Wilson loop and magnetic monopole through a non-Abelian Stokes theorem
aa r X i v : . [ h e p - t h ] F e b Chiba Univ. Preprint
CHIBA-EP-169
January 2008
Wilson loop and magnetic monopolethrough a non-Abelian Stokes theorem
Kei-Ichi Kondo, † , † Department of Physics, Graduate School of Science,Chiba University, Chiba 263-8522, Japan
Abstract
We show that the Wilson loop operator for SU ( N ) Yang-Mills gauge con-nection is exactly rewritten in terms of conserved gauge-invariant magnetic andelectric currents through a non-Abelian Stokes theorem of the Diakonov-Petrovtype. Here the magnetic current originates from the magnetic monopole de-rived in the gauge-invariant way from the pure Yang–Mills theory even in theabsence of the Higgs scalar field, in sharp contrast to the ’t Hooft-Polyakovmagnetic monopole in the Georgi-Glashow gauge-Higgs model. The resultingrepresentation indicates that the Wilson loop operator in fundamental repre-sentations can be a probe for a single magnetic monopole irrespective of N in SU ( N ) Yang-Mills theory, against the conventional wisdom. Moreover, weshow that the quantization condition for the magnetic charge follows from thefact that the non-Abelian Stokes theorem does not depend on the surface cho-sen for writing the surface integral. The obtained geometrical and topologicalrepresentations of the Wilson loop operator have important implications to un-derstanding quark confinement according to the dual superconductor picture. Key words: Wilson loop, magnetic monopole, Stokes theorem, quark confinement,Yang-Mills theory, Abelian dominance,PACS: 12.38.Aw, 12.38.Lg E-mail: [email protected] ontents SU (2) coherent state . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3 SU (3) coherent state . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 D.1 Maximal case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37D.2 Minimal case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
E Field strength 41
E.1 Maximal case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41E.2 Minimal case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42i
Introduction and main results
The Wilson loop operator [1] is a gauge-invariant observable of fundamental impor-tance in Yang-Mills gauge theories [2]. It is defined from the holonomy of the gaugeconnection around a given loop. Especially, it is well-known that the area law of theWilson loop average in Yang-Mills theory gives a criterion for quark confinement.In the first half of this paper, we give a pedagogic and thorough derivation of anon-Abelian Stokes theorem for the Wilson loop operator of SU(N) Yang-Mills gaugeconnection. The non-Abelian Stokes theorem (NAST) means that the non-AbelianWilson loop operator defined for a closed loop C is rewritten into a surface integralform over any surface bounding the loop C . In particular, we restrict our considera-tions to the Diakonov-Petrov (DP) type [3] among various types of NAST. The DPversion of NAST was originally derived in [3] for SU(2) case and later developed andextended to SU(N) case in [3, 4, 5, 6, 7] and [8, 9, 11, 10]. This is because the DP typecan give us a very useful tool for understanding quark confinement based on the dualsuperconductor picture [12] which is a promising scenario of quark confinement.In the latter half of this paper, some physical aspects obtained form the NASTwill be demonstrated. The following are the main results of this paper obtained byextending the previous works [9,11] for SU(N) Yang-Mills gauge connection, althoughsome of them have been known for the SU(2) case as will be mentioned in the relevantparts. We put our emphasis on the physical aspects rather than the mathematicalrigor.1) The DP version of NAST includes no ordering procedures, which enables oneto obtain an Abelian-like expression for the non-Abelian Wilson loop operator. Itis derived as a path-integral representation of the Wilson loop operator by usingthe coherent state for semi-simple Lie groups. The DP version of NAST is quiteuseful to consider the dual superconductivity as the electric-magnetic dual of ordinarysuperconductivity described by the Maxwell-like Abelian gauge theory. See section 2.2) The contribution to the Wilson loop average is separated into two parts originat-ing from the decomposition of the Yang-Mills potential A µ ( x ): A µ ( x ) = V µ ( x )+ X µ ( x )in such a way that only the V µ part is responsible for quark confinement and thatthe remaining X µ part decouples. This is an gauge invariant understanding of thephenomenon which was known so far as the infrared Abelian dominance [13, 14] inYang-Mills theory under a specific gauge fixing called the maximal Abelian gauge [15].In other words, this give a gauge-invariant “Abelian” projection and “Abelian dom-inance” for the Wilson loop. In fact, the 100% Abelian dominance for the Wilsonloop is a direct consequence derived in the gauge-invariant way from this construc-tion [18, 19], although it was confirmed by numerical simulations on a lattice underthe maximal Abelian gauge [16, 17]. See section 5.3) The Wilson loop operator is explicitly rewritten in terms of conserved currentsoriginating from the magnetic monopole which can be derived in the gauge-invariantway from the pure SU(N) Yang–Mills theory even in the absence of Higgs scalarfield [20, 21, 22, 23, 24, 25, 26, 27], in sharp contrast to the ’t Hooft-Polyakov magneticmonopole in Georgi-Glashow gauge-Higgs model. This implies that the Wilson loopoperator can be a probe for magnetic monopole whose condensation will cause thedual superconductivity in Yang-Mills theory. See section 6.4) The quark confinement in the sense of area law of the Wilson loop average, ifrealized, can be caused by a single ( non-Abelian U( N − C and a surface S bounded by C irrespective of N in SU( N ) Yang-Mills theory for N ≥
3, against the conventionalwisdom that N − U (1) N − are responsible for quark confinement. This was conjectured in [9, 11]. Seesection 4.5) The quantization condition for the magnetic charge of the single magneticmonopole is derived from the fact that the non-Abelian Stokes theorem should notdepend on the surface used to represent the surface integral. The quantization con-dition is the same as the Dirac type for SU(2), while it is similar to the Dirac type,but is different to that for N ≥
3. See section 7.6) The Wilson loop operator has a geometrical and topological meaning relatedto solid angle seen at the location of the magnetic monopole in three-dimensionalcase, and the winding number between the magnetic monopole loop and the surfacein four-dimensional case. This leads to a possibility that the area law can be derivedby calculating the geometrical configurations of the relevant topological objects. Seesection 6.Section 3 is devoted to explaining the coherent state as a technical material forderiving the non-Abelian Stokes theorem. Some of more technical details are collectedin Appendices.
The Wilson loop operator W C [ A ] along the closed loop C is defined as the trace of apath-ordered exponential of a gauge field A ( x ): W C [ A ] := N − tr (cid:20) P exp (cid:26) ig I C A (cid:27)(cid:21) , (2.1)where P denotes the path ordering defined precisely later, A is the Lie algebravalued connection one-form: A ( x ) := A A ( x ) T A := A Aµ ( x ) T A dx µ , (2.2)and the normalization factor N is the dimension of the representation R , in whichthe Wilson loop is considered, i.e., N := d R = dim( R ) = tr( R ) . (2.3)We wish to rewrite the non-Abelian Wilson loop operator defined by the lineintegral along a loop C into a surface integral form over the surface S having C as2he boundary: ∂S = C . See Fig. 1. A curve (path) L starting at x and ending at x is parameterized by a parameter s : x = x ( s ) and x = x ( s ). Then we define the parallel transporter W L [ A ]( s, s ) by W L [ A ] ab ( s, s ) = (cid:20) P exp (cid:26) ig Z L : x → x A (cid:27)(cid:21) ab = (cid:20) P exp (cid:26) ig Z ss ds A ( s ) (cid:27)(cid:21) ab , (2.4)where a matrix element is specified by two indices ab and we have defined A ( s ) = A µ ( x ) dx µ /ds = A Aµ ( x ) T A dx µ /ds. (2.5)The Wilson loop operator W C [ A ] for a closed loop C is obtained by taking thetrace of W L [ A ] for a closed path L = C : W C [ A ] = N − tr( W C [ A ]( s, s )) = N − N X a =1 W C [ A ] aa ( s, s ) . (2.6)The parallel transporter W L [ A ] satisfies the differential equation of the Schr¨odingertype: i dds W L [ A ]( s, s ) = − g A ( s ) W L [ A ]( s, s ) . (2.7)Therefore, W L [ A ] is regarded as the time-evolution operator of a quantum mechan-ical system with the Hamiltonian H ( s ) = − g A ( s ), if s is identified with the time.This implies that it is possible to write path-integral representations of the paralleltransporter and the Wilson loop operator according to the standard procedures:1. Partition the path L into N infinitesimal segments, P exp (cid:20) ig Z ss ds A ( s ) (cid:21) = P N − Y n =0 [1 + igǫ A ( s n )] , (2.8)where ǫ := ( s − s ) /N and s n := nǫ . We set s = 0 and s = s N .2. Insert the complete (normal) set of states at each partition point, = Z | ξ n , Λ i dµ ( ξ n ) h ξ n , Λ | ( n = 1 , · · · , N − , (2.9)where the state is normalized h ξ n , Λ | ξ n , Λ i = 1 . (2.10)3. Take the limit N → ∞ and ǫ → N ǫ = s is fixed.4. Replace the trace of the operator O with N − tr( O ) = Z dµ ( ξ N ) h ξ N , Λ | O | ξ N , Λ i . (2.11)3utting aside the issue of what type of complete set is chosen, we obtain N − tr (cid:26) P exp (cid:20) ig Z s ds A ( s ) (cid:21)(cid:27) = lim N →∞ ,ǫ → Z · · · Z dµ ( ξ ) h ξ , Λ | [1 + iǫg A ( s )] | ξ , Λ i× dµ ( ξ ) h ξ , Λ | [1 + iǫg A ( s )] | ξ , Λ i· · · dµ ( ξ N − ) h ξ N − , Λ | [1 + iǫg A ( s N − )] | ξ N , Λ i = lim N →∞ ,ǫ → N Y n =1 Z dµ ( ξ n ) N − Y n =0 h ξ n , Λ | [1 + iǫg A ( s n )] | ξ n +1 , Λ i = lim N →∞ ,ǫ → N Y n =1 Z dµ ( ξ n ) N − Y n =0 [ h ξ n , Λ | ξ n +1 , Λ i + iǫg h ξ n , Λ | A ( s n )] | ξ n +1 , Λ i ] , (2.12)where we have used ξ = ξ N .As a complete set to be inserted, we adopt the coherent state . As explainedin the next section, the coherent state | ξ n , Λ i is constructed by operating a groupelement ξ ∈ G to a reference state | Λ i : | ξ, Λ i = ξ | Λ i . (2.13)Note that the coherent states are non-orthogonal: h ξ ′ , Λ | ξ, Λ i 6 = 0 . (2.14)For taking the limit ǫ → O ( ǫ ) terms. Therefore, we find apart from O ( ǫ ) terms ǫ h ξ n , Λ | A ( s n ) | ξ n +1 , Λ i = ǫ h ξ n , Λ | A ( s n ) | ξ n , Λ i + O ( ǫ )= ǫ h Λ | ξ ( s n ) † A ( s n ) ξ ( s n ) | Λ i + O ( ǫ ) , (2.15)and h ξ n , Λ | ξ n +1 , Λ i = h ξ ( s n ) , Λ | ξ ( s n ) , Λ i + ǫ h ξ ( s n ) , Λ | ˙ ξ ( s n ) , Λ i + O ( ǫ )=1 + ǫ h ξ ( s n ) , Λ | ˙ ξ ( s n ) , Λ i + O ( ǫ ) , (2.16)where we have used the normalization condition, h ξ ( s n ) , Λ | ξ ( s n ) , Λ i = 1. Hence thedot denotes the differentiation with respect to s . Therefore, we obtain h ξ n , Λ | ξ n +1 , Λ i + iǫg h ξ n , Λ | A ( s n )] | ξ n +1 , Λ i =1 + igǫ h Λ | ξ ( s n ) † A ( s n ) ξ ( s n ) | Λ i + ǫ h ξ ( s n ) , Λ | ˙ ξ ( s n ) , Λ i + O ( ǫ )= exp[ igǫ h Λ | ξ ( s n ) † A ( s n ) ξ ( s n ) | Λ i + ǫ h ξ ( s n ) , Λ | ˙ ξ ( s n ) , Λ i + O ( ǫ )] . (2.17)Thus we arrive at the expression: W C [ A ] = Z [ dµ ( ξ )] C exp (cid:26) ig I C ds h Λ | [ ξ ( s ) † A ( s ) ξ ( s ) − ig − ξ ( s ) † ˙ ξ ( s )] | Λ i (cid:27) , (2.18)4here [ dµ ( ξ )] C is the product measure of dµ ( ξ n ) along the loop C :[ dµ ( ξ )] C := lim N →∞ ,ǫ → N Y n =1 dµ ( ξ n ) =: Y x ∈ C dµ ( ξ ( x )) . (2.19)Another form for the path integral representation of the Wilson loop operatorreads W C [ A ] = Z [ dµ ( ξ )] C exp (cid:18) ig I C h Λ | h ξ ( x ) † A ( x ) ξ ( x ) − ig − ξ ( x ) † dξ ( x ) i | Λ i (cid:19) , (2.20)where d denotes the exterior derivative: d = ds dds = ds dx µ ds ∂∂x µ = dx µ ∂∂x µ . (2.21)It is further rewritten into the form: W C [ A ] = Z [ dµ ( ξ )] C exp (cid:18) ig I C h m A A A + ω i(cid:19) , (2.22a)where we have introduced m A ( x ) := h Λ | ξ † ( x ) T A ξ ( x ) | Λ i , (2.22b) ω ( x ) := − h Λ | ig − ξ † ( x ) dξ ( x ) | Λ i . (2.22c)Now the argument of the exponential has been rewritten using Abelian quantities.Therefore, we can apply the (usual) Stokes theorem, I C = ∂S ω = Z S dω, (2.23)in the argument of the exponential. Thus we obtain a version of non-Abelian Stokestheorem (NAST): W C [ A ] = Z [ dµ ( ξ )] S exp (cid:18) ig Z S : ∂S = C h d ( m A A A ) + Ω K i(cid:19) , (2.24a)where we have defined a curvature two-form:Ω K := dω, (2.24b)and the product measure over the surface S :[ dµ ( ξ )] S := Y x ∈ S dµ ( ξ ( x )) . (2.24c)For the representation to be meaningful, the field ξ ( x ) must be continued over thesurface S inside the loop C , see [5] for detailed discussion on this issue.5 Coherent states
First, we construct the coherent state | ξ, Λ i corresponding to the coset representa-tives ξ ∈ G/ ˜ H . We follow the method of Feng, Gilmore and Zhang [30]. As inputs,we prepare the following:(a) The gauge group G has the Lie algebra G with the generators { T A } , whichobey the commutation relations[ T A , T B ] = if ABC T C , (3.1)where the f ABC are the structure constants of the Lie algebra. If the Lie algebrais semi-simple, it is more convenient to rewrite the Lie algebra in terms of the
Cartan basis { H j , E α , E − α } . There are two types of basic operators in theCartan basis, H j and E α . The operators H j may be taken as diagonal, while E α are the off-diagonal shift operators. They obey the commutation relations[ H j , H k ] =0 , (3.2)[ H j , E α ] = α j E α , (3.3)[ E α , E − α ] = α j H j , (3.4)[ E α , E β ] = N α ; β E α + β ( α + β ∈ R )0 ( α + β R, α + β = 0) , (3.5)where R is the root system, i.e., a set of root vectors { α , · · · , α r } , with r therank of G .(b) The Hilbert space V Λ is a carrier (the representation space) of the unitaryirreducible representation Γ Λ of G .(c) We use a reference state | Λ i within the Hilbert space V Λ , which can be normal-ized to unity: h Λ | Λ i = 1 . We define the maximal stability subgroup ( isotropy subgroup) ˜ H as a sub-group of G that consists of all the group elements h that leave the reference state | Λ i invariant up to a phase factor: h | Λ i = | Λ i e iφ ( h ) , h ∈ ˜ H. (3.6)Let H be the Cartan subgroup of G , i.e., the maximal commutative semi-simplesubgroup in G , and Let H be the Cartan subalgebra in G , i.e., the Lie algebrafor the group H . The maximal stability subgroup ˜ H includes the Cartan subgroup H = U (1) r , i.e., H = U (1) r ⊂ ˜ H. (3.7) Note that any compact semi-simple Lie group is a direct product of compact simple Lie group.Therefore, it is sufficient to consider the case of a compact simple Lie group. In the following weassume that G is a compact simple Lie group, i.e., a compact Lie group with no closed connectedinvariant subgroup. g ∈ G , there is a unique decomposition of g into a product oftwo group elements, g = ξh ∈ G, ξ ∈ G/ ˜ H, h ∈ ˜ H. (3.8)We can obtain a unique coset space G/ ˜ H for a given | Λ i . The action of arbitrarygroup element g ∈ G on | Λ i is given by | g, Λ i := g | Λ i = ξh | Λ i = ξ | Λ i e iφ ( h ) = | ξ, Λ i e iφ ( h ) . (3.9)The coherent state is constructed as | ξ, Λ i = ξ | Λ i . (3.10)This definition of the coherent state is in one-to-one correspondence with the cosetspace G/ ˜ H and the coherent states preserve all the algebraic and topological prop-erties of the coset space G/ ˜ H . The phase factor is unimportant in the followingdiscussion because we consider the expectation value of operators O in the coherentstate h g, Λ | O | g, Λ i = h ξ, Λ | O | ξ, Λ i . (3.11)If Γ Λ ( G ) is Hermitian, then H † j = H j and E † α = E − α . Every group element g ∈ G can be written as the exponential of a complex linear combination of diagonaloperators H j and off-diagonal shift operators E α . Let | Λ i be the highest-weight state,i.e., H j | Λ i = Λ j | Λ i , E α | Λ i = 0 ( α ∈ R + ) , (3.12)where R + ( R − ) is a subsystem of positive (negative) roots. Then the coherent stateis given by [30] | ξ, Λ i = ξ | Λ i = exp X β ∈ R − (cid:16) η β E β − ¯ η β E † β (cid:17) | Λ i , η β ∈ C , (3.13)such that(i) | Λ i is annihilated by all the (off-diagonal) shift-up operators E α with α ∈ R + , E α | Λ i = 0 ( α ∈ R + );(ii) | Λ i is mapped into itself by all diagonal operators H j , H j | Λ i = Λ j | Λ i ;(iii) | Λ i is annihilated by some shift-down operators E α with α ∈ R − , not by other E β with β ∈ R − : E α | Λ i = 0 (some α ∈ R − ); E β | Λ i = | Λ + β i (some β ∈ R − );and the sum P β is restricted to those shift operators E β which obey (iii).The coherent state spans the entire space V Λ . By making use of the the group-invariant measure dµ ( ξ ) of G which is appropriately normalized, we obtain Z | ξ, Λ i dµ ( ξ ) h ξ, Λ | = I, (3.14)which shows that the coherent states are complete, but in fact over-complete. Thecoherent states are normalized to unity: h ξ, Λ | ξ, Λ i = 1 . (3.15)7owever, the coherent states are non-orthogonal: h ξ ′ , Λ | ξ, Λ i 6 = 0 . (3.16)The resolution of identity (3.14) is very important to obtain the path integral formulaof the Wilson loop operator given later.The coherent states | ξ, Λ i are in one-to-one correspondence with the coset repre-sentatives ξ ∈ G/ ˜ H : | ξ, Λ i ↔ G/ ˜ H. (3.17)In other words, | ξ, Λ i and ξ ∈ G/ ˜ H are topologically equivalent.Here it is quite important to remark that for G=SU(2) the stability subgroupagrees with the maximal torus group, i.e., H = U (1) = ˜ H for G = SU (2) . (3.18)However, this is not necessarily the case for G = SU ( N ) ( N ≥ SU (2) coherent state In the case of SU (2), the rank is one, r = 1. The maximal stability group ˜ H agreeswith the maximal torus subgroup H = U (1) irrespective of the representation. Anirreducible spin- J representation {| J, M i} of SU (2) is characterized by the highestspin J : J = , , , , · · · . The root and weight diagrams are shown in Fig.2.First, we study the representation of the SU (2) coherent state by real numbers.Let | J, M i be an eigenvector of both the quadratic Casimir invariant J and thediagonal generator J (Cartan subalgebra) where M labels the eigenvalue of J : J | J, M i = J ( J + 1) | J, M i ( − J ≤ M ≤ J ) ,J | J, M i = M | J, M i . (3.19)Let | Λ i denote the highest weight state | Λ i = | J, J i of the spin- J representation .Spin coherent sates are a family of spin state {| n i} which is obtained by applyingthe rotation matrix R to the maximally polarized state | Λ i = | J, J i , | n i := R ( α, β, γ ) | J, J i = e − iJ α e − iJ β e − iJ γ | J, J i , (3.20)where J A ( A = 1 , ,
