˙Ibrahim S?ener

Seyh Samil Mahallesi 137.Cadde No:19 D:9,P.B.06824 Eryaman,Etimesgut-Ankara,Turkey

1 Introduction

The most important characteristic of the(anti-)self dual Yang–Mills theories on 4 dimension is the instanton solutions to the Yang–Mills equations,i.e.the BPST instanton solution of the SU(2)Yang–Mills theory on 4 dimension,[1]’t Hooft instanton solution.[2?5]The extending of the(anti-)self duality notions of the 2-forms to higher dimensional manifolds is studied by some authors.[6?8]The more general solutions on arbitrary dimensions are presented in Ref.[9].In addition to these duality concepts,there are another(anti-)self duality definitions of the 2-forms using an auxiliary exterior form,i.e.[10]on 6-dimensions. Relating to the 6-manifolds,one mentions from another works,i.e.instantons on 6-sphere,[11?12]the solutions with SO(6)gauge group dimension in Refs.[13–14],anti self dual SO(4)instanton,[15]anti self dual Yang–Mills-Higgs connection[16]and SU(3)structure 6-manifold.[17?18]Therefore,the gauge theories on 6 dimensions are mostly interpreted as an anti self dual SU(3)invariant Hermitian theories on 6-sphere,i.e.F+?(F∧Φ)=0.[19]However,according to the decomposition in present text,the “anti self dual”connections are R and so(4)valued while the “self dual” one is su(3)valued.Here we will use the abbreviations ASD and SD for the anti self and dual(ity)notions,respectively.On the other hand,from the Eqs.(48)and(49),the Lagrangians of both of the ASD and SD connections are bounded by the same topological bound.Also we see that,in a case of the Yang–Mills energy quantization,the ASD connection have negative integer topological charge or the quantum number,but the SD ones are positive integer,and so the radius of a 6-sphere in both quantized case is bounded by the coupling constant.

2 Decomposition of 2-Forms on 6 Dimensions

Given a duality mapon a realD=even manifold using an auxiliaryp-form Φ∈ΛD?4(M).For a 2-formη∈Λ2(M)we write following duality concept,i.e.similar to in Refs.[11],[12],[15],which we will call Φ-duality:

If the 2-formηsatisfies following relation,

then we call Φ-ASD/SD 2-form to this form,whereλ∈{Eigenvalues.In this context,we will use this duality notion on 6-manifoldal manifolds.

LetMbe a 6-manifold with local coordinates{xμ}∈R6.The volume element on this manifold is given by

Therefore,if we consider a 6-manifold as a coset space,i.e.M=SO(6)/(SU(3)×U(1)),for a local dual basis 1-forms(dx1,...,dx6)we can choose a well defined auxiliary 2-form Φ,i.e.which is invariant under the SU(3),such that

where

Then the Φ duality notion given in Eqs.(1)and(2)for any 2-formη∈Λ2(M),

presents following eigenvalues on the 6-manifolds

and the eigenspaces corresponding to these eigenvalues are

where

Thus we can mention from the decomposition of the bundle of the 2-forms with respect to these eigenspaces on the 6-manifolds.Therefore the decomposition of the bundle Λ2(R6)into three subbundles which are orthogonal each other is

and these subbundles are spanned by

This decomposition is also given with respect to the SU(3)structure on 6-dimension in Ref.[20].

On the other hand,the decomposition of the Lie algebra so(6)with respect to the coset space SO(6)/(SU(3)×U(1))as follow

Therefore,the subbundles decomposing the Λ2(R6)live following Lie algebras with respect to the decomposition of the so(6):

3Φ-ASD/SD Equations on 6-Dimensions

Since a connection and its curvature on any vector or principal bundle are matrix/Lie algebra valued 1 and 2-form,they are expressed within a matrix. Then avalued curvature matrix is written for some 2-forms,wherei,j=1,...,N.For each component of the curvature matrix the duality equation on 6-manifold reads

The expression of the duality equation(17)is written as follow

(i)Case λ=?1/2

In this case the curvature 2-form 2-formFbecomes only an ordinary 2-form spanned by the basison the subbundle

whereb∈C∞(M).Therefore,sincethe connection giving this curvature lives only Lie algebra R,that is it presents an Abelian gauge field on 6-manifold.

