Gauge-covariant decomposition and magnetic monopole for G(2) Yang-Mills field
aa r X i v : . [ h e p - t h ] J u l CHIBA-EP-217-v2, 2016
Gauge-covariant decomposition and magnetic monopole for G(2) Yang-Mills field
Ryutaro Matsudo ∗ and Kei-Ichi Kondo † Department of Physics, Graduate School of Science, Chiba University, Chiba 263-8522, Japan
We give a gauge-covariant decomposition of the Yang-Mills field with an exceptional gauge group G (2), which extends the field decomposition invented by Cho, Duan-Ge, and Faddeev-Niemi for the SU ( N ) Yang-Mills field. As an application of the decomposition, we derive a new expression of thenon-Abelian Stokes theorem for the Wilson loop operator in an arbitrary representation of G (2).The resulting new form is used to define gauge-invariant magnetic monopoles in the G (2) Yang-Millstheory. Moreover, we obtain the quantization condition to be satisfied by the resulting magneticcharge. The method given in this paper is general enough to be applicable to any semi-simple Liegroup other than SU ( N ) and G (2). PACS numbers: 12.38.Aw, 21.65.Qr
I. INTRODUCTION
Understanding the mechanism underlying quark confinement from the first principle of QCD is still a challengingproblem in theoretical particle physics [1]. As a possible step towards this goal, it will be efficient to extract thedominant field mode V responsible for confinement from the Yang-Mills field A to clarify the physics behind thephenomena of confinement. The well-known mathematical identity called the Cartan decomposition [2] is used todecompose the field variable A valued in the Lie algebra G = Lie( G ) of a gauge group G into the simultaneouslydiagonalizable part in the Cartan subalgebra H = Lie( H ) and the remaining off-diagonal part in the orthogonalcomplement of Lie( H ). However, the Cartan decomposition is not suited for studying the non-perturbative featuresof the gauge field theory with local gauge invariance, since the Cartan decomposition cannot retain the original formafter the gauge transformation, namely, the local rotation of the Cartan-Weyl basis for the Lie algebra.In view of these, the novel decomposition called the Cho-Duan-Ge-Faddeev-Niemi (CDGFN) decomposition [3–14]is quite attractive, since the CDGFN decomposition given in the form A = V + X is gauge covariant, namely, itkeeps its form under the gauge transformation or the local color rotation. In the CDGFN decomposition, the unit Liealgebra valued field n j called the color direction field or the color field for short plays the crucial role for retaining thelocal gauge covariance of the decomposition. For G = SU ( N ), the color field n j ( x ) is constructed from the maximallycommuting generators H j ( j = 1 , ..., rank G ) in the Cartan subalgebra H = Lie( H ) according to the local adjointrotation by a group element g of the gauge group G at every point x of spacetime: n j ( x ) = g ( x ) H j g † ( x ) , g ( x ) ∈ G. (1)The color direction field belongs to the subset of Lie( G ) which is topologically equivalent to G/ ˜ H , where a subgroup ˜ H called the maximal stability subgroup of G is specified from the degeneracy among the eigenvalues of a representationmatrix of H j . In other words, the color field n j is regarded as the local embedding of the Cartan direction H j in theinternal space of the non-Abelian group G . From this viewpoint, the Cartan decomposition is identified with a globallimit of the CDGFN decomposition, which urges us to consider that the Abelian projection method [15] is nothingbut a gauge-fixed version of the gauge-covariant CDGFN decomposition. The application of the novel decompositionto the Yang-Mills non-Abelian gauge field paves the way for understanding quark confinement in a gauge-independentmanner. In fact, this method has been extensively used to investigate quark confinement in the SU ( N ) Yang-Millstheory in the last decade, see e.g., [14] for a review.A promising mechanism for understanding quark confinement is well known as the dual superconductivity [16].It is a hypothesis based on the electro-magnetic dual analog of the type II superconductor in which the magneticfield applied to the bulk of the superconductor is squeezed to form the magnetic vortex due to the Meissner effectof excluding the magnetic field from the superconductor [17]. The color electric field created by a pair of a quarkand an antiquark would be squeezed to form an electric flux-tube or a hadron string with its ends on a quark and anantiquark. For the dual superconductivity to work, therefore, one needs magnetic objects, say magnetic monopoles to ∗ Electronic address: [email protected] † Electronic address: [email protected] be condensed in the vacuum of the Yang-Mills theory, which is supposed to be dual to the ordinary superconductivitycaused by condensation of electric objects, a pair of electrons called the Cooper pairs [17].The magnetic monopole in the pure Yang-Mills theory has been mostly constructed by the Abelian projectionmethod, which [15] breaks explicitly the original non-Abelian gauge group G to the maximal torus subgroup H .However, this is not the only way to define magnetic monopoles in pure Yang-Mills theory without the Higgs field. Infact, one can define gauge-invariant magnetic monopoles in the pure Yang-Mills theory without breaking the originalgauge symmetry by using the non-Abelian Stokes theorem [18–25] for the Wilson loop operator which is in itselfgauge invariant [1]. The gauge-invariant magnetic monopole is specified by the maximal stability subgroup ˜ H whichis uniquely determined for the highest-weight state of a given representation for a quark source. For quarks in thefundamental representation of SU ( N ), especially, the maximal stability subgroup ˜ H is given by ˜ H = U ( N − U (1) N − for N ≥ SU ( N ) Yang-Mills theory ( N ≥ SU (2) Yang-Mills theory. Whichever options we use, we need the fielddecomposition formula, which allows us to decompose an arbitrary element F of a Lie algebra G to the part F ˜ H inthe Lie algebra of ˜ H and the remaining part F G/ ˜ H . See e.g., [13, 23] and also [14] for a review.Another popular object of topological nature which is believed to be responsible for confinement