A review of the TN theory and its cousins

Progress of Theoretical and Experimental Physics, Nov 2015

The $T_N$ theory is a four-dimensional $\mathcal {N} = 2$ superconformal field theory that has played a central role in the analysis of supersymmetric dualities in the last few years. The aim of this review is to collect known properties of the $T_N$ theory and its cousins in one place as a quick reference.

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A review of the TN theory and its cousins

Prog. Theor. Exp. Phys. A review of the TN theory and its cousins Yuji Tachikawa 0 1 0 Kavli Institute for Physics and Mathematics of the Universe, University of Tokyo , Kashiwa, Chiba 277-8583 , Japan 1 Department of Physics, Faculty of Science, University of Tokyo , Bunkyo-ku, Tokyo 133-0022 , Japan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The TN theory is a four-dimensional N = 2 superconformal field theory that has played a central role in the analysis of supersymmetric dualities in the last few years. The aim of this review is to collect known properties of the TN theory and its cousins in one place as a quick reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Subject Index B10. B20, B33 - theory is the fundamental ingredient in understanding class S theories. Indeed, most of the newly found non-Lagrangian theories and most of the S-dualities among them are known to come from various properties of the TN theory. Due to this central role played by the TN theory, people devised various ways to obtain the properties of this theory, without relying on the classical Lagrangian description of the theory itself. For example, conformal and flavor central charges were studied in [5–8], various chiral ring relations were found in [9–12], and the superconformal indices have been intensively studied, e.g. in [13–16]. The properties of the TN theory are by now quite well understood, to the point that we can study a supersymmetry-breaking model that has the TN theory as an essential ingredient [17]. There are various cousins of the TN theory, either by starting from a 6d N = ( 2, 0 ) theory of type D or E , or by partially closing the punctures of the TN theory. We now have an extensive series of papers [18–24], pioneered by Chacaltana and Distler, describing these theories in detail. Somewhat unfortunately, these properties of the TN theory and its cousins were found gradually in the last several years using diverse techniques in various papers. The aim of this review article is to collect the most important of these properties, and give a short derivation for each of them from a uniform perspective. The author hopes that a person who would like to join the study of the TN theories can find this article an easy point of entry. The discussions in this article will be based on the following fact: Fact 1.1. The 6d N = ( 2, 0 ) theory of type G = An, Dn, or E6,7,8 on S1 is the 5d N = 2 supersymmetric Yang–Mills with gauge group G. We write this fact as an equation between quantum field theories: SG S1 = SYM5d N =2(G). (1.1) Here SG stands for the 6d N = ( 2, 0 ) theory of type G, and the bracket S1 denotes that the theory is compactified on S1. In the rest of the article, important results will be summarized similarly as Facts and given in italic. The derivation for each of the facts might not be quite complete and some of the facts presented might be better termed conjectures. Some of the facts are thus marked with question marks. It would be a great pleasure for the author if some of the readers got interested and establish these facts more rigorously. The rest of the article is organized as follows: In Sect. 2, we give a construction of the TN theory in terms of the 6d N = ( 2, 0 ) theory. Namely, we compactify the 6d theory on a Riemann surface of genus g without any punctures. We then split them into 2(g − 1) copies of the TN theory, corresponding to 2(g − 1) three-punctured spheres, and 3(g − 1) copies of an N = 2 vector multiplet with gauge group SU(N ). We then introduce the concept of the partial closure of punctures, and we detail the structure of the associated Nambu–Goldstone (NG) multiplets. We conclude the section by a discussion of the Argyres–Seiberg duality. In Sect. 3, we start from the known anomaly polynomials of the 6d N = ( 2, 0 ) theory to obtain the flavor and conformal central charges of the TN theory and its partially closed cousins. The discussion in this section improves upon previous discussions in the literature, by giving a logical derivation of the formulas conjectured in [8]. In Sect. 4, we discuss the superconformal indices of the TN theory and its cousins. By focusing on the so-called Schur limit, we give a rough derivation of its equivalence with the 2d q-deformed Yang–Mills. The technique is the same: we first consider the case corresponding to a genus-g surface without any punctures, and then we split them into contributions from copies of the TN theory and from the vector multiplets. In Sect. 5 we study the dimension of the moduli space of supersymmetric vacua of the TN theory and its cousins, again by starting from the case corresponding to a surface without any punctures. We give an explicit formula for the dimensions of the Higgs branch and the Coulomb branch. We then discuss the chiral ring relations of the operators on the Higgs branch. As our knowledge of these relations is not yet complete, the discussion here will be more schematic than other parts. We finish this section by discussing the Seiberg–Witten curves of the partially closed theories. In Sect. 6 we conclude by listing papers that describe the properties of the TN theory not described in this article, and by giving a short discussion on the future directions of research. Before proceeding, we pause here to mention that the statements in the first subsection of each section are applicable to all 4d N = 2 theories in general, whereas the other subsections are mainly for the class S theories. We should also mention here that there are other reviews [25–28] on the subjects surrounding the TN theory, and this article has some overlaps with them. 2. The TN theory and its cousins 2.1. Generalities on 4d N = 2 theories In this review, we often denote a quantum field theory by a letter such as Q, and we sometimes add curly brackets containing the flavor symmetries of the theory: Q{G} would be a theory with a flavor symmetry G. Unless otherwise mentioned, all quantum field theories we use are 4d N = 2 supersymmetric. Mostly we only discuss N = 2 superconformal theories, and they automatically have SU( 2 )R × U( 1 )R symmetry. We normalize the U( 1 )R charge of a supercharge to be ±1. We start with a universal fact that is used repeatedly in this review: Fact 2.1. A 4d N = 2 superconformal theory Q{G} has dimension-2 scalar operators μi=+,0,− in the adjoint of G and in the triplet of SU( 2 )R that are in the bottom component of a supermultiplet containing the conserved current of the flavor symmetry G. The operator μ+ is chiral in the language of 4d N = 1 supersymmetry, and μ− = μ+ ∗ is an anti-chiral operator. The details can be found, e.g., in [29] and references therein. As an example, for a free hypermultiplet of charge +1 consisting of chiral superfields Q and Q˜ , μ+ = Q Q˜ and μ0 = |Q|2 − Q˜ 2. The μ+,0,− are often called the moment map operators, since they are the moment maps of the G action on the Higgs branch under the three symplectic forms inherent in the hyperkähler structure. We will also use the following: Fact 2.2. Given a 4d N = 2 superconformal theory Q{G}, we can couple it to an N = 2 gauge multiplet with gauge group G. We denote the resulting gauged theory by Q{G}/τ G, where τ is the coupling constant defined at some renormalization scale. When the total one-loop beta function is zero, the coupling constant τ is exactly marginal. In this case we say that Q{G} can be conformally gauged. When the theory Q{G} is a free hypermultiplet, we can prove the equivalence of the vanishing of the one-loop beta function and the exact marginality of the coupling constant as follows. The Lagrangian of the gauge theory as an N = 1 theory roughly has the form c d2τ trWα W α + c.c. + c + u d4θ Q†eV Q + Q˜ eV Q˜ † . (2.1) Using the standard holomorphy arguments, we can show that τ is renormalized only at one-loop and the coefficient in front of Q Q˜ is not renormalized at all. We assume that the theory has zero one-loop beta function, so τ is not renormalized at all either. Now, the N = 2 supersymmetry fixes the ratios c : c and u : u and therefore nothing is renormalized. That was what we wanted to show. When Q is a strongly coupled field theory, we can use the method of [30] to show this fact. The contribution to the one-loop beta function from the theory Q{G} is given by the coefficient of the two-point function of the symmetry current. This is also called the flavor symmetry central charge, and we denote it by k(Q). We normalize it so that the contribution from a hypermultiplet in the adjoint representation to be k = 4h∨(G), which is 4N for G = SU(N ). Since the 4d N = 4 super Yang–Mills has zero one-loop beta function, the gauge multiplet has k = −4h∨(G). Then, any theory Q{G} with k = 4h∨(G) can be coupled to the N = 2 gauge multiplet with gauge group G to have zero one-loop beta function. We reiterate this as a fact since it is quite important: Fact 2.3. The theory Q{G} can be conformally gauged if and only if the flavor central charge k of Q{G} is 4h∨(G). Similarly, if two theories Q1{G} and Q2{G} have flavor central charge k1 and k2 such that k1 + k2 = 4h∨(G), we can conformally gauge the diagonal subgroup of two G flavor symmetries. We can denote the resulting theory by (Q1{G} × Q2{G})/τ Gdiag. (2.2) In general, we can characterize exactly marginal deformations of 4d N = 2 superconformal theories as follows: Fact 2.4. Exactly marginal deformations of a 4d N = 2 superconformal theory are in one-toone correspondence with dimension-2 scalar operators with U( 1 )R charge 4. That the bottom component of the supermultiplet containing a marginal deformation is a dimension-2 scalar chiral operator with U( 1 )R charge 4 is a simple consequence of the structure of short superconformal representations. The converse, that such a chiral operator always leads to an exactly marginal deformation, can be shown by the method of [30]. 