Infinitely many conservation laws in self-dual Yang--Mills theory
aa r X i v : . [ h e p - t h ] J u l Infinitely many conservation laws in self-dualYang–Mills theory
C. Adam a ) ∗ , J. S´anchez-Guill´en a ) † and A. Wereszczy´nski b ) ‡ a ) Departamento de Fisica de Particulas, Universidad de Santiago and Instituto Galego de Fisica de Altas Enerxias(IGFAE) E-15782 Santiago de Compostela, Spain b ) Institute of Physics, Jagiellonian University, Reymonta 4, Krakow, Poland
October 29, 2018
Abstract
Using a nonlocal field transformation for the gauge field known as Cho–Faddeev–Niemi–Shabanovdecomposition as well as ideas taken from generalized integrability, we derive a new family of infinitelymany conserved currents in the self-dual sector of SU (2) Yang-Mills theory. These currents may be relatedto the area preserving diffeomorphisms on the reduced target space. The calculations are performed in acompletely covariant manner and, therefore, can be applied to the self-dual equations in any space-timedimension with arbitrary signature. Keyword: Integrability, Conservation Laws, Self-dual Yang-Mills theoryPACS: 05.45.Yv
A powerful tool in the theory of topological solitons is the derivation of lower bounds for the energy (orEuclidean action) in terms of topological charges. Together with these bounds, in some cases one mayderive first order equations (so-called Bogomolny equations) such that any field configuration obeying theseBogomolny equations automatically saturates the topological lower bound and is a true minimizer of theenergy (or Euclidean action) functional. Obviously, any field configuration obeying the Bogomolny equationsautomatically obeys the original second order Euler–Lagrange equations, whereas the converse is not true ingeneral.In addition to providing true minimizers of the energy functional, these Bogomolny equations, due to theirmore restrictive nature, tend to enhance the number of symmetries and conservation laws. Sometimes, thereexist infinitely many symmetries and infinitely many conservation laws for the Bogomolny equations. Further,the Bogomolny equations are usually not of the Euler–Lagrange type, therefore for those symmetries whichare not symmetries of the original second order system, the issue of conservation laws has to be investigatedseparately, that is, Noether’s theorem does not apply. A theory where this happens is, for instance, theCP(1) model in 2+1 dimensions. For this theory both the infinitely many symmetries and the infinitelymany conservation laws of the Bogomolny sector have been calculated, e.g., in [1], and, indeed, they turnout to be different. Similar investigations for gauge theories have been performed recently. In the case ofthe Abelian Higgs model, an equivalent pattern has been found, i.e., there are infinitely many conservedcurrents in the Bogomolny sector, and Noether’s theorem does not apply, see [2]. A slightly different scenariois realized in the Abelian projection of Yang–Mills dilaton theory. There, too, exists a Bogomolny sector, butthis theory has infinitely many symmetries already on the level of the Lagrangian, therefore the symmetriesand conservation laws are related by Noether’s theorem, see [3]. ∗ [email protected] † [email protected] ‡ [email protected]
1n the case of SU(2) Yang–Mills theory, the solutions which minimize the Euclidean action functional areknown as instantons, and the Bogomolny type first order equations are the self-duality equations [4] - [7].The symmetries of the self-dual sector of SU(2) Yang–Mills theory have been studied by various authors [8]- [17]. The result is that the system posesses infinitely many symmetries and that almost all of them arenonlocal when expressed in terms of the original fields. A recent review of this issue can be found in [18], towhich we refer the reader for further information and additional references. Conservation laws of self-dualYang–Mills theory related to the non-local symmetries mentioned above have been studied, e.g., in [12] -[17].A slightly different, more geometric approach to the self-dual Yang–Mills (SDYM) equations focusingdirectly on their integrability has been initiated by R. Ward [19]. In that approach twistor methods areemployed, and the use of twistor methods in the investigation of the SDYM and their conservation laws hasplayed an important role subsequently (for some recent results, see [20] - [23]).Another approach to integrability