6d \( \mathcal{N}=\left(1,0\right) \) theories on T 2 and class S theories. Part I

Journal of High Energy Physics, Jul 2015

Abstract We show that the \( \mathcal{N}=\left(1,0\right) \) superconformal theory on a single M5 brane on the ALE space of type G = A n , D n , E n , when compactified on T2, becomes a class S theory of type G on a sphere with two full punctures and a simple puncture. We study this relation from various viewpoints. Along the way, we develop a new method to study the 4d SCFT arising from the T2 compactification of a class of 6d \( \mathcal{N}=\left(1,0\right) \) theories we call very Higgsable.

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6d \( \mathcal{N}=\left(1,0\right) \) theories on T 2 and class S theories. Part I

JHE = (1, 0) theories on T 2 and class S theories. 0 Kashiwa , Chiba 277-8583 , Japan 1 Bunkyo-ku, Tokyo 133-0022 , Japan 2 Kantaro Ohmori 3 Institute for the Physics and Mathematics of the Universe, University of Tokyo 4 School of Natural Sciences, Institute for Advanced Study 5 Department of Physics, Faculty of Science, University of Tokyo 6 Open Access , c The Authors 7 Princeton , NJ 08540 , United States of America We show that the N = (1, 0) superconformal theory on a single M5 brane on the ALE space of type G = An, Dn, En, when compactified on T 2, becomes a class S theory of type G on a sphere with two full punctures and a simple puncture. We study this relation from various viewpoints. Along the way, we develop a new method to study the 4d SCFT arising from the T 2 compactification of a class of 6d N = (1, 0) theories we call very Higgsable. Supersymmetric gauge theory; Duality in Gauge Field Theories; Field Theo- 1 Introduction Duality chain 2 3 Dimensions of the Coulomb branch Class S perspective Structure of the Higgs branch Seiberg-Witten curve Very Higgsable theories and the central charges Very Higgsable theories Structure of H and the central charges Properties to be recursively proved Rough structure of the proof Structure of H Central charges from measure factors General-rank E8 theories Central charges of minimal conformal matter on T 2 Conclusions and discussions In the last few years, we learned a great deal about the class S theories, i.e. the compacttherefore, is how much overlap there is between these two constructions. Main objective. As a first step in this direction, in this paper we show that a small but of type G on a sphere with two full punctures and a simple puncture. The 6d theories in question were called 6d (G, G) minimal conformal matters in [4], and the 4d class S theories can be called the generalized bifundamental theories. Using these terminologies, we can simply say that the T 2 compactification of the 6d minimal conformal matter gives generalized bifundamental theory in 4d. C2/ZN singularity is just a bifundamental hypermultiplet of SU(N )2, and the class S theory of type SU(N ) on a sphere with two full punctures and a single puncture is also a to the rank-1 E-string theory, as pointed out in [1, 4]. The class S theory of type SO(8) on a sphere with two full punctures and a single puncture was studied in [7], and it was found that it gives the E8 theory of Minahan and Nemeschansky. Therefore our objective is to show the relation in the other cases; but our analysis sheds new light even on the simplest of cases when G = SU(N ). Pieces of evidence. In the rest of the paper, we will provide other pieces of evidences: • In section 2, we follow the duality chain to show that the T 2 compactification of the 6d minimal conformal matter is a class S theory defined on a sphere with two full punctures and another puncture that cannot be directly identified with the present • In section 3, we compute and compare the dimension of the Coulomb branch both in 4d and in 6d. • In section 4, we show that the Higgs branch of the 4d generalized fundamentals, when the G2 flavor symmetry is weakly gauged, is given by the ALE space of type G. This is as expected from the 6d point of view. • In section 5, we compare the Seiberg-Witten curve of the 4d generalized bifundamental of type D and that of the 6d minimal conformal matter of type D in a certain corner of the moduli space and show the agreement. • In section 6, we develop a method to compute the 4d anomaly polynomial of the that to 6d minimal conformal matters and show that they agree with the known central charges of 4d generalized bifundamentals. We conclude with a short discussion in section 7. These sections are largely independent of each other and can be read separately. In particular, the analysis given in section 6 is quite general and applies to all 6d theories we call very Higgsable: these correspond, in the Ftheoretic language of [1, 2, 4], to theories whose configuration of curves C can be eliminated further complex structure deformation makes the theory completely infared free without turning on any tensor vevs. In other words, the theory has a completely Higgsed branch where no tensor multiplet remains. This explains our terminology very Higgsable.1 Let us first try to follow the duality chain to show that the 6d minimal conformal matter on T 2 is a class S theory on a sphere with two full punctures and a simple puncture. We will see that there