Comments on the twisted punctures of Aeven class S theory

Journal of High Energy Physics, Jun 2018

Abstract We point out that the USp symmetry associated to a full twisted puncture of a class S theory of type Aeven has the global anomaly associated to π4(USp) = ℤ2. We discuss manifestations of this fact in the context of the superconformal field theory R2,2N introduced by Chacaltana, Distler and Trimm. For example, we find that this theory can be thought of as a natural ultraviolet completion of an infrared-free SO(2N + 1) gauge theory with 2N flavors, whose USp(4N) symmetry clearly has the global anomaly.

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Comments on the twisted punctures of Aeven class S theory

Accepted: June Aeven class Yuji Tachikawa 0 1 3 Yifan Wang 0 1 2 Gabi Zafrir 0 1 3 Global Symmetries, Supersymmetric Gauge Theory 0 Princeton , NJ 08544 , U.S.A 1 University of Tokyo , Kashiwa, Chiba 277-8583 , Japan 2 Joseph Henry Laboratories, Princeton University 3 Kavli Institute for the Physics and Mathematics of the Universe We point out that the USp symmetry associated to a full twisted puncture of a class S theory of type Aeven has the global anomaly associated to discuss manifestations of this fact in the context of the superconformal eld theory R2;2N introduced by Chacaltana, Distler and Trimm. For example, we nd that this theory can be thought of as a natural ultraviolet completion of an infrared-free SO(2N + 1) gauge theory with 2N avors, whose USp(4N ) symmetry clearly has the global anomaly. Anomalies in Field and String Theories; Brane Dynamics in Gauge Theories - 2.1 2.2 2.3 2.4 1 Introduction and summary 2 Class S constructions Class S theories of type A2N with a twisted irregular puncture R2;2N from twisted irregular punctures Half-hypermultiplet from irregular twisted puncture 3 5d constructions 4 IIA constructions class of 4d N =2 superconformal eld theory obtained by compactifying 6d N =(2; 0) theory on a Riemann surface with punctures. This construction not only allowed a geometric understanding of various S-dualities, but also provided a huge variety of new 4d N =2 theories. This variety comes from various sources: we have the choice of the initial 6d N =(2; 0) theory, which comes with the ADE classi cation; then one can introduce punctures, which come roughly in two varieties, called the regular ones and the irregular ones; then one can further introduce twists by outer automorphisms. The class S theories with regular punctures have been systematically explored by Chacaltana, Distler and their collaborators [3{15], for almost all possible types of 6d N =(2; 0) theories with outer automorphism twists. The remaining two cases are the twisted punctures of type Aeven theories, and the case where Z2-twisted and Z3-twisted punctures of type D4 are combined. The aim of this paper is to make a small comment on the former case, namely the twisted punctures of type Aeven theories. More speci cally, we point out that the avor symmetry USp(2N ) of the full twisted puncture of A2N theory has the global anomaly associated to 4(USp(2N )) = Z2. This point can be seen most succinctly as follows. Consider splitting a Riemann surface on which the 6d N =(2; 0) theory of type A2N is compacti ed along a long tube around which we have Z2 outer-automorphism twist. This results in two twisted full punctures coupled by the 5d theory obtained by compactifying the A2N theory on S1 with Z2 outerautomorphism twist. This is the maximally supersymmetric 5d USp(2N ) theory with the discrete theta angle = [16, 17]. The 4d class S theory with a twisted full puncture, in this viewpoint, lives on a boundary of this 5d USp(2N ) theory. { 1 { First, this determines that the current algebra central charge of USp(2N ) symmetry of the twisted full puncture is k = 2N + 2 in the normalization where the half-hypermultiplet in the fundamental has k = 1.1 More importantly for us, this means that the USp(2N ) symmetry of the twisted full puncture has the global anomaly. This is because of the following. Note that the discrete theta angle is controlled