E 8 instantons on type-A ALE spaces and supersymmetric field theories

Journal of High Energy Physics, Sep 2017

Abstract We consider the 6d superconformal field theory realized on M5-branes probing the E 8 end-of-the-world brane on the deformed and resolved ℂ 2/ℤ k singularity. We give an explicit algorithm which determines, for arbitrary holonomy at infinity, the 6d quiver gauge theory on the tensor branch, the type-A class S description of the T 2 compactification, and the star-shaped quiver obtained as the mirror of the T 3 compactification.

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E 8 instantons on type-A ALE spaces and supersymmetric field theories

HJE E8 instantons on type-A Noppadol Mekareeya 0 1 2 5 Kantaro Ohmori 0 1 2 3 Yuji Tachikawa 0 1 2 4 Gabi Zafrir 0 1 2 4 0 Kashiwa , Chiba 277-8583 , Japan 1 Princeton , NJ 08540 , U.S.A 2 Piazza della Scienza 3 , I-20126 Milano , Italy 3 School of Natural Sciences, Institute for Advanced Study 4 Kavli Institute for the Physics and Mathematics of the Universe, University of Tokyo 5 Dipartimento di Fisica, Universita` di Milano-Bicocca , and INFN, sezione di Milano-Bicocca We consider the 6d superconformal field theory realized on M5-branes probing the E8 end-of-the-world brane on the deformed and resolved C2/Zk singularity. We give an explicit algorithm which determines, for arbitrary holonomy at infinity, the 6d quiver gauge theory on the tensor branch, the type-A class S description of the T 2 compactification, and the star-shaped quiver obtained as the mirror of the T 3 compactification. Supersymmetry and Duality; Field Theories in Higher Dimensions; Field - 3.1 3.2 3.3 4.1 4.2 4.3 5.1 5.2 5.3 5.4 5.5 5.6 1 Introduction and summary 2 Geometric preliminaries 3 Six-dimensional description 2.1 Topological data of the instanton configuration 2.2 Dimension of the instanton moduli space The general structure of the quiver The algorithm A subtlety concerning the 6d θ angle 3.4 Anomalies and the inflow 4 Lower dimensional incarnations 5 Examples Five-dimensional brane-web description Four-dimensional class S description Three-dimensional star-shaped quiver description Enhanced flavor symmetries from 3d quivers Theories differing by the 6d θ angle Massive E-string theories Higgsing the SU(k) flavour symmetry One of the many surprises during the second superstring revolution was the realization that the construction of SU(N ) instantons on R4 by Aityah-Drinfeld-Hitchin-Manin [1] and on asymptotically locally Euclidean (ALE) spaces by Kronheimer-Nakajima [2, 3] have a physical realization in terms of Dp-branes probing D(p+4)-brane on R4 [4] and/or ALE spaces [5]. There, we have a gauge theory with eight supercharges on Dp-branes such that its Higgs branch is given by the corresponding instanton moduli spaces: the equations of the ADHM and Kronheimer-Nakajima construction are the F-term and D-term conditions of the supersymmetric gauge theory. As a variation of this construction, we can consider M5-branes probing the E8 endof-the-world brane of the M-theory, either on R4 or on ALE spaces. The low-energy worldvolume theory on these M5-branes is a 6d N = 1 supersymmetric theory whose Higgs – 1 – branch is intimately related to the E8 instanton moduli spaces. These theories were studied already in the heyday of the second revolution, see e.g. [6], We did not, however, have many methods to understand the properties of these theories back then, since these theories are intrinsically strongly-coupled. Therefore we could not say anything new regarding the mathematics of the E8 instanton moduli spaces on R4 or ALE spaces, for which no constructions analogous to ADHM or Kronheimer-Nakajima are known even today. The situation has changed drastically since then, thanks to our improved understanding of strongly-coupled supersymmetric theories. Among others, we can count the class S construction in four dimensions initiated by [7], the determination of the chiral ring of the Coulomb branch in three dimensions starting with [8], and a new method to study N = 1 theories in six dimensions pioneered by [9]. Combining these developments, we believe there might be a chance that the physics might shed new lights on the mathematics of the structure of the E8 instanton moduli spaces on ALE spaces. The main target of our study in this paper is the 6d N = 1 theory on M5-branes probing the E8 9-brane on the A-type ALE singularity C2/Zk. Such a system can be labeled by the M5-brane charge Q and the asymptotic holonomy ρ : Zk → E8. For some simple choices of ρ, the structure of the 6d theory on generic points on its tensor branch was already determined in [10, 11], which was further extended in [12–14]. space C^2/Zk is given by Our first aim is to determine the tensor branch structure for an arbitrary choice of the asymptotic holonomy ρ. We give a complete algorithm determining the gauge group and the matter content in terms of ρ. Along the way, we encounter a subtle feature that there are two distinct ways to gauge su(2n + 8) symmetry of so(4n + 16) flavor symmetry of an usp(2n) gauge theory with Nf = 2n+8 flavors, due to the fact that the outer automorphism of so(4n + 16) is not a symmetry of the latter gauge theory. The Higgs branch MQ,ρ of our theory TQ6,dρ is not directly the instanton moduli space. In particular, MQ,ρ has an action of SU(k), which we do not expect for the instanton moduli space. Rather, by a small generalization of the argument in [15], we see that the E8 instanton moduli space MiQn,sρt,ξ of charge Q and asymptotic holonomy ρ on the ALE MiQn,sρt,ξ = (MQ,ρ × Oξ)///SU(k) (1.1) where ξ = (ξC, ξR) ∈ su(k) ⊗ (C ⊕ R) is an element in the Cartan of su(k) tensored by R3 specifying the hyperka¨hler deformation parameter of the ALE space, Oξ is the orbit of ξC in su(k)C with the hyperka¨hler metric specified by ξR as in [16], and the symbol /// denotes the hyperka¨hler quotient construction. This means that the space MQ,ρ knows the structure of the instanton moduli on the ALE space for arbitrary deformation parameter ξ. The existence of such a generating space was conjectured by one of the authors in [17], based on a study of SO(8) instantons on the ALE spaces. We then study the 4d theory which arises from the T 2 compactification of the 6d theory as in [15]. We find that they always correspond to a class S theory of type A, given by a sphere with three punctures. The 3d mirror of its S1 compactification is a star-shaped quiver, whose structure can be deduced from the class S description by the methods of [18]. We find that they have the form of an over-extended E8 quiver. In 3d, – 2 – the relation (1.1) can be physically implemented by realizing Oξ as the Coulomb branch of the T [SU(k)] theory. Using this, we will find that MiQn,sρt,ξ is the Higgs branch of an affine E8 quiver where ξ is now the mass parameter of an SU(k) flavor symmetry. For ξ = 0 this was already conjectured by mathematicians [19, 20] and by physicists [21, 22]. Organization of the paper. The rest of the paper is organized as follows. We start by recalling the geometric data characterizing our system in section 2. Then in section 3, we provide the algorithm determining the 6d quiver theory in terms of the asymptotic holonomy. In section 4, we discuss its dimensional reduction to 5d, 4d and 3d in turn. In 5d and 4d, we translate the Kac labels to the three Young diagrams characterizing the brane web and the class S description. In 3d, we give the star-shaped quiver. Finally in section 5, we provide many examples illustrating