# Self-dual Quantum Electrodynamics on the boundary of Bosonic

Symmetry Protected Topological States

###### Abstract

We study (or ) Quantum Electrodynamics (QED) realized on the boundary of (or ) bosonic symmetry protected topological (BSPT) states, using a systematic nonlinear sigma model (NLSM) field theory description of BSPT states developed in Ref. Bi et al., 2015. We demonstrate that many of these QED states have an exact electric-magnetic duality due to the symmetry of the BSPT states in the bulk. The gauge charge and Dirac monopole both carry projective representations of the bulk symmetry, and the emergent gapless photons of the QED phase also transform nontrivially under the bulk symmetry. Some of these QED boundary states can be further driven into a topological order, and the statistics and symmetry transformation of its point particle and vison loop excitations guarantee that this topological order cannot be driven into a trivial confined or Higgs phase. With a finite fourth dimension, the entire system becomes a lattice, the self-dual QED and the topological order can coexist on two opposite boundaries respectively, which together constitute an exotic self-dual “topological photon phase”.

###### pacs:

## I 1. Introduction

Symmetry protected topological (SPT) phases, a new type of quantum disordered phase pioneered in Ref. Chen et al., 2013, 2012, are intrinsically different from trivial direct product state, when and only when the system has certain symmetry . In terms of its phenomena, a SPT phase on a dimensional lattice should satisfy at least the following three criteria:

(i). On a dimensional lattice without boundary, this phase is fully gapped, and nondegenerate;

(ii). On a dimensional lattice with a dimensional boundary, if the Hamiltonian of the entire system (including both bulk and boundary Hamiltonian) preserves symmetry , then this phase is either gapless or gapped but degenerate.

(iii). The boundary state of this dimensional system cannot be realized as a -dimensional lattice system with the same symmetry .

Notice that the second criterion (ii) implies the following two possibilities: On a lattice with a boundary, the system is either gapless, or gapped but degenerate. For a SPT state, its boundary must be degenerate, which forms a projective representation of the symmetry group; for a SPT state, its boundary can be either gapless, or degenerate due to spontaneous discrete symmetry breaking; for a SPT state, the degeneracy of its boundary can correspond to either spontaneous breaking of , or correspond to certain topological degeneracy at the boundary. Which case occurs in the system will depend on the detailed Hamiltonian at the boundary of the system. For example, with strong interaction, the boundary of a 3d topological band insulator can be driven into a nontrivial topological phase Fidkowski et al. (2013); Chen et al. (2014); Bonderson et al. (2013); Wang et al. (2013); Metlitski et al. (2013). And one natural candidate boundary state of most bosonic SPT states is a topological order with and excitations both carrying fractional degrees of freedom Bi et al. (2015); Vishwanath and Senthil (2013).

In this paper we will investigate the boundary states of bosonic SPT states. All the possible boundary states discussed above can occur in our case, but there is one new possibility which does not occur in lower dimensions: the boundary can be a deconfined gapless photon state, which does not exist in lower dimensions with gapped matter fields due to the well-known fact that the compact QED with gapped matter fields is always confined due to the proliferation of Dirac monopole in the space-time. However in , a compact QED can have a deconfined photon phase with gapless photon excitations, and deconfined electric charge (denoted as ) and Dirac monopole (denoted as ) excitations. This boundary gapless photon must be very unusual, because based on the definition of an SPT state, this boundary state cannot be realized in without the bulk or the opposite boundary.

Please note that this photon phase only exists at the boundary; namely the bulk is still fully gapped and nondegenerate. Just like the topological order at the boundary of a SPT state, the and excitations of the photon phase must carry a nontrivial representation (or projective representation) of the symmetry groups, which implies that the system cannot be driven into a trivial confined or Higgs phase with a gapped and nondegenerate ground state by condensing or excitations. The quantum number of and excitations can be computed systematically using the NLSM field theory developed in Ref. Bi et al., 2015. Besides the quantum numbers carried by and , we are going to show that in many cases the boundary photon phase has an exact “self-dual” symmetry. In this work we are going to describe two examples in detail. In the first example, the self-dual symmetry is the physical time-reversal symmetry, and in the second example it is a symmetry.

