Euclidean Dilaton Black Hole Vortex and Dirac Fermions

Euclidean Dilaton Black Hole Vortex and Dirac Fermions

Łukasz Nakonieczny and Marek Rogatko Institute of Physics
Maria Curie-Sklodowska University
20-031 Lublin, pl. Marii Curie-Sklodowskiej 1, Poland
July 12, 2019
Abstract

We considered the behaviour of Dirac fermion modes in the background of Euclidean dilaton black hole with an Abelian Higgs vortex passing through it. Fermions were coupled to the fields due to the superconducting string model. The case of nonextremal and extremal charged black holes in the theory with arbitrary coupling constant between dilaton field and -gauge field were considered. We elaborated the cases of zero and non-zero Dirac fermion modes. One finds the evidence that the system under consideration can support fermion fields acting like superconducting hair on black hole in the sence that nontrivial spinor field configuration can be carried by Euclidean spherically symmetric charged dilaton black hole. It was revealed that the localization of Dirac fermion modes depended on the cosmic string winding number and the value of black hole surface gravity.

04.70. Bw

I Introduction

In the recent years, studies of a much more realistic case than scalar fields attracted more attention. Especially, solution of field equations describing fermions in a curved geometry is one of the theoretical tools of investigating the underlying structure of the spacetime. The better understanding of properties of black holes also acquires examination of the behaviour of matter fields in the vicinity of them chandra (). Dirac fermions behaviour was studied in the context of Einstein-Yang-Mills background gib93 (). Fermion fields were analyzed in the near horizon limit of an extreme Kerr black hole sak04 () as well as in the extreme Reissner-Nordström (RN) case loh84 (). It was also revealed fin00 ()- findir (), that the only black hole solution of four-spinor Einstein-dilaton-Yang-Mills equations were those for which the spinors vanished identically outside black hole. Dirac fields were considered in Bertotti-Robinson spacetime br1 (); br2 () and in the context of a cosmological solution with a homogeneous Yang-Mills fields acting as an energy source gib94 ().

The late-time decay of fermion fields in the background of various kinds of black holes is an important problem from the point of view of the uniqueness theorem for them. The late-time behaviour of massless and massive Dirac fermion fields were widely studied in spacetimes of static as well as stationary black holes jin04 ()-goz11 ().

Brane models in which our Universe is represented as -dimensional submanifold living in higher dimensional spacetime also attract the attention to brane black holes. The decay of massive Dirac hair on a brane black hole was considered in br08 ().

It could happened that at the beginning our Universe underwent several phase transitions. The mechanism of spontaneous symmetry breaking involved in the early Universe phase transitions might produce stable topological defects like cosmic strings, monopoles and domain walls vil (). Among them cosmic strings and cosmic string black hole systems acquire much interest. Assuming a distributional mass source the metric of this system was derived in ary86 () (the so-called thin string limit). In Ref.ach95 () the numerical and analytic evidences for the existence of an Abelian Higgs vortex on Schwarzschild black hole were given, while in Refs. cha98 ()-mod98 () the extensions of the aforementioned arguments to the case of charged RN black hole and dilaton black holes were performed. It was also found that an analog of Meissner effect (i.e., the expulsion of the vortex fields from the black hole) could take place. It happened that this phenomenon occurs for some range of black hole parameters bon98 (). On the contrary, extremal dilaton black holes always expel vortex Higgs fields from their interior mod99 (). The very similar situation takes place in the case of the other topological defect, domain wall which can be expelled from various kinds of black holes domainwalls ().

There are some cosmic strings which may become superconducting by the implementation of fermions. They may be responsible for various exotic astrophysical phenomena. For instance, closed superconducting loops, the so-called vortons vor () may constitute a fraction of cold dark matter in galactic halo, their slow quantum decays may be connected with the ultra-high energy cosmic rays (UHECRs) nag00 (). To ones dismay, it turned out that also the high-redshift gamma ray bursts could be a reasonable way to test superconducting string model che10 ().

