PHYSICAL REVIEW LETTERS 125, 211602 (2020)

String Universality and Non-Simply-Connected Gauge Groups in 8D

Mirjam Cvetič,1,2 Markus Dierigl ,1 Ling Lin ,3 and Hao Y. Zhang1 1Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6396, USA 2Center for Applied Mathematics and , University of Maribor, Maribor, Slovenia 3Department of Theoretical Physics, CERN, 1211 Geneva 23, Switzerland (Received 3 September 2020; revised 26 October 2020; accepted 4 November 2020; published 20 November 2020)

We present a consistency condition for 8D N ¼ 1 theories with nontrivial global structure G=Z for the non-Abelian gauge group, based on an involving the Z 1-form center symmetry. The interplay with other swampland criteria identifies the majority of 8D theories with gauge group G=Z, which have no realization, as inconsistent quantum theories when coupled to gravity. While this condition is equivalent to geometric properties of elliptic K3 surfaces in F-theory compactifications, it constrains the unexplored landscape of gauge groups in other 8D string models.

DOI: 10.1103/PhysRevLett.125.211602

Introduction.—One of the important lessons from string the 9D and 8D swampland considerably. (As g2 does not theory is that consistency conditions of are suffer similar anomalies, it remains an open question if it highly restrictive. In the low-energy limit, they result in a truly belongs to the 8D swampland.) small and possibly finite subset of effective descriptions, The goal of this work is to provide similar constraints for leaving behind a vast “swampland” of seemingly consistent the global structure of the gauge group of 8D N ¼ 1 quantum field theories coupled to gravity [1]. Recent theories, by deriving a field theoretic consistency condition attempts to specify the swampland’s boundary (cf. [2] for the gauge group to take the form G=Z, with Z ⊂ ZðGÞ a for reviews) have reinforced the idea of string universality: discrete subgroup of the center of G. Taking inspiration Every consistent quantum gravity theory is in the string from F theory [8], where the gauge group structure is landscape. encoded in the Mordell-Weil group of the elliptically Prototypical examples of string universality appear in 11 fibered compactification space [9–11], it appears that the and ten dimensions, where low-energy limits of M and allowed gauge groups G=Z are heavily restricted. For string theory give rise to the only consistent supergravity example, there are no 8D string compactifications, includ- theories. In ten dimensions (10D), this requires more subtle ing constructions beyond F theory, that have gauge group field theoretic arguments [3], or the incorporation of SUðnÞ=Zn, whereas SUðnÞ groups are ubiquitous. extended dynamical objects in the theory [4],to“drain” These restrictions are mathematically well known from the 10D supergravity swampland. the classification of elliptic K3 surfaces [12,13] (see also In lower dimensions, one observes a broader spectrum of [14]). Focusing on G a simply connected non-Abelian Lie string-derived supergravity theories, but these nevertheless group [more precisely, the most general gauge group is r r show some intricate structures not naively expected from ½G×Uð1Þ =Z×Zf, with Z ⊂ ZðGÞ, i.e., Z∩Uð1Þ ¼f1g— field theory considerations. For example, the rank rG of in this work, we consider constraints for Z exclusively, r the gauge group in known string compactifications is leaving a more detailed study including Zf ⊂Z½G×Uð1Þ ≅ r bounded by rG ≤ 26 − d in d dimensions and satisfies ZðGÞ×Uð1Þ , based on Ref. [11], for future work], the rG ≡ 1 mod 8 and rG ≡ 2 mod 8 in d ¼ 9 and d ¼ 8, geometry restricts Z; e.g., when Z ≅ Zl, then l ≤ 8. respectively. Likewise, not