Mass-deformed $T_N$ as a linear quiver

The $T_N$ theory is a non-Lagrangian theory with SU(N) flavor symmetry. We argue that when mass terms are given so that two of SU(N)'s are both broken to SU(N-1) x U(1), it becomes $T_{N-1}$ theory coupled to an SU(N-1) vector multiplet together with N fundamentals. This implies that when two of SU(N)'s are both broken to U(1)$^{N-1}$, the theory becomes a linear quiver. We perform various checks of this statement, by using the 5d partition function, the structure of the coupling constants, the Higgs branch, and the Seiberg-Witten curve. We also study the case with more general punctures.


Introduction and Summary
In the last few years, strongly-coupled superconformal field theories (SCFT) that do not admit any obvious Lagrangian description in the ultraviolet (UV) play more and more important roles in our understanding of the supersymmetric dynamics and dualities. In 4d, they are sometimes realized as a subcomponent of strongly-coupled limits of Lagrangian theories [1,2]; in 5d, they are often conjectured to exist as ultraviolet completions of Lagrangian theories [3,4]. They can often be constructed using superstring theory and M-theory.
Among these SCFTs, a central role is played by the so-called T N theory. The 4d version, originally introduced in [5,6], is an N =2 superconformal theory with SU(N ) 3 flavor symmetry, that arises as the four-dimensional limit of the 6d N =(2, 0) theory of type SU(N ) on a sphere with three full punctures. The 5d version was soon introduced in [7], as a superconformal theory living on the intersection of N D5-branes, N NS5-branes and N (1,1) 5-branes, and its compactification on S 1 gives back the 4d version.
Due to its intrinsic importance, the properties of the T N theory have been studied in earnest. For example, the partition function of the 4d version on S 1 × S 3 was found in [8,9] using the relation to the 2d topological quantum field theory; that of the 5d version on S 1 ×S 4 was found in [10,11] using the topological vertex formalism; many of the Higgs branch chiral ring relations were worked out in [12,13]. The theory can be deformed by giving vacuum expectation to the Higgs branch operators so that we have more general SCFTs labeled by three Young diagrams each with N boxes. The 4d versions are sometimes called the tinkertoys and extensively studied starting from [14], and some of their chiral ring relations have been analyzed [15].
In this paper, we study a different type of deformations, namely by mass terms. As T N theories have the flavor symmetry SU(N ) A ×SU(N ) B ×SU(N ) C , the mass terms take values in three traceless complex-valued N × N matrices m A,B,C , that are hermitian in the case of 5d version. The effect of the mass terms when they are nilpotent was studied in [12,13], and therefore our aim here is the case when they are diagonalizable.
We will claim that the mass deformation of the two SU(N ) flavor symmetries by diagonal mass matrices makes the T N theory flow to a linear quiver theory. This fact and its generalization were observed in [10,16], but it was unclear whether the gauge groups are unitary gauge groups or special unitary gauge groups. We will argue that the gauge groups are special unitary groups and there are additional hypermultiplets at the end of the quiver compared to [10,16].
Basic statement. Our basic claim, both in 5d and in 4d, is then the following: let us give mass terms to SU(N ) B and SU(N ) C such that they are both broken to SU(N −1) × U(1). More explicitly, take the mass terms to be Here, the T N −1 theory is coupled to an SU(N −1) gauge multiplet, that is also coupled to a bifundamental of SU(N −1) × SU(N ) A . In (1.2) the brackets are placed around SU(N ) A to emphasize that it is a flavor symmetry. In the 5d version, the mass m bif of the bifundamental and the gauge coupling 8π 2 /g 2 of the SU(N −1) are given by 3) and the Chern-Simons level of the SU(N −1) gauge group is zero. For example, take N = 3. This is the 5d version of the E 6 theory of Minahan and Nemeschansky. After the mass deformation, we have SU(2) coupled to T 2 and three flavors. Since T 2 is equivalent to two flavors of SU (2), the infrared theory is just SU(2) with five flavors. This is the setup originally found by Seiberg [3], where this class of 5d SCFTs was first discussed.
We can also consider an even simpler case of N = 2. Recall that the T 2 theory is just the tri-fundamental of SU(2) A × SU(2) B × SU(2) C . Giving masses (m B , −m B ) and (m C , −m C ) to SU(2) B,C , we have two flavors of SU(2) A , with masses m B + m C and m B − m C . In the limit |m B + m C | |m B − m C |, we just have one flavor of SU (2) A . This is the bifundamental of SU(2) × SU(1), with mass m B + m C .
Recursive application. Recursively applying this procedure, we immediately find that the infrared outcome of a more general mass deformation given by where groups enclosed in the brackets are flavor symmetries, other groups are gauged, and we have bifundamental hypermultiplets for each consecutive pair of groups. In the 5d version, all the Chern-Simons levels are zero. The same statement recently appeared in [17]. It turns out that "SU(1)" should be formally understood as an additional hypermultiplet charged under the SU (2) in addition to the bifundamental of SU(2) − SU (1). This can be seen by stopping the recursive process at T 2 . These statements can be easily generalized, by giving nilpotent vevs to the chiral operators in the adjoint of the SU(N ) A flavor symmetry. This process is often called the 'partial closing of the puncture' in the 4d class S theory, and we use the same terminology even in the 5d case.
In this language, the T N theory has three punctures, and in the more general case, we start from the theory with two full punctures and a puncture of type Y = [n 1 , n 2 , . . . , n k ] with n i = N . We still have the flavor symmetry SU(N ) B × SU(N ) C to which we give masses as in (1.4). Then we have a quiver theory of the form It is interesting to note here that the 3d quiver description of the 3d theory T Y (SU(N )), introduced originally in [18], has the same structure except that the groups are U(v i ). The reason will be uncovered in Sec. 4.4. The quiver (1.6) can also be realized as an A N −n 1 −1 class S theory with N simple punctures and one puncture of type Y = [n 2 , . . . , n k ].
Organization of the paper. In Sections 2, 3 and 4, we perform various tests to check these proposals. The checks given in those sections are mostly independent from each other, and can be read independently, depending on the taste of the reader.
We start in Sec. 2 by considering the 5d version of the story, where the T N theory has a construction by a web of branes and the relation to the linear quiver can be most easily seen. We recall the method to compute its Nekrasov partition function from the topological vertex, and use it to relate the mass parameters of the T N to the gauge couplings and the masses of the linear quiver theory.
Next, in Sec. 3, we perform a field-theoretical analysis to check that under the mass deformation preserving SU(N −1) B,C , the T N theory becomes the coupled theory (1.2). We consider the matching of the operators and of the vacuum moduli spaces, and speculate what happens when m C = 0.
Then, in Sec. 4, we study the field-theoretical analysis of the relation between the massdeformed T N theory and the linear quiver. In particular, we study the Seiberg-Witten curves and the Higgs branches. We also analyze the system when we replace the full puncture carrying SU(N ) A with a more general puncture. We also perform in Sec. 4.4 the analysis in the 3d version of the T N theory.
In Appendix A we summarize the Higgs branch operators of the T N theory and their chiral ring relations, some of which are new.
Note added: Recently there appeared a paper [17] where the relation of the massdeformed T N theory and the linear quiver of SU groups was also proposed, and their Sec. 2 and our Sec. 2 have a rather large overlap. Also, the relation to the linear quiver of U groups was already mentioned in [10] and further studied in detail in [16]. As our paper appears on the arXiv about two weeks later than [17] and half a year later than [16], we do not have any intention to claim the priority. That said, our checks are largely independent of those that they performed, and can be considered as more pieces of evidence for their proposal. The T N theory does not admit an obvious Lagrangian description, but the five-dimensional version can be explicitly realized in terms of a web of (p, q) 5-branes [7], shown in Fig. 1. It has N external D5-branes, N external NS5-branes, and N external (1, 1) 5-branes, and they are connected with each other in the internal part of the diagram. The 5d T N theory lives on the intersection of the 5-branes.
The global symmetry of the theory may be understood directly from the web diagram. For that, we put an orthogonal spacetime filling (p, q) 7-brane on the end of each external (p, q) 5-brane. This process does not break further supersymmetry. The lengths of the external 5-branes become finite and the global symmetry of the theory is realized on the (p, q) 7-branes [19]. In our case, we can end N D5-branes on N D7-branes, that gives SU(N ) symmetry. The same is true for NS5-branes and (1,1) 5-branes. In total, we see that the theory realized by the web of Fig. 1 has the flavor symmetry SU(N ) × SU(N ) × SU(N ).
At this point we can give a very crude argument relating the mass deformation of the T N theory and the linear quiver. The mass terms for a single SU(N ) correspond to the distance between N parallel 5-branes. Let us give equal masses for two SU(N )s so that they are far Q 1 (2) Figure 1: The web diagram for the 5d T N theory. In our convention, a horizontal, vertical and diagonal line denotes a D5-brane, an NS5-brane and a (1, 1) 5-brane respectively. P where L is the length of the corresponding internal line. They can be regarded as fugacities which appear in the computation of the partition function the 5d T N theory. larger than the mass terms for the third SU(N ). The web diagram now becomes the one shown in the upper row of Fig. 2. This configuration looks very much like a simple brane configuration given in the lower row of the same figure, which realizes the linear quiver We see that the U(1) parts of the gauge groups are frozen, since the two semi-infinite ends of a 'vertical' brane in the lower figure is in fact not parallel, as one can see in the web diagram. Therefore the dynamical part of the linear quiver is The objective of the rest of the section and of the paper is to make this rough argument more precise.

