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LETTERS 123, 161601 (2019)

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Anomaly of the Electromagnetic Duality of Maxwell Theory

Chang-Tse Hsieh ,1,2 Yuji Tachikawa ,1 and Kazuya Yonekura 3 1Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, Kashiwa, Chiba 277-8583, Japan 2Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan 3Department of Physics, Tohoku University, Sendai 980-8578, Japan (Received 19 August 2019; published 14 October 2019)

We consider the (3 þ 1)-dimensional Maxwell theory in the situation where going around nontrivial paths in the spacetime involves the of the duality transformation exchanging the electric and the magnetic field, as well as its SLð2; ZÞ generalizations. We find that the anomaly of this system in a particular formulation is 56 times that of a Weyl fermion. This result is derived in two independent ways: one is by using the bulk protected topological phase in (4 þ 1) dimensions characterizing the anomaly, and the other is by considering the properties of a (5 þ 1)-dimensional superconformal field theory known as the E- theory. This anomaly of the Maxwell theory plays an important role in the consistency of .

DOI: 10.1103/PhysRevLett.123.161601

Introduction.—Every physicist knows that the electro- chiral and the (2 þ 1)-dimensional Uð1Þ1 magnetic field is described classically by the Maxwell Chern-Simons theory and its generalization to the equation and that it is invariant under the electromagnetic [ð4n þ 1Þþ1]-dimensional self-dual form field and the duality S∶ ðE; BÞ ↦ ðB; −EÞ [1]. The properties of the [ð4n þ 2Þþ1]-dimensional bulk theory studied, e.g., in electromagnetic duality in quantum theory might not be as [20–26]. The essential point is that the (3 þ 1)-dimensional well known to physicists in general and, in fact, are not very Maxwell theory with a nontrivial background for its duality well understood in the literature. This is particularly true symmetry is a self-dual field, and we can utilize the when going around a nontrivial path in the spacetime techniques developed in the papers listed above to study results in a duality transformation [2]. In this Letter, we it. One of our main messages is that the subtle and interesting focus on a feature of the Maxwell theory and its duality issues concerning the self-dual fields studied in the past symmetry in such a situation, namely, the fact that it has a already manifest themselves in the case of the Maxwell quantum anomaly [14,15], which we explicitly determine. theory once the nontrivial background for its duality We recall that a quantum theory in d þ 1 dimensions symmetry is turned on. with a symmetry group G can have a quantum anomaly, in Before proceeding, we note that the electromagnetic the sense that its partition function has a controllable phase duality group in the quantum theory is, in fact, the 2- ambiguity. Our modern understanding is that such a theory dimensional special linear group SLð2; ZÞ over the integers is better thought of as living on the boundary of a symmetry acting on lattice Z2 of the electric and magnetic charges. Its ð þ 1Þþ1 protected topological (SPT) phase in the [ d ]- effect on the Maxwell theory on a curved manifold was dimensional bulk. It was noticed in the last few years in carefully analyzed in [27,28], and it was interpreted as a – [14,16 19] that a version of the Maxwell theory (often mixed SLð2; ZÞ- in [15]. Our result in called all-fermion electrodynamics, where all particles of this Letter can be considered as the determination of the odd charge are fermions) has a global gravitational pure SLð2; ZÞ part of the anomaly. anomaly and lives on the boundary of a certain bulk Our computation shows that the anomaly of the Maxwell SPT phase. As we will see, this result is a special case theory is 56 times that of a Weyl fermion in a certain precise of the anomaly and the corresponding bulk SPT phase that formulation of the duality. Where does this number 56 