On the anomaly of the electromagnetic duality of the Maxwell theory

We consider the 3+1 dimensional Maxwell theory in the situation where going around nontrivial paths in the spacetime involves the action of the duality transformation exchanging the electric field and the magnetic field, and its SL(2,Z) 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 SPT phase in 4+1 dimensions characterizing the anomaly, and the other is by considering the properties of a 5+1 dimensional superconformal field theory known as the E-string theory. This anomaly of the Maxwell theory plays an important role in the consistency of string theory.


INTRODUCTION
Every physicist knows that the electromagnetic field is described classically by the Maxwell equation, and that it is invariant under the electromagnetic duality S : (E, B) → (B, −E). The properties of the electromagnetic duality in the quantum theory might not be as well known to physicists in general, and in fact are not very well understood in the literature. This is particularly true when going around a nontrivial path in the spacetime results in a duality transformation. 1 In this letter, we uncover a feature of the Maxwell theory and its duality symmetry in such a situation, namely that it has a quantum anomaly.
We recall that a quantum theory in d+1 dimensions with a symmetry group G can have a quantum anomaly, in the sense that its partition function has a controllable phase ambiguity. Our modern understanding is that such a theory is better thought of as living on the boundary of a symmetry protected topological phase (SPT phase) in the (d+1)+1 dimensional bulk. It was noticed in the last few years in [10][11][12][13][14] that a version of the Maxwell theory, often called the all-fermion electrodynamics, where all particles of odd charge are fermions, has a global gravitational anomaly and lives on the boundary of a certain bulk SPT phase. As we will see, this result is a special case of the anomaly and the corresponding bulk SPT 1 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 particular setup was studied by O. Ganor and his collaborators [1][2][3][4], but what happens in a more general situation remains unanswered, to the authors' knowledge. There is also a series of interesting papers on the flux sectors of the Maxwell theory by G. W. Moore and his collaborators [5][6][7], which are related to the inherent self-dual nature of the Maxwell theory. Another intriguing scenario is to consider a Maxwell theory with dynamical "duality gauge fields", which might be thought of as a generalization of the Alice electrodynamics [8,9], where the charge conjugation C = S 2 is also gauged.
phase we find for the duality symmetry.
We study the anomaly and the bulk SPT phase by imitating the relationship between the 1+1d chiral boson and the 2+1d U(1) 1 Chern-Simons theory and its generalization to (4n+1)+1d self-dual form field and the (4n+2)+1d bulk theory, studied e.g. in [15][16][17][18][19][20][21]. The essential point is that the 3+1d Maxwell theory with a nontrivial background for its duality symmetry is a self-dual field, and we can utilize the techniques developed in the papers listed above to study it. One of our main messages is that the subtle and interesting issues concerning the self-dual fields studied in the past already manifest themselves in the case of the Maxwell theory, once the non-trivial background for its duality symmetry is turned on.
Before proceeding, we note that the electromagnetic duality group in the quantum theory is in fact SL(2, Z) acting on the lattice Z 2 of the electric and magnetic charges. Its effect on the Maxwell theory on a curved manifold was carefully analyzed in [22,23] and it was interpreted as a mixed SL(2, Z)gravitational anomaly in [24]. Our result in this paper can be considered as the determination of the pure SL(2, Z) part of the anomaly.
Our computation shows that the anomaly of the Maxwell theory is 56 times that of a Weyl fermion, in a certain precise formulation of the duality. Where does this number 56 come from? We will provide an answer using the property of a 5+1d superconformal field theory originally found in [25,26] known as the E-string theory; the name comes from the fact that it has E 8 global symmetry. The E-string theory has two branches of vacua, called the tensor branch and the Higgs branch. On the Higgs branch the E 8 symmetry is Higgsed to E 7 , which acts on 28 fermions via its 56 dimensional fundamental representation; this is possible since a pseudo-real representation R with dim R = 2k can act on k fermions in 5+1d because the spin representation S in 5+1d is pseudo-real and we can impose the Majorana condition on R ⊗ S. When one moves to the tensor branch, the E 8 symmetry is restored and arXiv:1905.08943v1 [hep-th] 22 May 2019 a self-dual tensor field appears. By compactifying this system on T 2 , one finds that one Maxwell field is continuously connected to 56 Weyl fermions, showing that they should have the same anomaly. The electromagnetic duality is formulated as the SL(2, Z) acting on this torus T 2 , and therefore is geometrized in this formulation. This means that both the purely SL(2, Z) part and the mixed gravitational-SL(2, Z) part of the 3+1d anomaly come from the purely gravitational anomaly of the 5+1d theory. These statements about the anomaly are valid if the E 8 background field is turned off.