The partition function
Formalism. Given a web of (p, q) 5-branes, we can compute the exact partition function of the 5d theory compactified on a circle. For that, we follow a chain of dualities, and view the web diagram as the toric diagram of a toric Calabi-Yau threefold [20]. In this picture, the 5d theory is realized as a low energy effective field theory of an M-theory compactification on this toric Calabi-Yau. For example, the web diagram corresponding to the 5d T N theory in Fig. 1 specifies a blow up of C 3 /Z N × Z N [7]. In this formulation, 5d BPS states come from M2-branes wrapping various two-cycles inside the toric Calabi-Yau threefold [21], and their index can be computed by the (refined) topological vertex [22][23][24][25], which can often be regarded as the 5d Nekrasov partition function of the corresponding 5d gauge theory [26][27][28][29]. This is not the end of the story, however. The refined topological vertex computation in fact automatically contains the contribution of some BPS states that do not carry gauge charges and are decoupled from the 5d theory. Such contributions come from strings between parallel external 5-branes (or M2-branes wrapping the corresponding two-cycles), and the web of (p, q) 5-branes (or a toric diagram) allows us to easily identify them and strip them [10,11,30,31]. These contributions appear as products of the plethystic exponentials, and we call them decoupled factors.
Explicit formulas. With the choice of the parameters (2.4), (2.6)-(2.8), we apply the refined topological vertex to the web in Fig. 1. In the computation, we choose the horizontal lines to be the preferred directions. The calculation was performed in [10,11]. The quantity assigned to the web diagram isZ 9) and the genuine partition function Z T N of T N is obtained by removing the decoupled factors: Let us explain the ingredients in turn. First, t and q are related to the Ω-deformation parameters ( 1 , 2 ) by t = e i 1 and q = e −i 2 . Then, M (t, q) is the refined MacMahon function given by This factor comes from the perturbative contribution of the Cartan part of the vector multiplets. From the topological string point of view, it comes from the constant maps and cannot be captured by the refined topological vertex. We put it by hand in (2.10) by adjusting its power by half of the dimension of the Coulomb branch moduli space of the T N theory. Let us next give Z pert and Z inst : Here, Y k = {Y k;a }, a = 1 · · · , k for k = 1, · · · , N −1 represent all possible Young diagrams. z bifund (k, l, m) is the contribution of bifundamental hypermultiplets of gauge groups U(k) × U(l) with mass m to the instanton partition function, and the explicit expression is where the function E(k, l, a, b, s) is defined as (2.15) γ 1 and γ 2 are again related to the Ω-deformation parameters by i 1 = γ 1 + γ 2 and i 2 = γ 1 − γ 2 . l Y (i, j) = Y i − j and a Y (i, j) = Y t j − i denote some lengths inside the Young diagram Y from a box specified by (i, j). Here Y i denotes the height of the i-th column of a Young diagram Y , and Y t means the transpose of the Young diagram Y . λ k;a , a = 1, · · · , k are the Coulomb branch moduli of the U(k) gauge group. The other functions in (2.13) can be written by (2.14): z vec (k) is the contribution of vector multiplets of a gauge group U(k) and it is z vec (k) = 1 z bif (k, k; iγ 1 ) .
Also z fund (k; m) is the contribution from a fundamental hypermultiplet of a gauge group U(k) with mass m z fund (k; m) = z bif (k, 0; m), (2.17) where the argument 0 in (2.17) means that we do not have the product of b and Y l,b in (2.14) and λ 0;b = 0. Finally, the decoupled factors are (2.20) They are for parallel external D5-branes, NS5-branes, and (1, 1) 5-branes, respectively. We defined m bif, Interpretation. From the explicit expression, it is now clear that the instanton part Z inst of the partition function (2.13) is exactly that of the following linear quiver theory with all the Chern-Simons levels being zero. But note that the sum of the Coulomb branch moduli of each gauge group is zero, because of (2.5). In (2.21), the group in the square brackets [·] denotes a flavor symmetry. The parameter u k can be regarded as the instanton fugacity for the U(k) gauge group. The partition function for the T N theory (2.10) involves the division by decoupled factors from the strings between the parallel external 5-branes in the T N . In the case of the T 3 theory, the E 6 ⊃ SU(3) × SU(3) × SU(3) flavor symmetry was reproduced only after the removal of the decoupled factors [10,11]. The decoupled factors (2.19) and (2.20) depend on the instanton fugacity, and therefore one cannot just say that the T N theory has the same partition function with the quiver (2.21). Instead we propose that (2.10) yields the partition function of the following linear quiver theory On "SU(1) instantons." Let us discuss the physics of "SU(1)" at the end of the quiver. As we recalled in the introduction, the T 2 theory, which is just the tri-fundamental of SU(2) A × SU(2) B × SU(2) C , flows to the bifundamental of SU(2) A × SU (1), under an appropriate choice of the mass terms. In this context, however, we can say even more. The Nekrasov partition function of the "SU(1) instantons" coupled to two fundamentals, i.e. the Nekrasov partition function of the U(1) theory after the removal of the decoupled factors, give back two fundamentals of SU (2). To see it, first consider the case N = 3. The instanton part of the partition function (2.10) reduces to Since the T 3 theory after the mass deformation can be thought of the SU(2) gauge theory with five flavors, (2.23) should be related to the partition function of the SU(2) gauge theory with five flavors if we redefine the parameters as [10] Here, m f 1 and m f 2 are the masses for the two fundamental hypermultiplets 1 , and u 2 is the instanton fugacity for the SU(2) gauge theory. Under this parameterization, we have Then the factor with (a, b) = (1, 2) in Z || (2.19) and the factor with (a, b) = (1, 2) in Z // (2.20) cancel the denominator of (2.25). What remains is the numerator of (2.25), which is the perturbative contribution of the fundamental hypermultiplet of the SU (2) with mass m f 2 . Combined with the perturbative contribution of the fundamental hypermultiplet with mass m f 1 already contained in (2.12), we obtain the perturbative partition function of the following linear quiver theory where T 2 simply denotes the two fundamental hypermultiplets with mass m f 1 and m f 2 . Since the part Y 2;1 = Y 2;1 = ∅ in (2.23) is now considered as the perturbative part, the genuine instanton part is in fact The result becomes the SU(2) ∼ = Sp(1) instanton partition function with five flavors [10]. An important property for establishing the identification is that the expansion associated with Y 1 in (2.27) stops at finite order with each Young diagram Y 2;1 and Y 2;2 fixed. This matches with the expectation from the field theory computation where the number of the terms involving the fugacity e im f 2 is finite at each instanton number. We have checked this until |Y 2;1 | + |Y 2;2 | = 3 for several orders of |Y 1 |. 2 Let us now discuss the situation with general N . Since the termination of the summation of Y 1 in (2.27) happens for each Young diagram Y 2;1 and Y 2;2 , the termination should also occur in .
(2.28) for fixed instanton numbers of u 2 , u k , k = 3, · · · , N −1. This implies that the instanton partition function for the SU(1) part in (2.22) should have an interpretation as the fundamental hypermultiplet with the mass m f 2 coupled the SU(2) gauge fields. To see one of 1 The signs of the masses are different from the ones used in [10]. However, the instanton partition function of an SU(2) gauge theory is invariant under the flip of two signs of mass parameters for fundamental hypermultiplets since that is a part of the Weyl symmetry of the perturbative flavor symmetry. 2 Until the order we have checked, the summation of Y 1 stops at the evidence, one can check the flavor symmetry associated to m f 1 and m f 2 . Since the fundamental hypermultiplets couple to the SU(2), the flavor symmetry is enhanced to SO(4). We have checked that the instanton partition function (2.28) is invariant under the Weyl symmetry of SO(4) for the T 4 theory until the order O(u 3 u 2 ).
Matching of the parameters. The argument so far strongly suggests that the partition function of the T N theory is exactly the partition function of the linear quiver theory (2.22) or (2.26). The global symmetry of the linear quiver theory is SU(N ) × U(1) 2N −2 . The physical interpretation of the equality is that, when one gives generic mass terms for the two SU(N ) flavor symmetries of the T N theory, the theory flows to the linear quiver theory (2.22). Let us study the relations between the mass parameters which break the SU(N )×SU(N ) into U(1) 2N −2 and the the parameters of the linear quiver theory. Each of the three SU(N ) flavor symmetries is associated with the 7-branes attached to each parallel external 5-branes. Regarding the web diagram in Fig. 1, all the 7-branes are separated from each other and the generic mass deformations are given to the three SU(N ) global symmetries. The mass deformation is characterized by the length between the parallel external 5-branes. Let us first denote the mass deformations by m A for the parallel external D5-branes, m B for the parallel external NS5-branes, and m C for the parallel external (1, 1) 5-branes. The three types of the mass deformations can be written by the fugacities that appear in Fig. 1 for k = 1, · · · , N −1. Then, combining (2.4), (2.6)-(2.8) with (2.29)-(2.31), we obtain the relations where we use the explicit form of the instanton fugacity u k = exp i 8π 2 g 2 k . We introduced the notation where there are k entries of 1 and tr H k = 0.