is the center vortex[26, 27], which is associated to the center subgroup of G . In fact, the confinement/deconfinement phase transitionat finite temperature in the SU ( N ) Yang-Mills theory is associated with the restoration/spontaneous breaking of thecenter symmetry Z ( N ), which is signaled by vanishing/nonvanishing of the Polyakov loop average. See e.g., [28] forreviews. We suppose, however, that magnetic monopoles and magnetic vortices cannot be independent topologicalobjects. They could be different views of a single physical object, just like two sides of a coin, to be simultaneouslydefined in a self-consistent way [14, 24]. However, this statement remains still a conjecture to be proved.The purpose of this paper is to extend the gauge-covariant field decomposition of the Yang-Mills field and thenon-Abelian Stokes theorem for the Wilson loop operator developed so far for SU ( N ) to the exceptional group G (2), which is a preliminary step toward reformulating the G (2) Yang-Mills theory [29] to discuss the mechanism forconfinement/deconfinement. Our interests of the exceptional group G (2) lie in a fact that the G (2) Yang-Mills theoryhas the linear potential [30–32] and that the center vortex confinement mechanism is argued to work for G (2) in [32],although G (2) has a trivial center subgroup, consisting only of the identity element [33, 34]. We want to define thegauge-invariant magnetic monopole in the G (2) Yang-Mills theory and then examine whether the magnetic monopoledefined in our framework can be a universal topological object responsible for confinement, irrespective of the gaugegroup. This investigation will help us to prove or disprove the above conjecture on the interrelation between magneticmonopoles and magnetic vortices.The present paper is organized as follows. In sec. II, we first determine the maximal stability subgroup for a givenrepresentation of G (2), after presenting some basic properties of G (2). In sec. III, we subsequently derive the gauge-covariant decomposition formula for the G (2) Yang-Mills field corresponding to each maximal stability subgroup. Weshow that the G (2) Yang-Mills field has different gauge-covariant decompositions depending on the maximal stabilitysubgroup, which are more complicated than those obtained for the SU ( N ) Yang-Mills field. In fact, it turns out thatthe decomposition formula for G (2) cannot be obtained as a simple extension of that for SU ( N ). This is becausethe fact that all roots of SU ( N ) have the same norm was used in deriving the decomposition formula for SU ( N ).However, some roots of G (2) have different norm from the other roots. Consequently, the relevant decomposition for G (2) cannot be obtained by using the double commutators, in sharp contrast to SU (3). Nevertheless, the multiplecommutators with the Cartan generators enable us to obtain the desired decomposition. Remarkably, we have foundthat the decomposition formula for G (2) can be obtained using sextuple commutators with the Cartan generators orthe color fields. Moreover, the method presented in this paper for obtaining the decomposition formula for G (2) canbe applied to any semi-simple Lie algebra, and therefore the reformulation of the Yang-Mills theory would be possiblefor an arbitrary semi-simple gauge group.In sec. IV, we derive a non-Abelian Stokes theorem for the Wilson loop operator of the G (2) Yang-Mills field,which is written using the sextuple commutators with the color direction fields, as an application of the decompositionformula. This enables us to define gauge-invariant magnetic monopoles in the G (2) Yang-Mills theory in sec. V. Wefind that which kind of magnetic monopoles can be defined is determined by the stability subgroup of G . We showthat the magnetic charge derived from the gauge-invariant magnetic monopole is subject to a novel quantizationcondition, which is similar to, but different from the quantization condition for the Dirac magnetic monopole and’tHooft-Polyakov magnetic monopole. The final section is devoted to conclusion and discussion. Some technicalderivations are collected in Appendices A, B and C. FIG. 1: The Dynkin diagram of G (2). II. AN EXCEPTIONAL GROUP G (2) A. Basic properties of G (2) In this section, we give some basic properties of an exceptional group G(2). We begin with the Dynkin diagram of G (2) given by FIG. 1. It indicates that G (2) has two simple roots (i.e., rank 2) with the opening angle 5 π/
3. In thispaper, we use α = ( 12 , − √
32 ) =: α (1) ,α = (0 , √ α (5) , (2)as simple roots. We see that the other positive roots are obtained as α + α = ( 12 , − √ α (6) ,α + 2 α = ( 12 , √ α (4) ,α + 3 α = ( 12 , √
32 ) =: α (3) , α + 3 α = (1 ,
0) =: α (2) . (3)FIG. 2 is the root diagram of G (2). Hence, there are two Cartan generators H k ( k = 1 ,
2) and twelve shift operators E α ( α ∈ R ), where R is the root system, i.e., the set of positive and negative root vectors. They satisfy the commutationrelation called the Cartan standard form: [ H j , H k ] = 0 ( j, k = 1 , , [ H k , E α ] = α k E α , [ E α , E − α ] = α · H [ E α , E β ] ∝ ( E α + β ( α + β ∈ R )0 (otherwise) (4)where α k denotes the k th component of the root vector α and α · H is the inner product defined by α · H := α k H k .In this paper we consider a unitary representation. Therefore representation matrices satisfy the Hermiticity: R ( H k ) † = R ( H k ) , R ( E α ) † = R ( E − α ) , (5)and the normalization: κ tr( R ( H j ) R ( H k )) = δ jk , κ tr( R ( E α ) R ( E β )) = ( β = − α )0 (otherwise) , (6)where the value of κ depends on the representation. Using this property, we can define the inner product in the Liealgebra as ( F , F ) = κ tr( R ( F ) R ( F )) , (7)which is independent of the representation R .The weight vector ~µ of a representation specified by the Dynkin index [ m , ..., m r ] of the Lie group with the rank r is obtained from the relation: 2 ~α k · ~µ~α k · ~α k = m k . (8) FIG. 2: The root diagram of G (2).FIG. 3: The weight diagrams of the fundamental representations. For G (2), the highest-weight vector µ j ( j = 1 ,
2) of a representation with the Dynkin index [1 ,
0] or [0 ,