by 4(USp(2N )) = Z2, which also controls the global anomaly on the 4d USp(2N ) symmetry, as originally discussed by Witten [19]. Therefore, there is an anomaly in ow from the bulk to the boundary, and the twisted full puncture needs to carry the global anomaly. This is analogous to the fact that if the bulk 5d theory has SU(n) gauge symmetry with the level k Chern-Simons term, the boundary 4d theory which is coupled to this bulk 5d theory should have 't Hooft anomaly di erence here is that this puncture does not have the global anomaly. In this paper, we discuss various manifestations of this fact, mainly using the 4d N =2 superconformal eld theory R2;2N introduced in [6, section 7.2] for N = 1 and in [10] for general N . This theory is obtained by compactifying the 6d N =2 theory of type A2N on a sphere with one simple puncture and two twisted full punctures. In [10], it was shown that the symmetry USp(2N )2 apparent in this construction enhances to USp(4N ), but Witten's anomaly could not be directly determined, since the diagonal combination USp(2N )diag USp(2N )2 USp(4N ) was gauged. Our main observation is that by turning on the mass term associated to the simple puncture, the theory becomes an SO(2N + 1) theory with 2N hypermultiplets in the vector representation, which is infrared free; the USp(4N ) symmetry of this theory clearly has the global anomaly. The rest of the paper consists of three sections, which can be read independently, and are written using di erent techniques. Namely, the section 2 uses the class S construction, the section 3 uses the dimensional reduction from 5d, and the section 4 uses a very traditional orientifold construction. In section 2, after brie y reviewing the original construction of R2;2N , we provide a di erent construction of the same R2;2N theory as a sphere of A4N theory with a full twisted puncture and an irregular puncture, generalizing the construction of [ 20 ]. This allows us to perform a consistency check of the global anomaly. In section 3, we point out that the mass deformation for the U(1) avor symmetry associated to the simple puncture gives rise to the SO(2N + 1) theory coupled to 2N avors. This will be done by constructing these theories by a twisted compacti cation of 5d N =1 theory, generalizing the work of [ 21 ]. In section 4, we revisit this mass deformation from the point of view of the old-fashioned type IIA construction. We will see that the standard Seiberg-Witten curve of the SO(2N + 1) theory with 2N avors, in the standard MQCD form, is literally equal to the Seiberg1We emphasize that the avor central charge here of the twisted full puncture is valid for class S of the 4d theory and thus change the avor central charge which is related to the U(1)R anomaly [18]. theory is simply the mass parameter of the R2;2N theory. Class S constructions Review of R2;2N Let us rst recall how the R2;2N non-Lagrangian SCFT was constructed in [10]. We start with a class S theory of A2N type with the UV curve given by a torus with minimal untwisted puncture decorated by an Z2 twist line along a handle of the torus. The theory has two S-dual frames with gauge theory descriptions: S dual SU(2N + 1) USp(2N ) R2;2N (2.1) punctures, see gure 1. SCFT to be and the conformal central charges are and the R2;2N SCFT emerges from the weak coupling limit of USp(2N ) gauge coupling. Motivated by the decoupling picture, one can engineer this SCFT directly using 3 regular A2N punctures on a sphere: one minimal untwisted puncture, and two maximal twisted The decoupling picture above determines the Coulomb branch spectrum of R2;2N 2 f3; 5; 7; : : : ; 2N + 1g a = 14N 2 + 19N + 1 24 ; c = 8N 2 + 10N + 1 12 : (2.2) (2.3) The theory has enhanced U(1) USp(4N )2N+2 avor symmetry where the USp(4N ) factor has the avor central charge kUSp(4N) = 2N + 2. Consequently, the 2d chiral algebra of the SCFT in the sense of [22] naturally contains the a ne Kac-Moody algebra of type C2N with level k2d = (N + 1) and a weight one current realizing the 4d U(1) symmetry. Furthermore kUSp(4N) saturates a avor central charge