Higgsing one full puncture
Brane construction. By tuning the lengths of 5-branes, we can put several parallel external 5-branes on one 7-brane. Then the fractionated 5-branes can move between the 7-branes. An example of the process is depicted in Fig. 3. This corresponds to the Higgs branch associated to one full puncture. Suppose the fractionated 5-branes between the 7branes are moved into infinity. Then, some 7-branes are disconnected from the web diagram and the number of the 7-branes gets reduced. In the infrared we obtain a different theory. In general, n i 5-branes may end on the i-th 7-brane where i n i = N . Hence, different IR theories are classified by the partition [n i ] of N . If the number of the bunches of n i 5-branes is w n i , the configuration carries a U(w n i ) global symmetry. The total global symmetry is then S ( i U(w n i )) since the diagonal U(1) does not appear in the IR theory [19]. This datum [n i ] is the same one introduced by Gaiotto [5]. In the case of the web diagram in Fig. 1, the partition is given by [1, 1, · · · , 1] where we have N 1's for all the three types of the parallel external 5-branes. Therefore, the web diagram corresponds to a sphere with three full punctures. We are interested in the partition function of this Higgsed system. The general prescription to compute the partition function has already been presented in [32]. Let us illustrate this by considering a specific example, namely the case where N −K upper parallel external D5-branes are put on one D7-brane. This corresponds to a puncture [N −K, 1, · · · , 1] where we have K 1's.
The tunings can be described by [10,32] for the fugacities in Fig. 1. Due to the constraints from the geometry of the web diagram. the tunings (2.36) induce another conditions for the fugacities in the interior of the web Perturbative part. Let us first look at the perturbative part (2.12). By using the relations (2.36) and (2.37), (2.12) reduces to Here, Z pert,1,2,3 , (2.39)-(2.41), are the perturbative partition function for the following linear quiver theory where an additional fundamental hypermultiplet whose contribution is (2.40) is coupled to the rightmost SU(K). The last factors (2.42) are the contributions of singlet hypermultiplets that are decoupled from the linear quiver theory (2.43).
Instanton part. We then apply the conditions (2.36) and (2.37) to the instanton partition function (2.13). First note that the tunings (2.36) and (2.37) trivialize some of the Young diagram summations. From the explicit form of (2.13) with (2.36) and (2.37), we obtain a non-zero result only when Y k,a = ∅, K + 1 ≤ a ≤ k for K + 1 ≤ k ≤ N −1. In other words, when horizontal internal lines become external and end on 7-branes, the Young diagram summations associated to the horizontal lines are trivialized. Physically, since the horizontal lines become external and semi-infinite, the instanton particles corresponding to M2-branes wrapping the horizontal lines become infinitely heavy and non-dynamical.
With this simplification of the Young diagrams as well as the conditions (2.36) and (2.37), there are various cancellations between the numerators and the denominators in the instanton partition function (2.13). The final result is (2.48) and similarly forz K vec (k) andz K fund (k; m) by (2.16) and (2.17) respectively with z bif (k, l; m) replaced withz K bif (k, l; m). Therefore, we obtain the instanton partition function of the following linear quiver theory The linear quiver theory can be also deduced from the web diagram that has one puncture of [N −K, 1, · · · , 1] type and two full punctures. Fig. 4 represents an example of such a web diagram. The original web diagram corresponds to the T 5 theory but now one of the punctures is Higgsed and three external D5-branes are put on one D7-brane. In analogy with Fig. 2, the web diagram on the right-hand side of Fig. 4  where an additional fundamental hypermultiplet is attached to the rightmost SU (2). The fundamental hypermultiplet is realized by strings between two D5-branes for that SU (2) and the D7-brane in the upper right part of the right figure of Fig. 4. We can also see that the case with [N −K, 1, · · · , 1] and two full punctures can give rise to the linear quiver theory of (2.43) by using the corresponding web diagram.
General puncture. We can see the general case by using a web diagram with a general puncture [n 1 , · · · , n k ], k i=1 n i = N , associated with the parallel external D5-brane. As in the right figure of Fig. 4, we consider moving the D7-brane to which n i D5-branes are attached. Due to the brane annihilation, the SU(N − n) gauge theory reduces to the group SU (N − n − i H (n i − n)). We introduced H(x) that satisfies H(x) = x for x ≥ 0 and H(x) = 0 for x < 0. Also the D7-brane provides a fundamental hypermultiplet coupled to SU(N −n i ). Hence, the general case with [n 1 , · · · , n k ] for one puncture yields a quiver theory where v n = N −n − k i=1 H(n i − n) for n = 1, · · · , N −1. Furthermore, w n fundamental hypermultiplets are coupled to the SU(v n ) where w n represents the number of n appearing in [n 1 , · · · , n k ]. One can also check that v n satisfies (1.7). From the viewpoint of the quiver theory (2.51), the global symmetry S ( n U(w n )) of the Higgsed T N theory is associated to the w n fundamental hypermultiplest coupled to the SU(v n ).

