Trifectas for $T_N$ in 5d

The trinions $T_N$ are a class of 5d $\mathcal{N}=1$ superconformal field theories (SCFTs) realized as M-theory on $\mathbb{C}^3/\mathbb{Z}_N \times \mathbb{Z}_N$. We apply to $T_N$, as well as closely-related SCFTs that are obtained by mass deformations, a multitude of recently developed approaches to studying 5d SCFTs and their IR gauge theory descriptions. Thereby we provide a complete picture of the theories both on the Coulomb branch and Higgs branch, from various geometric points of view - toric and gluing of compact surfaces as well as combined fiber diagrams - to magnetic quivers and Hasse diagrams.

1 Introduction 5d gauge theories are not renormalizable. Nevertheless, overwhelming evidence exists that they can have strongly-coupled UV completions by 5d SCFTs. Starting with the work by Seiberg [1], there have since been a multitude of approaches to study, and even potentially classify such 5d SCFTs. One approach is to characterize the SCFTs by M-theory on a Calabi-Yau threefold singularity [2,3], where the space of resolutions of singularities has an interpretation in terms of the Coulomb branch of the IR-description of the SCFT. In this approach the Higgs branch, which corresponds to the deformations of the singularity, is somewhat obscure. The advantage of this approach is however, that it does give evidence for SCFTs that do not admit a weakly coupled gauge theory description. Alternatively, 5d SCFTs are realizable in terms of 5-brane webs [4][5][6][7][8][9][10][11][12][13][14][15]. This approach is limited to theories that have a gauge theory description, but has the advantaged that both Coulomb and Higgs branch parameters are often manifest.
One of the key characteristics of 5d SCFTs is their flavor symmetry, which is generally enhanced compared to the IR flavor symmetry of any of its gauge theory descriptions. These features can be determined from both geometry and brane webs in a case-by-case analysis, but most systematically in the approach developed in [16][17][18][19] that encodes both the UV flavor symmetry, and the possible mass deformations in terms of a graph: each such combined fiber diagram (CFD) is associated with a 5d SCFT. One goal of the present paper is to make this approach more accessible, and to put it into the context of other, perhaps more widely known approaches.
From a geometric point of view, one particularly well-studied subset is the toric Calabi-Yau threefold singularity [20,21]. Nonetheless, the class of models that can be studied using the toric geometry toolbox is relatively limited. Alternatively, one can start with the geometries underlying the 6d classification of SCFTs in F-theory [22]. Although it remains an open problem, whether all 5d SCFTs arise from 6d by circle-reduction and mass deformations, this class of geometries reproduces all of the known 5d SCFTs -and more. The elliptically fibered Calabi-Yau geometries that are used in F-theory, have a partial resolution, which characterize 5d SCFTs and their Coulomb branch descriptions. This is the approach taken in [16][17][18][19][23][24][25][26].
Another recent development is the structure of the Higgs branch of 5d SCFTs. Its magnetic quiver (MQ) is defined to be a 3d N = 4 quiver gauge theory, whose Coulomb branch gives the Higgs branch of the 5d SCFT [33]. More recently, the toolkit of Hasse diagrams was developed from the magnetic quiver [34], which encodes the foliation structure of the Higgs branch as a hyper-Kähler singularity. In practice, the magnetic quiver and Hasse diagrams can be derived from the brane web description, for various classes of theories [35][36][37][38][39].
The goal of this paper is to study all of these approaches in the context of a single class of theories -the T N theories, which we believe will be useful in order to get an overview and understanding of the interconnection between various approaches. The T N theory was originally introduced in the context of 4d N = 2 class S theories [40], as N M5-branes wrapping 2-sphere with three full punctures. It has central status in the classification of 4d N = 2 theories, as it can be used as building blocks to glue more complicated theories [41][42][43].
The theories T N in 5d were first introduced in [44], where they were studied already in terms of toric geometry and brane webs. In the geometric picture, the T N SCFT is realized in terms of M-theory on the Calabi-Yau threefold singularity C 3 /Z N × Z N . Its UV flavor symmetry is which is enhanced to G F = E 6 in the N = 3 case. The Coulomb branch dimension (rank) is T 2 is a rank 0 theory and we will not further discuss it here. A weakly coupled description in terms of a linear quiver is known [45] [ and the partition functions of T N were studied in [46][47][48][49]. As a complementary approach, its holographic duals in AdS 6 were determined in [50][51][52][53], based on the constructions in [54][55][56].
In this paper, we will show how various approaches developed in the past two years are realized in this class of theories. In particular, we will not only present the definition of CFDs and its derivation from the toric description, but also the brane web approach can be used to determine the refined structure of the Higgs branch in terms of the Hasse diagram. Moreover, we provide new results on the BPS spectrum, the web of different SCFTs related by mass deformations and the Higgs branch structure of these theories. Beyond T N theories, and its descendants obtained by mass deformations -a substantial class of which have no weakly coupled description -we also study their 5d and 6d ancestors, i.e. SCFTs, from which T N descends after decoupling hypermultiplet matter.
The structure of this paper is as follows: in section 2, we present the definition of 5d T N theories in geometric language. We also review the geometric engineering rules of 5d SCFTs and gauge theories in M-theory, the toolkit of toric geometry and the definition of CFDs. In section 3, we discuss the descendants and ancestors of the 5d T N theories related by mass deformations, which is encoded in the transitions of the CFDs. Furthermore, we will present new results on the low spin BPS spectrum of the T N theories in terms of SU (N ) 3 representations. In section 4, we discuss the IR gauge theory descriptions of the 5d T N theories, from the perspectives of toric geometry, CFDs and box graphs. Finally, in section 5, we review the brane web descriptions, the magnetic quivers and present new results of the Hasse diagrams. We also show the magnetic quiver of some descendant theories of T N .