all gauge algebras have string Moreover, for each of the cases l ¼ 7, 8, there is exactly realizations. In particular, there are no string compactifi- one elliptic K3 on which F theory compactifies to an 8D 3 2 cations to 8D with soð2n þ 1Þ (n ≥ 3), f4, and g2. Again, theory with G¼SUð7Þ =Z7 and ½SUð8Þ ×SUð4Þ×SUð2Þ= novel swampland constraints [5,6] and refined anomaly Z8, respectively. Analogous restrictions on gauge groups arguments [7] reproduce these restrictions, thus downsizing also appear in heterotic compactifications [15]. A natural question is whether these restrictions reflect limitations of string theory or previously unknown con- sistency conditions of quantum gravity in 8D. Published by the American Physical Society under the terms of In this work, we show that the latter is the case. The key the Creative Commons Attribution 4.0 International license. G=Z Further distribution of this work must maintain attribution to is to realize a non-simply-connected group by gauging the author(s) and the published article’s title, journal citation, the Z 1-form center symmetry [16,17]. Thus, charting the and DOI. Funded by SCOAP3. swampland of gauge groups G=Z (in any dimension) can

0031-9007=20=125(21)=211602(6) 211602-1 Published by the American Physical Society PHYSICAL REVIEW LETTERS 125, 211602 (2020) be equivalently tackled by studying consistency conditions Besides the vector multiplets, 8D N ¼ 1 supergravity for gauging Z in gravitational theories. As we will discuss contains the gravity multiplet with a 2-form gauge field B2 below, in 8D N ¼ 1 theories, one such condition is the as one of its component fields [25]. The field strength H3 of absence of a mixed anomaly between the center 1-form this 2-form field obeys a modified Bianchi identity involv- symmetries and gauge transformations of higher-form ing the gauge fields of the theory: supergravity fields, which would obstruct the gauging of X Z. This rules out a vast set of seemingly acceptable 8D H3 ¼ dB2 þ miCSðAiÞ: ð1Þ N ¼ 1 theories without known string constructions and, in i particular, reproduce the geometric restrictions in models A with F-theory realization, thus providing further evidence Here, CSð iÞ are the Chern-Simons functionals for the G for string universality in 8D. gauge factor i. m The anomaly originates from a generalization of the The positive integers i associated with each gauge G “ ” G familiar θ term, θTrðF2Þ, in 4D. There, the fractional shift factor i, which we will refer to as the level of i, are of the instanton density TrðF2Þ, due to the presence of a a priori free parameters of the supergravity theory. They background field for the Z 1-form symmetry, breaks the 2π can be interpreted as the magnetic charge of gauge 2 instantons under B2—more apparent in the dual formu- periodicity of θ [16–19]. In higher dimensions, TrðF Þ lation, with B2 replaced by its magnetic-dual 4-form B4. couples to higher-form fields (e.g., to vector fields in 5D The most general Lagrangian contains the coupling [24] and tensors in 6D), which themselves possess gauge symmetries. These can lead to mixed anomalies with the Z X Z X Z 1-form center symmetry [20,21]. (See also [22] for recent B4 ∧ miTrðFi ∧ FiÞ≕ B4 ∧ miI4ðGiÞ; ð2Þ treatments of higher-form symmetries in higher-dimen- M8 i M8 i sional setups and [23] for an analysis of the global gauge group in 6D superconformal field theories.) where the trace is normalized such that the instanton density I4ðGiÞ¼1 for a one-instanton configuration of a The analogous coupling in 8D involves a 4-form B4. G M Crucially, while such a term is absent in a pure 8D i-bundle on a 4-manifold 4. G supersymmetric (as there are no appropriate The center 