our discussions. Accompanying Mathematica file. The paper comes with a Mathematica file which implements the algorithm to produce the 6d quiver given the asymptotic E8 holonomy. In addition, it allows the user to determine the 4d class S theory, and compute the anomalies from three different methods, namely the 6d field theory, the M-theoretic inflow, and the 4d class S technique. Summary of notations. • The asymptotic holonomy ρ : Zk → E8 is given by an element w ∈ e8 in the Cartan subalgebra, or equivalently in terms of the Kac label n := , a set of non-negative integers arranged on the affine E8 Dynkin diagram. For more details, see section 2.1. • We have closely related quantities Ninst, N3, NS, N6, and Q, which are all essentially the number of M5-branes or equivalently the instanton charge on the ALE space. They all increase by one when we add one M5-brane to the system. Their constant parts are however different. We could have used just one out of them, but any choice would make at least one of the formulas quite unseemly. We therefore decided to keep them and provide a summary here. – The integer Ninst is defined in terms of the instanton number as Z C^2/Γ tr F ∧ F ∝ Ninst − parameterize the ranks of groups in the 3d quiver, see (4.3). – Another integer NS defined by NS = N3 + n1 + · · · + n6 is useful to parameterize the class S data, see (3.5). – 3 – (1.2) (1.3) section 3.2. and satisfies see (3.12). – The integer N6 is the number of tensors of the 6d quiver. The difference between NS and N6 is determined by the Kac label and is described in the algorithm in – A rational number Q is the M5-charge which appears in the inflow computation, Q = Ninst − Topological data of the instanton configuration resolution C]2/Γ, where Γ ∈ SU(2). Here we recall the topological data necessary to specify a G-instanton on C2/Γ or its On C2/Γ, we first need to specify the holonomy at the origin and at the infinity. They determine the representation ρ0,∞ : Γ → G, which we consider as a linear action on the complexified adjoint representation g. On C]2/Γ, we specify the holonomy at infinity ρ∞. In addition, we need to specify the class in H2(C]2/Γ, π1(G)). This is the first Chern class when G = U(N ) and the second Stiefel-Whitney class when G = SO(N ). Finally we need to specify the instanton number, defined as the integral of tr F ∧F over the ALE space. Unless otherwise mentioned, we normalize the trace so that the instanton on R4 of the smallest positive instanton number satisfies Z tr F ∧ F = 1. On the ALE space, the instanton number is in general fractional. Our main interest lies in the case G = E8 and Γ = Zk. Since π1(E8) is trivial, we do not have to specify the class in H2. A holonomy ρ : Zk → E8 can be nicely encoded by its Kac label n := g of the generator of Zk in E8 be where g = e2πiw/k ∈ E8 w = X niwi ∈ e8. i6=0 – 4 – (2.1) (2.2) (2.3) (2.4) where wi are the fundamental weights of E8. Since g is of order k, ni are integers. We define n0 so that P dini = k, where the Dynkin marks d are given by d = It is known that by the Weyl reflections and the shifts, we can arrange ni ≥ 0 for all i and then the result is unique. This is the Kac label of the holonomy. The subalgebra of e8 left unbroken by the holonomy ρ can be easily read off from its Kac label. Namely, it is given by the subalgebra corresponding to the nodes i of the Dynkin diagram where ni = 0, together with an Abelian subalgebra making the total rank 8. On C^2/Zk, the instanton number modulo one is given by the classical Chern-Simons invariant evaluated on S3/Zk at infinity. One way to compute it is to introduce coordinates on S3/Zk using polar coordinates θ, φ on S2 and the angle ψ along the S1 fiber. The connection itself is ∝ w(dψ + · · · ). One finds that Z dimH MC^2/Γ,ρ∞ = 30Ninst − hw, ρi. Z Z – 5 – (2.5) (2.6) (2.7) (2.8) 2.2 Dimension of the instanton moduli space In this subsection we derive the formula (2.7) of the dimension of the moduli space. Those readers who trust the authors can skip this subsection. This computation is of course not new. It is provided here to make this paper more self-contained. The basic tool is the Atiyah-Patodi-Singer index theorem. Its explicit form on the orbifold of C2 was worked out e.g. in [24] for Γ ⊂ U(2). Here we quote the form used in Kronheimer-Nakajima [2] for Γ ⊂ SU(2). The formula for the orbifold is: dimH MC2/Γ,ρ∞,ρ0 = h∨(G) Here, h∨(G) is the dual Coxeter number of G, and the second and the third terms are the contributions from the η invariant of S3/Γ at the asymptotic infinity and at the origin, respectively, and Q is the standard two-dimensional representation of Γ from the defining embedding Γ ⊂ SU(2), On the ALE space C]2/Γ, we have: dimH MC^2/Γ,ρ∞ = h∨(G) , dimH MC^2/Γ,ρ∞ = h∨(G) Z reflecting the fact that if the holonomy at the origin of an instanton on C2/Γ is trivial, we can resolve/deform the instanton and the ALE at the same time to be on C]2/Γ. In the end, we find the formula e2πiw/k with the Kac label n. The eta invariant is Let us evaluate this formula when G = E8 with the holonomy ρ∞ specified by g = Δη = 1 X (χρ∞ (γ) − dim g) since 0 ≤ hα, wi ≤ k for positive roots α. Now we use and find X α:positive roots α = 2ρ, X α:positive roots hv1, αihα, v2i = h∨hv1, v2i h∨ 2k Δη = hw, wi − hw, ρi. To compute the dimension, we now plug in to (2.12) the formula for Δη found just above and the formula for the instanton number (2.6). The term proportional to hw, wi cancels out, and we indeed have the desired result (2.7). – 6 – (2.10) (2.11) (2.12) (2.13) (2.14) (2.15) (2.16) (2.17) (2.18) (2.19) Six-dimensional description After these geometrical preliminaries, we move on to the field theoretical analysis. We start with the six-dimensional quiver descriptions. As already mentioned in the introduction, for various simple choices of ρ, the six-dimensional quivers were already determined in [10–14]. By a series of trials and errors, and following the principle that the quiver should be determined in terms of the Kac label, the authors found the following algorithm. The general structure of the quiver Our 6d SCFT on the generic points on its tensor branch consists of a collection of N6 tensors, corresponding to a linear quiver of the form G1 × SU(m2) × SU(m3) × · · · × SU(mN6) × [SU(k)] where G1 is on the −1 curve, the rest is on −2 curves, and the final SU(k) is a flavor symmetry. In the notation of [11], we have 1 2 G1 su(m2) su(m3) 2 · · · su(mN6) 2 [SU(k)]. Below, we slightly abuse the notation and refer by G1 the combination of the group and the non-fundamental hypermultiplets on the −1 curve. The choices are: (3.1) (3.2) (3.3) • G1 = USp(m1), • G1 = SU(m1) with an antisymmetric hyper, or • G1 = SU(m1 = 6) with a rank 3 antisymmetric half-hyper. We consider the rank 1 E-string theory as USp(0), and furthermore, the rank 2 E-string theory is considered as a USp(0) connecting to an SU( 1 ) group. We have m1 ≤ m2 ≤ · · · ≤ mN6, and we define a1, . . . , a9 by as = #{i | mi+1 − mi = s}. We can reconstruct the whole of mi from m1, N6 and a1, . . . a9. For example, when the quiver is SU(3) × SU(9) × SU(13) × SU(17) × SU(18) × SU(19) × SU(20) (3.4) we have a8 = a7 = a5 = a3 = a2 = 0, a6 = 1, a4 = 2, a1 = 3. There are bifundamentals between two consecutive groups in the quiver, and finally fundamental hypers are added such that each group is anomaly free, that is: • Nf = 2N for SU(N ), • Nf = N + 8 for USp(N ) or SU(N ) with an antisymmetric hyper, and • Nf = 15 for SU(6) with a rank 3 antisymmetric half-hyper. – 7 – Now we present the algorithm to determine the structure of the quiver given the Kac label n and the number N6 of the groups. Along the way, we also define the quantity NS which will be used in the following. We will also need the quantity The algorithm is implemented in the accompanying Mathematica file, so that the reader can easily try it around. In general we have ai = ni for i = 1, 2, 3, 4, 5, 6. To specify a7,8,9, we need to consider various cases as summarized below: (3.5) (3.6) HJEP09(217)4  n′4 ≥ n′3             n′3 ≥ n′4 −→ −→  (n′4 − n′3 = even −→ Case 1, n′4 − n′3 = odd −→ Case 2, n′2 < (n′3 − n′4)/2 −→ Case 4,  n′2 ≥ (n′3 − n′4)/2 −→ (n′3 − n′4 = odd −→ Case 3, n′3 − n′4 = even −→ Case 5. For each case, the output of the algorithm is (a7,8,9, G1, NS) as shown below:1 1. n′4 ≥ n′3, n′4 − n′3 = even: • a7 = n′3, a8 = n′4−n′3 , a9 = 0. 