SCFTs, Geometry, and CFDs
We begin with the realization of 5d gauge theories and SCFTs using M-theory on a singular Calabi-Yau geometry. The Calabi-Yau resolutions of these singularities have various characterizations, for T N in particular in terms of the toric resolutions. We connect the toric description, which is well-known, to the new approach using CFDs.

M-theory, Calabi-Yau Singularities and 5d SCFTs
In this section we will give a brief introduction of the basics of 5d N = 1 superconformal field theories (SCFTs) and their descriptions in terms of M-theory on a non-compact Calabi-Yau threefold. Let us start with general properties of weakly coupled 5d N = 1 gauge theories.
In 5d the gauge coupling has negative mass dimensions, hence the Yang-Mills action is nonrenormalizable. However, the theory in the UV can be a superconformal fixed point that flows to said gauge theory. In other words, we can interpret a weakly coupled gauge theory in the IR as an effective description of an SCFT. In general, such an SCFT might have several inequivalent mass deformations, which can be interpreted as different "UV dual" gauge theory descriptions.
We consider a 5d N = 1 gauge theory with the gauge group N I=1 G I and matter content ⊕ f R f = ⊕ f (R 1 , . . . , R N ) f , where the matter hypermultiplets transform in representations R I of the simple factors G I . We only allow matter that is either charged under only one of the G I or in the bifundamental of G I × G J . The Coulomb branch of this theory can be described by the 1-loop IMS prepotential [3]. It is a function of the Coulomb parameters, i.e. the vacuum expectation values (vevs) of the real scalars in the vector multiplet. For a simple gauge group it is given by , with the T i the Cartan generators of G, and the α i s are the positive roots Φ + of G. Also, g 0 is the gauge coupling of G and k is the Chern-Simons level, which is only relevant for G = SU (N > 2). Finally, the λ a i are the weights of the representations R f and m f are the masses of matter hypermultiplets. For a quiver theory, in the simplest case given by two simple gauge groups coupled by bifundamental matter F , the prepotential is given by where each of the constituents has their individual gauge coupling and CS level as well as matter content and m B is the mass of the bifundamental. This naturally generalizes to longer quivers.
From the prepotential one can easily determine the 1-loop exact Lagrangian where the A i are the gauge fields with field strength F i . For consistency, the effective gauge coupling τ ij has to be positive definite and the effective Chern-Simons coefficients c ijk need to be integer-valued to ensure gauge invariance. We see from (2.1) that the prepotential depends on the relative values of the masses and the Coulomb branch parameters. We can choose the Weyl wedge of G such that α i φ i ≥ 0 for all roots, but the sign of λ a i φ i − m a depends on the phase in the extended Coulomb branch, which is parametrized by φ i and m a . This will be explained in more detail in section 4.3.
An important property of the gauge theories is their flavor symmetry. Let us consider the case where all the m f = 0, where the classical flavor symmetry G F,cl is maximal. The classical flavor symmetry generated by n fields in the representation R of G I is given by The other important matter type we will encounter is a single hypermultiplet in the bifundamental of SU (N 1 ) × SU (N 2 ) with classical flavor symmetry U (1). Furthermore, each simple gauge group factor G I provides a topological symmetry U (1) I T with current J T = 1 8π 2 Tr(F ∧ F ). In the strong coupling limit the flavor symmetry in the IR, G F, cl , can be enhanced by non-perturbative effects to the UV flavor symmetry with rank rk(G F ) = rk(G F,cl ) + N .
A way to provide evidence for the strongly-coupled UV fixed points is the construction of such 5d gauge theories and their moduli spaces in M-theory on a singular, non-compact CY 3 Y .
In this paper, we require that the singularity has a crepant, i.e. Calabi-Yau, resolution Y . The M2-branes on (−1)-curves in ruling flavor masses m f Volume of (−1)-curves Change of gauge theory phases Flops of (−1)-curves Table 1: Dictionary between 5d gauge theory data and the geometric description in M-theory on a CY 3 .
which usually characterize the flavor symmetry. The compact divisors are Poincaré dual to harmonic (1, 1) forms, which can be used to expand the M-theory C 3 form and yield r abelian gauge fields. If there is a non-abelian gauge theory description of the theory in question, the W-bosons arise from M2-branes wrapping collapsed curves in a particular singular limit of the surfaces S i . The key concept for this is a geometric ruling of the divisors by rational curves f i , i.e. P 1 s, over curves Σ In summary, M-theory on this geometry yields two types of vector multiplets: 1. The U (1) gauge bosons, from the expansion of C 3 in the (1, 1)-forms dual to the S i , such that i = 1, . . . , rk G.
2. The W-bosons are obtained from M2-branes wrapping the fibers f i of rulings, which are rational curves with self-intersection 0 inside S i .
The Cartan matrix of G is given by the intersection number We see that the W-bosons only become massless in the limit where the volume of the fibers f i goes to 0, and we obtain singularities of ADE-type over the curves Σ. This suggests that the volumes of the f i , i.e. the Kähler parameters, are proportional to the vevs of the φ i as the gauge group enhancement happens at the origin of the Coulomb branch. Different rulings of the same resolution lead to "dual" weakly coupled descriptions in terms of different gauge theories that however have the same UV completion. Finally, the Chern-Simons coefficients of the gauge theory depends on the triple intersection number Decoupling of hypermultiplet matter, and thus flowing to a different UV-fixed point is realized in terms of flops that take a (−1)-curve outside of ∪S i . In the singular limit, its volume, and thus the mass of the associated hypermultiplet stays finite, and thus decouples from the SCFT sector. We will consider in the following both internal flops, as well as flops that correspond to decoupling/mass-deformations. The latter will give rise to the complete decoupling or RG-flow tree. The theories obtained after decoupling will be referred to as descendant theories.

