An integral formulation of Yang-Mills on loop space
AAn integral formulation of Yang-Mills on loop space
L. A. Ferreira and G. Luchini
Instituto de F´ısica de S˜ao Carlos; IFSC/USP; Universidade de S˜ao PauloCaixa Postal 369, CEP 13560-970, S˜ao Carlos-SP, Brazil
It is proposed an integral formulation of classical Yang-Mills equations in the presence of sources,based on concepts in loop spaces and on a generalization of the non-abelian Stokes theorem for two-form connections. The formulation leads in a quite direct way to the construction of gauge invariantconserved quantities which are also independent of the parameterization of surfaces and volumes.Our results are important in understanding global properties of non-abelian gauge theories.
PACS numbers: 11.15.-q,11.15.Kc
The aim of the present paper is to propose an inte-gral formulation of the classical equations of motion ofnon-abelian gauge theories. Our approach is based on ageneralization of the non-abelian Stokes theorem for two-form connections, which allows to present the Yang-Millsequations as the equality of an ordered volume integralto an ordered surface integral on its border. The formu-lation leads in a quite simple way to the construction ofgauge invariant conserved quantities which are indepen-dent of the parameterizations of volumes and surfaces.The most appropriate mathematical language to phraseour results is that of generalized loop spaces. There isa quite vast literature on integral and loop space for-mulations of gauge theories [1]. Our approach differs inmany aspects of those formulations even though it sharessome of the ideas and insights permeating them. Wemake however concrete progress in relation to those ap-proaches. The main statement of this paper is:
Consider a Yang-Mills theory for a gauge group G ,with gauge field A µ , in the presence of matter currents J µ , on a four dimensional space-time M . Let Ω be anytridimensional (topologically trivial) volume on M , and ∂ Ω be its border. We choose a reference point x R on ∂ Ω and scan Ω with closed surfaces, based on x R , labelled by ζ , and we scan the closed surfaces with closed loops basedon x R , labelled by τ , and parametrized by σ , as we de-scribe below. The classical dynamics of the gauge fields isgoverned by the following integral equations, on any suchvolume Ω , P e ie (cid:82) ∂ Ω dτdσ (cid:2) αF Wµν + β (cid:101) F Wµν (cid:3) dxµdσ dxνdτ = P e (cid:82) Ω dζdτV J V − (1) where P and P means surface and volume ordered inte-gration respectively, (cid:101) F µν is the Hodge dual of the field ten-sor, i.e. F µν ≡ ε µνρλ (cid:101) F ρλ , with F µν = ∂ µ A ν − ∂ ν A µ + i e [ A µ , A ν ] , e is the gauge coupling constant, α and β are free parameters, and where we have used the notation X W ≡ W − X W , with W being the Wilson line definedon a curve Γ , parameterized by σ , through the equation d Wd σ + i e A µ d x µ d σ W = 0 (2) where x µ ( µ = 0 , , , ) are the coordinates on the fourdimensional space-time M . The quantity V is defined ona surface Σ through the equation d Vd τ − V T ( A, τ ) = 0 (3) with T ( A, τ ) ≡ ie (cid:82) π dσW − (cid:104) αF µν + β (cid:101) F µν (cid:105) W dx µ dσ dx ν dτ .and where J ≡ (cid:90) π dσ (cid:26) ieβ (cid:101) J Wµνλ dx µ dσ dx ν dτ dx λ dζ + e (cid:90) σ dσ (cid:48) (4) × (cid:104) (cid:16) ( α − F Wκρ + β (cid:101) F Wκρ (cid:17) ( σ (cid:48) ) , (cid:16) αF Wµν + β (cid:101) F Wµν (cid:17) ( σ ) (cid:105) × d x κ d σ (cid:48) d x µ d σ (cid:18) d x ρ ( σ (cid:48) ) d τ d x ν ( σ ) d ζ − d x ρ ( σ (cid:48) ) d ζ d x ν ( σ ) d τ (cid:19)(cid:27) where (cid:101) J µνλ is the Hodge dual of the current, i.e. J µ = ε µνρλ (cid:101) J νρλ . The Yang-Mills