1-form symmetry of i can be coupled to CðiÞ Z G fieldsP B4 in the N ¼ 1 vector multiplet), the coupling a 2-form background field 2 which takes values in ð iÞ. m B ∧ F2 ðiÞ i i 4 Trð i Þ necessarily exists if one includes a When C2 is nontrivial, it twists the Gi bundle into a gravity multiplet, which contains a unique tensor B2 that Gi=ZðGiÞ bundle with second Stiefel-Whitney class is dual to B4 [24]. Supersymmetry further demands that ðiÞ w2½Gi=ZðGiÞ ¼ C2 [16,17] that contributes to Eq. (2): mi ≠ 0 [25]. A mixed anomaly involving the symmetries of B4, which must begauged, and the center 1-form symmetry Z ðiÞ I4½Gi=ZðGiÞ ≡ αG PðC Þ mod Z; ð3Þ can, therefore, obstruct the gauging of the latter. The i 2 vanishing of this anomaly is, hence, a necessary condition G=Z with P the Pontryagin square. This contribution is, in to obtain a non-simply-connected gauge group . α general, fractional due to the coefficients Gi derived in Remarkably, in models with mi ¼ 1, this condition turns Ref. [18], which we reproduce here: out to reproduce geometric properties of elliptic K3 mani- G Z G α folds. In combination with other swampland criteria that i ð iÞ Gi constrain the coefficients mi, this anomaly restrictsQ possible SUðnÞ Zn ðn − 1=2nÞ combinations of simply connected G ¼ i Gi and Z ⊂ SpðnÞ Z2 n=4 Z G in 8D. With this, we can consequently “drain” large 1 ð Þ Spinð2n þ 1Þ Z2 portions of the 8D swampland and make predictions in 2 Spinð4n þ 2Þ Z4 ð2n þ 1=8Þ corners of theory space where the global gauge group Spin 4n ðLÞ ðRÞ n=4; 1 ð Þ Z2 × Z2 ð 2Þ structure in corresponding string models is yet to be explored. E Z 2 6 3 3 Mixed anomaly forQ center symmetries in 8D E Z 3 7 2 4 supergravity.—Let G ¼ i Gi be a non-Abelian group, G where i are simple simply connected Lie groups with Analogous to the situation in 6D [20], the coupling (2) N 1 A algebra gi.In8D ¼ , the gauge potential i,withfield combines the fractional instanton configuration with a large strength Fi,ofthegi gauge symmetry comes in a vector gauge transformation B4 → B4 þ b4, with b4 a closed multiplet with adjoint . There are no other ðiÞ 4-form with integer periods, into a phase 2πiA b4;C massless charged matter states, so at low energies one ð 2 Þ Q for the partition function expects a discrete ZðGÞ¼ i ZðGiÞ 1-form symmetry [17]. Moreover, since the only massless fermions transform in a X Z A b ;CðiÞ m α b ∪ CðiÞ : real representation, there are no pure gauge anoma- ð 4 2 Þ¼ i Gi 4 Pð 2 Þ ð4Þ lies [26]. i M8

211602-2 PHYSICAL REVIEW LETTERS 125, 211602 (2020) R b ∪ CðiÞ ∈ Z b rank [6]), we can exhaustively scan for all possible groups While M8 4 Pð 2 Þ for arbitrary 4, the whole α G that have an anomaly-free Zl ⊂ ZðGÞ with given l,by expression is, in general, fractional due to Gi . By general- finding s triples of integers ðni;ki;miÞ such that izing the arguments presented in Refs. [20,27], the electri- cally charged objects for B4 would acquire a fractional Xs ni − 1 2 charge if this anomalous phase is nontrivial. Since this ki mi ∈ Z; with ki · l ≡ 0 mod ni: ð6Þ 2ni violates charge quantization, the fractional shift (4) cannot i¼1 be compensated and can be understood as an anomaly Clearly, the levels mi play an important role. From an between the large gauge transformations of B4 and the center effective field theory perspective, these are free parameters 1-form symmetries. As the former symmetry is gauged, one that define the theory. However, these parameters them- CðiÞ cannot allow for background fields 2 where Eq. (4) is selves are constrained by swampland criteria. By the nontrivial. Similar to the 6D setting [20], we expect that the completeness hypothesis [29], the 2-form field B2 couples violation