2 • G1 = USp(2n′2). • NS = N6 − n′4+2 n′3 . 2. n′4 ≥ n′3 + 1, n′4 − n′3 = odd: • a7 = n′3, a8 = n′4−n′3−1 , a9 = 0. 2 • G1 = SU(2n′2 + 4) group with an antisymmetric hyper. • NS = N6 − n4+n′3−1 . ′ 2 3. n′3 ≥ n′4 + 1, n′3 − n′4 = odd, n′2 ≥ n′3−n′4−1 : 2 • a7 = n′4, a8 = n′3−n′4−1 , a9 = 0. 2 • G1 = SU(2n′2 + n′4 − n′3 + 4) group with an antisymmetric hyper. • NS = N6 − n′4+n′3−1 . 2 1When more than two cases apply to the same Kac label, they produce the same quiver. – 8 – 4. n′3 > n′4 + 2n′2 + ℓ, n′3 − n′4 − 2n′2 = 3x + ℓ, x ∈ Z, ℓ = 0, 1, 2: • a7 = n′4, a8 = n′2, a9 = n′3−n′4−2n′2−ℓ . 3 • G1 is – SU(3) for ℓ = 1, • NS = N6 − n′3+2n′4+n′2−l . 3 5. n′3 ≥ n′4, n′3 − n′4 = even, n′2 ≥ n′3−n′4 : 2 • a7 = n′4, a8 = n′3−n′4 , a9 = 0. 2 • G1 = USp(2n′2 + n′4 − n′3). • NS = N6 − n′4+n′3 . 2 – empty for ℓ = 0, in which case this node corresponds to a rank-1 E-string, – SU(6) with a half-hyper in the rank 3 antisymmetric for ℓ = 2. 3.3 A subtlety concerning the 6d θ angle Note that the quivers produced in Case 5 are the same ones as the ones produced by Case 1, as far as the data we described so far are concerned. This is perfectly fine when n′4 = n3, ′ since in this case we are just applying the different cases to the same Kac label. However, when n′4 6= n′3, or equivalently when a8 6= 0, the resulting quivers should however be subtly different, since e.g. they reduce to different 4d class S theories and 3d star-shaped quivers. We argue that the difference between them is how one embeds the SU(2N + 8) group into the SO(4N + 16) global symmetry group of USp(2N ). A relatively simple case is the following. Let us first consider the cases when n′4 = 2, n′3 = 0, n′2 = 0 versus n′4 = 0, n′3 = 2, n′2 = 1, with the rest of labels being zero n1,...,6 = 0. Both theories have the form of a long SU(8) quiver gauging an SU(8) subgroup of the rank 1 E8 theory. The two differ by the embedding of SU(8) inside E8 and in fact have different global symmetries. To see this, consider embedding SU(8) inside SO( 16 ) ⊂ E8. The adjoint of E8 decomposes under its SO( 16 ) maximal subgroup as 248 → 120 + 128. Now consider decomposing SO( 16 ) to its U( 1 ) × SU(8) maximal subgroup. Under this embedding the spinors of SO( 16 ) decompose to the rank x antisymmetric tensors of SU(8) for x = 0, 2, 4, 6, 8 for one spinor and x = 1, 3, 5, 7 for the other. However only one spinor appears in the adjoint of E8, and therefore there are two different embedding of SU(8) inside E8. In one of them the 128 contains gauge invariant contributions leading to the larger global symmetry. The general case corresponds to the situation where SU(2N + 8) is embedded in SO(4N + 16). There is no distinction in the perturbative sector of the theory. However the theory possesses instanton strings. The ones for USp groups will be in a chiral spinor of the SO group and so will decompose differently depending on the embedding. This then leads to theories with distinct spectrum of string excitations. Also note that this only occurs if the entire SO symmetry is gauged leaving only a U( 1 ) commutant. If – 9 – embedding. This agrees with the fact that the cases coincide when a8 = 0. We can understand this distinction from the existence of the discrete θ angle in 6d, due to the fact that π5(USp(2N ))5 = Z2. Suppose now that the USp group has 2n halfhypermultiplets in the fundamental. Classically it has an O(2n) flavor symmetry, but the parity part flips the discrete theta angle. Therefore the flavor symmetry is actually so(2n). The two embeddings of su(n) into so(2n) are related exactly by the parity part of O(2n), and therefore are inequivalent. The F-theoretical interpretation of these two inequivalent embeddings seems to be unknown. It would be interesting to work it out.2 Note that an analogous phenomenon exists in 5d, where given a pure USp group there are two distinct 5d SCFTs associated with this theory differing by the instanton spectrum of the 5d gauge theory. This is related to the existence of a Z2 valued θ angle originating from the fact that π4(USp(2N ))4 = Z2. 3.4 Anomalies and the inflow The anomaly of these 6d SCFTs can be computed from their quiver description using the technique of [25, 26]. We should be able to match it to the anomaly computed from the inflow using the M-theory description. The inflow computations of M5-branes probing the E8 end-of-the-world brane and of M5-branes probing the C2/Zk singularity was given in [27] and in an appendix of [25], respectively. We can combine the two computations into one and one finds the following contribution to the anomaly, excluding the most subtle contribution from the codimension5 singularity where the C2/Zk singularity hits the end-of-the-world brane: where Q is the M5-chage of the configuration, I8 = I4 = J4 = 1 48 1 4 1 48 1 4 (p2(N ) + p2(T ) − (p1(N ) − p1(T ))2), 1 k (p1(T ) − 2c2(R)), (k − )(4c2(R) + p1(T )) + 1 4 tr FS2U(k). Here I8 comes from the M-theory interaction R C ∧I8, I4 appears in the boundary condition G = I4 at the E8 wall, and J4 is the interaction on the C2/Zk singular locus R C ∧ J4. In this section the normalization of tr is as in [27]. Let n be the Kac label, and let w = P wini be the corresponding weight vector. By performing computations for many choices of n, we find that Iquiver(n, N3) = Iinnafliovwe(Q) + c(n) 2The authors thank D. R. Morrison for the correspondence on this point. (3.7) (3.8) (3.9) (3.10) (3.11) where c(n) is a constant depending on the Kac label n but independent of N3 and Recall that the instanton number as defined by the integral of tr F ∧ F was given by Q = N3 + k + 1 2 1 k − , see (2.6), and that k − 1/k is the Euler number of C]2/Γ, or equivalently of the integral of −p1/4 there, see (2.10). Then, assuming that Ninst = N3 + k, we can rewrite the effective M5-brane charge Q as Z Q = tr F ∧ F + Z p1 C^2/Zk 4 which is what we expect from the curvature coupling on the E8 end-of-the-world brane. The authors made a guess of the formula for c(n) by trial and error. It has the form c(n) = k (P0(n) + P2(n) + P4(n) + P6(n)) + 2 Ifree vector 1 where 1 384 1 11520 + 1 5760 1 288 P0 = P2 = P4 = − P6 = 1 240 1 1 5760 (3.12) (3.13) (3.14) (3.15) (3.16) (3.17) (3.18) (3.19) (3.20) (3.21) Ifree vector = (−240c2(R)2 − 120c2(R)p1(T ) − 7p1(T )2 + 4p2(T )) is the anomaly polynomial of a free vector multiplet and Pi(n) is a homogeneous polynomial of ni’s of degree i. Those polynomials are identified as (−88c2(R)2 + 32c2(R)p1(T ) − 5p1(T )2 + 4p2(T )) k2 2512c2(R)2 − 760c2(R)p1(T ) + 157p1(T )2 − 124p2(T ) (15hw, wi − khw, ρi) 112c2(R)2 − 40c2(R)p1(T ) + 7p1(T )2 − 4p2(T ) 9hw, wi2 + 15k2hw, wi − 2k4 − k 4c2(R)2 − c2(R)p1(T ) X hw, αi3 α∈Δ+ 5hw, wi3 + 15k2hw, wi2 − 5k4hw, wi + k6 − k X hw, αi5 c2(R)2, α∈Δ+ where Δ+ is the set of positive roots of E8. The authors have not been able to determine how this formula come from the correct anomaly inflow calculation. It would be interesting to understand it. 4.1 other way. Lower dimensional incarnations Five-dimensional brane-web description We can reduce the 6d theory on a circle to 5d. Roughly speaking, there are two different types of reductions. For example, starting from the E-string theory, one can obtain SU(2) theory with eight flavors in one way, or the 5d SCFT with E8 flavor symmetry in the First reduction. Keeping the radius of the circle non-zero the low-energy 5d theory is sometimes a 5d gauge theory. Specifically, the class of 6d theories we are considering can be realized by a brane construction involving a system of NS5-branes and D6-branes in the presence of an O8−-plane [28, 29]. Performing T-duality on this system results in a brane configuration involving NS5-branes and D6-branes in the presence of an O8−-plane. Alternatively, the system can also be described as D4-branes immersed in an O8−-plane and D8-branes, in the presence of a C2/Zk singularity [30]. Either way, the system can sometimes be deformed so as to describe a 5d gauge theory. Specifically, when compactifying we have a choice of the value of the radius as well as the freedom to turn on holonomies for the global symmetries. These then become mass parameters in the 5d theory. In specific ranges of these parameters the 6d theory may flow at low-energy to a 5d quiver gauge theory with the coupling constants of the gauge theory identified with the mass deformations. In general, a given 6d SCFT may have several different low-energy 5d gauge theory descriptions depending on the specific deformations used. Various 5d descriptions of 6d theories, including the type we are interested in, were studied in [12, 14, 31, 32]. We will not consider this problem here. Second reduction. Instead we shall take the limit of zero radius. In this case we argue that the 6d theory flows in the IR to a 5d SCFT. Furthermore, we claim that the 5d SCFT can be readily described in terms of the integer N and the Kac label n. To find the 5d theory, we first write down the 6d quiver following the algorithm presented in the last section.3 We realize this 6d quiver in type IIA using O8-planes, D8-branes, D6-branes and NS5-branes as in [28, 29]. We then compactify it on S1, T-dualize it to type IIB, and manipulate the branes. We will detail the procedure in slightly more detail below. The result can be conveniently represented by a brane web, which has a star shape form with a group of ( 1, 0 ), ( 0, 1 ) and ( 1, 1 ) 5-branes all intersecting at a point. The 5branes end on the appropriate 7-branes where some collection of 5-branes end on the same 7-brane. Specifying the configuration then is done by giving the distribution of 5-branes on the 7-branes. This is conveniently done by a Young diagram where each column represents a 7-brane, and the number of boxes in it represents the number of 5-branes ending on it. 3Purely field theoretically, the 6d quiver only contains the information on the low energy limit on the generic points on the tensor branch of a given 6d SCFT. Therefore further manipulations of the quiver such as dimensional reductions are not guaranteed to tell every detail of the original 6d SCFT. What we do is, instead, to realize the quiver using branes, and apply string dualities. This way we can keep all the ultraviolet information required in the process. The three Young diagrams for the SCFTs we are considering are given by: Y1 = (NS − n6, NS − n6 − n5, NS − n6 − n5 − n4, NS − n6 − n5 − n4 − n3, NS − n6 − n5 − n4 − n3 − n2, NS − n6 − n5 − n4 − n3 − n2 − n1, 2NS + n4′ + n2′ + n3′, 2NS + n4′ + n3′), Y3 = (3NS + 2n4′ + n2′ + 2n3′, 3NS + 2n4′ + n2′ + n3′). (4.1) More detail of the second reduction. For cases 1, 2 and 3, these results can be derived using the standard techniques. But there are some issues for cases 4 and 5. Case 4 naively does not have a brane construction of the type considered in [29] so this procedure appears to be inapplicable in this case. However, a conjecture for the 5d theories that lift to these types of 6d SCFTs was given in [12, 14], and we can use this conjecture to fill in this step for case 4. This leaves case 5. We can ask how does the 6d θ angle appears in the brane construction. In fact a similar issue arises in the analogue 5d system: D5-branes suspended between NS5-branes in the presence of an O7−-plane. In that case it was observed by [33] that accounting for the 5d θ angle seems to necessitate the introduction of two variants of the O7−-plane, where one is an SL(2, Z) T-transform of the other. This in particular means that they differ by their decomposition into a pair of 7-branes. Note that the distinction between the two cases vanishes when there are D7-branes on the O7−-plane. This becomes clear after we decompose the O7−-plane into 7-branes which can be moved through the monodromy lines of the 7-branes which will change them by a T-transformation. This of course agrees with the unphysical nature of the 5d θ angle once flavors are present. There should be a similar distinction for the O8−-plane, and so can account for the apparent 6d θ angle we observe. We will not pursue this here. However once we perform T-duality we end with a system with two O7−-planes, and we expect that we can accommodate this in the observed difference in O7−-planes. We have a discrete choice for each O7−-plane leading to four possibilities. However we are free to perform a global T-transformation. Since all the external branes are D7-branes, this will lead us to the same system, save for changing the types of both orientifolds. Thus we conclude that there are only two distinct choices: the same or differing types. These cases are expected to differ only when there are no 7-branes on the O7−-planes, and thus no D8-brane on the original O8−-plane. This exactly agrees with the two cases, which coincide once a8 = 0. We indeed find different 5d theories for these two choices, where the former is identified with case 1 while the latter with case 5. In this manner we can apply this procedure also to case 5. We can compactify on an additional circle to 4d. Using the results of [34], it is straightforward to write the 4d theory. It is just an A type class S theory given by the same set of Young diagrams as the 5d description, given above in (4.1). In fact it is also possible to motivate this class S description with the Young diagrams (4.1) directly from the 4d description, and then use the preceding discussion to connect the 6d quiver data to the Kac labels. We start from the observation that the class S theory whose Young diagrams are (4.1) can be thought of as generated by modifying the Young diagrams of the rank N E8 theory, which is given by a class S theory of type SU(6N ) with Young diagrams Y1 = (N 6), Y2 = (2N 3), Y3 = (3N 2). First the 4d theory needs to have the SU(k) global symmetry, coming from the C2/Zk singularity. This is given by the k boxes attached to the Young diagram Y1 of the E8 theory. That this is the correct way to account for it can be seen by comparing anomalies. For the type of 6d theories we are considering, there is a result due to [15] that allows for the computations of the central charges of the 4d result of the compactification of the 6d theory from the anomaly polynomial of the latter. Furthermore the anomaly polynomial of the 6d theories of the type we considered was