Monopoles, vortices, and kinks in the framework of noncommutative geometry

Edward TeoDepartment of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, England

and Department of Physics, National University of Singapore, Singapore 119260

Christopher TingDepartment of Computational Science, National University of Singapore, Singapore 119260

~Received 29 January 1997!

Noncommutative differential geometry allows a scalar field to be regarded as a gauge connection, albeit ona discrete space. We explain how the underlying gauge principle corresponds to the independence of physicson the choice of vacuum state, should it be nonunique. A consequence is that Yang-Mills-Higgs theory can bereformulated as a generalized Yang-Mills gauge theory on Euclidean space with aZ2 internal structure. Byextending the Hodge star operation to this noncommutative space, we are able to define the notion of self-duality of the gauge curvature form in arbitrary dimensions. It turns out that BPS monopoles, critically coupledvortices, and kinks are all self-dual solutions in their respective dimensions. We then prove, within this unifiedformalism, that static soliton solutions to the Yang-Mills-Higgs system exist only in one, two, and three spatialdimensions.@S0556-2821~97!06416-3#

PACS number~s!: 11.15.Ex, 02.40.2k, 11.30.Ly, 14.80.Hv

I. INTRODUCTION

A theory of scalar fields, possessing some symmetry, issaid to undergospontaneous symmetry breakingwhen itspotential acquires a family of degenerate minima. In thisevent, the vacuum of the theory is not unique, and possiblestates are related through the symmetry. Any specific choiceof a vacuum state, however, breaks this invariance.

Scalar fields with spontaneously broken symmetry play animportant role in modern Yang-Mills gauge theory. Throughthe Higgs mechanism@1#, they generate masses for the gaugebosons as required by phenomenology. Barring its experi-mental success, however, the form and content of the Higgssector lacks motivation from gauge principles so vital to thecorresponding Yang-Mills sector. Because of this, it is oftenregarded as anad hocand aesthetically unappealing featurein the otherwise geometrically beautiful backdrop of Yang-Mills theory, and there has been several attempts in the lit-erature to address this problem.

In the late 1970s, several authors@2,3# proposed a Kaluza-Klein unification of Yang-Mills and Higgs fields. Gaugetheory was formulated on a higher-dimensional space-time,and components of the gauge connection in the extra dimen-sions identified as Higgs fields. A dimensional reductionyielded four-dimensional Yang-Mills theory together with asymmetry-breaking potential for the Higgs fields. Thesemodels offered predictions for otherwise free parameterssuch as the Higgs boson mass, but failed to reproduce thestandard model of electroweak interactions.

It was more recently realized that one could replace thehigher-dimensional spaces of these Kaluza-Klein theories bydiscrete structures@4#, on which a generalized notion of dif-ferential geometry can be set up using noncommutative ge-ometry@5–9#. Already, the simplest case of space-time withan internal structure consisting of two points is sufficient to

exhibit the desired behavior@10,11#. Gauge theory formu-lated on this extended geometry yields a connection form,whose internal component between the two points can beinterpreted as a scalar field on space-time. We may thendefine an extended curvature form, and construct gauge-invariant actions from it. The usual choice leads to Yang-Mills theory and a Higgs field transforming in the adjointrepresentation, with the familiar symmetry-breaking poten-tial appearing naturally. In contrast with the Kaluza-Kleinmodels, an arbitrary truncation of continuous space degreesof freedom is now unnecessary.

More complicated Higgs sectors have been derived bygeneralizing the two-point space to other discrete structures@12#. There has also been much effort devoted to construct-ing a realistic model@13–15#, that would reproduce the stan-dard electroweak theory. It was hoped such a formulation ofthe Yang-Mills-Higgs system was constrained enough to pre-dict a classical value for the Higgs boson mass@5#, thusraising the possibility of experimental verification. However,it is now generally accepted that there is enough freedom inthe theory to make ‘‘different’’ numerical predictions. Asemphasized in Ref.@14#, what emerges is precisely the stan-dard electroweak model, with not one free parameter less. Itseems noncommutative differential geometry just furnishes anew and somewhat arcane way to rewrite models of particlephysics.

Yang-Mills-Higgs theories are also important in that theyadmit a variety of soliton solutions whose existence and sta-bility are due to topological factors. They are nondissipative,finite-energy field configurations possessing boundary condi-tions at infinity which are topologically different from thoseof a vacuum. The prototype is the ’t Hooft–Polyakov mono-pole in three spatial dimensions@16,17#. Its two-dimensionalanalogues are vortices, discovered by Nielsen and Olesen@18#. In one dimension where the Yang-Mills sector becomes

PHYSICAL REVIEW D 15 AUGUST 1997VOLUME 56, NUMBER 4

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trivial, an example is the kink solution ofw4 theory.1

Each of these solutions has an energy which is boundedfrom below by a nonzero quantity depending only on thetopology or boundary conditions of the system, hence itsstability against decay to the vacuum. This so-calledBogomol’nyi bound is saturated by field configurations sat-isfying certain first-order equations, which imply the fullequations of motion but are in general much easier to solveand analyze@20#. Monopoles in the Bogomol’nyi-Prasad-Sommerfield~BPS! limit @21# and vortices with critical cou-pling are examples when the energy is minimized.

The most well-known example of this in fact comes frompure Yang-Mills theory in a four-dimensional Euclideanspace. Denote byF the gauge curvature two-form and *F itsHodge dual. The non-negativity of the inner product(F7* F,F7* F) implies the inequality

~F,F !>u~F,* F !u, ~1.1!

the left-hand side being the action functional or energy of thesystem. Equality occurs if and only if the curvature is self-dual or anti-self-dual:

* F56F. ~1.2!

Solutions to this set of first-order equations are known asYang-Mills instantons@22#, and their energy is proportionalto the Pontryagin index of the geometry.