unitary bound of [22], which means that the stress tensor in the 2d chiral algebra is given by the Sugawara construction [10].2 2We emphasize here that the full chiral algebra of the R2;2N SCFT contains additional Virasoro primaries regular puncture irregular puncture tent if the USp(4N ) symmetry of R2;2N also carries the global anomaly; then the USp(2N ) diagonal gauging is non-anomalous. Thus if we have an alternative construction R2;2N that makes the enhanced avor symmetry manifest as a twisted puncture, this would o er a nontrivial consistency check. We show this is indeed the case in the next section. Our alternative construction involves irregular punctures. 2.2 Class S theories of type A2N with a twisted irregular puncture Consider general 4d N = 2 SCFTs in class S of type A2N from one regular maximal twisted puncture and one irregular twisted puncture, see gure 2. The twisted punctures correspond to codimension-2 defects in the (2,0) theory of A2N type and are speci ed by a local singularity of the Higgs eld at z = 0 on the UV curve. For the Higgs eld to be well-de ned, we require HJEP06(218)3 = T z2+ 4N+2 + { 4 { (e2 iz) = g[o( (z))]g 1 as one circles the singularity. Here o denotes the Z2 outer-automorphism associated to the A2N Dynkin diagram and g generates an inner automorphism of A2N . Among the twisted punctures, regular punctures are associated with simple poles for and are classi ed in [5], while the irregular punctures involve higher order poles and a detailed classi cation will appear in [23]. For our purpose here, we list two distinguished classes of twisted irregular punctures of A2N type and the relevant physical data from [23] below. Class I. has the Higgs eld of the form for odd, where T = diag(1; !2; !4; : : : ; !4N ); !4N+2 = 1: (2.4) (2.5) (2.6) We have the conjectural avor central charge formula Class II. has the Higgs eld of the form kUSp(2N) = 2N + 2 1 4N + 2 2 4N + 2 + : = T z2+ 2N + kUSp(2N) = 2N + 2 1 2N 2 2N + : where = T z2+ 1 4 N4N + T = diag(0; 1; !; : : : ; !4N 1); !4N = 1: This is chosen such that under z ! ze2 i, the Higgs eld transform by an automorphism of order 4N that is a product of the Z2 outer-automorphism o associated to the A4N Dynkin diagram and an inner automorphism of A4N as in (2.4). The singular Seiberg-Witten curve is where We have the conjectural avor central charge formula T = diag(0; 1; !; : : : ; !2N 1); !2N = 1: The conformal central charges for the corresponding 4d SCFTs are determined by for the above conjectures in [23]. 2.3 R2;2N from twisted irregular punctures Given the general construction of 4d N = 2 SCFTs from twisted irregular punctures in the last section, we now construct R2;2N from 6d (2; 0) A4N type theory on a sphere with two twisted punctures: one regular maximal puncture, and one irregular puncture of class II with = 1 4N in (2.8). The irregular twisted puncture is described by the following local singularity for the Higgs eld , x4N+1 + xz1 4N = 0 { 5 { (2.7) (2.8) (2.9) (2.10) (2.11) (2.12) (2.13) (2.14) and the Seiberg-Witten 1-form is = xdz. The scaling dimensions of the coordinates are Under the Z2 twist, the di erentials ` transform as Therefore ` for ` even (odd) involves integral (half-integral) powers of z. We can immediately read o the spectrum of Coulomb branch operators to be HJEP06(218)3 = f3; 5; 7; : : : ; 2N + 1g which all come from the odd-degree di erentials ` with ` = 2N + 3; 2N + 5; : : : ; 4N + 1. The USp(4N ) avor symmetry comes from the regular twisted puncture and its avor central charge is determined by (2.10) to be 2N + 2. The di erential 2N+1 contributes the additional mass parameter responsible for the U(1) factor in the avor symmetry. It is also easy to check that the central charges computed from (2.11) and (2.10) are consistent with the results in the previous section. Moreover we can directly see that the Seiberg-Witten curves from the two descriptions agree. For example let us look at the N = 1 case which is a rank-1 theory with a Coulomb branch operator u of dimension 3. In this case [x] = 3 and [z] = 4. The full Seiberg-Witten curve in the A4 description is (2.15) (2.16) (2.17) (2.18) (2.19) (2.20) Starting from the A4 curve, we perform a coordinate rede nition z ! which only changes the Seiberg-Witten di erential by an exact 1-form, and then the A4 curve becomes identical to the A2 curve, after throwing out irrelevant overall factors. 