The 6-sphereS6=SO(7)/SO(6)is also interpreted as a quotient space such thatTherefore,the solution on the subbundlecan be considered as a Yang–Mills instanton on the manifold R×G2/SU(3),i.e.in Ref.[21].Writing the components of the curvature in Eq.(20)asF14=F25=F36=b,the Hermitian–Yang–Mills equations onT3×T3with respect to the auxiliary form Φ=dx14+dx25+dx36in Ref.[22]reads the conditionF14+F25+F36=0.However,the connection is totaly anti self/self Φ-dual(shortly Φ-ASD/SD),then this part must vanish,and so the connection becomes a pure gauge for a smooth scalar?∈C∞(M):

Then the flat connection induced on Λ1(R)in Eq.(21)is a special case of the condition given in Ref.[22]for caseb=0.

(ii)Case λ=?1

In this case we say that the connection of this curvature is Φ-ASD.More general,the Φ-ASD 2-forms on a 6-manifold live on the subbundleThen a g-valued 2-formis rewritten as follow

Sinceand,then the 2-forms of the subbundlelive in the Lie algebra so(4).Thus the Φ-ASD 2-formFaon a 6-manifold is expressed with respect to the bases of thegiven in Eq.(10)as follow

Then an Φ-ASD so(4)-valued curvature 2-form is as follow

Therefore for this matrix one gets

Thus an so(4)-valued 2-form onis such that

(iii)

In this case we say that the connection is Φ-SD.More general,the Φ-SD 2-forms on a 6-manifold live on the subbundleSincethen the 2-forms on the subbundlelive in the Lie algebra su(3),and soThen an su(3)-valued 2-formis rewritten as follow

Thus the Φ-SD 2-formFaon a 6-manifold is expressed with respect to the bases of thegiven in Eq.(10)as follow

wheremeans without the indexThere is a nice accident here,so that,the Gell–Mann matrices of the generators of the group SU(3)have two diagonal elements:

Neglecting these generators,an su(3)-valued 2-form on the subbbundlebecomes a hermitian matrix-valued 2-form whose is zero diagonal elements,that is the generators of the su(3)except forτ3andτ8.We will show bywhose is such generators.Therefore we can write the generators of the seu(3),that of the su(3)as follow

The generatorsτ3andτ8are also given as that of the subgroup U(1)×U(1)on R×SU(3)/(U(1)×U(1))[23]and SU(3)-invariant solution to the Yang–Mills equation on the nearly K¨ahler manifold SU(3)/(U(1)×U(1)).Therefore,the subbundleare rewritten as follow

whereFor this matrix one gets also

Thus an su(3)-valued 2-form onis such that

4 Topological Invariants

LetEbe a complex vector bundle with typical fiber C3or a principalG?GL(3,C)-bundle on a 6-manifold.The curvature 2-form

of a connection on this bundle have as the following Chern classes

Chern number of the bundle is given by

and the total Chern number is then

Therefore,ifLm(F)is a Lagrangian 6-form including some gauge invariance terms of the curvature 2-form,i.e.(trF)3,trF3or trthen the total action integral which we will call the Φ-topological charge,such that

reads the field equation

As seen easily that,sinceLm(F)depends only on the curvature,this equation is reduced to the vacuum Yang–Mills,or instanton equation:

However,since we have the Φ-ASD/SD for the 2-forms on 6-manifolds,considering the total Φ-topological charge given in Eq.(39),a more general topological bound of the Yang–Mills action integral on 6-manifolds can be given such that

wherea1,a2,a3are some constants and the terms(ch1)3and ch2∧ch1are via the total Chern number.Here since the term tr[F2]relates to the second Chern class,we have

Therefore,we rewrite the topological bound as

Here one can see that the third Chern class ch3includes the terms trFand trF2,that is the first and second ones.

