More 5d KK theories

In this note, we discuss circle compactifications of 6d SCFTs for which a geometric M-theory construction is not known in previous literature.


Introduction
Recently, geometric constructions in M-theory have been very successful in progressing our understanding of 5d SCFTs and the 5d N = 1 gauge theories arising on the extended 1 Coulomb branch of 5d SCFTs [1][2][3][4][5][6][7][8][9][10][11][12][13][14] 2 . Thanks to this understanding, it has been possible to generate claims of obtaining a full classification of 5d SCFTs [2][3][4][10][11][12] which are backed by substantial evidence [2,14]. According to this classification proposal, all 5d SCFTs can be generated by performing special type of RG flows upon 5d theories obtained by compactifying 6d SCFTs on a circle of finite, non-zero radius. Such 5d theories are often referred to as 5d KK theories, and we will use this terminology throughout this paper. The special type of RG flows mentioned above can be understood as those processes that integrate out a set of BPS particles and strings [12] from the extended Coulomb branch of a 5d KK theory.
Such RG flows obtain a clean characterization when the 5d theories under discussion are constructed by compactifying M-theory on a local Calabi-Yau threefold (CY3) with an isolated singularity. The extended Coulomb branch of the 5d theory is obtained by resolving the singularity, and the resulting resolved CY3 can be described by a collection of intersecting compact Kahler surfaces. The RG flows under discussion are then 1 We define extended Coulomb branch to be the total space obtained by fibering the Coulomb branch over the space of supersymmetry preserving mass parameters.
2 See  for other recent related work on the subject of 5d N = 1 QFTs. mapped to motions on the extended Kahler cone of the resolved CY3 which decompactify a set of compact complex curves and surfaces. This map is a consequence of the fact that M2/M5 branes compactified on compact complex curves/surfaces produce BPS particles/strings in the resulting 5d theory, and the volumes of these curves/surfaces can be identified as the masses/tensions of the corresponding BPS particles/strings. Therefore, understanding the full set of 5d KK theories and the local CY3 associated to them is very important for the purposes of the classification program of 5d SCFTs based on the proposal described above. This task was undertaken by [3,4,11], but the work of [14] featured some 5d KK theories which did not appear in [3,4,11]. All such examples involve a twisted compactification of a 6d SCFT, which means that the observables in the 6d SCFT are acted upon by the action of a discrete global symmetry when transported around the circle. Twisted circle compactifications of 6d SCFTs and the CY3 associated to them were studied in [11]. This article is devoted to a study of 5d KK theories (and the associated CY3) that are missing from their analysis.
The twists discussed in this paper fall into the following two different classes: 1. In the first class are the twists whose associated discrete global symmetry acts by outer automorphism of a gauge algebra appearing in the low energy 3 6d N = (1, 0) gauge theory, and its action on the hypermultiplets cannot be represented as a permutation of the hypermultiplets. This is in contrast with the outer automorphism twists considered in [11], all of which could be represented as a permutation of hypermultiplets. For example, let us contrast the two 6d SCFTs which have one dimensional tensor branch and carry su(5) gauge algebra at low energies on the tensor branch. One of the theories carries 10 fundamental hypers and the other carries 13 fundamental hypers plus a hyper in two-index antisymmetric representation. The outer automorphism of su(5) exchanges fields transforming in fundamental/antisymmetric representation with fields transforming in complex conjugate of fundamental/antisymmetric representation inside each hypermultiplet. In the case of first theory, this action is equivalent to organizing the 10 fundamental hypers into 5 pairs and exchanging the hypers in each pair, thus representing the action of outer automorphism as a permutation on hypermultiplets. In the case of second theory, the number of fundamental and antisymmetric hypers are odd, and hence the action cannot be represented fully as a permutation of hypermultiplets.
2. In the second class are the twists whose associated discrete global symmetry acts only on the hypermultiplets, but not on the vector and tensor multiplets. This is in contrast to the twists considered in [11], all of which acted either on vector or on tensor multiplets. Such twists can arise when we have 2n half-hypermultiplets transforming in a pseudoreal representation of some gauge algebra. These halfhypers are rotated by an O(2n) global symmetry 4 , thus opening up the possibility of twisting the theory by a Z 2 element of determinant −1 inside O(2n). When we have 2n + 1 half-hypers, then a Z 2 element of determinant −1 inside O(2n + 1) acts as a central element of the gauge group, thus reducing the global symmetry group to SO(2n + 1).
