The effective action of a BPS Alice string

Recently a BPS Alice string has been found in a $U(1)\times SU(2)$ gauge theory coupled with a charged complex adjoint scalar field arXiv:1703.08971. It is a half BPS state preserving a half of supercharges when embedded into a supersymmetric gauge theory. In this paper, we study zero modes of a BPS Alice string. After presenting $U(1)$ and translational zero modes, we construct the effective action of these modes. In contrast to previous analysis of the conventional Alice string for which only large distance behaviors are known, we can perform calculation exactly in the full space thanks to BPS properties.


Introduction
Topological vortices are an important and interesting subject to study not only because of their mathematical elegance but also of practical purposes from superfluids, superconductors, ultracold atomic gases to quantum field theory, QCD, string theory and cosmology. In cosmological context, a symmetry breaking phase transition may have occurred in the early universe due to rapid cooling and expansion of universe, during which topological vortices known as 'cosmic strings' may be created [2,3]. Such classical vortex configurations become more interesting when interacting with quantum fields and these interactions generate massless excitations (zero modes) near the vortex core and dictate the low-energy dynamics of vortices. When a U (1) zero mode arises due to a breaking of bulk continuous symmetry inside the vortex core, the string core behaves as a superconducting wire [4]. In the case of local symmetry the excitation of zero mode inside the vortex may also excite massless gauge field in the bulk which generates logarithmically divergent energy.
Among those, Alice strings have very peculiar features. When a charged particle encircles an Alice string, its electric charge changes the sign [5], and therefore one cannot define an electric charge globally. The simplest example of an Alice string can be found in an SO(3) gauge theory with 5 representation, in which the SO(3) gauge group is spontaneously broken down to O(2) in the vacuum. So the vacuum manifold or order parameter space becomes G/H = SO(3)/O (2) RP 2 . This allows a non-trivial homotopy group π 1 (RP 2 ) Z 2 supporting an Alice string. The unbroken generator can be identified as the electromagnetic U (1) generator, and its sign changes while it encircles the Alice string once, as mentioned above. The bulk system of the Alice string behaves as 'Alice electrodynamics' where charge conjugation (Z 2 ) is taken as a local symmetry [6]. A change of the sign of electric charges around the Alice string implies a lost of the charge. This lost can be explained in terms of a nonlocal charge called the "Cheshire charge" [7][8][9]. 1 Many interesting global phenomena arise due to this exotic property, like the generalized Aharonov-Bohm effect [14], anyonic exchange statistics etc [15]. It was suggested that a nature of the Cheshire charge can be explained as follows; The U (1) gauge symmetry in the bulk is spontaneously broken in the presence of an Alice string, thereby giving rise to a U (1) zero mode, in addition to usual translational zero modes. This U (1) zero mode is non-normalizable and is charged under the U (1) gauge group. If there are two or more Alice strings, the Cheshire charge should be nonlocally defined over them in terms of a zero mode attached to each string. However, the unbroken bulk symmetries are multivalued in the case of Alice strings, and therefore the behavior of zero mode analysis becomes little subtle, in which case one needs to carefully take into account the multivalued nature of the unbroken group generator in the analysis, which is called "obstruction" [16][17][18][19]. 2 In general for vortices, this occurs due to the existence of local discrete unbroken symmetries [20].
One of difficulty to perform concrete calculation of zero mode analysis in a multi-string system may be due to the existence of the interaction between Alice strings. The homotopy group π 1 (RP 2 ) Z 2 implying that two Alice strings can annihilate each other and the existence of an attraction between them. One usually manipulate for instance an exchange of strings as adiabatic process but it is dynamically difficult or impossible. In contrast, there is no force between Bogomolnyi-Prasad-Sommerfield (BPS) solitons (strings) [24,25]. Therefore, one can place them at arbitrary positions, and n strings have the total energy (tension) exactly n times larger than that of a single string, thereby allowing a large moduli space of multi-string configurations. However, in contrast to the conventional strings in the Abelian-Higgs model [26,27], which can become BPS at the critical coupling, previously studied Alice strings were all non-BPS because two Alice strings annihilate each other as mentioned above. In our recent work [1], we have found that an SU (2) × U (1) gauge theory coupled with one charged complex adjoint scalar field admits BPS Alice strings, and have shown that it preserves a half supersymmetry if embedded into supersymmetric gauge theory. In this theory, one can expect to place as many Alice strings as one likes at arbitrary positions, thereby openings a possibility to study exotic phenomena of Alice strings more concretely.
In the present paper, as a first step, we systematically study zero mode analysis of a single BPS Alice string. A zero mode analysis for a conventional Alice string has been already discussed in literature but only at large distance since equations cannot be solved exactly. In this work we shall show that the zero modes of a BPS Alice string can be solved exactly as functions of the vortex profile functions, so that behaviors of the zero modes can be understood on the full space. We then construct the effective action for the U (1) and 1 The Alice string is also known to exchange magnetic charges from magnetic particles which creates the magnetic Cheshire charges [9]. Related to this, a ring of the Alice string can be thought of as magnetic monopole at large distances when the U (1) modulus is twisted along the ring, and this is supported by the non-trivial second homotopy group, π2(G/H) = 0 [10][11][12][13].
translational zero modes of a single BPS Alice string. This paper is organized as follows. In Sec. 2, we give a brief review of BPS Alice strings. In Sec. 3, we discuss the translational and U (1) moduli separately. In Sec. 4 we derive the effective action of zero modes, and finally we give a summary of our results and discussions in Sec. 5, in which we discuss a possible interaction of the U (1) mode and the U (1) gauge field in the bulk.

