Topological axion electrodynamics and 4-group symmetry

We study higher-form symmetries and a higher group in the low energy limit of a $(3+1)$-dimensional axion electrodynamics with a massive axion and a massive photon. A topological field theory describing topological excitations with the axion-photon coupling, which we call a topological axion electrodynamics, is obtained in the low energy limit. Higher-form symmetries of the topological axion electrodynamics are specified by equations of motion and Bianchi identities. We find that there are induced anyons on the intersections of symmetry generators. By a link of worldlines of the anyons, we show that the worldvolume of an axionic domain wall is topologically ordered. We further specify the underlying mathematical structure elegantly describing all salient features of the theory to be a 4-group.

Topological solitons are necessary ingredients to determine the dynamics and phases of systems. In the absence of the axion coupling, quantized magnetic vortices can give fractional Aharonov-Bohm (AB) effects. Gapped phases exhibiting these effects in the low energy limit are called topologically ordered phases [44][45][46][47], which are realized by anyons in fractional quantum Hall (FQH) states and s-wave superconductors in (2 + 1) and (3 + 1) dimensions, respectively. Once we take into account the axion cou-KEK-TH-2331, J-PARC-TH-0243 Email addresses: hidaka@post.kek.jp (Yoshimasa Hidaka), nitta@phys-h.keio.ac.jp (Muneto Nitta), ryokokur@post.kek.jp (Ryo Yokokura) pling, electromagnetic properties become richer: for instance, electric charges are induced by penetrating magnetic fluxes to the axionic domain wall [23,24,32,48]. Effects on the domain wall can be described by an Abelian Chern-Simons (CS) term on the domain wall, induced by the axion-photon coupling [49,50]. Since FQH states can be described by this CS term in the low energy limit [44][45][46], one can expect that the axionic domain wall is topologically ordered. However, topological objects giving the AB phases, which we will call topological order parameters in this Letter, have not been identified to the best of our knowledge.
In this Letter, we investigate higher-form global symmetries and a higher group in the low energy limit of the (3 + 1)-dimensional axion electrodynamics in the gapped phase, to show a topological order on the axionic domain wall. The higher-form symmetries can be specified by employing a dual topological quantum field theory for the massive photon [86][87][88][89][90] and massive axion [91][92][93][94][95], which we will call a topological axion electrodynamics. We find 0-, 1-, 2-, and 3-form symmetries and show the topological order on the axionic domain wall in terms of the higher-form symmetries, by identifying the topological order parameters as intersections of 0-and 1-form symmetry generators on which anyons are induced. On the axionic domain wall, the intersections can have a fractional phase determined by the groups of the higher-form symmetries. The fractional AB effect for the intersected symmetry generators implies nontrivial correlations between higher-form symmetries. We further find that the symmetries organize a 4-group symmetry. While a 4-group symmetry in a simpler case has been studied in Ref. [96], our 4-group is the first example where all the 0-, 1-, 2-, and 3-form symmetries are nontrivial.

Topological axion electrodynamics
Here, we give an action of the low energy limit of the axion electrodynamics with a massive axion and a massive photon. Using dual transformations, we derive a topological field theory describing this limit of the system. We begin with the effective action, where we use notations of differential forms: |ω p | 2 = ω p ∧ ω p for a p-form ω p , and is the Hodge star operator. M 4 denotes the (3 + 1)-dimensional spacetime spin manifold [97,98]. The quantities v, v are mass dimension 1 parameters, e is a coupling constant, and N is an integer. The axion φ is given by a 2π periodic pseudo-scalar field, φ(P) + 2π ∼ φ(P), for a point P in the spacetime. By this periodicity, the axion can have a winding number along a loop C: C dφ ∈ 2πZ. In addition, we have introduced a potential term V (φ) for the axion, which has a global symmetry under the shift V (φ + 2π/k) = V (φ). The potential has k of degenerated minima, V (2πn/k) = V (2πn/k) = 0 and V (2πn/k) > 0 for n ∈ Z mod k. The photon is described by a U (1) 1-form gauge field a with a gauge transformation, a → a + dλ. Here, λ is a U (1) 0-form gauge parameter normalized as C dλ ∈ 2πZ. The photon is subject to the Dirac quantization on a closed 2-dimensional subspace S (e.g., a sphere S 2 ), S da ∈ 2πZ. The 2π periodic scalar field χ is introduced as a phase component of a charge q(∈ Z) Higgs field. The field χ is shifted as χ → χ + qλ under the gauge transformation a → a + dλ.
To investigate topological properties of the system, we dualize the theory to a topological theory after neglecting the kinetic terms of the axion and photon at low energy. We dualize |dχ − qa| 2 and V (φ) to topological term given by 2-and 3-form gauge fields, respectively [86][87][88][89][90][91][92][93][94][95]. The dual topological action has the simple form, (2) Here, b and c are U (1) 2-and 3-form gauge fields, whose gauge transformation laws are given by 1-and 2-form gauge parameters λ 1 and λ 2 as b → b+dλ 1 and c → c+dλ 2 , respectively. They are normalized by the flux quantization conditions as V db, Ω dc, S dλ 1 , V dλ 2 ∈ 2πZ where V and Ω are closed 3-and 4-dimensional subspaces, respectively. We will call the dual theory the "topological axion electrodynamics," since it can be described by only topological terms.