1] satisfies2 α k · µ j α k · α k = δ kj , (9)and hence determined as µ = (1 , , µ = (cid:18) , √ (cid:19) . (10)The weight diagrams are determined by µ j ( j = 1 , µ correspondsto the 14-dimensional adjoint representation with the Dynkin index [1 , µ corresponds to the 7-dimensional fundamental representation with the Dynkin index [0 , G (2) is labeled by the two Dynkin indices [ n, m ] and its highest weight Λ can be written asΛ = nµ + mµ = (cid:18) n + m , m √ (cid:19) . (11)Notice that G (2) contains the Lie group SU (3) as a subgroup. We can see from the root diagram that the Liealgebra su (3) of SU (3), denoted as su (3) = Lie( SU (3)), is generated by a set of elements in su (3): { H , H , E α (1) , E α (2) , E α (3) , E − α (1) , E − α (2) , E − α (3) } ⊂ su (3) = Lie( SU (3)) . (12)Therefore, a representations of G (2) is written as direct sums of representations of SU (3). For example, the funda-mental representations of G (2) are written as = + ∗ + , (13) = + + ∗ . (14) B. Maximal stability subgroups
It is known [13, 14] that one can construct a number of the reformulations of the Yang-Mills theory which arediscriminated by the maximal stability subgroup. Therefore it is important to know which subgroup is identifiedwith the maximal stability subgroup for each representation. In view of this, we first derive a certain property tobe satisfied by the generators belonging to the Lie algebra of the maximal stability subgroup of G (2). By using thisproperty, then, we determine the maximal stability subgroup for each representation of G (2).The maximal stability subgroup ˜ H for the representation R of a group G is defined to be a subgroup whose element h ∈ ˜ H leaves the highest-weight state | Λ i of the representation R invariant up to a phase factor : h | Λ i = | Λ i e iφ ( h ) . (15)Hence, an element of its Lie algebra ˜ h = Lie( ˜ H ) can be written as a linear combination of the Cartan generators andshift-up and -down operators E α and E − α (where α is a positive root) such that E α | Λ i = 0 and E − α | Λ i = 0. Herenotice that, if there is E α in the linear combination, then there is also E − α , that is to say, the mutually Hermitian-conjugate generators E α and E − α must appear in pairs in the linear combination, since all matrices in a unitaryrepresentation of the Lie algebra are Hermitian.We show in the following that E α | Λ i = 0 and E − α | Λ i = 0 if and only if Λ · α = 0. Here, we should remember that( α · H ) /α and E ± α / | α | satisfy the commutation relations of su (2). We see from this fact that if α · H | µ i = 0 and E α | µ i = 0 then | µ i belongs to the space of the trivial representation of SU (2) and hence E − α | µ i = 0. Because | Λ i ishighest weight state, E α | Λ i = 0. Hence if Λ · α = 0 then E − α | Λ i = 0. In the same way, the converse can be proven.Thus we arrive at the conclusion that X ∈ ˜ h can be written as a linear combination of the Cartan generators H j and shift operators E ± α with positive root vectors α that are orthogonal to the highest-weight vector Λ: α · Λ = 0 . (16)Thus, it is easy to see that all representations of G (2) are classified into the following three categories.1. For the highest weight Λ = mµ , the positive root orthogonal to the highest weight is α = α (1) alone. Hence,the maximal stability subgroup is a U (2) with the generators H , H , E α (1) and E − α (1) :˜ H = U (2); Lie( U (2)) = u (2) ⊃ { H , H , E α (1) , E − α (1) } , (17)which agrees with a subset of SU (3) specified by (12).2. For the highest weight Λ = nµ , the positive root orthogonal to the highest weight is α = α (5) alone. Hencethe maximal stability subgroup is another U (2) with the generators H , H , E α (5) and E − α (5) :˜ H = U ′ (2); Lie( U ′ (2)) = u ′ (2) ⊃ { H , H , E α (5) , E − α (5) } , (18)which differs from a subset of SU (3) specified by (12).3. For the highest weight Λ = nµ + mµ ( n = 0 = m ), the maximal stability subgroup is equal to the maximaltorus subgroup U (1) × U (1) generated by the Cartan subalgebra { H , H } :˜ H = U (1) × U (1); Lie( U (1) × U (1)) ⊃ { H , H } . (19)This fact is confirmed as follows. We can write any positive root as kα + lα , where k and l are non-negativeintegers that are not zero simultaneously. Hence, the relation, Λ · α = ( nµ + mµ ) · ( kα + lα ) = nk + ml, implies that all positive roots are not orthogonal to the highest weight when n = 0 and m = 0. Thus, in thiscase, the generators of the maximal stability subgroup are given by H and H . Strictly speaking, we should write R ( h ) | Λ i = | Λ i e iφ ( h ) , but if we do so the presentation become rather cumbersome. Therefore we omit R ( · ) throughout this subsection. III. DECOMPOSITION FORMULA
Let F be an arbitrary element of the Lie algebra. To write the Wilson loop using the color direction fields, and toreformulate the G (2) Yang-Mills theory, we have to decompose F into the part F ˜ H belonging to ˜ h = Lie( ˜ H ), andthe remaining part F G/ ˜ H using its commutators with H k . This is achieved by using double commutators in the caseof SU ( N ), see [14]. But, in the case of G (2), we have to use sextuple commutators. Its proof is given in Appendix A.In this section, we give the explicit form of such a decomposition for any representation. A. Decomposing SU(3)
Before proceeding to the G (2) case, we reconsider the SU (3) case from the viewpoint of this paper. For SU (3), it isknown [14] that the maximal stability subgroup is U (2) or U (1) × U (1). In the case of the maximal stability subgroup U (2) with generators H , H , E α (2) and E − α (2) , the decomposition formula is written as F G/ ˜ H = 43 [ H , [ H , F ]] , (20)while in the case of the maximal stability subgroup U (1) × U (1), the decomposition formula is written as F G/ ˜ H = X j =1 , [ H j , [ H j , F ]] . (21)The derivation of these formulas is written in Appendix C and D in [14]. We rederive them using another methodwhich can be applied also to G (2). First, we consider the commutator of an arbitrary element of the Cartan subalgebrawith F . An arbitrary element of the Cartan subalgebra can be written as ν · H , where ν is an arbitrary 2-dimensionalvector. Using the Cartan decomposition, F = X j =1 , F j H j + X α ∈R + ( F ∗ α E α + F α E − α ) , (22)and the commutation relation (4), we can write the commutator as[ ν · H, F ] = X α ∈R + ( ν · α )( F ∗ α E α − F α E − α ) . (23)Here, we choose γ := ( √ / , /