2.2. The TN theory We introduce the TN theory using the theory SSU(N ), the 6d N =( 2, 0 ) superconformal field theory of type SU(N ). Let us put this 6d theory on a closed Riemann surface Cg of genus g. When g = 1, the curvature of the surface Cg breaks all of the supersymmetry. We can preserve some part of the supersymmetry by introducing a background R-symmetry gauge field on Cg that partially cancels 5d SYM where ωi is the spin connection of C. This preserves 4d N = 2 supersymmetry. Indeed, SO( 2 )R and SO( 3 )R of (2.3) can naturally be identified with U( 1 )R and SU( 2 )R symmetry of the 4d N = 2 theory. Finally, to isolate a genuine 4d theory, we take the limit where the area of Cg is zero. The shape, or equivalently the complex structure, of C remains a physical parameter of the 4d theory. We denote the resulting theory SSU(N ) Cg IR. Here, the bracket operation Cg stands for putting the theory on the manifold Cg, where we keep our choice of the R-symmetry background implicit in the notation. The final superscript IR is a reminder that we need to take the 4d limit by taking the area of Cg to be zero. Now we tune the shape of the surface C so that it is composed of 2(g − 1) three-punctured spheres and 3(g − 1) tubes connecting them. We choose almost all the area of the surface to be in the tubes. In this limit, each tube gives a segment of 5d N = 2 supersymmetric theory with gauge group SU(N ) with five adjoint scalars φi=1,2,3,4,5. Now, this segment of N = 2 super Yang–Mills is coupled to four-dimensional theories represented by two three-punctured spheres at the two ends, preserving N = 2 supersymmetry in 4d. There are 3(g − 1) complex structure deformations of a genus-g curve Cg, and in this description they correspond to the length and the twist of the tubes. In the 4d language they become 3(g − 1) exactly marginal deformations. See Fig. 1 for a schematic picture for g = 2. We define the TN theory to be the 4d limit of the 6d theory on a three-punctured sphere: TN = SSU(N ) C0,3 IR, (2.5) where C0,3 is the three-punctured sphere. We will introduce other types of punctures later, and this original type of puncture is called a full puncture. Each tube gives a segment of SU(N ) 5d N = 2 theory, and couples two SU(N ) flavor symmetries associated with two punctures. In the 4d limit, it reduces to a 4d N = 2 vector multiplet. To see this, we need to have a better understanding of the coupling of the segment to the 4d N = 2 theory at the boundary. Such a supersymmetry-preserving boundary condition is roughly described as follows; a similar half-supersymmetric condition of 4d N = 4 theory was first discussed in [31]. The boundary theory has an SU(N ) flavor symmetry and the bulk SU(N ) gauge field couples to it. We split the five scalars φi=1,2,3,4,5 of the 5d N = 2 vector multiplet according to (2.3) into a doublet φa=1,2 of SO( 2 )R and the triplet φi=1,2,3 of SO( 3 )R. Then we put a Neumann boundary condition for φa=1,2 and a modified version of the Dirichlet boundary condition for φi=1,2,3: Dnφa=1,2|boundary = 0, φi=1,2,3|boundary = μi=1,2,3. Here, the scalar operators μi=1,2,3 are the SU( 2 )R-triplet scalars associated with the SU(N ) flavor symmetry at the puncture, introduced in Fact 2.1. Now, suppose that a tube originally had a radius RS1 and a length Lsegment. First reducing it along the S1, we have the 5d N = 2 super Yang–Mills with gauge group SU(N ) on a segment, with 5d gauge coupling 1/gd2=5 ∼ 1/RS1 . We now take the limit where the length Lsegment of the segment is zero. We do this in such a way that the 4d coupling 1/gd2=4 ∼ Lsegment/RS1 is kept fixed. The three scalars φi=1,2,3 are eliminated due to the Dirichlet boundary condition, and the two scalars φa=1,2 together with the gauge field give rise to the 4d N = 2 vector multiplet. Summarizing, we see that the theory SSU(N ) C2 IR has a description as two copies of the TN theory coupled by three SU(N ) 4d N = 2 gauge multiplets: where τA,B,C are the complexified 4d gauge coupling constants associated with three tubes. Now we see that the 4d N = 2 SU(N ) vector multiplet coupling two TN theories via two punctures has the gauge coupling constant 1/gd2=4 as a tunable parameter. This means that the contribution to the one-loop beta function of a puncture is one half of that of an adjoint hypermultiplet. Summarizing, we have: Fact 2.5. The TN theory is a 4d N = 2 superconformal theory obtained by putting the 6d N = ( 2, 0 ) theory on a three-punctured sphere, with at least SU(N )3 flavor symmetry and taking the infrared limit: (2.6) TN := SSU(N ) C0,3 IR. (2.8) Each puncture carries an SU(N ) flavor symmetry, with the flavor symmetry central charge k = 2N . The TN theory does not have any exactly marginal deformations, or equivalently, it is an isolated superconformal theory. To show this, we just need to show that it does not have a chiral scalar operator with U( 1 )R charge 4. This is a direct consequence of Fact 5.7, which we will discuss later. We similarly define TG := SG C0,3 IR (2.9) for G = AN −1, DN , E6,7,8. In particular, TN = TSU(N ). Most of the discussions below apply equally well to TG theories for general G, but we often just discuss TN theories for notational brevity. In the IR limit we used to define the TN theory and the TG theory, the SU( 2 )R that remained unbroken by the background R-symmetry gauge field becomes the SU( 2 )R symmetry of the 4d N = 2 superconformal symmetry. Furthermore, this SU( 2 )R symmetry acts as an SO( 3 )R rotating φ3,4,5 of the 5d N = 2 theory that appeared in the intermediate theory. It should be noted, however, that it is not always the case that this unbroken SU( 2 )R in the compactification becomes the SU( 2 )R of the IR superconformal symmetry. For example, the compactification on S2 without any puncture gives rise to a hyperkähler sigma model with an intrinsic mass scale in the infrared, and does not lead to a nontrivial superconformal theory. Also, even when a compactification leads to a nontrivial superconformal theory, the SU( 2 )R in the IR can be different from the B A C D B A C D SU( 2 )R we just identified in the UV. For example, the compactification on T 2 without any puncture leads to the 4d N = 4 theory in the IR, but if viewed as an N = 2 theory in the standard manner, the SU( 2 )R in the IR is the subgroup of SU( 2 )R × SU( 2 )L SO( 4 )R rotating φ2,3,4,5 of the 5d N = 2 theory. Now, consider 6d N = ( 2, 0 ) theory of type SU(N ) on a sphere with four full punctures. Let us use a complex variable z to parametrize the sphere, and put the punctures A, B, C , and D at z = 0, z = q, z = 1, and z = ∞ respectively. When q is very small, the theory is given by taking a TN { A, B, G} and another TN {G, C, D} symmetry and by coupling them via a Gdiag = SU(N ) gauge multiplet with the exponentiated coupling constant q ∼ exp − 1/gd2=4 . Here we use an abbreviation where SU(N )A is written just as A, etc. Now, adiabatically change q to be close to 1; we can now perform the change of coordinates z = 1 − z so that the punctures A, B, C , D are now at z = 1, = 1 − q, = 0, and = ∞ respectively. Now the theory is given by taking a TN { A, D, G } and another TN {G , B, C } and by coupling them via a Gdiag = SU(N )G gauge multiplet with the exponentiated coupling constant 1 − q ∼ exp − 1/gd=42 , see Fig. 2 for an illustration. This is a strong–weak duality, or equivalently an S-duality. Rather than stating this in a sentence, let us write it as an equation: Fact 2.6. We have an S-duality where A, B, C, D and G, G are all SU(N ). TN A, B, G × TN G, C, D q Gdiag = TN A, D, G × TN G , B, C 1−q G diag, (2.10) Before proceeding, let us state what T2 and T3 are. We will have more support for these statements later in this review. Fact 2.7. The T2 theory is a theory of four N = 2 hypermultiplets. In the N = 1 language, it consists of eight chiral multiplets Qaiu, a, i, u = 1, 2, 3 with SU( 2 )3 flavor symmetry. Fact 2.8. The T3 theory has an enhanced symmetry SU( 3 )3 ⊂ E6, and is the E6-symmetric theory of Minahan and Nemeschansky, originally found in [1]. 2.3. Partial closure of punctures Let us consider a general situation again: take a 4d N = 2 superconformal theory Q{SU(N )} with flavor symmetry SU(N ). This has a chiral operator μ+ in the adjoint of SU(N ). We are going to give a nilpotent vev to μ+. A nilpotent matrix can be put into the Jordan normal form μ+ = JY := Jn1 ⊕ Jn2 ⊕ · · · , where N = ni , and Jn is an n × n Jordan block with zeros along the diagonal and n − 1 non-zero entries on one line above the diagonal. We use Y to denote ni collectively. We can and do order ni so that n1 ≥ n2 ≥ · · · without sacrificing generality. It is customary to identify Y with a Young diagram such that the i th column has height ni . It is also customary to abbreviate, e.g., the partition 8 = 3 + 2 + 2 + 1 as Y = 3221 . The Young diagram Y t transpose to Y is defined by exchanging the rows and the columns. Again, as an example, Y t = [431] if Y = 3221 . Let us first note that when Y = 1N , it is clearly a trivial operation. This is because μ+ = 0, so we do not do anything. Otherwise this is a nontrivial operation. The original flavor symmetry SU(N ) is broken to a subgroup GY , and there are Nambu–Goldstone modes and their superpartners associated with this breaking of the flavor symmetry. The subgroup GY is given by (2.11) where kn is the number of times n appears in the sequence [n1n2 · · · ]. Here, U kn acts by permuting the blocks GY = S n U kn , Jn ⊕ · · · ⊕ Jn . kn JY = ρY σ + , N = i ni . I 3 − 21 ρY σ 3 For example, for N = 9 and Y = 3213 , GY = S[U( 2 ) × U( 3 )]. We will detail the structure of the Nambu–Goldstone multiplets in Sect. 2.4. It turns out to be useful to regard where σ + is the raising operator of SU( 2 ) and ρY : SU( 2 ) → SU(N ) is an N -dimensional representation of SU( 2 ), so that we have Here and below, n is an n-dimensional irreducible representation of SU( 2 ). As μ+ = JY is the highest weight of the SU( 2 )R triplet and the highest weight of ρY (SU( 2 )) at the same time, the linear combination of the Cartan part I3 of the SU( 2 )R symmetry and a Cartan part of the flavor symmetry SU(N ) remains unbroken. The importance of this unbroken R-symmetry was pointed out in [16], for example. In total, we have the breaking pattern U( 1 )R × SU( 2 )R × SU(N ) → U( 1 )R × U( 1 )R × GY , (2.17) where the generator of U( 1 )R is (2.16). Note that the chiral supercharges Qi=1,2 have the charge α (1, ±1/2) under U( 1 )R × U( 1 )R. When the original theory Q{SU(N )1, . . . , SU(N )m } has SU(N )m symmetry, we can perform this operation for each SU(N )i , i = 1, . . . , m by setting μi+ = JYi . Let us denote the field theory governing everything except the NG modes by Q{Y1, . . . , Ym }: Set μi+ =JYi Q{SU(N )1, . . . , SU(N )m } −−−−−−−→ Q{Y1, . . . , Ym } + NG modes for Yi . (2.18) (2.19) (2.20) At this point, this theory Q{Y1, . . . , Ym } is a theory with mass scale set by the vev and with the symmetry U(1)R × U(1)R × i GYi ; now the generator of U(1)R is I 3 − 12 i ρYi σ 3 . We are interested mainly in the conformal theories, so let us take the infrared limit of this theory, take the IR limit Q{Y1, . . . , Ym } −−−−−−−−−→ Q{Y1, . . . , Ym }IR. The resulting theory Q{Y1, . . . , Ym }IR is by definition an N = 2 superconformal theory, but it can be free or empty in some special cases. The procedure of obtaining the new superconformal theory Q{Y1, . . . , Ym }IR from the theory Q is called the partial closure of punctures. Note that this operation, the partial closure of punctures, has mostly been applied only to class S theories in the literature so far, but it can in fact be performed on any 4d N = 2 theories. In favorable cases, this U( 1 )R symmetry is the Cartan subgroup of the SU( 2 )R symmetry of the low-energy N = 2 superconformal theory.2 Such partial closures are called good. Otherwise they are called not good.3 When the closures are not good, the low-energy theory Q{Y1, . . . , Ym }IR often has fewer flavor symmetries than i GYi . Using the partial closure of punctures, we introduce Fact 2.9. The theories TY1,Y2,Y3 := TN {Y1, Y2, Y3}IR, (2.21) when good, are N = 2 superconformal theories with flavor symmetry at least GY1 × GY2 × GY3 , obtained by the partial closures of punctures of the TN theory. When Y1 = Y2 = Y3 = 1N , we do nothing, so we obviously have TY1,Y2,Y3 = TN . Another fundamental fact is consisting of N = 1 chiral multiplets Qia, Q˜ ia, a, i = 1, . . . , N . Fact 2.10. The theory T 1N , 1N ,[N −1,1] is a theory of free bifundamental hypermultiplets We will justify this later. Note that when N = 2 this fact reduces to Fact 2.7. 