Unfortunately, this elegant geometrical interpretation hasfound limited application to Yang-Mills-Higgs systems. Inthe case of BPS monopoles, one can regard the Higgs fieldw as the last component of afour-dimensional gauge con-nection (A,w). The self-duality condition~1.2! is thenequivalent to the Bogomol’nyi equations governing thesemonopoles@23#. Such an argument has not been extended toother examples such as vortices, as this construction is pecu-liar to four-dimensional systems.

The aim of the present paper is to generalize this type ofself-duality property to Yang-Mills-Higgs systems in arbi-trary dimensions. To do so, we exploit the fact that scalarfields with a w4 potential can be reformulated as a two-dimensional Yang-Mills theory on a discreteZ2 geometry. AYang-Mills-Higgs theory in n Euclidean dimensions is,therefore, equivalent to a pure (n12)-dimensional Yang-Mills theory on the Euclidean space with aZ2 internal struc-ture. We define a generalized notion of Poincare´ duality onthis noncommutative geometry, as well as what is meant byself-duality of the gauge curvature form in arbitrary spatialdimensions. It is then shown that critically coupled vorticesand kinks are self-dual solutions in the extended sense. BPSmonopoles and instantons also emerge as self-dual examplesof this unified formalism.

We should point out there is a rather loose sense of ‘‘self-duality’’ in use in the literature~see, for example, Ref.@24#!.It refers to theories which have special interactions and cou-pling strengths, such that the second-order equations of mo-tion reduce to first-order ones. Solutions in general minimize

a Bogomol’nyi-type bound for the energy. It is in this sensethat critically coupled vortices have hitherto been referred toas self-dual. Our notion of self-duality is a more exacting onewhich requires the concept of a Hodge star operation, as inEq. ~1.2!, on the appropriate geometry.

We begin by reviewing how differential geometry on theZ2 geometry can be set up, as per Coquereauxet al. @11#. Indoing so, we shall recast the formalism, as far as possible, ina language that makes it~formally! similar to ordinary dif-ferential geometry. This is in line with the general philoso-phy of Madoreet al. @4,25#, and would lead to two newresults. The first is we are able to define a Hodge star opera-tor for this case, in direct analogy with the usual one. Thesecond is we can now identify the Maurer-Cartan one-formu of this geometry. Unlike the ordinary case, it has a nonva-nishing curvature which can ultimately be identified as theHiggs boson mass@26#. Using this interpretation, we explainhow the gauge principle underlying Higgs fields correspondsto the independence of physics on the choice of vacuumstate.

In the second part of the paper, we consider some explicitexamples to which this formalism can be applied. In particu-lar, kinks inn51 spatial dimension, vortices inn52, mono-poles inn53, and instantons inn54 are studied. We showthat the appropriate self-duality condition on the generalizedcurvature formV takes the general form

* V.6V`un22, ~1.3!

and that its solutions minimize the energy functional. Thecorresponding topological bound is calculated for each case,and verified to be in agreement with standard results. Wefinally prove, within our unified formalism, that static solitonsolutions to the Yang-Mills-Higgs system can only existwhen n<3 @27,28#. This is a generalization of the well-known result for pure Yang-Mills theory@29#, which statesthat n54 is the only dimension in which solitons are al-lowed.

II. DIFFERENTIAL GEOMETRY ON Z 2

A. Differential forms

Consider the cyclic group of order two,Z25$e,r ur 25e%, which has the explicit matrix representa-tion @30#

p~e!5S 1 0

0 1D , p~r !5S 1 0

0 21D . ~2.1!

The algebra of complex functions onZ2 can, therefore, berealized as the algebraM2

1 of diagonal232 matrices, withthe usual rules of addition and multiplication. It is a subal-gebra of the algebraM2 of complex 232 matrices gener-ated by the Pauli matricest i .

The exterior derivative of an elementaPM21 is defined

to be the commutator

da5 im@h,a#, ~2.2!

wherem is a mass scale andh a real linear combination oft1 andt2. It is actually sufficient to set

1Strictly speaking, the kink is not a soliton but a ‘‘solitary wave’’~see, for example, Ref.@19#!. This difference need not concern ushere.

2292 56EDWARD TEO AND CHRISTOPHER TING

h5cosgt11sin gt2 , ~2.3!

for some angle parameterg, since a global factor can beabsorbed intom @11#. Note that the usual Leibniz rule issatisfied by this definition.

A one-form onZ2 has the general formadb, wherea,bPM2

1 , and is an off-diagonal 232 matrix in the presentrepresentation. A basis for the space of one-forms is suppliedby

u151

2dt3 , ~2.4!

although another possible choice is

u25 i t3u1. ~2.5!

Together, they span the space over the complex numbersC.

The exterior derivative of a one-forma on Z2 is given bythe anticommutator

da52m$h,a%. ~2.6!

The nilpotency conditiond250 follows from the identityh251. Further requiringdt3`dt35d(t3dt3) to hold im-plies that the wedge product between one-formsa andb istheir matrix product, with an extra factor ofi :

a`b5 iab. ~2.7!

The graded Leibniz rule, given byd(aa)5da`a1adaandd(aa)5daa2a`da applies.

The space of two-forms onZ2 consists of elementsadb`dc, wherea,b,cPM2

1 . Any two-form V can be writtenas

V5V12u1`u2, ~2.8!

with V12PM21 . Note thatu1`u252u2`u15m2t3. This

means the space of two-forms is isomorphic to the algebraM2

1 itself. The universal algebra of forms onZ2 is, there-fore,M2. It has a graded structure, whereby forms of evendegree are diagonal matrices and forms of odd degree areoff-diagonal. The exterior derivative operator takes evenforms into odd ones, and vice versa. We shall extend thedefinition of the wedge product~2.7! so that it denotes matrixmultiplication between an even and an odd form, or betweentwo even forms.