Higher-form global symmetries in topological axion electrodynamics
We show higher-form global symmetries in this system found by the equations of motion for the dynamical fields φ, a, b, and c: respectively. The corresponding symmetry generators with groups parametrizing them are Here, we have introduced n 0 , ..., n 3 ∈ Z, p := gcd(N, q), and m := gcd(N, k), where "gcd" represents the greatest common divisor. Hereafter, we assume that the subspaces V, S, and C do not have any self-intersection for simplicity. All symmetry generators have the standard form of generators U n = e iθnQn with Q n = j n , where the (3 − n)-form current j n is closed under the equations of motion (4), i.e., dj n = 0. Note that the phase θ n is constrained by the large gauge invariance of the dynamical fields [64,65,[99][100][101][102]. Therefore, there are the electric Z m 0-form, Z p 1form, Z q 2-form, and Z k 3-form global symmetries. Their physical interpretations are as follows: U 0 (e 2πin0/m , V) is a worldvolume of the axionic domain wall: the equation of motion of c in the presence of U 0 is dφ = (2πn 0 /m)δ 1 (V).
Here, δ 4−n (Σ n ) is a delta function (4 − n)-form satisfying Σn ω n = M4 ω n ∧ δ 4−n (Σ n ) for a n-dimensional subspace Σ n [63, [103][104][105][106]. Similarly, U 1 (e 2πin1/p , S) represents a worldsheet of a quantized magnetic flux or an impulse of a quantized electric field, because the equation of motion of b in the presence of U 1 is da = (2πn 1 /p)δ 2 (S). U 2 and U 3 are a Wilson loop and a 2-point local operator of the axion, respectively. All of them are topological, and the vacuum expectation values (VEVs) are trivial, U 0 = · · · = U 3 = 1 thanks to the equations of motion in Eq. (3). Here, the symbol ' · · · ' means the VEV. We remark that the 2-and 3-form global symmetries are emergent symmetries in the low energy limit, where the magnetic vortices and axionic domain walls can be treated as objects with infinitely large tensions. We summarize corresponding charged objects, which are not necessary in the following discussion in Appendix A.
In addition to these electric symmetries, there are four magnetic U (1) (−1)-, 0-, 1-, 2-form symmetries, which are given as the form The conservation laws are given by the Bianchi identities. In this Letter, we mainly focus on the electric symmetries.

Topological order in bulk
Before discussing the topological order on the axionic domain wall, we show the topological order in the bulk for p = gcd(N, q) = 1 with a fractional AB phase given by Z p . Topological order parameters can be identified as the symmetry generators U 1 and U 2 , since they are topological and have nontrivial fractional phases. First, we have seen that they are topological, developing nonzero VEVs U 2 (e 2πin2/q , C) = U 1 (e 2πin1/p , S) = 1. Second, they have a fractional AB phase characterized by Z p : Here, we have defined θ 12 = −(2πn 1 n 2 /p) Link (S, C) with the linking number between n-and (3 − n)dimensional subspaces Σ n and Σ 3−n as Link (Σ n , Σ 3−n ) = ΩΣ n δ n+1 (Σ 3−n ) by using (n + 1)-dimensional subspace Ω Σn satisfying ∂Ω Σn = Σ n (see Appendix B). By the linking phase, the topological axion electrodynamics is topologically ordered in the bulk. We remark that this fractional AB phase is different from a Z q AB phase of an Abelian Higgs model without an axion [47,53], due to the presence of the axion-photon coupling deforming the 1-form symmetry.