and conservation laws has been proposed in [24], where a generalizedzero curvature representation suitable for higher-dimensional field theories was developed, analogously tothe zero curvature representation of Zakharov and Shabat, which provides integrable field theories in 1+1dimensions. Among other results, it was demonstrated in that paper that the SDYM permit a generalizedzero curvature representation. But still only finitely many conservation laws have been provided for self-dual Yang–Mills theory in Ref. [24]. It is the purpose of the present paper to further develop the issueof integrability and conservation laws of the self-dual sector of SU(2) Yang–Mills theory along these lines.We will find another set of infinitely many conservation laws by explicit construction. The correspondingconserved currents are nonlocal in terms of the original Yang–Mills field, but they will be local in terms of awell-known nonlocal field redefinition which we shall use in the sequel. In contrast to the conserved currentsfound previously, the ones we shall present below are given by manifestly Lorentz covariant expressions andmay, therefore, easily be generalized to different space time metrics and dimensions. Given the relevance ofself-dual Yang–Mills theories both for strong interaction physics and in a more mathematical context, webelieve that the discovery of these additional conserved currents is of some interest.We want to remark that in a recent paper devoted to similar problems [25], an investigation of integrabilityin the sector of Z N string solutions of Yang–Mills theory has been performed. Z N string solutions areeffectively lower dimensional solutions, but, nevertheless, they also belong to the self-dual sector. Further,the integrability of self-dual Yang–Mills theories on certain four-dimensional product manifolds has been usedin [26], [27] to demonstrate the integrability of abelian and nonabelian Higgs models on general Riemanniansurfaces.Our paper is organized as follows. In Section 2 we present a brief overview of some known results onthe self-dual Yang–Mills (SDYM) equations. Specifically, we present infinitely many nonlocal conservedcurrents as constructed by Prasad et al and by Papachristou. This overview shall serve later on to relateour own findings to these already known results. In Section 3 we recapitulate how the self-dual sector ofSU(2) Yang–Mills theory may be recast into the form of the generalized zero curvature representation. InSection 4 we introduce the Cho–Faddeev–Niemi–Shabanov (CFNS) decomposition of the gauge field andre-express the self-dual equations using this decomposition. Next, we write down the currents in terms ofthe decomposition fields and prove that they are conserved. Section 5 contains our conclusions. In theappendix we display the canonical four momenta and field equations which we need in the main text. J formulation of the SDYM The self-dual sector of SU (2) Yang-Mills theory in Euclidean space-time is constituted by gauge fields A aµ satisfying the following equations F aµν = ∗ F aµν , (1)where F aµν ≡ ∂ µ A aν − ∂ ν A aµ + ǫ abc A bµ A cν , ∗ F aµν ≡ ǫ µνρσ F ρσ (2)It is convenient to rewrite them as F ayz = 0 , F a ¯ y ¯ z = 0 , F ay ¯ y + F az ¯ z = 0 , (3)2here the new independent variables are defined as y = 1 √ x + ix ) , ¯ y = 1 √ x − ix ) , z = 1 √ x − ix ) , ¯ z = 1 √ x + ix ) . Defining the self-dual gauge fields as A ay = g − ∂ y g , A az = g − ∂ z g , A a ¯ y = g − ∂ ¯ y g , A a ¯ z = g − ∂ ¯ z g (4)we identically fulfill the first two equations in (3). Here, g , g are arbitrary group elements in SU(2). Thenthe third expression leads to a nontrivial equation giving an equivalent formulation of the self-dual equations F [ J ] ≡ ∂ ¯ y (cid:0) J − ∂ y J (cid:1) + ∂ ¯ z (cid:0) J − ∂ z J (cid:1) = 0 , (5)where J = g g − . (6)In other words, solutions of the self-dual sector are defined by Eq. (5). There is a formulation of the SDYM equations in terms of a linear system [8]-[10]. Namely, consider anauxiliary matrix field ψ defined by the following set of equations ∂ ¯ z ψ = λ (cid:0) ∂ y ψ + J − J y ψ (cid:1) , − ∂ ¯ y ψ = λ (cid:0) ∂ z ψ + J − J z ψ (cid:1) . (7)In fact, this is just the Lax pair formulation. The SDYM equations (5) are derived as a consistency (inte-grability) condition ψ ¯ z ¯ y = ψ ¯ y ¯ z . Further, it is possible to find the related B¨acklund transformation (BT) [8]. It is given by J ′ − J ′ y − J − J y = λ ( J ′ − J ) ¯ z , J ′ − J ′ z − J − J z = λ ( J ′ − J ) ¯ y . (8)If J is a solution of SDYM then J ′ is a new solution of SDYM. Further, this BT is an infinitesimal BT i.e.,a new solution generated by the BT may be found performing an infinitesimal transformation which leavesthe SDYM equation invariant. The pertinent transformation reads J − δJ = − ψT a ψ − α a (9)where T a is a basis of the Lie algebra for the gauge fields and α a are infinitesimal parameters. Indeed, if weassume that J ′ = J + δJ then we get the BT.This symmetry transformation gives the following commutator[ δ α , δ β ] J = α a β b C cab ddλ ( λδ c J ) , (10)where C cab are the structure constants of the Lie algebra. If we expand ψ = P ∞ n =0 λ n ψ ( n ) then we get theKac-Moody algebra [ δ ( m ) α , δ ( n ) β ] J = α a β b C cab λδ ( m + n ) c J, (11)where δ ( n ) is defined as J − δJ = P ∞ n =0 λ n J − δ ( n ) J and δ ( n ) J = − J P ∞ n =0 ψ ( n ) T ψ ( m − n ) . In this way,one may explain the hidden (infinite) symmetries observed by L. Dolan [11]. It is precisely the symmetrytransformation (9) mentioned above. The SDYM in J formulation is an Euler–Lagrange system, and the Noether theorem applies. Indeed, Eq.(5) may be easily derived from the Euclidean action S = Z d x Tr[( J − ∂ µ J )( J − ∂ µ J )] . (12)3herefore, the derivation of an infinite set of symmetries indicates that there should exist infinitely manyconserved quantities, as is expected in any case for an integrable system. In fact, several families of nonlocalconserved currents have been found. All these constructions use in an essential way the J -formulation of theself-dual sector and therefore are unique for 4-dimensional Euclidean space-time. Moreover, manifest Lorentzcovariance is lost since we introduced the complex variables y, z . On the other hand, this formulation of theself-dual equations possesses the advantage that equation (5) has the form of a conservation law.The first set of conserved currents was discovered by Prasad et al [12], [13]. The construction reads asfollows. Let us rewrite the SDYM equation as( v ( n ) y ) ¯ y + ( v ( n ) z ) ¯ z = 0 and v (1) y = J − J y , v (1) z = J − J z , (13)where v ( n ) y , v ( n ) y , n = 1 , , ... are higher conserved currents, which can be constructed by induction (itera-tively). One has to define a set of potentials X ( n ) v ( n ) y = ∂ ¯ z X ( n ) , v ( n ) z = − ∂ ¯ y X ( n ) , X (0) = I. (14)Then, if the n -th current has been found, the next one is given by the formula v ( n +1) y = ( ∂ y + J − J y ) X ( n ) , v ( n +1) z = ( ∂ z + J − J z ) X ( n ) . (15)A different family of nonlocal conservation laws, nontrivially related to Prasad’s ones, was presented byPapachristou [14]. The basic idea was to reformulate the SDYM equation using the potential symmetries.At the beginning we have a SDYM field J obeying F [ J ] = 0 and introduce a potential X (similarly as inPrasad’s work) J − J y ≡ X ¯ z , J − J z ≡ − X ¯ y . (16)The consistency (integrability) condition ( X ¯ z ) ¯ y = ( X ¯ y ) ¯ z gives F [ J ] = 0, whereas the condition ( J y ) z = ( J z ) y leads to the potential SDYM equation (PSDYM) G [ X ] ≡ X y ¯ y + X z ¯ z − [ X ¯ y , X ¯ z ] = 0 . (17)The point is that this expression may also be written as a conservation law ∂ ¯ y (cid:18) X y −
12 [
X, X ¯ z ] (cid:19) + ∂ ¯ z (cid:18) X z + 12 [ X, X ¯ y ] (cid:19) = 0 . (18)Therefore, we arrive at a new current. This procedure may be repeated. We introduce a new potential Y tothe last formula X y −
12 [
X, X ¯ z ] = Y ¯ z , X z + 12 [ X, X ¯ y ] = Y ¯ y (19)and consider the consistency condition ( X y ) z = ( X z ) y . As a result we derive a new PSDYM equation whichhas the form of a conservation law, as well. One may continue with this procedure and, at least in principle,derive an infinite set of conserved quantities. There is some similarity between the two sets of currents,however, the relation between them is non-trivial [14].The importance of the PSDYM equation originates in the observation that there is a one-to-one correspon-dence between symmetries of the SDYM and PSDYM, as it was formulated in the theorem by Papachristou[15] δX = α Φ is a symmetry of G [ X ] ⇔ δJ = αJ Φ is a symmetry of F [ J ] , (20)where X → X ′ = X + α Φ is a transformation which leaves the PSDYM invariant: G [ X ′ ] = 0 if G [ X ] = 0,or in other words δG ≡ H (Φ) = Φ y ¯ y + Φ z ¯ z + [ X ¯ z , Φ ¯ y ] − [ X ¯ y , Φ ¯ z ] = 0 . (21)The next step is to find a B¨acklund transformation generating the symmetries of the PSDYM [15] λ Φ ′ ¯ z = Φ y + [ X ¯ z , Φ] , λ Φ ′ ¯ y = − Φ z + [ X ¯ y , Φ] , (22)provided X is any given solution of the PSDYM equation, for example (16). Then starting with anylocal symmetry of the PSDYM Φ (0) (or the SDYM as they are in one-to-one correspondence) one is ableto construct an infinite tower of symmetries { Φ ( n ) } ∞ n =0 . Moreover, as the B¨acklund transformation (22)4mmediately provides a conservation law we get an infinite series of the conserved quantities, each based ona particular local symmetry of the SDYM equations.The extensive analysis of such families of conserved quantities has been performed by Papachristou [16]. Heintroduced a recursion operator ˆ R which transforms one symmetry of the PSDYM equation into anotherone and is given by a formal integration of the B¨acklund transformation (22)ˆ R ≡ ∂ − z ( ∂ y + [ X ¯ z , ]) . (23)To be precise, he constructed an infinite set of Lie derivatives ∆ ( n ) X = Φ ( n ) , where Φ ( n ) is a symmetry ofthe PSDYM equation as ∆ ( n ) k X = R ( n ) L k X. (24)Here L k is a symmetry operator for the PSDYM equation corresponding to a given local symmetry. Theresults may be summarized as follow.For internal symmetries Φ ≡ ∆ k X ≡ L k X = [ X, T k ], where T k is a basis for the su (2) Lie algebra of thegauge fields, we get that the infinite set of transformations∆ ( n ) k X = R ( n ) L k X = R ( n ) [ X, T k ] (25)obeys the Kac-Moody algebra [ ∆ ( m ) i , ∆ ( n ) j ] X = C kij ∆ ( m + n ) k X. (26)Once again, it is exactly the hidden symmetry of SDYM found by L. Dolan.Finally let us discuss the nine local (point) symmetries of the SDYM L = ∂ y , L = ∂ z , L = z∂ y − ¯ y∂ ¯ z , L = y∂ z − ¯ z∂ ¯ y , L = y∂ y − z∂ z − ¯ y∂ y + ¯ z∂ z , (27) L = 1 + y∂ y + z∂ z , L = 1 − ¯ y∂ y − ¯ z∂ z , L = yL + ¯ z ( y∂ ¯ z − z∂ ¯ y ) , L = zL + ¯ y ( z∂ ¯ y − y∂ ¯ z ) (28)The subset { L ...L } provides an infinite set of transformations∆ ( n ) k X = R ( n ) L k X, k = 1 .. L and L give two sets of infinitely many transformations∆ ( n ) X = R ( n ) LX, L = L or L (30)leading to two copies of the Virasoro algebra. Generators L and L probably do not result in any algebraicstructure.Obviously, conservation laws do not have to correspond to conserved charges. This happens, e.g., if thespatial integrals of the fluxes (charge densities) do not converge. As observed by Ioannidou and Ward [17],the nonlocal currents found by Prasad [12] and Papachristou [14], [15] lead to densities which diverge afterintegration. To be precise, it was discussed for the chiral model in (2+1) dimension but these results shouldhold also for SDYM. A general argument is the following. All nonlocal conserved currents of type [12],[14], [15], [16] are constructed using the integral operator ∂ − and, further, the instanton field is power-likelocalized. Thus, after a sufficient number of integrations we arrive at a divergent quantity. Here, following [24], we very briefly describe the self-dual sector of SU (2) Yang-Mills theory in the language ofgeneralized integrability. The basic step in this framework is the choice of a reducible Lie algebra ˜ G = G ⊕ H ,where G is a Lie algebra and H is an Abelian ideal (in practice, a representation space of G ), togetherwith a connection A µ ∈ G and a vector field B µ ∈ H . A system possesses the generalized zero curvaturerepresentation if its equations of motion may be encoded in two conditions. Namely, the flatness of theconnection ∂ µ A ν − ∂ ν A µ + [ A µ , A ν ] = 0 (31)5nd the covariant constancy of the vector field ∂ µ B µ + [ A µ , B µ ] = 0 . (32)Usually, one assumes a trivial connection i.e., A µ = g − ∂ µ g , where g ∈ G . In this case, one can easilyconstruct conserved currents J µ = g B µ g − . We say that a system is integrable if the number of currents is infinite. As it is equal to the dimension ofthe Abelian ideal H , the integrability condition is simply dim H = ∞ .Let us now express the self-dual equations of SU (2) YM in this manner. Again, we use the representationof the self-dual equations via the equation for the J matrix. In order to accomplish that we introduce a flatconnection A µ and a covariantly constant vector B µ taking values in an Abelian ideal in the following way A µ = J − ∂ µ J = A rµ T r (33) B ¯ y = A r ¯ y S r , B ¯ z = A r ¯ z S r , B y = 0 , B z = 0 , (34)where T r , S r form a basis