BPS Alice strings
In this section, we give a short review of BPS Alice strings, for details see Ref. [1]. We consider an SU (2) × U (1) b gauge theory with SU (2) and U (1) gauge fields A µ and a µ , respectively, coupled with charged complex scalar fields Φ in the adjoint representation. The corresponding action can be written as with gauge couplings g and e for SU (2) and U (1) b gauge fields, respectively. 3 The U (1) b symmetry gives the stability to a fractional vortex and we can have an unbroken Z 2 in the vacuum. We can choose the vacuum expectation value of the field Φ as with τ 1 = 1 2 σ 1 . This breaks the gauge symmetry group where stands for a semi-direct product. This can be understood as follows. Eq. (2.2) says that any rotation around τ 1 keeps Φ v invariant. The unbroken discrete group Z 2 is defined as a simultaneous π rotation around axes directed along any linear combination of τ 3 and τ 2 and in U (1) b . This keeps Φ v invariant since both the π rotations generate sign changes separately. The unbroken group elements can be written as where c 2 , c 3 are arbitrary real constants normalized to the unity as c 2 2 + c 2 3 = 1. The semi-direct product is arising here because Z 2 element changes the action of U (1) 1 . The fundamental group for this symmetry breaking can be written as This nontrivial fundamental group supports the existence of stable strings. The Z 2 element makes the generator of unbroken U (1) 1 globally undefined in the presence of a string: the generator of the unbroken U (1) changes sign as it encircles the string once, and this property of the vortex in this system identifies it as an Alice string.
Here we are studying BPS vortices and the BPS completion can be performed if we consider the critical couplings λ e = e 2 and λ g = g 2 . In this case, the theory can be embedded into a supersymmetric gauge theory in which Alice strings preserve a half of the supersymmetry [1], but we do not focus on such the aspect in this paper. The Bogomol'nyi completion of the tension, that is the static energy per a unit length, is found to be The saturation of the inequality in the above equation, implying the minimum tension of vortex solution in the same topological sector, yield BPS equations given by In order to solve these equations for a single vortex solution, we take the vortex ansatz of the scalar and gauge fields as where {r, θ} are radial and angular coordinates of the two dimensional space, respectively. The equations for the profile functions f 1 (r), f 2 (r), A(r) and a(r) depending only on the radial coordinate are the same with those of a non-Abelian vortices [29][30][31][32][33][34], and they can be solved numerically with the boundary conditions The numerical solution is displayed in the Fig. 1.
The vacuum configuration in Eq. (2.2) shows that it is invariant under the adjoint action of the U (1) subgroup generated by τ 1 = 1 2 σ 1 . However, in the presence of a string, the scalar field configuration in Eq. (2.10) depends on space coordinates even at very large distances as with Φ(R, θ = 0) = ξσ 1 . The holonomy Ω can be defined by So the embedding of the unbroken U (1) group becomes space dependent and the generator must be changed by the holonomies as it goes around the string as After a full encirclement it is easy to find that Ω 3 (2π) ∈ Z 2 and where the 'obstruction' parameter ζ = 1 2 . We shall see that the Z 2 affects the zero mode solution in next section.
One comment is in order. A global version of the Alice string in our theory was already found in Ref. [35] (see Refs. [36,37]) in the context of Bose-Einstein condensation of ultra cold atomic gases. In the same context, a global monopole as a twisted Alice ring was also found [38]. By gauging U (1) × SU (2) symmetry, we obtained our theory which allows BPS Alice strings for the critical coupling [1].