These two kind of twists lead to new building blocks for 5d KK theories and new ways of gluing these building blocks to produce 5d KK theories. We will enlist these building blocks and their possible gluings. We will provide data of resolved CY3 associated to each new building block, and rules for gluing the associated resolved CY3s for each new gluing between the corresponding building blocks. Our approach to the new resolved CY3s will differ from the approach employed in [11]. This is because the approach used in [11] relied on the knowledge of a low-energy 5d N = 1 non-Abelian gauge theory description of the 5d KK theory. The data of this low-energy 5d gauge theory was obtained by modding out the data of the associated 6d gauge theory by the action of discrete symmetry. So, for the 6d SCFT carrying su(5) with 10 fundamental hypers compactified using an outer automorphism twist, the associated low energy 5d gauge theory carries sp(2) with 5 fundamental hypers. However, this procedure of projecting the data of 6d gauge theory does not work for the first kind of twists discussed above, since in those cases the associated discrete symmetry does not act by permutation of hypermultiplets.
Due to this reason, we will instead simply propose the resolved CY3s associated to the new building block 5d KK theories. The proposed resolved CY3 satisfy necessary geometric consistency conditions implying that they are consistent geometric backgrounds to compactify M-theory. Moreover, the proposed resolved CY3s will be presented in a special form which makes it manifest that the 5d N = 1 theory resulting from M-theory compactification is actually a 5d KK theory. Such special presentations of the resolved CY3s associated to 5d KK theories were discussed at length in [11,14] and will be reviewed briefly in this paper. According to the analysis of [14], one can easily read the data of the associated 6d SCFT and the type of twist from the data of the resolved CY3 presented in this special form. In this way, we will identify the 5d KK theory associated to each proposed resolved CY3. This paper is organized as follows: In Section 2, we discuss the various new 5d KK theory building blocks that can arise by considering the two kinds of twists discussed above. These building blocks are 5d KK theories arising by compactifying 6d SCFTs having a one-dimensional tensor branch, or in other words carrying a single tensor multiplet. These building blocks are collected in Table 1. In Section 2.1, we propose resolved CY3 that describe the extended Coulomb branches of these new building blocks. In Section 2.2, we provide some checks of our proposal, where we review (following [14]) how some data of the associated 6d SCFT and twist can be read from the data of resolved CY3. In Section 2.3, we provide further arguments in favor of our proposal put forward in Section 2.1. In this section, we discuss how one can compute, using the data of associated CY3, the various low-energy effective 5d N = 1 non-abelian gauge theories arising upon compactifying a 6d SCFT (possibly with a twist) on a circle of finite, non-zero radius. These low-energy 5d gauge theories can be predicted by modding out the data of the 6d N = (1, 0) non-abelian gauge theory appearing on the tensor branch of the associated 6d SCFT by the action of the discrete symmetry generating the twist. We show that these predictions for new KK building blocks match the computations performed using the proposed associated resolved CY3s. In Section 3, we discuss the various ways in which the new 5d KK theory building blocks can be combined with other new/old 5d KK theory building blocks to produce more general 5d KK theories whose associated 6d SCFTs have a tensor branch of dimension more than one. Such combinations are collected in Table 4 where, due to reasons explained in Section 3, we have restricted our attention to new building blocks arising only from the twists of the first type discussed above. In Section 3.1, we propose rules for gluing the two CY3s associated to two building blocks, so that the combined CY3 describes the extended Coulomb branch of the KK theory produced by combining the two building blocks. In Section 3.2, we provide some checks of our proposal, where we review (following [14]) how data of the discrete symmetry (used for twisting) permuting tensor multiplets in the associated 6d SCFT can be read from the data of gluing rules. In Section 3.3, we describe how the gluing rules can be used to read the data of hypermultiplet content charged under multiple simple factors of the gauge algebra of 5d gauge theory appearing at low-energies. The discussion also supports some field-theoretic arguments made at the beginning of Section 3 and used to compile Table 4.