2 It can happen that U( 1 )R enhances to SU( 2 )R only in the infrared limit. It can also happen that U( 1 )R is already the Cartan of an SU( 2 )R symmetry before taking the infrared limit. 3 Note that even when U( 1 )R is a part of an SU( 2 )R symmery in the ultraviolet, it can happen that this SU( 2 )R does not survive in the infrared limit. 2.4. Structure of the NG bosons under the partial closures Let us study the structure of the Nambu–Goldstone modes that arise associated with the vev μ+ = JY by acting on them with SU(N ) generators and their superpartners. Using the complexified SU(N ) action, i.e. by using the SL(N ) action, the vev JY can be moved to any nilpotent matrix conjugate to JY . Let us call the set of all such matrices the nilpotent orbit OY of type Y . From this viewpoint, we picked the vev μ+ = JY ∈ OY , and the Nambu–Goldstone modes correspond to the tangent space at JY of OY . The directions along the tangent space arise from SU(N ) generators J a such that To find them, we just have to decompose the SU(N ) adjoint by regarding it as an SU( 2 ) representation by ρY , and taking non-highest-weight vectors under the SU( 2 ) action. Let us then say that we have the irreducible decomposition ρY σ + , J a under ρY . This decomposition can be obtained easily by plugging (2.15) into adj ⊕ C = N ⊗ N. Each direct summand mi above gives rise to ◦ mi − 1 complex scalars with U( 1 )R charge 0, and U( 1 )R charge mi 2− 1 , mi 2− 3 , . . . , 3 −2mi , ◦ and mi − 1 Weyl fermions with U( 1 )R charge −1 and U( 1 )R charge mi 2 − 1, m2i − 2, . . . , 1 − m2i . The complex dimension of the nilpotent orbit OY is then given by the sum combinatorial exercise to show that mi − 1 . It is a dimC OY = mi − 1 = N 2 − i i si2, where Y t = [s1s2 · · · ] is the Young diagram transpose to Y . They are free hypermultiplets, but with a slightly unusual assignment of the R-charges. Assuming that U( 1 )R enhances to SU( 2 )R in the infrared, one finds, therefore: Fact 2.11. When the partial closure μ+ = ρY σ + is good, the resulting Nambu–Goldstone modes consist of ◦ U( 1 )R neutral real scalars in mi − 2 ⊕ mi of SU( 2 )R and ◦ U( 1 )R-charge −1 Weyl fermions in mi − 1 of SU( 2 )R for each summand mi in the decomposition (2.23) of the SU(N ) adjoint under ρY . In total, there are 1 1 2 dimC OY = 2 N 2 − si2 i free hypermultiplets in the Nambu–Goldstone modes. Here, Y t = [s1s2 · · · ] is the Young diagram transpose to Y . (2.27) i u a i a u Note that the description at the first bullet point is not completely precise when mi is even, since mi and mi − 2 are not strictly real representations. Such mi ’s appear always in pairs, however, and therefore we have complex scalars in mi − 2 ⊕ mi for each such pair. 2.5. Complete closure Here let us explain why this operation is called the partial closure. Consider the 4d theory SSU(N ) Cg,n , obtained by putting the 6d theory on a genus g surface with n full punctures. Then we have the following statement: Fact 2.12. Choose one puncture from the theory SSU(N ) Cg,n , and perform the closure of type Y = [N ] to the SU(N ) symmetry associated to that puncture. Then the resulting theory is equivalent to SSU(N ) Cg,n−1 , where the chosen puncture that was originally full was completely closed and disappears: SSU(N ) Cg,n {[N ]} = SSU(N ) Cg,n−1 . (2.28) At this point we cannot justify this statement except for N = 2. We will see more justifications later in the review. So take N = 2, and assume further that there are n ≥ 2. We can modify the shape of the surface so that the theory is given as SSU( 2 ) Cg,n = SSU( 2 ) Cg,n−1 {SU( 2 )a} × trifundamental Qaiu SU( 2 )a; (2.29) see Fig. 3. Here we used Fact 2.7 that the T2 theory consists of trifundamental half-hypermultiplets of SU( 2 )3. Now μ+ associated with the indices i and u are given by μ(+i j) = Qaiu Qbjv ab uv, μ(+uv) = Qaiu Qbjv ab i j . We now want to close the SU( 2 )i puncture by Y = [2]. Equivalently, we would like to set μ+ij = 00 10 . Since μi+j = ik μ+k j , this amounts to setting μ(+11) = 1, keeping other components zero. This can be done by setting Qaiu = δi=1 au. This means that SU(2)a × SU(2)u is broken to the diagonal SU(2) subgroup. So, the SU(2)a gauge group is completely Higgsed, eating three hypermultiplets. Out of the four hypermultiplets in the trifundamental Qaiu, only one remains. Therefore, after setting μ+ = JY , the theory (2.29) becomes SSU( 2 ) Cg,n−1 {SU( 2 )a=i } + one free hypermultiplet. The one free hypermultiplet is the Nambu–Goldstone modes associated with the closure by Y = [2]. We conclude that the resulting theory from the closure is (2.30) (2.31) (2.32) SSU( 2 ) Cg,n {[2]} = SSU( 2 ) Cg,n−1 . This closure of a puncture of the T2 theory by Y = [2] is our first example of a non-good closure, so let us analyze it more closely. We start from the trifundamentals Qaiu, and close one puncture by setting μ(+11) = 1. To separate the Nambu–Goldstone mode and the rest, note that an infinitesimal complexified SU( 2 ) action by aσ + + bσ 3 + cσ − changes the vev μ+ by δ μ(+11) ∝ b, δ μ(+12) ∝ c, Therefore the Nambu–Goldstone mode can be eliminated by requiring keeping μ(+22) unspecified. In terms of Zau := uv Qv,i=1,a and Wau := uv Qv,i=2,a, Eq. (2.34) can be written as The first equation means that Z is on the SL( 2 ) = SU( 2 )C group manifold, and the second equation means that W can be identified as the coordinates of its cotangent bundle. Equivalently, the second equation can be more suggestively written as d det Z |d Z→W = 0, where we first apply the exterior derivative, and then we replace d Z by another commuting variable W . Summarizing, T2{[2]} is an N = 2 sigma model on T ∗SU( 2 )C, the cotangent bundle of SL( 2 ) = SU( 2 )C. In general, we have: Fact 2.13. The theory TN {[N ]} obtained by the complete closure of a full puncture of the TN theory is an N = 2 sigma model on T ∗SU(N )C. It has N × N chiral fields Z and W satisfying det Z = 1, d(det Z )|d Z→W = 0. (2.36) The SU(N )2 symmetry of this theory acts on the fields Z and W naturally by the left and the right actions. But they are not preserved intact at any point: even when W = 0 and Z = 1, SU(N )2 is broken down to the diagonal SU(N ). To take the low-energy limit, we need to set det Z = cN , expand Z = c + δ Z , and send c → ∞, keeping δ Z and W as the fluctuations. Only the diagonal SU(N ) is manifest in this limit. The fact above can itself be shown as follows: the TN theory was for the three punctured sphere, TN = SSU(N ) C0,3 IR. Therefore, we see that TN {[N ]} = SSU(N ) C0,2 . Now, a sphere with two punctures has an S1 isometry with two full punctures as two fixed points. Reducing around this S1, we find that TN {[N ]} is essentially the 5d N = 2 super Yang–Mills theory with SU(N ) gauge group on a segment. The boundary condition is in some sense the opposite of (2.6) on both ends: we have φa=1,2 = 0, Dnφi=1,2,3 = 0, (2.37) and the gauge transformations at the two boundaries are considered as flavor symmetries SU(N ) × SU(N ). We see that the 4d N = 2 vector multiplet part is killed by the boundary conditions. The Higgs branch can be found by studying the moduli space of the BPS equation with these boundary conditions, and turns out to be T ∗SU(N )C; see [32] for more details. 2.6. Argyres–Seiberg duality In Fact 2.6 we learned an S-duality of two TN theories coupled by an SU(N ) gauge group. When N = 2, this is the standard S-duality of SU( 2 ) gauge theory with N f = 4 flavors, as first beautifully B A C D B A C D B A C D demonstrated in [4]; see also [33]. However, when N > 2, this is a duality of non-Lagrangian theories coupled to gauge fields. Let us now use the partial closure to derive S-dual descriptions of Lagrangian gauge theories. First, we start from the duality of Fact 2.6. The basic point was to consider SSU(N ) C0,4 , the 6d theory put on a sphere with four punctures A, B, C, D, and split the sphere in two ways. Now, partially close the puncture B to type Y = [N − 1, 1], see Fig. 4. Considering the splitting in two ways, we have the duality TN A, [N − 1, 1]B , G × TN G, C, D q Gdiag = TN A, D, G × TN G , N − 1, 1 B , C 1−q G diag, where A, D, G, G are SU(N ) gauge or flavor groups. We stated previously in Fact 2.10 that TN {[N − 1]} is a theory of bifundamental hypermultiplets of SU(N )2. Therefore, this is a duality of the TN theory coupled to bifundamentals by an SU(N ) gauge group. We now further perform the partial closure of the puncture C to Y = [N − 1, 1]. On the left-hand side, we have TN A, [N − 1, 1]B , G × TN G, [N − 1, 1]C , D q Gdiag, which is just N f = N + N flavors of fundamental hypermultiplets coupled to the SU(N ) gauge group. On the right-hand side we have TN A, D, G × TN G , [N − 1, 1]B , [N − 1, 1]C /1−q G diag, which is more tricky to analyze. This is because the partial closure TN {[N − 1, 1], [N − 1, 1]} is not good, since TN {[N − 1, 1]} is already a free theory. At the same time, because TN {[N − 1, 1]} is free, we can study this partial closure explicitly. Denote the bifundamentals as Qia and Q˜ ia, where the indices i , a are for SU(N )C and SU(N )G respectively. We are setting Qia Q˜ aj = JN −1 ⊕ J1. One way to solve this is to take This forces us to have Q = diag(1, 1, . . . , 1, 0, 0), Q˜ = JN −1 ⊕ J1. Q˜ ib Qia = JN −2 ⊕ J1 ⊕ J1 = J N −2,12 , B A C D (2.38) (2.39) (2.40) (2.41) (2.42) (2.43) (2.44) which in turn force μ+ of SU(N )G of the first TN A, D, G to be set to μ+ = J N −2,12 (2.45) via the F-term equation of the adjoint scalar of SU(N )G . This means that the puncture G of TN A, D, G is partially closed to type Y = N − 2, 12 . Originally, there was SU(N )G × SU(N )C symmetry acting on the bifundamental and SU(N )G was