A system with infinite size is unrealistic. However, our study of the boundary of a SPT state can lead to exotic phases in as well. We can imagine making a thin slab of the SPT state, namely a system with a finite fourth dimension. Then the entire system becomes three dimensional, but one can realize two different states on the two opposite boundaries. For the first example state in which time-reversal plays the role of a self-dual symmetry, we can realize the self-dual photon phase on the top surface, but realize a fully gapped topological order on the bottom surface. Then at low energy only the photon phase on the top surface becomes visible, which by definition is a phase that cannot be realized in at all. Only at higher energy will the topological order on the bottom surface be exposed. By contrast, for the example where the symmetry plays the role as self-duality, it seems this photon phase cannot be driven into any gapped topological order without breaking the symmetry.

We note that in Ref. Kravec and McGreevy, 2013; Kravec et al., 2014, a QED state on the boundary of a bosonic short range entangled (BSRE) state was also studied, and this QED state can have a maximum duality symmetry. But in that case the BSRE state does not need any symmetry to be nontrivial because the boundary QED state is an “all fermion state”; namely its and excitations are both fermions Wang et al. (2014), which is a fact robust against any symmetry breaking. This bulk BSRE state is also an “invertible topological state” discussed in Ref. Kong and Wen, 2014; Wen, 2014; Kapustin, 2014a, b; Freed, 2014. However, in our case, the system is a nontrivial SPT state when and only when the system has certain symmetry. When the symmetry is broken, the system becomes a trivial bose Mott insulator.

## Ii 2. Self-dual photon phase with symmetry

Let us start with a SPT state with and time-reversal symmetry () symmetry only. This state can be described by the following NLSM field theory with a six component unit vector :

(2) | |||||

(3) |

Here , and is the volume of the five dimensional unit sphere. In Eq. 3, when the coupling constant is larger than some critical value, the system is in a quantum disordered phase with a fully gapped and nondegenerate bulk spectrum, and according to Ref. Bi et al., 2015, different SPT states correspond to different symmetry transformations on that keep the entire action, including the topological term invariant. In this work we primarily consider the state that corresponds to the following transformation as an example:

(5) | |||||

(6) |

In fact, even without the symmetry this state is already a nontrivial SPT state, and this state is just a generalization of the SPT with symmetry Levin and Gu (2012), which is described by a NLSM with a four component unit vector Xu and Senthil (2013). Also, this state can be viewed as a bosonic integer quantum Hall state with U(1) symmetry broken down to (see more details in appendix B).

In this work we will focus on the symmetry. But, Eq. 3 actually can also describe SPT states with much larger symmetries. Let us parameterize the six-component vector as , where and are both three-component unit vectors. We also tentatively introduce two more SO(3) symmetries to the system with and transforming as vectors under the two SO(3) symmetries respectively, although these exact SO(3) symmetries are unimportant to the main physics we are going to discuss. (Introducing extra symmetries and eventually breaking them has proved to be a very helpful trick for field theory analysis, as shwon in Ref. Vishwanath and Senthil, 2013.) At the boundary, Eq. 3 will reduce to a NLSM with a Wess-Zumino-Witten (WZW) term Bi et al. (2015) at level-1:

(8) | |||||

(9) |

Just like all WZW terms, the last term in Eq. 9 is equal to the volume of the target space enclosed by the trajectory of under a periodic evolution. is an extra parameter introduced and while .

The physical meaning of this WZW term becomes explicit when we reduce this WZW term on a hedgehog monopole of , which is a point defect ( is normalized to be a three-component unit vector). Since the hedgehog monopole is a singularity of , then at the hedgehog monopole the six component unit vector will reduce to another three-component unit vector , and this WZW term reduces to a WZW model for (for more details please see appendix C):

(10) |

With the extra SO(3) symmetries, the ground state of this field theory Eq. 10 is two fold degenerate:

(11) |

Here we have used the standard parametrization of the vector :

Now let us look at vector the and introduce the standard CP field parametrization of :

(12) |

As usual, a U(1) gauge field is introduced in this parametrization, and when the CP field is gapped and disordered, the U(1) gauge field is in its deconfined photon phase. In the deconfined photon phase, in addition to gapless photon excitations, there are also two types of basic gapped point particles. The first type is the electric charge , which is the CP field ; the second kind of gapped particle is the Dirac monopole , which is nothing but the hedgehog monopole of . This is because in the standard CP formalism, the U(1) gauge flux quantum through any closed surface is just the Skyrmion number . Therefore the Dirac monopole, which is the source of the gauge flux, is identified with the hedgehog monopole, which is the source of the Skyrmion number. Thus the Dirac monopole of the photon phase can also be represented by .