Building blocks
In this paper, we are going to use the notation developed in [11] to denote 5d KK theories. This captures the tensor branch data of the associated 6d SCFT and the action of discrete symmetry (used to twist the theory) on the tensor branch data. The notation used there for 5d KK theories arising from 6d SCFTs carrying a single tensor multiplet took the following form where g is a simple gauge algebra, q denotes the order of outer automorphism acting on g, and k (a positive integer) denotes the coefficient of Green-Schwarz term in the Lagrangian used for canceling 1-loop gauge anomaly. To incorporate twists of the second kind discussed in Section 1, we extend the above notation and use to denote that we have an extra twist by a Z 2 living inside an O(2n) flavor symmetry.
The new building block 5d KK theories arising from the two kinds of twists discussed in Section 1 have been collected in Table 1, where they are expressed in the notation reviewed above. Let us discuss each of the entries in the table: • The 6d SCFT denoted by 1 su(n) carries n + 8 hypers in fundamental and one hyper in two-index antisymmetric of su(n). Accordingly, a Z 2 outer automorphism of su(n) is a symmetry of the theory since it complex conjugates all fields inside hypermultiplets. For n = 3, 4, the action of the outer automorphism can be represented as a permutation of hypermultiplets (see Section 1) and thus the corresponding 5d KK theories appeared already in [11].
• The 6d SCFT denoted by 1 su( n) (2.4) carries n − 8 hypers in fundamental and one hyper in two-index symmetric of su(n). The hat on top of n in su(n) has been placed to distinguish this theory 1 su(n) (2) n ≥ 5 1 su( n) (2) n ≥ 8 from the theory (2.3). A Z 2 outer automorphism of su(n) is a symmetry of the theory since it complex conjugates all fields inside hypermultiplets.
• For su (6) with Green-Schwarz coupling k = 1, there is another possibility for matter content which we denote as 1 su(6) (2.5) and it carries 15 hypers in fundamental plus a half-hyper in three-index antisymmetric of su (6). A Z 2 outer automorphism of su(6) is a symmetry of the theory since it complex conjugates all fields inside fundamental hypermultiplets and leaves the half-hyper in three-index antisymmetric invariant.
• The 6d SCFT denoted by k so(10) (2.6) carries 6 − k hypers in fundamental representation and 4 − k hypers in an irreducible spinor representation of so (10). A Z 2 outer automorphism of so(10) leaves fundamental invariant but exchanges the spinor and cospinor representations. Since spinor and cospinor are complex conjugates for so (10), the outer automorphism acts as a symmetry of the theory. For k = 2, 4, the action of outer automorphism can be represented as a permutation of hypermultiplets and hence the corresponding 5d KK theories appeared already in [11].
• The 6d SCFT denoted by k e 6 (2.7) carries 6 − k hypers in 27 dimensional representation of e 6 . A Z 2 outer automorphism of su(n) is a symmetry of the theory since it complex conjugates all fields inside hypermultiplets. For k = 2, 4, 6, the action of outer automorphism can be represented as a permutation of hypermultiplets and hence the corresponding 5d KK theories appeared already in [11].
• The 6d SCFT denoted by 2 so(11) carries five hypers in fundamental and two half-hypers in spinor of so (11). The half-hypers in spinor are thus rotated by an O(2) global symmetry and we can twist by a Z 2 element of determinant −1 in O (2). The superscript (1) on so (11) denotes that there is no outer automorphism twist involved.
• The 6d SCFT denoted by 2 so(12) carries six hypers in fundamental representation and two half-hypers in irreducible spinor representation of so (12). Thus, we can twist by a Z 2 element of determinant −1 in O(2) rotating the two half-hypers in spinor representation.
• The 6d SCFT denoted by 1 so( 12) (2.10) carries seven hypers in fundamental, two half-hypers in spinor and one half-hyper in cospinor of so (12). Thus, we can twist by a Z 2 element of determinant −1 in O(2) rotating the two half-hypers in spinor representation. The hat on 12 in so (12) has been placed to distinguish it from the 6d SCFT denoted as 1 so (12) (2.11) which carries seven hypers in fundamental and three half-hypers in spinor of so (12).