gauged. Now the vev Qia and Q˜ ia breaks the symmetry down to SU( 2 ) × U( 1 )diag where SU( 2 ) × U( 1 ) is the natural symmetry for Y = N − 2, 12 on SU(N )G , U( 1 ) is the natural symmetry for Y = [N − 1, 1] on SU(N )C , and U( 1 )diag is the diagonal combination of these two U( 1 )s. In the end, only SU(2)G ⊂ SU(N )G remains gauged. Let us count how many hypermultiplets remain coupled to SU( 2 )G . Originally we had x = N 2 free hypermultiplets in the bifundamental. The partial closure of G to N − 2, 12 gives y = N (N − 1)/2 − 3 Nambu–Goldstone hypermultiplets, as can be found using Fact 2.11. SU(N )G is broken to SU( 2 )G and eats z = N 2 − 22 hypermultiplets. Therefore x + y − z = N (N − 1)/ 2 + 1 free hypermultiplets remain. Finally, the partial closure of C to [N − 1, 1] gives w = N (N − 1)/2 − 1 Nambu–Goldstone hypermultiplets, so only x + y − z − w = 2 hypermultiplets remain coupled to SU( 2 )G . This is a doublet of SU( 2 )G . Summarizing, we found that coupling TN G , N − 1, 1 B , N − 1, 1 C to an SU(N )G flavor symmetry via an SU(N )G gauge multiplet has the effect that the SU(N )G is spontaneously broken to SU( 2 )G , and there is a doublet hypermultiplet coupled to this unbroken SU( 2 )G . Therefore, we see that: Fact 2.14. We have an S-duality of the form 2N × N bifundamentals q SU(N ) = T 1N , 1N , N −2,12 × (a doublet)/1−q SU( 2 ), (2.46) where the SU( 2 ) gauge group on the right-hand side couples to the SU( 2 ) flavor symmetry of N − 2, 12 and to the SU( 2 ) of the doublet. For N = 3, the right-hand side slightly simplifies, since N − 2, 12 = 1N in that case. This is the Argyres–Seiberg duality [3]. For N ≥ 4 this was described in detail in [18]. Note that on the left-hand side there is an SU(2N ) flavor symmetry, while only SU(N ) × SU(N ) is manifest on the right-hand side. This means that the SU(N )2 flavor symmetry of T 1N , 1N , N −2,12 should enhance to SU(2N ). In particular, for T3, we see that T3 should be such that for any pair of two SU( 3 )s it should enhance to SU( 6 ). This is only possible when T3 has an E6 flavor symmetry, and supports Fact 2.8. Generalizing, it is common that when the punctures Y1,2 are rather small, the theory TN SU(N )A, Y1, Y2 has the effect that the first SU(N )A is spontaneously broken to a subgroup H and μ+A automatically is set to JY3 . To analyze the theory SSU(N ) Cg,n {Y1, Y2}, by taking out the two punctures as in Fig. 3, we have SSU(N ) Cg,n {Y1, Y2} = SSU(N ) Cg,n−1 {SU(N )A } × TN SU(N )A, Y1, Y2 q SU(N )A=A . (2.47) Now the μ+A = JY3 of the second factor causes μ+A = JY3 , partially closing the first factor and spontaneously breaking SU(N )A=A to some subgroup HY1,Y2 ⊂ GY3 . We end up with the gauge theory SSU(N ) Cg,n {Y1, Y2} = SSU(N ) Cg,n−1 {Y3} × Q(Y1, Y2) q HY1,Y2 , (2.48) where Q(Y1, Y2) is a theory determined by Y1, Y2 with a flavor symmetry HY1,Y2 . This point of view was explained, e.g., in [34]. To see which pair of Y1,2 leads to this phenomenon of the propagation of partial closure and what is the resulting Y3, HY1,Y2 and the remaining matter theory Q(Y1, Y2), the extensive set of tables in [18–22,24] is very useful. Note, however, that in these papers, such a pair Y1,2 is said to require an irregular puncture Y3∗ dual to Y3, Q(Y1, Y2) is listed as a theory of the three-punctured sphere with punctures of type Y1, Y2, Y3∗, and the group HY1,Y2 is listed as a cylinder connecting Y3 and Y3∗. Note also that the irregular punctures in their terminology are not the same concept as the irregular punctures as used, e.g., in [35–38]. 3. Central charges 3.1. Generalities on the central charges and anomalies Four-dimensional conformal theories have two conformal central charges a and c. For N = 2 superconformal theory, using nh and nv are more convenient, normalized so that (nh, nv) = ( 1, 0 ) for a free hypermultiplet and = ( 0, 1 ) for a free vector multiplet; a and c can be written as a = 214 nh + 254 nv, c = 112 nh + 61 nv. (3.1) The current two-point function of a flavor symmetry G is also characterized by a number k, called the flavor central charge. As already stated, we normalize it so that k = 4h∨(G) for an adjoint hypermultiplet. In this review, instead of computing the central charges directly, we use the following relations of the central charges and the ’t Hooft anomaly of the theory: Fact 3.1. The anomaly polynomial A6 of 4d N = 2 theory Q{G} with flavor symmetry G has the following form: A6 = (nv − nh) Here, c1 = c1 FU( 1 )R , c2 = c2 FSU( 2 )R are the Chern classes of the background U( 1 )R and SU( 2 )R gauge fields and p1 = p1(T X ) is the Pontrjagin class of the spacetime, and n(FG ) is the characteristic class for the background flavor symmetry gauge field proportional to trFG2 so that it integrates to 1 in the one-instanton background. In particular, we have n(FG ) = c2(FG ) when G = SU( 2 ). The essential analysis establishing this fact was performed in [39]. 3.2. The central charge of the TN theory Let us determine the central charge of the TN theory. As always our strategy is to start from the 6d theory. We first quote the known anomaly polynomial of the 6d N = ( 2, 0 ) theory of type G: Fact 3.2. The 6d N = ( 2, 0 ) theory SG of type G = AN −1, DN , E6,7,8 has the anomaly polynomial is the anomaly polynomial of the free N = ( 2, 0 ) tensor multiplet. As always, h∨, d, and r are the dual Coxeter number, the dimension, and the rank of G, and h∨ = N , d = N 2 − 1, and r = N − 1 for G = SU(N ). This formula was first found for G = AN in [40] using M-theory, conjectured generally in [41], and computed for G = DN in [42]. A field-theoretic derivation was given in [43,44]. From this we can easily find the anomaly polynomial in four dimensions [7,9]. We assume that the 6d spacetime is of the form Y6 = X4 × C2. Note that in this section the subscript of C2 stands for the dimensionality, not the genus. We arrange the SO( 5 )R bundle NY to partially cancel the curvature of C2 as specified in (2.3), (2.4). Then we just integrate the resulting I8 over C2, obtaining A6. In the actual computation, it is convenient to use the so-called splitting principle used in the algebraic topology when manipulating the characteristic classes. This principle says that the computation of the characteristic classes can be done assuming that the vector bundles are just direct sums of line bundles. The curvature of those constituent line bundles are called Chern roots. Let us denote the Chern roots of T X4, T C2, and NY6 by ±λ1,2, ±t , ±n1,2, 0 respectively. We also introduce the U( 1 )R bundle and the SU( 2 )R bundle on X4; let us denote their Chern roots by c1 and ±α respectively. We need to express n1,2 in terms of c1, α, and t . The cancellation of the curvature (2.4) in this language is to take n1|C2 = −t . Then we identify the 4d R-symmetries with the subgroup of the 6d R-symmetry via (2.3). Therefore we have n1 = 2c1 − t, The anomaly A8 can now be written in terms of the Chern roots by using the following facts: When an SO bundle B has the Chern roots ±λi , p1(B) = λi2 and p2(B) = i< j λi2λ2j . Also, for an SU( 2 ) bundle R with the Chern roots ±α, c2(R) = −α2. Plugging them into A8 and integrating over C2 using the Gauss–Bonnet theorem C2 t = 2 − 2g, we find A6 = (g − 1)r Therefore, the theory SG Cg has nv = (g − 1) 43 h∨d + r , nh = (g − 1) 43 h∨d. We know that this theory is composed of 2(g − 1) copies of the TG theory and 3(g − 3) tubes each representing a vector multiplet of gauge group G. We already determined the flavor central charge of three G3 flavor symmetries. Summarizing, we have: (3.5) (3.6) (3.7) Fact 3.3. The TG theory has the central charges where k1,2,3 are the current algebra central charges for three G symmetries. Take G = SU( 2 ). We find nv = 0 and nh = 4. In general, any N = 2 superconformal theory has nonnegative nv and nh [45,46]. The converse is a conjecture: Fact 3.4. An N = 2 superconformal theory with nv = 0 is a theory of free hypermultiplets. Similarly, when nh = 0, it is a theory of free vector multiplets. Assuming this, we find that the T2 theory consists of four free hypermultiplets with SU( 2 )3 symmetry, such that each of SU( 2 ) has k = 4. The trifundamental Qaiu is the only such multiplet, supporting our Fact 2.7. Using the central charges of the TG theory obtained above, it is easy to get the general formula of the central charges of the theory SG Cg,n . In the 4d language, it can be made from 2(g − 1) + n copies of the TG theory and 3(g − 1) + n vector multiplets with gauge group G. In total, we have nv = (g − 1) 34 h∨d + r + 23 h∨d + r −2 d n, nh = (g − 1) 43 h∨d + 23 h∨dn; (3.9) the flavor symmetry Gi associated with the i th full puncture has ki = 2h∨, as always. 3.3. Effect of the complete closure We would like to know the effect of the partial closures to these central charges, assuming that the closures are good. Let us first study the effect of the complete closure of a puncture. For definiteness take G = SU(N ). Originally, the contribution to the anomaly polynomial from a full puncture is 1 A6 = 2 (r − d) as can be seen from (3.9). We set μ+ = JN = ρ[N ] σ + . The SU( 2 )R of the infrared is essentially the diagonal combination of SU( 2 )R before the closure and ρ[N ](SU( 2 )). Stated differently, in terms of the Chern roots (+α, −α) of the infrared SU( 2 )R, the Chern roots of the original SU( 2 )R are (+α, −α) and those of SU(N ) are (N − 1)α, (N − 3)α, . . . , (1 − N )α. We then use that the instanton number n(FG ) of a bundle with Chern roots αi is n(FG ) = − 12 αi2. Plugging everything in to (3.10), and using r = N − 1, h∨ = N , and d = N 2 − 1, we find Note that this is the anomaly polynomial contribution from the puncture of type [N ] together with the Nambu–Goldstone bosons. To determine the contribution from the latter, we note that the decomposition of the adjoint (2.23) in this case is N adj = i=2 and therefore the Weyl fermions in the Nambu–Goldstone multiplets have U( 1 )R charge −1 and in the SU( 2 )R representation iN=−11 2i . So the contribution to the anomaly from the Nambu–Goldstone modes are 3.4. Effect of the partial closure to [N − 1, 1] Let us next consider the partial closure to [N − 1, 1]. We use the embedding ρ[N −1,1] : SU(2) → SU(N ). This preserves a U( 1 )B subgroup as the flavor symmetry. Now the Chern roots for the SU(N ) flavor symmetry are (N − 2)α + β, (N − 4)α + β, . . . , (2 − N )α + β, (1 − N )β, where ±α are the Chern roots for the SU( 2 )R in the infrared, and β is the Chern root of the U( 1 ) flavor symmetry. Then the total A6 is − The decomposition of the adjoint under SU( 2 )R is adj = 1 ⊕ N − 1 ⊕ N − 1 ⊕ The Weyl fermions in the Nambu–Goldstone modes therefore have the SU( 2 )R representations N − 2 ⊕ N − 2 ⊕ i=1 where two N − 2 terms have U( 1 )B charge ±N and the other terms are neutral. The contribution to the anomaly is then + 16 (N − 1)(N − 2) 2N 2 − 4N − 3 c1c2 − N 2(N − 2)c1β2. (3.20) Subtracting (3.20) from (3.17), we find that the contribution to the anomaly from the puncture of type [N − 1, 1] is − From this we can find the central charges of the theory TN {[N − 1, 1]} by adding three contribu tions: A6 = −(N − 1) Here the first line is the part proportional to 2 − 2g, the second line is from two full punctures, and the third line is from the puncture of type [N − 1, 1]. The final answer is rather simple: Therefore this theory has nv = 0, nh = N 2. This strongly suggests that the theory consists of N 2 free hypermultiplets. Then the term proportional to β2 says that the U( 1 )B charges of the hypermultiplets are ±1. In addition, two SU(N ) symmetries both have k = 2N . This supports the fact that this theory consists of free hypermultiplets in the bifundamental of SU(N ) × SU(N ), and the U( 1 )B charge carried by the puncture of type [N − 1, 1] can be identified with the baryonic symmetry of the bifundamental. 