Gauge and Integrable Theories in Loop Spaces
GGauge and Integrable Theories in Loop Spaces
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
Abstract
We propose an integral formulation of the equations of motion of a large class of fieldtheories which leads in a quite natural and direct way to the construction of conservationlaws. The approach is based on generalized non-abelian Stokes theorems for p -formconnections, and its appropriate mathematical language is that of loop spaces. Theequations of motion are written as the equality of an hyper-volume ordered integral toan hyper-surface ordered integral on the border of that hyper-volume. The approachapplies to integrable field theories in (1 + 1) dimensions, Chern-Simons theories in (2 + 1)dimensions, and non-abelian gauge theories in (2+1) and (3+1) dimensions. The resultspresented in this paper are relevant for the understanding of global properties of thosetheories. e-mail: [email protected] e-mail: [email protected] a r X i v : . [ h e p - t h ] S e p Introduction
Symmetries play a central role in the understanding of physical phenomena. The laws govern-ing the fundamental interactions in gauge theories and general relativity are strongly basedon symmetry principles. On the other hand the developments of non-perturbative methodsto study strongly coupled system rely on symmetries revealed by deep structures like theweak-strong coupling dualities in gauge theories. Even though the Noether symmetries ofLagrangians and equations of motion are very important in many aspects of a given theory, itis perhaps correct to say that the hidden symmetries are the ones that have proved to be mostefficient in the development of exact methods for non-linear and non-perturbative phenomena.The best examples of that are low dimensional theories with applications in many areas ofphysics like condensed matter, integrable field theories and solitons. The hidden symmetriesresponsible for the solvability of those (1 + 1)-dimensional theories appear in general as gaugesymmetries of an auxiliary flat one-form connection A µ . In fact, the connection is a functionalof the physical fields, and the zero curvature condition for A µ is equivalent to the classicalequations of motion of the theory. The crucial fact here is that the flatness condition implythat the path ordered integral of the connection between two given points is independent ofthe choice of the path joining them. That statement is a conservation law, and the conservedquantities are given by the eigenvalues of the operator obtained by the path ordered integralof A µ over the entire one dimensional space sub-manifold. The exact developments in solitontheories and in many non-linear phenomena in low dimensions, over the last decades, were adirect consequence of that very important and simple fact.If higher dimensional theories present similar structures or not is an open and interestingproblem. There have been several approaches to tackle that question, and we want to discusshere one which is a quite straightforward generalization of the ideas described above. Oneshould expect the conserved quantities of a d + 1 dimensional theory to be associated tointegrals of quantities on the d -dimensional space sub-manifold. However, that space can beseen as a path in a generalized loop space in the following way. Choose a reference point x R in the ( d + 1)-dimensional space-time M , and defined the space of maps LM from the ( d − S d − into M , such that the north pole of S d − is always mapped into x R ,i.e. LM = { γ : S d − → M | γ (0) = x R } . The images of those maps are ( d − M based at x R , and each one of them corresponds to a point of LM . Given a d -dimensional hyper-volume in M , one can scan it with a collection of thoseclosed hyper-surfaces Σ. Such collection is a path in LM , and so the hyper-volume in M canbe seen as a path in the loop space LM . The idea now is, for a given theory in M , to lookfor a one-form connection A in LM , such that the conditions for its curvature to vanish areequivalent to the classical equations of motion of that physical theory. The flatness condition1or A implies that its path ordered integral between two given points in LM (hyper-surfaces in M ) is independent of the choice of path (hyper-volume in M ) joining them. That would lead,in a similar way to (1 + 1)-dimensional theories, to conserved quantities as the eigenvaluesof the path ordered integral of A over the paths corresponding to the d -dimensional spacesub-manifold. That is the approach put forward in [1] and implemented in several examplesof field theories in ( d + 1) dimensions. See [2] for a review of the interesting results obtained.Among the difficulties of the approach are those associated to the non-locality and to thereparameterization invariance of the physical quantities. Note that a given hyper-volume in M corresponds in fact to an infinite number of paths in LM , which is a consequence of theinfinity of ways of scanning it with hyper-surfaces. So, the physical phenomena should notdepend upon the change of scanning. Despite those difficulties it was