Mass deformation from T N to T N −1
In this section, we study the RG flow from the T N theory to the T N −1 theory under a special form of mass deformations.
The masses of the T N theory take values in the Cartan subalgebra of SU(N ) A ×SU(N ) B × SU(N ) C . We denote them as m A , m B and m C respectively. Let us consider the following mass deformation; This mass deformation breaks the flavor symmetry as Furthermore, let us assume that the signs of the masses m B and m C are different, i.e., m B m C < 0. For concreteness we assume Then, under this mass deformation, we claim that the T N theory flows to the theory Consistency checks on this proposal will be discussed below. There is a U(1) bif which rotates the bifundamental of SU(N ) A × SU(N −1), and another U(1) inst coming from the instanton current of SU(N −1). These two U(1)'s are matched to the U(1) B × U(1) C of the original theory in the way indicated by the above formulas for m bif and 8π 2 /g 2 . We do not have many direct field-theoretical checks concerning the level of the Chern-Simons term. The ones we have are: (i) Recursively applying this procedure as we will do in Sec. 4, we see that SU(N − 1) has 2(N − 2) fundamentals. In [4] it was shown that in this case we can only have zero Chern-Simons level to have nontrivial UV fixed point. (ii) If there is nonzero Chern-Simons level k, the level −k should also be possible. Then there should be two subtly-different versions of the 5d T N theory, but this seems not to be the case.