The above definitions of the exterior derivative operatorhave been chosen so it satisfies

~da!†5da†, ~2.9!

for any aPM2, where the involution † denotes Hermitianconjugation. As a consequence,u1 andu2 are Hermitian. Wealso have

~a`b!†5~21!degadegbb†`a†, ~2.10!

wherea,bPM2. These Hermiticity properties are identicalto those obeyed by ordinary differential forms.

The one-formu52u2 will turn out to play an importantrole in the construction of a gauge theory below. The exteriorderivative ofaPM2 can be rewritten in terms ofu as @4#

da52 i @u,a#, ~2.11!

where @a,b#5a`b2(21)degadegbb`a henceforth denotesthe gradedcommutator. It respects the graded Leibniz rule

d~a`b!5da`b1~21!degaa`db. ~2.12!

Observe that Eq.~2.11! implies the identity

du1 iu`u5m2. ~2.13!

u is the analogue of the Maurer-Cartan form in ordinarydifferential geometry~see, for example, Ref.@31#!. We shallreturn to this fact in Sec. III C.

B. Metric structure

There is a natural metric structure onZ2 given by @4,25#

gab51

2m4Tr ~ua†ub!, ~2.14!

where a,b51,2. gab has been normalized so that it is thetwo-dimensional Euclidean metric, with physical dimensionsof inverse-mass squared.

A Hodge * operation can be defined on the universalalgebra of forms onZ2 by a straightforward application ofthe usual formula@31#

* ~ua1`•••`uap!

51

~n2p!!Agea1•••ap

ap11•••anuap11`•••`uan.

~2.15!

Here, n52 is the dimension of the geometry in question,ea1•••an

the n-dimensional anti-symmetric tensor, and

g51/m4 the determinant of the metric. Indices are raised andlowered with gab . Note that * maps p-forms into(n2p)-forms, hereby extending the notion of Poincare´ du-ality to this case. The even-odd grading of the universal al-gebra is preserved under this duality transformation. Explic-itly, we have *15Agu1`u2, * u15u2, and their inverserelations which follow from the property *25(21)p(n2p).

An invariant volume element onZ2 is provided byAgu1`u2. We define the integral of a two-formV to be itsmatrix supertrace@11#:

E V5STr V. ~2.16!

The appearance of the supertrace should not be surprising aswe are dealing with graded matrices. It is equivalent to theformula

E aAgu1`u25Tr a. ~2.17!

An inner product onM2, which immediately follows, is

56 2293MONOPOLES, VORTICES, AND KINKS IN THE . . .

~a,b!5E a†`* b. ~2.18!

It is identical to the one usually adopted for complex matri-ces.

C. Gauge theory

Let U5$gPM21ug†g51% be the group of unitary ele-

ments ofM21 . We would like to construct a gauge theory on

Z2, with U as the group of symmetry transformations. Mul-tiplying a function onZ2 by gPU corresponds to performingtwo global U(1) transformations, one for each element ofZ2.

This gauge symmetry is a local one, since, in general,dgÞ0. As in ordinary gauge theory, we have to introduce acovariant derivativeD5d1 i @v,•#, wherev is a Hermitianone-form known as the gauge connection. We require it togauge transform covariantly under the adjoint action ofU,namely,D→g21Dg, so that

v →g21vg2g21idg. ~2.19!

The curvature two-form is then defined to be

V5dv1 iv`v. ~2.20!

It is Hermitian and transforms covariantly under Eq.~2.19!.We shall write the gauge connection as@4#

v5u1f, ~2.21!

with u as the preferred origin. The gauge transformation ofuis defined to be

u →g21ug2g21idg. ~2.22!

But since the right-hand side equals the left-hand one by Eq.~2.11!, u is in fact agauge-invariantquantity.f, therefore,transforms covariantly as

f →g21fg. ~2.23!

In this case, the covariant derivative of an elementaPM2takes the form

Da5da1 i @v,a#5 i @f,a#. ~2.24!

Using the identity~2.13!, we see the curvature form~2.20! isexplicitly

V5m22f2. ~2.25!

Thus, the term on the right-hand side of Eq.~2.13! is nonzerobecauseu has curvature. Because it is gauge invariant, wecannot makeu vanish by a choice of gauge. Note that theBianchi identityDV50 is trivially satisfied.

The usual starting point for the study of Yang-Millstheory is the action, normally taken to be the norm square ofthe curvature form. In the present case, such a term is

~V,V!5Tr ~V12† V12!. ~2.26!

It is clearly invariant under the gauge transformation~2.19!.There is, however, another possible choice, absent in theYang-Mills case, given by2

u~u`u,V!u5Tr ~e12V12t3!. ~2.27!

That such a term should not be ignored in noncommutativegeometry was pointed out by Sitarz@32,33#.

III. YANG-MILLS-HIGGS THEORY

A. Differential geometry on M 3Z2

The geometry of interest in this paper is ann-dimensionalEuclidean spaceM with a Z2 internal structure. The algebraof functions on this extended geometry is the tensor productof the algebraM2

1 , introduced in the preceding section,with the algebraC of complex functions onM . An elementof M2

1^C has the explicit form

S f 1 0

0 f 2D , ~3.1!

where f 1 and f 2 are functions onM .Let a be a form onZ2, andA on M . We denote bya^ A

a generalized form on M3Z2, with total degreedega1degA. The space of generalized one-forms can bewritten as the direct sum@4#

L15LH1

% LV1 . ~3.2!

The so-called horizontal partLH1 5M2

1^ L1(C) consists of

diagonal 232 matrices with each component a one-form onM , while off-diagonal matrices with scalar entries make upthe vertical partLV

1 5L1(M21) ^C. Let $ua, a51, . . . ,n%

and $ua, a5n11,n12% be generators ofLH1 and LV

1 , re-spectively. We shall take the former to be the usual basis ofone-formsdxa on M , and latter to be given by Eqs.~2.4! and~2.5!. The complete set of generators ofL1 will be denotedby u i5$ua,ua%.