Now we can also turn on the symmetry . Based on the transformation in Eq. 6, interchanges the electric charge and Dirac monopole. Under the and symmetries, the electric charge and magnetic monopole transform as

(14) | |||||

(15) |

The electric and magnetic field will transform as

(17) | |||||

(18) |

The commutation relation , and hence the Maxwell equation, are invariant under the and symmetries. Thus in this photon phase the symmetry acts as a electric-magnetic duality. This photon phase cannot be driven into a trivial confined or Higgs phase because the condensate of either or will inevitably generate an order of a certain component of , which therefore breaks the and symmetry. Since plays the role as the duality symmetry, our photon phase is different from all the time-reversal symmetry enriched photon phases classified in Ref. Wang and Senthil, 2013.

The two extra SO(3) symmetries make the physical meaning of the
and particles transparent: the () particle is the
hedgehog monopole of ( ^{1}^{1}1This
opposite sign is due to the fact that in the WZW term of
Eq. 9, the monopole of is a O(3) WZW
model of with level , while the monopole of
is a WZW model of with level .),
which is also a fractionalized CP field of
(). Under the duality symmetry, the role of
and interchanges. But the nature of this photon phase does not
depend on the extra SO(3) symmetries. Thus after we establish this
photon phase, the SO(3) symmetries can be explicitly broken
without changing the physics of the photon phase.

Another point excitation of this photon phase is the dyon. A dyon is a bound state between and which forms a fermion (denoted as ). and view each other as a flux source. Therefore the effective theory that describes the internal degree of freedom of a dyon is precisely the O(3) NLSM with a WZW term at level-1, Eq. 10, except with replaced by where the vector that connects electric and magnetic charges. The ground state of this model is again a spin-1/2 doublet. Since now and interchange under , this means that in this model takes to , which implies that under , the dyon is a Kramers doublet fermion: , and . This is due to the fact that is effectively a rotation in space, and a spin-1/2 object acquires a minus sign under rotation. Thus the dyon transforms exactly the same under time-reversal as a physical electron.

In the bulk of the BSPT, the hedgehog monopole of , or equivalently the Dirac monopole of the gauge field in the CP formalism, is a loop defect. And Eq. 3 reduces to a O(3) NLSM of with a topological term with , which is an effective theory for a spin-1 antiferromagnet chain Haldane (1983a, b). Thus a closed loop of hedgehog (Dirac) monopole in the bulk is fully gapped and nondegenerate, and can therefore proliferate and drive the bulk into a gapped disordered SPT and gauge confined phase. But when a monopole line terminates at the boundary, its end cannot condense without breaking symmetry. Thus it is possible for a deconfined photon phase to exists at the boundary. The physical meaning of this boundary photon phase, including its emergent photon excitations, can also be understood in a different way, presented in the next section.

## Iii 3. topological order with a Kramers doublet fermion

A U(1) photon phase can usually be driven into a fully gapped topological order by condensing either a pair of or particles. In the condensate of , and are confined because they have nontrivial mutual statistics with , and the only deconfined point particle is . Besides the point particle, the system also has a gapped loop excitation (usually called the “vison” loop excitation), which has a mutual semionic statistics with ; when the closed trajectory of links with the vison loop by odd numbers, the system wave function will acquire a minus sign. However, the vison loop has trivial statistics with , and is therefore not confined in the condensate.

In our system, the condensate of either or will break the symmetry. In order to construct a topological order that preserves , we need to condense a Cooper pair of dyons (each dyon is a fermion). Since the dyon is a Kramers doublet, we will condense the singlet Cooper pair of the dyons, just to preserve all the symmetries. In this condensate, both and excitations of the original photon phase will be confined because they have nontrivial mutual statistics with the dyon Cooper pair (this is also called oblique confinement Cardy and Rabinovici (1982); Cardy (1982)), and the only deconfined but gapped point excitation is the fermionic dyon .

The condensate of a pair of dyons also has a loop excitation which has mutual semionic statistics with the dyon. To understand this loop defect, let us we first turn on two extra SO(2) symmetries where and transform as vectors under the first and second SO(2) group respectively:

(20) | |||||

(21) |

This means carries half charge of the first SO(2) symmetry and therefore must have a mutual semionic statistics with the vortex loop of ; likewise, must have a mutual semionic statistics with the vortex loop of . The dyon, which is a bound state of and , must have mutual semionic statistics with both types of vortex loops.