we find for the duality symmetry. come from? We will provide an answer using the property We study the anomaly and the bulk SPT phase by of a (5 þ 1)-dimensional superconformal field theory imitating the relationship between the (1 þ 1)-dimensional originally found in [29,30] and known as the E-string theory; the name comes from the fact that it has global symmetry. The E-string theory has two branches of vacua: Published by the American Physical Society under the terms of called the tensor branch and the Higgs branch. On the the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to Higgs branch, the E8 symmetry is Higgsed to , which the author(s) and the published article’s title, journal citation, acts on 28 fermions via its 56-dimensional fundamental and DOI. Funded by SCOAP3. representation; this is possible because a pseudoreal

0031-9007=19=123(16)=161601(6) 161601-1 Published by the American Physical Society PHYSICAL REVIEW LETTERS 123, 161601 (2019) representation R with dim R ¼ 2k can act on k fermions in gauge field A into a sum of the flat but topologically (5 þ 1) dimensions because the spin representation S in nontrivial part and the topologically trivial but nonflat part. (5 þ 1) dimensions is pseudoreal and we can impose the Assuming, for simplicity, that flat connections on M3 are Majorana condition on R ⊗ S. When one moves to the isolated, we have tensor branch, the E8 symmetry is restored and a self-dual Z R 2 tensor field appears. By compactifying this system on T , πi ðA=2πÞðF=2πÞ Z ð1Þ ðM3Þ¼ ½DA e one finds that one Maxwell field is continuously connected U CS top:trivial to 56 Weyl fermions, showing that they should have the X R πi ðA=2πÞðF=2πÞ same anomaly. The electromagnetic duality is formulated × e : ð1Þ as the SLð2; ZÞ acting on this torus T2, and therefore is A∶flat geometrized in this formulation. This means that both the Let us rewrite its phase. purely SLð2; ZÞ part and the mixed gravitational-SLð2; ZÞ The phase of the first term can be written in terms of the part of the (3 þ 1)-dimensional anomaly come from the eta invariant of the signature operator d: purely gravitational anomaly of the (5 þ 1)-dimensional Z R theory. These statements about the anomaly are valid if the 1 π ð 2πÞð 2πÞ 1 Arg ½DA e i A= F= ¼ − η : ð2Þ E8 background field is turned off. 2π top:trivial 8 signature The rest of the Letter is organized as follows. We start by recalling how the anomaly of a (1 þ 1)-dimensional chiral Here and below, the equality of the phase is modulo one boson is captured by the phase of the partition function of the and is simply denoted by the equal sign (¼). The phase of (2 þ 1)-dimensional Uð1Þ Chern-Simons theory at level 1. the second term can be rewritten as We outline the path integral computation of its phase, as well 1 X as how this can be matched with the anomaly of a (1 þ 1)- ð Þ ≕ ð Þ ð Þ 2π Arg q c Arf q ; 3 2 dimensional chiral fermion. We then adapt this discussion to c∈H ðM3;ZÞ the anomaly of the (3 þ 1)-dimensional Maxwell theory and the corresponding (4 þ 1)-dimensional bulk BdC theory. where c ¼ c1ðFÞ is theR first Chern class of the gauge We will see that the anomaly computed in this way ð Þ ≔ πi ðA=2πÞðF=2πÞ ð2 ZÞ bundle, and q c e . reproduces the known anomaly when the SL ; back- We note that qðcÞ is simply the exponentiated level-1 ground is trivial. We then consider the case of the nontrivial classical action evaluated at a flat A. As recalled above, ð2 ZÞ 5 Z ¼ 2 SL ; backgrounds on S = k for k , 3, 4, and 6; and defining it requires something more than an oriented we note that the resulting phase is equal to 56 times that of a manifold and the integration on it. Mathematically, q is charged Weyl fermion. This plays an important role in the − known as a quadratic refinement of the torsion pairing on consistency of the O3 plane and its generalizations. Finally, 2 H ðM3; ZÞ. The Arf invariant ArfðqÞ is defined by the we explain why the anomaly of the Maxwell theory has to be equation above and is known to take values in one eighth of 56 times that of a charged Weyl fermion in terms of the six- an integer. We end up with the formula dimensional superconformal field theory known as the E-string theory. More details will be provided in a longer 1 1 ð Þ¼− η þ ð Þ ð Þ version of the Letter [31]. 