studied in [12]. When applied to our case we find that kS4dU(k) = 2k + 12 independent of the details of the Kac label. This agrees with the anomaly of the class S theory. In addition to the SU(k) we also have the commutant of the orbifold in E8 as a global symmetry, which depends on the Kac labels. The E8 global symmetry is accommodated by the Young diagram structure of the starting E8 SCFT so it is natural to expect that modifying this will give the required global symmetry and take into account the Kac labels. The global symmetry which is manifest in the class S construction is SU(2)×SU(3)×SU(6) which can be identified with the three legs of the affine Dynkin diagram. This becomes more apparent once we compactify to 3d and consider the mirror dual, which we consider more extensively in the next subsection. The point is that we can associate a node in the legs of the affine E8 Dynkin diagram roughly with the difference between neighboring columns. The central node can be associated with the difference between the sum of the first columns of the three Young diagrams and the the total number of boxes in any of them. When that difference is zero, we get the E8 theory. It is now natural to associate that difference to the Kac label of the corresponding node. By the Kac prescription, this ensures that we get the correct global symmetry. This leads to the conjectured form. There is one ambiguity in determining the total number of boxes which is related to the rank of the initial E8 theory. This should be related to the number of tensors in 6d, but we need to determine the exact mapping. For this we use the relation outlined in the previous sections between the 6d and 4d theories. We can perform various consistency checks of this proposal. One check is to compare anomalies. We already mentioned that these can be computed from the 6d anomaly polynomial, and compare the SU(k) central charge. We can also compare the central charges a and c, and the dimension of the Coulomb branch. These can then be calculated from the 6d quiver on one side, and from the class S theory on the other, in terms of the Kac labels and NS. For the computations on the class S side, we use the standard results of [7, 35] and reviewed e.g. in [36]. The results themselves are rather complicated and not very illuminating, but we do find that all three objects agree between the two calculations. Any interested reader can play around with the Mathematica file which comes with this paper to confirm this point. Three-dimensional star-shaped quiver description Let us now move on to the three dimensions. We translate the Young diagrams Y1,2,3 given in (4.1) which specify the class S punctures to the 3d mirror description using the results of [18]. We find that the resulting theory is given by the quiver gauge theory 1 2 k ˜ N1 ˜ N2 ˜ N3 ˜ N4 ˜ N5 Xˆ := • − • − · · · − • − • − • − • − • − • − • − • − • . N˜4′ N˜2′ •|N˜3′ ˜ N6 Here, all nodes are unitary with the diagonal U( 1 ) removed, and the gray and the black blobs are used as a visual aid for the affine Dynkin part and the over-extended part. The ranks of the groups are specified by the vector (4.2) (4.3) (4.4) (4.5) (4.6) (4.7) (4.8) N˜ = N3d + X niqi q2 = q5 = q2′ = 3 3 1 (qi)j = didj − hwi, wj i CN˜ = k 0 0 0 0 0 0 0 0 + n 6 6 6 6 6 6 4 2 , where q1 = q4 = q4′ = 3 3 2 which is in fact given by a uniform formula where wi is the weight vector for the node i 6= 1 and w1 = 0. Another characterization of N˜ is where C is the affine Cartan matrix of E8. This determines N˜ mod d, since d is the only eigenvector of C of zero eigenvalue. The dimension of the Coulomb branch Mˆ is then dimH Mˆ = 30(N3 + k) − hw, ρi + k(k + 1) 2 − 1 (4.9) where w = P niwi is the Kac label as a weight vector and ρ = Pi wi is the Weyl vector. The Coulomb branch Mˆ of this system Xˆ is closely related to the instanton moduli space Minst on the ALE space C^2/Zk. To explain the relation, let us first recall that the resolution and deformation parameters of the ALE space can be specified by a parameter ξ = (ξC, ξR) ∈ su(k) ⊗ (C ⊕ R) (4.10) HJEP09(217)4 which takes values in the Cartan of su(k) tensored by R3. We now need an auxiliary hyperka¨hler space Oξ, which is the SU(k)C orbit of ξC in su(k) with the hyperka¨hler metric specified by ξR. Equivalently, Oξ is the Coulomb/Higgs branch of the T [SU(k)] theory whose quiver realization is given by T [SU(k)] = • − • − · · · − • − 1 2 k−1 k where the rightmost square node is a flavor symmetry and ξ is the SU(2)R triplet of mass parameters associated to it. argument given in [15]: We can now state the relation between Mˆ and and Minst by slightly modifying an Minst = (Mˆ × Oξ)///SU(k). This relation can be understood as follows. The resolution/deformation parameter ξ of the ALE space can be identified with the scalar vacuum expectation values of the 7d super SU(k) Yang-Mills theory supported on the M-theory singularity C2/Zk. The 6d SCFT on the M5-branes at the intersection of the E8 wall and the C2/Zk singularity couples to this 7d super Yang-Mills, via the standard coupling where the triplet moment map field of the 6d theory is identified with the limiting value of the triplet of scalars of the 7d bulk. The resulting hyperka¨hler manifold is then given by the hyperka¨hler reduction as in (4.12). Now, our system Xˆ can also be written using the theory X˜ Indeed, X˜ := k − • − • − • − • − • − • − • − • . ˜ N1 ˜ N2 ˜ N3 ˜ N4 ˜ N5 N˜4′ N˜2′ Xˆ = (T [SU(k)] × X˜ )///SU(k) •|N˜3′ ˜ N6 where the symbol T ///G means that we gauge the flavor symmetry G of the theory T . So the theory X whose Coulomb branch is Minst in (4.12) is given by X = (T [SU(k)] × T [SU(k)] × X˜ )///(SU(k) × SU(k)) But two T [SU(k)] gauged by a diagonal SU(k) is known to disappear, since it is the domain wall of 4d N = 4 SYM implementing the S-duality [37]. So we have, in fact, X = X˜ (4.11) (4.12) (4.13) (4.14) (4.15) (4.16) and the ALE deformation parameter ξ is now the mass parameter of the SU(k) flavor symmetry. We have dimH Minst = 30(N3 + k) − hw, ρi. This nicely agrees with the computation from the geometry (2.7) by the identification Ninst = N3 + k. (4.17) (4.18) This relation between N3 and Ninst is also consistent with what we found from the inflow, see (3.14). We note that the theory X = X˜ is the theory whose Higgs branch is the U(k) instanton moduli on C2/ΓE8 [2, 5]. From this reason, the Coulomb branch, at least when the mass parameter is zero, has been conjectured to be the E8 instanton moduli space on the singular space C2/Zk by various people. This follows, at least in a rough form, from the string duality: consider the theory on M2-branes on C2/Zk ×C2/ΓE8. It has two supersymmetric branches of vacua, one describing E8 instantons on C2/Zk and another describing U(k) instantons on C2/ΓE8. If the former is the Coulomb branch, then the latter is the Higgs branch. Note that we arrived at the quiver gauge theory X = X˜ from a totally different method, by first studying the 6d quiver and then by reducing on successively on circles. Therefore, this agreement can be thought of as an overall consistency check of our construction. Now, applying [2] and [5] in our case, we see that the U(k) holonomy at infinity of C2/ΓE8 is trivial, and the first Chern class c1 satisfies R Ei c1 = ni which can be read off from (4.8). It would be interesting to understand from M-theory point of view why the first Chern class on the C2/ΓE8 side is given by the asymptotic E8 holonomy on the C2/Zk. It seems important for the full story to consider a more general case where C2/Zk is replaced by the multicenter Taub-NUT space, see e.g. [20, 38]. 