The rest of the paper is organized as follows. We start by recalling how the anomaly of a 1+1d chiral boson is captured by the phase of the partition function of the 2+1d U(1) Chern-Simons theory at level 1. We outline the path integral computation of its phase, and how this can be matched with the anomaly of a 1+1d chiral fermion. We then adapt this discussion to the anomaly of the 3+1d Maxwell theory and the corresponding 4+1d bulk BdC theory. We will see that the anomaly computed in this way reproduces the known anomaly when the SL(2, Z) background is trivial. We then consider the case of nontrivial SL(2, Z) backgrounds on S 5 /Z k , for k = 2, 3, 4, 6, and note that the resulting phase is equal to 56 times that of a charged Weyl fermion. This plays an important role in the consistency of the O3 − -plane and its generalizations. Finally we explain why the anomaly of the Maxwell theory has to be 56 times that of a charged Weyl fermion, in terms of the six-dimensional superconformal field theory known as the E-string theory. More details will be provided in a longer version of the paper [27].

WARM-UP: ANOMALY OF 1+1D CHIRAL BOSON IN TERMS OF 2+1D U(1) CHERN-SIMONS
We start by recalling the well-understood case of the anomaly of the 1+1d chiral boson at the free fermion radius. This theory naturally lives at the boundary of the 2+1d U(1) Chern-Simons theory at level k = 1 whose Euclidean action is −S k=1 = πi (A/2π)(F/2π) [15,28,29]. The anomaly is then characterized by the partition function of this Chern-Simons theory on closed 3d manifolds M 3 .
Let us recall that the action at level 2, −S k=2 = 2πi (A/2π)(F/2π), is well-defined modulo 2πi when the manifold is oriented. However, there is a problem in dividing it by two. To make the action S k=1 well-defined modulo 2πi, it is known that we need to pick a spin structure [30]. Once this is done, the path integral can be performed explicitly, since the theory is free. The details are given e.g. in [28,[31][32][33]. Very roughly, we split the gauge field A into a sum of the flat but topologically-nontrivial part and the topologicallytrivial but non-flat part. Assuming for simplicity that flat con-nections on M 3 are isolated, we have Let us rewrite its phase. The phase of the first term can be written in terms of the eta invariant of the signature operator * d: Here and below, the equality of the phase is modulo one and is simply denoted by =. The phase of the second term can be rewritten as where c = c 1 (F ) is the first Chern class of the gauge bundle and q(c) := e πi (A/2π)(F/2π) . We note that q(c) is simply the exponentiated level-1 classical action evaluated at a flat A. As recalled above, defining it requires something more than an oriented manifold and the integration on it. Mathematically, q is known as a quadratic refinement of the torsion pairing on H 2 (M 3 , Z). The Arf invariant Arf(q) is defined by the equation above and is known to take values in one eighth of an integer. We end up with the formula Let us now recall that a chiral boson can be fermionized. Then the bulk theory can be taken to be the 2+1d fermion with infinite mass, whose partition function has the phase [ The values of η signature and η fermion on lens spaces are known in the literature, e.g. [35]. For example, on M 3 = S 3 /Z 2 , η signature = 0, while Arf(q) and η fermion can be either 1/8 or −1/8, depending on the spin structure. On M 3 = S 3 /Z 3 , η signature = 2/9, Arf(q) = 1/4, and η fermion = 2/9, as there is a unique spin structure. We indeed confirm which can be proved using a mathematical result [36]. We note 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+1d chiral boson we recalled above was generalized to the (4n − 3) + 1 dimensional self-dual form fields in [37] at the perturbative level. The study of the corresponding (4n − 2) + 1 dimensional theory in the bulk, generalizing the 2+1d Chern-Simons theory, was carried out in detail in [16][17][18][19][20][21]. The bulk theory has the action −S = πi (A/2π)d(A/2π), where A is now an (2n − 1)form gauge field. Assuming H 2n−1 (M 4n−1 , R) = 0, the phase of the partition function still has the form (4), where q is now a quadratic refinement of the torsion pairing on H 2n (M 4n−1 , Z), and its choice is not obviously related to the choice of the spin structure.