equations are recoveredfrom (1) in the case where Ω is taken to be an infinitesi-mal volume. Under appropriate boundary conditions theconserved charges are the eigenvalues of the operator Q S = P e ie (cid:82) ∂S dτdσ ( αF Wµν + β (cid:101) F Wµν ) dxµdσ dxνdτ = P e (cid:82) S dζdτV J V − (5) where S is the -dimensional spatial sub-manifold of M .Equivalently the charges are Tr Q NS . In order to prove that (1) does correspond to and inte-gral formulation of the classical Yang-Mills dynamics, weshall start by describing the generalization of the non-abelian Stokes theorem as formulated in [2, 3]. Con-sider a surface Σ scanned by a set of closed loops withcommon base point x R on the border ∂ Σ. The pointson the loops are parameterized by σ ∈ [0 , π ] and eachloop is labeled by a parameter τ such that τ = 0 corre-sponds to the infinitesimal loop around x R , and τ = 2 π to the border ∂ Σ. We then introduce, on each pointof M , a rank two antisymmetric tensor B µν taking val-ues on the Lie algebra G of G , and construct a quan-tity V on the surface Σ through (3), but with T ( A, τ )replaced by T ( B, A, τ ) ≡ (cid:82) π dσ W − B µν W d x µ d σ d x ν d τ ,and where the σ -integration is along the loop Γ labeled a r X i v : . [ h e p - t h ] S e p by τ , and W is obtained from (2), by integrating it alongΓ from the reference point x R to the point labeled by σ , where B µν is evaluated. By integrating (3), from theinfinitesimal loop around x R to the border of Σ, we ob-tain V = V R P e (cid:82) π dτ (cid:82) π dσW − B µν W dxµdσ dxνdτ , where P means surface ordering according to the parameteriza-tion of Σ as described above, and V R is an integrationconstant corresponding to the value of V on an infinites-imal surface around x R . If one changes Σ, keeping itsborder fixed, by making variations δx µ perpendicular toΣ then V varies according to (see sec. 5.3 of [2], sec. 2.3of [3], or the appendix of [4]) δV V − ≡ (cid:90) π dτ (cid:90) π dσ V ( τ ) { (6) W − [ D λ B µν + D µ B νλ + D ν B λµ ] W d x µ d σ d x ν d τ δx λ − (cid:90) σ dσ (cid:48) (cid:2) B Wκρ ( σ (cid:48) ) − ieF Wκρ ( σ (cid:48) ) , B Wµν ( σ ) (cid:3) dx κ dσ (cid:48) dx µ dσ × (cid:18) d x ρ ( σ (cid:48) ) d τ δx ν ( σ ) − δx ρ ( σ (cid:48) ) d x ν ( σ ) d τ (cid:19)(cid:27) V − ( τ )where D µ ∗ = ∂ µ ∗ + i e [ A µ , ∗ ]. The quantity V ( τ ) ap-pearing on the r.h.s. of (6) is obtained by integrating(3) from the infinitesimal loop around x R to the the looplabelled by τ on the scanning of Σ described above. Notethat the two σ -integrations on the second term on ther.h.s. of (6) are performed on the same loop labelled by τ . Consider now the case where the surface Σ is closed,and the border of Σ is contracted to x R . The expression(6) gives then the variation of V when we vary Σ keeping x R fixed. Therefore, if one starts with an infinitesimalclosed surface Σ R around x R one can blows it up until itbecomes Σ. One can label all those closed surfaces usinga parameter ζ ∈ [0 , π ], such that ζ = 0 corresponds toΣ R and ζ = 2 π to Σ. The expression (6) can be seen asa differential equation on ζ defining V on the surface Σ,i.e. d Vd ζ − K V = 0 (7)where K corresponds to the r.h.s. of (6) with δx µ re-placed by d x µ d ζ . By integrating (7) from Σ R to Σ, oneobtains V evaluated on Σ, which is now an ordered vol-ume integral, over the volume Ω inside Σ, and the order-ing is determined by the scanning of Ω by closed surfacesas described above. But this result has of course to bethe same as that obtained by integrating (3) when thesurface is closed, namely ∂ Ω. Therefore, we obtain thegeneralized non-abelian Stokes theorem for a two-formconnection B µν , parallel transported by