5 Examples in the other sections. 5.1 The case of k = 2 Let us demonstrate the above general statement in various examples. In this section, we take N to be the number of tensor multiplets in the 6d theory, which was denoted by N6 There are three possibilities. We label the cases with the Kac label n and the group H ⊂ E8 left unbroken by the Kac label. The choice k = 2 is somewhat special, since the ALE space C^2/Z2, also known as the Eguchi-Hanson space, has an exceptional isometry SU(2). Then the generic flavor symmetry of the 6d SCFT should be H × SU(2)2, where one SU(2) comes from the 7d gauge field on the singularity and another SU(2) comes from the isometry. 1. The first case is n = 0 2 0 0 0 0 0 0 0 The corresponding 6d theory is The T 3 reduction of this theory gives the following 3d N = 4 theory: · · · N{−z3 [SU(2)]. • − • − • − • − • − • − • − • − • − • . 1 2 N 2N 3N 4N 5N 6N 4N 2N n = 0 · · · N{−z2 su(2) 2 } [SU(2)] 2. The second case is The corresponding 6d theory is where the number of su(2) gauge groups in the quiver is N − 1. The Higgs branch dimension of the UV fixed point of this theory is 29N + 4 + 4(N − 1) − 3(N − 1) = 30N + 3. The mirror of the T 3 compactification of this theory is 3. The third case is The corresponding 6d theory is The Coulomb branch dimension, which is the sum of the rank of the gauge groups minus one, is indeed 30N + 3. n = · · · su(2) 2 } [SU(2)] where the number of SU(2) gauge groups associated with the (−2) curves is N − 1. The Higgs branch dimension of the UV fixed point of this theory is 29N + 16 + 4 + 4(N − 1) − 3 − 3(N − 1) = 30N + 16. The mirror of the T 3 compactification of this theory is 1 2 • − • − • − N+2 2N+2 • − 3N+2 4N+2 • − • − 5N+2 • − − • − • . 4N+1 2N (5.9) The Coulomb branch dimension, which is the sum of the rank of the gauge groups minus one, is indeed 30N + 16. This is consistent with figure 45 of [12], namely the T 2 compactification of (5.8) yields the class S theory whose Gaiotto curve is a sphere with punctures: 0 | 2 [(3N + 1)2], [(2N + 1)2, 2N ], [N 6, 12] . (5.10) su(2) 2 } •3N Now let us comment on the flavor symmetry from the point of view of the 6d quiver. Since an SU(2)-SU(2) bifundamental has an SU(2) flavor symmetry, the three 6d quivers presented above have order N copies of SU(2) symmetries on the generic points of the tensor branch. In fact the same issue already appears in the case of N M5-branes probing the C2/Z2 singularity, which has the quiver [SU(2)] su(2) 2 | · · · {z N su(2) 2 } which naively has too many SU(2) flavor symmetries. The issue can be resolved by recalling the fact derived in appendix A of [39] that the basic 6d SCFT whose quiver on the tensor branch is given by SU(2) with Nf = 4 with a naive SO(8) symmetry, only has an SO(7) symmetry under which the flavors transform in the spin representation. In the quiver representation of the same theory as this means the following: regard the bifundamental hypermultiplets on the left and on the right of the gauge group as the trifundamental half-hypermultiplets. At the quiver level there are therefore the flavor symmetry SU(2)1 × SU(2)′1 × SU(2)2 × SU(2)′2 ⊂ SO(8). Under the SO(7) symmetry which is the flavor symmetry of the SCFT, only the diagonal subgroup of SU(2)′1 and SU(2)′2 survives. Applying this argument at every su(2) node in (5.2), (5.5), (5.8), and (5.12), we see that the number of SU(2) flavor symmetries is There are also some interesting special cases with enhanced flavour symmetries when reduced appropriately. N is small: 1. N = 2, H = E8. In this case the quiver (5.2) degenerates to [SU(2)1] su(2) 2 [SU(2)2], su( 1 ) 2 [SU(2)] which is just the rank-2 E-string theory with three decoupled hypermultiplets. The 3d quiver in this case is (5.3) for N = 2: 1 2 2 4 6 8 10 • − • − • − • − • − • − • − • − • − • . 8 4 •6 | 12 Its Coulomb branch is H3 × (the reduced moduli space of 2 E8 instantons on C2) and we indeed see the same decoupled structure. The explanation from the perspective of the Coulomb branch operators will be described below. 2. N = 3 with H = E8. The 6d quiver is The flavour symmetry is enhanced to G2 × E8. The explanation from the perspective of the Coulomb branch operators will also be described below. (5.16) (5.17) (5.18) (5.19) (5.20) 3. N = 2, H = E7 × SU(2). The 6d quiver for this case reduces to su(2) 2 On the tensor branch, there is an SO(8) symmetry acting on the four flavors of SU(2) gauge group. In the SCFT it is known that there is only SO(7). The total symmetry is then SO(7) × E7. In fact this 6d theory is the (E7, SO(7)) minimal conformal matter [10], which describes “half M5-branes” on the E7 singularity. The 3d quiver in this case is (5.6) for N = 2: 1 2 3 4 6 8 10 • − • − • − • − • − • − • − • − • − • . 8 4 •6 | 12 This theory is the mirror of the S1 reduction of the class S theory whose Gaiotto curve is a sphere with punctures [24, 14], [62], In [12, 15] the T 2 compactification was also identified with a class S theory of the E6 type associated with the sphere with punctures 0, 2A1 and E6(a1). For consistency, these two class S theories should in fact be the same. Let us compute the central charges of (5.20). We find that the effective numbers of vector multiplets and hypermultiplets are nH = 112 and nV = 49, respectively. Thus, 1 24 a = (5nV + nH ) = c = (2nV + nH ) = (5.21) 119 8 , 1 12 35 2 . This agrees with a and c of the aforementioned class S theory of the E6 type; see (7.1) Enhanced flavor symmetries from 3d quivers In fact the symmetry enhancement of each of the three cases above can be generalized to other over-extended Dynkin quivers in 3d, namely: 1. For the quiver consisting of a tail • − • attached to the affine Dynkin diagram of type g with gauge groups being unitary groups of the ranks given by 2 times the dual Coxeter labels, the Coulomb branch moduli space is H3 × Mf2,g, where Mf2,g denotes the reduced two-instanton moduli space of group g on C2. For example, the Coulomb branch of the quiver is H3 × Mf2, su(2), and the Coulomb branch of the quiver is H3 × Mf2, so(8). 2. For the quiver consisting of a tail • − • attached to the affine Dynkin diagram of type g with gauge groups being unitary groups of the ranks given by 3 times the dual Coxeter labels, the Coulomb branch moduli space has a symmetry G2 × g. For example, the Coulomb branch of the quiver has a symmetry G2 × SU(2), and the Coulomb branch of the quiver 1 1 1 2 • − • − • = • 2 2 1 2 has a symmetry G2 × SO(8). 1 2 3. For the quiver consisting of a tail • − • attached to the affine Dynkin diagram of type g with the affine node being U(3) and other gauge groups being unitary groups of the ranks given by 2 times the dual Coxeter labels, the Coulomb branch has a symmetry SO(7) × g˜, where g˜ is the commutant of su(2) in g. For example, the Coulomb branch of the following quiver 2 3 4 6 1 • − • − • − • − • − • − • 4 2 has a symmetry SO(7) × SU(6), where SU(6) is the commutant of SU(2) in E6. 1 2 k In each of the above examples, the quiver contains of a balanced affine Dynkin quiver diagram as a subquiver. If we consider only this subquiver, the R-charges of the monopole operators in this theory vanish, and hence this subquiver is indeed a bad theory. By attaching a quiver tail • − • − · · · − • to such a subquiver, the total quiver becomes good or ugly.4 We would like to consider the contribution of this quiver tail to the Coulomb branch of the total quiver. 