1 z3x4; x ! x3z2 2.4 Half-hypermultiplet from irregular twisted puncture Since we expect Witten's global anomaly to be a local property of the twisted full puncture, which is eventually a boundary condition for the 5d super Yang-Mills theory, it should not depend on the types of other punctures that are involved in a given Class S setup. In { 6 { x5 + x3 z22 + x2 m On the other hand, the full Seiberg-Witten curve in the A2 description is with [x] = 1 and [z] = 0. In both cases the Seiberg-Witten di erential is taken to be = xdz. Here 2; 4 label Casimirs of the USp(4) avor symmetries and 2; 20 label Casimirs of the SU(2) SU(2) subgroup. The additional U(1) mass is labelled by m. For simplicity, let us look at the curves with the mass parameters turned o : A4 : x5 + x z3 + u z 2 9 = 0; A2 : x3 iu 5 z 2 (z 1) = 0: other words, if we can tune the choice of the irregular puncture in gure 2 such that the 4d theory has a free/weakly-coupled description that clearly exhibits the anomaly, it strongly indicates that the twisted full puncture carries the global anomaly. Below we see this is indeed the case for a free half-hyper multiplet in the fundamental representation of USp(2N ). The half-hyper can be constructed using A2N twisted punctures: a Class I twisted irregular puncture with and one twisted full regular puncture with USp(2N ) avor symmetry. The spectral curve is x2N+1 + X z2i 2i + z 21 2N = 0: Here the scaling dimensions are determined by [x] = 1 4N and [z] = 4N + 2. Thus 2i with dimension 2i is a degree 2i Casimir for the USp(2N ) avor symmetry. The above is also consistent with (2.7) which gives as expected for a half-hyper. 3 5d constructions In the previous section we relied on class S methods to argue that the R2;2N SCFT has a global anomaly for the USp(4N ) group. Here we shall show further evidence for this by using a di erent realization of the R2;2N SCFT. The realization we employ is the compacti cation of 5d SCFTs with a global symmetry twist. In this manner, 4d N = 2 SCFTs can be engineered by the compacti cation of 5d SCFTs. This method can be used to engineer many 4d N = 2 SCFTs including non-Lagrangian theories appearing in class S constructions [24]. Many 5d SCFTs possess discrete global symmetries. It is then possible to consider a compacti cation with a twist under said discrete symmetry. In other words the compacti cation is done such that upon traversing the circle one comes back to the theory acted by the discrete symmetry element. To try to understand the results of such compacti cations it is useful to consider 5d SCFTs with a string theory construction that exhibits the global symmetry. A convenient way to realize 5d SCFTs in string theory is using brane webs [25, 26]. Discrete symmetries of the SCFTs are then manifested by discrete symmetries of the brane system. A particular interesting case is when the symmetry is manifested on the web as a combination of spacetime re ections and an SL(2; Z) transformation. The cases where the discrete symmetry is Z2 and Z3 were analyzed in [ 21 ] where it was argued that such a construction can realize the R2;2N SCFT. We next review some aspects of this construction that will be important for us. { 7 { (2.21) (2.22) (2.23) web after moving some of the 7-branes. This corresponds to a mass deformation of the 5d SCFT. As can be seen from the web, this mass deformation sends the 5d SCFT to an SU(2N + 1) gauge theory with 4N avors in the fundamental representation. Consider the bane web shown in gure 3(a). This describes a 5d SCFT which lives at the intersection of all the 5-branes. This SCFT can be de ned in eld theory as the UV xed point of an SU(2N + 1) gauge theory with 4N avors in the fundamental representation. This can be seen from gure 3(b) which shows the web after a mass