a one-form con-nection A µ V R P e (cid:82) ∂ Ω dτdσW − B µν W dxµdσ d xνd τ = P e (cid:82) Ω dζ K V R (8) where P means volume ordering according to the scan-ning described above, and V R is the integration constantobtained when integrating (3) and (7). It corresponds infact to the value of V at the reference point x R . Notethat such theorem holds true on a space-time of any di-mension, and since the calculations leading to it make nomention to a metric tensor, it is valid on flat or curvedspace-time. The only restrictions appear when the topol-ogy of the space-time is non-trivial (existence of handlesor holes for instance).Going back to (1) one notes that it can be ob-tained from (8) by replacing B µν by ie (cid:104) α F µν + β (cid:101) F µν (cid:105) ,and using the Yang-Mills equations, D ν F νµ = J µ and D ν (cid:101) F νµ = 0, to replace ( D λ B µν + D µ B νλ + D ν B λµ ) in(6) by ( − ieβ (cid:101) J µνλ ), and so K introduced in (7) is nowgiven by K = (cid:82) π dτ V J V − , with J given in (4).Therefore, (1) is a direct consequence of the Yang-Millsequations and the Stokes theorem (8). Note that V R in-troduced in (8), does not appear in (1) because it hasto lie in the centre Z ( G ) of G to keep the gauge co-variance of (1) (see [5]). On the other hand the integralequation (1) implies the local Yang-Mills equations. Inorder to see that, consider the case where Ω is a infinites-imal volume of rectangular shape with lengths dx µ , dx ν and dx λ along three chosen Cartesian axis labelled by µ , ν and λ . We choose the reference point x R to be at avertex of Ω. By considering only the lowest order contri-butions, in the lengths of Ω, to the integrals in (1), oneobserves that the surface and volume ordering becomeirrelevant. We have to pay attention only to the orien-tation of the derivatives of the coordinates w.r.t. theparameters σ , τ and ζ , determined by the scanning of Ωdescribed above. In addition, the contribution of a givenface of Ω for the l.h.s. of (1) can be obtained by evalu-ating the integrand on any given point of the face sincethe differences will be of higher order. Consider the twofaces parallel to the plane x µ x ν . The contribution to thel.h.s. of (1) of the face at x R is given by − ie ( αF µν + β (cid:101) F µν ) x R dx µ dx ν , with the minus sign due to the orienta-tion of the derivatives, and the contribution of the faceat x R + dx λ is ie ( W − ( αF µν + β (cid:101) F µν ) W ) ( x R + dx λ ) dx µ dx ν ,with W ( x R + dx λ ) ∼ − ieA λ ( x R ) dx λ . By Taylorexpanding the second term, the joint contribution is ieD λ ( αF µν + β (cid:101) F µν ) x R dx µ dx ν dx λ , with no sums in theLorentz indices. The contributions of the other two pairsof faces are similar, and the l.h.s. of (1) to lowest orderis 1l + ie ( D λ [ αF µν + β (cid:101) F µν ] + cyclic perm.) x R dx µ dx ν dx λ .When evaluating the r.h.s. of (1) we can take the in-tegrand at any point of Ω since the differences are ofhigher order. In addition, the commutator term in J given in (4) is of higher order w.r.t. the first term in-volving the current. Therefore, the r.h.s. of (1) to lowestorder is 1l+ ieβ (cid:101) J µνλ dx µ dx ν dx λ . Equating the coefficientsof α and β one gets the pair of (Hodge dual) Yang-Millsequations.Let us discuss some consequences of (1). In order towrite it for a given volume Ω, we had to choose a ref-erence point x R on its border, and define a scanning ofΩ with surfaces and loops. If one changes the referencepoint and the scanning, both sides of (1) will change.However, the generalized non-abelian Stokes theorem (8)guarantees that the changes are such that both sides arestill equal to each other. Therefore, one