satisfying[ T r , T s ] = C urs T u , [ T r , S s ] = C urs S u , [ S r , S s ] = 0 . Obviously, the connection is flat as it is a pure gauge configuration. Moreover the condition for the vectorfield i.e., D µ B µ = 0 is equivalent to the self-dual equation (5). One can construct conserved currents J ¯ y = A r ¯ y J S r J − J ¯ z = A r ¯ z J S r J − , J y = 0 , J z = 0 , (35)then, the conservation laws are just the self-dual equations (5). More conservation laws may be derived asdiscussed in the previous section.Of course, the obtained result is not surprising. A system which possesses the standard zero curvature repre-sentation admits also the generalized zero curvature formulation. However, there is a simple prescription howto construct an infinite family of additional conserved currents for a model with generalized zero curvatureformulation. In general, they are spanned by the canonical momenta conjugated to the field degrees of free-dom. It is important to check whether such currents can be also found for the self-dual sector of the SU (2)YM theory, and, if the answer is positive, what is their relation with the standard non-local conservationlaws described before. In order to derive such conserved quantities in an exact form we perform a nonlocal change of variablesknown as the Cho–Faddeev–Niemi–Shabanov decomposition [28] - [33]. The decomposition ~A µ = C µ ~n + ∂ µ ~n × ~n + ~W µ (36)relates the original SU (2) non-Abelian gauge field with three fields: a three component unit vector field ~n pointing into the color direction, an Abelian gauge potential C µ and a color vector field W aµ whichis perpendicular to ~n . The fields are not independent. In fact, as we want to keep the correct gaugetransformation properties δn a = ǫ abc n b α c , δW aµ = ǫ abc W bµ α c , δC µ = n a α aµ (37)under the primary gauge transformation δA aµ = ( D µ α ) a = α aµ + ǫ abc A bµ α c (38)one has to impose the constraint ( n bµ ≡ ∂ µ n b etc.) ∂ µ W aµ + C µ ǫ abc n b W cµ + n a W bµ n bµ = 0 . (39)6n the subsequent analysis we assume a particular form for the valence field W aµ . It is equivalent to a partialgauge fixing where one leaves a residual local U (1) gauge symmetry. Namely, W aµ = ρn aµ + σǫ abc n bµ n c , (40)where ρ, σ are real scalars. For reasons of convenience we combine them into a complex scalar v = ρ + iσ .Then the Lagrange density takes the form ( u µ ≡ ∂ µ u etc.) L = F µν − − | v | ) H µν + (1 − | v | ) H µν + 8(1 + | u | ) (cid:2) ( u µ ¯ u µ )( D ν vD ν v ) − ( D µ v ¯ u µ )( D ν vu ν ) (cid:3) , (41)where H µν = ~n · [ ~n µ × ~n ν ] = − i (1 + | u | ) ( u µ ¯ u ν − u ν ¯ u µ ) , H µν = 8(1 + | u | ) [( u µ ¯ u µ ) − u µ ¯ u ν ] (42)and the covariant derivatives read D µ v = v µ − ieC µ v , D µ v = ¯ v µ + ieC µ ¯ v and we expressed the unit vectorfield by means of the stereographic projection ~n = 11 + | u | (cid:0) u + ¯ u, − i ( u − ¯ u ) , | u | − (cid:1) . Further, F µν ≡ ∂ µ C ν − ∂ ν C µ is the Abelian field strength tensor corresponding to the Abelian gauge field C µ . Notice that only thecomplex field v couples to the gauge field via the covariant derivative. Now, we apply the CFNS decomposition to the self-dual equations. As we know the full field strength tensorreads ~F µν = (cid:2) F µν − (1 − | v | ) H µν (cid:3) ~n + 12 (cid:2) ( D µ v + D µ v ) ~n ν − ( D ν v + D ν v ) ~n µ (cid:3) +12 i (cid:2) ( D µ v − D µ v ) ~n ν × ~n − ( D ν v − D ν v ) ~n µ × ~n (cid:3) . (43)Therefore, using the self-dual equations (1) we get two expressions, one parallel and one perpendicular tothe color vector ~n ǫ µνρσ [ F ρσ − (1 − | v | ) H ρσ ] = F µν − (1 − | v | ) H µν (44)and12 ǫ µνρσ (cid:2)(cid:2) ( D ρ v + D ρ v ) ~n σ − ( D σ v + D σ v ) ~n ρ (cid:3) − i (cid:2) ( D ρ v − D ρ v ) ~n σ × ~n − ( D σ v − D σ v ) ~n ρ × ~n (cid:3)(cid:3) = (cid:2) ( D µ v + D µ v ) ~n ν − ( D ν v + D ν v ) ~n µ (cid:3) + i (cid:2) ( D µ v − D µ v ) ~n ν × ~n − ( D ν v − D ν v ) ~n µ × ~n (cid:3) . (45)For later convenience we now want to derive some constraints which result from these two sets of equations.On the one hand, after projection on ~n µ , Eq. (45) gives( D µ v + D µ v ) ~n µ · ~n ν − ( D ν v + D ν v ) ~n µ − i ( D µ v − D µ v ) H µν = − iǫ µνλω ( D λ v − D λ v ) H µω . (46)On the other hand, if we multiply (45) by × ~n µ and project on ~n then we get( D µ v − D µ v ) ~n µ · ~n ν − ( D ν v − D ν v ) ~n µ − i ( D µ v + D µ v ) H µν = − iǫ µνλω ( D λ v + D λ v ) H µω . (47)Both equations lead to the simple expression D ν v ( u µ ¯ u µ ) − u ν ( D µ v ¯ u µ ) = ǫ µνρσ D ρ vu µ ¯ u σ (48)7nd its complex conjugate. This expression just constitutes a system of linear homogeneous algebraic equa-tions for the unknowns D µ v , M µν D µ v = 0 , M µν = ( u α ¯ u α ) δ µν − u µ ¯ u ν − ǫ µνρσ u ρ ¯ u σ . (49)In order to find all solutions of this system of equations we consider the corresponding eigenvalue problem M µν D µ v = λD µ v . Of course, a solution exists if and only if the determinant vanishesDet( ˆ M − λI ) = 0 . (50)On the other hand one can find thatDet( ˆ M − λI ) = λ ( λ − u µ ¯ u µ )( u µ ¯ u ν − λu µ ¯ u µ + λ ) . (51)Generically there is a single eigenvalue λ = 0 corresponding to the solution D µ v = f u µ , (52)where f is an arbitrary function. However, if the complex field u obeys the complex eikonal equation u µ = 0,then λ = 0 is a degenerate eigenvalue with degeneracy 2. In this case, there exists a second solution. Thissecond solution may be expressed more easily in terms of real vectors. Indeed, if we write u = a + ib thenthe complex eikonal equation corresponds to a µ b µ = 0 , a µ = b µ . (53)If we introduce analogously D µ v = c µ + id µ then the second solution is given by a µ c µ = b µ c µ = a µ d µ = b µ d µ = c µ d µ = 0 (54)and c µ = d µ . (55)The vector D µ v = c µ + id µ is unique up to a multiplication by an arbitrary complex function, as befits thesolution to a complex, homogeneous linear equation. Conditions (54), (55) imply that the complex vector D µ v has to obey u µ D µ v = ¯ u µ D µ v = 0 , D µ vD µ v = 0 (56)in order to be a solution of the second type. We remark that a wide class of explicitly known instantonconfigurations, like, e.g., the cylindrically symmetric solutions found by Witten [34], belongs to this secondcase.A further possibility, u α ¯ u α = 0, which would lead to a even higher degeneracy, is physically uninterestingsince it leads to the trivial solutions u = const.Taking into account formula (48) and its general solutions discussed above we find three constraints whichare satisfied by all self-dual configurations( D λ vu λ )( u β ¯ u β ) − ( D λ v ¯ u λ ) u β = 0 , (57)( D λ v ) ( u β ¯ u β ) − ( D λ v ¯ u λ )( D β vu β ) = 0 , (58)( D ν vD ν v ) u µ − ( D ν vu ν )( D µ vu µ ) = 0 . (59) Following considerations presented, e.g., in [35] - [37], the family of conserved currents may be constructedin the following form j Gµ = i (1 + | u | ) (cid:18) ¯ π µ ∂G∂u − π µ ∂G∂ ¯ u (cid:19) , (60)where G is an arbitrary real function of the complex field u i.e., G = G ( u, ¯ u ) and π µ is the canonicalmomentum (73). The four-divergence reads ( G u ≡ ∂ u G etc.) ∂ µ j Gµ = i (1 + | u | ) [ G u ∂ µ ¯ π µ − G ¯ u ∂ µ π µ + G uu u µ ¯ π µ + G u ¯ u ¯ u µ ¯ π µ − G ¯ uu u µ π µ − G ¯ u ¯ u ¯ u µ π µ ] +2 i (1 + | u | )( u ¯ u µ + ¯ uu µ )( G u ¯ π µ − G ¯ u π µ ) . (61)8r ∂ µ j Gµ = i (1 + | u | ) (cid:20) G u (cid:18) ∂ µ ¯ π µ + 2 u | u | ¯ u µ ¯ π µ (cid:19) − G ¯ u (cid:18) ∂ µ π µ + 2¯ u | u | π µ u µ (cid:19) + G u ¯ u (¯ u µ ¯ π µ − u µ π µ ) (cid:21) + i (1 + | u | ) (cid:20)(cid:18) G uu + 2¯ uG u | u | (cid:19) u µ ¯ π µ − (cid:18) G ¯ u ¯ u + 2 uG ¯ u | u | (cid:19) ¯ u µ π µ (cid:21) . (62)Taking into account that ¯ u µ ¯ π µ = u µ π µ and the pertinent field equations (1 + | u | ) ∂ µ π µ + 2¯ uπ µ u µ = 0 weget ∂ µ j Gµ = i (1 + | u | ) (cid:20)(cid:18) G uu + 2¯ uG u | u | (cid:19) u µ ¯ π µ − (cid:18) G ¯ u ¯ u + 2 uG ¯ u | u | (cid:19) ¯ u µ π µ (cid:21) . (63)Due to the arbitrariness of the function G the currents are conserved if u µ ¯ π µ = 0 and ¯ u µ π µ = 0. Theseso-called integrability conditions introduce some new relations between degrees of freedom and, in principle,do not have to be satisfied for all solutions of Yang-Mills theory. However, it turns out that in the self-dualsector both conditions hold identically. To prove it observe that¯ u µ π µ = 8(1 + | u | ) (cid:2) ( D ν vD ν v )¯ u µ − ( D ν v ¯ u ν ) D µ v ¯ u µ (cid:3) , (64)where we have used the antisymmetry of F µν and K µ ¯ u µ ≡ K µ is defined in (75)). The resultingexpression is just the complex conjugate of formula (59) and therefore equals zero for all configurations ofthe self-dual sector.The charges corresponding to the currents (60) are Q G ≡ Z d xj G (65)obey the algebra of area-preseving diffeomorphisms on the target space two-sphere spanned by the field u under the Poisson bracket, where the