In Ref.col91 () it was revealed that Euclidean vortex solution in the spacetime of a black hole led to the nonperturbative exponentially decay of electric field outside the event horizon of a Schwarzschild black hole. It was shown dow92 () that Euclidean Schwarzschild black hole could support vortex solution at the event horizon. In Ref.mod98a () the generalization of the above problem to the case of Einstein-Maxwell-dilaton gravity was proposed. On the other hand, in Ref.gre92 () it was observed that the Dirac operator in the spacetime of the system composed of Euclidean magnetic RN black hole and a vortex in the theories containing superconducting cosmic strings wit85 () possessed zero modes. In turn, the aforementioned zero modes caused the fermion condensate around magnetic RN black hole.

The motivation of our paper was to provide some continuity with the researches presented in Ref.gre92 () and to generalize them to the theory constituting the modification of Einstein-Maxwell theory, the so-called dilaton gravity being the low-energy limit of the heterotic string theory. In dilaton gravity one has to do with the non-trivial coupling of dilaton field with the -gauge field. Our considerations will be valid for an arbitrary coupling constant. In our researches we shall consider static spherically charged black hole solution in dilaton gravity which has quite different topological structure than the one studied in gre92 (). Contrary to the researches conducted in the aforementioned reference we shall not only pay attention to Dirac zero fermion modes but we elaborate the non-zero fermion modes in the underlying spacetime. To our knowledge, the problem of non-zero Dirac fermion modes in the spacetime of black hole cosmic string system has not been studied before. We also pay attention to the near-horizon behaviour of fermionic fields causing superconductivity in the case of extremal charged dilaton black hole. As was mentioned before, in the spacetime of extremal dilaton black hole one can observe the analog of Meissner effect.

The layout of our paper will be as follows. In Sec.II we start with the discussion of the vortex itself in the background of the Euclidean dilaton black hole being the static solution of dilaton gravity equations with an arbitrary -coupling constant. Sec.III will be devoted to the superconducting cosmic string piercing the black hole in question. We shall elaborate the behaviour of zero Dirac fermion modes both on non-extremal and extremal Euclidean dilaton black hole. In Sec.IV we take into account fermion modes for the case when on the same kinds of black holes. We find that Dirac fermion modes may be regarded as hair on the considered black holes. In the next section we conclude our studies.

Ii Euclidean dilaton black hole /Abelian Higgs vortex system

In the following section we shall consider Euclidean charged dilaton black hole/ string vortex configuration. One assumes the complete separation between the degrees of freedom of each of the objects in question. We shall treat static charged dilaton black hole line element as the background solution and numerically justify the existence of the vortex solution, for arbitrary coupling constant in the considered theory.

The system under consideration will be described by the action of the form as

 S=S1+Sbos, (1)

where is the dilaton gravity action being the low-energy limit of the string action with arbitrary coupling constant. It is provided by

 S1=∫√−g d4x[R−2(∇ϕ)2−e−2α ϕFμνFμν], (2)

where , is the dilaton field, is a coupling constant which determines the interaction between dilaton and Abelian gauge fields. In action , the corresponding Abelian gauge field can be thought as the everyday Maxwell one.

The other gauge field is hidden in the action and it is subject to the spontaneous symmetry breaking. Its action implies

 Sbos=∫√−g d4x[−(dμΦ)†dμΦ−14BμνBμν−λ4(Φ†Φ−η2)2], (3)

where is the field strength associated with -gauge field, is the energy scale of symmetry breaking and is the Higgs coupling. The covariant derivative has the form , where is the gauge coupling constant.