The moduli of a BPS Alice string
Zero modes arise due to the breaking of any continuous unbroken bulk symmetry in the presence of a vortex solution. We can have continuously degenerate BPS solutions which keeps BPS equation and tension invariant. So we have moduli space of solutions, and the motion on the moduli space generates zero modes. Here we would like to discuss translational and U (1) zero modes separately.

Translational moduli
In this subsection, we discuss translational modes arising due to breaking of translational invariance, which are are almost the same as other BPS vortices. The solution of the BPS equations can be shifted to any arbitrary point as center of vortex. So the zero modes can be found by expanding the BPS solutions around the origin which is taken as center of our vortex. In a particular gauge the zero mode solutions look like where f mi and F mi (x) are the field strengths computed from the BPS solutions and δX m is the displacement of the centre of the vortex from the origin, where we place the vortex.

U (1) modulus
As it is well known that zero modes arise when a soliton solution is not invariant under the action of a continuous unbroken symmetry group. In the case of the Alice string the unbroken U (1) symmetry in the bulk is spontaneously broken in the core of the vortex. It can be observed easily when applying U (1) transformation on the order parameter. On the x-axis (at θ = 0) the order parameter can be written as Any small change due to the action of the U (1) group elements e i ϕ 2 σ 1 for small ϕ is written as for a constant parameter ϕ. It is also true if ϕ is a function of the x, y-coordinates, in which case we have to add an inhomogeneous term with the gauge field [17]. Since the U (1) symmetry is broken inside the vortex we stick to a global transformation. This would generate an infinite set of solutions which are physically distinct. So ϕ can be treated as a U (1) modulus.

The effective action of a BPS Alice string
In the last section we have discussed the existence of the zero modes by heuristic arguments. In this section, we construct the effective action of these zero modes by the moduli approximation [39].

The effective action of the translational moduli
Let us start by shifting the center of the vortex to (X(z, t), Y (z, t)), and then the solution of BPS equations becomes Here, we assume a slow variation of the vortex center with the moduli space approximation. This slow variation generates non-zero current along the t-z directions, and we have to introduce t, z derivatives to solve the Gauss' law. We may write the α = {0, 3} derivative terms as Here the velocity is defined as V j α = ∂ α X j . Let us choose a gauge so that above expressions can be rewritten as Here we are considering only small fluctuations, and we have neglected all higher order terms. Now using the first two terms, we may write By using the BPS equations, we may rewrite above equations as Now using rotational invariance and BPS equation D + Φ = 0 we may rewrite, We now use the equations (4.5) , (4.6) and (4.7) and following Eq. (2.6) and the solutions of BPS equations the effective action is written as (4.8)