Matching of operators and states
First, let us recall the following basic facts. We denote the U(1) B × U(1) C charges of operators or states as q B and q C , respectively. The Hilbert space is decomposed as where H (q B ,q C ) is the subspace which has charge (q B , q C ). By the BPS bound, the energies of the states in H (q B ,q C ) are bounded as Then, operators charged under U(1) B × U(1) C can create only massive states. 4 We consider the case q B = q C and q B = q C separately. This is because, according to (3.6), the states with q B = q C have instanton charges in the IR theory. The IR effective theory description is particularly good if |m bif | 8π 2 /g 2 , i.e., the gauge coupling is small at the energy scale of m bif . Then the states with q B = q C are light or massless, while the states with q B = q C are very heavy and involve instantons.

Chiral operator matching for
The T N theory has the following chiral operators (in the language of 4d N = 1 supersymmetry) which correspond to Higgs branch: By comparing flavor charges, it is easy to find the following operator matching between the UV and IR theories. We often treat B andB as matrices. For the µ operators, where W α is the field strength superfield of the SU(N −1) gauge group. The last equation needs explanation. The µ operators are moment maps of the flavor symmetry, and they are in the same supermultiplets as the flavor symmetry currents. Since the flavor current for the instanton charge is F ∧ F , the corresponding operator with the correct mass dimension and flavor symmetry is W α W α . Actually, in the Language of 4d N = 1 supersymmetry, a moment map µ and a mass m appear as d 2 θmµ. For the symmetry corresponding to q B − q C , we take µ → W α W α and m → 1/g 2 . For the Q operators, and where all the indices i B and i C are ≤ N −1. These are the simplest operator matchings one can think of. For these equations to preserve the charges, the U(1) B × U(1) C charges of B must be (q B , q C ) = (1, 1). This supports claim (3.5).

State matching for q B = q C
In the T N theory, the chiral operators with charges q B = q C are given by where −∧ k formally means ∧ N −k . The cases k = 0 and N are given by µ operators, while 1 ≤ k ≤ N −1 are given by Q operators. Tensor products of these representations are also possible by considering products of the operators. Below, we reproduce these representations by performing the semi-classical quantization of instantons.
Semiclassical quantization of instantons. If we have an operator O (q B ,q C ) with charge (q B , q C ), we can consider states created by these operators, We also do not discuss the gauge charge carried by the instanton particles; they will be affected by the "zero modes" of the T N −1 , and we assume that they are always canceled by appropriately combining various zero modes. In a static instanton background, the action of the fermion ψ in the hypermultiplet B is given as where i = 1, 2, 3, 4 runs over spatial directions, and we have explicitly written the index i A of SU(N ) A . We take γ i (i = 1, 2, 3, 4) as the usual 4d gamma matrices, and take the gamma matrix in the time direction as γ t = −iγ 5 . Thenψ = ψ † γ 5 .
To perform semi-classical quantization of the zero modes, we assume ψ has the form where ψ 0 (x i ) is the zero mode of γ i D i in the fundamental representation of SU(N −1), and a i A only depend on the time coordinate. The zero mode has a definite chirality γ 5 ψ k = ψ k , and hence the action becomes Canonical quantization gives where H is the Hamiltonian. We have included the classical energy 8π 2 /g 2 of the instanton particles. The zero point energy −N/2 is required by the symmetry Let |0 be the state with a † i A |0 = 0 for all i A . Then, we obtain instanton particle states as |k = a i A,1 · · · a i A,k |0 . (3.24) We denote the instanton charge as q inst and the U(1) charge rotating the field B as q bif . Then the state |k has the charge This is in the representation ∧ k of SU(N ) A , so we want to identify these states with the states (3.18). The case of the minus sign of ± in (3.18) is realized by anti-instantons. This requires the identification of charges as The Hamiltonian (3.23) gives the masses This is equal to m B q B + m C q C if and only if (3.5) and (3.6) are satisfied. This is the basis of our claim (3.6).

Matching of the moduli space of vacua
In this subsection, we compare the moduli space of vacua of the UV and IR theories. We consider the case in which SU(N ) A also has the mass parameter of the form In this case, most of the B,B fields in the IR theory become massless, and hence we get a larger Higgs branch. The flavor symmetry is broken as SU(N )

IR theory
In the IR theory All the fields in this quiver are massless. This theory has a baryonic branch in which we give diagonal vevs to b andb. In terms of gauge invariant operators, we define The baryonic branch is given as This is a hyperkahler manifold C 2 /Z N −1 . On this branch, the SU(N −1) gauge group is Higgsed, and the low energy theory consists of the T N −1 theory and the neutral moduli fields (3.32). We will see that this result reproduces the moduli space of the UV T N theory deformed by the mass terms.