The line element of the general static spherically symmetric Euclidean dilaton black hole yields

 ds2=A2dτ2+B2dr2+C2(dθ2+sin2θdϕ2), (4)

where in order to Euclideanize the metric we set the Euclidean time as . For the case when , the explicit forms of the metric coefficients are as follows:

 A2=(1−r+r)(1−r−r)1−α21+α2, (5) C2=r2(1−r−r)2α21+α2, (6)

where are related to the mass and charge of the black hole due to the relations

 2M = r++1−α21+α2r−, (7) Q2 = r+r−1+α2. (8)

On the other hand, the dilaton field is given as

 e2αϕ=(1−r−r)2α21+α2. (9)

The location of the event horizon is , the is another singularity, but one can ignore it for .
Having in mind the above charged dilaton black hole solution one can see that the structure of the black hole in question is drastically changed due to the presence of dilaton field. Moreover, the arbitrary -coupling constant is the other non-trivial element in the studies. Recently, the numerical studies of the dynamical collapse of complex charged scalar field bor11 () reveal that due to the coupling between dilaton and -gauge field the collapse leads to the Schwarzschild black hole rather than the collapse of charged field in Einstein-Maxwell gravity. Though, when one puts coupling constant to zero we obtain the bahaviour leading to a black hole with a Cauchy horizon.

Treating a nonlinear system coupled to gravity is a very difficult problem but it is worth mentioning that it found in Refs.ach95 ()-mod99 () that the self-gravitating Nielsen-Olesen vortex can act as a long hair for various kinds of black holes. In what follows, we refine our attention to the vortex itself and elaborate its behaviour in the background of Euclidean dilaton black hole in the theory with arbitrary coupling constant . To begin with we choose and fields provided by the expressions

 Φ(xi)=ηX(r)eiNτκ, (10)
 Bμ(xi)=κeR[Pμ(r)−N∇μτ], (11)

where is the surface gravity of the Euclidean dilaton black hole. Further, we assume that is the only one non-vanishing coefficient of the gauge field which is subject to the spontaneous symmetry breaking.

Let us introduce quantities defined by

 √λη(M,QBH,r,r+,r−,κ)≡(¯M,¯QBH,¯r,¯r+,¯r−,¯κ). (12)

Taking the background solution as the spacetime of Euclidean charged dilaton black hole, one reaches to the following equations of motion for and fields:

 1C2(C2A2X,¯r),¯r − ¯κ2P2XA2−12X(X2−1)=0, (13) 1C2(C2P,¯r),¯r − 1νX2PA2=0, (14)

where we denoted . The above equations can be rearranged in the forms which imply

 (1−¯r+¯r)(1−¯r−¯r)1−α21+α2d2d¯r2X+{2¯r+2α21+α2¯r−¯r1¯r−¯r−+1¯r2(1−¯r−¯r)1−α21+α2[¯r++¯r−1−α21+α2¯r−¯r+¯r−¯r−]}dd¯rX+ −¯κ2(¯r¯r−¯r+)(¯r¯r−¯r+)1−α21+α2P2X−12X(X2−1)=0, (15)
 d2d¯r2P+{2¯r+2α21+α2¯r−¯r1¯r−¯r−}dd¯rP−(¯r¯r−¯r+)(¯r¯r−¯r−)1−α21+α2PX2ν=0. (16)

We solve this set of equations numerically using the relaxation technique press (). As in Ref.gre92 () we shall work in the so-called supersymmetric limit when . Behaviour of the fields in question are depicted in Fig.1 and Fig.2, respectively. For the completeness of the studies we also plotted in Fig.3 the -dependence of the surface gravity. It can be seen that the bounded solution at the event horizon for field integrated out the exponentially decaying at infinity while field tends to the constant value equal to at infinity. We take the coupling constant equal to , respectively. On the other hand, the charge of the considered black hole was taken by , where . In our numerical analysis we set as the radial coordinate for the Euclidean black hole in question. For the flat spacetime we chose the ordinary -coordinate. We obtained the perfect agreement with the previous numerical studies for the case , Ref.dow92 () and when , Ref.mod98a (). Our numerical investigations reveal that the larger coupling constant is the quicker field tends to zero. For the other vortex field the conclusion is similar, i.e., the larger value of one considers the quicker field tends to the constant value equal to .