2) as ν , which is orthogonal to α (1) , i.e., γ · α (1) = 0. Then the commutator reads[ γ · H, F ] = ( γ · α (2) )( F ∗ α (2) E α (2) − F α (2) E − α (2) ) + ( γ · α (3) )( F ∗ α (3) E α (3) − F α (3) E − α (3) ) , (24)where the terms corresponding to α (1) disappear. By taking the commutator once more, we can eliminate anotherterm. For the vector γ := (0 , α (2) , i.e., γ · α (2) = 0, the double commutator is written as[ γ · H, [ γ · H, F ]] = ( γ · α (3) )( γ · α (3) )( F ∗ α (3) E α (3) + F α (3) E − α (3) )= 34 ( F ∗ α (3) E α (3) + F α (3) E − α (3) ) . (25)Thus we obtain F ∗ α (3) E α (3) + F α (3) E − α (3) = 43 [ γ · H, [ γ · H, F ]] . (26)In this way, we can extract the element of F corresponding to a particular positive root by taking the doublecommutator. For the other positive roots, the similar identity holds: F ∗ α (1) E α (1) + F α (1) E − α (1) = −
43 [ γ · H, [ γ · H, F ]] , (27) F ∗ α (2) E α (2) + F α (2) E − α (2) = 43 [ γ · H, [ γ · H, F ]] , (28)where we have introduced γ := ( √ / , − / α (3) . Using these expressions, we can write thedecomposition formula for any case of the maximal stability subgroup. For ˜ H = U (2), the decomposition formula iswritten as F ˜ H = X j , ( F , H j ) H j + ( F ∗ α (2) E α (2) + F α (2) E − α (2) )= X j , ( F , H j ) H j + [ H , [ H , F ]] −
13 [ H , [ H , F ]] F G/ ˜ H = ( F ∗ α (1) E α (1) + F α (1) E − α (1) ) + ( F ∗ α (3) E α (3) + F α (3) E − α (3) )= 43 [ H , [ H , F ]] , (29)while for ˜ H = U (1) × U (1), the decomposition formula is written as F ˜ H = X j , ( F , H j ) H j F G/ ˜ H = ( F ∗ α (1) E α (1) + F α (1) E − α (1) ) + ( F ∗ α (2) E α (2) + F α (2) E − α (2) ) + ( F ∗ α (3) E α (3) + F α (3) E − α (3) )= X j =1 , [ H j , [ H j , F ]] , (30)where we have used the commuting property:[ H j , [ H k , F ]] = [[ H j , H k ] , F ] + [ H k , [ H j , F ]] = [ H k , [ H j , F ]] , (31)following from [ H j , H k ] = 0. B. Decomposing G(2)
Now we consider the G (2) case. We want to write the coset part F G/ ˜ H of F as a linear combination of multiplecommutators with the Cartan generators: F G/ ˜ H = X j ,...,j n ∈{ , } η j ··· j n [ H j , · · · , [ H j n , F ] · · · ] , (32)where the sum is over independent terms by taking account of the commuting property (31). We can obtain (32) forany choice of ˜ h if, for every positive root β , the relevant shift part R β is written in the form: R β := F ∗ β E β + F β E − β = X j ,...,j n ˜ η j ··· j n [ H j , · · · , [ H j n , F ] · · · ] . (33)In the following, we give a derivation of this fact (33). In the way similar to that written in the above for SU (3),indeed, we can obtain (33). By using the commutation relation (4), the commutator is calculated as[ ν · H, F ] = X α ∈R + ( ν · α )( F ∗ α E α − F α E − α ) . (34)If ν is chosen to be orthogonal to a particular α , then the corresponding terms of E α and E − α disappear from thisexpression. Thus, by taking the commutator repeatedly, we can eliminate all shift terms except one shift term R β that corresponds to a particular positive root β . Since there are six positive roots in G (2), we have to eliminate fiveshift terms. We can do so using the quintuple commutator:[ ν · H, [ ν · H, [ ν · H, [ ν · H, [ ν · H, F ]]]]] = ( ν · β )( ν · β )( ν · β )( ν · β )( ν · β )( F ∗ β E β − F β E − β ) , (35)where ν , . . . , ν are appropriate 2-dimensional vectors. In this expression, the sign of the term of E β is opposite tothat of E − β . To make both signs equal, we need to take the commutator once more. We choose an 2-dimensional The reason why we need quintuple commutator is that we can eliminate only one term by taking the commutator once. This is becausethe roots of G (2) is 2-dimensional. If the dimension of the root vectors is larger, we can eliminate more than one term by taking thecommutator once. FIG. 4: Unit vectors which is orthogonal to one of the positive roots. vector ν which is non-orthogonal to β to obtain the non-vanishing commutator of ν · H and (35):[ ν · H, (35)] = ( ν · β )( ν · β )( ν · β )( ν · β )( ν · β )( ν · β )( F ∗ β E β + F β E − β ) . (36)Thus we obtain the key relation: R β = 1 N [ ν · H, [ ν · H, [ ν · H, [ ν · H, [ ν · H, [ ν · H, F ]]]]]] ,N := ( ν · β )( ν · β )( ν · β )( ν · β )( ν · β )( ν · β ) . (37)Although this expression is nothing but the desired one (33), it should be remarked that the coefficients ˜ η j ··· j isnot uniquely determined. If we multiply ν by a constant, the coefficients ˜ η j ··· j do not change. This point will beobserved more concretely shortly.To obtain the expression (37) for each positive root concretely, we introduce six unit vectors γ a ( a = 1 , . . . ,
6) suchthat γ a is positive and orthogonal to one of the positive roots, say α ( a ) ( a = 1 , . . . , γ = ( √ ,
12 ) , γ = (0 , , γ = ( √ , −
12 ) , γ = ( 12 , − √
32 ) , γ = (1 , , γ = ( 12 , √
32 ) . (38)See FIG. 4. Consequently these vectors satisfy the following conditions: γ a ⊥ α ( a ) , γ k α (1) , γ k α (2) , γ k α (3) , γ k α (4) , γ k α (5) , γ k α (6) . (39)For example, R α (1) is obtained as R α (1) = N − [ γ · H, [ γ · H, [ γ · H, [ γ · H, [ γ · H, [ γ · H, F ]]]]]] ,N = ( γ · α (1) )( γ · α (1) )( γ · α (1) )( γ · α (1) )( γ · α (1) )( γ · α (1) ) , (40)where γ is an arbitrary 2-dimensional vector that is not orthogonal to α (1) .To obtain more explicit form, we put an arbitrary 2-dimensional vector γ in the form: γ = aγ + bγ (orthogonaldecomposition of γ ) where γ is orthogonal to α (1) and γ is parallel to α (1) : γ ⊥ α (1) and γ k α (1) . Here b = 0to avoid γ · α (1) ≡ bγ · α (1) = 0. Using γ · α (1) = 0, we have γ · α (1) = −√ | α (1) | / γ · α (1) = √ | α (1) | / γ · α (1) = | α (1) | , γ · α (1) = | α (1) | / γ · α (1) = −| α (1) | /