Toric Geometry
Here we briefly review toric Calabi-Yau threefold singularities and the geometry of of their divisors. For a general introduction to toric geometry, see [57][58][59]. For the notations for toric Calabi-Yau threefold singularities, see for instance [20].
A toric threefold X Σ is described by a toric fan Σ, which is a set of cones in the 3d lattice N = Z 3 with a common origin (0, 0, 0). The 1d 3d vectors with integral components, also called rays of the toric fan. Geometrically, each ray corresponds to a toric divisor (complex surface) D i of X Σ . Similarly, the 2d cone v i v j corresponds to the complete intersection curve D i · D j , and the 3d cone v i v j v k corresponds to Here v i , v j and v k are the boundary rays of these cones. It is required that the intersection of two cones is either empty or another cone in the toric fan.
Two toric fans Σ and Σ are equivalent if and only if the set of rays can be mapped to each other with an SL(3, Z) rotation, while keeping the cone structure unchanged. The equivalence of two toric fans also induces the topological isomorphism between X Σ and X Σ .
There are three linear relations on the divisors D i The anticanonical divisor of X Σ is given by the sum of all the toric divisors Hence if X Σ is Calabi-Yau, with K X Σ = 0, all the rays v i have to lie on the same plane in Z 3 .
In this paper, after an SL(3, Z) rotation, we take the form of all v i to be A toric threefold is compact if and only if the cones in Σ span the whole Z 3 . It is then easy to see that a toric Calabi-Yau threefold X Σ is always non-compact. The rays on the boundary of Σ correspond to the non-compact divisors of X Σ , which will be denoted by D α in the remainder of the paper. On the other hand, the rays in the interior of Σ correspond to compact divisors of X Σ , which will be denoted by S i . In this notation, the curves D α · D β are always non-compact, while the curves S i · D α and S i · S j are always compact.
A toric threefold is smooth if and only if each 3d cone has the form v i v j v k , and they all On the other hand, the toric threefold has a singularity if a 3d cone has more than three vertices, or its volume is greater than one. A crepant resolution of a toric Calabi-Yau threefold singularity exactly corresponds to a subdivision of the toric fan. After the resolution, all the 3d cones will have unit volume and X Σ will be smooth. We plot an example of a simple toric Calabi-Yau threefold singularity and its crepant resolution in figure 1.
In this picture, the topology of compact toric divisor S 1 can be easily read off. S 1 itself is always a toric surface, and its toric fan Σ(S 1 ) can be constructed as follows. Denote the 3d ray of S 1 in Σ by v, then each 2d cone v i v ∈ Σ gives rise to a ray v (2) in Σ(S 1 ). Similarly, each 3d cone v i v j v ∈ Σ gives rise to a 2d cone v (2) i v (2) j in Σ(S 1 ). From figure 1, it is easy to see that S 1 has the topology of Hirzebruch surface F 0 = P 1 × P 1 . Each toric divisor is presented as a point on the (x, y)-plane, and we label their coordinates in Z 3 . Each 2d cone is presented as a line segment between the points and each 3d cone is given by a polygon. On the left hand side, there is a singular toric Calabi-Yau threefold with four non-compact divisors D 1 , . . . , D 4 . On the right hand side, after the crepant resolution, the toric fan is subdivided and each 3d cone (presented as a triangle) has unit volume. There is a new compact divisor S 1 in the middle. More generally, this rule can be applied to any compact toric divisor, see figure 2 for another example.
For a smooth toric threefold, h 1,1 and the self-triple intersection number of a compact divisor S i are given by (2.14) The latter equation holds because S 3 i = K S i · S i K S i , and S i is always a rational surface. Then we consider the triple intersection number D 2 v · S i , where D v corresponds to the ray v ∈ Σ, and it can be either compact or non-compact. D 2 v · S i can be computed by is the complete intersection curve on S i . The self-intersection number of C i,v on S i can then be read off by a simple rule (see also [60]): denote the 1d ray of C i,v in Σ(S i ) by v (2) , and its two neighbor rays by v 2 . Then the self-intersection number C 2 i,j is given as a solution to For the example in figure 2, for the curves on S 1 (the blue node), we have C 2 1,v = −1, Finally, for the triple intersection number among three distinct divisors D i · D j · D k on a smooth X Σ (no matter they are compact or not) is given by Here we list all the non-zero triple intersection numbers among the compact divisors S i in figure 2:  Figure 3: The toric fan of each compact surface S i . The subscript i is labelled at the center of each toric fan, and the self-intersection number of each toric curve is written in the brackets. gdP n refers to the generalized del Pezzo surfaces.
A toric flop among compact surfaces changes the 2d and 3d cones in Σ, while leaving the rays unchanged, see figure 4 for an example. After this flop, the triple intersection numbers change as: As a point of reference, we also present the triple intersection numbers S i ·S j ·S k before and after the flop in the notations of [28][29][30] in figure 4 (b). Here each circle denotes a compact surface. For each compact surface S i , the label i k n in the center means that it is a Hirzebruch F n , blown up at k points. The Picard group elements of such a surface are denoted by h, e, f and x i (i = 1, . . . , k). They have the following intersection numbers: Note that the choice of n and the Picard group generators are not unique. Then two circles are connected by a line if the two surfaces S i and S j intersect at a curve C ij . At the ends of the line, the number in the square box inside the circle i denotes the intersection number We also pressent the Picard group element of the curve C ij over the line. Finally, if three surfaces intersect at a point, we label the triple intersection number at the center of the triangle.
Note that the diagrams in figure 4 (b) of the type used in [28][29][30] do not contain information about the non-compact divisors D α . Nonetheless, D α s are crucial for the determination of (a) flop 6 (-1) Figure 4: (a) An example of toric flop of the resolved T 5 geometry among the compact divisors. The triangulation in the picture changes, while the rays remain the same. We furthermore indicate how the geometry is glued from the rational surfaces such as in figure 3. The numbers (−n) at the external vertices indicate the self-intersection numbers of curves that are intersections between compact and non-compact divisors and encode the SU (5) 3 flavor symmetry.
(b) Another presentation of the geometry, where the ith internal vertex i k n is Bl k F n . In this presentation the flavor symmetry is not manifest.
(3) Figure 5: The labeling of T N singularity, with an example of N = 5.
the flavor symmetry G F and the CFD, which will be discussed later. Contrary to that, the description in terms of the surfaces as in figure 3 do contain this information -and will therefore be key in encoding the flavor symmetries (and thereby the CFDs).
More generally, the orbifold singularity C 3 /(Z N × Z N ) associated to T N theory can be described by a toric fan with rays (0, 0, 1), (N, 0, 1) and (0, N, 1). The fully resolved toric fan has rays (x, y, 1), The Cartan subalgebra of the superconformal flavor symmetry G F is given by the non-compact divisors D (i) . As there exists three linear equivalence relations (2.9), the non-compact divisors D (i) 0 are modded out, and the total rank of G F is The W-bosons of G F are given by M2/anti-M2 branes wrapping certain 2-cycles, which become massless in the singular limit. In the case of figure 5, these 2-cycles are (also see section 2.3 for more information): 4 · (S 2 + S 4 ) .