The wedge product between two forms on this extendedgeometry is defined to be@11#

~a^ A!`~a8^ A8!5~21!dega8degA~a`a8! ^ ~A`A8!.~3.3!

The wedge product betweenA andA8 is the ordinary wedgeproduct between horizontal forms, while the wedge producton Z2 is understood betweena and a8. Forms of higherdegree can then be systematically constructed fromu i . Forexample, a two-formV has the general expansion

2In the Yang-Mills case, the analogue of this term is the contrac-tion of the metric tensorgab with the Yang-Mills field strengthFab , which, of course, vanishes identically.

2294 56EDWARD TEO AND CHRISTOPHER TING

V5 12 V i j u

i`u j5 12 Vabua`ub1 1

2 Vaaua`ua1 12 Vaaua

`ua1 12 Vabu

a`ub, ~3.4!

where V i j PM21

^C. Vab is the horizontal component ofV, Vab the vertical component, andVaa andVaa the mixedcomponents. Note that Eq.~3.3! implies the relationua

`ua52ua`ua, and soVaa52Vaa .When written as a matrix, a generalizedn-form has the

form

S A1B C

C8 A81B8D , ~3.5!

where A, A8 are horizontal n-forms, B, B8 are(n22)-forms, while C, C8 are (n21)-forms. Thus, thecomponents of Eq.~3.5! need not have a homogeneous de-gree, although it can be written as a sum of matrices whichdo. The wedge product~3.3! in terms of such matrices is

S A C

D BD `S A8 C8

D8 B8D 5S A`A81~21!degCiC`D8 C`B81~21!degAA`C8

D`A81~21!degBB`D8 B`B81~21!degDiD `C8D . ~3.6!

The exterior derivative operator onM3Z2 can be decom-posed into a direct sum of its horizontal and vertical parts:

d5dH% dV , ~3.7!

wheredH5]adxa is the ordinary exterior derivative operatoron M , and dV is that corresponding toZ2, given by Eq.~2.11!. Demanding thatd250 requiresdH and dV to anti-commute. The exterior derivative of a generalized form isgiven by @11#

d~a^ A!5dVa^ A1~21!degaa^ dHA, ~3.8!

or

dS A C

D BD 5S dHA 2dHC

2dHD dHB D2mS eigC1e2 igD ie2 ig~A2B!

2 ieig~A2B! eigC1e2 igD D ,

~3.9!

in terms of matrices.A duality operation on the universal algebra of forms on

M3Z2 can also be constructed in a straightforward manner.The dual ofa^ A is

* ~a^ A!5~21!degadegA* a^ * A, ~3.10!

where * acting onA denotes the usual Hodge star operationon M , while that ona is defined by Eq.~2.15!. Explicitly, itreads

* S A C

D BD 5S * A ~21!degCi * C

2~21!degDi * D 2* B D .

~3.11!

On the other hand, the involution onM and that onZ2 ex-tend to this case by the formula

~a^ A!†5~21!degadegAa†^ A†, ~3.12!

or, equivalently,

S A C

D BD †

5S A† ~21!degDD†

~21!degCC† B† D . ~3.13!

An invariant volume element and inner product on this ex-tended geometry follow in the usual way.

The calculus developed in this subsection is similar to thatof Ref. @11#. There are some minor differences in the aboveformulas, because we start off with a slightly different defi-nition of the exterior derivative operator in Eq.~2.6!.

B. Gauge-invariant action

We shall construct a generalized Yang-Mills gauge theoryon M with unitary Lie groupG, tensored with aZ2 internalstructure. Let us write the combined connection one-form as@4#

v5A1u1f, ~3.14!

where the horizontal componentA can be identified as theusual~Hermitian! Yang-Mills gauge connection. The verticalcomponentu1f is the gauge connection corresponding toU, which has already been discussed in Sec. II C. Under ageneralized gauge transformation, we have

v →g21vg2g21idg, ~3.15!

where gPG3U. This can be decomposed into transforma-tions of the individual components ofv:

A→g21Ag2g21idHg,

u→g21ug2g21idVg,

f→g21fg. ~3.16!

Moreover, u is invariant under this gauge transformation.The curvature two-form is

V5dv1 iv`v5F1DHf1m22f2, ~3.17!

whereF5dHA1 iA`A is the usual Yang-Mills curvature,and DHf5dHf1 i @A,f# the Yang-Mills gauge-covariantderivative off.

Now, we shall write

56 2295MONOPOLES, VORTICES, AND KINKS IN THE . . .

A5S A 0

0 BD , f5S 0 w

w† 0 D , ~3.18!

whereA andB are one-forms onM , andw a scalar field onM . These component fields take values in the Lie algebra ofG. If we set

g5S g1 0

0 g2D , ~3.19!

for group elementsg1 ,g2PG, the gauge transformations~3.16! become

A→g121Ag12g1

21idHg1 ,

B→g221Bg22g2

21idHg2 ,

w→g121wg2 . ~3.20!

A is, therefore, the gauge connection associated with the leftaction ofG on w, while B is that associated with the rightaction ofG on w. The curvature form is@11#

V5S F1m22ww† 2DHw

2DHw† G1m22w†wD , ~3.21!

where F5dHA1 iA`A, G5dHB1 iB`B, andDHw5dHw1 i (Aw2wB). It can be checked to satisfy thegeneralized Bianchi identity

DV50, ~3.22!

where the covariant derivative is, as usual, given byD5d1 i @v,•#.

In the notation of tensors, then(n21)/21n11 nonvan-ishing components ofV can be taken to be

Vab5S Fab 0

0 GabD ,

Va~n11!5i

mS eigDaw 0

0 2e2 igDaw†D ,

V~n11!~n12!51

m2S m22ww† 0

0 2m21w†wD . ~3.23!

