e

one-halflike the

Physics Letters B 557 (2003) 303–308

www.elsevier.com/locate/np

Half-monopoles and half-vortices in the Yang–Mills theory

E. Harikumar, Indrajit Mitra, H.S. Sharatchandra

The Institute of Mathematical Sciences, C.I.T. Campus, Taramani P.O., Chennai 600 113, India

Received 16 January 2003; accepted 3 February 2003

Editor: M. Cvetic

Abstract

It is demonstrated that there are smooth Yang–Mills potentials which correspond to monopoles and vortices ofwinding number. They are the generic configurations, in contrast to the integral winding number configurations’t Hooft–Polyakov monopole. 2003 Elsevier Science B.V. All rights reserved.

PACS: 14.80.Hv; 11.15.-q; 11.15.Tk

Keywords: Monopole; Poincaré–Hopf index; One-half winding number

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In this Letter, we demonstrate Yang–Mills fieconfigurations of monopoles and vortices with hthe usual charges. We show that these are the gefield configurations, in contrast to the integral windinumber configurations such as the ’t Hooft–Polyakmonopole. In Refs. [1,2], the monopole configuratwas related to the singularities of the eigenvector fieof the real symmetric matrix

(1)Sij (x)= Bai (x)Baj (x),whereBai = εijk(∂jA

ak − 1

2εabcAbjA

ck) is the SO(3)

magnetic field. Such singularities arise due to indeminacy of the directions of the eigenvectors, andit is crucial that the eigenvalues become degenerathe points of singularity [3]. The topology of the cofiguration can be traced to these singularities. We r

E-mail addresses: hari@imsc.res.in (E. Harikumar),indrajit@imsc.res.in (I. Mitra), sharat@imsc.res.in(H.S. Sharatchandra).

0370-2693/03/$ – see front matter 2003 Elsevier Science B.V. All rigdoi:10.1016/S0370-2693(03)00203-X

to these points of singularities as the ‘centres’ oftopological configurations. For the ’t Hooft–Polyakmonopole [4,5],

(2)Sij = α(r2)δij + β(r2)xixjwhereα andβ are functions of the distancer fromthe origin only. One of the eigenfunctions is tradial vectorxi , with unit winding number. This haindeterminate direction at the originr = 0. But thereis no contradiction becauseS ∝ I (the identity matrix)at the origin, and any vector is an eigenvector.

Note that the entries of the matrixS are smooth inthe coordinatesxi at the origin. Singularities arise ispite of this, due to the eigenvalue equation.

The ’t Hooft–Polyakov monopole has some exctional features which are not generic. The first of this that two eigenvalues are degenerate everywhSecondly, the entries of the matrixS are quadratic inthe coordinates. Thus, in the Taylor series expanof Sij (x) about the origin, linear terms are missin

hts reserved.

304 E. Harikumar et al. / Physics Letters B 557 (2003) 303–308

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Both these features are a consequence of the rotatinvariance of the ’t Hooft–Polyakov monopole. (Throtational invariance is under simultaneous and eqrotations in physical and isospin spaces.)

In this Letter, we analyse the generic case, iwe consider Sij (x) with linear terms in the Taylorexpansion about the origin. We find the novel featureof half-integral winding number configurations aobtain the interpretation of such configurations.

As we are interested in the eigenfunctions, we mappropriately subtract a multiple of the identity matfrom Sij . Also an overall scale is irrelevant.We willrefer to the matrix after these changes as Tij .

We first illustrate the possibility and meaningconfigurations with half a winding number using2× 2 real symmetric matrix fieldTij (x, y). The par-adigm is provided by the matrix

(3)T =(x y

y −x).

The eigenvalues areλ± = ±r, wherer = √x2 + y2.

We denote the corresponding eigenfunctions byζ±i .

