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NORDITA-2000/65 HE Non-Abelian Stokes Theorems in Yang–Mills and Gravity Theories (cid:5)(cid:3) (cid:3) Dmitri Diakonov and Victor Petrov (cid:5) NORDITA, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark (cid:3) St.Petersburg Nuclear Physics Institute, Gatchina 188 350, Russia E-mail: [email protected], [email protected] Abstract We recall the non-Abelian Stokes theorem for the Wilson loop in the Yang –Mills theory and discuss its meaning. Then we move to ‘gravitational Wilson loops’, i.e. to holonomies in curved d = 2,3,4 spaces and derive ‘non-Abelian Stokes theorems’ for these quantities as well, which are similar to our formula in the Yang–Mills theory. In particular we derive an elegant formula for the holonomy in the case of a constant- curvature background in three dimensions and a formula for small-area loops in any number of dimensions. 1 Introduction One of the main objects in the Yang{Mills theory as well as in gravity is the parallel trans- porter along closed contours, or holonomy. In Yang{Mills theory it is conventionally called the Wilson loop; it can be written as a path-ordered exponent, (cid:73) 1 dx(cid:22) a a W = Tr P exp i d(cid:28) A T ; (1) r (cid:22) d(r) d(cid:28) where x(cid:22)((cid:28)) with 0 (cid:20) (cid:28) (cid:20) 1 parametrizes the closed contour, Aa is the Yang{Mills (cid:12)eld (or (cid:22) connection) and Ta are the generators of the gauge group in a given representation r whose dimension is d(r). In curved Riemannian spaces the ‘gravitational Wilson loop’ or holonomy for d-dimensional vectors can be also written as a trace of the path-ordered exponent of the connection, this time of the Christo(cid:11)el symbol, (cid:34) (cid:35) 1 (cid:73) dx(cid:22) (cid:20) WG = P exp − d(cid:28) Γ : (2) vector (cid:22) d d(cid:28) (cid:20) One can also consider parallel transporters of spinors in curved background: in this case the holonomy is de(cid:12)ned not by the Christo(cid:11)el symbols but by the spin connection which is not uniquely determined by the metric tensor, see the precise de(cid:12)nitions below. The Yang{Mills Wilson loop is invariant under gauge transformations of the background (cid:12)eld A ; the gravitational Wilson loop is invariant under general coordinate transformations (cid:22) or di(cid:11)eomorphisms, provided one transforms the contour as well. Itisgenerallybelieved thatinthreeandfourdimensions theaverage oftheWilsonloopin a pure Yang{Mills quantum theory exhibits an area behaviour for large and simple contours (like ﬂatrectangular). This shouldbetruenotforallrepresentations but thosewith‘N-ality’ nonequal zero; in the simplest case of the SU(2) gauge group these are representations with half-integer spin J. One of the di(cid:14)culties in proving the asymptotic area law for the Wilson loop in half- integer representations (and proving that in integer representations it is absent) is that the Wilson loop is a complicated object by itself: it is impossible to calculate it analytically in a general non-Abelian background (cid:12)eld. Meanwhile, it is sometimes easier to average a quantity over an ensemble than to calculate it for a speci(cid:12)c representative. However, in case of the Wilson loop the path-ordering is a serious obstacle on that way. A decade ago we have suggested a formula for the Wilson loop in a given background belonging to any gauge group and any representation [1]. In this formula the path order- ing along the loop is removed, but at the price of an additional integration over all gauge transformations of the given non-Abelian