2π ArgZUð1ÞCS M3 8 signature Arf q : 4 Warm-up: Anomaly of (1 þ 1)-dimensional chiral boson in terms of (2 þ 1)-dimensional U(1) Chern-Simons.—We Let us now recall that a chiral boson can be fermionized. start by recalling the well-understood case of the anomaly Then, the bulk theory can be taken to be the (2 þ 1)- of the (1 þ 1)-dimensional chiral boson at the free fermion dimensional fermion with infinite mass, for which the radius. This theory naturally lives at the boundary of the partition function has the phase [38] (2 þ 1)-dimensional Uð1Þ Chern-Simons theory at level ¼ 1 − ¼ 1 k R , for which the Euclidean action is Sk¼1 ArgZ ðM3Þ¼η : ð5Þ πi ðA=2πÞðF=2πÞ [20,32,33]. The anomaly is then char- 2π fermion fermion acterized by the partition function of this Chern-Simons The values of η and η on lens spaces are known theory on closed 3-dimensional manifolds M3. signature fermion 3 − ¼ in the literature, e.g., [39]. For example, on M3 ¼ S =Z2, LetR us recall that the action at level 2, Sk¼2 η ¼ 0 ð Þ η 1 8 2πi ðA=2πÞðF=2πÞ, is well-defined modulo 2πi when signature ; whereas Arf q and fermion can be either = 3 the manifold is oriented. However, there is a problem in or −1=8, depending on the . On M3 ¼ S =Z3, η ¼ 2 9 ð Þ¼1 4 η ¼ 2 9 dividing it by two. To make the action Sk¼1 well-defined signature = ,Arfq = , and fermion = because modulo 2πi, it is known that we need to pick a spin there is a unique spin structure. We indeed confirm structure [34]. Once this is done, the path integral can be 1 performed explicitly because the theory is free. The details − η þ ArfðqÞ¼η ; ð6Þ are given, e.g., in [32,35–37]. Very roughly, we split the 8 signature fermion

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2 which can be proved using a mathematical result [40].We where B; C ∈ H ðM5; R=2πZÞ are the background fields η note that signature is independent of the spin structure but for the electric and magnetic 1-form Uð1Þ symmetry of ArfðqÞ does depend on the spin structure. In other words, the Maxwell theory [43], whichR we chose to be flat. Its the spin structure provides us the quadratic refinement. Arf invariant is computed to be ðB=2πÞβðC=2πÞ, where β — 2 The anomaly of the Maxwell theory. The analysis is the Bockstein homomorphism β∶ H ðM5; R=ZÞ → 1 þ 1 3 of the anomaly of the ( )-dimensional chiral boson H ðM5; ZÞ; the Bockstein homomorphism β can roughly we recalled above was generalized to the [ð4n − 3Þþ1]- be regarded as the exterior derivative d when it acts on dimensional self-dual form fields in [41] at the perturbative torsion elements of cohomology groups. The end result level. The study of the corresponding [ð4n − 2Þþ1]- is that 2 þ 1 dimensional theory in the bulk, generalizing the ( )- Z dimensional Chern-Simons theory, was carried out in 1 B C ð Þ¼ β ð Þ detail inR [21–26]. The bulk theory has the action 2π ArgZBdC M5 2π 2π : 9 −S ¼ πi ðA=2πÞdðA=2πÞ, where A is now a (2n − 1)- 2n−1ð RÞ¼0 form gauge field. Assuming H M4n−1; , the This reproduces a known result. Indeed, the mixed phase of the partition function still has the form of anomaly is known to be of the form Eq. (4), where q is now a quadratic refinement of the Z 2n torsion pairing on H ðM4 −1; ZÞ, and its choice is not B C n 2π obviously related to the choice of the spin structure. i 2π d 2π ; M5 Here, we are more interested in the (3 þ 1)-dimensional Maxwell theory. The natural generalization in this case is to for which the mathematically precise formulation [44] consider the bulk theory with the action reduces to Eq. (9) when we only consider flat fields. Z Furthermore, we can take B=2π ¼ C=2π ¼ w2, where w2 − ¼ π B C − C B is the Stiefel-Whitney class of the spacetime, which is here S i d d ; 2 2π 2π 