(2.23)
As a result, the points in the interior of the boundary lines form the Dynkin diagram of G F .
In the T N case N > 3, we get exactly (2.24) Finally, we make a comment that the methodology introduced in this section can be generalized to any other toric configuration as well. The only subtlety is that the flavor symmetry read off from the toric diagram may only form a subalgebra of the full G F . For example, in the case of the T 3 theory, the flavor symmetry will be enhanced to This phenomenon will be explained from the BPS states counting in section 3.4.

CFDs for T N and other Toric CY
Combined fiber diagrams (CFDs) were introduced in [16][17][18][19] to provide an efficient graphical tool to characterize for a given 5d SCFT the following characteristics: • G F -UV flavor symmetry: subgraph of (−2)-vertices (marked in green) • Mass deformations: (−1)-vertices (marked in white) • weakly coupled descriptions and dualities As we discussed, in the M-theory-SCFT dictionary, (−2)-curves inside the compact surfaces ∪ i S i correspond to flavor symmetries. These are encoded in the marked subgraph of the CFD.
Likewise matter multiplets are encoded in (−1)-curves, which are precisely the vertices with label (−1). Each vertex of a CFD is labeled not only by an integer (the self-intersection number of the associated curve), but also by the genus g. Mass deformations correspond to flopping (−1)-curves outside of the collection of compact surfaces ∪ i S i . These will translate into operations on the CFDs which map a CFD for one SCFT to a descendent's CFD. Such flops are realized in terms of CFD transitions.
The initial framework of CFDs was not dependent on a toric description, and simply relied on the resolution of Calabi-Yau singularities (in particular elliptic models). Here we will apply this approach to toric Calabi-Yau three-folds, where it turns out the CFDs are also very natural objects to define, which encode all the above data.
We will start by discussing CFDs for a general toric geometry, which is given by a triangulation of a convex 2d polyhedron. In the polyhedron, the internal points are the compact divisors S i (i = 1, . . . , r) and the boundary points are non-compact divisors D α . If an internal point corresponding to S i is connected to a boundary point corresponding to D α , then the line segment corresponds to the intersection curve S i · D α . The general procedure of reading off CFDs from geometry was developed in [19], which will be applied here.
In the CFD associated to this geometry, the labels (n, g) for each node D α are given by: Geometrically, n is the self-intersection number of the intersection curve and g its genus. Here, ξ i,α is an integral multiplicity factor associated to the curve S i · D α , which is read off in the following way.
Given a fixed D α , we plot all the curves S i · D α with their self-intersection numbers S 2 i · D α in D α , which form a loop in the toric case: then after the blow down process, we get: We already labeled the multiplicity factor for each curve above them. In the toric picture, this process exactly corresponds to the following flops: One can check that the (n, g) = (−2, 0) for D 1 is indeed invariant during the process, using Because in the terminated geometry as in (2.30), the curve S 1 · D 1 always has normal bundle . For the vertices of the polyhedron, it corresponds to a CFD vertex with n ≥ −1 and g = 0. It always has genus zero in the toric case, since all the toric curves are topologically P 1 . Finally, the number of edges between two CFD vertices corresponding to D α and D β is given by: For the particular T N theory, we denote by D (k) α , k = 1, 2, 3 and α = 0, · · · , N − 1 the non-compact divisors that correspond to the SU (N ) 3 flavor symmetry. The label k is assigned in a counter-clockwise order. The α = 0 node can be identified with the affine node. The which will be identified with the marked/unmarked vertices of the CFD. The relative inter- figure 6. To obtain the complete CFD for the T N theory that has also the manifest SU (N ) 3 flavor symmetry, it is key that at least one compact divisor connects to each external vertex. Figure 6: CFD for the T N theory. The (−2)-curves are the marked vertices from which the SU (N ) 3 flavor symmetry can be read off. The (−1) curves correspond to unmarked vertices in the CFD, which are the starting points for CFD-transitions.
We consider an example, the T 5 theory, with the following toric triangulations and CFD: This is a rank 6 theory, and we label the 6 compact divisors by S i , from top to bottom, left to right. The reduced intersection matrix S i (D (k) α ) 2 with the Cartans of the flavor symmetry SU (5) 3 with Cartan divisors D (k) α , k = 1, 2, 3 and α = 0, · · · , 4 is In the left triangulation on the LHS of (2.33), the assignment is a = c = 0, b = 1. In the triangulation on the RHS of (2.33) the assignment is a = 1 = c, b = 0. Although the two triangulations are inequivalent, they both correspond to the same SCFT fixed point, and thus same CFD, and thus SCFT.
...   Note that for the left triangulation, there are non-trivial multiplicity factors while all the other multiplicity factors are trivially one.
The approach presented here is not limited to the T N theories, but is quite generally applicable to toric Calabi-Yau threefolds. In particular there is a class of geometries that are closely related to T N , which have the toric diagram given by a square with side lengths (n + 1) × (N + 1). For n = N this is simply two copies of T N glued along the diagonal. The associated CFD and SCFT was discussed in [19], which has SU (n) 2 ×SU (N ) 2 flavor symmetry and arises as the 5d avatar of the SU (n) − SU (n) conformal matter theory of rank N .