Topological order on axionic domain walls
Now, we show that the axionic domain walls are also topologically ordered with a nontrivial fractional linking phase in a different manner from the bulk. The topological order parameter can be identified as an intersection of the 0-and 1-form symmetry generators. The fractional linking phase is given by the group Z mp 2 / gcd(N,mp 2 ) , different from the one in the bulk.
A rough description is as follows. The worldvolume of the axionic domain wall can be understood as a FQH state, since U 0 in Eq. (4) has a level N/m ∈ Z CS term, N 4πm V a ∧ da. By intersecting U 1 to U 0 , we can have an anyon, whose worldline is a topological order parameter of FQH states. Linked anyons on the domain wall can be obtained by two U 1 's intersected in the bulk.
A more precise proof can be given as follows. First, we intersect U 0 to U 1 to create a worldline of an anyon on the domain wall. We begin with the correlation function U 0 (e 2πin0/m , V 0 )U 1 (e 2πin1/p , S 0 ) assuming V 0 ∩ S 0 = ∅ (the left panel of Fig. 1). We deform S 0 to S 1 that is intersected to V 0 (the right panel of Fig. 1). Here, we choose S 1 so that S 1 ∩ V 0 is a 1-dimensional closed subspace, and S 0 ∩ S 1 = ∅. We interpolate between S 0 and S 1 using a 3-dimensional subspace V 01 satisfying ∂V 01 = S 0 ∪S 1 . We assume that V 01 does not intersect with any singularity such as an 't Hooft line. Using U 1 (e 2πin1/p , S 0 ) = U 1 (e 2πin1/p , ∂V 01 )U 1 (e 2πin1/p , S 1 ), and absorbing U 1 (e 2πin1/p , ∂V 01 ) into the action by a redefinition a + (2πn 1 /p)δ 1 (V 01 ) → a, the correlation function can be written as Here, we have introduced an object on a 2-dimensional subspace, Ω V0 ∩ S 1 , with a boundary, where Ω V0 is a 4-dimensional subspace whose boundary is V 0 . It is straightforward to show that U 01 is topological under the deformation of S 1 or V 0 by the redefinition of a or φ, respectively. Therefore, we find that, when U 0 and U 1 are intersected, there should be an additional magnetic 1-form symmetry generator with the boundary U 01 . The boundary object expresses a worldline of an anyon on the domain wall. This anyon cannot solely exist in the bulk if N/mp ∈ Z; it is always trapped on the domain wall. From the phase factor in Eq. (7), the anyon line has a fractional electric charge N n 0 n 1 /(mp) [32]. The appearance of anyons trapped on the domain wall is one of the main results of this Letter. The anyons also have a fractional linking phase, which we will show in the following. We next consider intersections of 1-form symmetry generators, which are necessary to derive the fractional linking phase of anyons. We begin with the correlation function (left panel of Fig. 2), U 1 (e 2πin1/p , S 0 )U 1 (e 2πin 1 /p , S 0 ) , where 2-dimensional subspaces S 0 and S 0 satisfy S 0 ∩S 0 = ∅. We deform S 0 to S 1 that intersects with S 0 , where the deformation is characterized by a 3-dimensional subspace V 01 satisfying ∂V 01 = S 0 ∪S 1 . By the same procedure as Eq. (6), the correlation function becomes where we have introduced an object on a 1-dimensional subspace, V S0 ∩ S 1 , with boundaries, and V S0 is a 3-dimensional subspace whose boundary is S 0 . Thus, we should add the magnetic 2-form symmetry generator with boundaries U 11 if we intersect two U 1 's. Again, the intersected configuration is topological under deformations of S 0 or S 1 . Using the interpretation of U 1 as external electric or magnetic fluxes, the presence of U 11 can be physically understood as the fact that the E · B is a source of the axion, since the boundary objects of U 11 can be identified as local objects of the axion. Finally, we consider the intersection of three symmetry generators to show the fractional linking phase of anyons on the domain wall. We begin with the following cubic but trivial correlation function, where we choose the subspaces such that S 0 ∩ S 0 = S 0 ∩ V 0 = S 0 ∩ V 0 = ∅. We deform S 0 to S 1 using V 01 , where S 1 intersects with V 0 , but S 1 ∩S 0 = ∅. As shown in Eq. (6), we have U 01 (e 2πiN n 0 n 1 mp , Ω V0 ∩ S 1 ) in the correlation function. We then deform S 0 to S 1 interpolated by V 01 , where S 1 intersects with V 0 , and also with S 1 transversally. We thus obtain C 011 (e 2πin0/m , V 0 ; e 2πin1/p , S 1 ; e 2πin 1 /p , S 1 ) Here, θ 011 := −2πN n 0 n 1 n 1 /(mp 2 ) Link (S 1 , S 1 )| V0 , and we have used the linking number of S 1 and S 1 on V 0 Link (S 1 , S 1 )| V0 := V0 δ 1 (V S1 ) ∧ δ 2 (S 1 ), and V S1 is a 3dimensional subspace whose boundary is S 1 . The above relation means that the boundary line objects of U 01 are topological order parameters with the fractional linking phase. Thus, the axionic domain wall is topologically ordered, and the fractional phase, e iθ011 ∈ Z mp 2 / gcd(N,mp 2 ) , is different from that of the bulk. The linking phase comes from the fractionally quantized magnetic fluxes (1/p) 2 on the level N/m Abelian CS action.