background (cid:12)eld, or, more precisely, over a coset depending on the particular representation in which the Wilson loop is considered. Fur- thermore, the Wilson loop can be presented in a form of a surface integral [2], see the next section. We call this representation the non-Abelian Stokes theorem. It is quite di(cid:11)erent from previous interesting statements [3, 4, 5, 6] also called by their authors ‘non-Abelian Stokes theorem’ but which involve surface ordering. Our formula has no surface ordering. A classi(cid:12)cation of ‘non-Abelian Stokes theorems’ for arbitrary groups and their representations has been given recently by Kondo et al. [7] who used the naturally arising techniques of ﬂag manifolds. Though these formulae usually do not facilitate (cid:12)nding Wilson loops in particular back- grounds, they can be used to average Wilson loops over ensembles of Yang{Mills con(cid:12)gura- tions or over di(cid:11)erent metrics, and in more general settings, see. e.g. [8, 9, 7, 10]. The main aim of this paper is to present new formulae for the gravitational holonomies in curved d = 2;3;4 spaces: they are similar to our non-Abelian Stokes theorem for the Yang{Mills case. We get rid of the path ordering in eq. (2) and write down the holonomies as exponents of surface integrals. Instead of path ordering we have to integrate over certain covariantly unit vectors (in case of d = 3) or covariantly unit (anti)self-dual tensors (in case of d = 4). Remarkably, these formulae put parallel transporters of di(cid:11)erent spins on the same footing. In particular, holonomies for half-integer spins are presented in terms of the metric tensor (and its derivatives) only but not in terms of the vielbein or spin connection. 2 Apart from purely theoretical interest we have a practical motivation in mind. Quite recently we have shown, both in the continuum and on the lattice, that the SU(2) Yang{ Mills partition function in d = 3 can be exactly rewritten in terms of local gauge-invariant quantitiesbeingthesixcomponentsofthemetrictensorofthedualspace[11]. Thisrewriting may beuseful to investigate the spectrum and the correlation functions of the theory directly in a gauge-invariant way, but it is insu(cid:14)cient to study the interactions of external sources since they couple to the Yang{Mills potential and not to gauge-invariant quantities. The present paper demonstrates, however, that a typical source, i.e. the Yang-Mills Wilson loop can be expressed not only through the potential (or connection) but also through the metric tensor which is gauge-invariant. Thus, not only the partition function but also Wilson loops in the d = 3 Yang{Mills theory can be expressed through local gauge-invariant quantities. We leave a detailed formulation of the resulting theory for a forthcoming publication. Thoughthemaincontentofthepaperarethenon-AbelianStokestheoremsforholonomies in 3 and 4 dimensions, we have added three short sections with relevant material. We add for completeness the Stokes theorem in two dimensions, compute the holonomy in a special case of constant curvature with cylinder topology in three dimensions and give a general formula for the ‘gravitational Wilson loop’ for small loops in any number of dimensions. 