The eigenfunction(ζ+

1ζ+2

)hasζ+

1 /ζ+2 = y/(r − x). Thus

the normalised eigenfunction has the simple form

(4)

(ζ+

1

ζ+2

)=(

cosθ2sin θ2

)

in the polar coordinates. Hereθ = tan−1(y/x).The occurrence of half the polar angle in (4) is s

nificant. If we go round the origin once, the eigenvetor changes the sign. It is not possible to definevector fieldζ+

i (x) continuously everywhere. Therenecessarily a discontinuity (change of sign) acros“branch cut” starting from the origin. The choicethis branch cut is arbitrary, except that it starts atorigin. If we consider the complex vectorζ+

1 + iζ+2 =

exp(iθ/2), the phase changes byπ when we go aroundthe origin once. In this sense, the winding numbehalf. We call this configuration a half-vortex. It canchecked that such a phase change takes place foother eigenvectorζ−

i (x) too (Fig. 1).We emphasise that the entries of the matrixTij

are smooth even at the origin. In spite of this,eigenvalue equation gave a discontinuous eigenvefield.

It is easy to see that only half-integral windinnumber is possible in this case. The eigenve

l

Fig. 1. A winding number half configuration:ζ±i changes sign when

taken around any closed path enclosing the centre. The curvedrepresents the (arbitrary) line of discontinuity.

of a real symmetric matrix is real and hencenon-degenerate eigenvector, after normalisationambiguous only up to a sign. Therefore, when tacontinuously around a closed path, the only posschange in the eigenvector on return to the initial pois by an overall sign. This indeed happens inpresent case.

We now argue that this describes the situationthe generic case too. The most general 2× 2 realsymmetric linear in the coordinates is

(5)T =(ax + by cx + dycx + dy ex + fy

).

For the eigenvalue problem, we can subtract a multof identity matrix fromT given above. Subtractin12((a + e)x + (b+ f )y)I , we get a symmetric matrixWe now choose the oblique system of coordinates

(6)2x ′ = (ax + by)− (ex + fy), y ′ = cx + dy.(In the generic case these are linearly independenta valid choice of new coordinates.) With this we aback to the paradigm considered in (3).

Therefore, on considering the Taylor series expsion of the entriesTij about the point of degeneracsayx = 0, y = 0, it is clear that so long as the termlinear inx andy are present, we get the phenomenof one-half winding number described above.

The situation will be totally different in the caswhere the entries are quadratic in the coordinates.simplest example is the one analogous to the cas’t Hooft–Polyakov monopole:

(7)Tij = xixj .Now the eigenvectors are

r =(

cosθ

sinθ

)and θ =

(−sinθ

cosθ

).

E. Harikumar et al. / Physics Letters B 557 (2003) 303–308 305

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For both of these eigenvectors, the winding numaround the origin is one, and the vector fields cbe defined continuously everywhere (except forsingularity at the origin).

It is interesting to consider the case where

(8)T =(x2 − a2 xy

xy y2

).

Now the double degeneracy is at two points, vx0 = ±a, y0 = 0. Around each point the Taylor serieexpansion has the form

(9)T =(

2X Y

Y 0

)+ higher order terms inX, Y,

where X = x0(x − x0) and Y = x0(y − y0). Theleading term has precisely the form of the paradiwe considered (up to a multiple of the identity matriSo we get half a winding number around each poof degeneracy. We may conveniently choose thejoining the two centres as the branch cut. The windnumber along a curve enclosing both centres is oIndeed, asa → 0, we recover from (8) the windinnumber one configuration considered in (7). In tlimit of a → 0, the pair of half winding numbeconfigurations merge together to give winding numone configuration (Fig. 2).

Let us also consider the matrix

(10)T =(x2 − a2 ay

ay 0

).

This again has the same two points of degeneracthe matrix in (8). However, in the present case

Fig. 2. Two winding number half configurations give a windinumber one configuration at large distances.

winding numbers are±1/2, respectively. (The configuration around(−a,0) is related to our paradigm i(9) by reflection about theX-axis:Y → −Y . So it hasthe winding number−1/2.) In the limit a → 0, theeigenvectors are now

(10

)and

(01

), and each of them

has a vanishing winding number.We have considered 2× 2 matrices though theSij

relevant for the Yang–Mills theory are 3× 3 matrices.We regard the matrix in (3) as a block of the 3× 3matrix

(11)T = x y 0y −x 0

0 0 0

.