can say that (1)transforms “covariantly” under the change of scanningand reference point. In fact to be precise, the equation(1) is formulated not on Ω but on the generalized loopspace L Ω = (cid:8) γ : S → Ω | north pole → x R ∈ ∂ Ω (cid:9) . Theimage of a given γ is a closed surface Σ in Ω contain-ing x R . A scanning of Ω is a collection of surfaces Σ,parametrized by τ , such that τ = 0 corresponds to theinfinitesimal surface around x R and τ = 2 π to ∂ Ω. Suchcollection of surfaces is a path in L Ω and each one cor-responds to Ω itself. In order to perform each mapping γ we scan the corresponding surface Σ with closed loopsstarting and ending at x R , and each loop is parametrizedby σ , in the same way as we did in the arguments leadingto (8). Therefore, the change of the scanning of Ω corre-sponds to a change of path in L Ω. In this sense, the r.h.s.of (1) is a path dependent quantity in L Ω and its l.h.s.is evaluated at the end of the path. Of course, we do notwant physical quantities to depend upon the choice ofpaths in L Ω, neither on the reference point. Note that ifwe take, in the four dimensional space-time M , a closedtridimensional volume Ω c , then the integral Yang-Millsequation (1) implies that P e (cid:72) Ω c dζdτV J V − = 1l (9)since the border ∂ Ω c vanishes, and the ordered integralof the l.h.s. of (1) becomes trivial. On the loop space L Ω c , Ω c corresponds to a closed path starting and end-ing at x R . Consider now a point γ on that closed path,corresponding to a closed surface Σ, in such a way thatΩ corresponds to the first part of the path and Ω to thesecond, i.e. Ω c = Ω +Ω , and Σ is the common border ofΩ and Ω . By the ordering of the integration determinedby (7) one observes that the relation (9) can be splitas P e (cid:82) Ω2 dζdτV J V − P e (cid:82) Ω1 dζdτV J V − = 1l. However, byreverting the sense of integration along the path, one getsthe inverse operator when integrating (7). Therefore, Ω and Ω − are two different paths (volumes) joining thesame points, namely the infinitesimal surface around x R and the surface Σ, which correspond to their border. Onethen concludes that the operator P e (cid:82) Ω dζdτV J V − is in-dependent of the path, and so of the scanning of Ω, aslong as the end points, i.e. x R and the border ∂ Ω, arekept fixed.The path independency of that operator can be used toconstruct conserved charges using the ideas of [2, 3]. First of all, let us assume that the space-time is of the form
S ×
IR, with IR being time and S the spatial sub-manifoldwhich we assume simply connected and without border.An example is when S is the three dimensional sphere S . It follows from (9) that Q S ≡ P e (cid:72) S dζdτV J V − = 1l.That means that Q S is not only conserved in time, butalso that there can be no net charge in S . In fact, there isthe possibility of getting charge quantization conditionsin such case (see [6, 7]).Let us now assume the space-time is not bounded, butstill simply connected, like IR . We shall consider twopaths (volumes) joining the same two points, namely theinfinitesimal surface around x R , which we take to be atthe time x = 0, and the two-sphere at spatial infinity S , ( t ) ∞ , at x = t . The first path is made of two parts. Thefirst part corresponding to the whole space at x = 0, i.e.the volume Ω (0) ∞ inside S , (0) ∞ , the two-sphere at spatialinfinity at x = 0. The second part is a hyper-cylinder S ∞ × I , where I is the time interval between x = 0 and x = t , and S ∞ is a two-sphere at spatial infinity at thetimes on that interval. The second path is also madeof two parts. The first one corresponds to the infinitesi-mal hyper-cylinder S × I , where S is the