5D N=1 super QFT: symplectic quivers

We develop a method to build new 5D $\mathcal{N}=1$ gauge models based on Sasaki-Einstein manifolds $Y^{p,q}.$ These models extend the standard 5D ones having a unitary SU$\left( p\right) _{q}$ gauge symmetry based on $% Y^{p,q}$. Particular focus is put on the building of a gauge family with symplectic SP$\left( 2r,\mathbb{R}\right) $ symmetry. These super QFTs are embedded in M-theory compactified on folded toric Calabi-Yau threefolds $% \hat{X}(Y^{2r,0})$ constructed from conical $Y^{2r,0}$. By using outer-automorphism symmetries of 5D $\mathcal{N}=1$\textbf{\ }BPS quivers with unitary SU$\left( 2r\right) $ gauge invariance, we also construct BPS quivers with symplectic SP$\left( 2r,\mathbb{R}\right) $ gauge symmetry. Other related aspects are discussed. Keywords: SCFT$_{5}$, 5D $\mathcal{N}=1$ super QFT on a finite circle, Sasaki-Einstein manifolds, BPS quivers, outer-automorphisms.

1 Introduction N = 1 supersymmetric gauge theories in five space time dimensions (super QFT 5 ) are non renormalizable field theories with eight supercharges. They are admitted to have UV fixed points which can be deformed by relevant operators such that in the infrared they flow to 5D N = 1 super Yang-Mills (SYM 5 ) coupled to hypermultiplets [1,2]. A typical massive deformation generating this type of flow is given by the SYM 5 term tr F 2 µν /g 2 Y M where in 5D the inverse gauge coupling square 1/g 2 Y M has dimension of mass. These 5D gauge theories are somehow special compared to 6D gauge theories [3,4,5] including maximally supersymmetric Yang-Mills theory believed to flow to N = (2, 0) supersymmetric 6D theory in the UV [6,7]. In the few last years, super QFT 5 s and their compactification, in particular on a Kaluza-Klein circle with finite radius and to 3D, have been subject to some interest in connection with their critical behaviour and specific properties of their gauge phases [8]- [15]. Though a complete classification is still lacking [16,5], several examples of such gauge theories are known; and most of them can be viewed as deformations of 5D superconformal theories [17,18,19]. Simplest examples of SCFT 5 s are given by the so-called Seiberg family possessing a rich flavor symmetry [20]; many others are obtained through embedding in string theory. Generally speaking, this embedding can be achieved in two interesting ways; either by using 5-brane webs in type IIB string theory [21]- [26]; or by using M-theory compactification on Calabi-Yau threefolds [27]- [31]. Below, we comment briefly on these two methods while giving some references which certainly are not the complete list since the works in this matter are abundant.
The method of (p, q) 5-brane webs in type IIB string theory has led to several findings and has several features; in particular the following: First, it gives evidence for the existence of fix point of 5D gauge theories flowing to UV conformal points corresponding to collapsed webs; and as such permits to study conditions for existence of critical fix points. This web construction also indicates that not every 5D gauge theory can flow to a SCFT 5 [1]; the existence of a SCFT constraints the matter content of the theory. The 5-brane method allows also to study gauge theory dualities in 5D. This is because a given SCFT 5 can have several gauge theory deformations; thus generating different (but dual) gauge theories in infrared [21]. Also, the web method provides us with a tool to compute the instanton partition function that captures the BPS spectrum of the 5D theory by applying the topological vertex formalism [32]- [37]. It also allows the study the global symmetry enhancements of the SCFTs [7,38] and UV-dualities [21], [39]- [41]. More interestingly, the 5-brane webs approach give a way to elaborate families of 5D gauge models with fix points closely related to quivers with SU gauge in the shape of Dynkin diagrams. By introducing an orientifold plane like O5-plane, the 5-brane webs can describe 5D super QFTs with flavors and gauge groups beyond SU(N) such as SO(N) and Sp(2N) [42,43] as well as exceptional ones like G 2 [34].
Certain (p, q) 5-brane webs have interpretation in terms of toric diagrams [23] although, for 5D gauge theories with a large number of flavors, they lead to non-toric Calabi-Yau geometries [44]. This brane based method is not used in this paper; it is described here as one of two approaches to study 5D N = 1 super QFTs underlying SCFT 5 . For works using this method, we refer to rich the literature in this matter; for instance [21]- [23], [45]- [50].
Regarding the M-theory method, to be used in this study, one can also list several interesting aspects showing that it is a powerful higher dimensional geometric approach. First of all, the 5D gauge theories are obtained by compactifying M-theory on Calabi-Yau threefolds (CY3) X ( resolvedX). Then, the effective prepotential F 5D and its non trivial variations δ n F 5D /δφ n , characterising the Coulomb branch of the 5D super QFTs, have interesting CY3 interpretations; i.e. a geometric meaning in the internal dimensions. The F 5D is given by the volume vol(X) while its variations -describing magnetic string tensions amongst othersare interpreted as volumes of p-cycles. Moreover, the calculation of F 5D can be explicitly done for a wide class ofX's; in particular for the family of toric Calabi-Yau threefolds like those based on the three following geometries: (a) The toric del Pezzo surfaces dP n with n=1,2,3; these Kahler manifolds are toric deformations of the complex projective plane P 2 .
(b) The Hirzebruch surfaces F n given by non trivial fibrations of a complex projective line P 1 over a base P 1 [51]- [54]. (c) The familyX (Y p,q ) given by a crepant resolution of toric threefolds realised as real metric cone on Sasaki-Einstein Y p,q spaces labeled by two positive integers (p, q) constrained as p ≥ q ≥ 0 [55]- [59].