1. For this case, the node •, which is the affine node in the affine Dynkin diagram, 2 1 2 2 is over-balanced in the sense of [37]. Following [37], we can split the quiver into two parts, namely • − • − • and the rest of the Dynkin diagram. The R-charge of the monopole operators from the subquiver • − • − • receives the contribution from the hypermultiplets and vector multiplets in the way described in [37], except that there is no contribution from the vector multiplet of the rightmost node •, since this 1 1 2 2 2 T 2 1 2 2 T 2 1 1 2 2 3 T 3 1 2 T 4 was cancelled inside the affine Dynkin quiver. The contribution from the subquiver is therefore the same as that of the quiver • − • − •, where ∩ denotes an adjoint hypermultiplet of the U(2) rightmost node. The Coulomb branch of •−•− • contains 3 free hypermultiplets, which can be seen from the monopole operators with SU(2)Rspin 1/2. This explains the H3 factor in (5.15). The reduced moduli space of two E8 instantons on C2 can be realised as in [21]. 2. Similarly, for this case, the total quiver can be split into • − • − • and the rest of the Dynkin diagram. The contribution to the R-charge of the monopole operators from 1 2 3 the subquiver • − • − • can be realised from the quiver • − • − •, where ∩ denotes an adjoint hypermultiplet of the U(3) rightmost node.5 Indeed, it was pointed out in section 3.3.2 of [41] that the Coulomb branch of the latter model has a G2 symmetry. (Note that the corresponding 4d class S theory had been studied in [40]. The G2 symmetry on the Higgs branch of such a theory had also been pointed out in that reference.) This therefore explains the G2 symmetry in case 3. The E8 symmetry follows from the Dynkin subquiver. 4 3. Finally, for this case, • is the unbalanced node in the quiver. There are two contributions to the Coulomb branch operators with SU(2)R-spin 1. One contribution can be realised using the quiver • − • − • − • in a similar fashion to the above discussion. 1 2 3 3 1 2 3 T 4 4 3 This quiver has a Coulomb branch symmetry SU(4) and thus gives 15 operators with SU(2)R-spin 1 in the adjoint representation of SU(4). The other contribution can be seen as follows. Since the node •, which was originally a part of the affine Dynkin subquiver, now belongs to the tail •−•−•−•, we also need to take into account the contribution that arises from the removal of this node from such an affine Dynkin diagram. The second contribution thus comes from considering • − • − •. There are 6 Coulomb branch operators with SU(2)R-spin 1 in the latter. Therefore, we have in total 15+6 = 21 operators with SU(2)R-spin 1; this explains the enhancement to the SO(7) symmetry. The remaining symmetry is thus the commutant of SU(2), which arises from node •, in the original symmetry associated with the affine Dynkin diagram. 4See also [40] for a related consideration from the 4d point of view. 5The authors thank S. Cremonesi for this argument. 5.3 There are ten possibilities. The F-theory quiver for the 6d theories are listed on page 73 of [11]. Here are the mirrors of the T 3 compactification of them. and the 3d quiver is 2. The second case is with the 6d quiver The 3d quiver is 3. The third case is with the 6d quiver n = 0 n = n = 0 sp( 1 ) su(3) zsu(4) }| su(4{) [SO(14)] 1 2 The dimension of the SCFT Higgs branch is 29N + 2 + 6 + 12 + 4 + 16(N − 3) − 3 − 8 − 15(N − 3) = 30N + 10 . (5.32) • − • − • − • − • − • − • − • − • − • − • − • 1 2 3 4 N+1 2N 3N 4N 5N 6N 4N 2N and the dimension of the Coulomb branch is 30N + 10. (5.27) (5.28) (5.29) (5.30) (5.31) (5.33) (5.34) (5.35) HJEP09(217)4 The dimension of the SCFT Higgs branch is The Coulomb branch dimension is 30N + 11. n = 0 0 2 0 0 0 0 0 0 , [SU(2)] N−2 }| su(4) { 29N + 8 + 8 + 16(N − 2) − 3 − 15(N − 2) = 30N + 11 . • − • − • − • − • − • − • − • − • − • − • − • 1 2 3 4 N+2 2N 3N 4N 5N 6N 4N 2N •3N | n = 0 0 0 0 0 0 0 0 2 , H = SO( 16 ) sp(2) su(4) z 2 N−1 }| su(4) { [SO( 16 )] 1 29N + 14 + 6 + 12 + 4 + 16(N − 2) − 3 − 8 − 15(N − 2) = 30N + 23 . (5.36) • − • − • − • − 1 2 3 4 N+2 2N+2 3N+2 4N+2 5N+2 6N+2 4N+1 2N − • − • (5.37) and the Coulomb branch dimension is 30N + 23. HJEP09(217)4 The 3d quiver is 4. The fourth case is with the 6d quiver The 3d mirror is 5. The fifth case is with the 6d quiver •3N+1 | • •3N+2 (5.38) (5.39) (5.40) (5.41) (5.42) (5.43) (5.44) The dimension of the SCFT Higgs branch is The 3d quiver is 29N + 32 + 16 + 16(N − 1) − 10 − 15(N − 1) = 30N + 37 . 1 2 3 4 N+4 2N+4 3N+4 4N+4 5N+4 6N+4 4N+2 2N • − • − • − • − − • − • (5.45) and the Coulomb branch dimension is 30N + 37. 6. The sixth case is with the 6d quiver n = 0 0 1 0 0 0 0 0 1 , H = SO(12) × SU(2) × U( 1 ) [SO(12)] 1 su(4) 2 [SU(2)] N−1 }| su(4) { The dimension of the SCFT Higgs branch is 29N + 12 + 8 + 8 + 16(N − 1) − 3 − 15(N − 1) = 30N + 24 . The 3d quiver is 1 2 3 4 • 2N+2 − • 3N+2 − • 4N+2 − • 5N+2 − The Coulomb branch dimension is 30N + 24. •3N+1 | • 6N+2 − • 4N+1 − • 2N 7. The seventh case is with the 6d quiver The 3d mirror is 8. The eighth case is with the 6d quiver (5.46) (5.47) (5.48) (5.49) (5.50) (5.51) (5.52) (5.53) (5.54) (5.55) The dimension of the SCFT Higgs branch is and the Coulomb branch dimension is 30N + 12. 29N + 6 + 12 + 4 + 16(N − 2) − 8 − 15(N − 2) = 30N + 12 . 2 3 1 • 2N+1 − • − • − • − • − • − • 3N 4N 5N 6N 4N 2N •3N | n = 0 1 0 1 0 0 0 0 0 , H = E6 × SU(2) × U( 1 ). 2 z su(4) 2 N−2 }| su(4) { n = 1 , H = SU(8) × U( 1 ) su(4) { z su(4) 2 29N + 24 + 12 + 4 + 16(N − 1) − 8 − 15(N − 1) = 30N + 31 . (5.56) 2 3 1 and the Coulomb branch dimension is 30N + 31. The 3d quiver is 9. The ninth case is with the 6d quiver n = 0 , H = SO(10) × SU(4) su(4) 2 [SU(4)] N−1 }| su(4) { [SO(10)] 1 29N + 16 + 16(N − 1) − 15(N − 1) = 30 + 15 . The dimension of the SCFT Higgs branch is The 3d quiver is 2 3 1 • − • − • − • − • − • and the dimension of the Coulomb branch is 30N + 15. 10. The final tenth case is with the 6d quiver n = 0 , H = SU(8) × SU(2), [SU(8)] su(4) 1 [antisym] z su(4) 2 }| su(4) { 29N + 32 + 6 + 16N − 15N = 30N + 38 . The dimension of the SCFT Higgs branch is The 3d quiver is 2 3 1 • − • − • − • − • − 4 N+4 2N+4 3N+4 4N+4 5N+4 6N+4 4N+2 2N+1 • − • − • − • − − • − • (5.65) •3N+2 | • and the dimension of the Coulomb branch is 30N + 38. (5.58) (5.59) (5.60) (5.61) (5.62) (5.63) (5.64) HJEP09(217)4 In this subsection we look at the 4d and 3d theories generated from 6d SCFTs differing by the choice of 6d θ angle. The first case where this possibility occurs is for k = 8, where the two choices are given by Kac labels n′3 = 2, n′2 = 1 for one and n′4 = 2 for the other with the rest zero. These can be generalized to k = 2l + 8 with Kac labels n′3 = 2, n′2 = 1 + l for one and n′4 = 2, n′2 = l for the other with the rest zero. The 6d quiver in both cases is given by: 1 2 [SU(8)] usp(2l) szu(2l+8) }| su(2l+8{) N−1 . . . 2 (5.66) where we identify the case n′4 = 2, n′2 = l with θ = 0 and n′3 = 2, n′2 = 1 + l with θ = π. The associated 4d theories are different for the two cases. In the θ = 0 case we associate the class S theory given by: [(N − 1)6, 12l+8], [2N + l + 2, 2N + l, 2N ], [(3N + l + 1)2] , while the θ = π case is associated with: [(N − 1)6, 12l+8], [(2N + l + 1)2, 2N ], [3N + l + 2, 3N + l] . (5.67) (5.68) | • •3N+l+1 − • − • , 2N (5.69) •3N+l | • − • − • , 2N (5.70) The 3d quivers are: for the θ = 0 case, and 1 2 for the θ = π case. 1 2 •−•−. . .