deformation corresponding to the SU(2N + 1) coupling constant. This 5d SCFT has an SU(4N ) U(1)2 global symmetry. Additionally, it also has a Z2 discrete symmetry, which in the brane web is given by a rotation in the plane of the web combined with I in the SL(2; Z) transformation. In the low-energy SU(2N + 1) gauge theory it is manifested as charge conjugation which is a symmetry of the gauge theory. We can now consider compactifying the 5d SCFT with a twist under this discrete symmetry. It was argued in [ 21 ] that this should give the 4d R2;2N SCFT. There are several reasons for this identi cation. One is that they have the same symmetries. Here the SU(4N ) part is projected to USp(4N ) by the twist while one U(1) remains and another is projected out. This is readily seen in the 5d gauge theory description by considering how charge conjugation acts on these symmetries. Another reason is that one can argue from the brane web that the resulting theory needs to participate in the same duality (2.1). We can now use the brane construction to study various properties of the R2;2N SCFT. The particular properties, that are of interest to us here, are mass deformations. We have already encountered one such mass deformation, the one leading to the SU(2N + 1) gauge theory. This deformation is invariant under the Z2 discrete symmetry as can be seen from the low-energy gauge theory. We thus expect it to remain also in the R2;2N SCFT where it should correspond to a deformation in the U(1) global symmetry. We can infer the result of this deformation from the 5d construction, where it should just be the twisted compacti cation of the 5d gauge theory. As the twist acts on it by charge conjugation, we expect the SU(2N + 1) to be projected to SO(2N + 1) while the 4N avors should be split to two groups of 2N each mapping to the other. The end result is a 4d SO(2N + 1) gauge theory with 2N hypermultiplets in the vector representation. This theory indeed has a USp(4N ) global symmetry as required. Furthermore it is easy to see { 8 { same web after moving some of the 7-branes. This corresponds to a mass deformation of the 5d SCFT. As can be seen from the web, this mass deformation sends this 5d SCFT to the 5d SCFT in gure 3(a). that it has a global Witten's anomaly. By anomaly matching arguments this implies that the starting theory, the R2;2N SCFT, must also have the same anomaly. We can also consider mass deformations leading to the R2;2N SCFT. For instances consider the 5d SCFT shown in gure 4(a). This is a 5d SCFT that has the same discrete symmetry and so we can also consider its compacti cation to 4d with a twist. The result of such a compacti cation was considered in [ 21 ] where it was argued that it leads to a known 4d SCFT with SU(2) USp(4N + 2) global symmetry. An interesting property of this 4d SCFT is that it is dual to an SO(2N + 3) gauge theory with 2N + 1 hypermultiplets in the vector representation upon gauging its SU(2) global symmetry. This duality in particular implies that its USp(4N + 2) global symmetry has a global anomaly. This theory can also be engineered by a Class S of A4N+2 type involving one twisted irregular puncture and one twisted regular full puncture that realizes the USp(4N + 2) avor symmetry with the global anomaly [23]. As shown in gure 4(b) we can ow from this SCFT to the R2;2N SCFT via a mass deformation. Then anomaly matching arguments again suggest that the USp(4N ) global symmetry of the R2;2N SCFT has a global anomaly. 