Here we are more interested in the 3+1d Maxwell theory. The natural generalization in this case is to consider the bulk theory with the action −S = πi [(B/2π)d(C/2π) − (C/2π)d(B/2π)], where B and C are two 2-form gauge fields to be path-integrated over. This action has the SL(2, Z) symmetry acting on (B, C), which corresponds to the duality symmetry of the Maxwell theory. We can and will introduce the background gauge field ρ for this SL(2, Z) symmetry, which means that there is a nontrivial duality transformation when going around a nontrivial loop in spacetime. The phase of the partition function is then where the eta invariant is now for the signature operator * d acting on the differential forms tensored with (Z 2 ) ρ , and q is now the quadratic refinement of the natural torsion pairing on H 3 (M 5 , (Z 2 ) ρ ). Here (Z 2 ) ρ signifies the coefficient system twisted by the SL(2, Z) bundle ρ. The eta invariant of the signature operator with such a twist and its reduction from higher dimensions was considered earlier in the mathematical literature, see e.g. [38]. Let us first consider the case where we do not have the SL(2, Z) background. In this case, the signature eta invariant simply vanishes, and only the Arf invariant contributes. Recall that a quadratic refinement is simply the classical action evaluated on flat B and C. Then a general quadratic refinement can be written as where B, C ∈ H 2 (M 5 , R/2πZ) are the background fields for the electric and magnetic 1-form U(1) symmetry of the Maxwell theory [39], which we chose to be flat. Its Arf invariant is computed to be (B/2π)β(C/2π) where β is the Bockstein homomorphism β : H 2 (M 5 , R/Z) → H 3 (M 5 , Z); the Bockstein homomorphism β can roughly be regarded as  the exterior derivative d when it acts on torsion elements of cohomology groups. The end result is that This reproduces a known result. Indeed, the mixed anomaly is known to be of the form 2πi M5 (B/2π)d(C/2π), whose mathematically precise formulation [40] reduces to (9) when we only consider flat fields. Furthermore, we can take B/2π = C/2π = w 2 where w 2 is the Stiefel-Whitney class of the spacetime, here regarded as an element of H 2 (M 5 , R/Z) by using Z 2 → R/Z. The Maxwell theory with this coupling is also known as the all-fermion electrodynamics, and has the gravitational anomaly 2πi w 2 βw 2 = πi w 2 w 3 [13,14].
Let us next consider the case when a nontrivial SL(2, Z) background is present. We can choose the symmetry structure on M 5 to consider, such as spin × SL(2, Z) or spin-Mp(2, Z) (:= spin× Z2 Mp(2, Z)), distinguished by whether C 2 = +1 or = (−1) F . Here, C ∈ SL(2, Z) is the charge conjugation C : (E, B) → −(E, B) and the metaplectic group Mp(2, Z) is the double cover of the group SL(2, Z). We will focus on the latter case in this letter, as it has a natural connection to the 6+1d CdC theory on a spin 7-manifold as we will see. Canonical examples of M 5 associated with this symmetry structure are S 5 /Z k , k = 2, 3, 4, 6, where going around the generator of π 1 (S 5 /Z k ) = Z k comes with the duality action by an element g of order k in SL(2, Z); while S 5 /Z k is not spin for even k, it has a natural spin-Z 2k structure for any k by embedding S 5 /Z k ⊂ C 3 /Z k . Then we get spin-Mp(2, Z) structure by embedding Z 2k ⊂ Mp(2, Z). The results of explicit computations for (7) are tabulated in Table I. When there are multiple choices for g or q, we chose a particular one. Other quadratic refinements correspond to different background fields (B, C) for electromagnetic 1-form symmetries.
When k = 2, the relevant element in SL(2, Z) is just the charge conjugation symmetry C. This case has the anomaly 1 2π Arg Z = 1/2 on S 5 /Z 2 . This is responsible for the difference 1/2 of the RR charges of the O3 + -plane and O3 − -plane in Type-IIB string theory [41]. As explained in [42], for the consistency of the theory, the fractional part of the RR charge must be exactly negative of the anomaly of a D3-brane liv-ing on S 5 /Z 2 . The background (B, C) produced by O3 ± is such that only the O3 − leads to the anomaly of the Maxwell theory, explaining the difference of the RR charges; we note that the charge 1/4 of the O3 + -plane was already explained by the fermion anomaly [42]. We can also check that the resulting 1 2π Arg Z for other k is exactly what is necessary to reproduce the RR charge of the N =3 S-fold [43,44].