3.5. General formula From the examples above, it is clear that we can compute the superconformal central charges nv, nh and the flavor symmetry central charges ki of the theories with partially closed punctures, by identifying SU( 2 )R after the closure in the original variables and subtracting the contributions from the Nambu–Goldstone multiplets. Instead of giving a detailed derivation we just quote the facts: Fact 3.5. The central charges nv, nh of the 4d theory obtained by putting the 6d theory of type G on a genus-g surface with n punctures, labeled by Y1, . . . , YN , are given by 4 nv = (g − 1) 3 h∨d + r + i nv(Yi ), 4 nh = (g − 1) 3 h∨d + i nh(Yi ), (3.24) where nv(Y ) = 32 h∨d − 4ρW · hY + 21 (r − no(Y )), nh(Y ) = 32 h∨d − 4ρW · hY + 21 ne(Y ). (3.25) Here, ρW is the Weyl vector of G, hY is the highest element in the Weyl orbit of ρY (σ3), no,e(Y ) are the number of direct summands in the decomposition of the adjoint (2.23) under ρY , that are respectively odd and even dimensional. When G = SU(N ), ρW = (N − 1, N − 3, . . . , 1 − N )/2 and hY is the vector ρY (σ3) reordered so that the components are non-decreasing, e.g. h[3,1] = ( 1, 0, 0, −1 ). 19/38 (3.22) (3.23) This general form of the nv,h was originally derived in [8] using various string dualities. Here we instead gave a derivation using the Nambu–Goldstone multiplets.4 Similarly, we have the following facts concerning the flavor symmetry central charge: Fact 3.6. For a puncture of type Y = [n1n2 · · · ] and a factor of the flavor symmetry SU( ) associated with the columns of height h, its flavor symmetry central charge kSU( ) is given by (3.26) kSU( ) = 2 sh , h ≤h where Y t = [s1s2 · · · ] is the Young diagram transpose to Y . As an example, consider a puncture of type Y = [N − 2, 1, 1]. The SU( 2 ) symmetry associated with two columns of height 1 then has k = 6, since Y t = 3, 1N −3 . This is nicely consistent with the Argyres–Seiberg duality we reviewed as Fact 2.14. Indeed, in the second line, the SU( 2 ) gauge group couples to the SU( 2 ) flavor symmetry of a puncture of type [N − 2, 1, 1] and to a doublet. The one-loop beta function contribution from the matter sector is therefore 6 + 2 = 8, which means that this combined SU( 2 ) symmetry can be conformally gauged. 4. Superconformal index 4.1. Generalities on the superconformal index In this section we summarize the superconformal index of the TN theory and its cousins. The first study of this topic was in [13], and the full structure began to emerge in [14]. The main original reference is [15], and a nice review can be found in [28].5 We concentrate on the so-called Schur limit, which can be introduced most logically at the technology currently available. For a 4d N = 2 superconformal theory with flavor symmetry G F , its superconformal index is a Witten index with respect to a carefully chosen supercharge. Let us define a function on four variables s, p, q, t and g ∈ G F by I (s, p, q, t ; g) = trH S3 (−1)F s /2− j2−I3+r/4 p /2+ j1−I3−r/4q /2− j1−I3−r/4t I3+r/2g. (4.1) Here, H S3 is the Hilbert space of the theory on S3, or equivalently the space of operators; j1,2 are the spins of the spacetime SO( 4 ) SU( 2 )1 × SU( 2 )2, I3 is the spin under SU( 2 )R, r is the U( 1 )R charge normalized so that the supercharges have charge ±1. The exponent of s is Q1 −˙, (Q1 −˙)† , and the exponents of p, q, t together with g all commute with Q1 −˙. As such, it is invariant under all the exactly marginal deformations and independent of s. This defines the superconformal index that depends on three variables p, q, and t . The superconformal index with three variables is not completely understood, but the particular limit q = t is well understood. Let us set q = t and replace s by s/q in (4.1): I (s/q, p, q, q; g) = trH S3 (−1)F s /2− j2−I3+r/4 p /2+ j1−I3−r/4q −I3 g. (4.2) 4 The author should confess that he has not combinatorially proved that the formula (3.25) results from the analysis of Nambu–Goldstone multiplets. At least he checked the validity in numerous cases. 5 Note that I3 = Rthere and r = 2rthere. Also beware that the definitions of p, q, and t in their series of papers before [15] fluctuated greatly. The exponent of p is Q1+, Q1+ † , and now both Q1+ and Q1 −˙ commute with the exponent j2 − j1 + I3 of q. Therefore, the expression above is automatically independent of both s and p. This limit is often called the Schur limit. Summarizing, Fact 4.1. Given a 4d N = 2 superconformal theory Q{G} with symmetry G, the superconformal index in the Schur limit is defined by This is essentially the partition function of the theory Q on S1 × S3, and We keep the argument q to the index implicit. IQ(g) = trHQ S3 (−1)F q −I3 g. q = e−2π RS1 /RS3 . Take an N = 2 superconformal theory Q{G, H } whose flavor symmetry G can be conformally gauged. Then the gauge theory Q/G{H } is itself a superconformal theory with flavor symmetry H where the coupling constant is exactly marginal. The superconformal index of the resulting theory can then be computed in the limit where the vector fields are very weakly coupled. The result can be summarized as follows: Fact 4.3. When the theory Q has flavor symmetry SU(N ) × H and the SU(N ) can be confor mally gauged, the theory T /SU(N ) with flavor symmetry H has the superconformal index given by 1 IQ/SU(N )(h) = N ! N −1 Here we set s = p = q in (4.2), and determined the relation between q and the radii of S1, S3 by the conformal mapping. For free hypermultiplets we can easily compute this trace to obtain the following fact: Fact 4.2. A hypermultiplet containing N = 1 chiral multiplets in the representation R ⊕ R¯ of a symmetry G has the index I (g) = w n≥0 where w runs over the weights of R ⊕ R¯ , g is now regarded as a Cartan element g = z1, z2, . . . , gr of the symmetry group G, and gw := ziwi where w = w1, . . . , wr . For example, a trifundamental half-hypermultiplet of SU( 2 )3 has the index I (a, b, c) = where (4.8) K (zi )−1 = n≥0 ⎡ 1 − qn+1 N −1 i= j and z = diag(z1, . . . , zn) ∈ SU(N ) and h ∈ H . Here, in (4.7), the part iN=−11 d zi / 2π √−1zi i= j 1 − zi /z j is the standard Haar measure of the Cartan torus of the SU(N ) group manifold, and K (z)−2 are the contributions from the other components of the vector multiplets. Here the formulas are stated for simplicity for G = SU(N ), but can be easily generalized to arbitrary gauge groups. 4.2. The index of the TN theory and its cousins Now let us determine the index of the TN theory. Our strategy is always the same, and we start by considering the index of the theory SSU(N ) Cg . Almost by definition, this is the partition function of the 6d theory on S1 × S3 × Cg, with an R-symmetry background preserving an appropriate number of supersymmetry. Now we use the basic fact of the 6d theory, and reduce along S1 first. We have the N = 2 supersymmetric Yang–Mills theory with gauge group SU(N ) on S3 × C . The 5d coupling constant is 8π 2/g52 = 1/R6. Now we have a Lagrangian and can perform the localization computation to get the partition function. This computation was done in [47].6 The resulting theory is essentially the 2d SU(N ) gauge theory on Cg, but the Kaluza–Klein modes along S3 give the dressing. The final answer is the 2d q-deformed Yang–Mills, with the parameter q = exp − g52/ 4π RS3 = e−2π RS1 /RS3 . The 2d q-deformed Yang–Mills was introduced in [49–52]. This result allows us to write down the index Ig,n of the theory SSU(N ) Cg,n for the genus-g surface with n full punctures as follows: Ig,n ai = i N ai χλ ai λ N n+2g−2χλ qρ n+2g−2 0 , (4.9) where ai ∈ SU(N ) is the flavor symmetry element for the i th puncture, λ runs over the irreducible representations of SU(N ), χλ(a) is the character of the element a in the representation λ, and qρ := q(N −1)/2, q(N −3)/2, . . . , q(1−N )/2 . Here we already took the limit where the area of Cg is zero. Here, N0 and N (a) are renormalization factors the authors of [47] did not determine. N0 comes from the term of the form c C √g R = c(2 − 2g + n) in the action, and N (a) can come from the boundary term at the puncture. Both can be induced via renormalization, and the author does not know how to fix it by a direct computation. We can still determine N0 and N (a) using the compatibility under the gluing and the complete closure, as shown below. First, take two copies of TN = SSU(N ) C0,3 IR, pick two punctures, and gauge them by an SU(N ) vector multiplet. The index of the resulting theory with SU(N )a × SU(N )b × SU(N )c × SU(N )d 6 Strictly speaking, the background used in [47] is not the one that preserves N = 2 in 4d, but the one that preserves N = 1. Still, from the study of [48] it is guaranteed solely by the supersymmetry that the index in this particular case equals the N = 2 Schur-limit index. The author thanks T. Kawano for discussions. symmetry can be computed via Fact 4.3 and (4.9): 1 I (a, b; c, d) = N ! × × N −1 λ χλ(a)χλ(b)χλ(z) χλ qρ adj = d ⊗ Rd d 23/38 But the resulting theory is SSU(N ) C0,4 IR, and the index should have the form (4.9) with g = 0, n = 4. To have this, we need 1 N ! N −1 Compare this equation with the orthogonality of the characters of the irreducible representations: 1 N ! N −1 From this we see that the renormalization factors N (z) we wanted to determine are given by N (z) = K (z). This factor K (z) in the superconformal index can naturally be identified as the contribution from the conserved current multiplet in SU(N ), including μi=+1,0,−. Note, for example, that the definition (4.8) of K (z)−1 involves a product over a basis of the adjoint of SU(N ). Now, it is straightforward to obtain the superconformal index of theories with partially closed punctures. Originally, a full puncture has the contribution K (a)χλ(a) in the numerator of (4.9). Let us set μ+ = ρY σ + . The new SU( 2 )R symmetry in the infrared is the diagonal combination of the original SU( 2 )R and ρY (SU( 2 )). Denoting by b an element of the flavor symmetry GY , this means that