In this investigation, we focus on the particular class of 5D supersymmetric SU(p) q unitary field models based onX (Y p,q ) and look for a generalisation of these quantum field models to other gauge symmetries. Our interest into the Sasaki-Einstein (EM) based CY3s has been motivated by yet unexplored specific properties of Y p,q and also by the objective of generalizing partial results obtained for the unitary family. In this context, recall that the toric 5D super QFTs based onX (Y p,q ) have unitary SU(p) q gauge symmetries with Chern-Simons (CS) level q. Thus, it is interesting to seek how to generalize these unitary gauge models based onX (Y p,q ) for other gauge symmetries like the orthogonal and the symplectic. As a first step in this exploration, we show in this study that the 5D unitary gauge theories based on Y p,q have discrete symmetries that can be used to construct new gauge models. These finite groups come from symmetries of p-cycles inside theX (Y p,q ). By using specific properties of the unitary set and folding under outer-automorphisms of p-cycles, we construct a new family of 5D SQFTs having symplectic SP(2r, R) gauge invariance.
To undertake this study, it is helpful to recall some features of the Sasaki-Einstein based CY3: (i) They are toric and they extend theX (dP 1 ) and theX (F 0 ) . These geometries appear as two leading members in theX (Y p,q ) family. (ii) They have been used in the past in the engineering of 4D supersymmetric quiver gauge theories [60]- [63]; and have been recently considered in models building of unitary 5D N = 1 super CFTs [64]- [68]. (iii) Being toric, the threefoldsX (Y p,q )s and the unitary 5D super QFTs based on them can be respectively represented by toric diagrams ∆X (Y p,q ) and by BPS quivers QX (Y p,q ) describing the BPS particle states of the unitary supersymmetric theory.
The toric ∆X (Y p,q ) and the BPS QX (Y p,q ) are particularly interesting because they play a central role in our construction; as such, we think it is useful to comment on them here. We split the properties of these objects in two types: general and specific. The general properties, which will be understood in this investigation, are as in the geometric engineering of 4D super QFTs [69]- [73]. They also concern aspects of the Sasaki-Einstein manifolds and the brane tiling algorithms (a.k.a dimer model) [74]- [83]. Some useful general aspects for this study are reported in the appendices A, B, C. The specific properties ∆X (Y p,q ) and QX (Y p,q ) regard their outer-automorphisms and the implementation of the Calabi-Yau condition of X as well as a previously unknown property ofX (Y p,q ) that we describe for the leading members p = 2, 3, 4. By trying to exhibit manifestly the Calabi-Yau condition on the toric diagram ∆X (Y p,q ) , we end up with the need to introduce a new graph representingX (Y p,q ).
This new graph is denoted like G Ĝ X(Y p,q ) with G referring either to the gauge symmetry SU(p) or to SP(2r, R). The construction of G Ĝ X(Y p,q ) will be studied with details in this paper; to fix ideas, see eq(4.1) and the Figure 7  In the present paper, we contribute to the study of 5D N = 1 super QFT models based on conical Sasaki-Einstein manifolds and their compactification on a circle with finite radius.
Using the above mentioned discrete symmetries, we develop a method to build new 5D N = 1 Kaluza-Klein quiver gauge models based on Sasaki-Einstein manifolds Y p,q . For that, we first revisit properties of the internalX (Y p,q ) geometries which are known to host gauge models with SU(p) q gauge symmetry. Then, we show that some of these Sasaki-Einstein based threefolds have non trivial discrete symmetries that exchange p-cycles inX (Y p,q ) and which we construct explicitly. By using these finite symmetries and cycle-folding ideas, we build a new set of 5D supersymmetric gauge models based onX (Y p,q ) having symplectic SP(2r, R) gauge invariance; thus extending the set of unitary gauge models for this family of CY3. We also derive the associated BPS quivers encoding the data on the BPS states of the symplectic theory. We moreover show that the cycle-folding by outer-automorphisms generate super QFT models having no standard interpretation in terms of gauge phases. For a pedagogical reason, we mainly focus on the leading members of the symplectic SP(2r, R) family; in particular on the 5D N = 1 super QFT with SP(4, R) invariance. The first SP(2, R) member is isomorphic to the 5D N = 1 SU(2) model of the unitary series. To achieve this goal, we (i) revisit the toric Calabi-Yau threefoldX (Y 4,0 ) (p=4 and q=0), hosting a lifted SU(4) 0 gauge symmetry; and (ii) reconsider the BPS quiver Q SU 4 X(Y 4,0 ) of the underlying with 5D N = 1 super QFT compactified on a circle with finite size. After that we develop an approach to construct toric Calabi-Yau threefolds with symplectic symmetry and a method to build the BPS quiver Q SP 4 X(Y 4,0 ) with SP(4, R) invariance. The extension of this construction to other gauge symmetries is discussed in the conclusion section.
The organisation is as follows: In section 2, we review properties of the toric diagram . We show that ∆ SU 4 X(Y 4,0 ) has non trivial outerautomorphisms H outer ∆ SU 4 having a fix point. We also show that this discrete group H outer ∆ SU 4 can be interpreted as a parity symmetry in Z 2 lattice. In section 3, we investigate the properties of the BPS quiver Q SU 4 X(Y 4,0 ) associated with ∆ SU 4 X(Y 4,0 ) . Here we show that Q SU 4 X(Y 4,0 ) has also an outer-automorphism symmetry H outer Q SU 4 with fix points. This outer-automorphism group has two factors given by (Z 4 ) Q SU 4 × (Z outer 2 ) Q SU 4 . In section 4, we introduce a new diagram to represent the toricX (Y 4,0 ) . It is given by a graph G where the Calabi-Yau condition is manifestly exhibited. To avoid confusion, we denote this graph like G SU 4 X(Y 4,0 ) and refer to it as the unitary CY graph of the toricX (Y 4,0 ) with SU(4) gauge symmetry. To deepen the construction, we also give the unitary CY graphs G SU 2 X(Y 2,0 ) and G SU 3 X(Y 3,0 ) representing the toriĉ X (Y p,0 ) with p=2 and p=3. In section 5, we construct the symplectic CY graphG SP 4 X(Y 4,0 ) and the associated symplectic BPS quiverQ SP 4 X(Y 4,0 ) . In section 6, we give a conclusion and make comments. In the appendix, we give useful properties on the geometric properties of the Coulomb branch of M-theory on CY3s and describe the building of BPS quivers.