3) are three generators of su(2) and ( α, β, γ ) are Euler angles.It turns out that the spin coherent state is characterized by the unit vector n . It isshown that for J = , h n ( x ) | J A | n ( x ) i = J n A ( x ) , (3.21)where n ( x ) is a vector field of a unit length with three components: n ( x ) = ( n ( x ) , n ( x ) , n ( x )) = (sin β ( x ) cos α ( x ) , sin β ( x ) sin α ( x ) , cos β ( x )) . (3.22)We have the freedom to define γ arbitrary. This is a U(1) degrees of freedom.Therefore, the spin coherent states are in one-to-one correspondence with the (right)8oset SU (2) /U (1) ∼ = S where U (1) is generated by J (rotation about the third axisin the target space).Moreover, it is known that the coherent sates are not orthogonal. The overlap,i.e. the inner product of any two coherent states is evaluated as h n | n ′ i = n · n ′ ! J e − iJ Φ( n , n ′ ) , Φ( n , n ′ ) :=2 arctan ( cos[ ( β + β ′ )]cos[ ( β − β ′ )] tan α − α ′ !) + γ − γ ′ , (3.23)where γ and γ ′ correspond to the U(1) degrees of freedom mentioned above.The coherent states span the space of states of spin J . The measure of integrationis defined by dµ ( n ) := 2 J + 14 π δ ( n · n − d n = 2 J + 14 π sin βdβdα. (3.24)This is a Haar measure of the coset SU (2) /U (1), it is the area element on the two-sphere S .The state | n i can be expanded in a complete basis of the spin- J irreducible repre-sentation {| J, M i} . The coefficients of the expansion are the representation matrix, | n i = + J X M = − J | J, M i D ( J ) MJ ( n ) , (3.25)where D ( J ) MJ ( n ) is the Wigner D -function. The resolution of unity is given by Z dµ ( n ) | n ih n | = + J X M = − J | J, M ih J, M | = I, (3.26)where I is an identity operator. Hence the coherent state | n i forms the complete set,although it is not orthogonal. Thus the coherent states form an overcomplete basis.In particular, for the fundamental representation J = , an element in SU(2) iswritten in terms of three local variables ( α, β, γ ) corresponding to the Euler angles, R ( α, β, γ ) = e − iασ / e − iβσ / e − iγσ / = e − i ( α + γ ) cos β − e − i ( α − γ ) sin β e i ( α − γ ) sin β e i ( α + γ ) cos β ! ,α ∈ [0 , π ] , β ∈ [0 , π ] , γ ∈ [0 , π ] , (3.27)and (3.20) reads | n i = R ( α, β, γ ) ! = e − i γ e − i α cos β e i α sin β ! . (3.28)Making use of the explicit representation, we can make sure that the formulae (3.21),(3.22), (3.23), (3.24), (3.25) and (3.26) hold for J = 1 / - a L H - a a Figure 2: Root diagram and weight diagram of the fundamental representation of SU (2) where Λ is the highest weight of the fundamental representation.Next, we study the representation of the SU (2) coherent state by a complexnumber. The coherent state for F := SU (2) /U (1) is obtained as | J ; w i = ξ ( w ) | J, − J i = e ζJ + − ¯ ζJ − | J, − J i = 1(1 + | w | ) J e wJ + | J, − J i , (3.29)where | J, − J i := | J, M = − J i is the lowest state of | J, M i , J + = J + iJ , J − = J † + , w = ζ sin | ζ || ζ | cos | ζ | ∈ C , (3.30)and (1 + | w | ) − J is a normalization factor that ensures h J, w | J, w i = 1 , which isobtained from the Baker-Campbell-Hausdorff (BCH) formulas [29]. The invariantmeasure is given by dµ = 2 J + 14 π dwd ¯ w (1 + | w | ) . (3.31)For J A = σ A ( A = 1 , ,
3) with Pauli matrices σ A , we obtain J + = ! and ξ = e wJ + = w ! T ∈ F = CP = SU (2) /U (1) ∼ = S . (3.32)A representation of O (3) vector n is given by n A ( x ) = φ ∗ a ( x ) σ Aab φ b ( x ) ( a, b = 1 , , (3.33)which is equivalent to n = 2 ℜ ( φ ∗ φ ) , n = 2 ℑ ( φ ∗ φ ) , n = | φ | − | φ | . (3.34)The complex coordinate w obtained by the stereographic projection from the northpole is identical to the inhomogeneous local coordinates of CP when φ = 0, w = w (1) + iw (2) = n + in − n = 2 φ φ ∗ ( | φ | + | φ | ) − ( | φ | − | φ | ) = φ φ . (3.35)10he complex variable w is a CP = P ( C ) variable written as w = e iα cot β , (3.36)in terms of the polar coordinate ( β, α ) on S or Euler angles. While the stereographicprojection from the south pole leads to w = n + in n = φ φ ! ∗ = e iα tan β , (3.37)if φ = 0. The variable w is U (1) rotation invariant.Another representation of n is obtained by using the parameterization (3.32) ofthe F variable ξ : n A = h Λ | ξ ( w ) † σ A ξ ( w ) | Λ i = (cid:16) φ ∗ (cid:17) w ∗ ! σ A w ! φ ! . (3.38)This leads to n = | φ | ( w + w ∗ ) , n = − i | φ | ( w − w ∗ ) , n = | φ | (1 − ww ∗ ) . (3.39)Indeed, this agrees with (3.34) if w = ( φ φ ) ∗ . The entire space of F is covered by twocharts, CP = U ∪ U , U a = { ( φ , φ ) ∈ CP ; φ a = 0 } . (3.40)For more details, see Ref. [8, 9]. SU (3) coherent state The rank of SU(3) is two r = 2 and every representation is specified by the Dynkinindices [ m, n ]. The two-dimensional highest weight vector of the representation [ m, n ]is given by Λ = m h + n h , (3.41)using the highest weight h of a fundamental representation [1 ,
0] = and h ofanother fundamental (conjugate) representation [0 ,
1] = ∗ . The weight diagrams forfundamental representations and ∗ are given in Fig. 3. As the highest weight of we adopt the standard one: h = , √ ! =: ν . (3.42)As the highest weight of the conjugate representation ∗ , on the other hand, we chooseas in [9] h = , √ ! = − ν . (3.44) The following results hold also for other choices of h , e.g., we can choose h = (cid:18) , √ (cid:19) =: ν , h = (cid:18) , − √ (cid:19) = − ν , Λ = (cid:18) m + n , m − n √ (cid:19) . (3.43) H - a(1) - a(2) Ln n n - a(3) - a(2) (cid:13) H H L(cid:13)(cid:13)
Figure 3: The weight diagram and root vectors required to define the coherent statein the fundamental representations [1 ,
0] = , [0 ,
1] = ∗ of SU (3) where ~ Λ = ~h = ν := ( , √ ) is the highest weight of the fundamental representation, and the otherweights are ν := ( − , √ ) and ν := (0 , − √ ). H H - a(3)a(2) - a(1) - a(2) Figure 4: The root diagram of SU (3), where positive root vectors are given by α (1) =(1 , α (2) = ( , √ ), and α (3) = ( − , √ ). The two simple roots are given by α := α = α (1) , and α := β = α (3) .Then the highest weight of [ m.n ] is given by Λ = (Λ , Λ ) = m , m + 2 n √ ! . (3.45)The generators of SU(3) in the Cartan basis are written as { H , H , E α , E β , E α + β ,E − α , E − β , E − α − β } , where α = α and α = β are the two simple roots. (See Fig. 4for the explicit choice.)If mn = 0, ( m = 0 or n = 0), the maximal stability group ˜ H is given by ˜ H = U (2) with generators { H , H , E β , E − β } ( minimal case I, in which the coset G/ ˜ H is minimal). Such a degenerate case occurs when the highest-weight vector ~ Λ isorthogonal to some root vectors (see Fig. 3). If mn = 0 ( m = 0 and n = 0), H isthe maximal torus group ˜ H = U (1) × U (1) with generators { H , H } ( maximal caseII, in which the coset G/ ˜ H is maximal). This is a non-degenerate case (see Fig.5).Therefore, for the highest weight Λ in the minimal case (I), the coset G/ ˜ H is givenby SU (3) /U (2) = SU (3) / ( SU (2) × U (1)) = CP , (3.46)whereas in the maximal case (II), SU (3) / ( U (1) × U (1)) = F . (3.47)12 H - a(3) - a(1) - a(2)L Figure 5: The weight diagram and root vectors required to define the coherent statein the adjoint representation [1 ,
1] = of SU (3), where Λ = ( , √ ) is the highestweight of the adjoint representation.Here, CP n is the complex projective space and F n is the flag space [31]. Therefore, thetwo fundamental representations of SU(3) belong to the minimal case (I), and hencethe maximal stability group is U (2) , rather than the maximal torus group U (1) × U (1).The implications of this fact to the mechanism of quark confinement is discussedin subsequent sections.The coherent state for F = SU (3) /U (1) is given by | ξ, Λ i = ξ ( w ) | Λ i := V † ( w ) | Λ i , (3.48)with the highest- (lowest-) weight state | Λ i , i.e., | ξ, Λ i = exp X α ∈ R + ( ζ α E − α − ζ ∗ α E †− α ) | Λ i = e − K ( w,w ∗ ) exp X α ∈ R + τ α E − α | Λ i , (3.49)where e − K is the normalization factor obtained from the K¨ahler potential K ( w, w ∗ )(explained below): K ( w, w ∗ ) := ln[(∆ ( w, w ∗ )) m (∆ ( w, w ∗ )) n ] , ∆ ( w, w ∗ ) := 1 + | w | + | w | , ∆ ( w, w ∗ ) := 1 + | w | + | w − w w | . (3.50)The coherent state | ξ, Λ i is normalized, so that h ξ, Λ | ξ, Λ i = 1.It is shown [9] that the inner product is given by h ξ ′ , Λ | ξ, Λ i = e K ( w,w ∗′ ) e − [ K ( w ′ ,w ∗′ )+ K ( w,w ∗ )] , (3.51)where K ( w, w ∗′ ) := ln[1 + w ∗ ′ w + w ∗ ′ w ] m [1 + w ∗ ′ w + ( w ∗ ′ − w ∗ ′ w ∗ ′ )( w − w w )] n . (3.52)Note that K ( w, w ∗′ ) reduces to the K¨ahler potential K ( w, w ∗ ) when w ′ = w , inagreement with the normalization h ξ, Λ | ξ, Λ i = 1.13t follows from the general formula that the SU (3) invariant measure is given (upto a constant factor) by dµ ( ξ ) = dµ ( w, w ∗ ) = D ( m, n )[(∆ ) m (∆ ) n ] − Y α =1 dw α dw ∗ α , (3.53)where D ( m, n ) = ( m + 1)( n + 1)( m + n + 2) is the dimension of the representation.For the choice of shift-up ( E + i ) or shift-down ( E − i ) operators E ± := λ ± iλ √ , E ± := λ ± iλ √ , E ± := λ ± iλ √ , (3.54)with the Gell-Mann matrices λ A ( A = 1 , · · · , " X i =1 τ i E − i = w w w T ∈ F = SU (3) /U (1) , (3.55)where we have used the abbreviation E ± i ≡ E ± α ( i ) ( i = 1 , , w = τ √ , w = τ √ τ τ , w = τ √ , (3.56)or conversely τ = √ w , τ = √ (cid:18) w − w w (cid:19) , τ = √ w . (3.57)An element of CP can be expressed using two complex variables, e.g., w and w : exp " X i =1 τ i E − i = w w T ∈ CP = SU (3) /U (2) , (3.58)The complex projective space CP is covered by three complex planes C throughholomorphic maps [32] (see e.g., [9]).The parameterization of SU (3) in terms of eight angles is possible also in SU (3),just as SU (2) is parameterized by three Euler angles (see Ref. [33]).The SU(N) case can be treated in the similar way, see Ref. [9] for details. Here we introduce H given by H := Λ · H = r X j =1 Λ j H j , (4.1)where H j ( j = 1 , , · · · , r ) are the generators from the Cartan subalgebra ( r = N − G = SU ( N )) and r -dimensional vector Λ j ( j = 1 , , · · · , r )is the highest weight of the representation in which the Wilson loop is considered.14or fundamental representations, we can write e ff := 1tr( ) + 2 H , H = 12 e ff − ) ! , (4.2)where we introduce a matrix e ff which has only one non-vanishing f f diagonalelement with the value one:( e ff ) ab = δ fa δ fb (no sum over f ) . (4.3)A highest-weight state | Λ i is the common (normalized) eigenvector of H , H , · · · , H r with the eigenvalue Λ , Λ , · · · , Λ r , i.e., H j | Λ i = Λ j | Λ i ( j = 1 , · · · , r = N − . (4.4)For SU(2), every representation is specified by an half integer J :Λ = J = 12 , , , , · · · , H = σ , (4.5)and the fundamental representation J = of SU(2) leads to H = 12 (cid:18) e − (cid:19) = 12 diag (cid:18) , − (cid:19) = 12 σ . (4.6)For SU(3), the explicit form of H for the fundamental representations reads usingthe diagonal set of the Gell-Mann matrices λ and λ : H = 12 (Λ λ + Λ λ )= 12 m + n − m + n
