2.84):
This theory can be obtained by using the KK theory The geometry can be written as (3.158) along with the following gluing rules: • f, f, f, f, f, f in S 1 are glued to x 4 − x 5 , y 4 − y 5 , x 6 − x 7 , x 8 − x 9 , y 6 − y 7 , y 8 − y 9 in N 0 .
• f, f in S 3 are glued to x 3 − x 7 , x 6 − x 8 in N 1 .
• f, f in S 4 are glued to x 5 − x 6 , x 2 − x 3 in N 1 .
• f, f in S 5 are glued to The theory (2.84) is produced by integrating out f − x 5 from S 2 which integrates out N 1 , thus leaving an su(2) 2 flavor symmetry.

Derivation of (2.86):
This theory can be obtained from the KK theory for which the geometry can be figured out to be The fundamentals can be integrated out by successively integrating out x i living in F 8+8 6 . Integrating out x 1 integrates out N 1 thus leaving an su(2) ⊕ sp(7) flavor symmetry as claimed in (2.86). Further integrating out x 2 , integrates out N 6 and N 8 thus leading to sp(6) ⊕ u(1) flavor symmetry, which shows no enhancement. Thus, there is no enhancement as we integrate out even more fundamentals.

Derivation of (2.87):
These theories can be obtained from the KK theory  -118 -for which the geometry can be written as x-y 164) The theories in (2.87) are produced by successively integrating out y i living in F 2+2 0 .

Derivation of (2.88):
This theory can be produced using the KK theory = 1 so(9) (1) so(10) + 3S + 4F (3.165) The geometry for this theory can be figured out to be 2e-x i (3.166) along with the following gluing rules: The fundamentals are integrated out by successively integrating out f − z i living in S 2 .
Integrating out f −z 4 integrates out N 3 , M 0 and M 3 , thus leaving a u(3)⊕sp(3)⊕su(2) flavor symmetry as claimed in (2.88). Now, further integrating out f − z 3 further integrates out N 2 and N 4 , thus leaving a u(3) ⊕ sp(2) ⊕ u(1) flavor symmetry, which is just the classical flavor symmetry. Hence, integrating more than one fundamental out of the KK theory leaves only a classical flavor symmetry without any enhancement.

Derivation of (2.89):
These theories can be produced using the KK theory  for which the geometry can be figured out to be 168) The fundamentals are integrated out by successively integrating out x i living in F 6+6 8 .

Derivation of (2.90):
This theory can be constructed using the KK theory f - (3.170) -121 -along with the following gluing rules: The fundamentals are integrated out by successively integrating out x i living in S 5 .

Rank 6
Derivation of (2.91-2.93): These can be produced using the KK theory for which the geometry can be figured out to be along with the following gluing rules: • e in S 4 is glued to x − y in N 6 .
• y 5 , y 6 in S 3 are glued to f − x, y in N 6 .
• y i − y i+1 in S 3 is glued to f in N i for i = 1, · · · , 5.
• e − y 1 − y 2 in S 3 is glued to f in N 0 . • • e in S 5 is glued to x 2 − x 3 in M 1 .
• f, f in S 6 are glued to The theories in (2.91) are produced by integrating out y i living in S 3 . The theories in (2.92) are produced by integrating out y i after integrating out f − y 1 living in S 3 . The theories in (2.93) are produced by integrating out (at least two) y i after integrating out f − y 1 , f − y 2 living in S 3 .

Derivation of (2.94):
These can be produced using the KK theory so(13) + S + 5F = 3 for which the geometry can be figured out to be 2e-x Derivation of (2.95): These can be produced using the KK theory = 1 sp(0) (1) so(13) + 1 2 S + 9F 2 su(9) (2) (3.175) The associated geometry can be written as along with the following gluing rules: • f, f in S i are glued to x i − x i−1 , y i−1 − y i in N 0 for i = 2, 3, 4.
• e in S 5 is glued to y 4 − x 4 in N 0 .
The theories in (2.95) are produced by successively integrating out x i living in S 1 .

Derivation of (2.96):
These can be produced using the KK theory h -x 1 x 3 -x 2 , y 4 -y 1 4f 4f The theories in (2.96) can be produced by successively integrating out y i living in the right-most compact surface.

Derivation of (2.97):
This theory can be produced using the KK theory along with the following gluing rules: • h − x 1 − y 2 − z 2 − w 1 in S 6 is glued to f in N 0 .
• h − x 2 − y 1 − z 1 − w 1 in S 6 is glued to f in N 1 .
• w 1 − w 2 in S 6 is glued to f in N 2 .
The theory in (2.97) is produced by integrating out w 2 living in S 6 which integrates out N 2 and M 1 , thus leaving an su(3) ⊕ su(2) flavor symmetry as claimed in (2.97).

Derivation of (2.98):
These can be produced using the KK theory  along with the following gluing rules: • h − x 2 − y 1 − z 1 − w 1 in S 1 is glued to f in M 1 .
• e in S 2 is glued to x 4 − y 4 in N 0 .
• x i − x i+1 , y i+1 − y i in S 6 are glued to f, f in N i for i = 1, · · · , 7.
• x 8 − y 8 in S 6 is glued to f in N 8 .
The theories in (2.98) are produced by integrating out y i living in S 6 .