Conical Sasaki-Einstein threefoldX(Y 4,0 )
We begin by recalling that the Calabi-Yau threefoldX(Y 4,0 ), taken as a real metric cone over the 5d Sasaki-Einstein variety [84,85], is a toric complex 3d manifold whose toric diagram is a finite sublattice of Z 3 as in the Figure 1. This toric ∆ SU 4 X(Y 4,0 ) has seven points given by: (a) Four external points W 1 , W 2 , W 3 , W 4 defining the geometry on which rests the singularity of the SU(4) gauge fiber. (b) Three internal points V 1 , V 2 , V 3 describing the crepant resolution of the singularity. This resolution can be imagined as an intrinsic sub-geometry of the toricX(Y 4,0 ) to which we often refer to as the fiber geometry.

Toric diagram
The four above mentioned W i points (i=1,2,3,4) of the toric diagram ∆ SU 4 X(Y 4,0 ) can be also interpreted as associated with four non compact divisors D i of the toricX(Y 4,0 ). Similarly, the three internal points V a (a = 1, 2, 3) are interpreted as corresponding to three divisors E a of the toricX(Y 4,0 ); but with the difference that the three E a 's are compact complex 2d surfaces. In terms of the classes of these divisors, the Calabi-Yau condition of the toriĉ X(Y 4,0 ) is given by the vanishing sum; see also appendix A, This homological condition is implemented at the level of the toric diagram by restricting the seven points of ∆ SU 4 X(Y 4,0 ) to sit in the same hyperplane by taking the external like W i = (w i , 1) and the internal points as V a = (v a , 1) with w i and v a belonging to Z 2 . A particular realisation of the seven points of Table 1: Toric data of the Calabi-Yau threefoldX (Y 4,0 ) .
representing the resolved Calabi-Yau threefoldX(Y 4,0 ) is depicted by the Figure 1 where a triangulation the surface of ∆X (Y 4,0 ) is highlighted [86]. It describes the lifting of the A 3 ≃ SU (4) singularity. Notice that Table 1 is the data for (p, q) = (4, 0) saturating the we have the following data This toric ∆X (Y p,q ) has 3+p points and then 3+p divisors; p-1 of them are compact. They concern the divisor set {E a } 1≤a≤p−1 . Notice also that the three internal (red) points of ∆X (Y 4,0 ) represented by the Figure 1 form a (vertical) linear chain A 3 in the toric diagram with boundary points effectively given by the two (blue) external w 2 and w 4 . For convenience, we rename these two particular boundary points like w 2 = υ 0 and w 4 = υ 4 so that the above mentioned chain A 3 can be put in correspondence with the standard A 3 -geometry of the ALE space with resolved SU(4) singularity [87,88]. With this renaming, the Table 1 gets mapped to A similar description can be done for ∆X (Y p,0 ) . For simplicity of the presentation, we omit it. Having introduced the particular toric diagram ∆ SU 4 X(Y 4,0 ) hosting an underlying unitary SU(4) gauge symmetry, we turn now to explore one of its exotic properties namely its outer-automorphism symmetries. A careful inspection of the Figure 1 reveals that the toric diagram ∆ SU 4 X(Y 4,0 ) has outerautomorphism symmetries forming a discrete group H outer ∆ SU 4 . This is a finite symmetry group generated by the following transformations of the external points w i and the internal υ a s, Notice that the outer-automorphisms in the gauge fiber act by exchanging the two internal υ 1 ↔ υ 3 ; but fix the central point υ 2 . This property is interesting; it will be used later on to engineer a new gauge fiber. By using the parametrisation w i = (w x i , w y i ) and v a = (v x a , v y a ), we learn that the outer-automorphism group H outer ∆ SU 4 is given by the product of two reflections like Form these outer-automorphism transformations, we learn that (Z x 2 ) ∆ SU 4 acts trivially on the internal points v a of the A 3 -linear chain of ∆ SU 4 X(Y 4,0 ) . So the group (Z x 2 ) ∆ SU 4 leaves invariant the A 3 -gauge fiber within the toric Calabi-YauX(Y 4,0 ). It affects only the external points w 1 and w 3 which are associated with the transverse geometry shown in the table 3. Regarding the (Z y 2 ) ∆ SU 4 reflection, it acts non trivially on the points of the A 3 -chain; we have: Under this mirror symmetry, the A 3 -gauge fiber has then a fix point which is an interesting feature that we want to exploit to build a new gauge fiber by using folding ideas [89,90,69,70]. In this regards, recall that the (Z y 2 ) ∆ SU 4 action looks like a well known outerautomorphism symmetry group Z 2 that we encounter in the folding of the Dynkin diagrams of the finite dimensional Lie algebras A 2r−1 . Here, we are dealing with the particular A 3 ∼ SU (4) which is just the leading non trivial member of the A 2r−1 series. As an illustration; see the pictures of the Figure 2 describing the folding of the Dynkin diagram A 3 giving the Dynkin diagram of the symplectic C 2 ≃ sp (4, R) which, thought not relevant for our present study, it is also isomorphic to B 2 ≃ so (5). Recall as well that the Dynkin diagrams of finite dimensional Lie algebras g may be also thought of in terms of the Cartan matrices K (g) ij = α ∨ i .α j defined by the intersection of simple roots α i and co-roots For the examples of A 3 ≃ su (4) and C 2 ≃ sp (4, R) , we have the following matrices Notice that the picture on the left of the Figure 2 can be put in correspondence with the internal (red) points of the A 3 -linear chain of the Figure 1. At this level, one may ask what about toric diagrams with a C 2 type sub-diagram. We will answer this question later on after highlighting another property of ∆ SU 4 X(Y 4,0 ) . Before that, let us describe succinctly the BPS quivers associated with the toric diagram ofX(Y 4,0 ); and study its outer-automorphisms.

BPS quiver
In this section, we investigate two examples of unitary BPS quivers namely the Q SU 3 X(Y 3,0 ) and the Q SU 4 X(Y 4,0 ) . These unitary BPS quivers are representatives of the families Q SU 2r−1 X(Y 2r−1,0 ) and Q SU 2r X(Y 2r,0 ) with r ≥ 1. They have intrinsic properties that we want to study and which will be used later on. First, we consider the quiver Q SU 4 X(Y 4,0 ) with gauge symmetry SU(4) as this quiver is one of the main graphs that interests us in this study. Then, we turn to the BPS quiver Q SU 3 X(Y 3,0 ) with unitary symmetry SU(3). The Q SU 3 X(Y 3,0 ) quiver is reported here for a matter of comparison with Q SU 4 X(Y 4,0 ) . The results obtained for these quivers hold as well for the families Q SU 2r−1 X(Y 2r−1,0 ) and Q SU 2r X(Y 2r,0 ) .

BPS quiver
The construction of the unitary BPS quiver Q SU 4 X(Y 4,0 ) of the 5D N = 1 super QFTs, compactified on a circle with finite size and based onX(Y 4,0 ), follows from the brane tiling of the so called brane-web∆ SU 4 X(Y 4,0 ) (the dual of the toric diagram ∆ SU 4 X(Y 4,0 ) ) by applying the fast inverse algorithm [92,93,94]. Up to a Seiberg-type duality transformation, the repre- As shown by the Figure 3, the 8 nodes of the BPS quiver are linked by 4×4 = 16 quiver-edges j|l interpreted in terms of chiral superfields in the language of supersymmetric quantum mechanics (SQM) [65]. The unitary BPS quiver Q SU 4 X(Y 4,0 ) has been first considered in [66] (see figure 25-a, page 61). For later use, we re-draw the Figure 3 as depicted by the equivalent  They should be identified as they concern the same nodes' pair.
and (ii) on the Kronecker quivers like κ c → κ c+4 . These outer-automorphisms, which act also on the oriented arrows, have no fix node and no fix arrow. They play a secondary role in our construction.
In addition to (Z 4 ) Q SU 4 , the unitary BPS quiver Q SU 4 X(Y 4,0 ) has another outer-automorphism group factor namely (Z outer Second the (Z 4 ) Q SU 4 is also a subgroup of S 8 ; it generated by the product of two 4-cycles as follows, So, both (Z outer 2 ) Q SU 4 and (Z 4 ) Q SU 4 are subgroups of the enveloping S 8 . Similar outerautomorphism groups can be written down for the family Q SU 2r X(Y 2r,0 ) with r ≥ 2.

BPS quiver
Here, we study the BPS quiver Q SU 3 X(Y 3,0 ) and some of its outer-automorphisms in order to has a quite similar structure as Q SU 4 X(Y 4,0 ) ; but a different quiver dimension which is given by As such, the BPS quiver Q SU 3 X(Y 3,0 ) has six nodes {1} , ..., {6} interpreted in terms of 6 elementary BPS particles. They organise into three Kronecker quivers namely This BPS quiver has 12 oriented arrows as depicted by the Figure 5. Though not very important for our present study as it cannot induce a BPS quiver with symplectic gauge symmetry, notice that the quiver Q SU 3 X(Y 3,0 ) has also outer-automorphism symmetries forming a group H outer Q SU 3 with two factors as given below The factor (Z outer which are given by Figure 4. Recall that the H outer Q SU 4 has four fix nodes instead of two for Q SU 3 X(Y 3,0 ) ; i.e: two Kronecker quivers for H outer Q SU 4 , against one Kronecker quiver for H outer Q SU 3 . This difference holds as well for generic quivers Q SU 2r X(Y 2r,0 ) and Q with respective outer-automorphism groups H outer Q SU 2r and H outer Q SU 2r−1 . Regarding the factor (Z 3 ) Q SU 3 , it allows to represent the quiver Q SU 3 X(Y 3,0 ) as a periodic chain as depicted by the Figure 6.
. Its elementary BPS states are represented by the 6 nodes and are linked by 12 edges. The outer-automorphism group In this section, we introduce a new graph to deal with the toric diagram ∆ SU 4 X(Y 4,0 ) with p=4 representing the Calabi-Yau threefoldX(Y 4,0 ) with a resolved SU(4) gauge fiber. We refer to this new graph as the unitary Calabi-Yau graph and we denote it like G SU 4 X(Y 4,0 ) . This graph is explicitly defined by p − 1 vector q b with components given by the triple intersection where the label A = (i, a) with i = 1, 2, 3, 4, for non compact divisors D i , and a = 1, 2, 3 for the compact E a . Below, we refer to these q b 's as generalised Mori-vectors. Though this CY graph G SU 4 X(Y 4,0 ) looks formally different from the toric diagram, it is in fact equivalent to it. It is just another way to deal with ∆ SU 4 X(Y 4,0 ) where the Calabi-Yau condition is manifestly exhibited. As we will show below, this is useful in looking for solutions of underlying constraint relations required by the toric threefoldX(Y 4,0 ).