Then the interpretation is that we have a vorwith one-half winding number centred on thez-axis and extending indefinitely along it. To justithis interpretation, we have to exhibit an Yang–Mipotential which will give rise toTij as consideredin (11). It has been shown in Ref. [6] that in the genesituation where the 3× 3 matrixBai is invertible andsmooth, there exists a smoothAai which will reproducesuch aBai . So for the case here,Aai can be constructeas a Taylor series expansion about the origin. Wepresent such a series for a different example beWe will also discuss the finiteness of energy (per ulength) there.

We now show that monopoles of one-half windinumber also occur. The paradigm in this caseprovided by the 3× 3 real symmetric matrix

(12)T =

0 0 x

0 0 y

x y −2z

.

Here the eigenvalues areλ± = −r(cosθ ∓ 1) andλ0 = 0. In the spherical coordinates, the correspondeigenfunctions are

ζ+ =

cosθ2 cosφ

cosθ2 sinφ

sin θ2

, ζ− =

sin θ2 cosφ

sin θ2 sinφ

−cosθ2

,

(13)ζ 0 =

−sinφ

cosφ

0

.

Comparingζ±i with the radial vector and with th

normalised Higgs in the ’t Hooft–Polyakov monopo

306 E. Harikumar et al. / Physics Letters B 557 (2003) 303–308

eonto the

(a) (b)

Fig. 3. (a) A winding number half configuration in three dimensions. There is a vortex of winding number one along the positivz-axisterminating at the centre. (b) A winding number one configuration in three dimensions. The upper half of this configuration is mappedentire sphere in (a) to give one-half winding number.

-

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x

ng

er-ofthe

obeis

tedotble

of

leexis

tthe

pfhe

viz. (sinθ cosφ,sinθ sinφ,cosθ), we notice that essentially the angleθ is replaced byθ/2. This leadsto one-half winding number in the present case. Tphenomenon is illustrated forζ+

i in Fig. 3. In effect,the configuration in the upper half of the spherewinding number one is mapped onto the entire sphto give one-half winding number. We refer to suchconfiguration as a half-monopole. Note that the vtor field ζ+

i is singular (indeterminate in direction) aalong the positivez-axis. This is possible becauseTijhas a double degeneracy there. This has the intetation of a vortex (of winding number one) along tpositivez-axis terminating at the origin and giving risto a monopole. Because of this vortex, the vector fiis not continuous on the sphere, and therefore onewinding number is possible. If the vector field wesmooth on the sphere, the winding number would hbeen only integral.

In an analogous way,ζ−i corresponds to a vorte

of unit winding number along the negativez-axis,terminating at the origin. Finallyζ 0

i is again a vortexof winding number one extending indefinitely alothez-direction.

That the monopole centre (point of triple degenacy) is a terminating point of vortex centre (linedouble degeneracy) is a generic situation. In fact,generic situation is as follows. The configurationζAi ,for eachA, will have double degeneracy along twlines terminating at the centre. Each such line willthe centre of a vortex of winding number half. Th

-

will be elaborated elsewhere. It may also be nothat for the ’t Hooft–Polyakov monopole, which is ngeneric due to rotational invariance, we have doudegeneracy everywhere.

If we formally compute the Poincaré–Hopf indexthe vector fieldζ+

i , we get it to be−1/2. The index forζAi is given byM = ∮

SdSi kAi , where the integration

is over a surfaceS enclosing the centre andkAi is thePoincaré–Hopf current [2,5,7]

(14)

kAi = 1

2εijkεlmnζ

Al ∂j ζ

Am∂kζ

An (no sum overA).

We have in the present case (see Eq. (13))

(15)k+i = −xi 1

4r2cosec

θ

2.

The vector fieldζ+i is not smooth at the north po

of the sphere. Therefore, the definition of the indM is only formal. Nevertheless, this singularityof zero measure in the integration overS and weget the winding number to be−1/2. Note that the“magnetic field” k+i of this half-monopole is nospherically symmetric, in contrast to the case ofDirac monopole. It has only an axial symmetry.