Iii Fermions in the Euclidean Dilaton Black Hole Background.

In Ref.wit85 () it was shown that cosmic strings can behave like superconductors and are able to carry electric currents. In principle there are two varieties of superconducting strings, i.e., fermionic or bosonic. In fermionic case superconductivity takes place due to the appearance of charged Jackiw-Rossi jac81 () zero modes which effectively can be regarded as Nambu-Goldstone bosons in dimensions. They give a longitudinal component to photon field on the cosmic string, and may be trapped in the string as massless zero modes. On the other hand, bosonic superconductivity occurs when a charged Higgs field acquires an expectation value in the core of the cosmic string. In this case the current in the aforementioned object is carried by bosonic modes.

In this section we shall consider fermionic superconducting cosmic string piercing an Euclidean charged dilaton black hole. As was mentioned above it turned out that it is possible for the currents to be carried by fermionic degrees of freedom confined to the cosmic string core wit85 (). If one takes into account the electromagnetic one, then the cosmic string will behave as superconducting. It can be performed by extending Lagrangian by adding the following fermionic sector:

 SFE=∫√−g d4x[¯ψγμDμψ+¯χγμDμχ+i ~α (Φ ψTCχ−Φ∗ ¯ψC¯χT)], (17)

where is a coupling constant while Dirac operator satisfies the relation of the form as

 Dμ=∇μ+i R eRBμ+i Q eqAμ. (18)

We take that . The covariant derivative for spinor fields is given by the standard relation . Dirac gamma matrices forming the chiral basis for the problem in question yield

 γ0=(0II0),γa=(0iσa−iσa0), (19)

where the Pauli matrices are provided by

 σ0=(1001),σ1=(0110), (20) σ2=(0−ii0),σ3=(100−1). (21)

Moreover, the charge conjugation matrix implies

 C = (−iσ200iσ2), (22) C† = CT=−C. (23)

In what follows we consider the line element of the Euclidean spherically symmetric static black hole with a vortex passing through it. Its metric given by (4), with the line element on the -sphere defected by the presence of the vortex. Hence, it yields

 dΩ2=C2(r)(dθ2+~b2sinθ2dϕ2), (24)

where is a cosmic string parameter.
For the above metric the curved spacetime gamma matrices are related to those given by Eq.(19) by the relations as

 γτ = A−1(0II0),γr=B−1(0iσ1−iσ10), (25) γθ = C−1(0iσ2−iσ20),γϕ=1C~bsinθ(0iσ3−iσ30). (26)

In the spacetime under consideration spinors and and their complex conjugations must be regarded as independent fields. Following this idea, variation of the fermion action with respect to and fields, implies the following equations of motion:

 γμDμψ−i~α Φ∗C¯χT=0, γμDμ¯χ†−i~α Φ∗Cψ∗=0, (27)

plus the analogous relations achieved by conjugations of the adequate relations in question.

With the above definitions one finds the exact form of the Dirac operator. Accordingly, they yield the result that

 /D=γμDμ=(0D+D−0), (28)

where we have denoted by and the following parts of the Dirac operator defined above

 D+ = στDτ+iσk Dk, (29) D− = στDτ−iσj Dj. (30)

Consequently, with the remark that in Euclidean spacetime and must be treated as independent fields, we implicitly choose and as left-handed, while and as right-handed. Namely, they can be brought to the forms

 ψ = (ψL0),χ=(χL0), (31) ¯ψ = (0ψR),¯χ=(0χR). (32)

Returning to the equations of motion, they can be rewritten as

 D+ψR + i~α Φ∗(−iσ2χ∗L)=0, D−χL + i~α Φ∗(iσ2ψ∗R)=0. (33)

As in Ref.gre92 () we choose the following form of and spinors:

 ψL = fLξ−,χR=gRξ+, (34) ψR = fRξ−,χL=gLξ+, (35)

where we choose as the right and left-handed ones. They obey the relation of the form as , which assure that fermions propagate along the cosmic string. On the other hand, taking into account the normalization conditions , one arrives at the explicit form of . Namely, they may be written in the form as

 ξ+=1√2(1−1),ξ−=1√2(11). (36)

By virtue of the above one can readily verify that they satisfy the following:

 iσ1ξ± = ∓iξ±,−iσ2ξ±=±ξ∓, (37) σ3ξ± = ξ∓,σ0ξ±=ξ±. (38)

Iv s-waves

First of all, we shall elaborate the s-wave case. Behaviour of the Dirac operator acting on -sphere with magnetic monopole and pierced by an Abelian vortex will be crucial in this case. Namely, on evaluating the action of the Dirac operator on spinor we find that it is proportional to . Having in mind the orthogonality condition for one can draw conclusion that the only admissible eigenvalue of is . Taking into account the explicit for of the Dirac operators and Eqs.(34)-(37), after a little algebra, equations of motion (33) reduce to

 (39)

and

 A−1(−i∂τ+R κ (P−N)) gL+(B−1∂r+12A−1B−1∂rA+B−1C−1∂rC) gL+~α Φ∗ f∗R=0. (40)

In what follows we assume that the time-dependence of and will be of the form and , where . The explicit form of the field given by Eq.(10), as well as Eqs.(39)-(40), enables us to deduce that . Consequently, it leads to the condition

 ωg=N κ−ω∗f. (41)

As was argued in Ref.gre92 () the action of the operators and appearing in the definition of the covariant derivative given by Eq.(18), can be defined as and , while and . On this account it is customary to rewrite equations of motion as follows:

 A−1( − ω∗f+~r κ (P−N))f∗R+(B−1∂r−12A−1B−1ddrA+B−1C−1ddrC)f∗R+mfer XgL=0, (42) A−1( − N κ+ω∗f−(~r+1) κ (P−N))gL+(B−1∂r+12A−1B−1ddrA+B−1C−1ddrC)gL + mfer Xf∗R=0,

where for brevity we have denoted .

iv.1 Nonextremal Euclidean Dilaton Black Hole

First we elaborate the behaviour of Dirac fermions in the vicinity of the black hole event horizon. We shall begin with considering the nonextremal case of dilaton Euclidean black hole. Condition determines the outer black hole event horizon, while , is the area of the event horizon. Having in mind the form of the line element (4), we assume the regularity of the solution at . On this account, we can choose locally cylindrical coordinates in which the aforementioned metric is regular. Namely, one has that , where is the period of the Euclidean time. Moreover, regularity yields that To proceed further, let us suppose that and are provided by the following:

 f∗R = x+(^r) exp(∫(12(P−N)κA−mfer X)d^r+iω∗fτ), (44) gL = x−(^r) exp(∫(12(P−N)κA−mfer X)d^r−iω∗gτ). (45)

It helps us to rewrite Eqs. of motion (42)-(IV) in the form as

 dd^rx±±(~r+12)(P−N)^r x±−1^r (ω±±12)±mfer X (x−−x+)=0, (46)

where we set and .

One can draw a conclusion that and are of order of unity as we reach the event horizon, i.e., . They can be regarded as hair on the Euclidean dilaton black hole in the sense of the nontrivial field configurations maintained by black hole event horizon. The same observation was revealed in the case of Euclidean RN black hole solution pierced by superconducting string gre92 (). Returning to the relations (46), we observe that they resemble equations of motion obtained for cosmic string with fermion modes in flat Minkowski spacetime gre92 ().
An alternative way of treating the problem is to expand the metric coefficients of the considered line element in the nearby of the black hole event horizon. They will be given by the following:

 A2(r)≃a(r+)(r−r+),B2(r)≃b(r+)(r−r+)−1,C2(r)=C2(r+). (47)