Global 4-group symmetry in topological axion electrodynamics
The topological order on the axionic domain wall implies a so-called higher-group structure; as shown below, there is a 4-group symmetry which is a set of 0-, 1-, 2-, and 3-form symmetries with nontrivial correlations between them, where the 0-and 1-form symmetry generators lead to a 2-form symmetry generator, and two 1-form symmetry generators lead to a 3-form symmetry generator.
As shown in Eq. (6), the correlation of the 0-and 1-form symmetry generators induces a magnetic 1-form symmetry generator, whose boundary is the intersection of the 0-and 1-form symmetry generators. The intersection is a closed 1-dimensional object, which generate a 2-form symmetry. To see this, we again focus on Eq. (11), and evaluate it as follows: The 2-form symmetry and its symmetry group can be identified as follows. We remark that U 1 is charged under the 2-form symmetry with the charge −n 1 q/p, , S) , as in Eq. (5). Thus, the intersection of U 0 and U 1 can be regarded as a 2-form symmetry generator. By comparing θ 011 to the charge −n 1 q/p, we find that the intersection of U 0 and U 1 is parametrized by Z Q with Q = q·mp/ gcd(N, mp) being product of q and the denominator of N/mp. Thus, the 2-form symmetry group is identified as Z Q , which is transmuted from Z q .
Similarly, the intersection of two 1-form symmetry generators becomes the boundary of a magnetic 2-form symmetry (8), which behaves as a 3-form symmetry generator. We again focus on the correlation function in Eq. (11), and evaluate it as follows: Since the correlation function U 3 (e 2πin3/k , (P, P ))U 0 (e 2πin0/m , V) = e −2πi n 0 n 3 k · k m Link (V,(P,P )) U 0 (e 2πin0/m , V) implies the charge of U 0 under the 3-form symmetry is −n 0 k/m, we find that the intersection of two U 1 's becomes a symmetry generator of the 3-form symmetry with the symmetry group Z K , where K := k · p 2 /gcd(N, p 2 ) with the nontrivial denominator of N/p 2 .
The appearance of topological objects at intersections of symmetry generators is a signal of higher groups. Our discussion in this Letter can have potential applications to both of physics and mathematics. For the physics side, we may apply this discussion to, e.g., topological superconductors. To discuss 't Hooft anomalies would be an important issue to determine the ground state structures. For the mathematics side, clarifying the precise definition of the 4-group would lead to a construction of a new class of higher-groups. We will address these issues in the forthcoming paper [107].

Appendix A. Charged objects for higher-form symmetries
Here, we summarize charged objects for the higher-form symmetries. The corresponding charged objects are the Wilson loop and its analogues. For 0-, 1-, 2-, and 3-form symmetries, the charged objects are explicitly written as L(q 0 , P) = e iq0φ(P) , respectively. Here, the charges are quantized q 0 ∈ Z m , q 1 ∈ Z p , q 2 ∈ Z q , and q 3 ∈ Z k because of the large gauge invariance for the gauge fields. We remark that W (q 1 , C) is identical to the symmetry generator U 2 (e −2πiq1/q , C). The symmetry transformations are induced by the link of a charged operator and the symmetry generator (see Appendix B): U 0 (e 2πin0/m , V)L(q 0 , P) = e 2πiq0n0 Link (V,P)/m L(q 0 , P) , U 1 (e 2πin1/p , S)W (q 1 , C) = e 2πiq1n1 Link (S,C)/p W (q 1 , C) , U 2 (e 2πin2/q , C)V (q 2 , S) = e 2πiq2n2 Link (C,S)/q V (q 2 , S) , U 3 (e 2πin3/q , (P, P ))D(q 3 , V) = e 2πiq3n3 Link ((P,P ),V)/k D(q 3 , V) . (A.2)

Appendix B. Derivations of correlation functions
Here, we briefly explain derivations of the correlation functions. The derivations are based on the reparametrizations of dynamical fields in the path-integral formalism, which can be understood as finite versions of the Ward-Takahashi identity or Schwinger-Dyson equation.
As an example, we explain the derivation of Eq. (5). In the path-integral formalism, the correlation function can be written as U 1 (e 2πin1/p , S)U 2 (e 2πin2/q , C) = N D[φ, a, b, c]e iSTAE U 1 (e 2πin1/p , S)U 2 (e 2πin2/q , C), where N is the normalization factor such that 1 = 1. We can absorb the symmetry generator U 1 into the action by the reparametrization of a as follows. We take a 3dimensional subspace V S whose boundary is S = ∂V S , and express the integral in the symmetry generator by the Stokes theorem as Here, Link (S, C) = V S δ 3 (C) is the intersection number of V S and C, which is equal to the linking number of S and C.