Parents, Descendants, and BPS States of T N
CFDs not only encode the flavor symmetries of the UV fixed point, but also all massdeformations, i.e. decouplings of hypermultiplets. The resulting theory, the so-called descendants, have a description in terms of CFDs again, and for T N we relate the decoupling to specific toric flops. In addition to these descendants, obtained by mass deformation, we also identify the parent theories of T N , which are SCFTs in themselves, in 5d. Following up the 'genealogy tree' of T N to the KK-theory, we also identify the 6d uplift. To keep with the generic analogy, we also determine the key characteric of these theories, by computing their BPS states using the CFDs and toric resolutions.

CFD-Transitions and Flops
For weakly coupled gauge theories, the descendant theories are obtained after decoupling hypermultiplets and RG-flow. For an SCFT we refer to its descendants as the UV completions of the descendants of the gauge theory description. In the CFD language, this corresponds to applying CFD-transitions to the (−1) vertices. In the toric case, the CFD-transition rules are particularly simple. Suppose that the (−1)-vertex has two neighbors then after the middle (−1)-vertex is removed (or flopped out of the compact surfaces), this local configuration is replaced by The other parts of the CFD remain unchanged. As one can see, this transition is exactly the same as the blow down of a (−1)-curve on a single toric surface. For this reason, the vertex can also be refered to as a "(−1)-curve". For the more general set of rules, one can see section 2.4 of [19], but we will not use them in this paper.
An example of a CFD-transition tree is shown for T 3 in figure 8. Each diagram corresponds to a descendant SCFT obtained by mass deformation from T 3 . These are precisely the descendants of the rank one E 6 Seiberg theory. We will discuss the IR-description in the next section.
In the toric description the CFD-transitions correspond to flops. However, not each flop corresponds to a CFD-transition. CFD-transitions correspond to flops, which change at least one of the intersection numbers (D (k) α ) 2 · ( S i ). In particular, if curves are flopped between surfaces the associated geometries do not correspond to different SCFTs -these are merely different weakly coupled Coulomb branch phases, which have however the same UV fixed point.
The precise correspondence between Coulomb branch equivalence classes and resolutions was discussed in [18], and more importantly for the present context of toric triangulations in [62,63].
Again let us exemplify this in the T 5 theory. Figure 9 shows the first few flop transitions starting with the geometry (2.33) 1 . We denote the curves by their location in the (x − y)-axis.
By changing the triangulation of the bottom left face, the (−1)-curve located at the origin, which was contained in S 4 , gets flopped out. After the flop it is in fact no longer contained in ∪ i S i . This corresponds to a CFD-transition on the (−1)-curve. The resulting theory has SU (5)×SU (4) 2 flavor symmetry, and there are two new (−1)-curves located at (1, 0) and (0, 1). 1 We should note that there are several more equivalent descriptions of this geometry which are related by flop transitions that do not change (D

CFD-Transitions
Toric Triangulations with same CFD  theory -i.e. distinct SCFTs and thus CFDs that arise by mass deformation). However, each CFD will correspond to an equivalence class of toric triangulations, and in order to perform a CFD transition, one has to be in the right toric triangulation.

Descendant Trees and Non-Lagrangian Theories
At the very bottom of the T 3 -descendant tree in figure 8 is the theory with CFD given by three (+1) vertices connected in a triangle. This corresponds to the rank one theory, which does not admit an SU (2) weakly coupled description, and whose geometric realization is in terms of a P 2 . We will find that all T N have such a descendant with triangular CFD and all three vertices labeled by (N − 2). We will refer to these as the B N theory, and they are (a) x 2 -2  x 2 There are several endpoints of the RG-flow tree for T N , which all do not admit an IR gauge theory description. We list them in figure 13 including their strongly-coupled flavor symmetry G F . The vertices of the CFD are drawn such that the toric diagram can be easily read off from the figure.