Next making the change of the variables described by the relations

 ρ2=4 b(r+) (r−r+),T=12√a(r+)b(r+)τ, (48)

it can be shown that the line element of nonextremal Euclidean black hole yields

 ds2=ρ2dT2+dρ2+C2(r+) dΩ2. (49)

On the other hand, the asymptotic behaviour of the background solutions subject to the equations of motion for Abelian vortex fields are provided by gre92 (); ach95 ()

 X∼(r−r+)|N|/2=ρ|N|,P∼N−O(r−r+)=N+O(ρ2). (50)

Returning to the equations of motion for Dirac fermion, one can easily verify by the above relations that they reduce to the forms as

 ddρf∗R − (ω∗f+12ρ) f∗R+mfer ρ|N| gL=0, (51) ddρgL + (−N κ+ω∗f+12ρ) gL+mfer ρ|N| f∗R=0. (52)

It will be interesting to consider the influence of the winding number on the behaviour of the fermion modes in question. We shall elaborate two limiting cases of the aforementioned problem, i.e., the case when and the case for which the winding number tends to .

We shall begin with the case . The close inspection of formulae (51) and (52) reveals that the mass term proportional to can be neglected because of the fact that near the black hole event horizon. On this account, one has

 ddρf∗R − (ω∗f+12ρ) f∗R=0, ddρgL + (−Nκ+ω∗f+12ρ) gL=0. (53)

It can be easily checked that the solutions of the above set of differential equations imply

 f∗R = c1 ρω∗f+12, gL = c2 ρNκ−ω∗f−12, (54)

where and are constants.
Consistent with the requirement of the finiteness of the solutions in question on the black hole event horizon we arrive at the condition

 Nκ−12≥Re(ωf)≥−12. (55)

It could be also verified, by the direct calculations, that and belong to the square integrable class of functions, i.e., and .

In the case when equations of motion for the Dirac fermion fields are as follows:

 ddρf∗R − (ω∗f+12ρ) f∗R+mferρ|N|gL=0, ddρgL + (−ωg+12ρ) gL+mferρ|N|f∗R=0, (56)

where we have used relation (41) to eliminate -dependence in the second term on the left-handside of equation (IV.1). In order to solve Eqs.(IV.1), we assume the following ansatz for the spinors and :

 f∗R = ρω∗f+12 ~f, gL = ρNκ−ω∗f−12 ~g. (57)

Defining and after a little of algebra we get

 ddρ ~f+mferρβ ~g=0, ddρ ~g+mferρ2|N|−β ~f=0. (58)

From the first equation we find that . Substituting the expression for into the second equation of the underlying system we reach to the second order differential equation for , i.e., one gets

 ddρ(ρ−β ddρ ~f)−m2fer ρ2|N|−β ~f=0. (59)

Having in mind the explicit form of and given by (IV.1), and the condition for , one finds that

 Nκ−12≥Re(ωf)≥−12. (60)

After some further calculations in which we make use of the so-called Lommel’s transformation for Bessel functions, i.e., we look for the solution of Eq.(59) in the form as

 ~f=ρp Gν(λ ρa), (61)

where is Bessel function while are constants, it can be shown that the solution of (59) can be written in the form as

 ~f=C1 ρ1+β2 Iν(imN+1 ρN+1)+C2 ρ1+β2 Kν(imN+1 ρN+1), (62)

where and are constants while . stands for the modified Bessel function of the first kind while is the Macdonald’s function. Assuming that and having in mind behaviour of when one can conclude that the spinor function near the event horizon tends to the constant value described by . This condition leads us to and to the conclusion that for one gets the greatest value of hair near the black hole event horizon.

iv.2 Extremal Dilaton Black Hole

In the following section we shall discuss zero fermion modes in the background of the extremal Euclidean dilaton black hole with a vortex passing through it. In the extreme black hole case the outer event horizon coincides with the inner one, i.e., . In order to describe the system we use coordinates given by relation (47)with the only modification in coefficient. Because of the condition for the black hole be an extremal one, we have