UV theory
Now we study the moduli space of the T N theory. We use two different methods.
Field theory method. Generally in 4d N = 2 or 5d N = 1 theories, the potential on the Higgs branch under the mass deformation is [33] i m i v i 2 (3.33) where v i are the Killing vector of the i-th generator acting on the Higgs branch. This can be easily seen in 4d from the fact that, in the language of 4d N = 1 supersymmetry, the superpotential is given in terms of the holomorphic moment maps µ i as i m i µ i , and the derivative of the holomorphic moment maps µ i by moduli fields are given by the holomorphic killing vectors v i by definition. Therefore, after the mass deformation, we only have to keep operators uncharged under the Killing vector i m i v i to see the moduli space of vacua. The T N theory has operators (µ X ) i X j X (X = A, B, C) and 5 When the relation (3.29) is satisfied, the killing vector i m i v i acts trivially on the operators Q N N N and Q N N N . It also acts trivially on (µ X ) N N . Then we can give vevs to these operators. The chiral ring relations of the T N theory are summarized in Appendix A. The relation tr µ k A = tr µ k B = tr µ k C for any k, requires that the eigenvalues of the matrices µ X (X = A, B, C) are the same and we take their vevs as The chiral ring relation (A.5) requires that they satisfy the relation as was also discussed in [13]. If we identify then (3.35) is the same as (3.32). In fact, one can check that these identifications follow from the operator matchings (3.14), (3.15) and (3.12). Furthermore, by these vevs, the T N theory flows to the T N −1 theory as discussed in [13]. Therefore, the moduli spaces are matched between UV and IR description.
6d method. Here we consider the T N theory in four dimensions. The 4d T N theory is realized by the compactification of the N = (2, 0) theory on a Riemann sphere with three full punctures [5]. The Seiberg-Witten curve of the T N theory is given by where z is a coordinate of the Riemann surface and φ k (z)(dz) k are k-th differential, i.e., sections of the k-th power of the canonical bundle K = T * C of the Riemann surface.
Assuming that the punctures are at z = z X (X = A, B, C), these φ k are such that the N solutions of x near these punctures are given by the eigenvalues of m X , x ∼ m X z − z X + lower order. (3.38) Now, if the relation (3.29) is satisfied, the curve can be factorized by tuning some Coulomb moduli as (3.39) where F N −1 (x, z) = x N −1 + · · · is a curve of the T N −1 theory, and φ 1 is a one-form on the Riemann surface which has poles at z = z X with residues m X (X = A, B, C), respectively. This is possible if and only if the sum of the residues is zero, i.e., m A + m B + m C = 0, since the sum of residues of any meromorphic one-form φ 1 on a Riemann surface must be zero due to the equation C\{punctures} dφ 1 = 0. Intuitively, this branch is understood as follows. The theory may be realized by compactification of N coincident M5 branes on the Riemann surface. The above factorization corresponds to the case that one of the N M5 branes is separated from the rest of the N −1 M5 branes. See [34] for more systematic treatment. If the curve is factorized in this way, we get quaternionic dimension one contribution to the Higgs branch from the motion of the separated one M5 brane as explained systematically in [34,35] which generalize the earlier works [36,37]. This should be identified with the baryon branch of (3.32). Furthermore, it is clear that we get the T N −1 theory with the curve F N −1 on this branch. This is exactly as in the IR theory (3.4).

Strong coupling point and phase transition?
When m bif = 0, it is possible to integrate out the bifundamental of SU(N ) A − SU(N −1). By doing that, we will be able to understand why the condition (3.3) is necessary in five dimensions.
Suppose we have a simple gauge group G and a hypermultiplet H in some irreducible representation r of G. The gauge group G has a coupling g and the hypermultiplet H has a mass m. We would like to compute the low energy gauge coupling g after integrating out H.
By supersymmetry, we only need to compute it at the one-loop level. This is because the gauge coupling is directly related to the masses of BPS instanton particles, and masses of BPS particles are given by the central charge whose dependence on mass parameters is restricted. Another way of seeing the one-loop exactness is that if we extend the mass parameter to background vector superfield, the correction to the gauge coupling is related by supersymmetry to Chern-Simons couplings of the form (gauge) 2 (flavor).
The computation is straightforward, and we only write down the result. When we integrate out scalars or fermions or vectors in d-dimensions, the one-loop modification to the gauge coupling in spacetime dimension d is given by where t r is the Dinkin index normalized to be 1/2 for the fundamental representation of SU(N ), and C is given by where d f is the complex dimension of the spinor. One can check that this reproduces the usual result when d = 4. A single hypermultiplet contains two complex scalars and fermions of dimension d f = 4. By putting d = 5 and using Γ(−1/2) = −2 √ π, we get This result would also be obtained by comparing the lowest mass states of instanton particles before and after integrating out the hypermultiplet H. Now let us apply the above result to our case. By integrating out B,B, the SU(N −1) gauge coupling becomes This is positive as long as the condition (3.3) is satisfied. However, when one of the masses, say m B , goes to zero, the coupling becomes infinitely large. In that case, the description in terms of the effective theory (3.4) breaks down and the T N theory with m B = 0 should flow to some strongly coupled theory. Note that when m B = 0, the symmetry should be restored to SU(N ) B , thus it is a priori expected that something must happen at this point.
If we further take m B to be negative, the IR coupling formally becomes negative. So there must be some phase transition at m B = 0. It would be very interesting to study this phase transition and the theory at m B < 0.

Mass deformation to linear quivers
In the last section, we argued that the T N theory, when deformed by mass terms breaking SU(N ) B to SU(N −1) × U(1) and similarly for SU(N ) C , becomes in the IR the theory of the form [SU(N ) A ] − SU(N −1) − T N −1 . In this section, we study what happens when we give generic mass terms to SU(N ) B,C . We also study the case when we replace the full puncture for SU(N ) A by more general ones.