The effective action of the U (1) modulus
To understand the behavior of the U (1) modulus we need to excite the modes along the z-axis to write down the effective action. We set the fluctuation of the U (1) modulus of the scalar field as Φ(ϕ(t, z), r, θ = 0) = U ϕ (t, z)Φ(r)U † ϕ (t, z), U ϕ (t, z) = e iϕ(t,z)τ 1 , where ϕ(t, z) is taken as a slowly varying function on the t-z plane. Since the generator of unbroken group is not well defined globally as described in Ref. [1], we define the θ dependence of the scalar as This can be rewritten as Similarly we can introduce the gauge field fluctuation corresponding to U (1) modulus as The (t, z) dependence of the fluctuations may keep the BPS equations unchanged, but it would generate J 0 and J 3 in the equations of motion of the gauge field. This violates the Gauss' law and Biot-Savart-law, and we need to introduce A 0 and A 3 to solve these equations. Since F αβ = 0, we may write the effective action as Eq. (4.13) is a part of the full action, and the rests vanish for the BPS solutions. Now by variations of a α and A α , we may write the equations of motion as (4.14) To solve these equations of motion we introduce an ansatz of generated components A α of the gauge field as Here Ψ i (r, θ) are real and imaginary components of an arbitrary complex function on the x-y plane and can be found by solving equations of motion. To find the profile functions Ψ i (r, θ), we may insert the ansatz into the equations of motion in Eq. (4.14) and solve the equations for Ψ i (r, θ). Likewise we may insert the ansatz into the effective action in Eq. (4.13) and find equations for Ψ i after a variation with respect to Ψ i . Since both the cases give the same equations of Ψ i , we use the second calculation here. It is straightforward to show that the above ansatz in Eq. (4.15) solves the equations of motion in Eq. (4.14). Now we may use the zero mode ansatz in Eq. (4.15) and the vortex solution ansatz in Eq. (4.11) and (4.12) to express F iα and D α Φ in terms of Ψ i and ϕ as where the "obstruction" parameter is ζ = 1 z). Here Φ 1 and Φ 2 have been defined by After inserting the expressions in Eq. (4.16) into the effective action in Eq. (4.13) we find  By using this complex function, the above effective action in Eq. (4.18) can be rewritten in a simpler form as where D i Ψ = (∂ i − iζb i )Ψ and We should notice here that Ψ behaves as if a scalar field in 1 + 1 dimensions in the background of the vortex Abelian gauge field b i with a charge ζ = 1 2 . The important is that the (x, y) and (t, z) dependent parts are completely separated. So we may integrate over x-y plane and get the 1 + 1 dimensional vortex effective action. To evaluate the front factor we extremize I Ψ by varying Ψ and find the equation for Ψ as We may write the equation in the polar coordinates (r, θ) as where we have made the above equation dimensionless by defining a rescaled length as ρ = ∆ g r.
In order to find solutions of this equation, let us decompose the complex profile function Ψ into partial wave modes as a 0 (ρ) = 0, (4.24) for m = 0, Ψ would only be described by b 0 (ρ). After inserting expansion of Eq. (4.24) into Eq. (4.23) we find the coupled equations of the infinite tower of pairs {a m (ρ), b m (ρ)} as From now on, we would like to show the following expressions give a solution to the above equations (4.25) and (4.26): Inserting these solutions into the last terms of Eqs. (4.25) and (4.26) yields (by using 1 − ζ = ζ) where we have used the identity In order to show that these equations hold, let us use the BPS equations. Following Ref. [1] we may express the BPS equations for the profile functions f 1 (ρ), f 2 (ρ) and h A (ρ) = 1−A(ρ) as By defing a function f m (ρ) = ρ m−ξ f (ρ) ξ , this equation can be rewritten as where ξ is any real number. By applying 1 ρ ∂ ρ from the left and by using Eq. (4.30) again, this equation can be rewritten as So for ξ = ±ζ, this reduces to This equation is the same with Eq. (4.29) if we identify f m with a m or b m for ζ = 1 2 . We thus have found an exact solution to Eq. (4.29) is given by Eq. (4.27).
Let us discuss the asymptotic behaviors at large and short distances. At large distances the solutions diverge as This can also be understood by analyzing asymptotic behavior of Eq. (4.25). Since at large distances f 1 (ρ) = f 2 (ρ), the Eq. (4.25) has the solution b m (ρ) = a m+1 (ρ) ρ m+ζ . This behavior at large distance is known in the literature [17].
On the other hand, the short distance behavior was not known before. To describe the asymptotic behavior of the solution one should notice that at the center of the vortex the solution vanishes as ∼ ρ m except for b 0 . The ρ dependence cancels for b 0 in the limit ρ −→ 0 since f 1 (ρ) ∼ ρ as ρ −→ 0 and we find as ρ −→ 0, (4.35) where f 2 (0) = 0 can be observed from the solution plot in Fig. 1.
Finally, let us analyze the energy by using the solution we found. The energy of the fluctuation mode is determined with the front factor I Ψ defined in Eq. (4.20) as To derive the value of I Ψ we have to use the equations of motion of Ψ written in Eq. (4.22). We multiply Ψ * with Eq. (4.22) and add this with its complex conjugate equation to yield This relation would help us to compute I Ψ exactly as If we consider the system within a large loop of dimensionless radius R, the above integral gives So the energy of the zero mode can be expressed as For a zero mode of wavelength λ in the z-direction the energy (m = 0) behaves as So for fixed radius R energy of very long wavelength zero mode fluctuations vanish.