infinitesimaltwo-sphere around x R and I as before. The second partcorresponds to Ω ( t ) ∞ , the whole space at time x = t , i.e.the volume inside S , ( t ) ∞ . From the path independencyfollowing from (9) one has that the integration of (7)along those two paths should give the same result, i.e. V ( S ∞ × I ) V (Ω (0) ∞ ) = V (Ω ( t ) ∞ ) V ( S × I ), where we haveused the notation V (Ω) ≡ P e (cid:82) Ω dζdτV J V − , and whereall integrations start at the reference point x R taken tobe at x = 0, and at the border S , (0) ∞ of Ω (0) ∞ . In fact,one obtains V (Ω) by integrating (7), and so one has tocalculate K = (cid:82) π dτ V J V − , on the surfaces scanningthe volume Ω. We shall scan a hyper-cylinder S × I with surfaces, based at x R , of the form given in figure(1.b), with t (cid:48) denoting a time in the interval I . Eachone of such surfaces are scanned with loops, labelled by τ , in the following way. For 0 ≤ τ ≤ π , we scan theinfinitesimal cylinder as shown in figure (1.a), then for π ≤ τ ≤ π we scan the sphere S as shown in figure(1.b), and finally for π ≤ τ ≤ π we go back to x R withloops as shown in figure (1.c). The quantity K can thenbe split into the contributions coming from each one ofthose surfaces as K = K a + K b + K c . In the case of theinfinitesimal hyper-cylinder S × I , the sphere has in-finitesimal radius and so it does not really contribute to K b . We shall assume the currents and field strength van-ish at spatial infinity no slower than J µ ∼ /R δ , and F µν ∼ /R + δ (cid:48) , with δ, δ (cid:48) >
0, for R → ∞ . Thereforethe quantity J , given in (4), vanishes when calculatedon loops at spatial infinity. Consequently, in the case ofthe hyper-cylinder S ∞ × I , the contribution to K b comingfrom the sphere with infinite radius vanishes, and we havethat K calculated on the surfaces scanning S ∞ × I and S × I is the same, and so V (cid:0) S ∞ × I (cid:1) = V (cid:0) S × I (cid:1) .In fact there is more to it, since when we contract theradius of the cylinders in figure 1 to zero the loops infigures (1.a) and (1.c) become the same. Therefore, thequantities J calculated on them are the same except fora minus sign coming from the derivatives dx µ dτ , since theloops in figure (1.a) get longer with the increase of τ , andin figure (1.c) the opposite occurs. In addition, the quan-tity V inside the the expression K = (cid:82) π dτ V J V − is insensitive to that sign since it is obtained by inte-grating (3) starting at x R in both cases. Therefore,it turns out that K a + K c = 0. The loops scanningthe sphere in figure (1.b) have legs linking the referencepoint x R , at x = 0, to the same space point but at x = t (cid:48) , i.e. x t (cid:48) R . Therefore, when integrating (3) one gets V x R = W ( x t (cid:48) R , x R ) − V x t (cid:48) R W ( x t (cid:48) R , x R ), where W ( x t (cid:48) R , x R ) isobtained by integrating (2) along the leg linking x R to x t (cid:48) R , and where we have used the notation V x , meaning V obtained from (3) with reference point x . Using the samearguments and notation one obtains from (4) that, on theloops of figure (1.b), J x R = W ( x t (cid:48) R , x R ) − J x t (cid:48) R W ( x t (cid:48) R , x R ),and so K b,x R = W ( x t (cid:48) R , x R ) − K b,x t (cid:48) R W ( x t (cid:48) R , x R ). Thequantity V (Ω ( t ) ∞ ) is obtained by integrating (7) and byscanning the volume Ω ( t ) ∞ with surfaces of the typeshown in figure (1.b), and where the radius of S varies from zero to infinity keeping the point x tR fixed.Therefore, from the above arguments one gets that V x R (Ω ( t ) ∞ ) = W ( x tR , x R ) − V x tR (Ω ( t ) ∞ ) W ( x tR , x R ). Onethen concludes that such operator has an