00 0 − m − n = 12 diag (cid:18) m + n , − m + n , − m − n (cid:19) . (4.7)We enumerate all fundamental representations : [1,0 ]: Λ = , √ ! := ν , H = 12 diag (cid:18) , − , − (cid:19) , (4.8a)[-1,1 ]: Λ = − , √ ! := ν , H = 12 diag (cid:18) − , , − (cid:19) , (4.8b)[0,-1 ]: Λ = , − √ ! := ν , H = 12 diag (cid:18) − , − , (cid:19) = − √ λ , (4.8c)and their conjugates ∗ For another choice of h = − ν , the same results are obtained if the following replacement isperformed. [-1,1] → [0,-1], [0,-1] → [-1,1], [0,1] → [1,-1], [1,-1] → [0,1]. Λ = , √ ! = − ν , H = 12 diag (cid:18) − , − , (cid:19) = − √ λ , (4.9a)[1,-1 ]: Λ = , − √ ! = − ν , H = 12 diag (cid:18) − , , − (cid:19) , (4.9b)[-1,0 ]: Λ = − , − √ ! = − ν , H = 12 diag (cid:18) , − , − (cid:19) . (4.9c)For three fundamental representations (4.8a), (4.8b) and (4.8c), the eigenvectors(4.4) are found to be | Λ i = (1 , , T , | Λ i = (0 , , T , & | Λ i = (0 , , T , (4.10)respectively.Now we check that the coherent state indeed satisfies the desired properties. Inwhat follows, we restrict our consideration to the fundamental representations ofSU(N), N ≥
3. For SU(2), any representation will be discussed separately.As a reference state or a highest-weight state for a fundamental representation ofSU(N), we choose, e.g., f = N | Λ i = (0 , · · · , , T = or h Λ | = (0 , · · · , , , (4.11)which yields a projection operator: | Λ i h Λ | = e NN = · · · · · · · · · . (4.12)First, we show the completeness Z | ξ, Λ i dµ ( ξ ) h ξ, Λ | = /d R . (4.13) The following relationships hold even if we replace ξ ∈ G/ ˜ H by a general element g ∈ G , sincethey hold on a reference state | Λ i . ab element of the left-hand side reads (cid:18)Z | ξ, Λ i dµ ( ξ ) h ξ, Λ | (cid:19) ab = Z dµ ( ξ ) (cid:16) ξ | Λ i h Λ | ξ † (cid:17) ab = Z dµ ( ξ ) (cid:16) ξ e ff ξ † (cid:17) ab = Z dµ ( ξ )( ξ ) ac ( e ff ) cd ( ξ † ) db = 1 d R δ ab δ cd ( e ff ) cd = 1 d R δ ab , (4.14)where we have used the integration formula for the Haar measure dU = dµ ( G ) ofSU(N): Z dU U ac ( R ) U † db ( R ′ ) = 1 d R δ ab δ cd δ RR ′ . (4.15)Second, we show Z dµ ( ξ ) h ξ, Λ | O | ξ, Λ i = tr( O ) / tr( ) . (4.16)In fact, we have Z dµ ( ξ ) h ξ, Λ | O | ξ, Λ i = Z dµ ( ξ ) h Λ | ξ † O ξ | Λ i = Z dµ ( ξ )( ξ † O ξ ) ff = Z dµ ( ξ )tr( ξ † O ξ e ff )= Z dµ ( ξ )tr( O ξ e ff ξ † )=tr( O Z dµ ( ξ ) ξ e ff ξ † )=tr( O d R ) = tr( O ) d R , (4.17)where we have used the previous result (4.14).Third, the normalization condition is trivial: h ξ, Λ | ξ, Λ i = D Λ | ξ † ξ | Λ E = h Λ | Λ i = 1 . (4.18)Finally, an important relationship for the matrix element is derived. By using thecoherent state for fundamental representations of SU(N), the matrix element of anyLie algebra valued operator O in the coherent state is cast into the form of the trace: h ξ, Λ | O | ˜ ξ, Λ i = h Λ | ξ † O ˜ ξ | Λ i =( ξ † O ˜ ξ ) ff (no sum over f )=tr[ ξ † O ˜ ξ e ff ] (4.19a)=tr[ O ˜ ξ e ff ξ † ] . (4.19b)17his is further rewritten as h ξ, Λ | O | ˜ ξ, Λ i =tr[ O ˜ ξ e ξ † ]=tr " ξ † O ˜ ξ ) + 2 H ! (4.20a)=tr " O ) ˜ ξξ † + 2 ˜ ξ H ξ † ! . (4.20b)The diagonal matrix element is cast into [30, 31] h ξ ( x ) , Λ | O ( x ) | ξ ( x ) , Λ i = tr ( O ( x ) " ) + 2 m ( x ) . (4.21)where we have introduced a new field m ( x ) having its value in the Lie algebra G = su ( N ) by m ( x ) := ξ ( x ) H ξ ( x ) † = r X j =1 Λ j ξ ( x ) H j ξ ( x ) † . (4.23)We can introduce the normalized color field n ( x ) by n ( x ) := s NN − m ( x ) = s NN − ξ ( x ) H ξ ( x ) † = s NN − r X j =1 Λ j ξ ( x ) H j ξ ( x ) † . (4.24)In particular, we have h Λ | O ( x ) | Λ i = tr (" ) + 2 H O ( x ) ) . (4.25)Moreover, the traceless O ( x ) obeys more simple relations: h ξ ( x ) , Λ | O ( x ) | ξ ( x ) , Λ i =2tr { m ( x ) O ( x ) } , h Λ | O ( x ) | Λ i =2tr {H O ( x ) } . (4.26)Note that the m field defined from the coset element ξ ∈ G/ ˜ H is the same as thatdefined from the original group g ∈ G : m ( x ) = g ( x ) H g ( x ) † . (4.27)This is because g ( x ) H g ( x ) † = ξ ( x ) h ( x ) H h ( x ) † ξ ( x ) † = ξ ( x ) H ξ ( x ) † , (4.28)which follows from a fact: h ( x ) H h ( x ) − = H ⇐⇒ [ h ( x ) , H ] = 0 , (4.29) The m ( x ) field can be normalized by multiplying a factor q NN − , since2tr[ m ( x ) m ( x )] = 2tr( HH ) = 2Λ j Λ k tr( H j H k ) = Λ j = N − N . (4.22)
18y introducing the color fields n j defined by n j ( x ) := g ( x ) H j g † ( x ) , (4.30)the m field is written as a linear combination m ( x ) = g ( x ) H g ( x ) † = r X j =1 Λ j n j ( x ) . (4.31)For SU(2), every representation is specified by a half integer J and the color fieldis unique, n ( x ) = n ( x ) , (4.32)and m ( x ) = Λ n ( x ) = ± J n ( x ) = ± J ξ ( x ) σ ξ ( x ) † . (4.33)For J = , Using (3.21), we obtain A A n A = tr[ σ G † A G ] , (4.34)where σ is the third Pauli matrix. Indeed, we have12 tr[ σ G † A G ] = 12 [( G † A G ) − ( G † A G ) ] = ( G † A G ) = (cid:16) (cid:17) [ G † A G ] ! = h n | A A T A | n i = h n | T A | n i A A = 12 n A A A , (4.35)where we have used the fact that the matrix ( G † A G ) is traceless, i.e., ( G † A G ) +( G † A G ) = 0, since tr[ G † A G ] = tr[ A GG † ] = tr[ A ] = A A tr[ T A ] = 0. In otherwords, n has the adjoint orbit representation: n A ( x ) = tr( σ G † ( x ) T A G ( x )) , n A ( x ) T A = G ( x ) T G † ( x ) . (4.36)Using (3.27), we can see that the unit vector n ( x ) defined by (4.36) is indeed equalto (3.22). Indeed, the adjoint orbit representation leads to the consistent result:tr( σ G † A G ) =tr( Gσ G † A ) = 2tr( n A T A A B T B )=2 n A A B tr( T A T B ) = n A A B δ AB = n A A A . (4.37)For SU(3), the m field is a linear combination of two color fields: n ( x ) = g ( x ) λ g † ( x ) , n ( x ) = g ( x ) λ g † ( x ) . (4.38)The m field reads for [1 ,
0] and [ − , m ( x ) = ± " n ( x ) + 1 √ n ( x ) = ± √ " √ n ( x ) + 12 n ( x ) , (4.39a)19or [0 , −
1] and [0 , m ( x ) = ± " − n ( x ) + 1 √ n ( x ) = ± √ " − √ n ( x ) + 12 n ( x ) . (4.39b)In particular, for [ − ,
1] and [1 , − = 0 and hence the m field is written usingonly n ( x ): m ( x ) = ± " − √ n ( x ) = ± √ − n ( x )] . (4.39c)For SU(N), we can introduce N − n j ( x ) ( j = 1 , · · · , N −
1) corre-sponding to the degrees of freedom of the maximal torus group U (1) N − of SU(N).This is just the way adopted in the conventional approach. However, this is not nec-essarily effective to see the physics extractable from the Wilson loop. This is becauseonly the specific combination m ( x ) of the color fields n j ( x ) has a physical meaningas shown in the above and this nice property of m ( x ) will be lost once m ( x ) is sepa-rated into the respective color field, except for the SU(2) case in which m ( x ) agreeswith the only one color field n ( x ) of SU(2). In view of this, only the last color field n N − ( x ) is enough for investigating quark confinement through the Wilson loop infundamental representations of SU(N). Note that m ( x ) = m A ( x ) T A , m A ( x ) = 2tr( m ( x ) T A ) = 2tr( g ( x ) H g † ( x ) T A ) , (4.40)for the normalization tr( T A T B ) = δ AB . For three fundamental representations(4.8a), (4.8b) and (4.8c) of SU(3), m A ( x ) is equal to the first, second and thirddiagonal elements of g ( x ) T A g † ( x ) respectively: m A ( x ) = ( g † ( x ) T A g ( x )) ff ( f = 1 , , . (4.41)This is checked easily, e.g., for [1 , m A ( x ) =2tr( H g † ( x ) T A g ( x ))= 23 ( g † ( x ) T A g ( x )) −
13 ( g † ( x ) T A g ( x )) −
13 ( g † ( x ) T A g ( x )) =( g † ( x ) T A g ( x )) , (4.42)where we have used a fact that g † ( x ) T A g ( x ) is traceless. Therefore, we have m A ( x ) = D Λ | g † ( x ) T A g ( x ) | Λ E = D g ( x ) , Λ | T A | g ( x ) , Λ E = D ξ ( x ) , Λ | T A | ξ ( x ) , Λ E . (4.43)using the highest weight state | Λ i of the respective fundamental representation. Thestate | ξ ( x ) , Λ i is regarded as the coherent state describing the subspace corresponding This is called the minimal option proposed in [34]. Indeed, the above combinations (4.39a),(4.39b) and (4.39c) correspond to 6 minimal cases discussed in [34]. A unit vector n ( x ) introducedin (4.24) and ref. [34] is related to m as √ m ( x ) = n ( x ) = (cos ϑ ( x )) n ( x ) + (sin ϑ ( x )) n ( x ) where ϑ ( x ) denotes the angle of a weight vector in the weight diagram measured anticlockwise from the H axis. Here (4.39a), (4.39b) and (4.39c) correspond to ϑ ( x ) = π ( π ) π ( π ), and π ( π )respectively.
20o the subgroup G/ ˜ H = SU (3) /U (2) ≃ CP , the two-dimensional complex projectivespace. This is also assured in the following way. The component m A of m is rewrittenas m A ( x ) = φ ∗ a ( x )( T A ) ab φ b ( x ) ( a, b = 1 , , , (4.44)by introducing the CP variable φ a ( x ): φ a ( x ) := ( g ( x ) | Λ i ) a . (4.45)The complex field φ a ( x ) is indeed the CP variable, since there are only two inde-pendent complex degrees of freedom among three variables φ a ( x ) which are subjectto the constraint: φ † ( x ) φ ( x ) = X a =1 φ ∗ a ( x ) φ a ( x ) = h Λ | g † ( x ) g ( x ) | Λ i = h Λ | Λ i = 1 . (4.46)For more details, see [9]. This result suggests that the Wilson loop operator infundamental representations of SU(N) can be studied by the CP N − valued fieldeffectively, rather than F N − . Applying (4.26) to (2.22b) and (2.22c) in (2.22a), we obtain m A ( x ) =2tr n H ξ † ( x ) T A ξ ( x ) o = 2tr n m ( x ) T A o , (5.1) ω ( x ) = − n H ig − ξ † ( x ) dξ ( x ) o . (5.2)This leads to another representation of (2.22a): The path ordering P in the Wilsonloop operator W C [ A ] := tr (cid:20) P exp (cid:26) ig I C A ( x ) (cid:27)(cid:21) / tr( ) , A ( x ) := A Aµ ( x ) T A dx µ , (5.3)can be eliminated at the price of introducing integrations over all gauge transforma-tions along the loop C : W C [ A ] = Z D C G exp (cid:26) ig I C H [ G † ( x ) A ( x ) G ( x ) − ig − G † ( x ) dG ( x )]) (cid:27) , (5.4)where D C G is the product of the invariant Haar measure dG ( x ): D C G := Y x ∈ C dG ( x ) . (5.5)Then we can cast the line integral to the surface integral using the usual Stokestheorem: W C [ A ] = Z D C G exp (cid:26) ig I C A (cid:27) = Z