2 Non-Abelian Stokes theorem in Yang–Mills theory Let (cid:28) parametrize the loop de(cid:12)ned by the trajectory x(cid:22)((cid:28)) and A((cid:28)) be the tangent compo- nent of the Yang{Mills (cid:12)eld along the loop in the fundamental representation of the gauge group, A((cid:28)) = Aatadx(cid:22)=d(cid:28), Tr(tatb) = 1(cid:14)ab. The gauge transformation of A((cid:28)) is (cid:22) 2 d A((cid:28)) ! S((cid:28))A((cid:28))S−1((cid:28))+iS((cid:28)) S−1((cid:28)): (3) d(cid:28) Let H be the generators from the Cartan subalgebra (i = 1;:::;r; r is the rank of the gauge i group) and the r-imensional vector m be the highest weight of the representation r in which the Wilson loop is considered. The formula for the Wilson loop derived in ref. [1] is a path integral over all gauge transformations S((cid:28)) which should be periodic along the contour: (cid:90) (cid:90) (cid:104) (cid:105) W = DS((cid:28))exp i d(cid:28) Tr m H (SAS−1 +iSS_−1) : (4) r i i Let us stress that eq. (4) is manifestly gauge invariant, as is the Wilson loop itself. For example, in the simple case of the SU(2) group eq. (4) reads: (cid:90) (cid:90) (cid:104) (cid:105) W = DS((cid:28))exp iJ d(cid:28) Tr (cid:28) (SASy +iSS_y) (5) J 3 1 3 where (cid:28) is the third Pauli matrix and J = ; 1; ;::: is the ‘spin’ of the representation of 3 2 2 the Wilson loop considered. The path integrals over all gauge rotations (4,5) are not of the Feynman type: they do not contain terms quadratic in the derivatives in (cid:28). Therefore, a certain regularization is understood in these equations, ensuring that S((cid:28)) is su(cid:14)ciently smooth. For example, one canintroducequadratictermsintheangularvelocities iSS_y withsmallcoe(cid:14)cientseventually put to zero; see ref.[1] for details. In ref.[1] eq. (5) has been derived in two independent ways: 3 i) by direct discretization and ii) by using the standard Feynman representation of path integrals as a sum over all intermediate states, in this case that of an axial top supplemented by a ‘Wess{Zumino’ type of the action. Another discretization but leading to the same result has been used recently by Kondo [7]. A similar formula has been used by Alekseev, Faddeev and Shatashvili [12] who derived a formula for groupcharacters to which the Wilson loop is reduced in case of a constant A (cid:12)eld actually considered in [12]. In ref.[13] eq. (4) has been rederived in an independent way speci(cid:12)cally for the fundamental representation of the SU(N) gauge group. Finally, another derivation of a variant of eq. (5) using lattice regularization has been presented recently in ref. [14]. The second term in the exponent of eqs. (4,5) is in fact a ‘Wess{Zumino’-type action, and it can be rewritten not as a line but as a surface integral inside a closed contour. Let us consider for simplicity the SU(2) gauge group and parametrize the SU(2) matrix S from eq. (5) by Euler’s angles, S = exp(iγ(cid:28) =2) exp(i(cid:12)(cid:28) =2) exp(i(cid:11)(cid:28) =2) 3 2 3 (cid:32) (cid:33) cos (cid:12) ei(cid:11)+2γ sin (cid:12) e−i(cid:11)−2γ = 2 2 : (6) −sin (cid:12) ei(cid:11)−2γ cos (cid:12) e−i(cid:11)+2γ 2 2 The derivation of eq. (5) implies that S((cid:28)) is a periodic matrix. It means that (cid:11)(cid:6)γ and (cid:12) are periodic functions of (cid:28), modulo 4(cid:25). The second term in the exponent of eq. (5) which we denote by (cid:8) is then (cid:90) (cid:90) (cid:8) = d(cid:28) Tr((cid:28) iSS_y) = d(cid:28) (cid:11)_(cos(cid:12) +γ_) 3 (cid:90) (cid:90) = d(cid:28) [(cid:11)_(cos(cid:12) −1)+((cid:11)_ +γ_)] = d(cid:28) (cid:11)_(cos(cid:12) −1): (7) The last term is a full derivative and can be actually dropped because (cid:11)+γ is 4(cid:25)-periodic, therefore even for half-integer representations J it does not contribute to eq. (5). Notice that (cid:11) can be 2(cid:25)-periodic if γ (which drops from eq. (7)) is 2(cid:25); 6(cid:25);:::-periodic. If (cid:11)(1) = (cid:11)(0)+2(cid:25)k, (cid:11)((cid:28)) makes