Building the CY graph
, we start form the Calabi-Yau condition given by eq(2.1) namely relation is expressed in terms of the four non compact divisors D i and the three compact E a ; but it is not the only constraint that must be obeyed by the divisors. There are two other constraints that must be satisfied by the divisors. So, the seven divisors (D i , E a ) of the toric Calabi-Yau threefoldsX(Y 4,0 ) are subject to three basic constraints. They can be collectively expressed as 3-vector equation like where W i = (w i , 1) and V a = (v a , 1) are as in Table 1. To deal with the CY constraint eq(2.1), we bring it to a relation between triple intersection numbers Multiplying formally both sides of eq(2.1) by , we obtain the following relationships between the triple intersection numbers, These three relationships can be put into two convenient expressions; either as . The second expression is precisely the relation that we have in gauged linear sigma model (GLSM) realisation of toric Calabi-Yau threefolds [95]. Regarding the CY relation (4.5), notice that it is quite similar to the well known relation giving the CY condition we encounter in the study of complex 2d ADE surfaces describing the  (2), the expression the (Mori-) vectors A ADE can be written down. For the example of the complex A 3 surface, the three Mori-vectors read as follows where the Cartan matrix K(SU 4 ) of the Lie algebra of the SU(4) gauge symmetry appears as a square sub-matrix of the above (Q a ) SU 4 . Recall that K (SU 4 ) is given by For the case of the CY graph G SU 4 X(Y 4,0 ) we are interested in this study, and depicted by the Figure 7, the three generalised Mori-vectors (q a ) SU 4 are given by Calabi-Yau threefolds condition is ensured by the vanishing sum of the total charge at each red exceptional node. The underlying SQFT has an SU(4) gauge symmetry. Notice also that this graph has a remarkable outer-automorphism symmetry to used later on.
This is a 3 × 7 rectangular matrix that contains the square 3×3 sub-matrix I a b defined like E a .E 2 b and reading as follows The diagonal terms I a a (a = b) describe precisely the triple self intersections of the compact divisors namely E 3 a = −8. The off diagonal terms I b a (a = b) describes the intersection between the compact divisor E a and the compact curve E 2 b . As eqs(4.9) and (4.10) are one of the results of this study, it is interesting to comment them by describing their content and exploring their relationship with ADE Dynkin diagrams.
These comments are as listed below: (1) First, recall that E 3 = −8 is the triple self intersection of the Hirzebruch surface F 0 given by a complex projective curve P 1 fibered over another P 1 . So, eq(4.10) describes three (F 0 ) 1 , (F 0 ) 2 and (F 0 ) 3 intersecting transversally. The cross intersection described by (4.10) given by (4.1) describes the graph of the toric Calabi-Yau threefoldsX(Y 4,0 ). This quantity is interesting from various views; in particular the three following: A eq(4.7) associated with Calabi-Yau twofolds (CY2). Then the q b A , concerning 4-cycles, can be imagined as a generalisation of the Mori vector Q b A dealing with 2-cycles. As such q b A and Q b A can be put in correspondence. This link is also supported by the fact that both q b A and Q b A are based on SU(4) and both obey the CY condition namely q b A = 0 (4.5) and Q b A = 0 (4.6). (ii) As for q b A describing the toricX(Y 4,0 ), with graphic representation given by the Figure  7, the Q b A describes also a toric CY2 surfaceẐ SU 4 . This complex surface also has a graphic representation formally similar to the vertical line of the Figure 7; that is the line containing the red nodes. Recall thatẐ SU 4 is given by the resolution of ALE space C 2 /Z 4 . The compact part of the associated toric diagram is given by the Figure 2-a where the nodes describe three intersecting CP 1 curves.
(iii) The above comments done for SU(4) holds in fact for the full SU(p) family with p ≥ 2.
So, the correspondence between q b A and Q b A is a general property valid for SU(p) 0 gauge models in 5D. This correspondence holds also for the intersection matrices I (SU p ) b a and K (SU p ) b a associated with the compact parts in q b A and Q b A respectively. However, the graph of K ab is just the Dynkin diagram of the Lie algebra of SU (4). In this regards, recall that we have K ab = α ∨ a .α b where the α a 's stand for the simple roots and the α ∨ a = 2α a /α 2 a for the co-roots. Clearly for SU(p), we have α 2 a = 2. The Cartan matrix K ab has also an interpretation in terms of intersecting 2-cycles C (2) a in the second homology group H 2 ; that is C From this description, a natural question arises. Could the intersection matrix (4.10), also has a similar algebraic interpretation as K ab = α ∨ a .α b ? For example, could I ab be a generalized Cartan matrix K We end this section by noticing that eq(4.9) is a particular solution of the Calabi-Yau condition (4.5). It relies on the equality Other solutions of A q b A = 0 violating the above symmetric property can be also written down; they are omitted here. The first member of the G SUp X(Y p,0 ) family is given by G SU 2 X(Y 2,0 ) (p=2). It has four (external) non compact divisors D 1 , D 2 , D 3 , D 4 ; but only one internal compact divisor that we denote E 0 .

Leading members of the G
So, there is one generalised Mori-vector given by where the CY condition, given by the vanishing of the trace of (q) SU 2 , is manifestly exhibited.
The diagram representing the CY graph G SU 2 X(Y 2,0 ) is given by the picture on the right side of the Figure 8. On the left side of this figure, we have given the picture of the standard A 1 geometry of ALE space involving complex projective curves with self intersection −2.
Notice that the Calabi-Yau threefoldX(Y 2,0 ) is preciselyX(F 0 ), the toric threefold based On the right, the graph G SU 2 X(Y 2,0 ) geometry having one compact 4-cycle, with triple self intersection I 000 = −8, intersecting four non compact (blue) 4-cycles. The Calabi-Yau threefolds condition is ensured by the vanishing sum of the total charge. on the Hirzebruch surface F 0 which is known to have a triple self intersection (−8). Notice also that this graph has outer-automorphisms given by the mirror (Z x 2 ) ∆ SU 2 × (Z y 2 ) ∆ SU 2 fixing E 0 and acting by the exchange D 1 ↔ D 3 and D 2 ↔ D 4 .
Concerning the second member of the family namely G SU P X(Y P,0 ) with p = 3, it has four external non compact divisors D 1 , D 2 , D 3 , D 4 ; but two compact divisors E 1 and E 2 . For this case, there are two generalised Mori-vectors given by The representative CY graph G SU 3 X(Y 3,0 ) is depicted by the Figure 9. Notice that the graph Figure 9: The CY graph G SU 3 X(Y 3,0 ) of the toric threefoldsX (Y 3,0 ) exhibiting manifestly the Calabi-Yau condition at each internal point of the graph.

Symplectic graphs and quivers
In this section, we first build the symplectic CY graph G SP 4 X(Y 4,0 ) by starting from the unitary G SU 4 X(Y 4,0 ) and using folding ideas under (Z x 2 ) ∆ SU 4 ×(Z y 2 ) ∆ SU 4 . Then, we construct the symplectic quiver Q SP 4 X(Y 4,0 ) with symplectic SP(4, R) gauge symmetry by using the unitary BPS quiver Q SU 4 X(Y 4,0 ) and outer-automorphisms (Z outer 2 ) Q SU 4 .