fundamental Poisson bracket is (with x = y ) { u ( x ) , π ( y ) } = { ¯ u ( x ) , ¯ π ( y ) } = δ ( x − y ) , (66)as usual. Explicitly, the algebra of area-preserving diffeomorphisms is { Q G , Q G } = Q G , G = i (1 + | u | ) ( G , ¯ u G ,u − G ,u G , ¯ u ) . (67)Finally, let us remark that the currents (60) are invariant under the residual U(1) gauge transformationsthat remain after the partial gauge fixing implied by the CFNS decomposition, see Eq. (40). Using this method we are able to construct more families of infinitely many conserved quantities in self-dualYang–Mills theory, which are based on other canonical momenta. They are given by the expressions j Hµ = ¯ P µ ∂H∂v − P µ ∂H∂ ¯ v , (68) j ˜ Gµ = ω µν ∂ ˜ G∂u u ν + ∂ ˜ G∂ ¯ u ¯ u ν ! , (69) j ˜ Hµ = ω µν ∂ ˜ H∂v D ν v + ∂ ˜ H∂ ¯ v D ν v ! , (70)where the function H = H ( u, ¯ u, v ¯ v ) while the functions ˜ G, ˜ H depend on the moduli only ˜ G = ˜ G ( u ¯ u, v ¯ v ) , ˜ H =˜ H ( u ¯ u, v ¯ v ). However, all these currents are trivially conserved. To see this let us analyze the first family indetail. First of all observe that it may be written as j Hµ = H ′ (¯ v ¯ P µ − vP µ ) (71)9here the prime denotes the derivative w.r.t. v ¯ v , and P µ is defined in Eq. (76) . Using the self-dual equationswe find that j Hµ = 8 H ′ (1 + | u | ) (cid:16) ǫ αµβγ u α (¯ vD β v + vD β v )¯ u γ (cid:17) = 8 H ′ (1 + | u | ) (cid:0) ǫ αµβγ u α (¯ vv β + v ¯ v β )¯ u γ (cid:1) . (72)Therefore, these currents are conserved entirely due to the antisymmetry of the ǫ αµβγ tensor. Analogouslyone can check that the two remaining families are trivially conserved, as well. The main achievement of the present paper is the derivation of a new family of infinitely many conservedcurrents for the self-dual sector of classical SU (2) YM theory. This has been accomplished by a combinationof techniques developed in the so-called generalized integrability (generalized zero curvature) formulationwith a nonlocal transformation of the original gauge degrees of freedom (CFNS decomposition). This alter-native procedure provides currents with rather different properties than the previously known ones.First of all, all calculations are done in a completely covariant manner. Therefore, the obtained currents areconserved for the self-dual sector of SU (2) YM in space-times in any dimension with a completely arbitrarysignature.Secondly, these new currents have a more standard geometrical origin. They are the Noether currents cor-responding to the area preserving diffeomorphisms on the two dimensional target space. Therefore theyobey the classical diffeomorphism algebra instead of the Kac-Moody or Virasoro ones. Also, the relationbetween conservation laws and symmetries is different in our case. Although the currents we found generatearea-preserving diffeomorphisms on target space, this does not imply that these diffeomorphisms are sym-metries of the SDYM equations. The reason is that the SDYM equations in the CFNS decomposition arenot Euler–Lagrange, therefore the Noether theorem does not apply (observe that the canonical momentaare derived from the Lagrangian of the original Yang–Mills system, which gives rise to the full Yang–Millsequations).Thirdly, the currents derived here are given in an explicit form. This is an advantage in comparison withthe currents of Prasad and Papachristou, which are given in a more complicated, iterative way and are,therefore, not so easy to work with.Finally, let us briefly mention some possible generalizations and further directions of future investigations.On the one hand, the procedure employed here is based on the generalized zero curvature condition of Ref.[24], which is not restricted to the SDYM. It has been and will be used to detect further integrable sectorsin different field theories. On the other hand, recently other nonlocal decompositions of Yang–Mills theoryhave been proposed, like, e.g., the spin-charge separation of [38] - [39]. It is an interesting question whetherthese decompositions allow to detect further conservation laws in SDYM. This problem is under currentinvestigation. Acknowledgements
C.A. and J.S.-G. thank MCyT (Spain) and FEDER (FPA2005-01963), and support from Xunta de Galicia(grant PGIDIT06PXIB296182PR and Conselleria de Educacion). A.W. acknowledges support from theFoundation for Polish Science FNP (KOLUMB programme) and Ministry of Science and Higher Educationof Poland (grant N N202 126735).