is one step we can not quite follow, due to our lack of knowledge of the 6d N =(2, 0) theory. conformal matter of type G weakly coupled to G2 gauge fields in 7d. By putting it on a torus, we should have a 4d theory with G2 flavor symmetry, which is weakly coupled to G2 gauge fields in 5d. of T 2 and taking the T-dual of the other, we have Type IIB string theory on R1,3 × S1 × now take the limit to isolate the low-energy degrees of freedom and ignore the center-ofthe tension of the D3-brane becomes effectively infinite. Therefore we should have a BPS defect of codimension-2. With the class S technology currently available to us, we do not see how to directly identify this defect; let us call it X. We now take the limit where S1 is small. Then we have a localized degrees of freedom, that is the class S theory of type G on a sphere with two full punctures and a puncture S1× semi-infinite lines. Therefore we conclude that the 6d minimal conformal matter of type G, when compuncture X. At present, the most we can say just using the duality chain is that we know that the puncture X is the simple puncture when G is either SU(N ) or SO(8), and that the only statement that naturally generalizes this is that the puncture X is always the simple puncture for arbitrary G. Dimensions of the Coulomb branch In this section, we compute the dimension of the Coulomb branch both in 4d and in 6d, and show that the results indeed agree. First, we take the 6d point of view. In section 2 we followed the duality chain to map the 1The authors thank D.R. Morrison for the suggestion that led to this naming. study the Coulomb branch in 4d, let us put the theory on another SR1 of radius R and directly identify the hyperka¨hler structure of the 3d Coulomb branch. Take the T-dual of this SR1, and call it S˜11/R. Then lift the whole system back to M-theory. Here we are following the analysis of appendix A.3 of [8]. R1,2. The singularity has G gauge multiplet on its singular loci, and the M2-brane can be absorbed into an instanton configuration. We conclude that the 3d Coulomb branch of the This gives an interesting new perspective on the tensor branch of the 6d minimal conformal matter. We consider an instanton configuration on T 3 × R. By restricting CS(t). In our case, a single M5 gives a single M2 that becomes one instanton. Let us say CS(−∞) = 0, then we have CS(+∞) = 1. three edges of T 3 should commute. We take them to be in the Cartan of G. By following the duality chain, we see that they can be identified with the original Wilson lines of G2 used in the compactification. It is known that the Chern-Simons invariant of this flat gauge The quaternionic dimension of the moduli space including the center-of-mass motion theorem [9] to be dT 3,G = h∨(G) − rank(G) term is from the boundary contribution.2 Therefore, this is the dimension (plus one, due to the center-of-mass motion) of the Coulomb branch of the 4d theory we obtain by putting the 6d minimal conformal matter on T 2: dT 2,(G,G) min. conf. matter = h∨(G) − rank(G) − 1 . Let us see these degrees of freedom in more detail below. These details can be skipped in a first reading. G = SU(N ). mass motion as the tensor branch degree of freedom. That condition is satisfied by instanton configurations when the holonomies g1,2,3 are generic so that the gauge group is broken to its Cartan. Then the equation (3.1) follows from the fact that the 3d Dirac the zero modes) of flat connections is zero. By continuity, (3.1) should be valid even if we take g1,2,3 → 1, although a direct analysis of this case is complicated. singularity can split into two fractional M5-branes, and the emerging gauge group between So we want to identify these degrees of freedom in the instanton moduli space. (g1∗, g2, g3∗) that cannot be simultaneously conjugated into the Cartan; they can be chosen ∗ to be in a common Spin(7) subgroup, see appendix I of [10]. The Chern-Simons invariant is one-instanton configuration on T 3 × R: (1, 1, 1). CS(t) stays almost constant close to 0. CS(t) to jump to 1. • Again, for t0 < t < t1, the configuration remains almost constant. from the Cartan so(2N − 7). In total, we have N − 2. We can now identify the parameters t0 and t1 as the positions of the two fractional super Yang-Mills in these two descriptions are inversely proportional to each other, and therefore the groups we see are related by S-duality. For G = E6, we have the following commuting triples: value v of CS 0 commutant Gv e6 Then the one-instanton configuration can go through these commuting triples. The degrees of freedom in the instanton moduli space are now the “time” of the jump from one commutin the Cartan of Gv. In total, the equality (3.1) is reproduced if h∨(G) = possible value v of CS (1 + rank Gv) and indeed this is satisfied. We also see that this is the sequence of gauge groups when the M5-brane gets fractionated on the E6 singularity found in [4]. For G = E7, the list of the commuting triples are value v of CS commutant Gv e7 su(2) usp(6) su(2) and for G = E8, these are value v of CS commutant Gv e8 ∅ f4 ∅6 . In both cases, we can check that indeed the crucial equality (3.4) is satisfied, and the sequence of the groups are the S-dual of the ones that appear in the fractionation of