Recursive application of the basic deformation
To analyze the effect of generic mass terms, we can first give the mass terms preserving SU(N −1) B,C , and then add masses which break SU(N −1) B,C . For convenience, we define generators of Cartan subalgebra of SU(N ) as H k = diag(1, · · · , 1, −k, 0, · · · , 0) (k = 1, 2, · · · , N −1) (4.1) where there are k entries of 1 so that tr H k = 0. Then, consider the mass matrices  We denote the gauge couplings of SU(N −1) and SU(N −2) as g N −1 and g N −2 , respectively.
There is one point one should be careful about. When the theory flows from (4.3) to (4.4), some of the heavy "fields" in the T N −1 theory are integrated out. We have seen in the previous subsection 3.3 that the coupling constants receive quantum corrections when massive fields are integrated out. However, because the T N −1 is non-Lagrangian, we cannot perform the one-loop calculation to determine the corrections.
Here we simply write down the correction which are consistent with other analysis we performed. We denote the couplings of SU(N −1) in (4.3) and (4.4) as g N −1,b and g N −1,a , respectively. The coupling g N −1,b of (4.3) is just given by (3.6), and the difference g −2 N −1,a − g −2 N −1,b is expected to depend only on m X,N −2 (X = B, C). It is given by The choice of the coefficient (N − 2)/2 can be explained as follows. Let us consider an instanton of the SU(N −1) gauge group. This gauge group is coupled to the bifundamentals of SU(N ) A × SU(N −1) and SU(N −1) × SU(N −2), and we can perform semi-classical quantization as in subsection 3.1.2. The mass spectrum is given by where m bif,N −1 = m B,N −1 + m C,N −1 and m bif,N −2 = m B,N −2 + m C,N −2 are bifundamental masses, and k = 0, · · · , N and k = 0, · · · , N −2. For example, the state k = k = 0 has m inst = −N m C,N −1 + (N −2)m C,N −2 . This is the same as the BPS mass created by the operator (µ C ) N N −1 . For more detailed comparison, it is necessary to determine which instanton states do or do not have gauge charges. This can be done by putting the theory on S 4 times the time direction and compute the index. This type of analysis was performed in [17]. Here we preferred to present a more elementary semi-classical analysis, that does not require the full machinery of the index computation.
Repeating the above procedure, we get the following result. We give masses of the form Then, we get a linear quiver where T 2 is just two fundamental hypermultiplets. Let m bif,k be the mass of the bifundamental of SU(k + 1) × SU(k), m f 1 and m f 2 the masses of the fundamentals in T 2 , and g k the gauge coupling of SU(k). Then, we get m bif,k = m B,k + m C,k , (k = 2, · · · , N −1) (4.9) This has the following interpretation. By using the formula (3.42), we can see that the coupling (4.14) is precisely the one obtained by integrating out the fundamental with mass m f 2 in T 2 . This is the T 2 version of what has happened in (4.5). Furthermore, the "SU(1)" coupling 8π 2 /g 2 1 is equal to m f 2 . So we may formally think of this fundamental as the "SU(1) instanton". We discussed that indeed, this fundamental hypermultiplet comes from the instanton of U(1) in the computation of partition functions in Sec. 2.2.
Next let us consider the case when we replace the full puncture giving SU(N ) A with a more general puncture of type Y = [n 1 , . . . , n p ], with n i = N . This can be realized by giving a vev to µ A of the T N theory of the form µ A = J n 1 ⊕ · · · ⊕ J np (4.16) where J n is the nilpotent Jordan block of size n. After the general mass deformation (4.7), we have the linear quiver (4.13), where µ A is given by the quadratic combination of the leftmost bifundamental. The effect of a nilpotent vev of the form (4.16) to µ A to a linear quiver whose gauge groups are all SU(N ) was studied in detail in Sec. 12.5 of [38]. There, it was shown that the rank of the i-th gauge group from the left is reduced by rank(µ A ) i , and the additional hypermultiplets in the fundamental of the i-th gauge group is given by the number of times i appears in [n 1 , . . . , n p ]. The same argument can be applied verbatim when we start from the quiver (4.13).
Therefore, the resulting linear quiver is of the form so that every node has zero beta function when considered as a 4d gauge group. Let K = N −n 1 . We have v N −i = i for i ≤ K, since w i = 0 for i > N −K = n 1 . Then v N −K = K, and after that the gauge groups are decreasing as K ≥ v N −K−1 ≥ · · · ≥ v 1 .

Seiberg-Witten curves in 4d
Here we derive the Seiberg-Witten curve of linear quiver gauge theory from the curve of the mass-deformed T N theory. More generally, we consider a theory realized by a Riemann sphere with two full punctures and one arbitrary puncture Y . Then we introduce generic masses to the SU(N ) B × SU(N ) C flavor symmetry of the full punctures. Let z be a coordinate of the Riemann sphere. We put the full punctures at z = ±1 and the puncture Y at z = 0. The curve is given by where the k-th differential φ k is given by  (4.22). Right: Removing the leftmost column. The numbers inside the boxes represent the = 1, 2, · · · , K. and then take → 0. The scaling of x is determined so that the Seiberg-Witten differential λ = xdz is fixed.
Let K = p N = N −n 1 as before. Also, define q such that The explicit form of q will be obtained later. Then, (4.19) becomes we get the curve Furthermore, the Seiberg-Witten differential is λ = xdz ∼ = −zdx up to a total derivative term. Therefore, we get a curve of a class S theory of A K−1 type by changing the roles of z and x.
There are N simple punctures at the solutions of x N + N k=2 c k x N −k = 0. The parameters c k are related to the positions of the simple punctures, and µ k are related to the mass parameters of these simple punctures. From the results of the previous sections, the condition that the bifundamentals are massless is given by m B + m C = 0. In that case, we get µ k = 2 k−1 (M B,k − (−1) k M C,k ) = 0, consistent with the fact that µ k are related to the mass parameters at the simple punctures.
We also have a puncture at x = ∞. Let x = 1/x. Then we get Therefore, this puncture, which we denote as Y , is given by singularities of order p = 2 −q .
It turns out that Y is obtained from the Young diagram Y by removing the leftmost column with hight n 1 as in the right of Fig. 5. That is, if Y = [n 1 , n 2 , · · · , n p ], then Y = [n 2 , · · · , n p ]. The number of boxes in Y is N −n 1 = K. This is understood as follows. By removing the leftmost column, each value of p k = 1, 2, · · · , K appears precisely once as is clear in table 5. Note that p k gives the upper bound of for each fixed k. Now we reinterpret these numbers inside the boxes of Y as the values of . Then, for each fixed , the value of k is bounded as k ≥ + a , where a is the height of the -th box in Y . Then we get p = 2 − q = − a . This is exactly the rule which associates the degrees of poles to the Young diagram Y . Now that we know K = p N and the puncture Y , it is easy to reconstruct the linear quiver, using the standard class S technology. When the original puncture Y is the full puncture, we find the linear quiver (4.30) In the more general case, we can check that it indeed reproduces the quiver given in (4.17).
Examples where the number of SU(K) is N −K, and the rightmost SU(K) has an additional fundamental hypermultiplet to make the theory conformal. This quiver can also be derived from the general form given above: w 1 = K, w N −K = 1 and other w i are zero. This datum determines v i . The flavor symmetry of the full puncture is SU(K) when K < N −1. When K = N −1, the bifundamentals of [SU(N −1)] − SU(N −1) and the additional fundamental of SU(N −1) are combined and the symmetry is enhanced to SU(N ). Let us give vevs to the adjoint scalars of the vector multiplets of SU(k) (k = 2, 3, · · · , N −1) so that the maximal number of bifundamentals becomes massless. By appropriate vevs which break the gauge groups as SU(k) → U(k − 1), the theory can be reduced to a quiver (4.33) where all the bifundamentals are massless. It is standard that tr µ k A = 0 and µ N −1 A = 0 follow from the F-term conditions, see e.g. Sec. 3.3 of [18]. Thus µ A is a nilpotent N × N matrix which, by complexified [SU(N ) A ] transformations, is conjugate to a block diagonal matrix J N −1 ⊕ J 1 .