Appendix
Here we calculate the canonical momenta π µ = ∂L∂u µ = 8 i − | v | (1 + | u | ) F µν ¯ u ν + 16 (1 − | v | ) (1 + | u | ) K µ + 8(1 + | u | ) (cid:2) ( D ν vD ν v )¯ u µ − ( D ν v ¯ u ν ) D µ v (cid:3) , (73)¯ π µ = ∂L∂ ¯ u µ = − i − | v | (1 + | u | ) F µν u ν + 16 (1 − | v | ) (1 + | u | ) ¯ K µ + 8(1 + | u | ) (cid:2) ( D ν vD ν v ) u µ − ( D ν vu ν ) D µ v (cid:3) (74)10here K µ = ( u ν ¯ u ν )¯ u µ − ¯ u ν u µ (75)and P µ = ∂L∂v µ = 8(1 + | u | ) (cid:2) ( u ν ¯ u ν ) D µ v − ( D ν vu ν )¯ u µ (cid:3) , (76)¯ P µ = ∂L∂ ¯ v µ = 8(1 + | u | ) [( u ν ¯ u ν ) D µ v − ( D ν v ¯ u ν ) u µ ] (77)and finally ω µν = ∂L∂ ( ∂ µ C ν ) = 4 (cid:0) F µν − (1 − | v | ) H µν (cid:1) . (78)The pertinent equations of motion for the complex u field read ∂ µ π µ = L u = − i ¯ u − | v | (1 + | u | ) F µν u µ ¯ u ν − · u (1 − | v | ) (1 + | u | ) (cid:2) ( u µ ¯ u µ ) − u µ ¯ u ν (cid:3) − u (1 + | u | ) (cid:2) ( u µ ¯ u µ )( D ν vD ν v ) − ( D µ v ¯ u µ )( D ν vu ν ) (cid:3) (79) ∂ µ ¯ π µ = L ¯ u = − iu − | v | (1 + | u | ) F µν u µ ¯ u ν − · u (1 − | v | ) (1 + | u | ) (cid:2) ( u µ ¯ u µ ) − u µ ¯ u ν (cid:3) − u (1 + | u | ) (cid:2) ( u µ ¯ u µ )( D ν vD ν v ) − ( D µ v ¯ u µ )( D ν vu ν ) (cid:3) , (80)while for the complex v field we get ∂ µ P µ = L v = − i ¯ v (1 + | u | ) F µν u µ ¯ u ν + 2 · v (1 + | v | )(1 + | u | ) (cid:2) ( u µ ¯ u µ ) − u µ ¯ u ν (cid:3) + − ie (1 + | u | ) (cid:2) ( u µ ¯ u µ )( C ν D ν v ) − ( C µ ¯ u µ )( D ν vu ν ) (cid:3) (81) ∂ µ ¯ P µ = L ¯ v = − iv (1 + | u | ) F µν u µ ¯ u ν + 2 · v (1 + | v | )(1 + | u | ) (cid:2) ( u µ ¯ u µ ) − u µ ¯ u ν (cid:3) +8 ie (1 + | u | ) [( u µ ¯ u µ )( C ν D ν v ) − ( C µ u µ )( D ν vu ν )] . (82)The equation for the Abelian gauge field has the form ∂ µ ω µν = ∂L∂C ν = − ie (1 + | u | ) (cid:8) ( u µ ¯ u µ ) (cid:2) vD ν v − ¯ vD ν v (cid:3) − v ¯ u ν ( D µ vu µ ) + ¯ vu ν ( D µ v ¯ u µ ) (cid:9) . (83) References [1] Adam C and S´anchez-Guill´en J 2005 JHEP
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