we perform the replacement a → bqρY (σ3)/2 in χλ(a) and K (a)−1. The latter still contains the contributions from the Nambu–Goldstone modes that need to be removed: (4.10) (4.11) (4.12) (4.13) (4.14) (4.15) (4.16) where a→bqρ(σ3)/2 K (a) −−−−−−−→ KY (b) × (contrib. from the NG modes), KY (b)−1 = (1 − qn+(d+1)/2bw). d w:weights of Rd n≥0 Here, we refined the decomposition (2.23) of the adjoint under ρY (SU( 2 )) to the decomposition under ρY (SU( 2 )) × GY , where d is the d-dimensional irreducible representation of SU( 2 ) as always, and Rd is a representation of GY . In particular, when we completely close a puncture, we change a factor of K (a)χλ(a) in the numerator by K[N ]χλ qρ . This should be equivalent to having one less puncture. Therefore, we should have N0 = K[N ] = N Now we completely determined the superconformal index. Summarizing, we have: Fact 4.4. The superconformal index in the Schur limit of the 4d theory SSU(N ) Cg,n {Y1, . . . , Yn} obtained from the 6d theory on a genus-g surface with punctures of type Y1, . . . , Yn is given by i KYi ai χλ ai qρYi λ K n+2g−2χλ qρ n+2g−2 0 , where ai ∈ GYi , qρY := qρY σ 3 /2, KY (a) is defined in (4.15), and K0 := K[N ], qρ := qρ[N] . This general result was first found in [14]. The theory TY1,Y2,Y3 with a suitable choice of Y1,2,3 can be a free hypermultiplet. In these cases, Fact 4.2 together with Fact 4.4 implies an identity between an infinite sum and an infinite product. As examples we have the following equalities: Fact 4.5. The theory T 1N , 1N ,[N −1,1] is a free theory of the bifundamental hypermultiplet. We then have the equality 1 1 u,i n≥0 1 − qn+1/2 ai /bu α 1 − qn+1/2 bu/ai /α = K (a)K (b)K[N −1,1](α) K0 χλ(a)χλ(b)χλ q(N −2)/2α, · · · , q(2−N )/2α, α1−N χλ q(N −1)/2, · · · , q(1−N )/2 where λ ⎡ N −1 K[N −1,1](α)−1 = ⎣ ⎤ (1 − qd+n)⎦ 1 − qn+N/2 α±N d=1 n≥0 ± n≥0 and K (z) and K0 were defined above. When N = 2 the formula further simplifies and we have 1 − qn+1/12a±b±c± = K (a)KK(0b)K (c) d χd (aχ)dχdq(1b/)2χd (c) , (4.21) where χd (a) = ad−1 + ad−3 + · · · + a1−d is the SU( 2 ) character in the d-dimensional irreducible representation. The proof of the case N = 2 can be found in Appendix E of [15]. 24/38 (4.17) (4.18) , (4.19) (4.20) Also, the T3 theory has an enhanced E6 symmetry that is not manifest from the construction. Therefore, we have the following fact: Fact 4.6. The T3 theory has the E6 symmetry. Therefore, its superconformal index has an expansion of the form K (a)K (b)K (c) K0 = n qnχRn a1,2, b1,2, c1,2 , (4.22) where Rn is a representation of E6. Let us check this to O q2 . We have K (a) = 1 + χ8(a)q + O q2 , K0 = 1 + O q2 , (4.23) and in the sum, only λ = 1, 3, and 3¯ contribute. Then we have I (a, b, c) = 1 + q(χ8(a) + χ8(b) + χ8(c) + χ3(a)χ3(b)χ3(c) + χ3¯(a)χ3¯(b)χ3¯(c)) + O q2 . (4.24) We now see the decomposition under E6 ⊃ SU( 3 )A × SU( 3 )B × SU( 3 )C . adj of E6 = 8A ⊕ 8B ⊕ 8C ⊕ 3A ⊗ 3B ⊗ 3C ⊕ 3¯ A ⊗ 3¯ B ⊗ 3¯C (4.25) 4.3. Comments on further refinements In this review we wrote down the explicit formula (4.18) of the superconformal index only for the special case q = t . As we argued above, this choice makes the superconformal index automatically independent of p. This choice is called the Schur limit. When p = 0 with q and t generic, we still have an explicit formula generalizing (4.18), obtained in [15]. One crucial change is to replace the characters χλ(a) by the Macdonald polynomials Pλq,t (a), that now depend on q and t . An interesting subcase of the Macdonald limit is to take p = q = 0. It is then conventional to use the variable τ = t 1/2. The Macdonald polynomials reduce to the Hall–Littlewood polynomials Hλτ (a), and therefore this limit is called the Hall–Littlewood limit. This limit is particularly useful to study the Higgs branch of the theory, since it is known that, when the genus is 0, the Hall–Littlewood limit of the superconformal index agrees with the Hilbert series of the Higgs branch. The superconformal index with three general parameters p, q, t can also be written in a form similar to (4.18) by replacing the characters χλ(a) by suitable functions ψ p,q,t (a) [16], but the functions λ ψ p,q,t (a) are not well understood; see, e.g., [53] for G = SU( 2 ). λ The superconformal index in the Schur limit is also important from another point of view. In [54], it was shown that a 2d chiral algebra can be extracted from any 4d N = 2 superconformal theory by restricting operators to lie on a 2d plane in the 4d space, and that the partition function of the vacuum module of this 2d chiral algebra equals the index in the Schur limit. The Schur index of the TN theory was further studied from this point of view in [54,55]. The indices of the theories SG Cg,n for G = Dn, En can of course be studied similarly; for explicit formulas, see [56,57]. Also, the TY1,Y2,Y3 theory for suitable choices of Y1,2,3 is a higher rank version of En theories of Minahan and Nemeschansky and their superconformal indices are studied in detail in, e.g., [58–60]. 25/38 5. Moduli spaces of supersymmetric vacua and chiral ring relations 5.1. Generalities on the moduli spaces of supersymmetric vacua A 4d N = 2 superconformal theory has a moduli space of supersymmetric vacua. The part of the moduli space where SU( 2 )R is unbroken is called the Coulomb branch, whereas the Higgs branch is where U( 1 )R is unbroken. The other parts are called the mixed branch. The chiral primary operators, in the N = 1 sense, that parametrize the Coulomb/Higgs branch are called the Coulomb/Higgs branch operators. Their scaling dimensions are fixed by the R-charges: Fact 5.1. The scaling dimension Higgs branch operator is = 2I3. of a Coulomb branch operator is = r/2, and that of a Higgs branch operators have intricate chiral ring relations, some of which will be discussed below. As for the Coulomb branch operators, a Lagrangian N = 2 gauge theory clearly does not have any relations among them, since it is a classic mathematical theorem that the gauge-invariant polynomials constructed from an adjoint operator are free of relations. For example, for SU(N ), they are generated by trφk for k = 2, . . . , N . The Coulomb branches of the theories SG Cg,n {Y1, . . . , Yn} have been studied in detail, and no chiral ring relations have been found so far. Generalizing, we have: Fact 5.2. The Coulomb branch operators are free of chiral ring relations. Assuming this, we have [46]: Fact 5.3. The central charge nv and the spectrum of the Coulomb branch operators are related as follows: u where u runs over the generators of the Coulomb branch operators. nv = (2 (u) − 1), (5.1) Now, take a theory Q{G} whose G flavor symmetry can be conformally gauged. Then the Coulomb branch and the Higgs branch of the gauge theory Q/G are related in a simple manner to those of the original theory Q: Fact 5.4. The Coulomb branch operators of Q/G consist of those of the theory Q plus the gauge-invariant polynomials of the adjoint scalar φ in the N = 2 gauge multiplet. When G = SU(N ), those polynomials are trφk , k = 2, . . . , N . In particular, dimC Coulomb(Q/G) = dimC Coulomb(Q) + r, (5.2) where r is the rank of G. Fact 5.5. The Higgs branch operators of Q/G can be obtained by taking the Higgs branch operators of the theory T , setting μ+ = 0 where μ+ is the adjoint chiral operator in the G current 26/38 multiplet, and keeping only the G-invariant part. As manifolds, this operation is often written as Higgs(Q/G) = Higgs(Q)///G (5.3) and called the hyperkähler quotient construction, introduced in [61]. In particular, the dimension of the Higgs branch of Q/G is given by dimH Higgs(Q/G) = dimH Higgs(Q) − (dim G − dim H ), (5.4) where H is the unbroken gauge group at the generic point of the Higgs branch. Also, given a theory Q{G} with flavor symmetry G = SU(N ), we can partially close it by setting μ+ = ρY σ + and removing the Nambu–Goldstone modes, which are of the form y = ρY σ + , x where x is an adjoint element of SU(N ). Those y that are not in this form can be singled out by imposing the constraint ρY (σ −), y = 0. Summarizing [62]: Fact 5.6. The Higgs branch of the Q{Y } theory obtained from the theory Q by a partial closure is defined by μ+ ∈ SY , SY = ρY σ + + y ρY σ − , y = 0 . The subspace SY is called the Slodowy slice at ρY σ + . In particular, dimH Higgs(Q{Y }) = dimH Higgs(Q) − 21 dimC OY /2. (5.5) (5.6) We do not yet have a good general way to understand the Coulomb branch of the partially closed theory Q{Y }, when Q is not a class S theory of type A. It would be desirable to have a method to understand this problem that applies to all 4d N = 2 theories. 5.2. The moduli space of the TN theory Let us study the moduli space of the TN theory. Again, the strategy is the same: we first consider the theory SSU(N ) Cg,0 for the genus-g surface without punctures. Then we decompose it into copies of the TN theory and the contributions from the vector multiplets. To use the Lagrangian formalism, it is useful to compactify the 4d theory further on S1. The resulting 3d theory is the 5d N = 2 theory with gauge group SU(N ) on a genus-g surface C . Let us denote by φ1,2,3,4,5 the five adjoint fields in 5d. Due to the R-symmetry background, C = φ1 + i φ2 transforms as a one-form on C with U( 1 )R charge 2, while φ3,4,5 is an SU( 2 )R triplet scalar on C . To make the 3d N = 2 structure manifest, we combine φ3,4 into H = φ3 + i φ4. The other complex scalar in the hypermultiplet is a combination of φ5 and the scalar that is dual to the gauge field in 3d. The supersymmetric vacua correspond to the case when C and H commute, are holomorphic on C , considered up to complexified gauge transformations. The Coulomb branch corresponds to the situation C = 0 while H = 0, while the Higgs branch is the case where C = 0 and H = 0. The Coulomb branch of the 3d theory is described by the so-called Hitchin system on C , and a nice review can be found in [27]. This is a hyperkähler manifold whose complex dimension is twice that of the Coulomb branch in 4d. The 4d Coulomb branch operators are encoded in terms of tr C (z)d , which is a holomorphic d-differential on C , for d = 2, . . . , N . There are (2d − 1)(g − 1) linearly 27/38 independent holomorphic d-differentials. As tr C (z)d has U( 1 )R charge 2d, we find that there are (2d − 1)(g − 1) Coulomb branch operators of scaling dimension d. As this theory is composed of 2(g − 1) copies of the TN theory together with 3(g − 1) copies of the SU(N ) N = 2 vector multiplet, one finds: Fact 5.7. The TN theory has d − 2 Coulomb branch operators of scaling dimension d, for each d = 3, 4, . . . , N . We can now combine this fact and Fact 5.3 to derive nv of the TN theory. This nicely agrees with Fact 3.3. Note also that we have now deduced that the TN theory does not have any Coulomb branch operator of scaling dimension 2, U( 1 )R charge 4. This means that the TN theory does not have any exactly marginal