k windings. Integration over all possible (cid:11)((cid:28)) implied in eq. (5) can be divided into distinct sectors with di(cid:11)erent winding numbers k. Introducing a unit 3-vector 1 a a y n = Tr (S(cid:28) S (cid:28) ) = (sin(cid:12)cos(cid:11); sin(cid:12)sin(cid:11); cos(cid:12)) (8) 3 2 we can rewrite (cid:8) as (cid:90) 1 abc a b c (cid:8) = d(cid:28)d(cid:27)(cid:15) (cid:15) n @ n @ n ; i;j = (cid:28);(cid:27); (9) ij i j 2 where one integrates over any surface spanned on the contour (we shall call it a ‘disk’), and n or (cid:11) and (cid:12) are continued inside the disk without singularities. We denote the second coordinate by (cid:27) such that (cid:27) = 1 corresponds to the edge of the disk coinciding with the contour and (cid:27) = 0 corresponds to the center of the disk. See ref. [14] for details on the continuation inside the disk. 4 Let us note that if the surface is closed or in(cid:12)nite the r.h.s. of eq. (9) is the integer topological charge of the n (cid:12)eld on the surface: (cid:90) 1 abc a b c Q = d(cid:27)d(cid:28) (cid:15) (cid:15) n @ n @ n : (10) ij i j 8(cid:25) Eq. (9) can be also rewritten in a form which is invariant under the reparametrizations of the surface. Introducing the invariant element of a surface, (cid:32) (cid:33) @x(cid:22) @x(cid:23) @x(cid:23) @x(cid:22) d2S(cid:22)(cid:23) = d(cid:27)d(cid:28) − = (cid:15)(cid:22)(cid:23) d(Area); (11) @(cid:28) @(cid:27) @(cid:28) @(cid:27) one can rewrite eq. (9) as (cid:90) 1 2 (cid:22)(cid:23) abc a b c (cid:8) = d S (cid:15) n @ n @ n : (12) (cid:22) (cid:23) 2 We get for the Wilson loop [1]: (cid:90) (cid:20) (cid:90) (cid:90) (cid:21) iJ a a 2 (cid:22)(cid:23) abc a b c W = Dn((cid:27);(cid:28)) exp iJ d(cid:28)(A n )+ d S (cid:15) n @ n @ n : (13) J (cid:22) (cid:23) 2 The interpretation of this formula is obvious: the unit vector n plays the role of the instant direction of the colour ‘spin’ in colour space; however, multiplying its length by J does not yet guarantee that we deal with a true quantum state froma representation labelled by J { that is achieved only by introducing the ‘Wess{Zumino’ term in eq. (13): it (cid:12)xes the representation to which the probe quark of the Wilson loop belongs to be exactly J. Finally, we can rewrite the exponent in eq. (13) so that both terms appear to be surface integrals [2]: (cid:90) (cid:90) (cid:16) (cid:17) iJ W = Dn((cid:27);(cid:28)) exp d2S(cid:22)(cid:23) −Fa na +(cid:15)abcna(D n)b(D n)c ; (14) (cid:22)(cid:23) (cid:22) (cid:23) 2 where Dab = @ (cid:14)ab+(cid:15)acbAc is the covariant derivative and Fa = @ Aa−@ Aa+(cid:15)abc Ab Ac (cid:22) (cid:22) (cid:22) (cid:22)(cid:23) (cid:22) (cid:23) (cid:23) (cid:22) (cid:22) (cid:23) is the (cid:12)eld strength. Indeed, expanding the exponent of eq. (14) in powers of A one observes (cid:22) that the quadratic term cancels out while the linear one is a full derivative reproducing the Aana term in eq. (13); the zero-order term is the ‘Wess{Zumino’ term (9) or (7). Note that both terms in eq. (14) are explicitly gauge invariant. We call eq. (14) the non-abelian Stokes theorem. We stress that it is di(cid:11)erent from previously suggested Stokes-like representations of the Wilson loop, based on ordering of elementary surfaces inside the loop [3, 4, 5, 6]. For a further discussion of eq. (14) see [14]. Let usbrieﬂy discuss gaugegroups higher than SU(2): forthat purposewe have toreturn to our eq. (4). Eq. (4) is valid for any group and any representation. It is easy to present it also in a surface form, see ref. [14]. Actually, eq. (4) depends not on all parameters of the gauge transformation but only on those which do not commute with the Cartan combination H = m H . In the SU(2) case one has m H = J(cid:28) ; J = 1=2;1;3=2;:::, since SU(2) is of r i i i i 3 rank 1, and there is only one Cartan generator. Therefore, in the SU(2) case one integrates over the coset SU(2)=U(1) for any representation; this coset can be parametrized by the n (cid:12)eld as described above. 