Besides their mutual statistics with point excitations, these vortex loops also have a nontrivial spectrum. The WZW term in Eq. 9 will decorate each vortex line of with a O(4) WZW-term at level-1:

(23) | |||||

(24) |

where is a SU(2) matrix. The condensate of will break part of the symmetries, but it preserves another new symmetry which is the combination of a rotation of SO(2) and the symmetry. Then this symmetry guarantees that the vortex loop of must be either gapless or degenerate, due to the WZW term in Eq. 24. The nature of the vortex loop can also be understood as the following: In the bulk, a vortex of is a membrane, and according to the bulk action Eq. 3, this membrane is decorated with a SPT phase with symmetry, which implies that when the vortex membrane terminates at the boundary, it becomes a vortex loop, and it is also the boundary of a SPT state; thus it must be either gapless or degenerate. The degeneracy of the vortex line corresponds to spontaneous symmetry breaking of the symmetry: for instance or along the vortex loop. Then there are two flavors of vortex loops, and the domain wall between the two flavors is precisely the hedgehog monopole of .

Although a single vortex loop of must have nontrivial spectrum, a double strength vortex loop (a vortex loop of vorticity) can be gapped and nondegenerate. One way to see this is that, because the SPT phase with symmetry has a classification Levin and Gu (2012); Chen et al. (2013, 2012), two copies of such a state becomes trivial; their boundary can be rendered gapped and nondegenerate.

Many interesting phases can be obtained by manipulating the dynamics of the vortex loops. For example, consider a superfluid phase with spontaneous SO(2) symmetry breaking, a superfluid phase with condensation of complex boson (this phase also spontaneously breaks ). Then according to Ref. Motrunich and Senthil, 2005, if the vortex loop has two flavors and both flavors proliferate, then this phase is precisely the photon phase described in section 2. The topological order with condensate discussed at the beginning of this section can be realized when the strength-2 (fully gapped and nondegenerate) vortex loop of proliferates.

Now let us start with a superfluid phase where the bound state of bosons and condense. Under , and ; thus the condensate of bound state does not break ( the real and imaginary parts of are both invariant under ). In this phase there are two types of vortex loops: vortex loops of and . Now we argue that the topological order with condensate can be constructed by proliferating the bound state of these two types of vortex loops. First of all, since and particles both have mutual semionic statistics with this “bound vortex loop”, they will both be confined in this condensate; the only deconfined point particle is the dyon , which views this “bound vortex loop” as a flux instead of a flux loop. Since in the topological order the bound state of the two vortex loops already proliferate, there is only one type of well-defined loop excitation, and it can be viewed as the remnant of either the or vortex loop, either of which has the correct semionic statistics with the dyon. Thus this vortex loop can be identified as the vison loop excitation of the desired topological order.

More systematically, the condensate of bound state can be described by the following the effective action in the Euclidean space-time:

(25) |

where , and is a 1-form gauge field. In this equation, when both and condense, the only gauge invariant order parameter is .

We can take the standard Villain form of the action by expanding the cosine function at its minimum and introducing the 1-form fields and ():

(26) |

In the last line, we introduce the 2-form fields () on the dual space-time manifold, such that resolves the constraint . Summing over will require to take only integer values, which could be imposed by adding a term, and the theory now becomes

(27) |

Integrating out the gauge field will impose the constraint , which can be resolved by , . Therefore the final action takes the form of

(28) |

and are both 1-form vector fields. creates a segment of vortex line of along the direction, while creates a unit vortex loop. If we are going to proliferate the bound state of the two types of vortex loops, say , then the effective action for reads

(29) |

When proliferates, can take two inequivalent minima: ; therefore this state is a topological order.

Once we establish the existence of this topological order, the extra SO(2) symmetries can be broken, which will not affect the statistics between vison loops and the dyon . In this topological order, since there is no spontaneous symmetry breaking at all, the original symmetry already guarantees that the vison loop must be either gapless or degenerate because the vison loop is effectively the boundary of a SPT state with symmetry. However, after we break the two SO(2) symmetries, the bound state between the and vortex loops become gapped and nondegenerate, allowing it to safely proliferate. Because the deconfined point particle excitation of this topological order is a fermion, it cannot condense and drive the system into a trivial Higgs phase; similarly, because the vison loop is gapless or degenerate, it also cannot proliferate and drive the system into a gapped and nondegenerate confined phase.