In particular,Va(n12)50. The norm square ofV is

12 E dnx Tr ~V i j

† V i j !5E dnx$ 12 FabFab1 1

2 GabGab

12Daw†Daw12~m22w†w!2%,

~3.24!

where the integral is overM and trace over the Lie algebraof G is implied. This functional is extremized when the con-dition

D* V50 ~3.25!

is satisfied. It translates into the component equations

DbFab5 i ~Daww†2wDaw†!,

DbGab5 i ~Daw†w2w†Daw!,

DaDaw522~m22ww†!w. ~3.26!

Supposeg15g2 andAa5Ba . Following the usual case,we may take the action to beI 5(1/4e2)(V,V), where thedimensionless parametere plays the role of a gauge couplingconstant. Under the rescalingv→ev, it becomes

14 ~V,V!5E dnx$ 1

4 FabFab1 12 Daw†Daw1V~w!%,

~3.27!

where now Fab5]aAb2]bAa1 ie@Aa ,Ab#,Daw5]aw1 ie@Aa ,w#, while V (n11)(n12) has acquired anextra factor ofe. The potential is

V~w!52m2w†w11

2l~w†w!2. ~3.28!

Here, m5em and l5e2 are the two parameters of thetheory. This familiar action describes Yang-Mills theorycoupled to a scalar field possessing a quartic self-interaction,with the latter transforming under the adjoint representationof G. The usual field equations arise naturally from Eq.~3.25!.

Observe that the potential acquires a minimum at a non-vanishing valuew0 of w, which satisfies

w†w5m2

l. ~3.29!

This traces out a sphere of minima in the space of complexw, which corresponds to an infinity of degenerate vacuumstates. Any particular choice of a vacuum statew0 on thissphere, however, breaks the gauge symmetry.

As anticipated earlier, there is another term which can beadded to the action, proportional to

u~u`u,V!u5em2E dnx~m22w†w!. ~3.30!

Let us denote this constant of proportionality by14 r. We

obtain an action again of the form~3.27!, but with the pa-

rameterm25(e1 14 r)em2 no longer constrained to being

positive. When it is negative, the scalar field acquires a massand its potential has a unique minimum. There is no sponta-neous symmetry breaking.

We close this subsection by pointing out when the gaugegroup G is Abelian, we have to takeBa52Aa , and soGab52Fab . The action remains unchanged, although thecovariant derivative is now given byDaw5]aw12ieAaw.These facts will be needed when we consider the AbelianHiggs model below.

C. Gauge principle underlying Higgs fields

If the groupG were trivial in the foregoing analysis, theaction

2296 56EDWARD TEO AND CHRISTOPHER TING

I 5 14 $~V,V!1ru~u`u,V!u% ~3.31!

reduces to

E dnx$ 12 ]aw* ]aw2m2w* w1 1

2 l~w* w!2%. ~3.32!

It describes pure scalar field theory with a quartic self-interaction. The gauge symmetry in question is given by thegroup U, which translates into the global U~1! symmetryw→eiLw. When r is chosen such thatm2 is positive, wemay absorbr into e to leave an action purely of the form14 (V,V). Thus, scalar field theory with a symmetry-breakingpotential is a gauge theory onM3Z2 with the usual qua-dratic action. In this subsection, we will show that the under-lying gauge principle corresponds to the requirement thatphysics does not depend on the choice of vacuum out of adegenerate infinity of possible states.

Observe that the curvature form~2.25! is minimized when

f52u. ~3.33!

u thus parametrizes the nontrivial vacuum of the theory,through the angle parameterg in Eq. ~2.3!. As pointed out inSec. II A, it resembles a Maurer-Cartan form. Recall that inordinary Yang-Mills theory of a Lie groupG, the Maurer-Cartan form can be written asu52h21idHh, for an elementhPG @31,25#. It satisfies the equationdHu1 iu`u50, andso has a zero curvature. In the present case, however,u has anonvanishing curvature~2.13!, which is ultimately respon-sible for the appearance of a circular vacuum in the potential,of radiusm in the complex-w plane. Fixing a value ofg istantamount to choosing a particular vacuum state on thiscircle, which breaks the U~1! symmetry, sinceu is not in-variant under the transformationu→g21ug.

It, therefore, follows from Eq.~2.21! that v describes thephysical excitation off about a chosen vacuum state2u.Like u, it does not respect the U~1! symmetry. However,local variations ofv, given by

dv52 iD V«5@f,«#, ~3.34!

for some infinitesimal«PU, coincide with those off, as tobe expected.

Having identified the physical meanings of the fieldsuand v, let us summarize the general situation. We first de-mand a field theory off which has the symmetry

f →g21fg. ~3.35!

Supposef has a nontrivial vacuum structure. We then en-counter the problem that any choice of a vacuum state2udoes not respect this symmetry. However, physics has to beindependent of the choice of vacuum. To ensure this, weextendthe transformation~3.35! to one of the form

v →g21vg2g21idVg, ~3.36!

wheredVg52 i @u,g#, and demand that physical quantitiesbe invariant under this transformation. In particular,u re-

spects the new symmetry. We may regard a fieldv trans-forming under Eq.~3.36! as a fluctuation off about itsvacuum state:

f52u1v. ~3.37!

A field theory of f with symmetry ~3.35! has thus beenturned into one ofv with a symmetry given by Eq.~3.36!.

Written in this form, the symmetry~3.36! looks uncannilylike the generic transformation law of a Yang-Mills gaugeconnection. Indeed, as we have seen, such an interpretationcan be realized on some noncommutative geometry, wheredV is the exterior derivative operator on this geometry, andv a connection one-form. Fromv, we may construct thecurvature two-formV, which transforms asV→g21Vg.Gauge-invariant candidates for the action of the theory, suchas (V,V) and u(u`u,V)u, then follow immediately.