the minimal conformal matter, see [4]. Actually, we can do a refined check of the above picture. Consider instanton configwith the commutant Gi (Gi+1) and Chern-Simons number vi (vi+1). The dimension of the moduli space of these configurations is given by the Atiyah-Patodi-Singer theorem as di,i+1 = h∨(G)(vi+1 − vi) − rank(Gi) + rank(Gi+1) , class in the adjoint representation. Using the above tables for the values of vi and the for adjacent commuting triples in the tables. This is interpreted as the fact that a fractional M5-brane has only the center-of-mass degrees of freedom. Here, it is interesting to note that the equality (3.4) is exactly the one that guarantees both in the infrared using the gaugino condensation and in the ultraviolet using the semiclassical quantization. For more, see e.g. [12]. Class S perspective Before moving to the class S theory side, let us recall the necessary notions of the nilpotent orbits. A nilpotent orbit for an nilpotent element e in g is the set of elements in g that are GC-conjugate to e. We denote the nilpotent orbit containing the nilpotent element e homomorphisms are classified by Young diagrams as is well-known in the class S theory of so in the following we will assume this. Note that the Spaltenstein map is order-reversing, so that Oe′ ≥ Oe if O¯e′ ⊃ Oe. The maximal orbit under this partial ordering is called the principal orbit Oprin and is equal to d(O0), the Spaltenstein dual to the zero orbit O0. The dimension of the principal dimC Oprin = dim(G) − rank(G) . The next-to-maximal orbit is called the subregular orbit Osubreg and is equal to d(Omin), where Omin is the minimal nilpotent orbit.3 The dimension of the minimal orbit is dimC Omin = 2 h∨(G) − 1 . surface of genus g with regular and untwisted punctures Xi which correspond to homo4d N =2 theory is [13] One of the important ingredients in the relationship between the theory of nilpotent orbits and the class S theory is the Spaltenstein map d, defined for any simple Lie algebra g. This is a map d : {nilpotent orbits of g} → {nilpotent orbits of g∨} , ∨ cases, Let us apply this formula to the class S theory we are considering, namely, (2,0) theory of type G on a sphere with two full punctures and a simple puncture. The full puncture and the simple puncture are defined so that the corresponding nilpotent orbits are O0 and Osubreg, respectively. Then, the Coulomb branch dimension is dclass S = dimC d(O0) + dimC d(Osubreg) − dim(G) = dimC Oprin + dimC Omin − dim(G) = h∨(G) − rank(G) − 1 , where in the last line we used (3.9) and (3.10). This result agrees with (3.2). instanton minimally into the group G. The dimension (3.10) is the same as the dimension of the oneinstanton moduli space of G minus the dimension of the center-of-mass of the instanton. As the Higgs branch should remain identical under the T 2 compactification, the 6d theory and the 4d theory should have the same Higgs branch. We will check this below, at the level of complex manifolds. It would be interesting to extend the analysis to the level of holomorphic symplectic varieties or hyperka¨hler manifolds. Type SU(N ). When the type G of the theory we consider is SU(N ), both the 6d minimal conformal matter and the 4d generalized bifundamental of type SU(N ) are just a bifundamental hypermultiplet of SU(N )2. It naively seems there is not much to see here. However, we can still have some fun in this case, as we will see momentarily. Consider a single M5 brane on the C2/ZN singularity. The 6d theory consists of the bifundamental of SU(N )2, weakly coupled to the 7d vector multiplet on the singular loci on the left and on the right of the M5 brane. The Higgs branch of the system should describe the motion of the M5-brane on the C2/ZN singularity. Therefore, we should be able to obtain C2/ZN as the Higgs branch of the weakly-gauged bifundamental. Let us BB˜ = M N . µ C = µ = µ˜ = diag(m1, . . . , mN ) , BB˜ = Y(M + mi) , B = det Q , B˜ = det Q˜ , M = QiaQ˜ia/N . Note also that the bifundamental couples to the 7d gauge field via the moment maps They satisfy an important relation tr µ k = tr µ˜k for any k. K¨ahler parameters for the resolution and µ C the complex deformation. Therefore we can det QiaQ˜ja = det Q det Q˜ = BB˜ and 0 = µ ij = QiaQ˜ja − M δij , we find This is indeed the equation of the C2/ZN singularity. More generally, when we have QiaQ˜ja ∼ diag(m1 + M, . . . , mN + M ). Therefore, we have which is again the equation of the deformed C2/ZN singularity. General type. Let us proceed to the general case. The 6d minimal conformal matter of type G, with the G2 flavor symmetry weakly gauged, should have the Higgs branch of branch should be independent under the T 2 compactification, we should be able to check this using the class S description of the 4d generalized bifundamental. The Higgs branch of the class S theory in general was studied e.g. in [14]. As discussed there, the Higgs branch of the class S theory of type G on a sphere with two full punctures and a single regular puncture of arbitrary type is described as follows. We start from the Higgs branch XG of the TG theory, i.e. the class S theory of type G on a sphere with three full punctures. The hyperka¨hler space XG has actions of G3, and correspondingly has