D Σ G exp (cid:26) ig Z Σ: ∂ Σ= C dA (cid:27) , (5.6) The factor 2 in front of the trace is due to the normalization tr( T A T B ) = δ AB adopted in thispaper. V is the one-form defined by A := A µ dx µ , A µ ( x ) := 2tr( m ( x ) A µ ( x )) − H ig − G † ( x ) ∂ µ G ( x )) . (5.7)Therefore, the Wilson loop operator originally defined in the line integral formalong the closed loop C is cast into the surface integral form over the surface Σbounded by C . This version of the non-Abelian Stokes theorem is called the Diakonov-Petrov (DP) type, since this form was for the first time derived by Diakonov andPetrov for the SU(2) gauge group [3]. The DP version of the non-Abelian Stokestheorem is regarded as the path integral representation of the Wilson loop operator,which enables us to extend the non-Abelian Stokes theorem to more general gaugegroups by using the coherent state representation, as demonstrated in [8, 9].The curvature two-form is calculated as F = dA = 12 ( ∂ µ A ν − ∂ ν A µ ) dx µ ∧ dx ν = F (1) + F (2) . (5.8)Here the first term reads( ∂ µ A ν − ∂ ν A µ ) (1) ( x ) = ∂ µ m ( x ) A ν ( x )) − ∂ ν m ( x ) A µ ( x )) . (5.9)It can be shown [34] that the original SU(N) gauge field A µ ( x ) is decomposed as A µ ( x ) = V µ ( x ) + X µ ( x ) , (5.10)such that V µ ( x ) transforms under the gauge transformation just like the original gaugefield A µ ( x ), while X µ ( x ) transforms like an adjoint matter field: V µ ( x ) → V ′ µ ( x ) = Ω( x )( V µ ( x ) + ig − ∂ µ )Ω † ( x ) , (5.11) X µ ( x ) → X ′ µ ( x ) = Ω( x ) X µ ( x )Ω † ( x ) , (5.12)by way of a single m or n field which transforms as m ( x ) → m ′ ( x ) = Ω( x ) m ( x )Ω † ( x ) . (5.13)Moreover, the field V µ ( x ) can be further decomposed as V µ ( x ) = C µ ( x ) + B µ ( x ) , (5.14)such that C µ ( x ) and B µ ( x ) are the parallel and perpendicular to m ( x ) in the sensethat tr( m ( x ) C µ ( x )) = tr( m ( x ) A µ ( x )) , tr( m ( x ) B µ ( x )) = 0 , (5.15)in addition to tr( m ( x ) X µ ( x )) = 0 . (5.16)The decomposed fields C µ ( x ), B µ ( x ) and X µ ( x ) are explicitly written in terms of A µ ( x ) and m ( x ). They are regarded as those obtained by the (non-linear) change ofvariables from the original gauge field. For the SU(2) case, this is well-known as theCho-Faddeev-Niemi-Shabanov (CFNS) decomposition [20, 21, 22]. However, SU(N)22ase needs further discussions. See [34] and Appendix D for the details. Consequently,we obtain( ∂ µ A ν − ∂ ν A µ ) (1) ( x ) = ∂ µ m ( x ) C ν ( x )) − ∂ ν m ( x ) C µ ( x )) . (5.17)On the other hand, the second term reads( ∂ µ A ν − ∂ ν A µ ) (2) ( x ) =2tr( H{ ∂ µ [ ig − G † ( x ) ∂ ν G ( x )] − ∂ ν [ ig − G † ( x ) ∂ µ G ( x )] } )=2tr( H{ ig − ∂ µ G † ( x ) ∂ ν G ( x ) − ig − ∂ ν G † ( x ) ∂ µ G ( x ) } )+ 2tr( H ig − G † ( x )[ ∂ µ , ∂ ν ] G ( x ))=2tr( H ig [ ig − G † ( x ) ∂ µ G ( x ) , ig − G † ( x ) ∂ ν G ( x )]) , (5.18)or ( ∂ µ A ν − ∂ ν A µ ) (2) ( x ) =2tr( H ig [ ig − G † ( x ) ∂ µ G ( x ) , ig − G † ( x ) ∂ ν G ( x )])=2tr( m ( x ) ig [ ig − ∂ µ G ( x ) G † ( x ) , ig − ∂ ν G ( x ) G † ( x )])=2tr( m ( x ) ig [ B µ ( x ) , B ν ( x )])=2tr( m ( x ) F µν [ B ]( x )) . (5.19)where we have used B µ ( x ) = − ig − ∂ µ G ( x ) G † ( x ) = ig − G ( x ) ∂ µ G † ( x ) , (5.20)and F µν [ B ]( x ) = ig [ B µ ( x ) , B ν ( x )] . (5.21)Therefore, we have arrived at the final expression: F µν ( x ) := ∂ µ A ν ( x ) − ∂ ν A µ ( x )=2tr( m ( x ) F µν [ V ]( x ))=2tr( m ( x ) F µν [ C ]( x )) + 2tr( m ( x ) F µν [ B ]( x ))= ∂ µ m ( x ) C ν ( x )) − ∂ ν m ( x ) C µ ( x )) + 2tr(4 ig − m ( x )[ ∂ µ m ( x ) , ∂ ν m ( x )]) , (5.22)where we have used the relation shown in (E.21):tr( m ( x ) F µν [ B ]( x )) = 4tr( ig − m ( x )[ ∂ µ m ( x ) , ∂ ν m ( x )]) . (5.23)Thus the Wilson loop operator originally defined in terms of A µ ( x ) has been rewrittenin terms of only the variable V µ ( x ) and the variable X µ ( x ) has disappeared in thefinal expression: W C [ A ] = Z D Σ G exp (cid:26) ig Z Σ: ∂ Σ= C m ( x ) F [ V ]( x )) (cid:27) , m ( x ) = G ( x ) H G ( x ) † . (5.24) Hereafter, we omit the last term 2tr( H ig − G † ( x )[ ∂ µ , ∂ ν ] G ( x )). m fieldagrees with the color field n up to a numerical factor: m ( x ) = − n ( x ) , (5.25)which reproduces the well-known field strength of the form: F µν ( x ) = −
12 tr(2 n ( x ) F µν [ V ]( x ))= −
12 [ ∂ µ n ( x ) C ν ( x )) − ∂ ν n ( x ) C µ ( x )) + 2tr( ig − n ( x )[ ∂ µ n ( x ) , ∂ ν n ( x )])] . (5.26)This is rewritten into another manifestly gauge-invariant form by using the covariantderivative D [ A ] µ n ( x ) := ∂ µ n ( x ) − ig [ A µ ( x ) , n ( x )]: F µν ( x ) = −
12 2tr { n ( x ) F µν [ A ]( x ) + ig − n ( x )[ D [ A ] µ n ( x ) , D [ A ] ν n ( x )] } , (5.27)since the gauge transformation is given by n ( x ) → Ω( x ) n ( x )Ω † ( x ) , F µν [ A ]( x ) → Ω( x ) F µν [ A ]( x )Ω † ( x ) ,D [ A ] µ n ( x ) → Ω( x ) D [ A ] µ n ( x )Ω † ( x ) . (5.28)Under the identification between the color field and the (normalized) Higgs field: n A ( x ) ↔ ˆ φ A ( x ) := φ A ( x ) / | φ ( x ) | ( | φ ( x ) | := q { φ ( x ) φ ( x ) } ) , (5.29)the antisymmetric tensor of rank two f µν = 2 F µν in the Yang-Mills theory has thesame form as the ’t Hooft–Polyakov tensor in the Georgi-Glashow model which leadsto the SU(2) gauge-invariant ’t Hooft–Polyakov magnetic monopole. This fact sug-gests that the gauge-invariant magnetic monopole is defined even in the pure Yang-Mills theory without the Higgs field, as shown explicitly in the next section.For SU(3) in the fundamental representation, the simplest choice is m ( x ) = − √ h ( x ) h ( x ) := n ( x ) = G ( x ) λ G † ( x ) . (5.30)This leads to the field strength F µν ( x ) = − √ h ( x ) F µν [ V ]( x ))= − √ ∂ µ h ( x ) C ν ( x )) − ∂ ν h ( x ) C µ ( x )) + 2tr( 43 ig − h ( x )[ ∂ µ h ( x ) , ∂ ν h ( x )])] . (5.31)24or SU(N) in the fundamental representation, we can choose m ( x ) = − s N − N h ( x ) , h ( x ) := n N − ( x ) = G ( x ) H N − G † ( x ) . (5.32)This leads to the field strength F µν ( x ) = − s N − N tr(2 h ( x ) F µν [ V ]( x ))= − s N − N { ∂ µ h ( x ) C ν ( x )) − ∂ ν h ( x ) C µ ( x ))+ 2tr( 2( N − N ig − h ( x )[ ∂ µ h ( x ) , ∂ ν h ( x )]) } . (5.33)Finally, we consider the Abelian limit G → U (1) N − and the Abelian case G = U (1). In the Abelian case G = U (1), we need neither taking the trace nor insertingthe complete sets. The Haar measure disappears from the representation. The off-diagonal elements do not exist. Therefore, the non-Abelian Stokes theorem reproducesthe usual Stokes theorem. Let σ = ( σ , σ ) = ( τ, σ ) be the world sheet coordinates on the two-dimensionalsurface Σ C which is bounded by the Wilson loop C , while let x ( σ ) be the targetspace coordinate of the surface Σ C in R D . First of all, we rewrite the surface integral R Σ C dS µν F µν into the volume integral: Z Σ C : ∂ Σ C = C F = Z Σ C dS µν ( x ( σ )) F µν ( x ( σ ))= Z d D xF µν ( x )Θ Σ C µν ( x ) , (6.1)where we have introduced an antisymmetric tensor of rank two,Θ Σ C µν ( x ) := Z Σ C dS µν ( x ( σ )) δ D ( x − x ( σ )) . (6.2)We call Θ Σ C µν ( x ) the vorticity tensor with the support on the surface Σ C spannedby the Wilson loop C . Here the surface element dS µν of Σ C is rewritten using theJacobian J from x µ , x ν to σ , σ = τ, σ as dS µν ( x ( σ )) = 12 d σǫ ab ∂x µ ∂σ a ∂x ν ∂σ b = 12 d σJ µν ( σ ) , J µν ( σ ) := ǫ ab ∂x µ ∂σ a ∂x ν ∂σ b = ∂ ( x µ , x ν ) ∂ ( σ , σ ) . (6.3)Second, it is rewritten in terms of two conserved currents, the “magnetic-monopolecurrent” k and the “electric current” j , defined by k := δ ∗ f = ∗ df,j := δf, (6.4)25here f is the two-form defined by f = q NN − F (5.33): f µν ( x ) = ∂ µ h ( x ) A ν ( x )) − ∂ ν h ( x ) A µ ( x ))+ 2tr( 2( N − N ig − h ( x )[ ∂ µ h ( x ) , ∂ ν h ( x )]) . (6.5)In fact, we find Z Σ: ∂ Σ= C f = Z d D x Θ µν Σ ( x ) f µν ( x ):=(Θ Σ , f )=( ∗ Θ Σ , ∗ f )=( ∗ Θ Σ , ∆ − ( dδ + δd ) ∗ f )=( ∗ Θ Σ , ∆ − dδ ∗ f ) + ( ∗ Θ Σ , ∆ − δd ∗ f )=( δ ∆ − ∗ Θ Σ , δ ∗ f ) + (Θ Σ , ∗ ∆ − δ ∗ δf )=( δ ∆ − ∗ Θ Σ , δ ∗ f ) + (Θ Σ , ∆ − dδf )=( δ ∆ − ∗ Θ Σ , k ) + (∆ − δ Θ Σ , j ) , (6.6)where we have used ∗∗ = ( − p ( D − p ) and δ = ( − p ∗ d ∗ ( D =odd), δ = − ∗ d ∗ ( D =even) when applied to p -form in D -dimensional Euclidean space and ∆ is the D -dimensional Laplacian (d’Alembertian)∆ := dδ + δd .In this way we obtain another expression of the NAST for the Wilson loop operatorin the fundamental representation of SU(N): W C [ A ] = Z [ dµ ( ξ )] Σ exp ig s N − N ( k, Ξ Σ ) + ig s N − N ( j, N Σ ) , (6.7)where Ξ Σ and N Σ are defined byΞ Σ := ∗ d Θ Σ ∆ − = δ ∗ Θ Σ ∆ − , N Σ := δ Θ Σ ∆ − . (6.8)For SU(2), in particular, arbitrary representation is characterized by J = , , , , , · · · .The Wilson loop operator in the representation J for SU(2) obey the non-AbelianStokes theorem: W C [ A ] = Z [ dµ ( ξ )] Σ exp { igJ ( k, Ξ Σ ) + igJ ( j, N Σ ) } . (6.9)This agrees with (6.7) for a fundamental representation J = of SU(2). Thus, theWilson loop can be expressed by the electric current j µ and the magnetic monopolecurrent k µ which depend on the group G , while N Σ and Ξ Σ do not depend on thegroup G and depend only on the geometry of the surface Σ.Note that k is ( D − j is one-form in D = d + 1 dimensions. N Σ isone-form in any dimension having the component: N µ Σ ( x ) = ∂ xν Z d y Θ µν Σ ( y )∆ − D ) ( y − x )= 12 ∂ xν Z Σ d S µν ( x ( σ ))∆ − D ) ( x ( σ ) − x ) . (6.10)26igure 6: The magnetic monopole q m at x and the solid angle Ω Σ ( x ) at x subtendedby the surface Σ bounding the Wilson loop C .Whereas, Ξ Σ is ( D − D = d + 1 dimensional case. The explicit form is ob-tained by using the D -dimensional Laplacian (d’Alembertian) ∆ ( D ) in the spacetimedimension D in question as follows.We show below that the factor W mC := exp[ ig q N − N ( k, Ξ Σ )] has geometrical andtopological meanings.For D = 3, Ξ Σ is the zero-form with the component:Ξ Σ ( x ) = 12 