As we shall only be interested in Yang-Mills-Higgstheory for the rest of this paper, our starting action would begiven by Eq.~3.31!, with the conditionr.24e implicitlyassumed.

IV. TOPOLOGICAL SOLITONS

A. Kinks

Let us first consider the casen51. Because Yang-Millstheory is trivial in one spatial dimension, we are left with apure scalar field theory

I 5E dxH 1

2

dw*

dx

dw

dx1

1

2e2~m22w* w!2J , ~4.1!

from an action of the form14 (V,V). We may neglect the

linear term in Eq.~3.31!, since its presence simply corre-sponds to a rescaling ofe. The only two nonvanishing com-ponents of the curvature form are3

V12521

m

d

dx~Re w1 i Im w t3!,

V2351

m2e~m22w* w!t3 , ~4.2!

where we have fixedg5p/2 for definiteness.Since this is a three-dimensional system, the usual notion

of a self-dual or anti-self-dual curvature form does not hold.But as we shall see, it has an appropriate generalization.Consider the equation between two-forms:

* V`u56i

A4 gV, ~4.3!

where the fourth root ofg is present to ensure that both sidesof the equation have the same physical dimensions. Solutions

3As explained in Sec. III A, we setu15dx. The formsu1 andu2 introduced in Sec. II are, in this case, denoted byu2 and u3,respectively. This shift in the discrete geometry indices generalizesappropriately when we consider cases with highern below.

56 2297MONOPOLES, VORTICES, AND KINKS IN THE . . .

to Eq. ~4.3! would automatically satisfy the field equation~3.25!, provided Du50. This condition holds whenw isreal.4

Using the relations *(u1`u2)5u3 and *(u2`u3)5(1/Ag)u1, and recalling thatu52 i t3u2, we have

* V`u52 iV12t3u2`u32i

AgV23t3u1`u2. ~4.4!

Equation~4.3! in component form then reads

V12571

A4 gV23t3 , ~4.5!

which is equivalent to the first-order equation

dw

dx56e~m22w2!. ~4.6!

If we obtain the equation of motion by varyingI with respectto w, we would find that it is integrable and in fact identicalto Eq. ~4.6!. It admits the kink~antikink! solution @19#

w~x!56m tanh~emx!, ~4.7!

which approaches6m asx→6`, and so has finite energy.Now, it can be checked that

Ag~* V`u,* V`u!5~V,V!, ~4.8!

by explicit expansion in terms of the curvature components.This identity, together with the non-negativity of the innerproduct (V7A4 g* V`u,V7A4 g* V`u), implies the follow-ing lower bound for the energy functional:

~V,V!>A4 gu~V,* V`u!u. ~4.9!

It is saturated for the kink solution~4.7!, and its energy isgiven by

14 A4 gu~V,* V`u!u5 2

3 em2uw~`!2w~2`!u5 43 em3.

~4.10!

As this quantity depends only on the asymptotic values ofw, the kink is stable against decay to the vacuum.

We, therefore, conclude that Eq.~4.3! provides a suitablenotion of self-duality ofV. Like the case of Yang-Mills in-stantons described in the introduction, the energy is topologi-cally bounded from below. This bound is reached when andonly when the curvature form is self-dual.

B. Vortices

The Yang-Mills-Higgs system inn52 spatial dimen-sions, known as the Abelian Higgs model, describes a scalarfield interacting with a U~1! gauge field. Nielsen and Olesenhave found finite-energy solutions to this model@18#, whichthey interpret to be vortices or magnetic flux tubes.

The action of the Abelian Higgs model is@28#

I 5E d2x$ 14 FabFab1 1

2 Daw* Daw1 12 l~m22w* w!2%,

~4.11!

for some positive coupling constantl. The vortex solutionsare recovered by solving the field equations, together withthe asymptotic conditions

Daw50, m22w* w50 ~4.12!

for finite energy. They have a nontrivial magnetic fieldB5F12, whose total flux

F5E d2xB ~4.13!

is quantized in units ofp/e, with chargek.In the special case whenl5e2, Bogomol’nyi has shown

that the action functional of the system satisfies@20#

I>em2uFu5pm2uku, ~4.14!

with equality if and only if

D1w56 iD 2w, B56e~m22w* w!. ~4.15!

It has been established that these first-order equations admitsolutions for each value of integerk @28#. They describekvortices~or antivortices! in equilibrium, balanced by the neteffect of the repulsive magnetic field and the attractive Higgsfield.

We would like to recast this model in the language ofnoncommutative geometry. As we have seen,I is recoveredfrom the action~3.31!, where the four nonvanishing gaugecurvature components on this four-dimensionalM3Z2 ge-ometry are

V125Bt3 ,

V13521

mD1~Re w1 i Im wt3!,

V23521

mD2~Re w1 i Im wt3!,

V3451

m2e~m22w* w!t3 . ~4.16!

Whenr50, we obtain the critical casel5e2. That the ac-tion is bounded from below is then a consequence of theinequality

~V,V!>u~V,* V!u. ~4.17!

It is minimized when and only when the curvature form isself-dual~anti-self-dual!:

* V56V. ~4.18!

Such field configurations obey the field equations by virtueof the Bianchi identity~3.22!. In components, the self-dualitycondition becomes

4Should we preferw to be imaginary, the appropriate choice ofg is 0. This freedom lies in the choice of vacuum.

2298 56EDWARD TEO AND CHRISTOPHER TING

V1356 iV23t3 , V12561

AgV34, ~4.19!

which is easily seen to be precisely Eqs.~4.15!. Also, ob-serve that

~V,* V!5E d2x$4eB~m22w* w!12i ~D1w* D2w

2D2w* D1w!%

5E d2x$4em2B12i ]1~w* D2w!