4 IIA constructions In this section we provide another way to see that the R2;2N SCFT can be mass deformed to be the SO(2N +1) gauge theory with 2N avors, using a traditional brane construction [ 27 ] using orientifolds [16, 28]. Let us rst recall the brane construction of 4d SO(2N + 1) gauge theories with hypermultiplets in the vector representation using O4, D4 and NS5 branes, see gure 5(a). f + As shown in the gure, this requires the use of Of4 -plane to realize the SO(2N + 1) gauge group. Then, across a half-NS5-brane, the type of the orientifold plane switches to O4 -plane, and we have a half-hypermultiplet in the bifundamental of USp and SO(2N +1) at the intersection of the D4-branes and the half-NS5-brane. Note that this bifundamental half-hypermultiplet has the global anomaly for USp symmetry. From the anomaly in ow { 9 { + n D4 USp(2n) + N D4 SO(2N+1) (a) ½NS5 ½NS5 + nʹ D4 USp(2nʹ) + N D4 USp(2N) + N D4 USp(2N) full NS5 (b) + requires the use of two half-NS5-branes. (b) When we collapse the two half-NS5-branes, we obtain the IIA reduction for the R2;2N theory. = xdz=z. This becomes, for n = n0 = N , argument, this implies that the 5d USp gauge symmetry on the Of4 -plane has to have the discrete theta angle = . This is a much simpler argument for this theta angle than the one given in [17]. Instead, consider reducing the three-punctured sphere de ning the R2;2N theory given + in gure 1 from M-theory to type IIA. A twisted full puncture corresponds to a semi-in nite segment of N D4-branes on the Of4 -plane, and the untwisted simple puncture corresponds to a full NS5 brane, see gure 5(b). Clearly, the two setups shown in gure 5(a) and 5(b) are related by the motion of the two half-NS5-branes, when n = n0 = N . Since the distance between the two half-NS5-branes specify the squared inverse gauge coupling of the SO(2N + 1) gauge theory at the string scale, this clearly means that the R2;2N theory is at the in nite coupling limit of the SO(2N + 1) gauge theory with 2N avors. What remains is to show that how the gauge coupling looks like from the point of view of the R2;2N theory. Here we encounter a mild surprise: the Seiberg-Witten curve of the SO(2N + 1) theory with 2N avors is the same with the Seiberg-Witten curve of the R2;2N theory, without making any approximation. To see this, recall the Seiberg-Witten curve of SO(2N + 1) theory with n + n0 avors, as determined by lifting the brane setup shown in gure 5(a) to M-theory: x " 2N 1 2n 1 z1=2 n Y(x2 i=1 mi2) + 2N 1 2n0z1=2 Y(x2 m~ i20) # = x2N + u2x2N 2 + + u2N : (4.1) # n0 i=1 m~i20) = x2N + u2x2N 2 + + u2N : (4.2) We now compare this to the Seiberg-Witten curve of the R2;2N theory. We put two full twisted punctures at z = 0 and z = 1, and the untwisted simple puncture at z = 1. The Seiberg-Witten curve is given by x2N+1 + 1(z)x2N + 2(z)x2N 1 + + 2N+1(z) = 0; (4.3) = xdz=z. Note that we allow 1(z) to be nonzero to simplify the description of the mass-deformed untwisted simple puncture, where the conditions just become that for all k, k(z) have at most a simple pole at z = 1. Combining with the conditions from the twisted full punctures, we see that the Seiberg-Witten curve of the R2;2N theory to be x2N+1 + x2N + mz1=2 z 1 z 1 2z + 02 x2N 1 + u^3z1=2 z 1 x2N 2 + + 2N z + 0 z 1 2N x + u^2N+1z1=2 z 1 where m is the mass parameter for the untwisted simple puncture, 0 ; 2 ; 02N are the mass parameters for the two twisted full punctures, and u^3; ; u^2N+1 are the Coulomb branch parameters of the R2;2N theory. We see that the curve (4.2) for the SO(2N + 1) theory with 2N avors and the curve (4.4) for the R2;2N theory are the same up to a minor relabeling of the parameters as one can see by dividing (4.2) by (z1=2 + z 1=2)= to make the coe cient of x2N+1 to be one. In particular, we see that the scale of the Landau pole of the infrared gauge theory can simply be equated to the mass deformation parameter m of the R2;2N theory. It might be interesting to look for similar phenomena in other 4d N =2 gauge theories with infrared-free matter content, such that its Seiberg-Witten curve is the same as that of some = 0; (4.4) 2; : : : ; 2N and Acknowledgments YT is partially supported by JSPS KAKENHI Grant-in-Aid (Wakate-A), No.17H04837 and JSPS KAKENHI Grant-in-Aid (Kiban-S), No.16H06335, and also by WPI Initiative, MEXT, Japan at IPMU, the University of Tokyo. YW is supported in part by the US NSF under Grant No. PHY-1620059 and by the Simons Foundation Grant No. 488653. 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Yuji Tachikawa, Yifan Wang, Gabi Zafrir. Comments on the twisted punctures of Aeven class S theory, Journal of High Energy Physics, 2018, 163, DOI: 10.1007/JHEP06(2018)163