Anomaly of the Electromagnetic Duality of Maxwell Theory
IIPMU-19-0068TU-1088
Anomaly of the Electromagnetic Duality of Maxwell Theory
Chang-Tse Hsieh,
1, 2
Yuji Tachikawa, and Kazuya Yonekura Kavli Institute for the Physics and Mathematics of the Universe (WPI),University of Tokyo, Kashiwa, Chiba 277-8583, Japan Institute for Solid State Physics, University of Tokyo, Kashiwa, Chiba 277-8581, Japan Department of Physics, Tohoku University, Sendai 980-8578, Japan
We consider the ( )-dimensional Maxwell theory in the situation where going around nontrivial paths inthe spacetime involves the action of the duality transformation exchanging the electric field and the magneticfield, as well as its
SL(2 , Z ) generalizations. We find that the anomaly of this system in a particular formulationis 56 times that of a Weyl fermion. This result is derived in two independent ways: one is by using the bulksymmetry protected topological phase in dimensions characterizing the anomaly, and the other is by con-sidering the properties of a ( )-dimensional superconformal field theory known as the E-string theory. Thisanomaly of the Maxwell theory plays an important role in the consistency of string theory. INTRODUCTION
Every physicist knows that the electromagnetic field is de-scribed classically by the Maxwell equation, and that it isinvariant under the electromagnetic duality S : ( E , B ) (cid:55)→ ( B , − E ) . The properties of the electromagnetic duality inquantum theory might not be as well known to physicists ingeneral and, in fact, are not very well understood in the litera-ture. This is particularly true when going around a nontrivialpath in the spacetime results in a duality transformation. Inthis Letter, we focus on a feature of the Maxwell theory andits duality symmetry in such a situation, namely the fact thatit has a quantum anomaly [12, 13], which we explicitly deter-mine.We recall that a quantum theory in d +1 dimensions with asymmetry group G can have a quantum anomaly, in the sensethat its partition function has a controllable phase ambigu-ity. Our modern understanding is that such a theory is betterthought of as living on the boundary of a symmetry protectedtopological (SPT) phase in the [( d +1)+1] -dimensional bulk.It was noticed in the last few years in [12, 14–17] that a ver-sion of the Maxwell theory (often called the all-fermion elec-trodynamics, where all particles of odd charge are fermions) For some of the early contributions to the study of the duality transforma-tion, see e.g., [1, 2]. One example is a periodic boundary condition twisted by duality: E ( x + L, y, z ) = B ( x, y, z ) and B ( x + L, y, z ) = − E ( x, y, z ) . This partic-ular setup was studied by O. Ganor and his collaborators [3–6], but whathappens in a more general situation remains unanswered, to the authors’knowledge. There is also a series of interesting papers on the flux sectorsof the Maxwell theory by G. W. Moore and his collaborators [7–9], whichare related to the inherent self-dual nature of the Maxwell theory. Anotherintriguing scenario is to consider a Maxwell theory with dynamical “du-ality gauge fields”, which might be thought of as a generalization of theAlice electrodynamics [10, 11] where the charge conjugation C = S isgauged. has a global gravitational anomaly and lives on the boundaryof a certain bulk SPT phase. As we will see, this result is aspecial case of the anomaly and the corresponding bulk SPTphase that we find for the duality symmetry.We study the anomaly and the bulk SPT phase by imitat-ing the relationship between the (1+1) -dimensional chiral bo-son and the (2+1) -dimensional U(1) Chern-Simons theoryand its generalization to the [(4 n +1)+1] -dimensional self-dual form field and the [(4 n +2)+1] -dimensional bulk theorystudied, e.g., in [18–24]. The essential point is that the (3+1) -dimensional Maxwell theory with a nontrivial background forits duality symmetry is a self-dual field, and we can utilize thetechniques developed in the papers listed above to study it.One of our main messages is that the subtle and interesting is-sues concerning the self-dual fields studied in the past alreadymanifest themselves in the case of the Maxwell