22i ]2~w* D1w!%, ~4.20!

where we have made use of the identityi (D1w* D2w2 D2w* D1w) 5 2e Bw* w 1 i ]1 (w* D2w) 2 i ]2 (w* D1w ) .

The latter two terms become line integrals at infinity, whichvanish by the boundary conditions~4.12!. Thus14u(V,* V)u5em2uFu, and so the inequality~4.17! is equiva-lent to that in Eq.~4.14!.

The self-duality condition~4.18! is identical to that forYang-Mills instantons~1.2!, which is not surprising as bothare four-dimensional systems. Indeed, this formal similaritybetween the two systems was at the heart of how these re-sults were originally discovered.

C. Monopoles

We move on ton53 spatial dimensions, and again con-sider a Yang-Mills-Higgs action of the form~3.31!. Theseven nonvanishing components ofV can be written as

Vab5Fab ,

Va4521

mS Daw 0

0 Daw†D ,

V4551

m2AlS m22ww† 0

0 2m21w†wD ~4.21!

for some positive coupling constantl. The action then re-duces to1

4(V,V), and is explicitly

I 5E d3x$ 14 FabFab1 1

2 Daw†Daw1 12 l~m22w†w!2%.

~4.22!

Following the three-dimensional kink case, we take thecondition of self-duality~anti-self-duality! of V in this five-dimensional system to be

* V56A4 gV`u, ~4.23!

both sides of this equation being three-forms. The field equa-tion ~3.25! follows if Du50, namely, if w is Hermitian.Now, Eq. ~4.23! is equivalent to

A4 gVab57eabgVg4 , V4550. ~4.24!

The former equation implies that

Fab56eabgDgw, ~4.25!

while the latter equation ensures the vanishing of the cou-pling constantl.

Now, we have the identity

Ag„* ~V`u!,* ~V`u!…5~V,V!. ~4.26!

As before, that the norm square ofV7A4 g*( V`u) is non-negative then implies

~V,V!>A4 gu„V,* ~V`u!…u, ~4.27!

with equality if and only if Eq.~4.23! is satisfied. The lowerbound for the action is explicitly

14 A4 gu„V,* ~V`u!…u5E d3x$F12D3w2F13D2w1F23D1w%.

~4.28!

Using the Bianchi identityeabgDaFbg50, and partial inte-gration, the right-hand side can be turned into a surface in-tegral on the two-sphere at infinity. It is thus a topologicalquantity, and is in fact quantized in units of magnetic charge.

It is well known that the action~4.22! admits a monopolesolution, discovered in approximate form by ’t Hooft@16#and Polyakov@17# for the case ofG5SU(2). In the BPSlimit l→0, exact solutions are known@21,20#. These BPSmonopoles satisfy the first-order equation~4.25!. They haveenergy given by Eq.~4.28!, and are topologically stable.

The latter equation in Eqs.~4.24! actually imposes thatone of the two discrete dimensions is trivial. In this sense,BPS monopoles are solutions of a generalized four-dimensional gauge theory, while ’t Hooft–Polyakov mono-poles belong to the full glory of the five-dimensional system.

D. Instantons

The case ofn54 is in many ways similar to the previousone, with 11 nontrivial curvature componentsVab , Va5,andV56. The natural generalization of the self-duality~anti-self-duality! condition onV is

* V57AgiV`u`ut3 , ~4.29!

where the extra factor oft3 is needed for consistency. Itimplies the field equation~3.25!, providedDu5D(ut3)50.The former ensures thatw is Hermitian, while the latter thatw is anti-Hermitian. Together, they imply that the Higgs fieldmust vanish. Indeed, Eq.~4.29! in components reads

Fab56 12 eabgdFgd , Va55V5650. ~4.30!

Thus, the Higgs sector becomes trivial, leaving a four-dimensional Yang-Mills theory whose curvature formFab

56 2299MONOPOLES, VORTICES, AND KINKS IN THE . . .

satisfies the familiar self-duality constraint. Solutions to Eq.~4.29! are but Yang-Mills instantons@22#, which were brieflydiscussed in the introduction.

There is another possible definition of self-duality in thiscase, of the form

* V.6V`V, ~4.31!

which has previously been considered by various authors@34,35# in the context of six-dimensional Yang-Mills theory.However, this over-constrained set of equations does not ad-mit any nontrivial finite-energy solutions, as discovered inRef. @36# by analyzing the asymptotics of these equations.This is a consequence of a virial theorem for Yang-Mills-Higgs solitons that we shall prove in the following subsec-tion.

E. A virial theorem

A well-known theorem of Derrick@37# states that thereare no static soliton solutions to pure scalar field theory,except in one spatial dimension where we have the kink so-lutions, amongst others. On the other hand, Deser@29# hasshown that there are no such solutions in pure Yang-Millstheory, apart from the instanton in four spatial dimensions.These nonexistence results are due to the attractive nature ofscalar fields and the repulsive one of Yang-Mills fields, re-spectively. When the two interactions are combined, wewould expect these conditions to be relaxed. It turns out thatstatic solitons in Yang-Mills-Higgs theory only exist in one,two, and three spatial dimensions, a result previously provedin Refs. @27,28#. Kinks, vortices, and monopoles are, there-fore, a complete list of the different types of soliton solutionsto this theory. We shall briefly show how this result can beobtained, by directly extending Deser’s arguments to thegeneralized Yang-Mills case on space-time with aZ2 inter-nal structure.

We shall extend the Euclidean Yang-Mills-Higgs system,given by Eq. ~3.27!, to (n11)-dimensional Minkowskispace-time, with signature (21,1, . . . ,1) and thespace-timeindex given by m,n50,1, . . . ,n. We also denote bya,b51, . . . ,n the spatial index, andi , j 50, . . . ,n12 thecombined space-time and discrete geometry index. The gen-eralized energy-momentum tensor is

Qmn5 12 $2V~m

† iVn) i114 gmnV†i j V i j %, ~4.32!

whose trace is equal to the usual energy-momentum tensor ofYang-Mills-Higgs theory.