• The 6d SCFT denoted by The last four entries in Table 1 involve twists of the second type discussed in Section 1. Such twists require the presence of matter in pseudo-real representations of the gauge algebra. For 6d SCFTs, along with the cases discussed above, there is another case carrying pseudo-real representations. This 6d SCFT is denoted as 1 sp(n) (2.13) and carries 4n + 16 half-hypers in fundamental of sp(n). However, the global symmetry associated to these half-hypers is Spin(4n + 16) rather than O(4n + 16) since an instanton string tranforms in irreducible spinor representation of the so(4n + 16) global symmetry algebra. This obstructs the existence of a Z 2 element of determinant −1 in the associated global symmetry group since it exchanges spinor and cospinor representations of so(4n + 16), and thus is not a symmetry of the 6d SCFT. Consequently (2.13) does not lead to any new 5d KK theory building blocks.

Associated CY3
In this subsection we will propose the resolved CY3 associated to new 5d KK theory building blocks appearing in Table 1. The data for resolved CY3 will be presented in terms of a graph. The vertices of the graph denote different irreducible compact Kahler surfaces and the edges indicate intersections between these surfaces. An intersection between two surfaces S 1 and S 2 can be described as a gluing of a curve C 1 in S 1 to some curve C 2 in S 2 . We indicate the data of C 1 and C 2 at the two ends of the corresponding edge. We refer the reader to Sections 5.1, 5.2 and Appendix A of [11] for further geometric details used throughout the rest of this paper. The proposed resolved CY3 are described below. We will use an integer ν to parametrize different CY3s associated to a single KK theory building block, with the CY3s for different values of ν related by flop transitions. This parameter ν will be helpful for us when we discuss the gluing rules in Section 3.1.
e-x iy i where n ≥ 3, and let us clarify that there is an edge between S i and S n gluing For n = 2, we have for n ≥ 4. For n = 3, we have 2e+f e e f 5 e (2) 6 f -x-y (2.28) (2.30) (2.31) Let us also collect CY3s associated to some KK theory building blocks already discussed in [11]. We are doing so because we will need some flop frames of these CY3s (in Section 3.1) which were not described in [11].

Reading the data of 5d KK theory from CY3
In this subsection, we review the method of [14] which allows one to identify the 5d KK theory if the associated CY3 is presented in a specific form. In this form, all the compact surfaces S i are presented as Hirzebruch surfaces, with fibers f i , such that the intersection matrix takes the form of Cartan matrix of an (untwisted or twisted) affine Lie algebra g (q) . Then, g is identified as the gauge algebra appearing on the tensor branch of the associated 6d SCFT and q captures the order of outer automorphism acting on g while compactifying the 6d SCFT on circle. q = 1 indicates that there is no outer automorphism twist. All of the proposed CY3 appearing in Section 2.1 have been presented in this specific form and the reader can check that the intersection matrix reproduces the associated affine Lie algebra displayed there. Moreover, every blowup x must satisfy where d ∨ i are dual Coxeter numbers associated to g (q) . This condition captures and generalizes the "shifting of prepotential" proposal of [11]. The reader can check that every blowup appearing in every CY3 proposed in Section 2.1 satisfies this condition.
The GS coupling of the associated 6d SCFT is captured by whereẽ denotes the e curve of a specific Hirzebruch surface. This can be taken to be the surface denoted by S n for su(2n + 1) (2) , su(2n) (2) ; the surface denoted by S 3 for so(10) (2) , e 6 ; and the surface denoted by S 1 for so (11) 7 . The reader can now verify that the GS coupling for each proposed CY3 in Section 2.1 matches the GS coupling displayed there. For more details, we refer the reader to Sections 3.3 and 3.4 of [14].
Using the above information, one can determine a lot of information about the 5d KK theory associated to a resolved CY3. This includes, as we have seen, the gauge algebra arising on the tensor branch of the associated 6d SCFT and the GS coupling associated to the 6d SCFT. In many cases, these two data are sufficient to uniquely determine the matter content arising on the tensor branch of the 6d SCFT. However, in some cases, there exist multiple 6d SCFTs with the same associated GS coupling and gauge algebra, but having different hypermultiplet content. To distinguish between such 6d SCFTs, we need further analysis to which we turn in the next subsection.