theory oncethe non-trivial background for its duality symmetry is turnedon.Before proceeding, we note that the electromagnetic dual-ity group in the quantum theory is, in fact, the 2-dimensionalspecial linear group SL(2 , Z ) over the integers acting on thelattice Z of the electric and magnetic charges. Its effect onthe Maxwell theory on a curved manifold was carefully ana-lyzed in [25, 26] and it was interpreted as a mixed SL(2 , Z ) -gravitational anomaly in [13]. Our result in this Letter can beconsidered as the determination of the pure SL(2 , Z ) part ofthe anomaly.Our computation shows that the anomaly of the Maxwelltheory is 56 times that of a Weyl fermion in a certain pre-cise formulation of the duality. Where does this number 56come from? We will provide an answer using the property ofa (5+1) -dimensional superconformal field theory originallyfound in [27, 28] and known as the E-string theory; the namecomes from the fact that it has E global symmetry. TheE-string theory has two branches of vacua, called the tensorbranch and the Higgs branch. On the Higgs branch the E a r X i v : . [ h e p - t h ] O c t symmetry is Higgsed to E , which acts on 28 fermions viaits 56-dimensional fundamental representation; this is possi-ble since a pseudo-real representation R with dim R = 2 k can act on k fermions in dimensions because the spinrepresentation S in dimensions is pseudo-real and wecan impose the Majorana condition on R ⊗ S . When onemoves to the tensor branch, the E symmetry is restored anda self-dual tensor field appears. By compactifying this systemon T , one finds that one Maxwell field is continuously con-nected to 56 Weyl fermions, showing that they should havethe same anomaly. The electromagnetic duality is formulatedas the SL(2 , Z ) acting on this torus T , and therefore is ge-ometrized in this formulation. This means that both the purely SL(2 , Z ) part and the mixed gravitational- SL(2 , Z ) part of the (3+1) -dimensional anomaly come from the purely gravita-tional anomaly of the (5+1) -dimensional theory. These state-ments about the anomaly are valid if the E background fieldis turned off.The rest of the Letter is organized as follows. We startby recalling how the anomaly of a (1+1) -dimensional chi-ral boson is captured by the phase of the partition function ofthe (2+1) -dimensional U(1) Chern-Simons theory at level 1.We outline the path integral computation of its phase, as wellas how this can be matched with the anomaly of a (1+1) -dimensional chiral fermion. We then adapt this discussion tothe anomaly of the (3+1) -dimensional Maxwell theory andthe corresponding (4+1) -dimensional bulk BdC theory. Wewill see that the anomaly computed in this way reproduces theknown anomaly when the
SL(2 , Z ) background is trivial. Wethen consider the case of nontrivial SL(2 , Z ) backgrounds on S / Z k , for k = 2 , , , and , and we note that the resultingphase is equal to 56 times that of a charged Weyl fermion.This plays an important role in the consistency of the O − -plane and its generalizations. Finally, we explain why theanomaly of the Maxwell theory has to be 56 times that of acharged Weyl fermion, in terms of the six-dimensional super-conformal field theory known as the E-string theory. Moredetails will be provided in a longer version of the Letter [29]. WARM-UP: ANOMALY OF (1+1) -DIMENSIONAL CHIRALBOSON IN TERMS OF (2+1) -DIMENSIONAL U(1)CHERN-SIMONS
We start by recalling the well-understood case of theanomaly of the (1+1) -dimensional chiral boson at the freefermion radius. This theory naturally lives at the bound-ary of the (2+1) -dimensional U(1) Chern-Simons theory atlevel k = 1 , for which the Euclidean action is − S k =1 = πi (cid:82) ( A/ π )( F/ π ) [18, 30, 31]. The anomaly is then charac-terized by the partition function of this Chern-Simons theory on closed 3-dimensional manifolds M .Let us recall that the action at level 2, − S k =2 =2 πi (cid:82) ( A/ π )( F/ π ) , is well-defined modulo πi when themanifold is oriented. However, there is a problem in divid-ing it by two. To make the action S k =1 well-defined modulo πi , it is known that we need to pick a spin structure [32].Once this is done, the path integral can be