5d Parents of T N and their 6d Origin
There is a close connection between 5d and 6d SCFTs. In particular, there is ample evidence that all 5d SCFTs arise as dimensional reductions with holonomies in the flavor symmetry/mass deformations, of 6d SCFTs -at least to this date there are no counter examples to this claim. It is thus natural to ask for the 6d avatar of the T N theories.  Figure 13: CFDs (and toric diagrams) for all RG-flow tree endpoints for T N . G F is the flavor symmetry and each of these models has no further descendant, and also no IR-gauge theory description.
The 6d (1, 0) SCFT uplift for 5d T N theories are given in [12], with the following tensor branch: (3.5) The corresponding 5d marginal theory, i.e. the 5d theory, which UV completes in the above 6d theories, has the following 5d quiver description which has three more fundamental flavors than the 5d T N quiver.
The CFD of the marginal theory (3.5) is shown in figure 14 and has exactly affine flavor symmetry SU (3N ). As we apply the transition rules on the CFDs, the first descendant G N has SU (3N ) flavor symmetry, and after two more flops we obtain the T N CFD. These intermediate ... . . .
... has to be k = 0, as k → −k leads to equivalent theories. This phenomenon was observed in the (D k , D k ) theory as well [16,17]. Furthermore, there exists the following IR duality between two different quiver theories:  . . .
...    Figure 15: Geneaological tree of T N : the theory at the top is the KK-theory, obtained from 6d by circle reduction. The first mass deformation (up to permutation) is the 5d SCFT G N , with flavor symmetry G F = SU (3N ). It has two descendants, with one of them the theory P N with G F = SU (2N ) × SU (N ). Further mass deformation yields T N . Figure 16: (a) T 3 as a descendant of the (E 6 , SU (3)) Rank one conformal matter theory. (b) T 4 is a rank three theory which is realized as a descendant of the (E 7 , SO (7)) conformal matter theory.
Similarly, for the P N theory in figure 15, there are the following dual IR descriptions For the lower rank theories, there exist other marginal descriptions. The rank one T 3 theory is a descendant of the rank one E-string theory. Namely, the quiver 1 − [SU (9)] in (3.5) and the CFD in figure 14 are an equivalent description of the rank one E-string theory with explicit flavor symmetry SU (9) ⊂ E 8 . On the other hand, the T 3 theory is also realized as a descendant of the (E 6 , SU (3)) conformal matter theory, which is another realization of the rank one E-string [17,19]. The mass deformations are depicted in figure 16(a).

BPS States
The BPS states of the 5d SCFTs have contributions from two sources: a subset is determined by the CFD, i.e. curves that fall into representations of the flavor symmetry, and curves, which are intersections between compact surface components. For T N we will now determine both types of contributions.

BPS states from the CFD
Here we briefly review the procedure of reading off BPS states from the geometry and CFD [16,17]. In the Calabi-Yau threefold geometry, M2-branes wrapping complex curves give rise to particle BPS states on the Coulomb branch. The spin of such BPS states is given by the moduli space of the curve C. In this paper, we only consider the cases where C is a genus 0 complete intersection curve C = D 1 · S 1 , and where C is a linear system on both D 1 and S 1 .
Hence the spin-0 hypermultiplet is given by a In the CFD, each vertex is associated to a curve Then a linear combination of the vertices can be defined to be  N, 1, N) , (N, N, 1) , (1, N, N) .  They can be exactly combined into a single irreducible representation 27 of E 6 . Similarly, the states in (3.14) can be combined into 27 of E 6 . This is consistent with the flavor symmetry

Additional BPS States from Geometry
From the CFD vertices, we have enumerated the BPS states from a curve in the form of (3.12). After this criterion is applied, the other BPS states are generated by the cyclic permutation of (Λ k , Λ l , 1). The additional spin-1 BPS states charged under a single SU (N ) are For the spin-1 states charged under two SU (N )s, we have the following criterion: for a representation (Λ k , Λ l , 1), we draw the same line L k,l passing through (k, 0) and (N − l, l).
Then such a representation exists if and only if L k,l pass through at least one integral interior point. Similarly, the other BPS states are generated by the cyclic permutation of (Λ k , Λ l , 1).
Note that this criterion covers the BPS states from the CFD in (3.14) as well.
Combining these with the BPS states from the CFD, the total spin-0 BPS states from (3.20) The states charged under two SU (N )s are determined by the spin-0 L k,l criterion.
The total spin-1 BPS states from O(0) ⊕ O(−2) curves are The states charged under two SU (N )s are determined by the spin-1 L k,l criterion, and we have included the adjoint of SU (N ) 3 as well.