5 In case of higher groups the particular coset depends on the representation of the Wilson loop. For example, in case the Wilson loop is considered in the fundamental representation of the SU(N) group the combination m H is proportional to one particular generator of the i i Cartan subalgebra, which commutes with the SU(N−1)(cid:2)U(1) subgroup. [In case of SU(3) this generator is the Gell-Mann (cid:21) matrix or a permutation of its elements.] Therefore, the 8 appropriatecosetforthefundamentalrepresentationoftheSU(N)groupisSU(N)=SU(N− 1) =U(1) = CPN−1. A possible parametrization of this coset is given by a complex N-vector u(cid:11) of unit length, uyu(cid:11) = 1. To be concrete, the Cartan combination in the fundamental (cid:11) representation can be always set to be m H = diag(1;0;:::;0) by rotating the axes and i i subtracting the unit matrix. In such a basis u(cid:11) is just the (cid:12)rst column of the unitary matrix S while uy is the (cid:12)rst row of Sy. Unitarity of S implies that uyu(cid:11) = 1. (cid:11) (cid:11) In this parametrization eq. (4) can be written as (cid:90) (cid:90) dx(cid:22) WSU(N) = DuDuy(cid:14)(uyu(cid:11)−1) expi d(cid:28) uy (ir )(cid:11) u(cid:12); (r )(cid:11) = @ (cid:14)(cid:11)−iAa (ta)(cid:11): fund (cid:11) d(cid:28) (cid:11) (cid:22) (cid:12) (cid:22) (cid:12) (cid:22) (cid:12) (cid:22) (cid:12) (15) Using the identity, (cid:16) (cid:17) (cid:104) (cid:105) (cid:15) @ uyr u = (cid:15) (r u)y(r u)+uyr r u ij i i ij i j i j (cid:20) (cid:21) i = (cid:15) − (uyF u)+(r u)y(r u) ; (16) ij ij i j 2 we can present eq. (15) in a surface form: (cid:90) (cid:90) (cid:20) (cid:21) 1 WSU(N) = DuDuy(cid:14)(juj2−1) expi dS(cid:22)(cid:23) (uyF u)+i(r u)y(r u) ; (17) fund (cid:22)(cid:23) (cid:22) (cid:23) 2 whereF isthe(cid:12)eldstrengthinthefundamentalrepresentation. Eq.(17)hasbeen(cid:12)rstpub- (cid:22)(cid:23) lished in ref.[13] however with an unexpected overall coe(cid:14)cient 2 in the exponent. Eq. (17) presents the non-Abelian Stokes theorem for the Wilson loop in the fundamental represen- tation of SU(N). In the particular case of the SU(2) group transition to eq. (14) is achieved by identifying the unit 3-vector: na = uy((cid:28)a)(cid:11)u(cid:12) where (cid:11) (cid:12) (cid:32) (cid:33) u(cid:11) = cos (cid:12)2 e−i(cid:11)+2γ ; 2iuy@ u = (cid:11)_(cos(cid:12) −1)+((cid:11)_ +γ_): (18) sin (cid:12) ei(cid:11)−2γ (cid:28) 2 It should be mentioned that the quantity (cid:90) y (cid:11) d(cid:27)d(cid:28)(cid:15) i@ u @ u = 2(cid:25)Q (19) ij i (cid:11) j appearing ineq. (17) is the topologicalcharge of the2-dimensional CPN−1 model. For closed or in(cid:12)nite surfaces Q is an integer. Incase theWilson loopistaken inthe adjoint represention ofthe SU(N) gaugegroupthe combination m H is the highest root. Only group elements of the form exp(i(cid:11) H ) commute i i i i with this combination, belonging to the maximal torus subgroup U(1)N−1. Hence, in case of the adjoint representation one in fact integrates over the maximal coset SU(N)= U(1)N−1 = FN−1, i.e. over ﬂag variables [15, 7]. 