2, and γ · α (1) = bγ · α (1) = b | α (1) | , which yields N = b | α (1) | = 3 b . (41)Combining the result N − = b with γ · H = aγ · H + bγ · H , therefore, we obtain R α (1) = 163 [ γ · H, [ γ · H, [ γ · H, [ γ · H, [ γ · H, [ γ · H, F ]]]]]]+ c [ γ · H, [ γ · H, [ γ · H, [ γ · H, [ γ · H, [ γ · H, F ]]]]]] , (42)where c = 16 a/ b . We can see from this expression that the non-uniqueness of an expression of R α (1) comes fromthe fact that the following sextuple commutator is identically vanishing: Z := [ γ · H, [ γ · H, [ γ · H, [ γ · H, [ γ · H, [ γ · H, F ]]]]]]= X j ,...,j ∈ , ¯ ζ j ··· j [ H j , [ H j , [ H j , [ H j , [ H j , [ H j , F ]]]]]] = 0 , ¯ ζ = 316 , ¯ ζ = − , ¯ ζ = 316 . (43)Thus, the non-uniqueness of the decomposition formula is attributed to degree of freedom due to one parameter c .In the same way as the above, we obtain R α (2) = 163 [ γ · H, [ γ · H, [ γ · H, [ γ · H, [ γ · H, [ γ · H, F ]]]]]] + c Z , (44) R α (3) = −
163 [ γ · H, [ γ · H, [ γ · H, [ γ · H, [ γ · H, [ γ · H, F ]]]]]] + c Z , (45) R α (4) = 163 ( √ [ γ · H, [ γ · H, [ γ · H, [ γ · H, [ γ · H, [ γ · H, F ]]]]]] + c Z , (46) R α (5) = 163 ( √ [ γ · H, [ γ · H, [ γ · H, [ γ · H, [ γ · H, [ γ · H, F ]]]]]] + c Z , (47) R α (6) = −
163 ( √ [ γ · H, [ γ · H, [ γ · H, [ γ · H, [ γ · H, [ γ · H, F ]]]]]] + c Z , (48)where c , . . . , c are arbitrary constants. Here, we have used the commuting property (31).Collecting an appropriate set of R α , we obtain the desired decomposition formula corresponding to each categoryof representations given in (17)(18)(19):1. For ˜ H = U (2) ⊂ SU (3) with the generators H , H , E α (1) and E − α (1) , the highest weight is Λ = ( m/ , m/ (2 √ , m/ √
3) in order to obtain simpler form. This is possible because newone is obtained by acting a Weyl group element on an old one. Thus ˜ H has the generators H , H , E α (2) and E − α (2) . The ˜ H -commutative part F ˜ H and the coset part F G/ ˜ H of F are given by F ˜ H = X j =1 , ( F , H j ) H j + R α (2) , F G/ ˜ H = R α (1) + R α (3) + R α (4) + R α (5) + R α (6) . (49)The explicit form is given by F ˜ H = X j =1 , ( F , H j ) H j + X j ,...,j ˜ ζ j ··· j [ H j , [ H j , [ H j , [ H j , [ H j , [ H j , F ]]]]]] + c Z , ˜ ζ = 1 , ˜ ζ = − , ˜ ζ = 1 , F G/ ˜ H = X j ,...,j ζ j ··· j [ H j , [ H j , [ H j , [ H j , [ H j , [ H j , F ]]]]]] + c Z ,ζ = 7213 , ζ = − , ζ = 27 , (50)where the other ζ j ··· j s and ˜ ζ j ··· j are zero and c := c + c + c + c + c . The simplest choice is c = 0 and c = 0.2. For ˜ H = U ′ (2)( SU (3)) with the generators H , H , E α (5) and E − α (5) , F ˜ H = X j =1 , ( F , H j ) H j + R α (5) , F G/ ˜ H = R α (1) + R α (2) + R α (3) + R α (4) + R α (6) . (51)We obtain F ˜ H = X j =1 , ( F , H j ) H j + X j ,...,j ˜ ζ ′ j ··· j [ H j , [ H j , [ H j , [ H j , [ H j , [ H j , F ]]]]]] + c Z , ˜ ζ ′ = 27 , ˜ ζ ′ = − , ˜ ζ ′ = 27 , F G/ ˜ H = X j ,...,j ζ ′ j ··· j [ H j , [ H j , [ H j , [ H j , [ H j , [ H j , F ]]]]]] + c ′ Z ,ζ ′ = 1 , ζ ′ = 210 , ζ ′ = − , (52)where the other ζ ′ j ··· j s and ˜ ζ j ··· j are zero. We can take the simplest choice c ′ = 0 and c = 0.03. For ˜ H = U (1) × U (1), F ˜ H = X j =1 , ( F , H j ) H j , F G/ ˜ H = R α (1) + R α (2) + R α (3) + R α (4) + R α (5) + R α (6) . (53)We obtain F G/ ˜ H = X j ,...,j ζ ′′ j ··· j [ H j , [ H j , [ H j , [ H j , [ H j , [ H j , F ]]]]]] + c ′′ Z ,ζ ′′ = 1 , ζ ′′ = 237 , ζ ′′ = − , ζ ′′ = 27 , (54)where the other ζ ′′ j ··· j s are zero. We can take the simplest choice c ′′ = 0.Using the decomposition formula, we can define the field decomposition in the similar way to the case of the gaugegroup SU ( N ). For this purpose, we define the color direction field for G (2) as n j ( x ) := Ad g ( x ) ( H j ) , (55)where Ad g ( x ) is the adjoint representation of g ( x ), where g ( x ) is an arbitrary group-valued field. For any Lie algebravalued field F ( x ), by applying the decomposition formula to Ad g − ( x ) ( F ( x )) and operating Ad g ( x ) on the both sides,we can decompose F ( x ) into the part F ˜ H ( x ) belonging to Ad g ( x ) (˜ h ) and the remaining part F G/ ˜ H ( x ): F ( x ) = F ˜ H ( x ) + F G/ ˜ H ( x ) , F ˜ H ( x ) = X j =1 , ( F ( x ) , n j ( x )) n j ( x ) + X j ,...,j ξ j ...j [ n j ( x ) , · · · , [ n j ( x ) , [ n j ( x ) , F ( x )]] · · · ] , F G/ ˜ H ( x ) = X j ,...,j η j ··· j [ n j ( x ) , · · · , [ n j ( x ) , [ n j ( x ) , F ( x )]] · · · ] , (56)where ξ j ··· j and η j ··· j are appropriate coefficients specified by the maximal stability subgroup. We decompose theYang-Mills field A µ ( x ) into two pieces, V µ ( x ) and X µ ( x ): A µ ( x ) = V µ ( x ) + X µ ( x ) , (57)where the decomposed fields V µ ( x ) and X µ ( x ) are obtained as the solution of the defining equations:0 = D µ [ V ] n j ( x ) := ∂ µ n j ( x ) − ig YM [ V µ ( x ) , n j ( x )] , (58)0 = X µ ( x ) ˜ H ⇔ X µ ( x ) = X j ,...,j η j ··· j [ n j ( x ) , · · · , [ n j ( x ) , [ n j ( x ) , X µ ( x )]] · · · ] . (59)Using the first defining equation (58), we find D µ [ A ] n j ( x ) = D µ [ V ] n j ( x ) − ig YM [ X µ ( x ) , n j ( x )] = ig YM [ n j ( x ) , X µ ( x )] . (60)By substituting this relation into the second defining equation (59), X µ ( x ) is rewritten as X µ ( x ) = − ig − YM X j ,...,j η j ··· j [ n j ( x ) , · · · , [ n j ( x ) , D µ [ A ] n j ( x )] · · · ] . (61) This definition of the color direction field is consistent with the definition adopted in the previous works for the gauge group SU ( N )because R ( n j ( x )) = R (Ad g ( x ) ( H j )) = R ( g ( x )) R ( H j ) R ( g ( x )) † . Here we have used the same notation R to denote the group representation and the corresponding algebra representation, which doesnot cause the confusion because the domains are different. V µ ( x ) is written as V µ ( x ) = A µ ( x ) − X µ ( x )= A µ ( x ) + ig − YM X j ,...,j η j ··· j [ n j ( x ) , · · · , [ n j ( x ) , D µ [ A ] n j ( x )] · · · ] . (62)Thus V µ ( x ) and X µ ( x ) are written in terms of the original Yang-Mills field A µ ( x ) and the color fields n j ( x ).Notice that V µ ( x ) is further cast into V µ ( x ) = A µ ( x ) + X j ,...,j η j ··· j [ n j ( x ) , · · · , [ n j ( x ) , [ A µ ( x ) , n j ( x )]] · · · ]+ ig − YM X j ,...,j η j ··· j [ n j ( x ) , · · · , [ n j ( x ) , ∂ µ n j ( x )] · · · ]= X j =1 , ( A µ ( x ) , n j ( x )) n j ( x ) + X j ,...,j ξ j ...j [ n j ( x ) , · · · , [ n j ( x ) , [ n j ( x ) , A µ ( x )]] · · · ]+ ig − YM X j ,...,j η j ··· j [ n j ( x ) , · · · , [ n j ( x ) , ∂ µ n j ( x )] · · · ] , (63)where we have applied the the formula (56) to A µ ( x ) in the last