While the topological order itself cannot be driven into either a trivial Higgs or confined phase, two copies of this topological order can indeed be trivialized. The reason is that, for two copies of the topological order (labeled and ), one can first condense the bound state of dyons from both copies ( condensate of ) to break the two copies of topological order down to one topological order. Then in this residual topological order the only well-defined point particle is the dyon (or equivalently because of the background pair condensate). The vison loop is the bound state of vison loops from both copies: ; this is because and individually have semionic statistics with , and hence must be confined in the condensate. Since and both carry a WZW term at level-1, their bound state is fully gapped, and hence can further proliferate and drive the entire system into a trivial confined phase without any symmetry breaking. This implies that two copies of the BSPT states Eq. 3 with symmetry is a trivial state, which is consistent with the classification based on the NLSM itself given in appendix A.

Having understood the self-dual photon phase and the topological order, we can realize an exotic self-dual topological photon phase by making a thin slab of BSPT state with symmetry, and realizing the self-dual photon phase on the top boundary and the topological order on the bottom boundary. At low energy, only the photon phase at the top boundary will be detectable while only at higher energy will the bottom boundary be exposed.

## Iv 4. Self-dual photon phase with symmetry

Another BSPT phase that leads to a self-dual photon phase at its boundary is a state with symmetry, which is still described by Eq. 3, but now the the vector transforms as

(34) | |||||

where , with . Using the formalisms introduced in section 2, we can demonstrate that the boundary of this BSPT is a self-dual photon phase with the following transformation of its and excitations:

(35) |

Note here and are hedgehog monopoles of three component vectors and respectively. The emergent electric and magnetic fields transform under the symmetry as

(36) |

Again the Maxwell equation and the commutation relation between and fields are invariant under this transformation.

Unlike the previous case, it is not obvious whether we can drive this this self-dual photon phase into a gapped topological order with full symmetry. As we discuss in appendix B, this BSPT state can be constructed by breaking the U(1) symmetry of the bosonic integer quantum Hall state down to . In Ref. You and Xu, 2015, we argued that a BSPT state whose boundary has perturbative gauge anomaly after “gauging” the symmetry cannot be driven into a fully gapped topological order because the system must respond to infinitesimal external gauge field. The boundary of the bosonic integer quantum Hall (BIQH) state we discuss in appendix B has a perturbative gauge anomaly after the U(1) symmetry is gauged. Thus the boundary of the BIQH state cannot be driven into a symmetric topological order. It is possible that the BSPT state inherits this property from its BIQH parent state. More rigorous study will be given in the future.

## V 5. Summary

In this work we studied two examples of different self-dual photon phases that can be realized on the boundary of bosonic SPT states. Both states need certain symmetries to protect their boundaries, which is an important difference from the self-dual photon phase studied in Ref. Kravec et al., 2014. Understanding these symmetry protected self-dual photon phases at the boundary of systems can lead to exotic photon phases on a system as well, as was discussed in the end of section 3. In even higher dimensions, topological orders and photon states can still exist on the boundary of BSPT states; but more exotic boundary states can also be realized, such as deconfined spin liquid phases with nonabelian gauge fields. This is due to the fact that a nonabelian gauge fields usually lead to confinement in dimensions lower than , while in higher dimensions a stable deconfined phase exists.

The authors are supported by the the David and Lucile Packard Foundation and NSF Grant No. DMR-1151208.

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## Vi Supplemental Material

## Appendix A A. NLSM field theory description of BSTP states

Bosonic SPT phases in all dimensions with various continuous and discrete symmetries can be systematically described and classified by semiclassical nonlinear Sigma model (NLSM) field theories with a topological -term Bi et al. (2015).

In , it is well known that a spin-1 chain can be described by an NLSM with topological a -term Haldane (1983a, b); Affleck et al. (1987); Kennedy (1990); Hagiwara et al. (1990); Ng (1994) at where its disordered phase corresponds to a BSPT state protected by or time reversal symmetry. In general, a BSPT phase in dimensional space-time can always be formulated by an O() NLSM with a topological -term, assuming the symmetry group of the BSPT state is a subgroup of and other discrete symmetries such as time reversal:

(37) | ||||

The boundary theory of the -dimensional theories with are described by dimensional O() NLSMs with a Wess-Zumino-Witten (WZW) term at level-1:

(38) | ||||

To define the WZW-term we need to extend the order parameter field to with the following condition:

(39) | |||||

(40) |

The spectrum of the boundary theory above is in general non-trivial: either degenerate or gapless, provided we have enough symmetry. For example, when , the boundary is a 0+1d O(3) NLSM with a WZW term at level-1. If the theory has full symmetry or time reversal symmetry, the ground state is a doublet with protected two-fold degeneracy. When , the boundary is a 1+1d WZW-term at level-1, which is conformal assuming the full symmetry is preserved Witten (1984); Knizhnik and Zamolodchikov (1984). The spectrum can also be degenerate if we only have discrete symmetry and the degeneracy is precisely due to the symmetry breaking.