ǫ νρσ ∂ xν Z d y Θ Σ ρσ ( y )∆ − ( y − x )= 12 ǫ νρσ Z Σ C d S ρσ ( x ( σ )) ∂ xν ∆ − ( x ( σ ) − x )= 12 ǫ νρσ Z Σ C d S ρσ ( x ( σ )) ∂ xν π | x − x ( σ ) | , (6.11)while k is also zero-form, i.e, the magnetic monopole density function: k = ρ m = 12 ǫ νρσ ∂ ν f ρσ . (6.12)It is known that the solid angle Ω Σ ( x ) under which the surface Σ shows up to anobserver at the point x is written asΩ Σ ( x ) = ∂∂x µ Z Σ C d S µ ( y ) 1 | x − y | = 12 ǫ µαβ ∂∂x µ Z Σ C d S αβ ( y ) 1 | x − y | =4 π Ξ Σ ( x ) , (6.13)where d S µ := 12 ǫ µαβ d S αβ . (6.14)In the case when Σ is a closed surface surrounding the point x , we have Ω Σ = 4 π ,since due to the Gauss lawΩ Σ ( x ) = I Σ d S µ ( y ) ∂∂x µ | x − y | = Z V : ∂V =Σ d y ∂∂x µ ∂∂x µ | x − y | = Z V d y πδ ( x − y ) = 4 π. (6.15)27 Figure 7: Quantization of the magnetic charge. The difference between the surfaceintegrals of f over two surfaces Σ and Σ with the same boundary C is equal to thesurface integral R Σ f of f over one closed surface Σ = Σ + Σ . Here the direction ofthe normal vector to the surface Σ must be consistent with the direction of the lineintegral over C . The magnetic charge Q m is non-zero if the magnetic monopole existsinside Σ, otherwise it is zero.This is a standard result for the total solid angle in three dimensions.Thus, for D = 3, Ξ Σ is the normalized solid angle (Ω Σ divided by the total solidangle 4 π ) and the exponential factor in SU(2) NAST reads W mC = exp (cid:20) ig Z d xk ( x )Ξ Σ ( x ) (cid:21) = exp " ig Z d xρ m ( x ) Ω Σ ( x )4 π . (6.16)See Fig. 6.We examine the relationship between the magnetic charge and its quantizationcondition. In order to extract the information on the magnetic charge through thenon-Abelian Stokes theorem for the Wilson loop operator, we must consider the in-tegration of the curvature two-form f over the closed surface Σ. In fact, the Diracquantization condition q m = 4 πg − n for the magnetic charge q m is obtained for SU(2)from the condition of the non-Abelian Stokes them does not depend on the surfacechosen for spanning the surface bounded by the loop C , remembering that the originalWilson loop is defined for the specified closed loop C .2 πn = g Z d xρ m ( x ) Ω Σ ( x )4 π − g Z d xρ m ( x ) Ω Σ ( x )4 π = g Z d xρ m ( x ) Ω Σ ( x ) − Ω Σ ( x )4 π = g Z d xρ m ( x )= g q m , (6.17)where we have used Ω Σ ( x ) − Ω Σ ( x ) = 4 π . See Fig. 7.For an ensemble of point-like magnetic charges located at x = z a ( a = 1 , · · · , n ) k ( x ) = ρ m ( x ) = n X a =1 q am δ (3) ( x − z a ) , q am = 4 πg − n a , n a ∈ Z , (6.18)we have a geometric representation: W mC = exp ( i g π n X a =1 q am Ω Σ ( z a ) ) = exp ( i n X a =1 n a Ω Σ ( z a ) ) , n a ∈ Z . (6.19)28igure 8: The color field n in SU(2) case.For D = 4, the magnetic current one-form k in the continuum SU(N) Yang-Millstheory is defined as k µ = 12 ǫ µνρσ ∂ ν f ρσ . (6.20)The magnetic current k is conserved, ∂ µ k µ = 0. Then the magnetic charge is definedby Q m = Z d xk = Z d x ǫ jkℓ ∂ ℓ f jk ( x ) . (6.21)Whereas, for D = 4, Ξ Σ is one-form with the component:Ξ µ Σ ( x ) = 12 ǫ µνρσ ∂ xν Z d y Θ ρσ ( y )∆ − ( y − x )= 12 ǫ µνρσ Z S d S ρσ ( x ( σ )) ∂ xν ∆ − ( x ( σ ) − x )= 12 ǫ µνρσ Z S d S ρσ ( x ( σ )) ∂ xν π | x − x ( σ ) | = ǫ µνρσ Z S d S ρσ ( x ( σ )) ( x ( σ ) − x ) ν π | x ( σ ) − x | . (6.22)For SU(2), the rank is one r = 1.Λ = J, H = 12 σ , H = J σ J diag( 12 , −
12 ) , m = J G † σ G. (6.23)Thus we recover the SU(2) case investigated so far [3, 8]. The SU(2) gauge-invariantmagnetic current is obtained from the SU(2) gauge-invariant field strength: f µν =2tr( n F µν [ V ])= ∂ µ c ν − ∂ ν c µ − g − n · ( ∂ µ n × ∂ ν n ) = E µν + H µν . (6.24)We can show that the color field generates the magnetic charge subject to the Diracquantization condition. The color field n parameterized by two polar angles ( α, β )on the target space S ≃ SU (2) /U (1), see Fig. 8: n ( x ) := n ( x ) n ( x ) n ( x ) := sin β ( x ) cos α ( x )sin β ( x ) sin α ( x )cos β ( x ) , (6.25)yields n · ( ∂ µ n × ∂ ν n ) = sin β ( ∂ µ β∂ ν α − ∂ µ α∂ ν β ) = sin β ∂ ( β, α ) ∂ ( x µ , x ν ) . (6.26)29aking into account the fact that ∂ ( β,α ) ∂ ( x µ ,x ν ) is the Jacobian from ( x µ , x ν ) ∈ S phy to( β, α ) ∈ S int ≃ SU (2) /U (1) parameterized by ( β, α ), we obtain the Dirac quantiza-tion condition: Q m = Z d x ∂ ℓ ǫ ℓjk f jk = I S phy dS ℓ ǫ ℓjk f jk = I S phy dS jk g − n · ( ∂ j n × ∂ k n )= − g − I S phy dS jk ∂ ( β, α ) ∂ ( x j , x k ) sin β = − g − I S int dβdα sin β =4 πg − n ( n = 0 , ± , · · · ) , (6.27)since dβdα sin β is the surface element on S int and a surface of a unit radius has thearea 4 π . Hence n gives a number of times S int is wrapped by a mapping from S phys to S int . This fact is understood as the Homotopy group: Π ( SU (2) /U (1)) = Π ( S ) = Z .For SU(2), the explicit configuration yielding the non-zero magnetic monopole isgiven as follows. We consider the Wu-Yang configuration n A ( x ) = x A /r, r := | x | = √ x A x A ( A = 1 , , . (6.28)This leads to the magnetic-monopole density ρ m = 12 ǫ jkℓ ∂ ℓ H jk = 12 ǫ jkℓ ∂ ℓ (cid:20) − g − ǫ jkm x m r (cid:21) = − g − ∂ ℓ " x ℓ r = − πg − δ ( x ) . (6.29)This corresponds to a magnetic monopole with a unit magnetic charge q m = 4 πg − located at the origin. Hence, we obtain W mC = exp " − ig Z d x πg − δ ( x ) Ω Σ ( x )4 π = exp (cid:20) − i
12 Ω Σ (0) (cid:21) . (6.30)Therefore, exp[ ig ( k, Ξ Σ )] gives a non-trivial factor exp[ ± iπ ] = − q m = 4 πg − at the origin, since Ω Σ (0) = ± π for the upper or lower hemisphere Σ. Indeed, this result does not depend on whichsurface bounding C is chosen in the non-Abelian Stokes theorem.For D = 4, Ξ agrees with the four-dimensional solid angle given byΩ µ Σ ( x ) = 18 π ǫ µνρσ ∂∂x ν Z Σ d S ρσ ( y ) 1 | x − y | = Ξ µ Σ ( x ) . (6.31)30onsequently, for D = 4 we have W mC = exp (cid:20) ig Z d xk µ ( x )Ξ µ Σ ( x ) (cid:21) = exp (cid:20) ig Z d xk µ ( x )Ω µ Σ ( x ) (cid:21) . (6.32)For an ensemble of magnetic monopole loops C ′ a ( a = 1 , · · · , n ): k µ ( x ) = n X a =1 q am I C ′ a dy µa δ (4) ( x − x a ) , q am = 4 πg − n a , (6.33)we obtain W mC = exp ( i g n X a =1 q am L ( C ′ a , Σ) ) = exp ( πi n X a =1 n a L ( C ′ a , Σ) ) , n a ∈ Z , (6.34)where L ( C ′ , Σ) is the linking number between the curve C ′ and the surface Σ [39]: L ( C ′ , Σ) := I C ′ dy µ ( τ )Ξ µ Σ ( y ( τ )) , (6.35)where the curve C is identified with the trajectory of a magnetic monopole and thesurface Σ with the world sheet of a hadron string for a quark-antiquark pair. By remembering the relationship: F µν = tr(2 m ( x ) F µν [ V ]( x )) = − s N − N tr(2 h ( x ) F µν [ V ]( x )) = − s N − N f µν ( x ) , (7.1)the magnetic charge is given by Q m = Z d xk = Z d x ǫ jkℓ ∂ ℓ f jk ( x )= Z d x ǫ jkℓ ∂ ℓ ( h ( x ) , F jk [ V ]( x ))= s NN − Z d x ǫ jkℓ ∂ ℓ ( m ( x ) , F jk [ V ]( x ))= Z d x ǫ jkℓ ∂ ℓ ( n ( x ) , F jk [ V ]( x )) . (7.2)The magnetic charge Q m appears in the factor W mC in the Wilson loop operator forthe closed surface Σ for which Ω Σ ( x ) = 4 π as W mC → exp ig s N − N Q m = 1 . (7.3)31n the case of SU(3), it is known that one can define two gauge-invariant conservedcharges Q (1) and Q (2) from the respective color field n and n , which obey thedifferent quantization conditions [40, 41, 42, 43, 44, 28]: Q (1) := Z d x ǫ jkℓ ∂ ℓ ( n ( x ) , F jk [ V ]( x )) = 2 πg (2 n − n ′ ) ,Q (2) := Z d x ǫ jkℓ ∂ ℓ ( n ( x ) , F jk [ V ]( x )) = 2 πg √ n ′ , n, n ′ ∈ Z , (7.4)where √ m ( x ) = n ( x ) = (cos ϑ ( x )) n ( x ) + (sin ϑ ( x )) n ( x ). However, we have shownthat the Wilson loop in the fundamental representation does not distinguish Q (1) and Q (2) , and can probe only the specific combinations Q m represented through m in(7.2). For fundamental representations of SU(3), therefore, the magnetic charge Q m obeys the following quantization condition. For [1 ,
0] and [ − , Q m = ± " √ Q (1) + 12 Q (2) = ± πg [(2 n − n ′ ) + n ′ ] √
32 = ± πg √ n, (7.5)for [0 , −
1] and [0 , Q m = ± " − √ Q (1) + 12 Q (2) = ± πg [ − (2 n − n ′ ) + n ′ ] √
32 = ± πg √ n ′ − n ) , (7.6)and, in particular, for [ − ,
1] and [1 , − Q m = ± h − Q (2) i = ± πg [ − n ′ ] √
32 = ∓ πg √ n ′ . (7.7)These quantization conditions for Q m are reasonable because they guarantee thatthe Wilson loop operator defined originally by the closed loop C does not depend onthe choice of the surface Σ bounded by the loop C when rewritten into the surfaceintegral form in the non-Abelian Stokes theorem, just as in the SU(2) case: W mC = exp ( ig √ Q m ) = 1 → Q m = 2 π √ g − n, n ∈ Z . (7.8)Thus, we have shown that the SU(N) Wilson loop operator can probe a single(gauge-invariant) magnetic monopole in the pure Yang-Mills theory, which can bedefined in a gauge invariant way even in the absence of any scalar field. The magneticcharge is subject to the quantization condition which is analogous to the Dirac type: Q m = 2 πg − s NN − n, n ∈ Z . (7.9)Therefore, one need not to introduce N − U (1) N − of SU(N).Thus, calculating the Wilson loop average reduces to the summation over the contri-butions coming from the distribution of magnetic monopole charges or currents withthe geometric factor related to the solid angle or the linking number. This issue willbe discussed in a future publication. 32 cknowledgments The author is appreciative of continuous discussions from Takeharu Murakami, ToruShinohara and Akihiro Shibata. Part of this work was done while the author stayed asan invited participants of the programme “Strong Fields, Integrability and Strings”held at Isaac Newton Institute for Mathematical Sciences in Cambridge U.K. for theperiod from 6 Aug to 24 Aug, 2007. The author would like to thank Nick Dorey andSimon Hands who enabled him to attend the programme for hospitality. This workis financially supported by Grant-in-Aid for Scientific Research (C) 18540251 fromJapan Society for the Promotion of Science (JSPS).