of charge quantization is tied to the lack of to strings which carry localized degrees of freedom sensitive counterterms that could absorb this anomaly. Moreover, to the gauge group. These left-moving, charged excitations we expect that arguments developed in Ref. [19] suggest that on the string have to cancel the world volume anomalies there cannot be a topological Green-Schwarz mechanism arising due to anomaly inflow [4,30]. However, in d that cancels the above anomaly. [Note that Ref. [19] dis- dimension the left-moving central charge for such a string cusses precisely the 4D analog of the anomaly (4) involving is bounded by cL ≤ 26 − d. While each U(1) gauge factor the θ angle instead of B4.] c c 1 contributes to L with Uð1Þ ¼ , each non-Abelian simple In general, while the individual centers ZðGiÞ are gauge factor Gi with level mi contributes ci ¼½mi dimðGiÞ= anomalous, there can be a nontrivial subgroup Z ⊂ Q mi hi , with hi the dual Coxeter number of Gi. Hence, we Z G þ i ð iÞ that is anomaly-free. Assuming that there are have no other obstructions to switch on a background for this Z X m G subgroup of the center, or other breaking mechanisms, i dimð iÞ n ≤ 18: m h þ Uð1Þ ð7Þ this combination should be gauged, in line with common i i þ i lore that in quantum theories of gravity no global sym- metries (including discrete and higher-form symmetries) Combined with the constraint that the rank of the total gauge are allowed [2,28]. In turn, this leads to the gauge group of the 8D supergravity theory can be only 2, 10, or 18 group G=Z. [6], the mi are considerably restricted. In particular, it is Condition for anomaly-free center symmetries.—In the easily shown that, in the rank-18 case, all mi must be 1 and following, we will discuss how to determine subgroups all non-Abelian factors must have simply laced algebras (see Zl ≅ Z ⊂ ZðGÞ, for which a 1-form symmetry background Supplemental Material [31], Sec. A SM, for more details). has no fractional contribution (4)—a necessary condition to This is well known in string compactifications, where mi are gauge Z. Q Q the levels of the world sheet current algebra realizations of s s Z G Z k ; …;k ∈ Z spacetime gauge groups and are all mi 1 on the rank-18 Let ð Þ¼ i¼1 ni and ð 1 sÞ i¼1 ni be the ¼ generator for Z ≅ Zl. This means that l is the smallest branch of the N ¼ 1 moduli space. As we will see now, integer such that kil ≡ 0 mod ni for all i. The generic the anomaly matches known geometric limitations in the background for the ZðGÞ 1-form symmetry consists of F-theory realization of 8D rank-18 theories, which restricts CðiÞ Z Z G the possible global gauge group structures. In the lower-rank fields 2 for each ni factor of ð Þ. Specifying a specific background for a subgroup then amounts to correlating the cases, these conditions can constrain gauge groups, whose CðiÞ’ algebras and levels fit in constructions such as the a priori independent 2 s [18]. In particular, the back- Chaudhuri-Hockney-Lykken (CHL) string [32] but whose C Z ≅ Z CðiÞ k C ground 2 for l correspondsQ to setting 2 ¼ i 2. global structure is yet to be explored. G s n For concreteness, let ¼ i¼1 SUð iÞ. Then, the Anomaly-free centers in theories of rank 18.—All rank- anomalous phase (4) in a nontrivial C2 background of 18 N ¼ 1 supergravity theories with a known string origin the subgroup Z ⊂ ZðGÞ is have a construction via F theory [8], where physical Z features, including the global gauge group structure, are Xs n − 1 ðiÞ i 2 K3 Aðb4;C2 Þ¼ ki mi b4 ∪ PðC2Þ; ð5Þ encoded in the geometry of elliptically fibered surfaces 2ni M i¼1 8 [26,33,34]. In particular, there are beautiful arithmetic results [12] which asserts that F-theory compactifications kC k2 C where we used Pð Þ¼ Pð Þ. Thus, the anomaly with non-Abelian gauge group G=Z, where G consists only vanishes if the coefficient is integral. of SUðniÞ factors, must satisfy Note that the anomaly contribution of non-SU groups can be written as a sum of contributions from SUðnÞ Xs ni − 1 2 subgroups [18]. Therefore, by further restricting ourselves ki ≡ 0 mod Z ð8Þ 2ni to rank ðGÞ ≤ 18 (which is the 8D bound for the total gauge i¼1