three holomorphic moment maps µ 1, µ 2, µ 3 taking values in gC. The hyperka¨hler dimension of dimH XG = rank G + (dim G − rank G) . A puncture is specified by a homomorphism In our case we take e to be the subregular element, since we want to have a simple dimH Ye = dimH XG − dimH Osubreg = dim G + 1 dimC Osubreg = dim G − rank G − 2 . We would like to study the Higgs branch where the G2 flavor symmetry is coupled to the left (GL) and the right (GR). Therefore the Higgs branch of the combined system is Ze = Ye///(GL × GR) , where /// denotes the hyperka¨hler quotient. On a generic point of Ze, GL × GR is broken by the M5 brane. The breaking from GL × GR to Gdiag should eat dim G hypermultiplets. Then the class S theory of type G, on a sphere with two full punctures and a puncture specified by e, has the Higgs branch of the form Se := {x + e | [x, f ] = 0} ⊂ gC . Ye = µ 1−1Se Such homomorphisms up to conjugation is known to be classified by the nilpotent element where we regarded µ 1 as a map XG → gC. puncture. The dimension is then where we used (4.6) and To see this, we use the following fact: let us say the TG theory has G1 × GL × GR flavor symmetry, and let us call the respective moment map operators as µ 1, µ L and µ R. Then the Higgs branch operators of the TG theory, invariant under GL × GR are just polynomials of µ [15]. We also know that µ , µ L and µ R satisfy the crucial relation (Oh,L × Ye × Oh,R)///(GL × GR) where Oh,L and Oh,R are two copies of the orbit Oh of elements of gC conjugate to h, and parameterize the vevs of the adjoint scalars in the 7d vector multiplets on the left and the right.5 Since GL × GR is now broken to U(1)rank G, the dimension of the resulting Higgs Ze = Se ∩ N . The simple puncture corresponds to e being the subregular element, and it is a classic mathematical fact by Brieskorn and Slodowy [16, 17] that this space is the singularity manifold. Such a smooth deformation can be parameterized by a generic element h in the Cartan of gC. The Higgs branch describing the motion of the M5-brane is then tr µ 1k = tr µ kR = tr µ kL which forces µ to be nilpotent via (4.13). Therefore the image of µ 1 in gC is the variety N of nilpotent elements, and the final Higgs branch is therefore again the expected answer. and the Higgs branch is now 2 dimH Oh + dimH Ye − (2 dim G − rank G) = 1 , Se ∩ Oh . Again, it is a classic result of Brieskorn and Slodowy [16, 17] that this space precisely gives Kronheimer [18] is more familiar to string theorists. But this result of Brieskorn and Slodowy was found much earlier in the mathematics literature. More precisely, they arise as follows. To obtain supersymmetric configurations of the 7d gauge field, we have to solve Nahm’s equations on the half space x7 > 0 (x7 < 0) for GL (GR) as in [19], where x7 is the direction perpendicular to the M5-brane. The solution (at the complex structure level) is that a complex In this section, we compare the Seiberg-Witten curve of the 4d generalized bifundamental and that of the 6d minimal conformal matter on T 2 when the type is Dn. In principle we should be able to analyze the curves of arbitrary type G in a uniform fashion, but the authors have not been able to do that. theory with 2N flavors. Therefore, we should be able to reproduce the 4d curve of this theory by giving a suitable Coulomb branch vev to the 4d generalized bifundamental of There is also another limit in which we can check the curve. Instead of going to the 6d tensor branch, we can first reduce the 6d minimal conformal matter to 5d and add B-fields to the ALE space. This makes the system one D4 brane on the DN orbifold, which is given by the quiver of the form where a circle enclosing i stands for an SU(i) gauge symmetry, and the edge between two explicitness. Adding B-fields corresponds to giving mass terms to the DN × DN flavor symmetries. Thus the 4d generalized theory should also realize this quiver by the mass with 2N flavors and this D-type quiver theory, was called “a novel 5d duality” in [4], and is the type D version of the “base-fiber duality” of [20]. What we find here is that the corresponding 4d theories are both a deformation of a single class S theory, providing a 4d realization of these dualities. The curve of the generalized bifundamental. The generalized bifundamental of type DN is a class S theory of type DN on a sphere with two full punctures and a single simple puncture. Therefore, it has the following Seiberg-Witten curve 0 = λ2N + φ2(z)λ2N−2 + · · · + φ2N−2(z)λ2 + φ2N (z) From this we find that the curve is given by z i=1 1 YN(x2 − mi2) + 2c + z Y(x2 − m˜i2) = N satisfied, M2 and M4 are quadratic and quartic polynomials of mi and m˜ i such that (5.3) The USp theory. Let us next recall the curve of USp(2n) with Nf + Nf′ flavors: Nf Λ2n+2−2Nf YNf (x2 − mi2) + 2c + Λ2n+2−2Nf′ z Y(x2 − m˜i2) = x2(x2n + u2x2n−2 + u4x2n−4 + · · · + u2n) z i=1 N′ where c2 = Λ4n+4−2(Nf +Nf′ ) QiN=f1(−m2) Qi=f1(− m˜i2). The differential is λ = xdz/z. This i curve in a hyperelliptic form was first found in [22]. The form given above follows easily from the brane construction, see e.g. [23]. Setting 2n = 2N − 8, Nf = N f′ = N , the curve becomes 1 YN(x2 −mi2) + 2c + z Y(x2 − m˜i2) = Λ6x2(x2N−8 + u2x2N−10 + · · · + u2N−8) N