Summary and Discussion
In this paper, we have studied the zero modes of a single BPS Alice string in the U (1) b × SU (2) gauge theory with one charged complex adjoint scalar field. In this system, the U (1) b × SU (2) symmetry is spontaneously broken to Z 2 U (1) O(2) and this symmetry breaking can create Alice strings. This Alice string has a peculiar property that the generator of the unbroken U (1) symmetry changes its sign as it encircles the string once, due to the Z 2 factor of a semi-direct product in the unbroken symmetry. We constructed this vortex after the BPS completion in the previous paper. We have discussed the U (1) zero mode as well as the translational modes of a single BPS Alice string. The translational modes are found to be similar to other conventional vortices. The U (1) mode is generated due to the spontaneous breaking of the unbroken U (1) bulk symmetry inside the vortex core. By promoting the moduli parameters as fields in the (t, z) plane and by solving the Gauss law equation, we have written down the effective action. To solve the equation, we have expanded the profile function in partial wave modes and have found that the equations are exactly solvable in terms of vortex profile functions f 1 (r) and f 2 (r). The partial wave modes have been found to be divergent in powers of radius. After inserting the zero mode profile functions into the effective action, we have written down the effective energy, which is found to be also divergent. The minimum mode is divergent linearly as expected from the derivation in Ref. [16]. In our calculation we have not taken into account the massless bulk gauge field fluctuation effect. In principle, to write down an effective action by using renormalization group, one should not integrate it out since it is massless. In this situation, by following Ref. [4], one may conjecture to write down a 2d-4d action as (see Ref. [19]) where M r is the regularized mass and the field strength can be defined with a branch cut as Here, Σ µν is a branch cut orthogonal to string world-sheet whose direction is determined by a singular gauge and can be changed by a singular gauge transformation. The electric charge changes its sign when it passes across the cut. The coefficeint 2πζ is an Aharonov-Bohm (AB) phase and in our case ζ = 1 2 . Then, the AB scattering of photons off from an Alice string and AB scattering of two Alice strings should be able to be studied. In particular for the latter, it can be dealt within the moduli approximation due to the BPS nature.
In this paper, we have discussed bosonic zero modes a bosonic theory. There appear fermion zero modes in the string core, once we couple the theory to fermions, such as supersymmetric extensions. Fermions zero modes will give rise to a non-Abelian statistics for both Majorana [40,41] and Dirac fermions [42]. It will be novel since there are two origin of non-Abelian statistics: bosons and fermion zero modes.
In this paper we have discussed zero mode excitations of a single Alice string and have found that the U (1) zero mode is non-normalizable. It would be more interesting to study multi-string configurations because for even number of strings there will be no obstruction globally. So one may expect to have a normalizable U (1) mode for even number of strings, and that the system may possess finite electric field which is expected to explain the existence of a delocalized Cheshire charge [17]. Since our Alice strings are BPS, we can consider a stable multi-vortex system. A similar zero mode whose wave functions are spread between solitons are known for BPS semi-local vortices [43].
It is also important to prove the index theorem for these vortices. It would be different from that of conventional vortices in the Abelian-Higgs model, since there is a massless field present in the bulk. So the vanishing theorem does not work in this case. We keep these issues as our future problems.