Symplectic CY graph
We start by the toric data of ∆ SU 4 X(Y 4,0 ) given by Table 3. Because these data are defined up to a global shift; we translate the points of ∆ SU 4 X(Y 4,0 ) by (0, −2). So the values of the w i and υ a points - Table 3 is invariant under the outer-automorphism symmetry group H outer Table 4: Toric data exhibiting manifestly outer-automorphism symmetry the property w ∓i = −w ±i and υ ∓a = −υ ±a , the outer-automorphism H outer ∆ SU 4 acts as a parity symmetry of the toric diagram, Notice that the outer-automorphism parity H outer ∆ SU 4 is isomorphic to the group product (Z x 2 ) ∆ SU 4 × (Z y 2 ) ∆ SU 4 generated by the reflections in x-and y-directions acting as follows where the (n x , n y )'s stand for the values of the external and the internal points of the toric diagram. So, the triangulated ∆ SU 2r X(Y 2r,0 ) is invariant under the outer-automorphism symmetry group with the central point υ 0 being the unique fix point of H outer ∆ SU 4 . By folding the CY graph G SU 4 X(Y 4,0 ) under the parity symmetry (Z x 2 ) ∆ SU 4 × (Z y 2 ) ∆ SU 4 , we end up with a new CY graphG having 2 + 2 = 4 points given, up to identifications, by w −1 ≡ w +1 , w −2 ≡ w +2 ; and υ 0 as well as υ −1 ≡ υ +1 . The CY graphG SP 4 X(Y 4,0 ) is depicted by the Figure 10. The generalised Mori-vectorsq 1 andq 2 associated with the symplectic CY graphG SP 4 X(Y 4,0 ) have each four componentsq a β . They are given bỹ Lie algebras. Here, we have The 2 × 2 square submatrix of the above rectangularq a β , associated with the triple intersections of the compact divisors E 0 and E ±1 , is given by Remarkably, this intersection matrix I a b = E 2 a .E b between the compact divisors is non symmetric. It can be put in correspondence with the non symmetric Cartan matrix K (C 2 ) of the symplectic C 2 Lie algebra given by The construction we have done for the particularG SP 4 X(Y 4,0 ) can be straightforwardly generalized toG SP 2r X(Y 2r,0 ) with r ≥ 2. The generic intersection matrix I ab with a,b=1, ....r; has a quite similar form as (5.5). It is non symmetric E 2 a .E b = E 2 b .E a and can be put in correspondence with the Cartan matrix of the symplectic Lie algebra C r ; see the discussion given after eq(4.10).
We end this subsection by making a comment on the folding of the family of Calabi-Yau graphs G SU p−1 X(Y p,0 ) with respect to the factor (Z x 2 ) ∆ SUp . It may be imagined as a partial folding in the transverse geometry represented by the points w 1 and w 3 of the toric diagram as indicated by Table 3. Recall that (Z x 2 ) ∆ SUp fixes all internal points υ a of the toric diagrams ∆ SU p−1 X(Y p,0 ) as well as the two external w 2 and w 4 ; but exchanges the two other external w 1 and w 3 . The folded gives an exotic Calabi-Yau diagram; which for the example p=3 is given by the Figure 11.
For this exotic folding, there are still two generalised Mori vectors that are associated with Figure 11: The folding CY graphs G SU 3 X(Y 3,0 ) under the partial outer-automorphism symmetry group (Z x 2 ) ∆ SU 3 . Because of the folding, the two divisors D 1 and D 3 merge. Here, we have the compact divisors E 1 and E 2 . These vectors are given bỹ The intersection matrix I ab = E 2 a .E b concerning the compact divisors is given by it is symmetric as in the unitary case.

BPS quiver with SP(4, R) invariance
The BPS quiver Q SP 4 X(Y 4,0 ) with SP(4, R) gauge invariance is obtained by folding the unitary Q SU 4 X(Y 4,0 ) by its outer-automorphism symmetry group H outer Q SU 4 whose action on the quiver nodes and the arrows is constructed below. The BPS quiver Q SU 4 X(Y 4,0 ) has 8 nodes {j} with j = 1, ..., 8; and 16 oriented arrows j|l as depicted by the Figure 4. Clearly, the BPS quiver has a non trivial outer-automorphism symmetry group with two factors given by The factor (Z 4 ) Q SU 4 has no fix quiver-node and no fix quiver-arrows; while (Z outer showing that four quiver-nodes amongst the eight ones are fixed. They concern the pair {3} is as depicted in the Figure 12. In addition to the nodes, the folded BPS quiver has 16 oriented arrows distributed as in the Table 5 where the complex X αa 12 with α = 1, 2 and a = 1, 2 form a quartet; and where the U α * and the U a * are doublets with U standing for X, Y and Z. With these complex superfields, one can write down the SQM superpotential of the theory; it will not be discussed here.

Conclusion and comments
In this paper, we have developed a method to construct a new family of 5D N = 1 supersymmetric QFT models compactified on a circle with finite radius. This family of gauge    This new graph, denoted as G SU 4 X(Y 4,0 ) , is given by a generalisation of the Mori-vectors of the ADE geometries of ALE spaces. It is defined by eq(4.1) and, to our knowledge, it has not been used before. We qualified the graph G SU 4 X(Y 4,0 ) as a unitary CY graph, first because of the unitary SU (4) symmetry of the gauge fiber withinX (Y 4,0 ); and second to distinguish it from the CY graph G SP 4 X(Y 4,0 ) having a symplectic SP (4, R) gauge symmetry. The use of G SU 4 X(Y 4,0 ) has the merit to (i) highlight the CY condition of the toricX (Y 4,0 ); (ii) extend the usual complex A 3 surface describing the resolution of an ALE space with an SU(4) singular-ity; and (iii) to study non trivial outer-automorphisms H outer ∆ SU 4 of the toric diagram ∆ SU 4 X(Y 4,0 ) . The outer-automorphism group H outer ∆ SU 4 has a fixed internal point (a compact divisor); and is used to build the symplectic CY graph G SP 4 X(Y 4,0 ) by using the folding G SU 4 X(Y 4,0 ) /H outer ∆ SU 4 . After having set the basis for the CY graphs to represent the toric threefoldsX (Y p,q ), we turned to investigating the BPS particles by constructing the symplectic BPS quiver Q SP 4 X(Y 4,0 ) that is associated with the symplectic CY graph G SP 4 X(Y 4,0 ) . This BPS quiver is obtained by folding the unitary BPS Q SU 4 X(Y 4,0 ) with respect to outer-automorphisms (Z outer 2 ) Q SU 4 . Recall that the X(Y 4,0 ) and exchanges the four others. It fixes four arrows and exchanges the 12 others. We end this conclusion by making two more comments regarding extensions of the analysis done in this paper.
The first extension concerns the building of symplectic BPS quivers Q SP 2r X(Y 2r,0 ) with generic rank. This is achieved by starting from the unitary quiver Q SU 2r X(Y 2r,0 ) with rank 2r-1 and use folding ideas. The resulting symplectic quivers Q SP 2r X(Y 2r,0 ) are associated with the toric threefolds obtained by folding the unitary Q SU 2r X(Y 2r,0 ) with respect to the outer-automorphism group (Z outer 2 ) Q SU 2r . The quiver series Q SP 2r X(Y 2r,0 ) is also related to the symplectic CY graphs G SP 2r X(Y 2r,0 ) obtained from the folding of the unitary G SU 2r X(Y 2r,0 ) under the outer-automorphism symmetry H outer ∆ SU 2r . The explicit expression of the generalised Mori-vectors and representative graph G SP 2r X(Y 2r,0 ) as well as the associated quivers have been omitted for the sake of simplifying the presentation of the underlying idea.
The second extension regards 5D super QFT models, based on conical Sasaki-Einstein manifolds Y p,q , with gauge symmetries beyond the unitary SU(r + 1) and the symplectic SP(2r, R) groups. These gauge symmetries concern the orthogonal SO(2r) and SO(2r + 1) groups; and eventually the three exceptional Lie groups E 6 , E 7 and E 8 . For 5D super QFT models with SO(2r) gauge symmetry embedded in M-theory onX (Y p,q ), one needs engineering toric Calabi-Yau threefoldsX p,q (D r ) with an SO(2r) gauge fiber. This might be nicely reached by using the technique of the CY graphs G SO 2r X(Dr) used in this study although an explicit check is still missing. This series of G SO 2r X(Dr) could be constructed by taking advantage of known results from the so-called complex D r surfaces describing the resolution of ALE space with SO(2r) singularity. The family of the CY graphs G SO 2r X(Dr) might be also motivated from the correspondence between eq(4.8) and eq(4.10) for simply laced case; see also the correspon-dence between eq(5.5) and eq(5.6) for non simply laced diagrams. If this SO(2r) study can be rigourously performed, one can also use outer-automorphisms of G SO 2r X(Dr) , inherited for the Dynkin diagram of so(2r) Lie algebra, as well as the outer-automorphisms of the associated BPS Q SO 2r X(Dr) to construct 5D supersymmetric QFT models with SO(2r − 1) gauge invariance. Progress in these directions will be reported elsewhere.