deformations.7 More precisely, the Seiberg–Witten curve of the 4d theory SSU(N ) Cg,0 is given by det λ − C (z) = 0, (5.7) where λ is the Seiberg–Witten one-form. Note that z is the coordinate of C and λ can be thought of as the coordinate along the cotangent direction of T ∗C . Then the equation (5.7) determines an N -fold cover of the base C embedded in T ∗C . As for the Higgs branch, we just give vevs to H . They are zero-forms on C , so there are just r = (N − 1)-dimensional Higgs branch, where we can put H to a diagonal form. Note that this is independent of the genus g of the curve. To add n full punctures, we consider the theory without punctures as the theory with n completely closed punctures. When we completely close a full puncture, we lose the Higgs branch dimension by 21 dimC O[N ] = 21 (d − r ) = N 2 2− N . (5.8) Then, the dimension of the Higgs branch of the theory with n punctures is n(N 2 − N )/2 + (N − 1). Specializing to the case g = 0, n = 3, we find that the dimension of the Higgs branch of the TN theory is 3(d − r )/2 + r . Note that this agrees with nh − nv of the TN theory, as can be checked using Fact 3.3. This means that at the generic point on the Higgs branch of the TN theory, no free U( 1 ) vector multiplet remains, because of the following analysis. Recall the anomaly polynomial of the TN theory, which contains a term of the form (nv − nh)c1 p1(T X ). Giving a Higgs branch vev does not break U( 1 )R symmetry, and therefore this term should be reproduced on a generic point on the Higgs branch. On a generic point, we just have free hypermultiplets whose number is given by the dimension of the Higgs branch, together with free U( 1 ) vector multiplets. That nv − nh agrees with the dimension of the Higgs branch then means that there is no free U( 1 ) vector multiplet. Now, take the theories SSU(N ) C0,n and SSU(N ) C0,n , and connect them to form SSU(N ) C0,n+n −2 . By comparing the dimension of the Higgs branch before and after connecting them, we learn that the G gauge symmetry is completely broken. 7 The TN theory comes from a three-punctured sphere, which does not have any complex structure deformation. This fact alone does not guarantee that it does not have any exactly marginal deformations. Indeed, many examples are now known where a 4d theory obtained from a three-punctured sphere has exactly marginal deformations; see [20,21]. Next, take a theory SSU(N ) Cg,n with n ≥ 2 punctures. Pick two punctures and connect them, to form the theory SSU(N ) Cg+1,n−2 . Again, by comparing the dimension of the Higgs branch before and after connecting the punctures we learn that the G gauge symmetry is broken to a subgroup of rank r = (N − 1). This is in fact the Cartan subgroup U( 1 )r of G. Repeating the procedure, we find that on the generic point on the Higgs branch of the theory SSU(N ) Cg,n , we have U( 1 )rg vector multiplets. This fact, when n = 0, can be checked by another method. Without punctures, in 3d, we are giving awigtehnier=ic 1d,ia.g..o,nNalizwaibthle veiv toC(i) =H .0, wCessheoeutlhdaatlesoacbhe dC(iia)giosnaalo.nCea-lfloinrmg eaancdhgdiivaegsorniasel etontrgyU(C(1i)) vector multiplets in 4d. In total there are (N − 1)g U( 1 ) multiplets in 4d, as we already found from a slightly different perspective. Summarizing, we have: Fact 5.8. The Higgs branch of the TG theory has dimension 3(d − r )/2 + r . On a generic point on the Higgs branch, there remain no free vector multiplets. The action of the flavor symmetry G3 is such that when a diagonal G subgroup of G2 is gauged by a G gauge multiplet, the Cartan subgroup U( 1 )r remains unbroken. 5.3. Chiral ring relations of the TN theory Now let us discuss the chiral ring relations. The Coulomb branch operators do not have any nontrivial relation, so let us just discuss the Higgs branch chiral ring relations. They have been gradually being uncovered [9–12,63], but we still do not have a complete understanding. The single most important one is: Fact 5.9. The operators μ+A,B,C in the adjoint of SU(N )A,B,C , in the SU(N )3 flavor symmetry multiplet, satisfy tr μ+A k = tr μ+B k = tr μC+ k (5.9) for k = 2, . . . , N , and therefore for arbitrary k. For this reason we often drop the subscript A, B, C in tr(μ+)k . This was first understood via dualities involving Lagrangian gauge theories in [9]. Here we will use a version of the argument given in [63] that is applicable to the TG theory for arbitrary G. We start from a three-punctured sphere with finite nonzero area, where most of the area is concentrated at the tubes around the punctures; see Fig. 5. We can reduce this theory along the S1 around three tubes. The result is the TG theory coupled to three segments of 5d N = 2 theory. When we take the strictly 4d limit, this setup just goes back to the TN theory. One nice thing about this modification is that the operators μ+A,B,C are directly visible. Indeed, three scalars φ3,4,5 of a segment transform as a triplet under SU( 2 )R, and three G flavor symmetries act as gauge transformations at the boundaries of three segments. The μ+A,B,C fields for the G A,B,C flavor symmetry can then be identified as the boundary values of H = φ3 + i φ4 at the ends of the segments. Now, let us compactify the entire setup further on another S1. Then we can compare with the compactification of the 5d N = 2 super Yang–Mills on a Riemann surface we used in the last subsection. Comparing the two descriptions involves 3d mirror symmetry as detailed in [64], but on a generic 29/38 TN point on the Higgs branch where the dynamics are Abelian we can just identify H used there and here. In particular, in a supersymmetric vacuum configuration, H should be holomorphic, but H is a section of a trivial bundle, and therefore it is a constant, up to complexified gauge transformations. Therefore, three μ+ fields should be conjugate to each other, on a generic point on the Higgs branch, and we find tr μ+A k = tr μ+B k = tr μC+ k for arbitrary k. Note that this argument breaks down when μ+ is non-generic due to the subtlety in the 3d mirror operation. For example, when we perform the partial closure, we often take μ+A = ρY σ + = 0 while trying to keep the other two vevs unchanged, μ+B = μC+ = 0. Let us move on to the analysis of Higgs branch operators other than μ+ in the flavor symmetry multiplet. The analysis is only applicable to the TN theory and not to the TG theory of general type. From the superconformal index in the Schur limit, it is easy to isolate the operators with lowest powers of q in each N -ality of the SU(N ) flavor symmetry, i.e. the charge under ZN ⊂ SU(N ). As the K (a) factors only contain representations with zero N -ality, they come from the numerator χλ(a)χλ(b)χλ(c) with smallest possible λ for each N -ality, and its power in q is determined by the denominator χλ qρ . In the sector with N -ality k, the smallest possible λ corresponds to the kth antisymmetric tensor representation of SU(N ) we denote by λ = ∧k . As χ∧k qρ = q−k(N −k)/2 (1 + O(q)), we see that the leading contribution to the N -ality k to the superconformal index is trH S3 ,N -ality k (−1)F q −I3 abc = qk(N −k)/2χ∧k (a)χ∧k (b)χ∧k (c) + higher. (5.10) By studying the superconformal index, we can check that this contribution indeed comes from a scalar operator with = 2I3 = k(N − k), transforming in ∧kA ⊗ ∧kB ⊗ ∧Ck . Summarizing, Fact 5.10. The TN theory has Higgs branch operators ◦ Qaiu with dimension 1(N − 1), ◦ Q[ab][i j][uv] with dimension 2(N − 2), . . . , ◦ Q[a1···ak][i···ik][u1···uk] with dimension k(N − k), . . . , ◦ Q[a1···aN−1][i1···iN−1][u1···uN−1] with dimension (N − 1)1, where a, i , u are the indices for SU(N )A, SU(N )B , SU (N )C , respectively. When k > N /2, it is often more convenient to raise the indices using epsilon symbols. For example, the last operator would become Q˜ aiu. In particular, for N = 2, we just have dimension 1 operators Qaiu, and for N = 3, we just have dimension 2 operators Qaiu and Q˜ aiu. When N = 3, the T3 theory is the E6 Minahan–Nemeschansky theory, for which the structure of the Higgs branch is known from different means. The fact above can then be checked [65]. 30/38 The Q operators introduced above, together with the operators μ+A,B,C , are believed to generate the Higgs branch chiral ring. To study the chiral ring relations, we first note that on a generic point on the Higgs branch, we can use flavor symmetry rotations to set μ+A = μ+B = μC+ = diag(μ1, μ2, . . . , μN ), μi = 0. When we gauge a diagonal subgroup SU(N ) of any two of SU(N )A × SU(N )B × SU(N )C , the U( 1 )N −1 Cartan subgroup should remain unbroken, as we saw in Fact 5.8. This means that the operator Q[a1···ak][i1···ik][u1···uk] can be nonzero only when the multi-indices a1 · · · ak , i1 · · · ik , and u1 · · · uk are the same up to the antisymmetry: Q[a1···ak][i1···ik][u1···uk] = q[a1···ak]δ[a1···ak],[i1···ik],[u1···uk], where we do not sum over the indices. This is consistent with a chiral ring relation we know from the superconformal index, that we describe now. Looking at the term of order q(N +1)/2 in the superconformal index, we see that there Qaiv μC+ vu. From the symmetry permuting three SU(N )s, we find that the relations are issiojnusNto+ne1H.8igTghsisbrmanecahnsoptheartattohrertreanarsefotrwmoinlginienarthreeltartiifounnsdaammeonntgal Q∧bAiu⊗μ∧+AB ab⊗, Q∧aCjuwμith+B dijim,aennd Qbiu μ+A ab = Qaju μ+B ij = Qaiv μC+ vu. On a generic point on the Higgs branch where we have (5.11), these relations (5.13) mean that μa Qabc = μb Qabc = μc Qabc. Therefore Qabc can be nonzero only when a = b = c. Next, consider the two operators Q[a[i(u Qb] j]v) and Q[ab][i j][w(u] μ+ w C v). They both have scaling dimension 2N − 2 and transform in ∧A ⊗ ∧B ⊗ SymC2 . From the superconformal index, we can 2 2 check that there is only one such Higgs branch operator, and therefore Q[a[i(u Qb] j]v) = Q[ab][i j][w(u] μC+ vw). qaqb = q[ab] μa − μb , qa1 · · · qak = q[a1···ak] μai − μa j , i< j q1 · · · qN = μi − μ j . 