2π 2π regarded as an element of H ðM5; R=ZÞ by using Z2 → R=Z. The Maxwell theory with this coupling is also where B and C are two 2-form gauge fields to be path known as the all-fermionR electrodynamics,R and it has the integrated over [14]. This action has the SLð2; ZÞ sym- gravitational anomaly 2πi w2βw2 ¼ πi w2w3 [18,19]. metry acting on ðB; CÞ, which corresponds to the duality Let us next consider the case when a nontrivial SLð2; ZÞ symmetry of the Maxwell theory [14]. We can and will background is present. We can choose the symmetry introduce the background gauge field ρ for this SLð2; ZÞ structure on M5 to consider, such as spin × SLð2; ZÞ or symmetry, which means that there is a nontrivial duality ð2 ZÞ ≔ ð2 ZÞ spin-Mp ; [ spin ×Z2 Mp ; ], distinguished by transformation when going around a nontrivial loop in whether C2 ¼þ1 or ¼ð−1ÞF. Here, C ∈ SLð2; ZÞ is spacetime. The phase of the partition function is then the charge conjugation C∶ ðE; BÞ ↦ −ðE; BÞ, and the metaplectic group Mpð2; ZÞ is the double cover of the 1 1 ð Þ¼− η þ ð ÞðÞ group SLð2; ZÞ. We will focus on the latter case in this ArgZBdC M5 signature Arf q 7 2π 4 Letter because it has a natural connection to the (6 þ 1)- dimensional CdC theory on a spin seven manifold, as we where the eta invariant is now for the signature operator d 2 will see. Canonical examples of M5 associated with this acting on the differential forms tensored with ðZ Þρ, and q 5 symmetry structure are S =Zk, k ¼ 2; 3; 4, and 6, where 5 is now the quadratic refinement of the natural torsion going around the generator of π1ðS =Z Þ¼Z comes with 3( ðZ2Þ ) ðZ2Þ k k pairing on H M5; ρ . Here, ρ signifies the the duality action by an element g of order k in SLð2; ZÞ, ð2 ZÞ ρ 5 coefficient system twisted by the SL ; bundle . The whereas S =Zk is not spin for even k; it has a natural spin- eta invariant of the signature operator with such a twist and 5 3 Z2k structure for any k by embedding S =Zk ⊂ C =Zk. its reduction from higher dimensions were considered Then, we get the spin-Mpð2; ZÞ structure by embedding earlier in the mathematical literature; see, e.g., [42]. Z2k ⊂ Mpð2; ZÞ. The results of explicit computations for Let us first consider the case where we do not have the Eq. (7) are tabulated in Table I. When there are multiple ð2 ZÞ SL ; background. In this case, the signature eta choices for g or q, we choose a particular one. Other invariant simply vanishes, and only the Arf invariant quadratic refinements correspond to different background contributes. Recall that a quadratic refinement is simply fields ðB; CÞ for electromagnetic 1-form . the classical action evaluated on flat B and C. Then, a When k ¼ 2, the relevant element in SLð2; ZÞ is just the general quadratic refinement can be written as charge conjugation symmetry C. This case has the anomaly 5 Z Z Z ð1=2πÞArgZ ¼ 1=2 on S =Z2. This is responsible for the B dC dB C B dC 1 2 þ þ ; ð8Þ difference of = of the Ramond-Ramond (RR) charges of 2π 2π 2π 2π 2π 2π the O3þ plane and the O3− plane in type-IIB string theory

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5 TABLE I. Partition functions and related data on S =Zk. SLð2; ZÞ bundle is the dimensional reduction of the 2 (6 þ 1)-dimensional CdC theory on M7, which is the T 5 Z 5 Z 5 Z 5 Z S = 2 S = 3 S = 4 S = 6 bundle over M5. η 1 1 14 We now embed this theory of a self-dual tensor field into signature 0 − 9 − 2 − 9 3 2 2 H (M5; ðZ Þρ) ðZ2Þ Z3 Z2 Z1 the tensor branch of the E-string theory [29,30]. The E- ð Þ 1 1 1 string theory is a (5 þ 1)-dimensional theory realized in M Arf q þ 2 − 4 þ 8 0 1 Z þ 1 − 2 þ 1 þ 7 theory, with two continuous families of vacua. One family 2π Arg 2 9 4 18 5 η − 1 − 1 − 5 − 35 of vacua is called the tensor branch, which describes an M fermion 16 9 32 144 close to the spacetime boundary of M theory carrying the E8 gauge symmetry [52,53]. On the tensor branch, the low energy theory consists of the self-dual tensor field with [45]. As explained in [46], for the consistency of the