Higgs branches
Next we study the Higgs branch of the T N theory under general mass deformations m B,C . We use two different methods.
Higgs branch from chiral rings. As already discussed in Sec. 3.2.2, the mass deformations kill all operators charged under the Cartan of SU(N ) B,C . In particular, Q i A i B i C and its cousins with more indices are all set to zero. The adjoint operators µ B,C are required to be diagonal, and we do not see conditions on µ A yet.
There are the well-known chiral ring relations for all k. Therefore, if we can show µ B,C = 0 we get tr µ k A = 0. We already know that µ B,C are diagonal. Then the relations (4.34) mean that we can assume µ B = µ C . Let us denote their N eigenvalues to be µ 1,...,N . With this, we have the chiral ring relation that follows from (A.10) in Appendix A.
We already know that all Q operators are zero. Since k is arbitrary in the relations above, we see that all µ i = 0, forcing µ B,C = 0. Therefore tr µ k A = 0. Furthermore, from the relation (A.5) applied to A and B and using µ B = 0, we get µ N −1 A = 0. These are what we wanted.
Higgs branch from SW curve. We can also obtain the same result from the Seiberg-Witten curve of the T N theory (4.19) by using the method in [34,39]. Let us make the curve the least singular at the puncture z = 0. From (4.23), this is achieved when all the Coulomb moduli are tuned to be zero, and we get φ k ∼ z −1 . This singularity is the one allowed by a simple puncture corresponding to the partition [N −1, 1]. Then, we can go to the Higgs branch where the puncture is Higgsed by a nilpotent vev conjugate to J N −1 ⊕ J 1 . This branch is exactly given by tr µ k A = 0 and µ N −1

From star shaped quiver to linear quiver in 3d
In three dimensions, the T N theory has a Lagrangian description in terms of a star-shaped quiver if we take mirror symmetry [40]. First, we define a superconformal theory T [SU(N )] as the low energy limit of the quiver [18] [SU(N )] − U(N − 1) − U(N − 2) − · · · − U(1). (4.36) In addition to the visible flavor SU(N ) symmetry of the Higgs branch, the T [SU(N )] has another flavor SU(N ) symmetry associated to the Coulomb branch. The Cartan subalgebra of this Coulomb branch SU(N ) symmetry is generated by the topological currents j k = tr F k associated to U(k) (k = 1, 2, · · · , N − 1) gauge groups, where F k is the field strength of U(k). We also define the moment map of the Higgs branch SU(N ) as When we add mass terms to the Cartan of SU(N ) X (X = A, B, C) of the T N theory, they become FI parameters of the gauge groups U(k) (k = 1, · · · , N − 1) of the corresponding T [SU(N )] X in the mirror side. If these FI parameters are generic, the moment map M X gets a diagonal vev (see Sec. 3.3 of [18]). This vev breaks the SU(N ) gauge group at the center of the star-shaped quiver to U(1) N −1 . Therefore, by generic deformation of SU(N ) B × SU(N ) C and integrating out massive degrees of freedom, we get a system in which the T [SU(N )] A survives and the Cartan of its Higgs branch SU(N ) is gauged by the U(1) N −1 .
By taking the mirror of the above system again, we get the low energy theory of the mass-deformed T N in the original description. The T [SU(N )] A is self-dual under the mirror symmetry, but its Higgs and Coulomb branches are exchanged. The gauging of the Higgs branch symmetry by U(1) N −1 becomes the gauging of the Coulomb branch symmetry by the mirror symmetry. As mentioned above, this Coulomb branch symmetry is generated by the topological currents j k = tr F k , so gauging this symmetry gives Chern-Simons couplings A k ∧ tr F k , where A k are the gauge fields of U(1) N −1 . These Chern-Simons couplings make all the U(1) gauge fields massive, including the U(1) k ⊂ U(k) subgroups. Therefore, we finally get a quiver in which all the gauge groups in (4.36) become special unitary SU groups instead of unitary U groups. This is exactly what we wanted.
The mass terms of the bifundamentals are generated by integrating out the massive fields in the superpotential which is schematically given by where Φ = (Φ k ) 1≤k≤N −1 and Φ k are the adjoint chiral fields of U(1) N −1 and U(k), respectively. The first term in the above superpotential is the supersymmetric partner of the Chern-Simons terms, while the second and third terms are the usual couplings of the adjoint chiral fields to hypermultiplets. It is easy to generalize this analysis to the 3d theory corresponding to the three-punctured sphere with two full punctures and one puncture of type Y . The 3d mirror to this theory is obtained by taking two copies of T [SU(N )] theory and one theory T Y [SU(N )] as introduced in [18], and gauging the common SU(N ) flavor symmetry. Now we give FI terms to the SU(N ) 2  where X can be either A, B or C. Then the following relation was described in [13]: Here we contracted the index i A ; we of course have the corresponding identities when the indices i B or i C are contracted. Suppose that the vevs of µ A,B,C are given as where µ 1 , · · · , µ N are generic. The complex dimension of the subspace of the Higgs branch under this condition on µ A,B,C is N −1.
It is strongly believed that when µ A,B,C are generic and diagonal as above, the only nonzero components of Q To reproduce the correct dimensions, q [i 1 ···i k ] need to be given by q i and µ i , and there must be one relation among q i . A sensible guess is then This equation was obtained for k = N − 1 and k = N in [13] with q [i 1 ···i N ] interpreted to be constant, and hence this is a natural generalization for arbitrary k. Combining (A.9) and (A.8), and assuming µ 1 , · · · , µ N are generic, we get i∈I, j ∈I (µ i − µ j ), (A.10) where I = {i 1 , · · · , i k }.
A candidate chiral ring relation that reduces to (A.10) when µ A,B,C are generic can be written down as follows: the left hand side is, in general, given by This has the desired properties that; (i) it is nonzero if and only if {j B,1 · · · j B,N −k } is the complement of {i B,1 · · · i B,k } (and similarly for C) and {i B,1 · · · i B,k } = {i C,1 · · · i C,k } as required by (A.7), and (ii) it has the correct anti-symmetric properties of the indices.