A Formulae for Cartan subalgebras
The Cartan algebras are written as H k = 1 q k ( k + 1) diag(1 , , · · · , , − k, , · · · , q k ( k + 1) k X j =1 e jj − k e k +1 ,k +1 , (A.1)where we have defined the matrix e AB whose AB element has the value 1 and otherelements are zero, i.e., ( e AB ) ab = δ Aa δ Bb . The the ab element reads( H k ) ab = 1 q k ( k + 1) k X j =1 δ aj δ bj − kδ a,k +1 δ b,k +1 . (A.2)The unit matrix with the element δ ab is written as = diag(1 , , · · · ,
1) = N X j =1 e jj . (A.3)The product of diagonal generators is decomposed as H j H k = √ j ( j +1) H k ( j > k ) √ k ( k +1) H j ( j < k ) N + − k √ k ( k +1) H k + P N − m = k +1 1 √ m ( m +1) H m ( j = k ) . (A.4)The first and second relations are easily derived from the definition. The third relationis derived as follows. H k H k = c + N − X m =1 c m H m , (A.5)where c m = 2tr( H k H k H m ) , c = tr( H k H k ) / tr( ) , (A.6)33ere the coefficients are calculated as c m =2tr " k ( k + 1) diag(1 , , · · · , , k , , · · · , H m = ≤ m ≤ k − − k √ k ( k +1) ( m = k ) √ m ( m +1) ( k + 1 ≤ m ≤ N − , (A.7)and c = tr( H k H k ) / tr( ) = 12 1 N . (A.8)
B Decomposition formulae
For any Lie algebra valued function V ( x ), the identity holds [37, 45]: V = N − X j =1 n j ( n j , V ) + N − X j =1 [ n j , [ n j , v ]] = N − X j =1 V n j ) n j + N − X j =1 [ n j , [ n j , V ]] . (B.2)This identity is equivalent to the identity [37] δ AB = n Aj n Bj − f ACD n Cj f DEB n Ej . (B.3)The identity is proved as follows. By using the adjoint rotation, V ′ = U V U † , wehave only to prove V ′ = N − X j =1 H j ( H j , V ′ ) + N − X j =1 [ H j , [ H j , V ′ ]] . (B.4)The Cartan decomposition for V = V ′ reads V = N − X k =1 V k H k + ( N − N ) / X α =1 ( W ∗ α ˜ E α + W α ˜ E − α ) , (B.5)where the Cartan basis is given by ~H =( H , H , H , · · · , H N − ) = ( T , T , T , · · · , T N − ) , ˜ E ± = 1 √ T ± iT ) , ˜ E ± = 1 √ T ± iT ) , · · · , ˜ E ± ( N − N ) / = 1 √ T N − ± iT N − ) , (B.6) The SU(2) version of this identity is v = n ( n · v ) − n × ( n × v ) = v k + v ⊥ , (B.1)which follows from a simple identity, n × ( n × v ) = n ( n · v ) − ( n · n ) v . W = 1 √ V + iV ) , W = 1 √ V + iV ) , · · · , W ( N − N ) / = 1 √ V N − + iV N − ) . (B.7)Now we calculate the double commutator as[ H j , [ H j , V ]]= N − X j =1 V k [ H j , [ H j , H k ]] + ( N − N ) / X α =1 ( W ∗ α [ H j , [ H j , ˜ E α ]] + W α [ H j , [ H j , ˜ E − α ]])= ( N − N ) / X α =1 ( W ∗ α [ H j , α j ˜ E α ] + W α [ H j , − α j ˜ E − α ])= α j α j ( N − N ) / X α =1 ( W ∗ α ˜ E α + W α ˜ E − α ) . (B.8)On the other hand, we have( H j , V )= N − X k =1 V k ( H j , H k ) + ( N − N ) / X α =1 ( W ∗ α ( H j , ˜ E α ) + W α ( H j , ˜ E − α ))= V j , (B.9)since ( H j , ˜ E α ) = tr( H j ˜ E α ) = 0. Thus the RHS of (B.4) reduces to N − X j =1 V j H j + N − X j =1 α j α j ( N − N ) / X α =1 ( W ∗ α ˜ E α + W α ˜ E − α ) . (B.10)This is equal to the Cartan decomposition of V itself, since N − X j =1 α j α j = 1 . (B.11) C More decomposition formulae
For any Lie algebra valued function M ( x ), the identity holds: V = N − X A =1 V A T A = ˜ V + h ( h , V ) + 2 N − N [ h , [ h , V ]] , (C.1)where we have defined the matrix ˜ V in which all the elements in the last column andthe last raw are zero:˜ V = ( N − − X A =1 V A T A = ( N − − X A =1 ( V , T A ) T A = ( N − − X A =1 V T A ) T A , (C.2)35r V = ˜ V + 2tr( V h ) h + 2 N − N [ h , [ h , V ]] . (C.3)Note that [ ˜ V , h ] = 0.The Cartan decomposition for V N = v ′ ∈ su ( N ) reads V N = N − X k =1 V k H k + ( N − N ) / X α =1 ( W ∗ α ˜ E α + W α ˜ E − α )= ˜ V N + M N − H N − + ( N − N ) / X α =[( N − − ( N − / ( W ∗ α ˜ E α + W α ˜ E − α ) . (C.4)Now we calculate the double commutator ( r = N −
1) as[ H r , [ H r , V N ]]= N − X j =1 V k [ H r , [ H r , H k ]] + ( N − N ) / X α =1 ( W ∗ α [ H r , [ H r , ˜ E α ]] + W α [ H r , [ H r , ˜ E − α ]])= ( N − N ) / X α =[( N − − ( N − / ( W ∗ α [ H r , α r ˜ E α ] + W α [ H r , − α r ˜ E − α ])= α r α r ( N − N ) / X α =[( N − − ( N − / ( W ∗ α ˜ E α + W α ˜ E − α ) . (C.5)On the other hand, we have( H r , V N )= N − X k =1 V k ( H r , H k ) + ( N − N ) / X α =1 ( W ∗ α ( H r , ˜ E α ) + W α ( H r , ˜ E − α ))= V r , (C.6)since ( H j , ˜ E α ) = tr( H j ˜ E α ) = 0.For any Lie algebra valued function V N ( x ), we obtain the identity: V N = N − X A =1 V A T A = ˜ V N + ( V N , H r ) H r + 1 α r [ H r , [ H r , V N ]]= ˜ V N + ( V N , H r ) H r + 2( N − N [ H r , [ H r , V N ]] . (C.7) D CFNS decomposition
This section is a summary of the results obtained in [34], which is added just for theconvenience of readers. 36 .1 Maximal case
In the maximal case, we introduce a full set of color fields n j ( x ) ( j = 1 , · · · , r = N − n j ( x ) = U † ( x ) H j U ( x ) , j ∈ { , , · · · , r } , (D.1)where r = rank SU ( N ) = N − H j are Cartan subalgebra. The fields n j ( x )defined in this way are unit vectors, since( n j ( x ) , n k ( x )) = 2tr( n j ( x ) n k ( x )) = 2tr( U † ( x ) H j U ( x ) U † ( x ) H k U ( x ))= 2tr( H j H k ) = ( H j , H k ) = δ jk . (D.2)These unit vectors mutually commute,[ n j ( x ) , n k ( x )] = 0 , j, k ∈ { , , · · · , r } , (D.3)since H j are Cartan subalgebra obeying[ H j , H k ] = 0 , j, k ∈ { , , · · · , r } . (D.4)Once such a set of color fields n j ( x ) is given, the original gauge field has thedecomposition: A µ ( x ) = V µ ( x ) + X µ ( x ) , (D.5)where the respective components V µ ( x ) and X µ ( x ) are specified by two defining equa-tions (conditions):(I) all n j ( x ) are covariant constant in the background V µ ( x ):0 = D µ [ V ] n j ( x ) := ∂ µ n j ( x ) − ig [ V µ ( x ) , n j ( x )] ( j = 1 , , · · · , r ) , (D.6)(II) X µ ( x ) is orthogonal to all n j ( x ):( X µ ( x ) , n j ( x )) := 2tr( X µ ( x ) n j ( x )) = X Aµ ( x ) n Aj ( x ) = 0 ( j = 1 , , · · · , r ) . (D.7)First, we determine the X µ field by solving the defining equations. We apply theidentity (B.2) to X µ and use the second defining equation (D.7) to obtain X µ = r X j =1 ( X µ , n j ) n j + r X j =1 [ n j , [ n j , X µ ]] = r X j =1 [ n j , [ n j , X µ ]] . (D.8)Then we take into account the first defining equation: D µ [ A ] n j = ∂ µ n j − ig [ A µ , n j ]= D µ [ V ] n j − ig [ X µ , n j ]= − ig [ X µ , n j ] = ig [ n j , X µ ] . (D.9)Thus X µ ( x ) is expressed in terms of A µ ( x ) and n j ( x ) as X µ ( x ) = − ig − r X j =1 [ n j ( x ) , D µ [ A ] n j ( x )] . (D.10)37ext, the V µ field is expressed in terms of A µ ( x ) and n j ( x ): V µ ( x ) = A µ ( x ) − X µ ( x )= A µ ( x ) + ig − r X j =1 [ n j ( x ) , D µ [ A ] n j ( x )]= A µ ( x ) − r X j =1 [ n j ( x ) , [ n j ( x ) , A µ ( x )]] + ig − r X j =1 [ n j ( x ) , ∂ µ n j ( x )] . (D.11)Now we apply the identity (B.2) to A µ to obtain V µ ( x ) = r X j =1 ( A µ ( x ) , n j ( x )) n j ( x ) + ig − r X j =1 [ n j ( x ) , ∂ µ n j ( x )] . (D.12)Thus, V µ ( x ) and X µ ( x ) are written in terms of A µ ( x ), once n j ( x ) is given as afunctional of A µ ( x ).It should be remarked that the background field V µ ( x ) contains a part C µ ( x ) whichcommutes with all n j ( x ):[ C µ ( x ) , n j ( x )] = 0 ( j = 1 , , · · · , r = N − . (D.13)Such a commutative part (or a parallel part in the vector form) C µ ( x ) in V µ ( x ) isnot determined uniquely from the first defining equation (D.6) alone. But it wasdetermined by the second defining equation as shown above. In view of this, wefurther decompose V µ ( x ) into C µ ( x ) and B µ ( x ): V µ ( x ) = C µ ( x ) + B µ ( x ) . (D.14)Applying the identity (B.2) to C µ ( x ) and by taking into account (D.13), we obtain C µ ( x ) = r X j =1 ( C µ ( x ) , n j ( x )) n j ( x ) . (D.15)If the remaining part B µ ( x ) which is not commutative [ B µ ( x ) , n j ( x )] = 0 is perpen-dicular to all n j ( x ):( B µ ( x ) , n j ( x )) = 2tr( B µ ( x ) n j ( x )) = 0 ( j = 1 , , · · · , r ) , (D.16)then we have ( A µ ( x ) , n j ( x )) = ( V µ ( x ) , n j ( x )) = ( C µ ( x ) , n j ( x )) . (D.17)Consequently, the parallel part C µ ( x ) reads C µ ( x ) = r X j =1 ( A µ ( x ) , n j ( x )) n j ( x ) . (D.18)and the perpendicular part B µ ( x ) is determined as B µ ( x ) = ig − r X j =1 [ n j ( x ) , ∂ µ n j ( x )] . (D.19) The SU(2) version in the vector form reads B µ ( x ) = g − ∂ µ n ( x ) × n ( x ).