right-hand-side of (5.4) can be neglected,7 and becomes (5.6). The identification of U6 This is consistent with the guess that this class S theory is the T 2 compactification of the minimal conformal matter of type DN . Also, we learn that the tensor branch scalar becomes U6, of scaling dimension 6, independent of N , and is the coupling constant of the The D-type quiver. This is a completely different limit than the above USp limit. Note gauge group, whose curve is well known. We start from the curve (5.4) of the class S theory, and focus on the neighborhood of 7This scaling limit is a little subtle due to the fact that our USp theory is not asymptotically free. For example, we throw away the term x2N but retain both zx2N and z−1x2N . curve is given, up to terms of O(t3), by where we have defined a little further effort, it can be checked that the residues of the double poles of ϕ2 are masses of the quiver. k′ ∼ hypermultiplets in the quiver and mi + m˜i may correspond to gauge couplings in 5d. relation of the pole coefficients at the simple puncture (5.3). This nonlinear relation is called a c-constraint in [7]. Now, rewrite the curve as (ξN + c2ξN−1 + · · · + c2N ) λ2 + 2 (µ 2ξN−1 + · · · + µ 2N )(dx)λ identified, and therefore this is a natural choice. x2N + c2x2N−2 + · · · + c2N = Y(x2 − mi2) + Y(x2 − m˜i2) , Uk′ = − 4 Uk + ck , b2N = (−1)N 1 Y mi − Y m˜i . µ 2x2N−2 + · · · + µ 2N = − Y(x2 − mi2) − Y(x2 − m˜i2) , In this section, we study the T 2 compactification of the class of 6d SCFTs that we call very Higgsable. We will determine the structure of the part of the Coulomb branch of the T 2 compactification that comes from the 6d tensor branch, and show in particular that there is a point where one has a 4d SCFT. We will also show that the central charges a, c and k of the 4d SCFT can be written as a linear combination of the coefficients of the anomaly polynomial of the 6d SCFT. Since the 6d minimal conformal matters are very Higgsable, we can apply the methods developed here to provide another check of our identification. Let us summarize the contents of this section. In section 6.1, we introduce the class of the 6d SCFTs of our interest, namely the very Higgsable theories. In section 6.2, we recursively prove that • the T 2 compactification of a very Higgsable theory gives a 4d SCFT, and • the central charges of the resulting 4d SCFT can be written as a linear combination of coefficients of the anomaly polynomial of the 6d theory. In 6.3, we compute the central charges of the minimal conformal matter on T 2 by using the relationship with the anomaly polynomial of the minimal conformal matter. We will see that the resulting central charges indeed agree with the known central charges of the class S theory involved. Very Higgsable theories Let us first define the class of 6d very Higgsable theories. In terms of the F-theoretic language of [1, 2, 4], a 6d SCFT can be characterized by the configuration C of curves on the complex two-dimensional base. We define a 6d SCFT to be very Higgsable if successive, Then a further complex structure deformation removes the singularity completely. In other words, there is a Higgs branch where the tensor multiplet degrees of freedom are completely eliminated, thus the word very Higgsable. As examples, the 6d (G, G) minimal conformal theory and the worldvolume theory of Q > 1 M5 branes probing an ALE singularity are not very Higgsable. We can also re-phrase the very Higgsable condition without referring to the F-theory construction, in the following recursive fashion: • Free hypermultiplets are very Higgsable. • An SCFT is very Higgsable if – it has a one-dimensional subspace of the tensor branch on which the low-energy degrees of freedom consist of a single tensor multiplet, one or more very Higgsable theories, possibly with a gauge multiplet G, – such that the Chern-Simons coupling SCS of the self-dual two-form field of the tensor multiplet B, and its associated Green-Schwarz term IGS in the anomaly polynomial is8 2 I4 , I4 ⊃ Tr FF2 − where the term Tr FF2 /4 is for the flavor symmetry, and the term Tr FG2 /4 is absent if there is no gauge multiplet. The condition (6.1) is a consequence of the fact that the tensor multiplet comes from a absorbed into FG. Therefore, this means that the instanton-string has charge 1 under the tensor multiplet, which is the minimal consistent value under the Dirac quantization condition. The p1(T ) etc. are the usual Pontryagin densities of the background metric. We would like to study the T 2 compactification of a very Higgsable theory. Consider the torus b = RT 2 B: In general, the quantum corrections mix this variable u with all the other Coulomb dimension-1 subspace H of the Coulomb branch parametrized by it. This is because if a gauge multiplet is present on the minimally-charged tensor branch, the 4d gauge coupling of the gauge field is infrared free, as we will prove below. Before proceeding, let us see two examples of this infrared freedom: • First, the one-dimensional tensor branch of the (Dk, Dk) minimal conformal matter therefore the system is infrared free as a 4d gauge theory. • Second, the F-theory realization of the (E6, E6) minimal conformal matter has three This still supports the SU(3) gauge