Appendices
In this section, we give three appendices: A, B and C. They collect useful tools and give some details regarding the study given in this paper. In appendix A, we recall general aspects of the families of CY3s used in the geometric engineering of 5D N = 1 super QFTs and the 5D N = 1 super CFTs. We also describe properties of the Coulomb branch of the 5D SQFTs.
In appendix B, we illustrate the derivation of the formula (3.1). In appendix C, we describe through examples the relationship between the 5D Kaluza-Klein BPS quivers and their 4D counterparts.

Appendix A
We begin by reviewing interesting aspects of M-theory compactified on a smooth non compact Calabi-Yau threefoldX. Then, we focus on illustrating these aspects for the class of CY3s given byX (Y p,q ) used in present study. We also use these aspects to comment on the properties of the BPS particle and string states of the 5D gauge theory.

Two local CY3 families
Generally speaking, we distinguish two main families of local Calabi-Yau threefoldsX depending on whether they have an elliptic fibration or not. These two families are used in the compactification of F-theory/M-theory/ type II strings leading respectively to effective gauge theories in 6/5/4 space time dimensions. These compactifications have received lot of interest in recent years in regards with the full classification of superconformal theories in various dimensions and their massive deformations. Because of dualities and due to the biggest 6D, the classification of 6D effective gauge theories has been conjectured to be the mother of the classifications in the lower dimensional theories. What concerns us in this appendix is not the study of the classification issue; but rather give some mathematical tools developed there and which can also be applied to our study.
• Family of local CY3s admitting an elliptic fibration.
These local Calabi-Yau threefoldsX are complex 3D spaces given by the typical fibration E → B with building blocks as: (i) B a complex 2D base; this is a Kahler surface. (ii) a complex 1D fiber E given by an elliptic curve. This genus zero curve is expressed by the Weirstrass equation where (x, y, z) are homogeneous coordinates of P 2 . Moreover, z is a function on the base B and (x, y, f, g) are sections with K B the canonical divisor class of B. Depending on the nature of the base, one can preserve either preserve 16 supersymmetric charges for bases B type T 2 → P 1 ; or eight supercharges in the case of bases B like for example P 1 ×P 1 and in general Hirzebruch surfaces F n . These elliptically fibered CY3 geometriesX ∼ E × B have been used recently in the engineering of superconformal theories in dimensions bigger than 4D. Regarding the SCFTs in 4D, the classification has been obtained a decade ago by using type II strings. For the classification of the 5D SCFTs using M-theory on elliptically fibered CY3 we refer to [4]. The graphs representing these theories are intimately related with the Dynkin diagrams of affine Kac-Moody Lie algebras.
• Family of local CY3s not elliptically fibered.
As examples of local Calabi-Yau threefoldsX, we cite the orbifolds of the complex 3-dimension space; i.e C 3 /Γ with discrete group Γ contained in SU(3). These orbifolds include the conical Sasaki-Einstein threefoldsX (Y p,q ) we have considered in this paper. The local CY3 geometries which are not elliptically fibered are used in the engineering of massive supersymmetric QFTs. The graphs representing these theories are related with the Dynkin diagrams of ordinary Lie algebras.
In what follows, we focus on M-theory compactified onX (Y p,0 ) considered in this study and on the corresponding U(1) p−1 Coulomb branch.
M-theory onX (Y p,0 ) The local threefoldsX (Y p,0 ) has four non compact divisors {D i } 1≤i≤4 and p-1 compact divisors {E a } 1≤a≤p−1 . These divisors are not completely free; they obey some constraint relations; in particular the Calabi Yau condition ofX (Y p,0 ) . They also obey gluing properties through holomorphic curves. The CY condition reads in terms of the divisor classes as in eq(2.1). For a generic positive integer p; it reads as follows [64] In our study, this condition has been transformed as in eq(4.1); and has been used to introduce the graphs given in section 4. Notice that the union of the compact divisors S = ∪ p−1 a=1 E a is important in this investigation; it is a local surface made of a collection of irreducible compact holomorphic surfaces E a . The irreducible holomorphic surfaces intersect each other pairwise transversally; this intersection is important and will be described below with details. Notice also that the Kahler parameters of the E a 's are identified as the Coulomb branch moduli φ a ; they appear in the calculation through the linear combination φ a E a which also plays an important role in the construction.
Regarding the gluing properties of the compact divisors and their consequences; they need introducing some geometric tools of the CY3. For a shortness and self contained of the presentation, we restrict to giving only those main tools that are interesting for this study.
However, we take the occasion to also describe some particular geometric objects that are relevant for the investigation of the Coulomb branch of the gauge theory. These geometric objects are introduced through the four following points (a), (b), (c) and (d).