Notice that all components of in Eq. A must have a nontrivial transformation under the symmetry group . Otherwise one can turn on a linear “Zeeman” term that polarizes some component of which will trivially gap out the edge states. In this case, the -term has no effect, and the bulk state is trivial.

Eq. 3 with symmetry is a nontrivial BSTP state when . However, two copies (layers) of Eq. 3 can be trivialized after turning on symmetry allowed interlayer couplings. For instance, starting with two copies of Eq. 3 (labeled and ), the following coupling is allowed by the symmetry:

(42) | |||||

(43) |

When is positive and large,

(44) | |||

(45) | |||

(46) |

As a result, the two terms of copies and will cancel out, and effectively the coupled system has , and is therefore a trivial state. This conclusion is consistent with the analysis based on the boundary topological orders in section 3.

## Appendix B B. bosonic integer quantum Hall state as parent state

In this section we discuss bosonic integer quantum Hall (BIQH) states and their relation with the two states discussed in this work. The BIQH state is a straightforward generalization of the BIQH state discussed in Ref. Levin and Senthil, 2013. It is described by a O(6) NLSM with (Eq. 3), where the six component vector transforms under the U(1) symmetry as

(51) | |||||

If we couple the U(1) charge to an external U(1) gauge field , then after integrating out the matter field , a Chern-Simons term is generated for :

(52) |

which is a CS theory at level 6.

Directly integrating out the boson field is technically difficult. But alternatively we can start with 8 copies of fermionic integer quantum Hall model:

(53) |

, , , , where . We focus on the phase with , where each fermion copy (labeled by ) gives rise to a chiral fermion at the boundary. Therefore there are in total 8 chiral fermions at the boundary of Eq. 53:

(54) |

Now we can couple the boundary chiral fermions to a six component vector :

(55) |

The same WZW term as Eq. 9 will be generated after integrating out the fermions.

This fermion model (Eq. 53) has at most a U(8) symmetry, which contains three U(1) symmetries as a subgroup. The three U(1) symmetries are generated by , , and so that , , and transform as two-component vectors under these three U(1) symmetries, respectively. Now let us couple the fermion model Eq. 53 to three U(1) gauge fields: , , . We give the fermions charge because we want the bosons to carry charge under these gauge fields. Then after integrating out the fermions, the following Chern-Simons field theory is generated:

(57) | |||||

(59) | |||||

(60) |

After breaking these three U(1) gauge symmetries down to a single U(1) gauge symmetry, Eq. 52 is generated.

The two BSPT states we discussed in this paper can be obtained by breaking the U(1) symmetry down to either or symmetry. Notice that the BIQH state with U(1) symmetry has a classification with U(1) symmetry and where each integer corresponds to a different BIQH state. Notice that the coupling Eq. 43 explicitly breaks the U(1) symmetry.

## Appendix C C. Dimensional Reduction of Topological Terms

In this Appendix, we are going to derive the effective field theory of a monopole core of an O(6) NLSM, namely Eq. 10. In a monopole configuration of an O(3) order parameter, for instance , can be understood as an intersection point of the domain walls of the three order parameter fields respectively. So we can derive the theory on a monopole core by three domain wall projections, which is described below.

To derive the theory on the domain wall of one of the order parameter fields, e.g. , we’ll first construct a domain wall configuration of . Consider the following configuration of the vector :

(61) |

By inserting this parametrization of into Eq. 9 and integrating along the direction, the WZW-term reduces to an WZW-term with the same level. More explicitly, the theory on the domain wall is:

(62) | |||||

(63) |

A domain wall projection reduces both the spatial dimension and the dimension of the order parameter field by one, and the effective field theory on the domain wall inherits the topological term from the original theory.

We can repeat this domain wall projection procedure once more. On the domain wall we just made, consider a domain wall of along the -direction. We can integrate over the -direction, and the resulting theory is an WZW-term with level-1:

(64) | |||||

(65) |

This field theory can be thought of as the effective field theory on a -vortex of components. Notice that this field theory is equivalent to an principle chiral model by introducing matrix field . The principle chiral model is precisely written as in Eq. 24.

Based on the configuration we already have, if we further make a domain wall of on the vortex core, then the whole configuration of the order parameter field corresponds to a monopole configuration of the order parameter. And right on the core of the monopole, the effective field theory is precisely an WZW-term at level-1 as in Eq. 10.