Derivation of (2.99):
These can be produced using the KK theory along with the following gluing rules: • 2e + f − x i − y in S 1 is glued to x 5 − y 5 in N 0 .
• x i − x i+1 , y i+1 − y i in S 6 are glued to f, f in N i for i = 1, · · · , 8.
• x 9 − y 9 in S 6 is glued to f in N 9 .
The theories in (2.99) are produced by integrating out y i living in S 6 .

Derivation of (2.100):
This theory can be produced using the KK theory
• e − x 6 in S 2 is glued to x 2 in N 6 .
• x i+1 − x i in S 2 is glued to f in N i for i = 1, · · · , 5.
• x 1 in S 2 is glued to x 4 in N 0 .
• f in S 2+i is glued to x 4−i − x 5−i in N 0 for i = 1, 2, 3.
• f in S 6 is glued to f − x 1 − x 2 in N 0 .
It was shown in [20] that the above geometry is flop equivalent to the following geometry which manifests the so(12) + S + 1 2 C + 6F frame and we have again omitted the noncompact surfaces corresponding to su(2) (1) . The gluing rules between S i and N j are the same as above. Now, the gluing of flavor su(2) (1) to a geometry for the KK theory 1 so(11) (1) (3.190) was presented in Part 1. The geometry presented there can be turned into the above geometry (3.189) by first performing some perturbative flops and finally applying S upon the surface labeled as S 2 in (3.189). In this way, the coupling of flavor su(2) (1) to the compact surfaces S i in (3.189) can be figured out.
The F are integrated out from so(12) + S + 1 2 C + 6F if we successively integrate out f − x i living in S 2 of (3.189). We claim that the coupling of flavor su(2) (1) is such that both the non-compact P 1 fibered surfaces comprising su(2) (1) are integrated out. Thus, the non-abelian contribution to the flavor symmetry for 5d SCFTs so(12) + S + 1 2 C + (6 − n)F comes purely from the surfaces N i in (3.189). This leads to the result presented in (2.102). Now, to obtain the coupling of flavor su(2) (1) to compact surfaces in (3.188) (starting from the coupling of flavor su(2) (1) to the KK theory (3.190) presented in Part 1) requires performing a lot of non-trivial, complicated flops. Fortunately, the knowledge of precise coupling is not required to deduce the flavor symmetry for 5d SCFTs so(12) + 3 2 S + (6 − n)F. For this deduction, first note that the non-abelian part of the flavor symmetry of a 5d SCFT must be given by a finite semi-simple Lie algebra. Thus, as we integrate out the first F, at least one of the two non-compact surfaces comprising su(2) (1) must be integrated out.
The first F is integrated out by integrating out f − x 6 living in S 2 of (3.188) which integrates out N 5 leading to an su(2) ⊕ sp(5) contribution to the non-abelian part of the flavor symmetry of 5d SCFT so(12) + 3 2 + 5F. If this process integrates out only one of the non-compact surfaces comprising su(2) (1) , then the full flavor symmetry for so(12)+ 3 2 +5F would be sp(5)⊕su(2) 2 since an extra su(2) would be contributed to the non-abelian part of the flavor symmetry. If, on the other hand, this process integrates out both of the non-compact surfaces comprising su(2) (1) , then the full flavor symmetry for so(12) + 3 2 + 5F would be sp(5) ⊕ u(2) since no other factor would be contributed to the non-abelian part of the flavor symmetry.
We claim that only one of the su(2) (1) surfaces is integrated out and that (2.101) shows the correct flavor symmetry for this theory. To show this, let us assume, to the contrary, that the flavor symmetry for so(12) + 3 2 + 5F is sp(5) ⊕ u (2), that is the only non-compact P 1 fibered surfaces arising in the geometry for so(12) + 3 2 + 5F are N i for i = 0, · · · , 4 and N 6 . Let us integrate out another fundamental to obtain so(12)+ 3 2 +4F. This is done by integrating out f − x 6 , f − x 5 (in that order) living in S 2 of (3.188). We see that this process only leaves non-compact surfaces N i for i = 0, · · · , 3 intact thus implying that the non-abelian part of the flavor symmetry for so(12) + 3 2 + 4F is sp(4), but this is a contradiction since the non-abelian part of the classical flavor symmetry for so(12) + 3 2 + 4F is sp(4) ⊕ su(2).

Derivation of (2.103):
These can be produced using the KK theory   (3.194) along with the following gluing rules: • e in S 2 is glued to x 4 − y 4 in N 0 .
• x i − x i+1 , y i+1 − y i in S 6 are glued to f, f in N i for i = 1, · · · , 7.
• x 8 − y 8 in S 6 is glued to f in N 8 .
The theories in (2.104) are produced by integrating out y i living in S 6 .

Rank 7
Derivation of (2.105): These can be produced using the KK theory