group. This gauge group is now coupled to two copies of the 4d version of rank-1 E-string theory, i.e. the E8 theory of Minahan It may not be completely rigorous to write a Lagrangian like (6.1) for the self-dual 2-form B. But we will only need dimensional reduction of that Chern-Simons term under the compactification on T therefore two copies are worth 12 flavors of SU(3) fundamentals. Therefore the SU(3) gauge coupling is infrared free. Thanks to the infrared freedom of G, it is meaningful to talk about the origin of the Coulomb branch of G even quantum mechanically. This determines the subspace H. Structure of H and the central charges Properties to be recursively proved Now, we use the mathematical induction to prove the following properties of very Higgsable • The topology of H is always the same as that of the rank-1 E-string theory, namely, there are three singularities. Here, – two of them are the points where a single hypermultiplet becomes massless, and – the third of them is a point at which the non-trivial SCFT appears, with the R-charge of the Coulomb branch operator u being 12. We call the resulting 4d SCFT as T4d. • Writing the anomaly polynomial I8 of the 6d theory T6d as9 I8 ⊃ αp1(T )2 + βp1(T )c2(R) + γp2(T ) + X κi p1(T ) Tr Fi2 , the central charges a, c and flavor central charges ki of i-th flavor symmetry of the 4d theory T4d are Rough structure of the proof As the discussions will be rather intricate, here we provide the schematic structure of the inductive proof. The first step is to check the relations (6.4) for the free hypermultiplets. In addition, we can check that free vector multiplets and free tensor multiplets both satisfy the relations (6.4). The inductive step is to study the system of a minimally-charged tensor multiplet, with a very Higgsable theory. There are two subcases: i) when there is no gauge multiplet, and ii) when there is a single gauge multiplet G. The subcase i) corresponds to the appearance of an E-string, for which the structure of H was studied long time ago [27]. In the subcase ii), 9Our normalizations and notations of 6d anomaly polynomials follows those in [26]. a nontrivial 4d SCFT T4d. From this, we will show that there will be two and only two additional singularities on H, and that these are points where one massless hypermultiplet In both subcases, we see that the structure of H is the same. Once this is known, we can employ the method of [28] to determine the central charges a, c and k of T4d in terms of the 6d anomaly polynomial. This then confirms the general relation (6.4), completing the inductive process. Structure of H Now let us start the full discussion of the inductive step. We first would like to establish the singularity structure of H. When there is no gauge multiplet on the tensor branch, we have the E-string theory, for which the structure of H is known [27]. There is a point where we have a 4d E8 theory of Minahan and Nemeschansky, where the R-charge of the Coulomb branch operator u is 12 and therefore the scaling dimension is 6. This is true for higher-rank E-string theory too. Let us next consider the case with a gauge multiplet with gauge group G. Denote the very Higgsable theory on this tensor branch by S. The low energy theory on this branch consists of S, the non-abelian gauge multiplet G, and a U(1) (or tensor) multiplet containing u, and we want to show that there is a point at which they are combined into a single strongly interacting superconformal theory T . is gauged by the non-abelian gauge group. The commutant F of G in H is the flavor symmetry of the total system. The term proportional to Tr FG2 p1(T ) in the total 6d anomaly polynomial is given by IS + Itensor + Igaugino + IGS ⊃ G − which means that the one-loop beta function is positive and the G gauge coupling in the 4d theory is infrared free. This guarantees that we can isolate the subspace H as we repeatedly In addition, away from the singularities on H, we can safely introduce the exponentiThe gauge group G is anomaly free in 6d, therefore Using the inductive assumption (6.4), we see that ated complexified coupling of the 4d G gauge field, defined at an arbitrary (but sufficiently small) renormalization The Green-Schwarz coupling (6.1) in 6d gives the 4d coupling ∞, the G coupling becomes extremely weakly coupled there, but we do not know of any physics to explain it. the G gauge multiplet. The U(1)RG2 is anomalous by the amount (6.7). At the SCFT point this U(1)R symmetry should be restored and it must be anomaly free. By the anomaly interpreted as the phase of u in the small u region. This can be done by assigning the U(1)R charge R[u] = 12 to the u near u = 0. Then the total U(1)RG2 anomaly is cancelled. We now show that there are two more singularities on H and that these two points are associated with an additional massless hypermultiplet. The proof goes as follows: consider the Seiberg-Witten curve on H given by y2 = x3 + f (u)x + g(u) . This curve is for describing the effective action of the U(1) gauge field coming from the non-abelian gauge group G. Using the special coordinate on H related to the curve (6.11) via daD = where A and B are the two independent cycles of the torus, the metric on H is ds2 = Im(da∗daD) = Im da ∗ daD