Geometry and Rulings
In this section, we present the geometric construction of the IR non-Abelian gauge theory description of T N theory, following the methodology of [3]. For each compact surface S i , we find a ruling by a P 1 = f i over curves (the sections). The ruling curve is an irreducible rational The fully resolved geometry X Σ has an IR non-Abelian gauge theory description if and only if: for any curve S i · S j , it is either assigned as a ruling curve on both S i and S j , or assigned as a section curve on both S i and S j . If this condition can be satisfied, then after the volume of all the ruling curves Vol(f i ) → 0, the M2-brane wrapping modes over f i become massless. They exactly correspond to the W-bosons of the enhanced non-Abelian gauge group G gauge . From the triple intersection numbers S 2 i · S j , one can read off the gauge group G gauge as follows: 1. Take each S i as a Cartan node in the Dynkin diagram of G gauge , which has the Cartan matrix C ij .
2. For each intersection curve S i · S j assigned as a section curve, draw a connection in the Dynkin diagram. More precisely, the off-diagonal entries of C are exactly given by Here the two fundamentals attached to the SU (2) gauge node are given by The five fundamentals attached to the SU (4) gauge node are given by This analysis can be generalized to arbitary T N theories, and the quiver gauge theory description is always (3) Figure 18: The ruling structure of compact surfaces in the example of T 5 theory that realizes the SO(4) 3 IR flavor symmetry. Each red line denotes the section curve of the ruling. The quiver description is not weakly coupled.
The total flavor rank is  This gauge theory description exactly has SO(4) 3 IR flavor symmetry. However, for N > 4, the other compact surfaces do not have consistent section/ruling assignments. Hence this "quiver gauge theory" description is ... . . . ...

IR Descriptions and BG-CFDs
Weakly coupled descriptions can be obtained either, as in the last section, from the ruling of the surfaces in the geometry. Alternatively, as developed in [18] they are constrained by the embedding of IR-versions of CFDs -called box graph CFDs (BG-CFDs). The BG-CFDs are subgraphs of the CFD, which encode the classical flavor symmetries G IR F for gauge theories with matter (or more generally quivers). Here we will briefly review some BG-CFDs used in this paper. Note that the (−2)-vertices are torquoise colored and the (−1)-vertices are gray. In [18] we showed that embedding of a BG-CFD into the CFD for a 5d SCFT is a necessary requirement for the existence of the associated IR-description.
The BG-CFDs for the T N theories can be read off in the usual way from the T N CFD and are shown in figure 19: • The linear quiver (4.5) corresponds to the embedding of the G IR F = U (N ) × SO(4) classical flavor symmetry.
• Finally, there is for N > 2 an embedding of SO(4) 3 , which corresponds to gluing the T N theory out of T 2 theories. This corresponds to the quiver with a strongly-coupled sector (4.8), as derived from the geometry.

Extended Coulomb Branch and Box Graphs
Associated to the linear quiver we can consider the extended Coulomb branch phases for the gauge groups and IR flavor symmetries. Extended here refers to the fact that we include mass deformations into the Coulomb branch description, which will allow us to also consider the decouplings of hypermultiplets within the framework of the Coulomb branch.
The most efficient way to characterize Coulomb branch phases is in terms of the box graphs [62,63,70,71], which comprise the data of the matter representations (R gauge , R IR F ). For quiver gauge theories, we need to also include bi-fundamental matter representations  To each box we associate a weight L i,j , which is the sums of fundamental weights of SU (N c ) and SU (N F ) respectively and the roots α act between the boxes. A consistent extended Coulomb branch phase is given by a sign assignment that satisfies the following basic rules (for an in depth analysis and generalizations see [18,71]  (4.14) We already colored/sign-assigned two of the boxes as these are fixed from the get-go. In addition there are the bifundamentals, which are simply box graphs for (N − m, N − m − 1) What we are interested in are those gauge group phases, that have the same flavor symmetries in the UV -in [18] these were called flavor equivalence classes of Coulomb branch phases. be combined in all possible ways. In the next section we will describe the corresponding toric triangulations.

Higgs Branch, Magnetic Quivers, and Hasse Diagrams
In this section we study the Higgs branch of T N and its descendants using magnetic quivers and associated Hasse diagrams. For 5d gauge theories, the magnetic quivers (MQs) can be motivated from various points of view. They are the 3d mirror dual quivers to the 3d N = 4 gauge theories obtained by T 2 -compactification [72]. It was found in [36] that the MQs can also be read off from (p, q) 5-brane webs, which we briefly review. This approach allows us to study the Higgs branch of both the strongly coupled theory and its quiver description.
Then, there is an effective 5d N = 1 gauge theory in the (x 0 , . . . , If the 5d theory has a description in terms of toric geometry, see section 2.2, then the brane web is the dual graph of the toric fan. Only the configuration of the external 7-branes is fixed by the geometry -the internal web is determined by the chosen triangulation. These correspond to different gauge theory phases. For example, the brane web for T 5 , with the triangulation as in figure 2, is shown in figure 22 (a). This generalizes naturally to any N [44]. Clearly, the web has much more information than the toric diagram as the latter does not show all the length scales explicitly. Nonetheless, these length scales parametrize the extended Coulomb branch of the gauge theory. In the brane web picture, the Coulomb branch parameters are proportional to the heights of the compact faces, which correspond to the internal points of the dual toric diagram. The mass parameters and gauge couplings are given by the positions of the external 7-branes, where three of them are fixed by Lorentz-invariance and supersymmetry. In this language, we confirm that From the alignment of compact faces in the brane web, we can also find the weakly coupled quiver description This was shown in detail in [7]. We see that at a certain point on the extended Coulomb branch, corresponding to the ruling in figure 17, the brane web of T 5 is given in figure 22 (b).
In this description we can clearly see the weakly coupled U (N ) × SO (4)  The SCFT sits at the origin of the extended Coulomb branch, i.e. all the parameters are set to zero. In terms of toric geometry, this corresponds to the singular limit, where the toric fan is untriangulated. The corresponding web is depicted in figure 22 (c). We can immediately see the classical SU (N ) 3 flavor symmetry with matter coming from the string modes between the N coincident 5-branes.