In the case of Φ-ASD the 2-forms are on the subbundleand are so(4)-valued.On the other hand,we have the isomorphismThen we can consider su(2)-valued curvature on the subbundleon 6-manifolds.Therefore,the first Chern classes vanishes and the third one is unde fined.On the other hand,for the Φ-SD the 2-forms are on the subbundleand they are su(3)-valued.Considering the curvature 2-formgiven in Eq.(31),this form satisfies Eq.(32):tr[F3]=0.Also,already the first characteristic class vanishes.We can easily see that both duality concepts on a 6-manifold have the same characteristic class and topological bound:

5ΦASD/SD Solutions on 6 Dimensions

Now take a connection as follow

whereis a constant anti-symmetric object,ω=is the Maurer-Cartan 1-form satisfying the Maurer-Cartan equationa constant and

andημνis the metric on the base manifold.Therefore,the curvature of the connectionAis written as

whereTo preserve the tensorial structure of the curvature it must be

that is the connection is SD,i.e.BPST instanton.[1]The solution to this equation is

Therefore,the instantonlike connection and its curvature become

On the other hand,the curvature(so(4))given locally in Eq.(22)for Φ-ASD connection is written as

In the easiest way,the coefficients can be chosen as

Other curvaturegiven locally in Eq.(27)for Φ-ASD connection is

and in the easiest way,its coefficients can be chosen as

Hence one gets following gauge invariant terms

wherem=odd andl=4 for the Φ-ASD connection defined on the bundleandm=even andl=1 for the Φ-SD connection defined on the bundle

The most known quadratic invariance of the curvature 2-form on all dimensions is the Yang–Mills invariance.Furthermore,one can construct some quadratic terms depending on the dimension of the base manifold.For the curvature 2-formF∈Λ2(R6,g)of a g-valued connection we can give a quadratic invariance term tr[F2∧F2].In this concept we can construct a Lagrangian 6-form as follow

wheregis the coupling constant.Using the curvature given in Eqs.(54),(61),(62),and(63),for the Φ-ASD/SD connection the LagrangianLmis written as

On the 6-sphere,since the volume element is dVol=π3r5drand the total volume of the six sphere with radiusR0is

the pseudo energy of this Lagrangian is

On the other hand,the pseudo energy of the Yang–Mills invariance and the Φ-topological charge,respectively,are

Removingφ,with respect to the lower bound condition on a 6-manifold given in Eqs.(48)and(49)we have following topological lower boun relation for the pseudo energy of the Yang–Mills action on the 6-sphere

At the originR0=0 of the 6-sphere,we see that

Considering the instantonlike connection(55)connection(55)and 56,the connection at this point becomes flat.

6 Quantization

and we find on 6-sphere

Therefore,for anyk=n∈Z,then it must be

On the other hand

and so the connection and its curvature then become

where

and so a quantization case is written as follow

whereδnmis the Kronecker delta,Jis a 3-form and the Planck’s constant is~=1.Therefore we rewrite the pseudo energies of the Yang–Mills invariance,Lmand the Φ-topological charge,respectively,as

where

As seen easily that if the connection is Φ-ASD,that ism=odd,then n∈Z?;otherwise,if the connection is Φ-SD,that ism=even,then n∈Z+.Therefore,in both cases the Φ-topological charge is always

Remark thatl=1 for Φ-SD andl=4 for Φ-ASD.Then the topological bound for both cases becomes

7 Conclusion and Discussion

The subbundles in Eq.(13)of the decomposition(12)give the Φ-ASD equations on theandand the Φ-SD equations onAs seen easily that the su(3)-valued connection on 6-manifold is SD in our ansatz.On the other hand,from Eqs.(48)and(49),the Lagrangians in both of the Φ-ASD and SD on a 6-manifold are bounded by the same topological bound.In addition to these,in the quantization condition in Eq.(75),for the Φ-ASD the Φ-topological charge or the quantum number is negative integer,but for the Φ-SD they are positive integer.Thus,considering Eqs.(82)and(85),the radius of a 6-sphere in the quantized case becomes such that

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