38n fact, it is easy to check that this expression indeed satisfies (D.16) and D µ [ B ] n j ( x ) = ∂ µ n j ( x ) − ig [ B µ ( x ) , n j ( x )] = 0 ( j = 1 , , · · · , r ) , (D.20)Thus, once full set of color fields n j ( x ) is given, the original gauge field has thedecomposition in the Lie algebra form: A µ ( x ) = V µ ( x ) + X µ ( x ) = C µ ( x ) + B µ ( x ) + X µ ( x ) , (D.21a)where each part is expressed in terms of A µ ( x ) and n j ( x ) as C µ ( x ) = N − X j =1 ( A µ ( x ) , n j ( x )) n j ( x ) = N − X j =1 c jµ ( x ) n j ( x ) , (D.21b) B µ ( x ) = ig − N − X j =1 [ n j ( x ) , ∂ µ n j ( x )] , (D.21c) X µ ( x ) = − ig − N − X j =1 [ n j ( x ) , D µ [ A ] n j ( x )] . (D.21d)In what follows, the summation over the index j should be understood when it isrepeated, unless otherwise stated. D.2 Minimal case
Now we consider the minimal case for SU ( N ). In this case, A µ is decomposed as A µ ( x ) = V µ ( x ) + X µ ( x ) , (D.22)using only a single color field n r ( r = N − h ( x ) := n r ( x ) = U † ( x ) H r U ( x ) , (D.23)where H r is the last diagonal matrix.The respective components V µ ( x ) and X µ ( x ) are specified by two defining equa-tions (conditions):(I) h ( x ) is covariant constant in the background V µ ( x ):0 = D µ [ V ] h ( x ) := ∂ µ h ( x ) − ig [ V µ ( x ) , h ( x )] , (D.24)(II) X µ ( x ) is orthogonal to h ( x ):( X µ ( x ) , h ( x )) := 2tr( X µ ( x ) h ( x )) = X Aµ ( x ) h A ( x ) = 0 . (D.25)(II) X µ ( x ) does not have a part ˜ X µ ( x ) which commutes with h ( x ):˜ X µ ( x ) = 0 . (D.26)First, we apply the identity (C.1) to X µ ( x ) and use the second and third definingequations to obtain X µ ( x ) = ˜ X µ ( x ) + ( X µ ( x ) , h ) h + 2( N − N [ h , [ h , X µ ( x )]]= 2( N − N [ h , [ h , X µ ( x )]] . (D.27)39hen we take into account the first defining equation: D µ [ A ] h = ∂ µ h − ig [ A µ , h ]= D µ [ V ] h − ig [ X µ , h ]= − ig [ X µ , h ] = ig [ h , X µ ] . (D.28)Thus X µ ( x ) is expressed in terms of A µ ( x ) and h ( x ) as X µ ( x ) = − ig − N − N [ h ( x ) , D µ [ A ] h ( x )] . (D.29)Next, the V µ field is expressed in terms of A µ ( x ) and h ( x ): V µ ( x ) = A µ ( x ) − X µ ( x )= A µ ( x ) + ig − N − N [ h ( x ) , D µ [ A ] h ( x )]= A µ ( x ) − N − N [ h ( x ) , [ h ( x ) , A µ ( x )]]+ ig − N − N [ h ( x ) , ∂ µ h ( x )] . (D.30)Thus, V µ ( x ) and X µ ( x ) are written in terms of A µ ( x ), once h ( x ) is given as a func-tional of A µ ( x ).Now we apply the identity (C.1) to A µ to obtain V µ ( x ) = ˜ A µ ( x ) + ( A µ ( x ) , h ( x )) h ( x ) + ig − N − N [ h ( x ) , ∂ µ h ( x )] . (D.31)We further decompose V µ ( x ) into C µ ( x ) and B µ ( x ): V µ ( x ) = C µ ( x ) + B µ ( x ) . (D.32)where C µ ( x ) commutes with h ( x ):[ C µ ( x ) , h ( x )] = 0 ( j = 1 , , · · · , r = N − , (D.33)and the remaining part B µ ( x ) which is not commutative [ B µ ( x ) , n j ( x )] = 0 is per-pendicular to h ( x ): ( B µ ( x ) , h ( x )) = 2tr( B µ ( x ) h ( x )) = 0 , (D.34)which leads to ( A µ ( x ) , h ( x )) = ( V µ ( x ) , h ( x )) = ( C µ ( x ) , h ( x )) . (D.35)Thus, once a single color field h ( x ) is given, we have the decomposition: A µ ( x ) = V µ ( x ) + X µ ( x ) = C µ ( x ) + B µ ( x ) + X µ ( x ) , (D.36a) C µ ( x ) = A µ ( x ) − N − N [ h ( x ) , [ h ( x ) , A µ ( x )]] , (D.36b) B µ ( x ) = ig − N − N [ h ( x ) , ∂ µ h ( x )] , (D.36c) X µ ( x ) = − ig − N − N [ h ( x ) , D µ [ A ] h ( x )] . (D.36d)40 Field strength
This section is a summary of the results obtained in [34], which is added just for theconvenience of readers.
E.1 Maximal case
The field strength is rewritten in terms of new variables as F µν [ A ]:= ∂ µ A ν − ∂ ν A µ − ig [ A µ , A ν ]= F µν [ V ] + ∂ µ X ν − ∂ ν X µ − ig [ V µ , X ν ] − ig [ X µ , V ν ] − ig [ X µ , X ν ]= F µν [ V ] + D µ [ V ] X ν − D ν [ V ] X µ − ig [ X µ , X ν ] , (E.1)Here F µν [ V ] is further decomposed as (omitting the summation symbol over j ): F µν [ V ]:= ∂ µ V ν − ∂ ν V µ − ig [ V µ , V ν ]= F µν [ B ] + ∂ µ C ν − ∂ ν C µ − ig [ B µ , C ν ] − ig [ C µ , B ν ] − ig [ C µ , C ν ]= F µν [ B ] + n j ∂ µ c jν − n j ∂ ν c jµ + c jν ∂ µ n j − ig [ B µ , c jν n j ] − c jµ ∂ ν n j + ig [ B ν , c jµ n j ] − ig [ c jµ n j , c kν n k ]= F µν [ B ] + n j ∂ µ c jν − n j ∂ ν c jµ := H µν + E µν . (E.2)Here we have used (D.3) and (D.20) in the last step in simplifying E µν . Therefore weobtain the decomposition of the field strength: F µν [ A ] = E µν + H µν + D µ [ V ] X ν − D ν [ V ] X µ − ig [ X µ , X ν ] , (E.3a)where we have defined E µν := N − X j =1 n j E jµν , E jµν := ∂ µ c jν − ∂ ν c jµ , (E.3b) H µν := F µν [ B ] = ∂ µ B ν − ∂ ν B µ − ig [ B µ , B ν ] . (E.3c)Here H µν is simplified for B µ ( x ) = ig − P N − j =1 [ n j ( x ) , ∂ µ n j ( x )] as (omitting the sum-mation symbol over j ): ∂ µ B ν − ∂ ν B µ = ig − [ ∂ µ n j , ∂ ν n j ] − ( µ ↔ ν ) (E.4a)= ig − ( ig ) [[ B µ , n j ] , [ B ν , n j ]] − ( µ ↔ ν ) (E.4b)= ig [[ n j , [ B ν , n j ]] , B µ ] + [[[ B ν , n j ] , B µ ] , n j ] − ( µ ↔ ν ) (E.4c)= ig [ B µ , [ n j , [ n j , B ν ]]] + [ n j , [[ n j , B ν ] , B µ ]] − ( µ ↔ ν ) (E.4d)= ig [ B µ , [ n j , [ n j , B ν ]]] − ( µ ↔ ν ) − ig [ n j , [ n j , [ B µ , B ν ]]] (E.4e)= ig ([ B µ , B ν ] − [ B ν , B µ ]) − ig [ n j , [ n j , [ B µ , B ν ]]] (E.4f)=2 ig [ B µ , B ν ] − ig [[ B µ , B ν ] + n j ( n j , ig [ B µ , B ν ]) (E.4g)= ig [ B µ , B ν ] + ig n j ( n j , [ B µ , B ν ]) , (E.4h)41here we have used (D.19) in (E.4a), (D.20) in (E.4b), the Jacobi identity for B µ , n j and [ B ν , n j ] in (E.4c), interchanged the commutator in (E.4d), the Jacobi identityin (E.4e), the algebraic identity (B.2) with (D.16) to obtain the first term of (E.4f)and again the algebraic identity (B.2) in (E.4g). Therefore we obtain H µν := ∂ µ B ν − ∂ ν B µ − ig [ B µ , B ν ] = ig n j ( n j , [ B µ , B ν ]) . (E.5)Thus we find H µν is written as the linear combination of all the color fields just like E µν : H µν = F µν [ B ] = N − X j =1 n j H jµν , H jµν = ig ( n j , [ B µ , B ν ]) . (E.6) E.2 Minimal case
The field strength F µν [ V ] reads F µν [ V ]:= ∂ µ V ν − ∂ ν V µ − ig [ V µ , V ν ]= F µν [ B ] + ∂ µ C ν − ∂ ν C µ − ig [ B µ , C ν ] − ig [ C µ , B ν ] − ig [ C µ , C ν ]= F µν [ B ] + D µ [ B ] C ν − D ν [ B ] C µ − ig [ C µ , C ν ] , (E.7)where F µν [ B ] = H µν := ∂ µ B ν − ∂ ν B µ − ig [ B µ , B ν ] . (E.8)We focus on the parallel part of the field strength F µν [ V ]:tr( h F µν [ V ])=tr( h F µν [ B ]) + tr( h D µ [ B ] C ν ) − tr( h D ν [ B ] C µ ) − ig tr( h [ C µ , C ν ])=tr( h F µν [ B ]) − tr(( D µ [ B ] h ) C ν ) + ∂ µ tr( h C ν )+ tr(( D ν [ B ] h ) C µ ) − ∂ µ tr( h C µ ) − ig tr( C ν [ h , C µ ])=tr( h F µν [ B ]) + ∂ µ tr( h C ν ) − ∂ µ tr( h C µ ) , (E.9)where we have used the defining equations for B µ and C µ : D µ [ B ] h = 0 and [ h , C µ ] =0. Then we focus on the parallel part of the field strength F µν [ B ] = H µν :tr( h H µν ) =tr( h F µν [ B ])=tr( h ∂ µ B ν − h ∂ ν B µ − ig h [ B µ , B ν ]) (E.10a)=tr( h ∂ µ B ν − h ∂ ν B µ + ig B ν [ B µ , h ]) (E.10b)=tr( h ∂ µ B ν − h ∂ ν B µ + B ν ∂ µ h ) (E.10c)= ∂ µ tr( h B ν ) − tr( h ∂ ν B µ ) (E.10d)=tr( B µ ∂ ν h ) − ∂ ν tr( h B µ ) (E.10e)=tr( B µ ∂ ν h ) , (E.10f)42here we have used tr( h B µ ) = 0 twice.On the other hand, we findtr( h [ ∂ µ h , ∂ ν h ]) =tr( h ∂ µ h ∂ ν h − h ∂ ν h ∂ µ h ) (E.11a)=tr(( h ∂ µ h − ∂ µ hh ) ∂ ν h ) (E.11b)=tr([ h , ∂ µ h ] ∂ ν h ) . (E.11c)By using the explicit form of B µ = ig − N − N [ h , ∂ µ h ] , (E.12)therefore, both expressions are connected astr( h H µν ) =tr( h F µν [ B ])= 2( N − N tr( ig − [ h , ∂ µ h ] ∂ ν h )= 2( N − N tr( ig − h [ ∂ µ h , ∂ ν h ]) . (E.13)Note that h F µν [ B ] = N − N ig − h [ ∂ µ h , ∂ ν h ].The same result is obtained by calculating explicitly the field strength F µν [ B ] := ∂ µ B ν − ∂ ν B µ − ig [ B µ , B ν ] using the property of h : hh = 12 N + 2 − N q N ( N − h , (E.14)which follows from the relation for the last Cartan generator: H N − H N − = 12 N + 2 − N q N ( N − H N − . (E.15)For a fundamental representation, a weight vector is given by Λ = ν N ≡ , , · · · , , − N q N ( N − . (E.16)Using H N − = 1 q N ( N −
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