Magnetic Quiver and Hasse Diagram of T N
From the brane web it is possible to read off the magnetic quiver [36]. The SU (2) R R-symmetry of the 5d N = 1 theories can be interpreted in the brane web as a rotation in R 3 R spanned by (x 7 , x 8 , x 9 ). The R-symmetry is broken along the Higgs branch. In the brane web picture, this is done by displacing individual 5-branes away from the origin in R 3 R . However, we can only do this as long as the web at each point in R 3 R is consistent. To map out the full Higgs branch, we thus need to divide the brane web into its possible subwebs. Each subweb corresponds to a node in the magnetic quiver, representing a unitary gauge group whose rank equals the multiplicity of the subweb. The edges between nodes, corresponding to bifundamental matter, have contributions from the intersection between the subwebs as well as 7-branes: 1. The intersection between a (p 1 , q 1 )-and a (p 2 , q 2 ) 5-brane contributes as + det p 1 p 2 q 1 q 2 .
2. The contribution from 7-branes is +1 if two 5-branes end on opposite sides of a 7-brane and −1 if they end on the same side.
Each node in the magnetic quiver has a balance, which can be computed from the node multiplicities n i and the edge multiplicities k ij Usually, the balanced nodes form the Dynkin diagram of the flavor group of the theory. 5 We will denote unbalanced nodes with a cross.
The Hasse diagrams describes the foliation structure of the Higgs branch into symplectic singularities. It is derived from the magnetic quiver by an application of simple combinatorial rules, namely, a repeated quiver subtraction [35] of ADE affine Dynkin diagrams, corresponding to nilpotent orbits, and subsequent rebalancing of the nodes [34,38]. 6 The magnetic quivers for the T N theories were determined in [44] and are shown in figure   23. The subgraph of balanced nodes reproduces the flavor symmetry G F,T N = SU (N ) 3 . For N = 3 the central node with label N is also balanced (the magnetic quiver is in fact given by the affine E 6 Dynkin diagram), and therefore the flavor symmetry is correctly reproduced as      Of course we can also determine the magnetic quiver for the weakly coupled quiver description with brane web shown in figure 22 (b). For general N it is shown in figure 26, and we also give the Hasse diagram for N = 4.

Magnetic Quivers for Ancestors and Descendants
In this section we state the results for the magnetic quivers for the ancestors and descendants of T N . First, let us start with the (grand)parent theories G N and P N introduced in figure 15.
The starting point is the quiver description (3.6) with one or two decoupled flavors respectively.
We can write down the brane web and perform some easy Hanany-Witten moves that allow us to take the strong-coupling limit. Using this method we can also confirm the dualities in (3.7) and (3.8). The results are summarised in figure 27. We clearly see that G N has flavor symmetry SU (3N ) whereas P N has flavor symmetry SU (2N ) × SU (N ). For N = 3 these are enhanced to E 8 and E 7 respectively.
As explained above the first descendant of T N is obtained by flopping out one curve, i.e. changing a triangulation in the toric diagram. In this description, it corresponds to the first arrow in figure 9. In the brane web picture, we can describe this procedure by turning on the corresponding mass parameter. The resulting brane web and magnetic quiver are shown in be reliably computed from the MQ using the Hilbert series [75]. 6 Another allowed subtraction are Kleinian singularities AN−1, represented by two U (1) nodes connected by N bifundamentals. We can continue this logic to find the magnetic quivers for the entire descendant tree of T N . In practice, this is quite arduous and depends heavily on N . However, the first few descendants are completely universal, and we now present the first three descendants and their magnetic quivers explicitly.
There are two different second descendants, which are generated by two mass deformations from the T N theory. In the toric diagram these are obtained by flopping two curves either on the same or different corners. For N = 3 both descendants are equivalent and the flavor

Summary, Conclusions and Outlook
The main purpose of this paper was to explore the physics of T N and closely related SCFTs, using all recently developed approaches to 5d gauge theories and SCFTs. The exposition should give a largely self-contained explanation of the advances in the Coulomb branch and geometry, as well as the application of brane webs to study the Higgs branch, magnetic quivers and Hasse diagrams. The T N and related theories that we discussed are particularly useful for such a comprehensive approach, as they have realizations in a multitude of different frameworks: both toric geometry, brane web, CFDs and gauge theories.
We hope that by discussing the CFDs in the context of toric Calabi-Yau singularities, their properties and uses are more accessible to some readers than the approach in the original work, where the geometries arose from non-flat resolutions of elliptic Calabi-Yau threefolds with non-minimal singularities. In comparison, the toric approach is limited to a subclass of theories, but the concepts of CFDs and BG-CFDs have hopefully been put into a more familiar framework to most readers.
We determined the descendant trees for T N , using both toric flops and CFD-transitions, and showed that the bottom of the CFD tree has in fact several theories that have no weaklycoupled gauge theory descriptions, much like the rank one P 2 theory. By utilizing brane web techniques, we also studied the magnetic quivers of the theories related to T N by mass    deformations. One of the goals would be to combine these approaches to obtain a global picture of the Higgs and Coulomb branches.
The focus of this paper has been the theories related to T N , but a similar analysis can be performed for any theory realized in terms of a toric Calabi-Yau singularity. We exemplified this by gluing two T N theories to get a new toric model. For a general convex toric diagram, the CFDs will be comprised of the boundary. Likewise the web can be obtained from the toric diagram and thereby the magnetic quiver. It would interesting to develop this complete approach, both in the Coulomb branch and Higgs branch for all toric models.