Low energy effective 5d gauge theories from CY3
At certain locations in the extended Coulomb branch, the 5d KK theories under discussion reduce at low-energies to non-Abelian 5d N = 1 gauge theories. The hypermultiplet content of a particular such low-energy 5d gauge theory encodes the hypermultiplet content of the parent 6d SCFT. This particular low energy description is obtained by contracting a maximal set of fibers and blowups to zero size when the CY3 associated to a KK theory is presented in the form described in Section 2.2.
The (inverse of the) radius of compactification captures the volume of the curve associated to the KK mode of the 5d KK theory. This curve f is a genus-one fiber that can be written in terms of the fibers f i of S i as [11] where d i are the Coxeter numbers associated to g (q) . For a finite value of the radius, the volume of f must be strictly positive, which implies that not all f i can be shrunk to zero volume simultaneously.
Thus, the maximum number of fibers that can be contracted to zero size must be one less than the total number of fibers in the CY3. Consequently, the Dynkin diagram of the gauge theory obtained after contracting a maximal set of fibers can be obtained by deleting one node from the Dynkin diagram of the affine algebra g (q) . Any such deletion leads to the Dynkin diagram of a finite semi-simple Lie algebra, thus guaranteeing that the gauge algebra for the low-energy 5d gauge theory must be finite and cannot be affine. This is a crucial consistency check since a gauge theory with an affine gauge algebra would have a troublesome positive semi-definite kinetic matrix 5 .
Let us now study an example of 5d KK theory which was mentioned in Section 1. Consider the 6d SCFT carrying su(5) with 10 fundamental hypers compactified with an outer automorphism twist. According to [11], the associated resolved CY3 can be expressed as which let us rewrite into the following form for our convenience where the low-energy effective theory is a 5d N = 1 gauge theory with gauge algebra sp(2) and 5 fundamental hypers. The volumes of the e curves of S 1 and S 2 capture the gauge coupling of the low-energy sp(2) gauge theory. Notice that sp(2) is the gauge algebra left invariant by the action of an outer automorphism of su(5), and projecting out by the action such an outer automorphism on 10 fundamental hypers of su(5) in the 6d theory, we indeed are left with 5 fundamental hypers (see Section 1) of su(5) which descend to 5 fundamental hypers of sp(2) under the projection. Thus, the low-energy theory associated to (2.44) can be determined by projecting out the tensor branch data of the associated 6d SCFT by the action of the discrete symmetry used for twisting. In fact, this was true for all KK theories studied in [11], where this fact was used to obtain the prepotential for the 5d KK theory starting from the prepotential of the associated low-energy 5d theory.
However, such a projection is not neatly defined (upon the hypermultiplet spectrum) for the twists of first type discussed in Section 1. An example for such a twist is provided by the KK theory 1 su(5) (2) (2.45) as discussed in Section 1. Let us start from the CY3 (2.15) proposed for this KK theory in Section 2.1, which we reproduce below after a flop and an isomorphism, and determine the associated low-energy 5d gauge theory. Again, we would like to contract f 1 and f 2 while keeping f 0 at non-zero size. However, this is not possible when CY3 is presented in the form (2.46) since, according to one of the gluings which implies that f 2 must remain at a non-zero size as well. However, this problem can be alleviated if we flop the blowup y living in S 2 to obtain the following form of the associated CY3 Now, we can provide a volume to x, y living in S 0 which is equal to the volume of f 0 , thus contracting the curves f 0 − x, f 0 − y living in S 0 . According to the gluings, this implies that the curves f 2 − x and x living in S 2 must have zero volume, which can be consistently achieved if both the curves f 2 and x living in S 2 have zero volume. We can also contract all the blowups x i living in S 1 without any obstruction. Thus, the low-energy 5d gauge theory associated to the KK theory (2.45) is that is, sp(2) with 7 fundamental hypers. Six of the hypers arise from the six blowups living in S 1 and one hyper arises from the blowup x living in S 2 [8,14]. Comparing (2.49) with the data of the 6d theory su(5) + Λ 2 + 13F (2.50) (where Λ 2 denotes the irreducible two-index antisymmetric representation) we can see that (2.49) is not a neat projection of (2.50). However, it is still possible to understand (2.49) morally as a projection of (2.50). The action of outer automorphism exchanges fields (in pairs) living inside hypermultiplets valued in Λ 2 + 13F, thus projecting it to " 1 2 (Λ 2 + 13F)", which descends to 6 " 1 2 Λ 2 " plus 7F of sp(2). Since Λ 2 is a strictly real representation for sp(n), the degrees of freedom in " 1 2 Λ 2 " should be completely projected out, leaving us only with the matter content shown in (2.49). The finite volume blowups x, y living in S 0 can be thought of as the remnant of " 1 2 Λ 2 " since they lead to massive BPS particles transforming in Λ 2 of sp (2).