a) Gluing the compact divisors
The compact holomorphic surfaces {E a } are complex surfaces inX. Neighboring surfaces E a and E b are glued to each others while satisfying consistency conditions. Before giving these conditions, recall that in our study, we have solved the CY condition by thinking of the E a 's as given by (F 0 ) a . As the holomorphic surface F 0 is given by a projective line P 1 f trivially fibered over a base P 1 B , then we have Notice that this is a particular solution of the CY on 4-cycles; it has been motivated by looking for a simple solution to exhibit the CY condition as in the Figures 7-8-9 of section 4.
However, general solutions might be worked out by using other type of holomorphic compact surfaces like the Hirzebruch surfaces (F n ) a of degree n and their blow ups at generic points.
To fix the ideas, we focus below on the surfaces F n and on two lattices associated with F n namely: (1) the lattice Λ l (F n ) of complex curves l in F n ; and (2) the Mori cone of curves M l (F n ); this is a particular sublattice of Λ l (F n ) . To that purpose, recall that holomorphic curves l in the compact surface F n are generated by two basic (irreducible) curves e and f.
The base curve e is the zero section of the fibration; and the f is the fiber P 1 f . The intersection numbers of these generators are given by From the above relations, we can perform several computations. For example, we have and where g is the genus of l. Because the genus g ≥ 0, the above quantity is greater than −2 due to the constraint 2 (g − 1) ≥ −2. (iii) Holomorphic curves in Mori cone M l (F n ) of the surface F n are given by the linear combination l ne,n f = n e e + n f f with positive integers n e and n f . These are particular curves of Λ l (F n ) corresponding to n e and n f arbitrary integers.
Notice that with this notation, we have h =l 1,n ; and the particular curve l 1,1 = e + f has a self intersection l 2 1,1 = 2 − n. (iv) If considering several surfaces F na with a=1,...,p-1; then eq(A.4) extend as follows e 2 a = −n a and f 2 a = 0 as well as e a .f a = 1. Quite similar relationships can be written down for the holomorphic curves in Λ a l = Λ l (F na ) and curves in M a l = M l (F na ). Returning to the gluing of curves l a and l b inside two compact surfaces S a and S b ; say the divisor E a and the divisor E b . It is defined by using the following restrictions The compact holomorphic curves C inX (Y p,0 ) are 2-cycles in the local Calabi-Yau threefolds. A subset of these curves is given by the e a 's and the f a 's generating the curves in the divisors E a when realised in terms of (F 0 ) a . In general, the compact curves C are given by linear combinations of generators C τ of compact holomorphic curves inX (Y p,0 ) ; they can be denoted like C n where n is an integer vector. As we have done above for the irreducible gauge divisors E a = (F 0 ) a , these CY3 holomorphic curves can be expressed as integer linear combinations like with n τ ∈ Z. From this expansion, we learn: (i) the set of compact holomorphic curves in In the case where all n τ integers are positive (n τ ∈ Z + ); the corresponding holomorphic curves belong to Mori cone M C (X).

c) Curves intersecting surfaces
This is an interesting intersection product defined in the CY3. Given the two following : (i) a holomorphic curve l belonging to the Mori cone M(X). (ii) a holomorphic surface S with canonical class K S sitting in the local Calabi-Yau threefoldsX. Then, the intersection between l and S is given by For the interesting case where the holomorphic surface S is given by the compact divisors E a , the above intersection reads as (l.K S )| Ea . The value of this intersection depends on two possibilities: (α) The case where l lives inside M a (X); then, we have l.E a = (l.K S )| Ea .
(β) The case where l lives inside another surface; say S = E b ; then we have where l ba is the curve participating in the gluing between E a and E b . The curve l ab also sits in M a (X). From these relations, we learn that the intersections l.E a can be recovered from the intersection products on the Mori cones M a .

d) Triple intersections
The triple intersections E a .E b .E c of the holomorphic surfaces are numbers that can be expressed as intersection products of gluing curves inside any of the three surfaces. For that, we use the typical curves L ab = E a .E b ; these intersection curves appear as irreducible curves l ab from the E a side; and as irreducible curves l ba from the side of E b . The intersection of E a and E b is obtained as described before; that is by the identification l ba = l ab . Similar identifications hold for the intersections of E b .E c and E c .E a . By taking the intersection curve l αβ as the diagonal sum of the the generators namely l αβ = e α + f α , we obtain in agreement with eq(4.9).

5D Coulomb branch and BPS states
To deal with the Coulomb branch of the 5D effective gauge theory and its BPS states, we need, in addition to the algebraic geometric objects given above, other basic quantities. One of these quantities concerns the metric ds 2 = τ ab dφ a dφ b of the Coulomb branch. It turns out τ ab derives from the effective scalar potential F (φ) of the low energy theory; it reads as follows Given F (φ) , one also has two other interesting quantities associated to it. (i) the gradient ∂F (φ) ∂φ a which give the tensions T a of BPS string states. (ii) the third derivatives as ∂ 3 F (φ) /∂φ a ∂φ b ∂φ c giving coefficient of the Chern-Simons term κ abc = kd abc . The higher derivatives vanish identically because F (φ) is a cubic function. Recall that the effective potential of the 5D effective theory is exactly known; it reads as follows This function has the properties: (i) It is a cubic function of the gauge scalar field moduli Finally, the volume of ofX (Y p,0 ) ; it is given by the triple intersection number of the divisor J. This is the prepotential of the low energy 5D theory Notice that by putting (A.19) back into T a , τ ab , κ abc and using (A.16), we end up with the following interpretation in terms of intersections The BPS states of the 5D theory In this effective gauge theory, we distinguish two kinds of BPS states: (i) Massive particle states (M2/C) given by M2-branes wrapping the compact holomorphic curves C. The masses of these particle states are given by V ol (C) . For the particular compact curves C a ; it is associated p-1 electrically charged elementary BPS particles given by the wrapping M2/C a . The masses of these particles are given by υ a .
(ii) String states M5/S arising from M5-brane wrapping the compact holomorphic surfaces S. The tensions of these strings are given by V ol (S) . For the particular compact surfaces given by the p-1 divisor E a ; it is associated p-1 magnetically charged elementary BPS strings M5/E a with tensions given byυ a .
Notice that the BPS spectrum of 5D N=1 theories include gauge instantons I in addition to the electrically charged particles and the magnetically charged monopole strings. The central charges of these particles are given by where n (elc) a , n (mag) a are integers. Notice that not every choice of these integers corresponds to the central charge of a physical state whose mass or tension has to be positive. The values of these n's are obtained using BPS quivers and their mutations. Notice also that by compactifying the 5D gauge theories on a finite circle; we generate a Kaluza Klein particle states as described in the core of the paper.
(d) Dirac pairing The intersection numbers C a .E b of compact curves C a and compact surfaces E b describe the Dirac pairing between the BPS particles and the BPS strings.