theory, some additional fields. The other family of vacua is called the fractional part of the RR charge must be exactly the Higgs branch, which describes an of the E8 3 5 Z negative of the anomaly of a D brane living on S = 2. E8 E7 gauge symmetry. The instanton breaks to , and it The background ðB; CÞ produced by O3 is such that only produces some chiral fermions as zero modes of the − the O3 leads to the anomaly of the Maxwell theory, instanton. A nontrivial fact in M theory is that these two explaining the difference of the RR charges; we note families of vacua are continuously connected; an M5 brane 1 4 3þ that the charge = of the O plane was already explained put on the spacetime boundary can become an E8 instanton. by the fermion anomaly [46]. We can also check that The transition point is a strongly coupled conformal field the resulting ð1=2πÞArgZ for other k is exactly what theory. is necessary to reproduce the RR charge of the N ¼ 3 We start from the M5 brane close to the spacetime S-fold [47,48]. boundary, and we transform it into an E8 instanton of Let us now consider the infinitely massive fermions nonzero size. In this process, the low energy theory is encoding the anomaly of a (3 þ 1)-dimensional Weyl changed from that of the tensor branch, containing the self- Z fermion of unit charge under 2k, which was studied in dual tensor field, into that of the Higgs branch, containing 5 [49–51]. The corresponding eta invariants on S =Zk are 28 ¼ 56=2 chiral fermions in (5 þ 1) dimensions trans- also tabulated in Table I. We can check that the relation forming under the fundamental 56-dimensional represen- tation of E7. Because this is a continuous process, the 1 anomaly at the start and the anomaly at the end should be − η þ ArfðqÞ¼56η ð10Þ 4 signature fermion the same; previously, the same argument was used to compute the anomaly polynomial of the E-string theory holds for the choices of the Arf invariants given in in [54] (which reproduced earlier results in [55–57]), but Table I. the same statement is true, even for the subtler anomalies Why 56?—Relation (10) about the anomaly of the we are discussing now. There are also some additional Maxwell theory and 56 Weyl fermions in (3 þ 1) fields on the tensor and Higgs branches, but their anomalies dimensions reminds us of the relation [Eq. (6)] about are manifestly the same on both sides, and so we can match the anomaly of a chiral boson and a chiral fermion in the anomaly of the self-dual tensor field and the 28 chiral (1 þ 1) dimensions. In the latter case, the equality should fermions. evidently hold because a chiral fermion can be bosonized Because one chiral fermion in (5 þ 1) dimensions gives to a chiral boson in (1 þ 1) dimensions. It also explained rise to two chiral fermions in (3 þ 1) dimensions, we how and why the spin structure could be used to define conclude that the anomaly of the Maxwell theory, formu- the quadratic refinement necessary to formulate the lated as the T2 compactification of the (5 þ 1)-dimensional integrand of the Uð1Þ Chern-Simons theory. In (3 þ 1) self-dual field with the trivial E8 background, should be dimensions, however, the Maxwell theory and 56 Weyl equal to that of the 56 Weyl fermions. See Fig. 1 for a fermions are two clearly different theories. What is summary of what we have described. the relation? How and why does the spin (or, more precisely, the spin-Z2k) structure provide the necessary quadratic refinement? One explanation is provided, somewhat surprisingly, by supersymmetric physics in Tensor branchE-string Higgs branch (5 þ 1) dimensions. Consider a self-dual tensor field in (5 þ 1) dimensions. (5+1)D Self-dual tensor 28 fermions 2 T 2 Its dimensional reduction on T gives rise to the Maxwell continuous 3 þ 1 ð2; ZÞ (3+1)D Maxwell 56 fermions theory in ( ) dimensions, geometrizing the SL deformation duality symmetry of the Maxwell theory. Correspondingly, the (4 þ 1)-dimensional BdC theory on M5 coupled to an FIG. 1. Maxwell to 56 fermions via E-string theory.