iso-spectraltime evolution V x tR (Ω ( t ) ∞ ) = U ( t ) V x R (Ω (0) ∞ ) U ( t ) − , with U ( t ) = W ( x tR , x R ) V (cid:0) S × I (cid:1) . Therefore, its eigenval-ues, or equivalently Tr( V x tR (Ω ( t ) ∞ )) N , are constant in time.Note that from the Yang-Mills equations (1) one has thatsuch operator can be written either as a volume or sur-face ordered integrals, and so we have proved (5). Wehave shown that, as a consequence of (9), such opera-tors are independent of the scanning of the volume. Thereference point x tR is on the border of the volume andso at spatial infinity. Then when we change the refer-ence point on the border to (cid:101) x tR , the operator V x tR (Ω ( t ) ∞ )changes under conjugation by W ( (cid:101) x tR , x tR ). However, ourboundary conditions implies that the field strength goesto zero at infinity and so the gauge potential is asymptot-ically flat, and consequently W ( (cid:101) x tR , x tR ) is independentof the choice of path joining the two reference points.Therefore, the conserved quantities are also independentof the base points. In addition, they are gauge invari-ant since, as shown in [5], V x tR (Ω ( t ) ∞ ) → g R V x tR (Ω ( t ) ∞ ) g − R ,with g R being the group element, performing the gaugetransformation, at x tR . Note in addition that if V x tR (Ω ( t ) ∞ )has an iso-spectral evolution so does g c V x tR (Ω ( t ) ∞ ), with g c ∈ Z ( G ). That fact has to do with the freedom we have to choose the integration constants of (3) and (7)to lie in Z ( G ), without spoiling the gauge covariance of(1) (see [5]). x R x t R x t R x R S , ( t ) x t R x R terça-feira, 12 de julho de 2011 (a) x R x t R x t R x R S , ( t ) x t R x R terça-feira, 12 de julho de 2011 (b) x R x t R x t R x R S , ( t ) x t R x R terça-feira, 12 de julho de 2011 (c) FIG. 1: Surfaces of type (b) scan a hyper-cylinder S × I . As an example consider a gauge theory for a gaugegroup G spontaneously broken to a subgroup H bya Higgs field φ in the adjoint representation. For aBPS dyon solution one has E i = sin θ D i φ , and B i =cos θ D i φ , with E i = − F i and B i = − ε ijk F jk , i, j, k = 1 , ,
3, with θ being an arbitrary constant an-gle. At spatial infinity one has that D i φ → ˆ r i πr G (ˆ r ),with ˆ r = 1, r → ∞ , and G (ˆ r ) being an elementof the Lie algebra of H , which is covariantly con-stant, i.e. D µ G (ˆ r ) = 0 [8]. We have that the gaugefield is asymptotically flat at spatial infinity, and soup to leading order one has A µ = ie ∂ µ W W − , andso on S , ( t ) ∞ one has G (ˆ r ) = W G R W − , with G R being the value of G (ˆ r ) at x tR . Therefore, one hasthat V x tR (Ω ( t ) ∞ ) = P e ie (cid:82) S , ( t ) ∞ dτdσ (cid:2) αF Wµν + β (cid:101) F Wµν (cid:3) dxµdσ dxνdτ =exp [ − ie ( α cos θ + β sin θ ) G R ]. Consequently, the con-served charges are given by the eigenvalues of G R , whichcontain among them the magnetic and electric charges ofthe dyon solution. Note that, even though we take G (ˆ r )at x tR , the eigenvalues are independent of the choice of x tR , since G (ˆ r ) at different points at infinity are relatedby conjugation. Acknowledgements
The authors are grateful tofruitful discussions with O. Alvarez, E. Castellano, P. Kli-mas, M.A.C. Kneipp, R. Koberle, J. S´anchez-Guill´en, N.Sawado and W. Zakrzewski. LAF is partially supportedby CNPq, and GL is supported by a CNPq scholarship. [1] S. Mandelstam, Annals Phys. , 1 (1962); C. N. Yang,Phys. Rev. Lett. , 445 (1974); T. T. Wu and C. N. Yang,Phys. Rev. D , 3845 (1975); A. M. Polyakov, Phys. Lett.B , 247 (1979); T. Eguchi, Y. Hosotani, Phys. Lett. B96 , 349 (1980); A. A. Migdal, Phys. Rept. , 199-290 (1983); Y. .M. Makeenko, A. A. Migdal, Phys. Lett.