those multi-values of the coupling constant. For example, in Seiberg-Witten theory of a massless U(1) field, position of the Landau pole of the infrared free gauge field, and in particular it is a dimensionful parameter. There seems to be no duality transformation which sends one value of the Landau pole to another, and have multivalued coupling constant on the moduli space due to S-duality of the conformal gauge group. Such a situation indeed appears in other theories and will be discussed elsewhere. structure of the T 2 used in the compactification from 6d to 4d. Then f and g should the compactification of a free tensor multiplet. Substituting the asymptotic behavior of f and g to (6.11), (6.12) and (6.13), we obtain n = 1. R[x] = 2rR[u] , R[y] = 3rR[u] , Using (6.14), the relation (1 − r)R[u] = 2 f ∼ u4, g ∼ u5 + u6. Central charges from measure factors Before proceeding, let us very briefly recall the method of [28] to compute the central in 4d on a curved manifold with a non-trivial metric and a background gauge field for the flavor symmetry F via the twisting of the SU(2)R R-symmetry with one of the SU(2)’s monodromy invariant coordinates on the Coumlomb branch. The path integral of the twisted theory is given as follows Z = Here [du] and [dq] are the path integral measures for the massless vector multiplets and other massless multiplets on the generic point of the Coulomb branch. The A(u), B(u) and C(u) are factors induced by the non-minimal coupling of u to the non-trivial backquires that they are holomorphic. See [30] for details. On a singular point on the Coulomb branch, we can have nontrivial superconformal field theory. Then there must be an enhanced U(1)R symmetry at each of these points, although U(1)R need not be defined globally on the Coulomb moduli space. The coefficients of the anomaly of U(1)R under background fields are related to the central charges a, c, k by supersymmetry as was used to derive (6.10), the central charges a, c and k are obtained as [28] a = c = R[B] + ageneric , R[B] + cgeneric , k = R[C] + kgeneric C|G ∼ exp(2πiτG(u)) ∼ u−1 and R[C|G] = −R[u] = −12. where R[A, B, C] are the U(1)R-charges of the measure factors A(u), B(u), C(u), and (a, c, k)generic are the central charges at a generic point on the Coulomb branch. The terms proportional to R[A, B, C] are the contributions from U(1)R Nambu-Goldstone bosons near each superconformal point. For the gauge group G, what we have found in the previous subsection may be rephrased as k|G = 0, kgeneric|G = kS4d − 4h∨G = 12, G 6d anomalies. Suppose that the 4-form appearing in (6.1), now including the second Chern class c2(R) of the SU(2)R background field, is given by I4 = dc2(R) + 4 p1(T ) + Tr FF2 − The explicit value of d can be determined by the method explained in [26] but it is not important here. The contribution to the 6d anomaly polynomial from (6.22) is 21 I42 ⊃ dc2(R)p1(T ) + 4d central charges. We now would like to determine the changes in a, c, k in 4d. To do this, we use the method of [28] recalled above. Putting the theory on a curved manifold via twisting leads to the path-integral (6.17). where we used the fact that c2(R) = − 1 χ − 14 p1(T ) due to the topological twist, and 2 log(u)I4 = (u− d2 )χ u 43 (1−d) σunF the theory on a generic point of H. This is just the anomaly matching of the U(1)R anomaly (6.18) discussed above. topologically twisted theory, the I4 of (6.22) becomes 1 = (1 − d) = AχBσCn → exp iα − χ + (1 − d)σ + nF Now consider a circle S1 going once at a large value of |u|. The phase shift is given points. First, for C we get It is known that B and C are single valued functions of u [30]. Then the phase shift around the large circle should be the same as the sum of the phase shifts around the singular appears which is not charged under the non-abelian flavor group F . Therefore we can determine the change in the flavor central charges: Next, for B we get Finally, we consider A. In this case, A is not a single valued function [30]. However, the nontrivial monodromy of A is fixed by the Seiberg-Witten curve of the U(1) multiplet of u. The equation (6.16) implies that the Seiberg-Witten curve is completely the same as that of rank-1 E-string theory on T 2. Therefore, the ratio A(u)/AE (u) is single-valued, where AE (u) is the A-factor of the rank-1 E-string theory on T 2. of the E8 Minahan-Nemeschansky theory [28] or from the fact that the study of the 6d and hence [AE(u)] = 5. Thus Combining (6.30), (6.32) and (6.35) with the assumption of the induction, the proof of (6.4) A(u)/AE (u) ∼ u−(d+1)/2. where Q is the rank. a = c = 3 Q2 + kE8 = 12Q , Q − Q − kSU(2)L = 6Q2 − 5Q − 1 , General-rank E8 theories As first examples of our general analysis, let us first consider the E-string theory of general rank. When put on T 2, this is known to reduce to the general-rank version of the E8 theory of Minahan and Nemeschansky. The central charges a, c and k of these theories were found The anomaly polynomial of 6d higher-rank E-string theories was obtained in [32]. The relevant coefficients in the anomaly polynomial are 7(30Q − 1) −Q(6Q + 5) Q2 − Q − 1 − 30Q We can check that the formulas (6.4) are indeed satisfied. Central charges of minimal conformal matter on T 2 As second examples, let