performed explic-itly, because the theory is free. The details are given e.g. in[30, 33–35]. Very roughly, we split the gauge field A into asum of the flat but topologically nontrivial part and the topo-logically trivial but non-flat part. Assuming, for simplicity,that flat connections on M are isolated, we have Z U(1)CS ( M ) = (cid:20)(cid:90) [ DA ] top.trivial e πi (cid:82) ( A/ π )( F/ π ) (cid:21) × (cid:34) (cid:88) A : flat e πi (cid:82) ( A/ π )( F/ π ) (cid:35) . (1)Let us rewrite its phase.The phase of the first term can be written in terms of the etainvariant of the signature operator ∗ d : π Arg (cid:90) [ DA ] top.trivial e πi (cid:82) ( A/ π )( F/ π ) = − η signature . (2)Here and below, the equality of the phase is modulo one andis simply denoted by the equal sign ( = ). The phase of thesecond term can be rewritten as π Arg (cid:88) c ∈ H ( M , Z ) q ( c ) =: Arf( q ) (3)where c = c ( F ) is the first Chern class of the gauge bundleand q ( c ) := e πi (cid:82) ( A/ π )( F/ π ) .We note that q ( c ) is simply the exponentiated level-1 clas-sical action evaluated at a flat A . As recalled above, definingit requires something more than an oriented manifold and theintegration on it. Mathematically, q is known as a quadraticrefinement of the torsion pairing on H ( M , Z ) . The Arf in-variant Arf( q ) is defined by the equation above and is knownto take values in one eighth of an integer. We end up with theformula π Arg Z U(1)CS ( M ) = − η signature + Arf( q ) . (4)Let us now recall that a chiral boson can be fermion-ized. Then the bulk theory can be taken to be the (2+1) -dimensional fermion with infinite mass, for which the parti-tion function has the phase [36] π Arg Z fermion ( M ) = η fermion . (5)The values of η signature and η fermion on lens spaces are knownin the literature, e.g., [37]. For example, on M = S / Z , η signature = 0 , whereas Arf( q ) and η fermion can be either / or − / , depending on the spin structure. On M = S / Z , η signature = 2 / , Arf( q ) = 1 / , and η fermion = 2 / , as there isa unique spin structure. We indeed confirm − η signature + Arf( q ) = η fermion , (6)which can be proved using a mathematical result [38]. Wenote that η signature is independent of the spin structure but Arf( q ) does depend on the spin structure. In other words,the spin structure provides us the quadratic refinement. THE ANOMALY OF THE MAXWELL THEORY
The analysis of the anomaly of the (1+1) -dimensionalchiral boson we recalled above was generalized to the [(4 n − -dimensional self-dual form fields in [39] atthe perturbative level. The study of the corresponding [(4 n − -dimensional theory in the bulk, generalizing the (2+1) -dimensional Chern-Simons theory, was carried out indetail in [19–24]. The bulk theory has the action − S = πi (cid:82) ( A/ π ) d ( A/ π ) , where A is now a (2 n − -form gaugefield. Assuming H n − ( M n − , R ) = 0 , the phase ofthe partition function still has the form of Eq. (4), where q is now a quadratic refinement of the torsion pairing on H n ( M n − , Z ) , and its choice is not obviously related to thechoice of the spin structure.Here, we are more interested in the (3+1) -dimensionalMaxwell theory. The natural generalization in this caseis to consider the bulk theory with the action − S = πi (cid:82) [( B/ π ) d ( C/ π ) − ( C/ π ) d ( B/ π )] , where B and C are two 2-form gauge fields to be path-integrated over [12].This action has the SL(2 , Z ) symmetry acting on ( B, C ) ,which corresponds to the duality symmetry of the Maxwelltheory [12]. We can and will introduce the background gaugefield ρ for this SL(2 , Z ) symmetry, which means that there isa nontrivial duality transformation when going around a non-trivial loop in spacetime. The phase of the partition functionis then π Arg Z BdC ( M ) = − η signature + Arf( q ) (7)where the eta invariant is now for the signature operator ∗ d acting on the differential forms tensored with ( Z ) ρ , and q isnow the quadratic refinement of the natural torsion pairing on H ( M , ( Z ) ρ ) . Here, ( Z ) ρ signifies the coefficient systemtwisted by the SL(2 , Z ) bundle ρ . The eta invariant of the sig-nature operator with such a twist and its reduction