Let us now consider the infinitely massive fermions encoding the anomaly of a 3+1d Weyl fermion of unit charge under Z 2k , which was studied in [45][46][47]. The corresponding eta invariants on S 5 /Z k are also tabulated in Table I. We can check that the relation − 1 4 η signature + Arf(q) = 56η fermion (10) holds for the choices of the Arf invariants given in Table I.
The relation (10) about the anomaly of the Maxwell theory and 56 Weyl fermions in 3+1 dimensions reminds us of the relation (6) about the anomaly of a chiral boson and a chiral fermion in 1+1 dimensions. In the latter case, the equality should evidently hold because a chiral fermion can be bosonized to a chiral boson in 1+1 dimensions. It also explained the reason how and why the spin structure could be used to define the quadratic refinement necessary to formulate the integrand of the U(1) Chern-Simons theory. In 3+1 dimensions, however, the Maxwell theory and 56 Weyl fermions are two clearly different theories. What is the relation? How and why does the spin (or more precisely the spin-Z 2k ) structure provide the necessary quadratic refinement? One explanation is provided, somewhat surprisingly, by supersymmetric physics in 5+1 dimensions.
Consider a self-dual tensor field in 5+1 dimensions. Its dimensional reduction on T 2 gives rise to the Maxwell theory in 3+1 dimensions, geometrizing the SL(2, Z) duality symmetry of the Maxwell theory. Correspondingly, the (4+1)dimensional BdC theory on M 5 coupled to an SL(2, Z) bundle is the dimensional reduction of the (6+1)-dimensional CdC theory on M 7 , which is the T 2 bundle over M 5 .
We now embed this theory of a self-dual tensor field into the tensor branch of the E-string theory [25,26] which describes an M5-brane close to the spacetime boundary carrying the E 8 gauge symmetry [48,49]. We can now bring the M5brane close to the spacetime boundary, and transform it into an E 8 instanton of nonzero size. This corresponds to the Higgs branch of the E-string theory, on which an E 7 subgroup of E 8 remains unbroken. In this process, one self-dual tensor field is converted into 28 = 56/2 chiral fermions in 5+1 dimensions, transforming under the fundamental 56-dimensional 5+1d 3+1d Maxwell representation of E 7 . Since this is a continuous process, the anomaly at the start and the anomaly at the end should be the same; previously the same argument was used to compute the anomaly polynomial of the E-string theory in [50] (which reproduced earlier results in [51][52][53]), but the same statement is true even for subtler anomalies we are discussing now. Since one chiral fermion in 5+1 dimensions gives rise to two chiral fermions in 3+1 dimensions, we conclude that the anomaly of the Maxwell theory, formulated as the T 2 compactification of the (5+1)d self-dual field with the trivial E 8 background, should be equal to that of the 56 Weyl fermions. See Fig. 1.
If we turn on a nontrivial E 7 background A E7 on the fermion side, the data is translated on the self-dual tensor side into the background 3-form field C which couples to the dynamical self-dual tensor field, which is basically given by the Chern-Simons term constructed from A E7 . When A E7 is flat, this determines a quadratic refinement required to define the 6+1d CdC theory. In particular, the trivial E 7 background which is available on any manifold provides a canonical quadratic refinement for the 6+1d CdC theory, and this construction only requires the spin structure. This point was already essentially made in [54].
Since this explanation of (10) requires a lot of information from string and M-theory, it would be of independent interest to check the equality (10) by a direct analysis in 3+1 and 4+1 dimensions. To translate the analysis in 5+1 dimensions to the study of the Maxwell theory, we need to require that the T 2 bundle over M 5 specified by the SL(2, Z) background is equipped with a spin structure. This means that the symmetry structure we consider is a spin-Mp(2, Z) structure. According to the cobordism classification theorem [55][56][57][58], the anomaly of any system with this symmetry is classified by the dual of Ω spin-Mp(2, Z) 5 = Z 9 ⊕Z 32 ⊕Z 2 , which is the bordism group for closed 5-manifolds with spin-Mp(2, Z) structures and is generated by S 5 /Z 3 , S 5 /Z 4 , and [(S 5 /Z 4 ) +9(S 5 /Z 4 )], respectively, where (S 5 /Z 4 ) and (S 5 /Z 4 ) both have the spin-Z 8 structure coming from the embedding S 5 /Z 4 ⊂ C 3 /Z 4 but with different actions of Z 4 given by diag(i, i, i, ±i). We have not directly determined which quadratic refinement comes from the trivial E 7 field, but we have checked that for a suitable choice we have the equality (10) for each case, providing a strong check of our identification (10).