6 3 ‘Gravitational Wilson loops’ An object similar to the Wilson loop of the Yang{Mills theory exists also in gravity theory. It is the parallel transporter of a vector on a Riemannian manifold along a closed contour, else called a holonomy. The holonomy is trivial if the space is ﬂat but becomes a non-trivial functional of the curvature in case it is nonzero. In the remaining sections we shall present new formulae for the parallel transporters on d = 2;3;4 Riemannian manifolds. Letus(cid:12)rstremindnotationsfromdi(cid:11)erentialgeometry. Weuse[16]asageneralreference book. Let g = g ((cid:22);(cid:23) = 1;:::d) be the covariant metric tensor, with the contravariant g(cid:22)(cid:23) (cid:22)(cid:23) (cid:23)(cid:22) being its inverse, g g(cid:23)(cid:20) = (cid:14)(cid:20). The determinant of the covariant metric tensor is denoted by (cid:22)(cid:23) (cid:22) g. The Christo(cid:11)el symbol is de(cid:12)ned as g(cid:22)(cid:21) @ g Γ(cid:22) = g(cid:22)(cid:21)Γ = (@ g +@ g −@ g ); Γ(cid:20) = (cid:23) : (20) (cid:23)(cid:20) (cid:21);(cid:23)(cid:20) (cid:23) (cid:21)(cid:20) (cid:20) (cid:21)(cid:23) (cid:21) (cid:23)(cid:20) (cid:23)(cid:20) 2 2g The action of the covariant derivative on the contravariant vector is de(cid:12)ned as (r )(cid:20)v(cid:21) = (@ (cid:14)(cid:20) +Γ(cid:20) )v(cid:21): (21) (cid:26) (cid:21) (cid:26) (cid:21) (cid:26)(cid:21) The commutator of two covariant derivatives determine the Riemann tensor: [r r ](cid:20) = R(cid:20) = g(cid:20)(cid:20)0R = @ Γ(cid:20) −@ Γ(cid:20) +Γ(cid:20) Γ(cid:28) −Γ(cid:20) Γ(cid:28) : (22) (cid:26) (cid:27) (cid:21) (cid:21)(cid:26)(cid:27) (cid:20)0(cid:21)(cid:26)(cid:27) (cid:26) (cid:27)(cid:21) (cid:27) (cid:26)(cid:21) (cid:26)(cid:28) (cid:27)(cid:21) (cid:27)(cid:28) (cid:26)(cid:21) A contraction of the Riemann tensor gives the symmetric Ricci tensor, (cid:20) (cid:20) (cid:20) (cid:21)(cid:27) R = R ; R = R g : (23) (cid:21)(cid:27) (cid:21)(cid:20)(cid:27) (cid:26) (cid:21)(cid:26)(cid:27) Its full contraction is the scalar curvature: (cid:21)(cid:27) (cid:20) R = R g = R : (24) (cid:21)(cid:27) (cid:20) The parallel transporter of a contravariant vector along a curve x(cid:22)((cid:28)) is determined by solving the equation, dx(cid:22) (r )(cid:20)v(cid:21)((cid:28)) = 0; (25) (cid:22) (cid:21) d(cid:28) whose solution can be written with the help of the evolution operator, (cid:104) (cid:105) (cid:20) (cid:20) G (cid:21) v ((cid:28)) = W ((cid:28)) v (0) (26) (cid:21) where v(cid:21)(0) is the vector at the starting point of the contour and v(cid:21)((cid:28)) is the parallel- transported vector at the point labelled by (cid:28). The evolution operator can be symbolically written as a path-ordered exponent of the Christo(cid:11)el symbol: (cid:34) (cid:35) (cid:104) (cid:105)(cid:20) (cid:90) (cid:28) dx(cid:22) (cid:20) WG((cid:28)) = P exp − d(cid:28) Γ : (27) (cid:22) (cid:21) 0 d(cid:28) (cid:21) Wede(cid:12)nethe‘gravitationalWilsonloop’asthetraceoftheparalleltransportingevolution operator along the closed curve x(cid:22)((cid:28)) with x(cid:22)(1) = x(cid:22)(0): 7 (cid:104) (cid:105) 1 (cid:20) G G W = W (1) : (28) vector d (cid:20) This quantity is di(cid:11)eomorphism-invariant: if one changes the coordinates x(cid:22) ! x0(cid:22)(x) the metric is transformed, but if one changes the contour accordingly, that is x(cid:22)((cid:28)) ! x0(cid:22)(x((cid:28))) the gravitational Wilson loop or the holonomy remains the same. In this respect the gravitational holonomy is di(cid:11)erent from the Yang{Mills Wilson loop which is invariant under gauge transformation, without changing the contour. The parallel transporter of a covariant vector is given by the transposed matrix; its trace coincides with that of the contravariant vector. 