from higherdimensions were considered earlier in the mathematical liter-ature; see, e.g., [40].Let us first consider the case where we do not have the SL(2 , Z ) background. In this case, the signature eta invariant S / Z S / Z S / Z S / Z η signature − − − H ( M , ( Z ) ρ ) ( Z ) Z Z Z Arf( q ) + − + π Arg Z + − + + η fermion − − − − TABLE I. Partition functions and related data on S / Z k . simply vanishes, and only the Arf invariant contributes. Re-call that a quadratic refinement is simply the classical actionevaluated on flat B and C . Then a general quadratic refine-ment can be written as (cid:90) B π dC π + (cid:90) dB π C π + (cid:90) B π dC π , (8)where B , C ∈ H ( M , R / π Z ) are the background fieldsfor the electric and magnetic 1-form U(1) symmetry of theMaxwell theory [41], which we chose to be flat. Its Arf in-variant is computed to be (cid:82) ( B / π ) β ( C / π ) where β is theBockstein homomorphism β : H ( M , R / Z ) → H ( M , Z ) ;the Bockstein homomorphism β can roughly be regarded asthe exterior derivative d when it acts on torsion elements ofcohomology groups. The end result is that π Arg Z BdC ( M ) = (cid:90) ( B / π ) β ( C / π ) . (9)This reproduces a known result. Indeed, the mixed anomalyis known to be of the form πi (cid:82) M ( B / π ) d ( C / π ) , for whichthe mathematically precise formulation [42] reduces to Eq. (9)when we only consider flat fields. Furthermore, we can take B / π = C / π = w , where w is the Stiefel-Whitney class ofthe spacetime and regarded as an element of H ( M , R / Z ) byusing Z → R / Z . The Maxwell theory with this coupling isalso known as the all-fermion electrodynamics, and it has thegravitational anomaly πi (cid:82) w βw = πi (cid:82) w w [16, 17].Let us next consider the case when a nontrivial SL(2 , Z ) background is present. We can choose the symmetry structureon M to consider, such as spin × SL(2 , Z ) or spin- Mp(2 , Z ) [ := spin × Z Mp(2 , Z ) ], distinguished by whether C = +1 or = ( − F . Here, C ∈ SL(2 , Z ) is the charge conjugation C : ( E , B ) (cid:55)→ − ( E , B ) , and the metaplectic group Mp(2 , Z ) is the double cover of the group SL(2 , Z ) . We will focus onthe latter case in this Letter, as it has a natural connection tothe (6+1) -dimensional CdC theory on a spin 7-manifold aswe will see. Canonical examples of M associated with thissymmetry structure are S / Z k , k = 2 , , , and , where go-ing around the generator of π ( S / Z k ) = Z k comes with theduality action by an element g of order k in SL(2 , Z ) . While S / Z k is not spin for even k , it has a natural spin- Z k struc-ture for any k by embedding S / Z k ⊂ C / Z k . Then, we getthe spin- Mp(2 , Z ) structure by embedding Z k ⊂ Mp(2 , Z ) .The results of explicit computations for Eq. (7) are tabulatedin Table I. When there are multiple choices for g or q , wechoose a particular one. Other quadratic refinements corre-spond to different background fields ( B , C ) for electromag-netic 1-form symmetries.When k = 2 , the relevant element in SL(2 , Z ) is just thecharge conjugation symmetry C . This case has the anomaly π Arg Z = 1 / on S / Z . This is responsible for the differ-ence / of the Ramond-Ramond (RR) charges of the O + -plane and O − -plane in Type-IIB string theory [43]. As ex-plained in [44], for the consistency of the theory, the fractionalpart of the RR charge must be exactly negative of the anomalyof a D3-brane living on S / Z . The background ( B , C ) pro-duced by O ± is such that only the O − leads to the anomalyof the Maxwell theory, explaining the difference of the RRcharges; we note that the charge / of the O + -plane wasalready explained by the fermion anomaly [44]. We can alsocheck that the resulting π Arg Z for other k is exactly whatis necessary to reproduce the RR charge of the N =3 S-fold[45, 46].Let us now consider the infinitely massive fermions encod-ing the anomaly of a (3+1) -dimensional Weyl fermion of unitcharge under Z k , which was studied in [47–49]. The corre-sponding eta invariants on S / Z k are also tabulated in Table I.We can check that the relation − η signature + Arf( q ) = 56 η fermion (10)holds for the choices of the Arf invariants given in Table I. WHY 56?