1≤i< j≤N On a generic point on the Higgs branch, we then find where we use this relation to fix the relative normalizations of Qaiu and Q[ab][i j][uv]. From the consideration of the scaling dimensions and the remaining SN Weyl group action, it is natural to guess the general relation (5.11) (5.12) (5.13) (5.14) (5.15) (5.16) (5.17) fixing the normalization of Q[a1···ak][i1···ik][u1···uk]. In particular, we have This fixes the normalization of Qaiu itself. Then we have (N − 1) complex degrees of freedom in μ1, . . . , μN due to μi = 0 and (N − 1) complex degrees of freedom in q1, . . . , qN because the product q1 · · · qN is fixed. Thus we find 8 The coefficient of q(N+1)/2 of the Schur limit index is 2. One is a contribution from the descendant of Qaiu itself, and there is another that is a Higgs branch operator. The structure is clearer if we use the Hall–Littlewood limit instead, for which the descendants do not contribute. N − 1 hypermultiplet degrees of freedom in μi and qi . In addition, to fix μ+A,B,C to the diag onal form, we have used 3(N 2 − N )/2 hypermultiplet degrees of freedom. In total, there are 3(N 2 − N )/2 + (N − 1) hypermultiplet degrees of freedom, matching with the dimension of the Higgs branch of the TN theory. The non-generic points on the Higgs branch are characterized by the vanishing of the discriminant of μ+A,B,C , and its square root is given by the right-hand side of (5.17). From these analyses, it is very likely that the μ operators for the three SU(N ) flavor symmetries and the Q operators generate all the Higgs branch operators. Summarizing, Fact 5.11. The operators Q[a1···ak][i1···ik][u1···uk] together with the operators μ+A,B,C generate the Higgs branch chiral ring. Some of the relations involving the Q operators are given in (5.13) and (5.14). More generally, a necessary and sufficient condition for a candidate chiral ring relation is that it should be satisfied on generic points on the moduli space, i.e. when we substitute (5.11), (5.12), (5.16), and (5.17) into the relation. The explicit forms of the chiral ring relations for the TN theory, known in March 2015, can be found in Sect. 2 of [11] and in the Appendix of [12]. A different method to obtain the chiral ring relation by studying the 2d chiral algebra associated with the theory was pursued in [55]. 5.4. The moduli space of the partially closed theories As a final topic let us discuss the moduli space of the partially closed theories. As for the Higgs branch, we just apply the general method of Fact 5.6 to the Higgs branch of the TN theory. We do not have much to say about the detailed structure of the chiral ring relations, but at least the dimension of the Higgs branch is easy to determine: we already computed the dimension for the theory SSU(N ) Cg,n where all punctures are full. Then we perform the partial closures. We find: Fact 5.12. The 4d theory SSU(N ) Cg,n {Y1, . . . , Yn} obtained by putting the 6d theory on a genus-g surface with n punctures, labeled by Y1, . . . , Yn, has a Higgs branch of dimension dimH Higgs SSU(N ) Cg,n Y1, . . . , Yn = (r − 1) + 1 i 2 d − r − dimC OYi . (5.18) For N = 2, n = 0, and g arbitrary, we see that the dimension is just 1. The precise form of the Higgs branch was determined in [66] to be the asymptotically locally Euclidean space of type Dg+1. In order to study the Coulomb branch, it is useful to revisit Fact 5.7 from a slightly different point of view. There, we obtained the dimension of the Coulomb branch of the TN theory by first counting the dimension of the Coulomb branch of the theory for a genus-g surface without any puncture, and then decomposing them into the contributions from the TN theory and from the tubes. Combining them again, we find that the number of Coulomb branch operators of scaling dimension d of a theory for a genus-g surface with n full punctures is (5.19) The contribution proportional to (g − 1) counts the dimension of the space of holomorphic d-differential on a genus-g surface, describing the degrees of freedom in tr C (z)d . The contribution 32/38 ed z C (z) = z − z0 + regular, 0d z C (z) = z − z0 + regular, eY t d z C (z) = z − z0 + regular, proportional to n can be accounted for by allowing tr C (z)d to have a pole at each full puncture, of order d − 1. This can be achieved if C (z) itself has a pole of the form where z0 is the coordinate of a puncture, and e is a generic nilpotent element. Indeed, considering tr C (z)d , we see that the term proportional to (z − z0)−d drops out because e is nilpotent, while the lower-order terms are generically nonzero. Note that the full puncture has the type 1N , whereas this nilpotent element e has the type [N ], which is the transpose of 1N . In the other extreme, if a puncture at z = z0 has the type [N ], it is equivalent to having no puncture at all, therefore the local form of the field C (z) is to make no changes to the system. Now the residue 0 is a nilpotent element of type 1N , and is given by the transpose of [N ]. In general, when the puncture at z = z0 has the type Y , it is known that the field C (z) has the form where the residue eY t is a nilpotent element e ∈ OY t of type Y t . This can be argued in many ways, but one goes as follows. Consider partially closing a full puncture to the type Y = [n1, n2, n3, . . . , nk ], by setting μ+ = JY . This is compatible with the mass deformation associated with the original SU(N ) symmetry of the full puncture given by mY = diag(m1, . . . , m1, m2, . . . , m2, . . . , mk , . . . , mk ). n1 n2 nk (5.20) (5.21) (5.22) (5.23) (5.24) Under this deformation, C (z) should have a residue at the puncture of the form MY d z C (z) = z − z0 + regular, where MY is a matrix whose eigenvalues agree with mY . This is due to the following: the Seiberg– Witten curve is still given by (5.7), and the mass terms are given by the residues of λ. At z = z0, the residues of λ are clearly given by the eigenvalues of MY , and this should be identified with mY . Up to the action of SU(N )C , we can always write MY ∼ mY + e. Here, A ∼ B means two matrices are conjugate, and e is a nilpotent matrix within the Lie algebra of GY , the subgroup of SU(N ) left unbroken by the mass deformation mY . This is just the standard Jordan decomposition of a matrix into the sum of a diagonal matrix plus an upper-triangular matrix. Within GY , the vev μ+ = JY is the maximal possible one, i.e. we are completely closing the puncture within GY . Therefore, the residue of C (z) at z = z0, when restricted to GY , should be zero, following the discussion around (5.21). So we have e = 0 and MY ∼ mY . Now we turn off the eigenvalues mi of mY to zero. Under this process, MY does not necessarily tend to zero, but rather tends to a nilpotent matrix conjugate to JY t , as explained in detail e.g. in [67]. As an example, take 33/38 Y = 12 . Then MY ∼ diag(m, −m). So take a one-parameter family of such MY given by MY = MY → m 0 0 1 0 0 1 −m = J[2]. det(λ − C (z)) = 0, ei d z C (z) = z − zi + regular, and take m → 0. We find Summarizing, Fact 5.13. The Seiberg–Witten curve of the 4d theory SSU(N ) Cg,n {Y1, . . . , Yn} for the genus-g surface with n punctures of type Y1, . . . , Yn is given by where C (z) is a meromorphic differential on Cg,n, in the adjoint of SU(N )C, such that it has a pole at the i th puncture at z = zi of the form where eYit is a nilpotent element of type Yit , i.e. ei ∈ OYit . Given this, it is easy to count the dimension of the Coulomb branch. We can check that tr C (z)d at z = zi has a pole of order pd (Yi ), where with the sequence νd defined by where Y t = [s1, s2, . . .]. Then: pd (Y ) = d − νd (Y ), (ν1, ν2, . . .) = (1, . . . , 1, 2, . . . , 2, . . .), s1 s2 Fact 5.14. The number of Coulomb branch operators of dimension d of the theory SSU(N ) Cg,n {Y1, . . . , Yn} for genus g and n punctures of type Y1, . . . , Yn is (2d − 1)(g − 1) + i pd (Yi ). For example, take the theory T 1N , 1N ,[N −1,1]. We have p 1N = (1, 2, 3, . . . , N − 1), p([N − 1, 1]) = (1, 1, 1, . . . , 1), (5.32) and therefore the number of Coulomb branch operators of this theory of scaling dimension d is zero for all d = 2, . . . , N . This is as it should be, since this theory is a theory of free bifundamental hypermultiplets. Still, this analysis emphasizes an issue that was not clearly understood until several years ago, that a free hypermultiplet can still have a meaningful Seiberg–Witten curve (5.27) that can be used in the analysis of the BPS geodesics, etc. 34/38 (5.25) (5.26) (5.27) (5.28) (5.29) (5.30) (5.31) Lastly, let us note that these numbers pd satisfy two relations: d pd (Y ) = dimC OY t /2, (2d − 1) pd (Y ) = nv(Y ), d where nv(Y ) was introduced in (3.25). The first means that the contribution from a puncture to the total dimension of the Coulomb branch is half the local degrees of freedom from the residue. This is reasonable, since the residue at a puncture contributes to the Coulomb branch dimension of the 3d theory, and the 4d Coulomb branch has half the dimension of that. The second equation means that the local contribution to nv and the local contribution to the number of the Coulomb branch operators satisfy the sum rule in Fact 5.3. Conclusions In this article, we recalled the construction of the TN theory and its partially closed cousins, studied their flavor and conformal central charges, determined their superconformal indices in the Schur limit, and described the Coulomb and the Higgs branches of these theories. There are many topics on the TN theory the author could not cover in this review. Let us at least list those properties where some work has been done: ◦ There are works on the line operators, the surface operators, and the boundary conditions of the TN theory; see, e.g., [68–72]. See also the review [73]. ◦ There is also a 5d version [74,75] and a 3d version of the TN theory [64] and a closely related 3d theory called T [SU(N )] theory that captures the physics at a single puncture [31]. ◦ The Nekrasov partition function of the 5d TN theory can be computed by the topological string theory technique, and the way to take the 4d limit is now being analyzed in earnest [76,77]. ◦ The TN theory itself is N = 2 supersymmetric, but we can couple it to N = 1 gauge and matter multiplets and add superpotential terms, and study the strongly coupled dynamics there. The combined system can be studied from the 6d point of view, by considering the R-symmetry background on the Riemann surface that only preserves N = 1 supersymmetry in 4d [10,11,48, 78–88]. ◦ The holographic dual of the TN theory was already found in the original paper [5], whereas the holographic dual for the TDn theory was found in [89]. The probes of the holographic duals for N = 1 theories were studied, e.g., in [90,91]. Finally, we should remember that the TN theory has various non-supersymmetric correlation functions, whereas so far we have only mentioned quantities that are protected either by topology (such as the anomaly) or by supersymmetry (such as the superconformal index or the Nekrasov partition function). Hopefully, one day, using a generalization of the superconformal bootstrap of [92], we might be able to compute the complete set of correlation functions of the TN theory. But we are still far from that goal. The author hopes that we make steady progress in the next couple of years. Acknowledgements It is a pleasure for the author to thank D. Gaiotto, G. Moore, and A. Neitzke; F. Benini, B. Wecht, and D. Xie; O. Chacaltana and J. Distler; N. Mekareeya and J. Song; H. Hayashi, A. Gadde, K. Maruyoshi, W. Yan, and K. Yonekura for fruitful collaborations on the topic covered in this review. 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Yuji Tachikawa. A review of the TN theory and its cousins, Progress of Theoretical and Experimental Physics, 2015, DOI: 10.1093/ptep/ptv098