Appendix B
Here, we consider M-theory compactified onX (Y 2,0 ) with SU(2) gauge symmetry and look for the derivation of the quiver dimension d bps = 2 (p − 1) + 2 of eq(3.1). Because of the choice p=2, we have d bps = 4 indicating that the BPS quiver Q SU 2 X(Y 2,0 ) has four nodes as shown by the Figure 13-(b). Recall that the BPS quiver Q SU 2 X(Y 2,0 ) is related to the toric diagram ∆ SU 2 X(Y 2,0 ) by the so-called fast inverse algorithm [92,93,94]. This algorithm involves two main steps summarized as follows: • Brane tiling BT This step maps the toric ∆ SU 2 X(Y 2,0 ) into a brane tiling in the 2-torus to which we refer to as BTX (Y 2,0 ) . It uses the brane web∆ SU 2 X(Y 2,0 ) (the dual of the toric diagram) to represent it by the tiling as given by the Figure 13-(a). Recall that the toric graph representing ∆ SU 2 X(Y 2,0 ) is Figure 13: (a) The brane tiling of ∆ SU 2 X(F 0 ) . (b) the BPS quiver Q SU 2 X(F 0 ) with SU(2) gauge symmetry.
(c) the BPS subquiver of the 4d N = 2 pure SU(2) 0 gauge theory. a standard diagram; it can be drawn by using the Table 2 with p=2 and q=0. It has four external points (n ext = 4), describing the four non compact divisors; and one internal point (n int = 1) describing the compact divisor associated with the SU(2) gauge symmetry. For a short presentation, we have omitted this graph.
• The BPS quiver The second step maps the brane tiling BTX (Y 2,0 ) into the BPS quiver Q SU 2 X(Y 2,0 ) as shown by the picture Figure 13-(b). This mapping is somehow technical, we propose to illustrate the construction by giving some details.
Dimension d bps of the quiver Q SU 2 X(Y 2,0 ) The Figure 13-(a) is a bipartite graph on the 2-torus with two kinds of nodes white and black. So, half of the nodes are white and the other half are black. This tiling is characterised by three positive integers (N W , N E , N F ) related amongst others by the following relation where χ xg = 2g − 2 is the well known Euler characteristics relation of discretized real genus- Before proceeding, notice that the mapping between a brane tiling BTX (S) and a toric diagram ∆X (S) is not unique. To a given toric diagram one may generally associate several brane tiling. So the brane tiling is a 1 → many. This diversity has an interpretation in terms of quiver gauge dualities of Seiberg-type. Notice also that for the 2-torus, we have g = 1; and then the quiver dimension can be also expressed as follows Building the quiver Q SU 2 X(Y 2,0 ) To build the quiver Q SU 2 X(Y 2,0 ) from BTX (Y 2,0 ) , we one proceed in steps as follows: (i) pick up a representative 2-torus unit cell (in green color in the Figure 13-a).
(ii) draw the corresponding BPS quiver given by the Figure 13-(b) by using the following method.
• To each face F i within the 2-torus unit cell of the BT-tiling, we associate a quiver-node {i} in the gauge quiver Q SU 2 X(Y 2,0 ) . As there are N F = 4 faces in the unit cell of BTX (Y 2,0 ) , then the Q SU 2 X(Y 2,0 ) has four nodes {1; 2; 3; 4} . Notice that the number N F can be presented in different, but equivalent, ways; for instance like N F = 2 (p − 1) + 2, or where we have set n ext − 2 = f + 1, and where for SU (2) the rank r=1. The number N F is precisely the dimension d bps given by eq(3.1).
• To each edge E ij of the brane tiling, separating the faces F i and F j , we associate a quiverarrow ij with direction determined by a traffic rule. In this rule, the circulation goes clockwise around white BT-nodes; and counter-clockwise around black BT-nodes. In the example Q SU 2 X(Y 2,0 ) ; we have 8 quiver-arrows organized into four pairs. The are given by arrows i (i + 1) with i = 1, 2, 3, 4 mod 4.
• To each BT-node in the brane tiling corresponds a superpotential monomial. So, the full superpotential W SU 2 X(Y 2,0 ) associated with the BPS quiver has four monomials; this is because N W = 4. For simplicity, we omit the explicit expression of W SU 2 X(Y 2,0 ) . In the end of this appendix, notice that the four nodes of the quiver Q SU 2 X(Y 2,0 ) are interpreted in type IIA string as the elementary BPS particles. The particles sitting at the nodes {1, 2} correspond to the electrically D2/C 2 and the magnetically D4/C 4 charged BPS particles where C n refers to n-cycles inX (Y 2,0 ). These two nodes form together a sub-graph of the SU(2) gauge quiver Q SU 2 X(Y 2,0 ) ; it is given by the usual Kronecker diagram depicted by the Figure 13-

Appendix C
In this appendix, we describe briefly helpful tools regarding the structure of BPS quivers in 4D N = 1 Kaluza-Klein while focussing on those relevant aspects for our study. The material given below aims facilitating the reading of section 3. For a rigorous and abstract formulation of BPS quivers using amongst others the central charges and the Coulomb branch moduli, we refer to literature in this matter. For instance, the section 2.3 of [66] for 4D KK quivers and [99]- [101], [90], [69]- [73] for 4D.

ADE gauge models
A short way to introduce the 4D KK BPS quivers is to go through the well studied BPS quivers Q Ĝ X (4d) of 4D N = 2 gauge theories with ADE gauge symmetries. The use of these 4D quivers may be motivated from various views in particular from the three following: (1) Q Ĝ X (4d) ⊂ Q Ĝ X (5d) . The quivers Q Ĝ X (4d), which are described below and mention in the text, appear as sub-quivers of the 4D KK BPS quivers Q Ĝ X (5d) . For example, compare the two pictures of the Figure 14 with the Figures 5 and 6 in the main text. This feature, which implies that the BPS states of 4D N = 2 belong also to 4D KK N = 1, can be explained by the fact that the 4D N = 2 theory corresponds just to the zero mode of 4D Kaluza-Klein N = 1 theory.
(2) Type II strings on CY3. The Q Ĝ X (4d) quivers deal with 4D N = 2 gauge theories with gauge symmetry G. These SQFTs can be remarkably embedded in type IIA string on CY3s.
However, because of the relationship between type IIA and M-theory, the 4D N = 2 theories can be also embedded in M-theory on CY3×S 1 which is the mother of 4D N = 1 KK theory.
(3) Quiver mutations and duality. The BPS quivers have been widely employed in the case of four-dimensional N = 2 theories. There, several techniques have been developed to handle them. Some of these techniques like quiver mutation algorithm apply also to Q Ĝ X (5d) ; these mutation have an interpretation in terms of Seiberg-like duality. For an explicit study, see [102,71,73].
In what follows, we focus on the 4D BPS quivers of pure 4D N = 2 gauge theories with gauge invariance G; say of type ADE. A general description of BPS quivers would also involve flavor matter; but for convenience, we ignore them here. The determination of the full set of BPS states of the N = 2 SQFT is a complicated issue; but nicely formulated in terms of BPS quivers Q ADÊ X (4d). So, the BPS quivers encode the relevant data on the BPS states of the N = 2 SQFT. Their properties depend on the coordinates of the moduli space of the theory. Depending on the gauge coupling regime, we distinguish two sets {Q ADÊ X (4d) n } I,II of BPS quivers termed as strong and weak chambers: • Strong chamber {Q ADÊ X (4d) n } str . This is a finite set of BPS quivers describing the BPS particle states in the strong chamber. For the derivation of the full list o3f BPS states for ADE Lie algebras; see for instance [71] and references therein.
• Strong chamber {Q ADÊ X (4d) n } weak . This is an infinite set of BPS quivers describing the BPS states in the weak chamber. For a description of this set; see for instance [73].
The full content of these BPS chambers can be obtained by constructing all BPS quivers using mutation algorithm (mutation symmetry group). One of these BPS quivers is given by the so-called primitive BPS quiver denoted below likeQ ADÊ X . This is a basic BPS quiver made of the elementary BPS states. By applying the mutation algorithm onQ ADÊ X , one generates new quivers made of BPS states given by composites of the elementary ones. By repeating this operation several times, one can generate all BPS particles of the theory. For the strong chambers, the mutation group is finite; however is it infinite for weak chambers.
There, one obtains recursive relations for the EM charges of the BPS states.

Primitive quiver
As far as the primitive quiver of pure gauge theories is concerned, its 2r BPS states have electric/magnetic (EM) charge vectors given by b 1 , ..., b r and c 1 , ..., c r ; they appear inQ ADÊ X as depicted by the pictures of the Figure 14 for SU(2) and SU (3)  giving the Cartan matrix of the Lie algebra. Notice that these charge vectors can be denoted collectively like b i = γ 2i and c i = γ 2i−1 with i = 1, ..., r. These EM charge vectors obey the Dirac pairings γ m • γ n that splits as Notice also that for simply laced Lie algebras, the primitive BPS quiverQ ADÊ X consists of 2r nodes and 3r − 2 links as described below: A) nodes inQ ADÊ X The 2r nodes ofQ ADÊ • 2 (r − 1) oblique links l ij joining two nodes of different pairs (b i , c i ) and (b j , c j ). This reduced number of links is due to the constraint eqs(C.6-C.7). So, the intersection matrix A G 0 describing the primitive quiverQ ADÊ X is related to the Cartan matrix of the Lie algebra as follows This construction extends to the BPS quivers with non simply laced gauge symmetries; see for instance [90,69,70].