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’ If we turn on a nontrivial E7 background AE7 on the situation remains unanswered to the authors knowledge. fermion side, the data are translated on the self-dual tensor There is also a series of interesting papers on the flux sectors side into the background 3-form field C, which couples to of the Maxwell theory by Moore and co-workers [9–11], the dynamical self-dual tensor field, which is basically which are related to the inherent self-dual nature of the given by the Chern-Simons term constructed from A . Maxwell theory. Another intriguing scenario is to consider a E7 Maxwell theory with dynamical “duality gauge fields,” When A is flat, this determines a quadratic refinement E7 which might be thought of as a generalization of the Alice 6 þ 1 required to define the ( )-dimensional CdC theory. In electrodynamics [12,13], where the charge conjugation particular, the trivial E7 background, which is available on C ¼ S2 is also gauged. any manifold, provides a canonical quadratic refinement for [3] S. Deser and C. Teitelboim, Duality transformations of the (6 þ 1)-dimensional CdC theory; and this construction Abelian and non-Abelian gauge fields, Phys. Rev. D 13, only requires the spin structure. This point was already 1592 (1976). essentially made in [58]. [4] S. Deser, Off-shell electromagnetic duality invariance, J. Because this explanation of Eq. (10) requires a lot of Phys. A 15, 1053 (1982). information from string and M theories, it would be of [5] O. J. Ganor and Y. P. Hong, Selfduality and Chern-Simons independent interest to check the equality [Eq. (10)]bya theory, arXiv:0812.1213. 3 þ 1 4 þ 1 [6] O. J. Ganor, Y. P. Hong, and H. S. Tan, Ground states of direct analysis in ( ) and ( ) dimensions. To N ¼ 4 translate the analysis in (5 þ 1) dimensions to the study S-duality twisted super Yang-Mills theory, J. High 2 Energy Phys. 03 (2011) 099. of the Maxwell theory, we need to require that the T [7] O. J. Ganor, Y. P. Hong, R. Markov, and H. S. Tan, Static ð2 ZÞ bundle over M5 specified by the SL ; background is charges in the low-energy theory of the S-duality twist, J. equipped with a spin structure. This means that the sym- High Energy Phys. 04 (2012) 041. metry structure we consider is a spin-Mpð2; ZÞ structure. [8] O. J. Ganor, N. P. Moore, H.-Y. Sun, and N. R. Torres- According to the classification theorem [59–62], Chicon, Janus configurations with SLð2; ZÞ-duality twists, the anomaly of any system with this symmetry is classified strings on mapping tori and a tridiagonal determinant spin-Mpð2;ZÞ formula, J. High Energy Phys. 07 (2014) 010. by the dual of Ω ¼ Z9 ⊕ Z32 ⊕ Z2, which is the 5 [9] D. S. Freed, G. W. Moore, and G. Segal, The uncertainty of bordism group for closed 5-manifolds with spin-Mpð2; ZÞ 5 5 fluxes, Commun. Math. Phys. 271, 247 (2007). structures and is generated by S =Z3, S =Z4, and 5 0 5 5 [10] D. S. Freed, G. W. Moore, and G. Segal, Heisenberg groups ½ðS =Z4Þ þ 9ðS =Z4Þ, respectively, where ðS =Z4Þ and and noncommutative fluxes, Ann. Phys. (Amsterdam) 322, 5 0 ðS =Z4Þ both have the spin-Z8 structure coming from the 236 (2007). 5 3 embedding S =Z4 ⊂ C =Z4 but with different actions of Z4 [11] A. Kitaev, G. W. Moore, and K. Walker, Noncommuting given by diagði; i; i; iÞ. We have not directly determined flux sectors in a tabletop experiment, arXiv:0706.3410. [12] A. S. Schwarz, Field theories with no local conservation of which quadratic refinement comes from the trivial E7 field, but we have checked that, for a suitable choice, we have the the electric charge, Nucl. Phys. B208, 141 (1982). [13] M. Bucher, H.-K. Lo, and J. Preskill, Topological approach equality [Eq. (10)] for each case, providing a strong check of to Alice electrodynamics, Nucl. Phys. B386, 3 (1992). our identification [Eq. (10)]. [14] S. M. Kravec and J. McGreevy, A The authors thank Nati Seiberg for useful comments on a Generalization of the Fermion-Doubling Theorem, Phys. draft of this Letter. Y. T. also thanks Yu Nakayama for Rev. Lett. 111, 161603 (2013). pointing out the relevance of papers by Ganor and co- [15] N. Seiberg, Y. Tachikawa, and K. Yonekura, Anomalies of duality groups and extended conformal manifolds, Prog. workers [5–8] to the authors. C.-T. H. and Y. T. are in part Theor. Exp. Phys. (2018), 073B04. supported by the WPI Initiative, MEXT, Japan at IPMU, [16] C. Wang, A. C. Potter, and T. Senthil, Classification of and the University of Tokyo. C.-T. H. is also supported in interacting electronic topological insulators in three part by JSPS KAKENHI Grant-in-Aid (Early-Career dimensions, Science 343, 629 (2014). Scientists) No. 19K14608. Y. 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(World Scientific, duality: Eðx þ L; y; zÞ¼Bðx; y; zÞ and Bðx þ L; y; zÞ¼ Singapore, 2005), Vol. 2, pp. 1606–1647. −Eðx; y; zÞ. This particular setup was studied by Ganor [21] D. Belov and G. W. Moore, Holographic action for the and co-workers [5–8], but what happens in a more general self-dual field, arXiv:hep-th/0605038.

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