4 Relation of gravity quantities to those of the Yang– Mills theory We shall show now that the ‘gravitational Wilson loop’ is not just analogous but directly expressible through the Yang{Mills Wilson loops of the SU(2) group. To that end we introduce the standard vielbein eA and its inverse eA(cid:22) such that (cid:22) p A A A B(cid:22) AB A(cid:22) A(cid:23) (cid:22)(cid:23) A e e = g ; e e = (cid:14) ; e e = g ; dete = g: (29) (cid:22) (cid:23) (cid:22)(cid:23) (cid:22) (cid:22) Let us decompose the vector experiencing the parallel transport in vielbeins, (cid:21) A A(cid:21) A A (cid:20) v = c e ; the reciprocal being c = e v ; (30) (cid:20) and put it into eq. (25) de(cid:12)ning the parallel transport. We have dx(cid:22) dx(cid:22) (cid:104) (cid:105) dx(cid:22) 0 = (r )(cid:20)cAeA(cid:21) = eA(cid:20)@ cA +cA(@ eA(cid:20) +Γ(cid:20) eA(cid:21)) = eB(cid:20)(@ (cid:14)BA +!BA)cA; (cid:22) (cid:21) (cid:22) (cid:22) (cid:22)(cid:21) (cid:22) (cid:22) d(cid:28) d(cid:28) d(cid:28) (31) where we have introduced the spin connection, 1 1 1 !AB = −!BA = eA(cid:20)(@ eB −@ eB)− eB(cid:20)(@ eA−@ eA)− eA(cid:20)eB(cid:21)eC(@ eC −@ eC); (32) (cid:22) (cid:22) (cid:22) (cid:20) (cid:20) (cid:22) (cid:22) (cid:20) (cid:20) (cid:22) (cid:22) (cid:20) (cid:21) (cid:21) (cid:20) 2 2 2 and used the fundamental relation: @ eA(cid:20) +Γ(cid:20) eA(cid:21) = −!ABeB(cid:20); (33) (cid:22) (cid:22)(cid:21) (cid:22) @ eA −Γ(cid:21) eA = −!ABeB: (34) (cid:22) (cid:20) (cid:22)(cid:20) (cid:21) (cid:22) (cid:20) One can introduce the SO(d) ‘(cid:12)eld strength’, FAB = [@ +! ;@ +! ]AB = @ !AB −@ !AB +!AC!CB −!AC!CB; (35) (cid:22)(cid:23) (cid:22) (cid:22) (cid:23) (cid:23) (cid:22) (cid:23) (cid:23) (cid:22) (cid:22) (cid:23) (cid:23) (cid:22) related to the Riemann tensor as FABeAeB = −R ; FAB = −R eA(cid:20)eB(cid:21); FABeA(cid:22)eB(cid:23) = R: (36) (cid:22)(cid:23) (cid:20) (cid:21) (cid:20)(cid:21)(cid:22)(cid:23) (cid:22)(cid:23) (cid:20)(cid:21)(cid:22)(cid:23) (cid:22)(cid:23) 8 The above material is common for any number of dimensions. To proceed further we need to consider separately the cases d = 3 and d = 4. The case d = 2 is considered in section 6. 4.1 d=3 In three dimensions one can immediately identify the spin connection with a Yang{Mills (cid:12)eld in su(2), 1 Ac = − (cid:15)abc!ab: (37) i i 2 Speaking about three dimensions we shall denote Lorentz indices by i;j;::: = 1;2;3 and the ﬂat triade indices by a;b;::: = 1;2;3. Recalling the generators in the J = 1 representation, (Tc)ab = −i(cid:15)cab; [TcTd] = i(cid:15)cdfTf; (38) we can rewrite the last parenthesis in eq. (31) as @ (cid:14)ab +!ab = @ (cid:14)ab −iAc(Tc)ab (cid:17) (D )ab; (39) i i i i i which is the standard Yang{Mills covariant derivative in the adjoint representation. In the fundamental (spinor) representation the Yang{Mills covariant derivative is (cid:18)(cid:27)c(cid:19)(cid:11) 1 (cid:104) (cid:105)(cid:11) (r )(cid:11) = @ (cid:14)(cid:11) −iAc = @ (cid:14)(cid:11) + !ab (cid:27)a(cid:27)b ; (cid:11);(cid:12) = 1;2; (40) i (cid:12) i (cid:12) i i (cid:12) i 2 (cid:12) 8 (cid:12) which coincides with the known expression for the covariant derivative in the spinor repre- sentation in curved space. The standard Yang{Mills (cid:12)eld strength is directly related to that of eq. (35): 1 Fa = @ Aa −@ Aa +(cid:15)abcAbAc = − (cid:15)abcFbc (41) ij i j j i i j ij 2 Hence from eq. (36) one has abc a b c (cid:15) F e e = R : (42) ij k l ijkl Let us consider the parallel transporter of a 3-vector in curved space, as de(cid:12)ned by eq. (25). According to eqs. (31,39) solving eq. (25) is equivalent to