Relation (10) about the anomaly of the Maxwell theory and56 Weyl fermions in dimensions reminds us of the rela-tion Eq. (6) about the anomaly of a chiral boson and a chi-ral fermion in dimensions. In the latter case, the equal-ity should evidently hold because a chiral fermion can bebosonized to a chiral boson in dimensions. It also ex-plained the reason how and why the spin structure could beused to define the quadratic refinement necessary to formulatethe integrand of the
U(1)
Chern-Simons theory. In di-mensions, however, the Maxwell theory and 56 Weyl fermionsare two clearly different theories. What is the relation? Howand why does the spin (or, more precisely, the spin- Z k ) struc-ture provide the necessary quadratic refinement? One expla-nation is provided, somewhat surprisingly, by supersymmetricphysics in dimensions.Consider a self-dual tensor field in dimensions. Its di-mensional reduction on T gives rise to the Maxwell theoryin dimensions, geometrizing the SL(2 , Z ) duality sym-metry of the Maxwell theory. Correspondingly, the (4+1) - T ∪ Tensor branch Higgs branch28 fermions56 fermions ∪ E-stringcontinuousdeformation
FIG. 1. Maxwell to 56 fermions via E-string theory dimensional
BdC theory on M coupled to an SL(2 , Z ) bun-dle is the dimensional reduction of the (6+1) -dimensional CdC theory on M , which is the T bundle over M .We now embed this theory of a self-dual tensor field intothe tensor branch of the E-string theory [27, 28]. The E-stringtheory is a (5+1) -dimensional theory realized in M-theory,with two continuous families of vacua. One family of vacuais called the tensor branch, which describes an M5-brane closeto the spacetime boundary of M-theory carrying the E gaugesymmetry [50, 51]. On the tensor branch, the low energy the-ory consists of the self-dual tensor field with some additionalfields. The other family of vacua is called the Higgs branch,which describes an instanton of the E gauge symmetry. Theinstanton breaks E to E , producing some chiral fermions aszero modes of the instanton. A nontrivial fact in M-theory isthat these two families of vacua are continuously connected;an M5-brane put on the spacetime boundary can become an E instanton. The transition point is a strongly coupled con-formal field theory.We start from the M5-brane close to the spacetime bound-ary and transform it into an E instanton of nonzero size. Inthis process, the low energy theory is changed from that of thetensor branch, containing the self-dual tensor field, into thatof the Higgs branch, containing
28 = 56 / chiral fermionsin dimensions transforming under the fundamental 56-dimensional representation of E . Because this is a continu-ous process, the anomaly at the start and the anomaly at theend should be the same; previously the same argument wasused to compute the anomaly polynomial of the E-string the-ory in [52] (which reproduced earlier results in [53–55]), butthe same statement is true even for the subtler anomalies weare discussing now. There are also some additional fields onthe tensor and Higgs branches, but their anomalies are mani-festly the same on both sides, so we can match the anomaly ofthe self-dual tensor field and the chiral fermions.Since one chiral fermion in dimensions gives rise totwo chiral fermions in dimensions, we conclude that theanomaly of the Maxwell theory, formulated as the T com-pactification of the (5+1) -dimensional self-dual field with thetrivial E background, should be equal to that of the 56 Weylfermions. See Fig. 1 for a summary of what we have de-scribed.If we turn on a nontrivial E background A E on thefermion side, the data are translated on the self-dual tensorside into the background 3-form field C which couples to thedynamical self-dual tensor field, which is basically given bythe Chern-Simons term constructed from A E . When A E is flat, this determines a quadratic refinement required to de-fine the 6+1d CdC theory. In particular, the trivial E back-ground, which is available on any manifold, provides a canon-ical quadratic refinement for the (6+1) -dimensional CdC the-ory, and this construction only requires the spin structure. Thispoint was already essentially made in [56].Since this explanation of Eq. (10) requires a lot of infor-mation from string and M-theory, it would be of indepen-dent interest to check the equality (10) by a direct analysisin and dimensions. To translate the analysis in dimensions to the study of the Maxwell theory, we need to re-quire that the T bundle over M specified by the SL(2 , Z ) background is equipped with a spin structure. This meansthat the symmetry structure we consider is a spin- Mp(2 , Z ) structure. According to the cobordism classification theorem[57–60], the anomaly of any system with this symmetry isclassified by the dual of Ω spin- Mp(2 , Z )5 = Z ⊕ Z ⊕ Z ,which is the bordism group for closed 5-manifolds with spin- Mp(2 , Z ) structures and is generated by S / Z , S / Z , and [( S / Z ) (cid:48) + 9( S / Z )] , respectively, where ( S / Z ) and ( S / Z ) (cid:48) both have the spin- Z structure coming from theembedding S / Z ⊂ C / Z but with different actions of Z given by diag( i, i, i, ± i ) . We have not directly determinedwhich quadratic refinement comes from the trivial E field,but we have checked that, for a suitable choice, we have theequality (10) for each case, providing a strong check of ouridentification of Eq. (10). ACKNOWLEDGMENTS
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