 C2(r)=c(r+)(r−r+). (63)

By virtue of this relation the line element describing the near-horizon geometry of the extremal Euclidean dilaton black hole yields

 ds2=ρ2dT2+dρ2+c(r+)4b(r+)ρ2dΩ2. (64)

Thus, in this picture, Eqs. of motion are provided by

 ddρ f∗R + (−ρ−1 ω∗f−12ρ−1+ρ−1) f∗R+mferρNgL=0, ddρ gL + (ρ−1(−Nκ+ω∗f)+12ρ−1+ρ−1) gL+mferρNf∗R=0. (65)

As in the previous case of nonextremal Euclidean dilaton black hole, we first we consider case. The mass term proportional to will tend to zero. Then, the solutions of (IV.2) may be written in the form as follows:

 f∗R = d1 ρω∗f−12, gL = d2 ρNκ−ω∗f−32, (66)

where and are constants. The finiteness and the regularity conditions on the event horizon imply that the following conditions are satisfied:

 Nκ−32≥Re(ωf)≥12. (67)

On the other hand, in the case when Eqs. of motion may be rewritten in the form as

 ddρ f∗R + (1−ω∗f−12ρ) f∗R+mferρNgL=0, ddρ gL + (32−Nκ+ω∗fρ) gL+mferρNf∗R=0. (68)

Introducing the ansatz for fermion zero modes given by

 f∗R = ρω∗f−12 ~f, gL = ρNκ−ω∗f−32 ~g, (69)

we obtain the same set of equations as described by relations (IV.1). Having in mind formula (IV.2) and the requirement of the finiteness of and on the black hole event horizon we reach the condition for . Thus, the result is provided by

 Nκ−32≥Re(ωf)≥12. (70)

For the completeness of our research we remark that by the same procedure as we followed in the case of nonextremal Euclidean black hole we can cast the set of the first order ordinary differential equations into the second order one for , which has solution in terms of generalized Bessel functions. The range of the parameters and conclusions about hair on extremal Euclidean dilaton black hole are the same as in the nonextremal case.

V k>0 Dirac Fermion Modes

Now, we turn our attention to the case when . Our main aim will be to solve Eqs.(33) and to discuss the behaviour of Dirac fermions in the case in question. Spinors and are singled out in such way to correspond with the case . Namely, one has that , where is a linear combination of . It can be done by preferring the basis in the form

 γ0 = (0II0),γ1=(0iσ3−iσ3), γ2 = (0iσ2−iσ2),γ3=(0iσ1−iσ1). (71)

Thus, the zero mode condition can be cast into , which provides the following relation:

 ~ξ+=(10),~ξ−=(01). (72)

In the case under consideration, the exact form of the Dirac operators will be given by the expressions

 D+ = A−1(∂τ+iR (P−N)κ)σ0+(B−1∂r−12B−1A−1∂rA+B−1C−1∂rC)iσ3+DS2C, (73) D− = A−1(∂τ+iR (P−N)κ)σ0+(−B−1∂r−12B−1A−1∂rA−B−1C−1∂rC)iσ3+DS2C, (74)

where the Dirac operator on -sphere with magnetic monopole pierced by an Abelian vortex yields

 DS2=iσ2(∂θ+12cotθ)+σ1(i∂ϕ~bsinθ+QeM~b−1cotθ). (75)

Moreover, the action of the operator on and spinors implies

 iDS2ψ=k ψ,iDS2χ=k χ, (76)

where are eigenvalues of the operator in question. As far as spinors and is concerned, we set them to be linear combinations of

 ψR=(f+f−),χL=(g+g−). (77)

By virtue of the above equations of motion are provided by the following relations:

 A−1(∂τ + iR(P−N)κ) f++(B−1∂r−12A−1B−1∂rA+B−1C−1∂rC) if+−ikCf+−iαΦ∗g∗−=0, (78) A−1(∂τ + iR(P−N)κ) f−−(B