211602-3 PHYSICAL REVIEW LETTERS 125, 211602 (2020) Q with ðk1; …;ksÞ ∈ i Z½SUðniÞ the generator of any arguments [6]. Groups in Eq. (11) have no known 8D Zl ⊂ Z subgroup. string realization at m ¼ 1. However, while Spð12Þ=Z2 and While we defer a more detailed explanation of the Spð16Þ=Z2 are excluded at any m due to the bound (7) [in geometric origin to this formula to Supplemental Material particular, (7) provides a physical explanation to the [31], Sec. B SM, it is obvious that it fully agrees limitation k ≤ 10 for spðkÞ gauge algebras known in 8D with the cancellation condition for every Zl subgroup string constructions], Spinð16Þ=Z2 does arise at m ¼ 2 as a of the center 1-form symmetry (6), as for rank-18 theories Wilson line reduction of the E8 CHL string. all levels are fixed to mi ¼ 1. We therefore find a deep More generally, at level m ¼ 2, the center anomaly in connection between the mixed anomaly of the super- conjunction with the bound (7) can rule out all G=Z gravity theory and the geometrical properties of F-theory theories with simple G except for compactifications. The constraint is particularly powerful when the order l 4 8 9 2 4 E SUð Þ; SUð Þ; SUð Þ; Spð Þ; Spð Þ; 7 ; of the gauged center subgroup is the power of a prime Z Z Z Z Z Z number. For such l ≥ 9, one can show that there are no 2 2 3 2 2 2 n ;k Spinð8Þ Spinð16Þ possible sets fð i iÞg for which the anomaly vanishes ; ; SOð2nÞ with 2 ≤ n ≤ 9; ð12Þ with gauge groups of rank ≤ 18.Forl ¼ 7, there is exactly Z2 Z2 one configuration with three simple non-Abelian factors, n1 ¼ n2 ¼ n3 ¼ 7 and ðk1;k2;k3Þ¼ð1; 2; 3Þ, correspond- all of which could, in principle, arise in CHL compacti- 3 ing to an SUð7Þ =Z7 theory. This agrees with the classi- fications [35]. We will leave an explicit verification and fications of K3 surfaces [12] for F-theory constructions analysis of the global gauge group in these types of 8D as well as possible heterotic realizations [15]. Likewise, in string models for future works. Note that SOð2nÞ (n odd) 3 the case l ¼ 8 ¼ 2 , the 1-form anomaly (8) allows for and Spð2Þ=Z2 seem to be ruled out in 8D by independent only G ¼ SUð8Þ1 ×SUð8Þ2 ×SUð4Þ ×SUð2Þ, into which swampland arguments [6], indicating mechanisms beyond Z k ;k ;k ;k the 8 subcenter embeds as ð 1 2 SUð4Þ SUð2ÞÞ¼ the anomaly (4) that break the 1-form center symmetry. It ð1; 3; 1; 1Þ. Furthermore, if we also take inspiration from would be interesting to find an explicit description for these geometric properties of K3 surfaces—there always is one breaking mechanisms. SUðniÞ factor with l dividing nj—we can show that there Discussion and outlook.—Using a mixed anomaly (4), N 1 are no possible configurations ðni;kiÞ for all l ≥ 10. This we have presented a necessary condition for an 8D ¼ also matches the dual heterotic constructions [15]. theory with given non-Abelian gauge algebras Qgi at level mi Predictions for simple groups.