step. Therefore, V µ ( x ) is decomposed into C µ ( x ) and B µ ( x ): V µ ( x ) = C µ ( x ) + B µ ( x ) C µ ( x ) := X j =1 , ( A µ ( x ) , n j ( x )) n j ( x ) + X j ,...,j ξ j ...j [ n j ( x ) , · · · , [ n j ( x ) , [ n j ( x ) , A µ ( x )]] · · · ] , B µ ( x ) := ig − YM X j ,...,j η j ··· j [ n j ( x ) , · · · , [ n j ( x ) , ∂ µ n j ( x )] · · · ] . (64)For the sake of convenience, we define the field m ( x ) for the highest weight Λ = (Λ , Λ ) by m ( x ) := Λ j n j ( x ) . (65)Here C µ ( x ) commutes with m ( x ): [ C µ ( x ) , m ( x )] = 0 , (66)while B µ ( x ) is orghogonal to n j ( x ): ( B µ ( x ) , n j ( x )) = 0 . (67)The first term in the right-hand side of C µ ( x ) corresponds to the element of the Cartan subalgebra Lie( H ) andthe second term to the remaining part Lie( ˜ H ) − Lie( H ) which vanishes when the maximal stability group coincideswith the maximal torus group ˜ H = H (This is the case for the maximal option of SU ( N )). Notice that B µ ( x )is the extension of the SU ( N ) Cho connection to G (2). An appropriate set of the above fields will be used in thereformulation of the G (2) Yang-Mills theory.We suppose that the dominant mode for quark confinement is the restricted field V µ ( x ) extracted from the original G (2) Yang-Mills field A µ ( x ) through the decomposition given in the above. In fact, this observation is exemplified forthe G (2) Wilson loop operator by using the non-Abelian Stokes theorem in the same manner as in SU ( N ), as givenin the next section. IV. NON-ABELIAN STOKES THEOREM
In this section, we derive the non-Abelian Stokes theorem for the Wilson loop operator in an arbitrary representationof G (2) gauge group using the color direction fields n k .2 A. General gauge group
Before proceeding to the case of the gauge group G (2), we discuss the general case.It is known [14, 23, 25] that the Wilson loop operator defined for any Lie algebra valued Yang-Mills field A andthe irreducible (unitary) representation R is cast into the following (path-integral) representation : W C [ A ] = Z [ dµ ( g )] Σ exp (cid:20) − ig YM Z Σ: ∂ Σ= C F g (cid:21) ,F g = 12 F gµν dx µ ∧ dx ν F gµν ( x ) = κ { ∂ µ tr( m ( x ) A ν ( x )) − ∂ ν tr( m ( x ) A µ ( x )) + ig YM tr( m ( x )[Ω µ ( x ) , Ω ν ( x )]) } , Ω µ ( x ) := ig − YM g ( x ) ∂ µ g † ( x ) , g ( x ) ∈ G, (68)where [ dµ ( g )] Σ is the product measure of the Haar measure on the gauge group G over Σ and Λ in m := Λ j n j is thehighest weight vector of the representation R . Here the gauge-invariant field strength F gµν is equal to the non-Abelianfield strength F µν [ V ] := ∂ µ V ν − ∂ ν V µ − ig YM [ V µ , V ν ] of the restricted field V µ (in the decomposition A = V + X )projected to the color field m : F gµν = tr { m F µν [ V ] } = Λ j f ( j ) µν , f ( j ) µν = tr { n j F µν [ V ] } . (69)Therefore, the restricted field V µ is regarded as the dominant mode for quark confinement, since the remaining field X µ does not contribute to the Wilson loop operator. The derivation of this fact is given in Appendix C.Let F be an arbitrary element of the Lie algebra G = Lie( G ). Suppose that F is decomposed as F = F ˜ H + F G/ ˜ H , F ˜ H = r X j =1 ( F , H j ) H j + X j ,...,j n ξ j ...j n [ H j , · · · , [ H j n , F ] · · · ] , F G/ ˜ H = X j ,...,j n η j ··· j n [ H j , · · · , [ H j n , F ] · · · ] , (70)where r is the rank of the gauge group. At least, this relation for the decomposition has already been proved for G (2)in the previous section, and the method is applicable to any semi-simple compact Lie group.In order to complete the non-Abelian Stokes theorem, we can follow the same procedures as those for SU ( N ) givenin [25], if g † ∂ µ m g does not have the part belonging to the Lie( ˜ H ):( g † ∂ µ m g ) ˜ H = 0 . (71)This enables us to rewrite [Ω µ ( x ) , Ω ν ( x )] in terms of the color fields n i ( x ) := Ad g ( x ) ( H j ), which is indeed shown inAppendix B.The relevant relation (71) is indeed verified as follows. By applying (70) to Ad g − ( ∂ µ m ), we obtain the decompo-sition: ( g † ∂ µ m g ) ˜ H = r X j =1 κ tr( g † ∂ µ m gH j ) H j + X i ,...,i n ξ i ...i n [ H i , · · · , [ H i n , g † ∂ µ m g ] · · · ] . (72) Strictly speaking, we should write F gµν in (68) as F gµν = κ { ∂ µ tr (cid:0) R ( m ( x )) R ( A ν ( x )) (cid:1) − ∂ ν tr (cid:0) R ( m ( x )) R ( A µ ( x )) (cid:1) + ig YM tr (cid:0) R ( m ( x ))([Ω µ ( x ) , Ω ν ( x )]) (cid:1) } ,R ( m ( x )) = Λ j R ( g ( x )) R ( H j ) R ( g ( x )) † = Λ j R (Ad g ( x ) ( H j )) , Ω µ ( x ) = ig − YM R ( g ( x )) ∂ µ R ( g ( x )) † . To simplify the notation, we omit the symbol R ( · ) throughout this section, Appendix B and C. g † ∂ µ m gH j ) = Λ i tr( g † ∂ µ ( gH i g † ) gH j )= Λ i tr( g † ∂ µ gH i H j + H i ∂ µ g † gH j )= Λ i tr( g † ∂ µ gH i H j − H i g † ∂ µ gH j )= Λ i tr( g † ∂ µ gH i H j − g † ∂ µ gH j H i )= Λ i tr( g † ∂ µ g [ H i , H j ])= 0 , (73)where we have used g † g = 1 in the second equality, the relation ∂ µ g † g = − g † ∂ µ g following from ∂ µ ( gg † ) = 0 in thethird equality and the cyclicity of the trace in the fourth equality. In addition, by taking account of g † ∂ µ m g = g † ∂ µ ( g Λ · Hg † ) g = g † ∂ µ g Λ · H + Λ · H∂ µ g † g = − ∂ µ g † g Λ · H + Λ · H∂ µ g † g = [Λ · H, ∂ µ g † g ] , (74)the second term is rewritten as X i ,...,i n ξ i ...i n [ H i , · · · , [ H i n , g † ∂ µ m g ] · · · ] = X i ,...,i n ξ i ...i n [ H i , · · · , [ H i n , [Λ · H, ∂ µ g † g ]] · · · ]= X i ,...,i n ξ i ...i n [Λ · H, [ H i , · · · , [ H i n , ∂ µ g † g ] · · · ]]= [Λ · H, ( ∂ µ g † g ) ˜ H ] , (75)where we have used the commuting property (31) in the second equality and (70) for F = ∂ µ g † g in the third equality.By substituting the Cartan decomposition of ( ∂ µ g † g ) ˜ H in ˜ h given by( ∂ µ g † g ) ˜ H = r X j =1 κ tr( ∂ µ g † gH j ) H j + X α ∈R + : E ± α ∈ ˜ h (( ∂ µ g † g ) ∗ α E α + ( ∂ µ g † g ) α E − α ) , (76)into (75), we find that the second term also vanishes,(75) = [Λ · H, X α ∈R + : E ± α ∈ ˜ h (( ∂ µ g † g ) ∗ α E α + ( ∂ µ g † g ) α E − α )]= X α ∈R + : E ± α ∈ ˜ h Λ · α (( ∂ µ g † g ) ∗ α E α − ( ∂ µ g † g ) α E − α )= 0 , (77)where we have used Λ · α = 0 for α satisfying E α ∈ ˜ h . Thus we have confirmed (71).4Thus we