Consider static solutions with finite energy, other than thevacuum. To ensure*dnxQ00,`, we need allV i j to fall offfaster thanr 2n/2. This, together with the fact that]0Q0a50,implies the identity

E dnx Tr Qaa5E dnx Tr ]b~xaQa

b!50. ~4.33!

Now, it can be checked that

Qaa5

1

2H 22n

2Va0

† Va01n24

4Vab

† Vab2n

2Va0

† Va0

1n22

2Vaa

† Vaa1n

4Vab

† VabJ . ~4.34!

In the static gauge where the solution is time independent,we haveVa050 @29#. A similar argument can be used toshow thatVa050. In this gauge,D0w5 ie@A0 ,w#. It followsfrom the equation of motionDaF0a5 i (D0ww†2wD0w†)5 i (D0w†w2w†D0w) that

Tr@~D0ww†2wD0w†1D0w†w2w†D0w!A0#50,~4.35!

or

Tr@~D0w!†D0w#50. ~4.36!

Thus, D0w50 and we obtain the desired result. Equation~4.33! reduces to the spatial integral

E dnx TrH n24

4Vab

† Vab1n22

2Vaa

† Vaa1n

4Vab

† VabJ50. ~4.37!

Observe that Eq.~4.37! is a weighted sum of positive-definite terms whose coefficients depend onn. In the case ofpure Yang-Mills theory,Vaa and Vab both vanish, and son54 in order for Eq.~4.37! to hold.Vab vanishes for purescalar field theory, and we requiren,2 so that the remainingtwo terms have a chance of canceling each other. This isanother way of proving Derrick’s theorem, at least for aw4

potential. In the general case, at least one of the coefficients(n24)/4 or (n22)/2 has to be negative. Nontrivial solutionscan, therefore, exist only whenn,4.

V. DISCUSSION

By adding a two-dimensionalZ2 structure to space-time,it is possible to recover scalar fields as components of agauge connection along the discrete direction. In particular,Yang-Mills-Higgs theory inn spatial dimensions emergesfrom a generalized Yang-Mills theory on this(n12)-dimensional noncommutative geometry. It is in thissense that kinks are solutions to a three-dimensional gaugetheory, vortices to a four-dimensional one, and monopoles toa five-dimensional one.

The case of vortices is formally very similar to that ofordinary four-dimensional Yang-Mills theory, and the usualself-duality condition yields critically coupled vortices. Wehave further defined the notion of self-dual gauge fields inthree and five dimensions, and showed that kinks and BPSmonopoles obey the respective self-duality equations.

It is possible to generalize the notion of self-duality to sixdimensions~and even higher!, but it imposes the conditionthat the Higgs sector is trivial. We thus recover ordinaryYang-Mills instantons. This, we have shown, is the conse-quence of a virial theorem which states that static solitonsolutions of Yang-Mills-Higgs theory exist only when thenumber of spatial dimensions isn<3.

2300 56EDWARD TEO AND CHRISTOPHER TING

This formalism is equally well suited to pure scalar fieldtheory. Consider, for example, the two-dimensionalCPN

model@19#. It consists ofN11 complex scalar fields, with aU(1) gauge symmetry, and which are subject to an orthonor-mality condition. By introducing an auxiliary gauge field, itis possible to write the action in the form~4.11!, where, ofcourse,Fab vanishes. Like the Abelian Higgs model, theCPN model enjoys many topological properties. In the caseof critical coupling l5e2, the energy functional has thesame lower bound~4.17!, which is saturated by analyticfunctions satisfying the self-duality condition~4.18!.

The reader may have discerned some connections be-tween the results of this paper and certain supersymmetrictheories, in particular, the fact that self-dual field configura-tions are precisely those which admitN52 supersymmetricextensions. Recall that inN52 supersymmetric Yang-Millstheory, the chiral multiplet has, in addition to a vector gaugefield, scalar and pseudoscalar components which can beidentified as Higgs fields. This provides an alternative super-space unification of Yang-Mills and Higgs fields. TheBogomol’nyi-type bound also appears naturally as a conse-quence of the supersymmetry algebra@38#. What then is therelationship between supersymmetry and noncommutativegeometry?

In fact, the bosonic part ofN52 supersymmetric Yang-Mills theory has been shown to arise from considering pureYang-Mills theory on anM3Z23Z2 geometry@39#. Thisresult should not be surprising as two copies ofZ2 are

needed to give the scalar and pseudoscalar fields. The non-commutative geometry we consider is embedded in thislarger geometry. Thus our results, at the level of the action,should be consistent withN52 supersymmetry. More pre-cisely, it corresponds to the special case when the pseudos-calar field is set to zero. The key difference is, of course, ourapproach builds upon the geometrical foundations of Yang-Mills theory, while that of supersymmetry revolves aroundits algebra. Our formalism is also a minimal one in the sensethat just one scalar field and no fermions are required for thetheory to be self-consistent and exhibit the desired proper-ties.

We hope to have conveyed to the reader, a sense of thepower and elegance of a conceptual unification of Yang-Mills and Higgs fields, as afforded by noncommutative dif-ferential geometry. By using a simple and well-defined com-putational procedure, it is possible to extend some veryimportant topological ideas of Yang-Mills theory to theYang-Mills-Higgs system. No doubt, we have only justscratched the surface; the potential of this formalism may befurther realized in areas such as Chern-Simons models@24#and quantum gauge theories@40#.

ACKNOWLEDGMENTS

E.T. wishes to acknowledge helpful discussions with JohnMadore and Andrzej Sitarz.

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