B88 , 135 (1979); I. Y. Arefeva, Phys. Lett.
B95 , 269-272(1980); Karpacz 1980, Proceedings, Developments In TheTheory Of Fundamental Interactions*, 295-330; R. Gam-bini, A. Trias, Phys. Rev.
D22 , 1380 (1980), Nucl. Phys.
B278 , 436 (1986); S. G. Rajeev, AIP Conf. Proc. , 41-48 (2003), [hep-th/0401215]; R. Loll, Theor. Math. Phys. , 1415 (1992) [Teor. Mat. Fiz. , 481 (1992)].[2] O. Alvarez, L. A. Ferreira and J. Sanchez Guillen, Nucl.Phys. B , 689 (1998) [arXiv:hep-th/9710147].[3] O. Alvarez, L. A. Ferreira and J. Sanchez-Guillen, Int. J.Mod. Phys. A , 1825 (2009) [arXiv:0901.1654 [hep-th]].[4] L. A. Ferreira and G. Luchini, [arXiv:1109.2606 [hep-th]].[5] Consider a gauge transformation A µ → g A µ g − + ie ∂ µ g g − and so F µν → g F µν g − and J µ → g J µ g − .From (2) W → g f W g − i , with g i and g f being the valuesof g at the initial and final points respectively of the pathdetermining W . Consequently, J defined in (4) transformsas J → g R J g − R , with g R being the value of g at x R .One also has T ( A, τ ) → g R T ( A, τ ) g − R , and so from (3) V → g R V g − R . Similarly, one sees that K → g R K g − R , andso (7) also implies that V transforms as V → g R V g − R .Note however that if V is a solution of (3) so is V = k V with k being a constant element of G . Similarly, if V satisfies (7) so does V = V h , with h ∈ G being con-stant. Under a gauge transformation V → g R V g − R , and V → g R V g − R = g R k V g − R . But k is any chosen con-stant group element and it should not depend upon thegauge field, and so it should not change under gauge trans-formations. In fact, the arbitrariness associated to k corre- sponds to the choice of integration constants in (3) and (7).From this point of view we should have V → k g R V g − R .The only way to establish the compatibility is to have k g R = g R k , i.e. k should be an element of the centre Z ( G )of G . A similar analysis applies to V and V . Therefore,the transformation law V → g R V g − R , and so the gaugecovariance of (1), is only valid when the integration con-stants in (3) and (7) are taken in Z ( G ). In such case, V R cancels out of (8) and that is why it does not appear in(1). Consequently (1) transforms covariantly under gaugetransformations.[6] If for some reason at the quantum level α and β arenot free parameters, then one gets quantization condi-tions. Indeed, take for instance Maxwell theory, where G = U (1), and so the commutators in (4) drop, the sur-face and volume ordering are irrelevant, and Q S is unityif (cid:82) S dζdτ dσ (cid:101) J µνλ dx µ dσ dx ν dτ dx λ dζ = πneβ , with n integer.[7] M. Alvarez, D. I. Olive, Commun. Math. Phys. , 13-28 (2000). [hep-th/9906093]; Commun. Math. Phys. ,331-356 (2001). [hep-th/0003155]; Commun. Math. Phys. , 279-305 (2006). [hep-th/0303229].[8] P. Goddard, D. I. Olive, Rept. Prog. Phys.41