obtain the final form of Wilson loop operator as W C [ A ] = Z [ dµ ( g )] Σ exp (cid:20) − ig YM Z Σ: ∂ Σ= C F g (cid:21) ,F g = 12 F gµν dx µ ∧ dx ν ,F gµν ( x ) = κ { ∂ µ tr( m ( x ) A ν ( x )) − ∂ ν tr( m ( x ) A µ ( x ))+ ig − YM X i ,...,i n η i ··· i n tr( m ( x )[ ∂ µ n i ( x ) , [ n i , · · · , [ n i n − ( x ) , ∂ ν n i n ( x )] · · · ]]) } . (78)The detail of the derivation of (78) is given in Appendix B, which is almost the same as that given in [25] for SU ( N ),once (71) is established. B. G (2) case In each case of representations, we can write the new form for the Wilson loop operator using the decompositionformula based on the above general consideration:1. For ˜ H = U (2) ∈ SU (3), the Wilson loop operator is written as (78) where n = 6 and η j ··· j = ζ j ··· j (79)where ζ j ··· j is defined in (50).2. For ˜ H = U (2) SU (3), the Wilson loop operator is written as (78) where n = 6 and η j ··· j = ζ ′ j ··· j (80)where ζ ′ j ··· j is defined in (52).3. For ˜ H = U (1) × U (1), the Wilson loop operator is written as (78) where n = 6 and η j ··· j = ζ ′′ j ··· j (81)where ζ ′′ j ··· j is defined in (54).By using m = Λ i n i = (2 n + m ) n / m n / (2 √ F gµν ( x ) = 2 n + m F (1) µν ( x ) + m √ F (2) µν ( x ) F (1) µν ( x ) = κ { ∂ µ tr( n ( x ) A ν ( x )) − ∂ ν tr( n ( x ) A µ ( x ))+ ig − YM X j ,...,j ζ ′ j ··· j tr( n ( x )[ ∂ µ n j ( x ) , [ n j , · · · , [ n j ( x ) , ∂ ν n j ( x )] · · · ]]) } F (2) µν ( x ) = κ { ∂ µ tr( n ( x ) A ν ( x )) − ∂ ν tr( n ( x ) A µ ( x ))+ ig − YM X j ,...,j ζ j ··· j tr( n ( x )[ ∂ µ n j ( x ) , [ n j , · · · , [ n j ( x ) , ∂ ν n j ( x )] · · · ]]) } . (82) We can rewrite F gµν of (78) using the inner product instead of using the trace as F gµν = ∂ µ ( m , A ) − ∂ ν ( m , A ) + ig YM X i ,...,i n η i ··· i n ( m ( x ) , [ ∂ µ n i ( x ) , [ n i ( x ) , · · · , [ n i n − ( x ) , ∂ ν n i n ( x )] · · · ]]) , so that the Wilson loop depends on the representation only through the highest weight vector Λ. V. MAGNETIC MONOPOLES
We can define magnetic-monopole current k as the co-differential of the Hodge dual of F g : k = δ ∗ F g . (83)In the D -dimensional spacetime, k is expressed by a differential form, ( D − D = 4, especially, themagnetic monopole current reads k µ = 12 ǫ µνρσ ∂ ν F gρσ . (84)Then, the magnetic charge q m is defined by q m := Z d xk = Z d x ǫ jkl ∂ l F gjk = Z d S l ǫ jkl F gjk . (85)We examine the quantization condition for the magnetic charge. The magnetic charge can have nonzero value becausethe map defined by m : S → G (2) / ˜ H = ( G (2) /U (2) G (2) / ( U (1) × U (1)) , (86)has the nontrivial homotopy group: π ( G (2) / ˜ H ) = π ( ˜ H ) = ( π ( S (2) × U (1)) = π ( U (1)) = Z π ( U (1) × U (1)) = Z + Z . (87)Because the value of the magnetic charge depends only on the topological character of n i , we can use specific groupelements g to obtain the quantization condition for the magnetic charge. Now, we consider a case in which g ( x )belongs to SU (3). In this case, F gµν reduces to F gµν = 2 n + m F (1) µν + m √ F (2) µν ,F (1) µν = κ { ∂ µ tr( n A ν ) − ∂ ν tr( n A µ ) − ig − YM tr( n [ ∂ µ n , ∂ ν n ] + n [ ∂ µ n , ∂ ν n ]) } ,F (2) µν = κ { ∂ µ tr( n A ν ) − ∂ ν tr( n A µ ) − ig − YM tr( n [ ∂ µ n , ∂ ν n ] + n [ ∂ µ n , ∂ ν n ]) } = κ (cid:26) ∂ µ tr( n A ν ) − ∂ ν tr( n A µ ) − ig − YM tr( n [ ∂ µ n , ∂ ν n ]) (cid:27) . (88)Here notice that two field strengths F (1) µν and F (2) µν appear in the non-Abelian Stokes theorem for SU (3). It is shown[14, 23] that the two kinds of the gauge-invariant charges q (1) m and q (2) m obey the different quantization conditions: q m = 2 n + m q (1) m + m √ q (2) m ,q (1) m := Z d x ǫ jkℓ ∂ ℓ F gjk = 4 πg YM (cid:18) ℓ − ℓ ′ (cid:19) ,q (2) m := Z d x ǫ jkℓ ∂ ℓ F gjk = 4 πg YM √ ℓ ′ , ℓ, ℓ ′ ∈ Z . (89)Thus, we obtain the quantization condition for the magnetic charge in G (2): q m = 4 πg YM (cid:16) n ℓ − ℓ ′ ) + m ℓ (cid:17) = 2 πg YM ( nk + mℓ ) , (90)where we have defined k := 2 ℓ − ℓ ′ , which can take an arbitrary integer. The observation based on the homotopygroup (87) that there need to be two integers in q m . There exist already two integers in q m . Therefore, it is enoughto consider a case g ( x ) ∈ SU (3) for deriving the quantization condition for the magnetic charge in G (2).6 VI. CONCLUSIONS AND DISCUSSIONS
For the exceptional group G (2), we have first shown that there exist three cases of the maximal stability subgroup.Then, we have derived the gauge-covariant decomposition formula which is written using the multiple commutatorswith the color direction fields, in accord with each stability group. Moreover, we have obtained the non-AbelianStokes theorem for the Wilson loop operator that is written in terms of the relevant color direction fields. Theseresults indicate that there exist three options for the reformulation of the G (2) Yang-Mills theory. In any option,we need the two kinds of color fields, since the two Cartan generators are inevitably required in the decompositionformula, in marked contrast to the minimal option of SU ( N ) group. Nevertheless, each option would be utilized fordescribing confinement of quarks in the relevant representation of G (2). This is because the the non-Abelian Stokestheorem for the Wilson loop operator is attributed to the decomposition formula available to a given representation.This would be confirmed more explicitly when the reformulation is ready to be checked.The method we have used in this paper for obtaining the decomposition formula would be so general that thedecomposition formula is written for any semi-simple Lie group using the multiple commutators with the Cartangenerators. In addition, once the decomposition formula given in the above is obtained, we can immediately obtainthe expression of the Wilson loop operator written in terms of the color direction fields, because we derived thenon-Abelian Stokes theorem in a general way. This observation suggests that the reformulation of the Yang-Millstheory with an arbitrary semi-simple gauge group would be possible. Acknowledgments
This work is supported by Grants-in-Aid for Scientific Research (C) No.24540252 and (C) No.15K05042 from theJapan Society for the Promotion of Science (JSPS).