—To further showcase the to have a non-simply-connected gauge group ½ i Gi=Z.In constraining power of the field-theoretic anomaly argu- combination with a set of swampland criteria that restrict ment, we use Eq. (4) to rule out 8D N ¼ 1 theories with the gauge rank and the levels mi, this condition rules out a gauge group G=Z, where G is a simple Lie group and vast set of possible gauge groups for 8D theories. The Z ⊂ ZðGÞ a nontrivial subgroup. For G with m ¼ 1 and constraints are especially powerful for theories of rank 18, rank ðGÞ ≤ 18,anyG=Z is inconsistent except where they reduce to known geometric properties of elliptic K3 surfaces. As these properties control the global gauge SUð16Þ SUð18Þ Spinð32Þ group structure in F-theory compactifications, the anomaly ; ; ; ð9Þ Z2 Z3 Z2 provides a purely physical explanation for the intricate patterns of realizable gauge groups in F theory. The Spð4Þ Spð8Þ SUð8Þ SUð9Þ anomaly can further make predictions for inconsistent ; ; ; ; ð10Þ Z2 Z2 Z2 Z3 models in lower-rank cases, where the global gauge group structure in the corresponding string compactifications is Spinð16Þ Spð12Þ Spð16Þ yet to be explored systematically. ; ; : ð11Þ Z2 Z2 Z2 We stress that the absence of the anomaly (4) is only a necessary, but not sufficient condition for the gauge group to The groups (9) indeed correspond to the only cases with be G=Z. Indeed, for F-theory constructions of the non- simple G realizable via F theory on elliptic K3’s. The simply-connected gauge groups (9), there also exist K3 groups in Eq. (10) are subgroups of Sp(10), which at m ¼ 1 surfaces that realize the simply connected versions in F can be constructed from the CHL string [32]. Note that this theory [13]. There are also other instances where both G and rules out all other SpðkÞ=Z2ðk<10Þ theories, which G=Z are realized in different compactifications; this is also seemed consistent based on the perturbative CHL spectrum confirmed in the heterotic picture [15]. As the center Z in all 35] ]. As we are not aware of any systematic study of the these cases is nonanomalous, this is consistent with our global gauge group structure in CHL compactifications, findings. At the same time, it is pointing toward additional we view this as a prediction based on the 1-form anomaly breaking mechanisms, e.g., in terms of massive states (4), which is also consistent with other swampland charged under Z. It would be interesting to investigate if

211602-4 PHYSICAL REVIEW LETTERS 125, 211602 (2020) these mechanisms are captured by an effective description between higher-form anomalies and the swampland ideas involving the 1-form center symmetry. [6] that also rule out certain non-simply-connected gauge There are also nonanomalous cases that have no reali- groups. These insights can hopefully lead to a complete zation in known classes of 8D string models. A particular understanding of the global gauge group structure and set of such cases are products G1=Z1 × G2=Z2 of anomaly- prove full string universality for non-simply-connected free factors, which would again be anomaly-free. For groups in 8D. 2 example, the anomaly-free gauge group ½SUð5Þ =Z5 × 4 2 4 We thank Miguel Montero and Cumrun Vafa for helpful SU 2 =Z2 SU 5 ×SU 2 =Z10 as the non- ½ ð Þ ¼½ ð Þ ð Þ comments. M. D. and L. L. further thank Fabio Apruzzi for Abelian part of a rank-18 theory (and, thus, all mi 1) ¼ valuable discussions and collaboration on related work has no string realization. As we have mentioned above, the [20]. M. D. is grateful to Miguel Montero for valuable F-theory geometry would forbid this case, because there is discussions. We further thank the referee for pointing out no SUðnÞ factor with 10 dividing n. Currently, we do not the importance of the parameters mi. The work of M. C. is know an adequate physical argument providing the same supported in part by the Department of Energy (HEP) Grant restriction. In terms of identifying gauged Zl center No. DE-SC0013528, the Fay R. and Eugene L. Langberg symmetries, one plausible possibility is the