Similarly, we would expect the following low energy descriptions for those KK theories in Table 1 that involve twist of the first type (see intoduction): 1 su(2n + 1) (2) sp(n) + (n + 5)F Here, we emphasize that we have made a choice of outer automorphisms so that the gauge algebras left invariant by them coincide with the algebras appearing on the right hand side of the above equations. We can also compute the low-energy theories associated to these KK theories according to their respective resolved CY3 proposed in Section 2.1. The results are collected in Table 2. Notice that the results match the above expectations except for the case of su (6). This mismatch can be explained if we recall that an sp(3) N = 1 gauge theory in 5d cannot contain 1 2 Λ 3 + nF, but can carry 1 2 Λ 3 + 2n+1 2 F. Thus an extra 1 2 F should be projected out from the expectation (2.55).
Out of these cases, the case of Z  Table 3. Low-energy 5d non-abelian gauge theories associated to 5d KK theory building blocks involving twists of second type. C denotes the conjugate spinor representation of so(2n) and F for e 7 denotes the 56 dimensional irreducible representation of e 7 .
Let us first choose to contract all fibers except f 0 . This will require that both x and y in S 0 remain at non-zero volume, which in turn implies that f 4 remains at non-zero volume. So, it is not possible to perform this contraction in the frame (2.28). However, we can perform some flops and write (2.28) in the following form (2.70) in which it is possible to perform this contraction. We can contract all the blowups except for the four blowups x, y, z, w living in S 0 . This limit leads to (2.68) as the low-energy description for (2.67). Now, let us choose to contract all fibers except f 1 . For similar reason as above, it is not possible to do so when the CY3 is expressed in the form (2.28). But, after doing some flops, and representing it as (2.71) allows us to take this limit. We are also able to contract all the blowups except for the eight blowups x i , y i , z i , w i living in S 1 . This limit leads to (2.69) as the low-energy description for (2.67).
Similarly, choosing to contract all fibers except f 6 leads to the low-energy theory (2.68), and choosing to contract all fibers except f 5 leads to the low-energy theory (2.69), as the reader can check. We propose that this existence of multiple low-energy limits is simply a reflection of the fact that while twisting the 6d SCFT 1 so ( 12) (2.72) carrying 7F + S + 1 2 C (in 6d) by the Z 2 element of determinant −1 in O(2) symmetry rotating S, one can also include a holonomy in the O(1) ≃ Z 2 symmetry rotating 1 2 C. If this holonomy is included, we expect the low-energy description to be (2.69); while, if this holonomy is not including, we expect the low-energy description to be (2.68). The fact that these two low-energy theories are continuously connected inside the extended Coulomb branch of the 5d KK theory Z (2) 2 1 so( 12) (1) (2.73) means that the inclusion of this holonomy does not lead to a physically distinguishable twist, as we already argued in Section 1.
We can test this proposal in other similar cases. For example, consider the CY3 for 3 so(11) (1) (2.74) which is [3,4,11]  This verifies the arguments presented in Section 1 in relation to the twists of second type.