solving the Yang{Mills equation for the parallel transporter, dxi ab b (D ) c = 0; (43) i d(cid:28) whose solution is (cid:34) (cid:35) (cid:104) (cid:105)ab (cid:104) (cid:105)ab (cid:90) dxi ab a YM b YM c c c ((cid:28)) = W ((cid:28)) c (0); W ((cid:28)) = P exp i d(cid:28) A T (44) 1 1 i d(cid:28) where the subscript \1" refers to that the path-ordered exponent is taken in the J = 1 representation. The parallel transport of a contravariant vector is therefore 9 (cid:104) (cid:105) ab k a ak ak YM b l v ((cid:28)) = c ((cid:28))e ((cid:28)) = e ((cid:28)) W ((cid:28)) e (0)v (0); (45) 1 l from where we immediately get the needed relation between the ‘gravitational’ and Yang{ Mills parallel transporters: (cid:104) (cid:105) (cid:104) (cid:105) k ab G a YM bl W ((cid:28)) = e ((cid:28)) W ((cid:28)) e (0): (46) 1 k 1 l The relation becomes especially neat for the Wilson loops, i.e. for the traces of parallel transporters along closed contours. Since for the closed contour the vielbeins at the end points are the same, ea(1) = ea(0), we get k k (cid:104) (cid:105) (cid:104) (cid:105) 1 k 1 aa G G YM YM W = W = W = W : (47) vector 1 1 1 3 k 3 In a similar way one can show that the same equation holds true for the gravitational parallel transporter of covariant vectors and, more generally, for parallel transporters of any integer spin J. In this case the Yang{Mills Wilson loop should be taken in the same representation as the gravitational one: G YM W = W : (48) J J In eq. (48) it is understood that the r.h.s. is expressed through the Yang{Mills (cid:12)eld equal to the spin connection according to eq. (37), while the l.h.s. is expressed through the Christo(cid:11)el symbols, that is through the metric. It should be stressed that the spin connection is de(cid:12)ned via the vielbein which is not uniquely determined by the metric tensor. Nevertheless, the Wilson loop, being a gauge-invariant quantity, is uniquely determined by the metric tensor and its derivatives. This is the meaning of eq. (48). For half-integer J there is no way to de(cid:12)ne the parallel transporter other than through the spin connection. Nevertheless, as we show in section 8 where we present the holonomy for any spin in a surface form, the ‘gravitational Wilson loop’ is also expressible through the metric tensor and its derivatives, even for half-integer spins. 4.2 d=4 In four Euclidean dimensions the rotational group is SO(4) = SU(2)(cid:2)SU(2), therefore all 1 irreducible representations of SO(4) can be classi(cid:12)ed as (J ;J ) where J = 0; ;1;::: are 1 2 1;2 2 the representations of the two SU(2) subgroups. For example, the 4-vector representation whose parallel transporter has been considered in the beginning of this section, transforms as the (1; 1) representation of SU(2)(cid:2)SU(2). 2 2 Because of this, it is convenient to decompose the spin connection !AB into self-dual and (cid:22) anti-self-dual parts using ’t Hooft’s (cid:17) and (cid:17)(cid:22) symbols. They are de(cid:12)ned as 1 (cid:17)aAB = Tr(cid:27)a((cid:27)A+(cid:27)B− −(cid:27)B+(cid:27)A−); (cid:27)A(cid:6) = ((cid:6)i(cid:27);1); (49) 2i 1 (cid:17)(cid:22)aAB = Tr(cid:27)a((cid:27)A−(cid:27)B+ −(cid:27)B−(cid:27)A+): (50) 2i We use capital Latin characters to denote ﬂat 4-dimensional vierbein indices, A;B;::: = 1;2;3;4, while a;b;::: = 1;2;3; (cid:27)a are the three Pauli matrices. The spin connection !AB (cid:22) 10

We recall the non-Abelian Stokes theorem for the Wilson loop in the Yang –Mills theory and discuss its meaning. ∂iya/|y| and hence. √ g is itself a

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