existence of Endowed Chair, the Slovenian Research Agency (ARRS some mechanism that forces the presence of a U(1) gauge No. P1-0306), and the Simons Foundation Collaboration factor into which, similarly to the hypercharge in the on “Special Holonomy in Geometry, Analysis, and , that Zl embeds. Such a theory would Physics,” ID: 724069. The work of M. D. is supported not be in contradiction to F-theory models, as center by the individual Deutsche Forschungsgemeinschaft Grant symmetries embedded in U(1)’s have a different geometric No. DI 2527/1-1. origin [11] (see [36] for direct implications for 4D particle physics models) not subject to the restriction (8). Moreover, in 8D F theory, there are additional sources for U(1) factors K3’ [harmonic (1,1)-forms on s that are not algebraic], [1] C. Vafa, arXiv:hep-th/0509212. whose center mixing with non-Abelian gauge factors needs [2] T. D. Brennan, F. Carta, and C. Vafa, Proc. Sci., TASI2017 further investigation. To complete the geometric picture (2017) 015 [arXiv:1711.00864]; E. Palti, Fortschr. Phys. 67, from the field-theoretic side, one must also extend the 1900037 (2019). discussion of anomalies to include U(1) gauge sectors, [3] A. Adams, W. Taylor, and O. DeWolfe, Phys. Rev. Lett. 105, which we defer to future studies. 071601 (2010). We further suspect that other discrete symmetries of the [4] H.-C. Kim, G. Shiu, and C. Vafa, Phys. Rev. D 100, 066006 theory can interact nontrivially with 1-form center sym- (2019). [5] H.-C. Kim, H.-C. Tarazi, and C. Vafa, Phys. Rev. D 102, metries, leading to further constraints on the gauge group 026003 (2020). structure. For example, it has been pointed out [37] that the E E [6] M. Montero and C. Vafa, arXiv:2008.11729. gauge symmetry of the 8 × 8 heterotic string should be [7] I. García-Etxebarria, H. Hayashi, K. Ohmori, Y. Tachikawa, augmented by an outer automorphism Z2 exchanging the and K. Yonekura, J. High Energy Phys. 11 (2017) 177. E8 factors, so that the gauge group is ðE8 × E8Þ⋊Z2.In [8] C. Vafa, Nucl. Phys. B469, 403 (1996); D. R. Morrison and fact, the 9D CHL string arises as the S1 reduction with C. Vafa, Nucl. Phys. B473, 74 (1996); B476, 437 (1996). holonomies in this Z2. Such an identification would also be [9] P. S. Aspinwall and D. R. Morrison, J. High Energy Phys. 4 4 07 (1998) 012. possible for, e.g., ½SUð2Þ =Z2 × ½SUð2Þ =Z2, all at [10] C. Mayrhofer, D. R. Morrison, O. Till, and T. Weigand, mi ¼ 1, which in 8D is free of the anomaly (4), but not J. High Energy Phys. 10 (2014) 016. realized in terms of a string compactification. If one could [11] M. Cvetič and L. Lin, J. High Energy Phys. 01 (2018) 157. establish other field theory or swampland arguments for [12] R. Miranda and U. Persson, Math. Z. 201, 339 (1989); Z why the 2 outer automorphism must be gauged in this in Problems in the Theory of Surfaces and Their Classi- case, there could be other mixed anomalies involving the fication (Academic Press, New York, 1991), pp. 167–192; 1-form symmetries such that only a diagonal Z2 center Sympos. Math., XXXII (Academic Press, London, 1991). 8 survives, leading to the realizable ½SUð2Þ =Z2 theory. [13] I. Shimada, arXiv:math/0505140. Finally, to fully classify the global gauge group structure [14] N. Hajouji and P.-K. Oehlmann, J. High Energy Phys. 04 in 8D N ¼ 1 theories based on the 1-form anomaly (4),it (2020) 103. will be important to have more stringent constraints on the [15] A. Font, B. Fraiman, M. Graña, C. A. Núñez, and H. Parra, arXiv:2007.10358. possible levels mi for given simple gauge factors Gi. While m 1 [16] A. Kapustin and N. Seiberg, J. High Energy Phys. 04 (2014) for rank-18 theories, Eq. 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