Combining the building blocks
In this section, we describe how the new building blocks can be combined with new/old building blocks to produce more general 5d KK theories. Let us first consider a 6d SCFT of the form k l su(2m) su(2n) which makes sense for k, l ∈ {1, 2} and k + l ≥ 3. The GS coupling is a 2 × 2 matrix whose diagonal entries are captrued by k and l, while a single edge in (3.1) denotes that both off-diagonal entries are −1. The mixed hypermultiplet content is a bifundamental of su(2n) ⊕ su(2m). We would like to perform an outer automorphism twist on su(2n), which should project the degrees of freedom living inside the bifundamental by a factor of half. If there is no outer automorphism twist acting on the su(2m) factor, then we would expect a half-hyper in bifundamental of sp(n) ⊕ su(2m) in the associated low-energy effective 5d gauge theory. However, a half-hyper in bifundamental is not allowed for this set of gauge algebras.
We can achieve a consistent projection if we also act by an outer automorphism on su(2m), but we choose this outer automorphism such that it projects su(2m) to so(2m) rather than sp(m). Then, we would expect the associated low-energy effective 5d gauge theory to contain a half-hyper in bifundamental of sp(n) ⊕ so(2m), which is indeed an allowed matter content. This expectation is verified geometrically where we observe that there is a consistent resolved CY3 associated to k l su(2m) (2) su(2n) (2) (3.2) but no consistent resolved CY3 associated to k l su(2m) (1) su(2n) (2) (3.3) We can compute from the CY3 associated to (3.2), presented later in this section, that the associated low-energy theory indeed contains a half-hyper in bifundamental of sp(n) ⊕ so(2m).
Let us now consider the 6d SCFT su(2m) (2) su(2n) (2) 2 su(2n) (2) (3.5) Second, we can exchange the two su(n) (along with the two bifundamentals). The corresponding KK theory was denoted in [11] as which represents a folding of the graph (3.4), and describes that no outer automorphism is acting on either of the two algebras (the projection by the exchange operation identifies the two su(2n)). Another possibility is to perform the outer automorphism on su(2m) alone, while representing the action of outer automorphism as an exchange of the two bifundamentals. Since the action of outer automorphism is a permutation on the hypermultiplet spectrum, this twist was already considered in [11] where the corresponding KK theory was denoted as which represents a folding of the graph (3.4), and describes that an outer automorphism is acting upon the su(2m) gauge algebra. We claim that there is yet another possible twist which can be represented as 2 2 where along with an exchange, both the su(2m) and su(2n) algebras have an outer automorphism acting upon them. This can be thought of as first reducing to (3.5) whose associated low-energy theory contains where each edge denotes a half-hyper in bifundamental. We can now add a further exchange symmetry to the twist, whose action on the above low-energy theory exchanges the two half-bifundamentals and the two sp(n).
In light of the above discussion, let us revisit the case of (3.1) when m = n and k = l. The 6d SCFT can then be represented as 2 2 su(2n) su(2n) (3.10) Are there any twists which involve the exchange of two su(2n)? If there is no outer automorphism involved, then this twist was already discussed in [11] and the corresponding KK theory was denoted as 2 su(2n) (1) (3.11) However, if there is an outer automorphism involved, we don't expect that the exchange is a symmetry since the low-energy theory associated to 2 2 su(2n) (2) su(2n) (2) (3.12) contains sp(n) so(2n) (3.13) which does not admit any exchange symmetry. That is, we do not expect the existence of a KK theory of the form 2 su(2n) (2) (3.14) Correspondingly, we do not find any consistent CY3 that could be associated to such a KK theory.
Using similar arguments, we can compile a list of possible ways in which the new KK theory building blocks can be combined with other KK theory building blocks. We present this list in Table 4. We note that we only need to study the possible combinations of the new KK building blocks arising from twists of the first type (see Section 1). The possible combinations for a KK building block arising from a twist of the second type, say su(n β ) (2) e n α ≤ 2n β ; en β ≤ 2n α ; e = 2, 3 (3. 16) which does not involve the extra Z 2 twist. This is because the Z 2 only affects hypers transforming in spinor of so(11), so (12) and F of e 7 , but it is not possible to gauge the global symmetries associated to these hypers and obtain a 6d SCFT. So, the Z 2 twist acts on parts of the theory which does not affect possible combinations. is a consistent 6d SCFT. The reason for the inconsistency of (3.17) can be found in the itemized list appearing towards the end of Section 3.4 of [11].