− • − • − 2l+7 2l+8 N+2l+7 2N+2l+6 3N+2l+5 4N+2l+4 5N+2l+3 6N+2l+2 4N+l 2l+7 2l+8 N+2l+7 2N+2l+6 3N+2l+5 4N+2l+4 5N+2l+3 6N+2l+2 4N+l+1 We can now inquire as to how these theories differ from one another. In the l = 0 case they differ already at the level of the global symmetry, where the θ = 0 case has an SU(8)2 × SU(2) × U( 1 ) global symmetry while the θ = π case has an SU(8)2 × U( 1 )2 global symmetry. In this case we have an SU(8) gauging of E8 and the two choices differ by their commutant inside E8. We note that this difference is in accordance with the symmetry expected from the Kac labels. When l > 0 the symmetries of the two theories agree. We can calculate the 4d anomalies of the two theories and find that all of them agree between the two theories. Again this is consistent with our interpretation as the 4d anomalies can be computed from their 6d counterparts, which in turn are independent of the θ angle. From our 6d interpretation we expect the two to differ slightly in their operator spectrum. Particularly the θ angle should affect the USp gauge group instanton strings changing their charges under the global and gauge symmetries. Upon compactification to lower dimensions these should map to local operators. We can observe this from the 3d quivers. We get a tower of monopole operators from every node. The basic monopole operator from the balanced nodes leads to enhancement of symmetry. We also have a basic monopole operator from the unbalanced nodes. These provide operators with higher R-charges, and we can read of their R-charges and nonabelian global symmetry charges from the quiver. We have three unbalanced nodes. Two of them give the same contribution in both theories: one operator of SU(2)R spin N2 in the bifundamental of the SU(2l + 8) × SU(8) global symmetry, and one operator of SU(2)R spin 2 in the 28 of the SU(8) global symmetry. These can be readily identified with gauge invariants in the 6d quiver, where the former is the one made from N − 2 SU(2l + 8) × SU(2l + 8) bifundamentals and the flavors, and the later is made from two SU(8) flavors and the USp(2l) × SU(2l + 8) bifundamental. The last one differ slightly between the two theories. In the θ = 0 case it is a flavor singlet with SU(2)R spin l+22 . Particularly for l = 0 this gives the conserved current enhancing the U( 1 ) to SU(2). In the θ = π case, however, it is in the 8 of SU(8) with SU(2)R spin l+23 . We can interpret these states as coming from the USp gauge group instanton strings wrapped on the circle. These are in the spinor of SO(4l +16), and depending on the θ angle decompose to all the even or odd rank antisymmetric tensor representations of the gauge SU(2l + 8) connected to the USp gauge group. In the θ = 0 case we get the even rank representations, which contain a gauge invariant part which is a flavor symmetry singlet. In the θ = π case we get the odd rank representations, which do not contain any gauge invariants. However we can combine it with one of the SU(2l + 8) flavors to form an invariant. This should contribute a state in the 8 of SU(8) with SU(2)R spin which is greater by 21 from that of the singlet. This agrees with what we observe. It might be interesting to study more accurately the spectrum, particularly, the Higgs branch chiral ring, and compare against the 6d expectations. We will not pursue this here. 5.5 Massive E-string theories In this subsection, we consider the following 6d theory TE6d(̟, m0, N ) : [E9−m0 ] 1 2 2 . . . 2 sum0 su2m0 su(̟−1)m0 su̟m0 su̟m0 2 [Nf =m0] | 2 · · · , N−{̟z−1 su̟m0 2 [SU(̟m0)]. (5.71) These theories were studied in [10, 12, 13, 42]. They can be called the “massive E-string theories” as in the last reference, since they correspond to NS5-branes probing the O8-D8 combination in the presence of the Romans mass. The mirror of the T 3 compactification of (5.71) is 1 2 N+r1 2N+r2 3N+r3 4N+r4 5N+r5 4N+r4′ 2N+r2′ (5.72) where the values of ri and the Kac labels for each m0 are given in table 1. Note that •3N+r3′ | • 6N+r6 − X ri = i 1 2 ̟m0(m0 − 1) . (5.73) 1 2 3 4 5 6 7 8 E8 E7 E6 SO(10) SU(5) SU(3) × SU(2) SU(2) × U( 1 ) SU(2) 0 0 0 0 0 0 1 0 The SCFT Higgs branch dimension of (5.71) is dimSHCFT Higgs of TE6d(̟, m0, N ) = 30N + ̟m02(̟ + 1) − 1 ; (5.74) 1 2 this is equal to the Coulomb branch dimension of (5.72). Higgsing the SU(k) flavour symmetry In the theories we have discussed so far, there is always an SU(k) flavour symmetry which came from the gauge symmetry on the C2/Zk singularity. From the 3d quiver perspective, this symmetry arises from the topological symmetry associated with the nodes in the tail 1 2 partition of k. We can obtain another class of models by on nilpotent VEVs that Higgs the flavour symmetry SU(k).6 Suppose that such VEVs are in the nilpotent orbit of SU(k) given by Li Jsi where Js is a s × s Jordan block so that Y = [s1, s2, . . . , sℓ] is a corresponding Assuming that the 6d quiver theory before the Higgsing has a sufficiently long plateau of SU(k) gauge groups, this Higgsing can be performed exactly as in 4d class S theory e.g. as described in section 12.5 of [43]. Its effect in 6d quiver was studied in [44, 45]. In the end, we see that the tail on the right-hand side of the quiver to have the form · · · 2 su(k) su(k−uℓ′ ) 2 · · · su(u2+u1) 2 su(u1) 2 [Nf =uℓ′ ] [Nf =(uℓ′−1−uℓ′ )] [Nf =(u2−u3)] [Nf =(u1−u2)] , (5.75) 6The authors thank Alessandro Tomasiello for the discussion about this class of theories. HJEP09(217)4 dimSHCFT Higgs of (5.75) = 30(N3 + k) − hw, ρi + k(k + 1) − 1 − dimH OY (5.76) where OY is the nilpotent orbit labeled by Y . The mirror of the T 3 compactification of (5.75) is TY (SU(k)) × − • − • − • − • − • − • − • − • ˜ N1 ˜ N2 ˜ N3 ˜ N4 ˜ N5 U(k)/U( 1 ) N˜4′ N˜2′ . (5.77) 1 2 •|N˜3′ ˜ N6 (5.78) (5.79) (5.80) (5.81) HJEP09(217)4 In other words, we simply replace the tail • − • − · · · − for the theories discussed in the preceding sections by TY (SU(k)), where the latter is defined as in [37]. The Coulomb 1 2 branch dimension of (5.77) is dimH Coulomb of (5.77) 1 2 1 2 = [30(N3 + k) − hw, ρi] + {(k2 − 1) − (k − 1)} − dimH OY + (k − 1) = 30(N3 + k) + k(k + 1) − 1 − dimH OY − hw, ρi , where the terms in the second square brackets in the second line denote the Coulomb branch dimension of TY (SU(k)). This result is indeed in agreement with (5.76). As an example, let us consider TE6d(k, m0 = 1, N ) of the previous section and perform the Higgsing with Y = [k − 1, 1]. The resulting 6d theory is su( 1 ) su(2) su(k−1) su(k) su(k)N−2k su(k) su(k−1) 2 where the number of tensor multiplets is N . This theory is similar to that discussed in (36) of [6], (5.2) of [29], except that we have only one (−1)-curve in the quiver, instead of two. The mirror of the T 3 compactification of this theory is 1 • − • − • − • − • − • − • − • − • − • , N 2N 3N 4N 5N 6N 4N 2N •3N | This quiver is a “good” theory in the sense of [37] if N + 1 ≥ 2k and k ≥ 2. In this case, this quiver is the 3d mirror theory of the S1 reduction of the class S theory of type SU(6N ) associated a sphere with the punctures [N 5, N − k, k − 1, 1], [(3N )2], [(2N )3] . Acknowledgments The authors thank Hiroyuki Shimizu for the collaboration at the early stages. NM sincerely thanks Stefano Cremonesi, Amihay Hanany and Alessandro Tomasiello for a close collaboration, invaluable insights, and several useful discussions. He also grateful to the hospitality of the organisers of the Pollica Summer Workshop 2017, including Fernando Alday, Philip Argyres, Madalena Lemos and Mario Martone. 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Noppadol Mekareeya, Kantaro Ohmori, Yuji Tachikawa, Gabi Zafrir. E 8 instantons on type-A ALE spaces and supersymmetric field theories, Journal of High Energy Physics, 2017, 144, DOI: 10.1007/JHEP09(2017)144