Appendix A: Necessity of sextuple commutators in the decomposition formula for G (2) In section III, we have seen that the sextuple commutator is used to obtain the decomposition formula for G (2).In this appendix we show that such a formula for G (2) cannot be obtained by taking the commutator less than sixtimes. Taking the commutator odd number of times, we obtain[ H j , · · · , [ H j n +1 , F ] · · · ] = X α ∈R + ( α j · · · α j n +1 F ∗ α E α − α j · · · α j n +1 F α E − α ) . (A1)In this expression the sign of the term E α is different from the sign of the term E − α and therefore this is notappropriate. Thus we see that we just need to consider the cases of double and quadruple commutators.First, we consider the case of double commutators. We can decompose an arbitrary element F of the Lie algebraif and only if there are real numbers k , k and k that satisfy k [ H , [ H , F ]] + k [ H , [ H , F ]] + k [ H , [ H , F ]] = X α ∈R + : E ± α / ∈ ˜ h ( F ∗ α E α + F α E − α ) . (A2)Using the Cartan decomposition of F , we find that the left hand side of (A2) is equal to X α ∈R + (( α ) k + α α k + ( α ) k )( F ∗ α E α + F α E − α ) , (A3)where we put α = ( α , α ). Hence (A2) is equivalent to( α ) k + α α k + ( α ) k = 1 for E α / ∈ ˜ h, ( α ) k + α α k + ( α ) k = 0 for E α ∈ ˜ h. (A4)In the case of ˜ H = U (2) ∈ SU (3), the three equations (A4) for α (1) , α (2) and α (3) can be written in a matrix form as − √
34 34 √
34 34 k k k = . (A5)7The solution of this equation is k = k = 0, k = 4 /
3. This solution is consistent with the decomposition formulafor SU (3). But, these values of k , k and k do not satisfy the equation (A4) for α (4) , α (5) and α (6) . For example,the equation (A4) for α (5) is given by 13 k = 1 , (A6)which is however not satisfied by k = 4 /
3. Therefore, there is no solution for all of (A4).In the case of ˜ H = U (2) / ∈ SU (3) and of ˜ H = U (1) × U (1), the equation (A4) for α (1) , α (2) and α (3) reads − √
34 34 √
34 34 k k k = . (A7)The solution of this equation is k = k = 1, k = 0. This solution is also consistent with the decomposition formulafor SU (3). But this does not satisfy the equation (A4) for α (4) , α (5) and α (6) . Therefore, also in this case, there areno solutions for all of (A4). Thus we confirm that there are no decomposition formulae using double commutators forall representations.Next, we consider the case of quadruple commutators. There exits the decomposition formulae if and only if thereare real numbers k , k , k , k and k that satisfy k [ H , [ H , [ H , [ H , F ]]]] + k [ H , [ H , [ H , [ H , F ]]]] + k [ H , [ H , [ H , [ H , F ]]]] + k [ H , [ H , [ H , [ H , F ]]]]+ k [ H , [ H , [ H , [ H , F ]]]] = X α ∈R + : E ± α / ∈ ˜ h ( F ∗ α E α + F α E − α ) . (A8)This is equivalent to( α ) k + ( α ) α k + ( α ) ( α ) k + α ( α ) k + ( α ) k = 1 for E α / ∈ ˜ h, ( α ) k + ( α ) α k + ( α ) ( α ) k + α ( α ) k + ( α ) k = 0 for E α ∈ ˜ h. (A9)The equivalent matrix form is given as116 −√ −
916 0 0 0 01 √ √ − √ − k k k k k = , or . (A10)Calculating the rank of the matrix, we see that these equations do not have the solutions. Thus we have confirmedthat quadruple commutators are not enough to obtain the desired decomposition formula. Appendix B: Derivation of the non-Abelian Stokes theorem for general gauge group using (71)
From the fact (71) that the ˜ H part of g † ∂ µ m g is vanishing, we have g † ∂ µ m g = ( g † ∂ µ m g ) G/H = X i ,...,i n η i ··· i n [ H i , · · · , [ H i n , g † ∂ µ m g ] · · · ] . (B1)8Multiplying both sides of this equation by g from the left and by g † from the right, we obtain ∂ µ m ( x ) = X i ,...,i n η i ··· i n [ n i ( x ) , · · · , [ n i n ( x ) , ∂ µ m ( x )] · · · ]= X i ,...,i n η i ··· i n [ n i ( x ) , · · · , [ m ( x ) , ∂ µ n i n ( x )] · · · ]= X i ,...,i n η i ··· i n [ m , [ n i ( x ) , · · · , [ n i n − , ∂ µ n i n ( x )] · · · ]]= ig YM [ B µ ( x ) , m ( x )] , (B2)where we have used [ n i , ∂ µ m ] = [ m , ∂ µ n i ] following from ∂ µ [ n i , m ] = 0 in the second equality, the commutingproperty (31) in the third equality, and we have introduced B µ ( x ) := ig − YM X i ,...,i n η i ··· i n [ n i ( x ) , · · · , [ n i n − ( x ) , ∂ µ n i n ( x )] · · · ] . (B3)in last equality. On the other hand, we find ∂ µ m ( x ) = ig YM [Ω µ , m ( x )] . (B4)Combining (B2) and (B4), we obtain [Ω µ , m ( x )] = [ B µ ( x ) , m ( x )] . (B5)Using this relation, we can rewrite the third term in F gµν ( x ) as ig YM tr( m ( x )[Ω µ ( x ) , Ω ν ( x )]) = ig YM tr([ m ( x ) , Ω µ ( x )]Ω ν ( x ))= ig YM tr([ m ( x ) , B µ ( x )]Ω ν ( x ))= ig YM tr([Ω ν ( x ) , m ( x )] B µ ( x ))= tr( ∂ ν m ( x ) B µ ( x ))= ig − YM X i ,...,i n η i ··· i n tr( ∂ ν m ( x )[ n i ( x ) , · · · , [ n i n − ( x ) , ∂ µ n i n ( x )] · · · ])= ig − YM X i ,...,i n η i ··· i n tr([ ∂ ν m ( x ) , n i ( x )][ n i , · · · , [ n i n − ( x ) , ∂ µ n i n ( x )] · · · ])= − ig − YM X i ,...,i n η i ··· i n tr([ m ( x ) , ∂ ν n i ( x )][ n i , · · · , [ n i n − ( x ) , ∂ µ n i n ( x )] · · · ])= − ig − YM X i ,...,i n η i ··· i n tr( m ( x )[ ∂ ν n i ( x ) , [ n i , · · · , [ n i n − ( x ) , ∂ µ n i n ( x )] · · · ]])= ig − YM X i ,...,i n η i ··· i n tr( m ( x )[ ∂ µ n i ( x ) , [ n i , · · · , [ n i n − ( x ) , ∂ ν n i n ( x )] · · · ]]) , (B6)where we have used the cyclicity of the trace in first, third, sixth and eighth equality, (B4) in fourth equality,[ ∂ ν m , n i ] = − [ m , ∂ ν n i ] following from ∂ ν [ m , n i ] = 0 in seventh equality and the fact that the first expression isanti-symmetric in µ and ν in the last equality. This completes the proof of the non-Abelian Stokes theorem (78). Appendix C: Derivation of F gµν = κ tr( m F µν [ V ]) In this appendix, we show that the remaining field X does not contribute to the Wilson loop operator by derivingthe equality (69). Using the decomposition (64), we obtaintr( m F µν [ V ]) = tr( m ( ∂ µ C ν − ∂ ν C µ − ig Y M [ C µ , C ν ] − ig Y M [ C µ , B ν ] − ig Y M [ B µ , C ν ]+ ∂ µ B ν − ∂ ν B µ − ig Y M [ B µ , B ν ]) . (C1)From the fact (66) we see that the third, fourth and fifth terms vanish. Thus we obtaintr( m F µν [ V ]) = tr( m ( ∂ µ C ν − ∂ ν C µ + ∂ µ B ν − ∂ ν B µ − ig YM [ B µ , B ν ])) . (C2)9The first term of (C2) reads ∂ µ tr( m A ν ) = ∂ µ (tr( m C ν ))= tr( ∂ µ m C ν + m ∂ µ C ν )= tr( m ∂ µ C ν ) , (C3)where we have used ( g † ∂ µ m g ) ˜ H = 0 and C ν ∈ g Lie( ˜ H ) g † . The third term of (C2) readstr( m ∂ µ B ν ) = − tr( ∂ µ m B ν )= − ig YM tr([ B µ , m ] B ν )= ig YM tr( m [ B µ , B ν ]) , (C4)where we have used tr( m B ν ) = 0 in the first equality, (B2) in the second equality and the cyclicity of the trace inthe last equality. Thus we obtain( C
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