Gluing rules for associated CY3
The CY3 associated to two KK theory building blocks α and β connected by an edge is obtained by gluing some curves (comprised of fibers and blowups) in the CY3 associated α with some curves (comprised of fibers and blowups) in the CY3 associated β. The gluings are independent of the diagonal GS couplings associated to α and β, and depend only on the type of edge and the associated twisted affine algebras 7 [4,11]. In the following, we will present such gluing rules. Our notation would be to call the building block appearing on the left as α and the building block appearing on the right as β.
We will denote the surfaces coming from CY3 associated to α as S i,α and the surfaces coming from CY3 associated to β as S i,β , where i is the labeling of different surfaces which can be found in Section 2.1 for the new KK theory building blocks and Section 5.2 of [11] for the old KK theory building blocks.

2
: Same as the ones provided in [11].
or Dynkin graph translate to the off-diagonal entries Ω αβ = −p and Ω βα = −1 in the associated Cartan matrix, and p number of undirected edges from α to β translate to the off-diagonal entries Ω αβ = Ω βα = −p. The reader can similarly check that this correspondence between type of edge and off-diagonal entries Ω αβ , Ω βα holds true for other combinations of building blocks shown in Table 4. This provides a non-trivial consistency check on the proposed gluing rules.

Low energy effective gauge theory
Using the techniques of Section 2.3, it is also possible to study the impact of gluing KK theory building blocks upon the associated low-energy theories, which is the topic of discussion in this subsection. Consider the KK theory k l su(2n β ) (2) su(2n α ) (2) (3.29) According to the arguments at the beginning of this section, the gluing rules suggested in Section 3.1 above should imply that, if the low-energy gauge algebras chosen for the two building blocks are so(2n α ) and sp(n β ) respectively, then contracting a maximal set of blowups should lead to a half-hyper charged in bifundamental of so(2n α ) ⊕ sp(n β ). This can be easily verified. To obtain the above low-energy limit, we can contract all fibers to zero size except the fibers for S nα,α and S 0,β . This forces all blowups x i for i = 1, · · · , n α living in S 0,β (and participating in the gluing rules) to remain at nonzero volume, while the blowups y i for i = 1, · · · , n α living in S 0,β can be contracted to zero volume. On the other hand, all the blowups living in S 0,α (and participating in the gluing rules) can be consistently contracted to zero size. To read the matter content in the low-energy theory, we restrict our attention only to those surfaces whose corresponding fibers are contracted to zero volume. The gluing rules are then reduced to: • x i − x i+1 , y i+1 − y i in S 0,α are glued to f, f in S i,β for i = 1, · · · , n β − 1.
• x n β − y n β in S 0,α is glued to f in S n β ,β .
In other words, the above gluing rules are telling us that a total of n β fundamental hypers of so(2n α ) are gauged by an sp(n β ). Thus, these blowups (living in S 0,α ) must give rise to a half-hyper in bifundamental of so(2n α ) ⊕ sp(n β ), as expected.
What happens if instead we choose the low-energy gauge algebra to be sp(n α ) ⊕ sp(n β )? In this case, since a half-hyper in bifundamental is not possible, we would expect to obtain no matter degrees of freedom charged under a mixed representation of sp(n α ) ⊕ sp(n β ). This can again be verified using the gluing rules presented in Section 3.1. To obtain the above low-energy limit, we can contract all fibers to zero size except the fibers for S 0,α and S 0,β . The reader can verify that this limit forces all the blowups living in S 0,α and S 0,β (and participating in the gluing rules) to remain at positive, non-zero volume. Since none of the blowups participating in the gluing rules give rise to massless particles, the low-energy sp(n α ) ⊕ sp(n β ) has no hypers charged in a mixed representation of sp(n α ) ⊕ sp(n β ), as expected.
In a similar way, one can check that the low-energy mixed hyper content for other combinations of KK building blocks, as expected from the arguments presented at the beginning of this section, is concretely reproduced by the gluing rules proposed in Section 3.1, thus providing strong consistency checks between the arguments and proposals presented in this paper.