us consider the central charge of the 6d (G, G) minimal conformal matter on T 2. The anomaly polynomial of that theory was obtained in [26]. The relevant coefficients in the anomaly polynomial are 1 + dim(G) , 1 + dim(G) , a = c = kG = 2h∨G . Then, we compute the central charges of the class S theory of type G on a sphere with two full punctures and a simple puncture. The relevant formula [13] is a = asimple + 2afull − c = csimple + 2cfull − kG = kfull , 13 h∨G dim(G) − 31 h∨G dim(G) − where asimple and afull are the contribution from the simple and full puncture, respectively. The contributions from the punctures are given by [13] asimple = csimple = kfull = 2h∨G . afull = cfull = 4h∨G dim(G) − 2h∨G dim(G) − dim(G) + rank(G) , Substituting these equations into (6.44), (6.45) and (6.46), we obtain the same central charges as (6.43). This provides a non-trivial check both for the central charge formula in (6.4) and the duality between the minimal conformal matter on T 2 and the class S In this paper we found that the world volume theory of a single M5-brane on the tip of an a type G class S theory with a sphere accompanied by two full-punctures and a simple puncture, namely 4d generalized bifundamental, by means of T 2 compactification. We have given several evidences on this statement. We provided the matching of coulomb branch dimensions and the Higgs branch geometry, and we checked the agreement of the Seiberg-Witten curve in the case of type D in a certain corner of the moduli space, by exhibiting the “base-fiber duality” indicated by the 6d brane construction at the level of the 4d Seiberg-Witten curves. We also developed a new method to study the central charges of the T 2 compactification of a class of the 6d SCFTs that we call very Higgsable, and applied this technique to the minimal conformal matters. We again found agreement with the central charges of the class S theories. With these checks, we find that our proposed identification is well established. Let us discuss some of the future directions. Other very Higgsable theories. There are many very Higgsable theories which are neither (G, G) minimal conformal matters nor higher-rank E-string theories. For T 2 compactifications of all of those, we showed that the formula (6.4) holds. Some of these theories can be obtained by considering “fractional M5-branes” on ALE • The (E7, SO(7)) minimal conformal matter, namely a “half M5-brane” on top of E7 • the (E8, G2) minimal conformal matter which is a “third M5-brane” on E8 singularity, • and the (E8, F4) minimal conformal matter which is a “half M5-brane” on E8 singuFor the (E7, SO(7)) minimal conformal matter, we can find a candidate of the corresponding 4d theory in the list of E6 tinkertoys [33]. Conbining the method of [26] and the formula (6.4), we find the central charges of T 2 compactified (E7, SO(7)) minimal conformal a = c = kE7 = 24 , kSO(7) = 16 . Those numbers are exactly the same as the conformal central charges of E6 fixture with punctures E6(a1), 2A1 and the full puncture, where the notation of the punctures are Similarly, the candidates for the (E8, G2) and (E8, F4) minimal conformal matter might be found in E7 or E8 fixtures. But the list of E7 and E8 fixtures are not yet available. Another natural series of very Higgsable theories can be found by considering theories on M5-branes on the intersection of an end-of-the-world brane and an ALE singularity locus. In contrast to the minimal conformal matters, the theories are endpoint-trivial for all integer numbers of M 5-branes, and therefore there are infinitely many of them. It would be interesting to search 4d corresponding theories in known 4d SCFTs. Non very Higgsable theories. The worldvolume theories on multiple coincident M5branes on an ALE singularity locus, are not very Higgsable. Thus the approach of this paper cannot be directly applied and new methods need be introduced to investigate such as its base are other cases recently studied in [34]. Although the structure of the base Fn is the T 2 compactification of any 6d SCFT. Compactification with general Riemann surfaces and punctures. from studying on S-dualities between compactified theories. The T 2 compactified theories studied in this paper might be a clue to find out the tube theories if one can find an appropriate boundary conditions at the ends of the tube. The authors hope to come back to these questions in the future. KO and HS are partially supported by the Programs for Leading Graduate Schools, MEXT, Japan, via the Advanced Leading Graduate Course for Photon Science and via the Leading Graduate Course for Frontiers of Mathematical Sciences and Physics, respectively. KO is also supported by JSPS Research Fellowship for Young Scientists. YT is supported in part by JSPS Grant-in-Aid for Scientific Research No. 25870159, and in part by WPI Initiative, MEXT, Japan at IPMU, the University of Tokyo. The work of KY is supported in part by DOE Grant No. DE-SC0009988. Open Access. 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Kantaro Ohmori, Hiroyuki Shimizu, Yuji Tachikawa, Kazuya Yonekura. 6d \( \mathcal{N}=\left(1,0\right) \) theories on T 2 and class S theories. Part I, Journal of High Energy Physics, 2015, 14, DOI: 10.1007/JHEP07(2015)014