In Ref. [2], it was shown that the Poincaré–Hocurrent for the eigenvectorζAi can be expressed as tcurl of an Abelian vector potentialωAi :

(16)kAi = εijk∂jωAk − Dirac string contributions,

E. Harikumar et al. / Physics Letters B 557 (2003) 303–308 307

2ctor

tivehat

iveole

a

err anot

.

the

e

of

eset.the

niteue

eenr isioninIn

heof

ptes

ofsaryofnch

teralfry.ndthe

lf-in

where

(17)ωAi = 1

2εABCζBj ∂iζ

Cj .

Here the indicesA, B andC, having the values 1,and 3, label the three eigenvectors. The Abelian vepotential corresponding toζ+

i is

(18)w+i = −φi 1

2rsec

θ

2.

This potential has the Dirac string along the negaz-axis. This Dirac string is unphysical, in the sense tit does not contribute to the “magnetic field”kAi (seeEq. (16)). In contrast, the vortex line along the positz-axis is physical, and, because of it, the monopdoes not have spherical symmetry.

Similarly, we get the Poincaré–Hopf index forζ−i

as −1/2. In the case ofζ 0i , notice that it spans

two-dimensional vector space as we varyφ. Thereforethe index computed overS will be zero (three-dimensional winding number is zero). On the othhand, it makes sense to calculate the index ovetwo-dimensional surface. For any such surfacecontaining thez-axis, we get winding number one.

We now present the Taylor series expansion ofAaiabout the origin which leads toTij considered in (12)Consider first the matrix(B)ia = Bai . In the symmetricgauge(B)ia = (B)ai [2], we have(B2)ij = Sij , sothat, for the case given in (12),

(19)B = I + 1

2

0 0 x

0 0 y

x y −2z

+ · · · .

Here the ellipsis indicates terms of higher order incoordinates. The most general Taylor expansion ofAaiabout the origin is:

(20)Aai = aai + baij xj + caijkxjxk + · · · .To obtainBai as given in (19), it suffices to take

(21)aai = 0, baij = −1

2εaij ,

(22)caijk = 1

2

(εijpM

apk + εikpMa

pj

),

where

(23)M131 = −1

6, M2

32 = −1

6, M3

33 = 1

2,

and all otherMaij are zero. Thus our solution for th

gauge field is

(24)

A= 1

2

−xy/3 z− y2/3 −y + yz−z+ x2/3 xy/3 x − xz

y −x 0

+ · · · ,

where(A)ia =Aai .We now address the question of finiteness

the energy of the half-monopole, given byE =∫d3x Sii/2. As Sij can be expanded in Taylor seri

about the origin, the energy is finite in the ultraviolAlso the infrared finiteness of the energy resides inscale factors ofSij , such asα(r2) andβ(r2) in Eq. (2),and these can be chosen appropriately to get a fienergy. Note that the one-half winding number is dto the tensorial structure ofSij , the eigenvectorsζAibeing unaffected by the scale factors.

In both two and three dimensions, we have sthat the phenomenon of one-half winding numbedue to the generic linear terms in the Taylor expansof Sij . Nevertheless, there are crucial differencesthe origin of this phenomenon in the two cases.two dimensions, the ambiguity in the sign of teigenvector was the underlying reason. The linediscontinuity (the “branch cut”) was arbitrary, excefor the starting point. In three dimensions, linof double degeneracy terminating at the centrethe monopole were necessary to give the necesdiscontinuity in the form of a vortex. But these linesdouble degeneracy are rigid, in contrast to the bracuts in two dimensions.

To conclude, we have pointed out in this Letthat vortex and monopole configurations of one-hwinding number are present in the Yang–Mills theoThey arise from smooth Yang–Mills potentials, aare indeed the generic configurations in contrast to’t Hooft–Polyakov monopole.

Note added in proof

Some of the works which discuss vortices of hainteger winding number in other contexts are givenRef. [8].

308 E. Harikumar et al. / Physics Letters B 557 (2003) 303–308

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References

[1] R. Anishetty, P. Majumdar, H.S. Sharatchandra, Phys. LB 478 (2000) 373.

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A.M. Polyakov, JETP Lett. 20 (1974) 194.

[5] P. Goddard, D. Olive, Rep. Prog. Phys. 41 (1978) 1357.[6] P. Majumdar, H.S. Sharatchandra, Phys. Rev. D 63 (20

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