possible to impose localconditions in M which lead to the vanishing of the curvature of the connection A in the loopspace LM , and made the physical quantities reparameterization invariant [1, 2].In this paper we want to use the very same ideas proposed in [1, 2] to construct conservedquantities for a large class of theories, which include integrable field theories in (1 + 1) dimen-sions, Chern-Simons theories with sources in (2 + 1) dimensions, and Yang-Mills theories in(2 + 1) and (3 + 1) dimensions. However, instead of looking for a connection in loop spacewhich zero curvature condition is equivalent to the classical equations of motion, we proposean integral form of those equations, related to generalizations of the non-abelian Stokes the-orem, and which lead in a quite simple and direct way to the conservation laws. Consider aphysical theory in a ( d + 1)-dimensional simply connected space-time M , and let Ω be any(in some sense topologically trivial) d -dimensional hyper-volume in M , and suppose that thedynamics of such theory can be described by integral equations of the form P d − e (cid:82) ∂ Ω F = P d e (cid:82) Ω J (1.1)where ∂ Ω is the border of Ω, and where P d − and P d stand for hyper-surface and hyper-volumeordering integrations respectively. The quantities F and J are built out of d − d forms in M respectively, and which are functionals of the physical fields. The details of the constructionwill be given in the examples discussed in the next sections. However, we deal with local fieldtheories and the equations (1.1) are a direct consequence of the local differential equations ofmotion of the theory and of some generalization of the non-abelian Stokes theorem. On theother hand, since (1.1) is valid on any hyper-volume Ω, it turns out that (1.1) imply thoselocal differential equations when Ω is taken to be infinitesimally small. In order to define theordered integrations in (1.1) we scan Ω with ( d − x R on its border ∂ Ω. Therefore, the equations (1.1) are not really definedon each Ω, but on the generalized loop space L Ω = { γ : S d − → Ω | γ (0) = x R } , i.e. the space2f mappings from the ( d − S d − to Ω, such that its north pole is alwaysmapped into x R . Consequently, Ω can be seen as a path in L Ω, and the r.h.s. of (1.1) isdefined on such a path, with the l.h.s. of (1.1) being evaluated on its end points. But there isan infinite number of paths in L Ω corresponding to the same Ω. When one changes the choiceof path representing Ω, both sides of (1.1) change. However, the non-abelian Stokes theoremleading to (1.1) guarantees that the changes are such that both sides of (1.1) remain equal.Therefore, (1.1) transforms “covariantly” under change of parameterization of Ω. In addition,we show in the next sections that (1.1) transforms covariantly under gauge transformationsassociated to the differential forms leading to the quantities F and J .An important consequence of the integral form of the equations of motion (1.1) is that ifone considers a closed hyper-volume Ω c , i.e. without border, then the l.h.s of (1.1) becomestrivial and one gets that P d e (cid:82) Ω c J = 1l (1.2)Note that Ω c corresponds to a closed path in L Ω c . Then let us choose an intermediate point onthat path, i.e. a closed ( d − c = Ω + Ω , with Ω being the part of Ω c going from the infinitesimal hyper-surface Σ R around the reference point x R to Σ, and Ω to the part going from Σ back to Σ R . Then, the ordered integration impliesthat P d e (cid:82) Ω2 J P d e (cid:82) Ω1 J = 1l. The order may be reversed depending upon the definition of theordered integration. By reverting the sense of integration along the path one gets the inverseoperator, and so one can rewrite that relation as P d e (cid:82) Ω1 J = P d e (cid:82) Ω − J , with Ω − being thepath Ω in reversed order. Since that is valid for any closed path passing through Σ R and Σ,one concludes that the operator P d e (cid:82) Ω J is independent of the path Ω joining Σ R and Σ. Thatpath independency is a conservation law, and by choosing appropriate boundary conditionsas we explain in the next sections, one gets that the conserved charges are the eigenvaluesof the operator obtained by the path ordered integral P d e (cid:82) Ω J , with Ω corresponding to thewhole space sub-manifold. In the examples we discuss such conserved charges are shown tobe gauge invariant, and independent of the parameterization of the hyper-volumes as well asof the choice of the reference point x R . In the case where the space-time is of the form S ×
IR,with IR being the time, and S being a space sub-manifold without border, i.e ∂ S = 0, thenone gets from (1.2) that P d e (cid:82) S J = 1l. Therefore, such operator is not only constant in timebut trivial. In many cases, that topological property of the space-time leads to quantizationof charges. That is a very important consequence of our construction.Even though we have not introduced a one-form connection in the loop space L Ω, thatconcept is hidden in the quantity J . In addition, since we use generalizations of the non-abelian Stokes theorem, the quantity J corresponds to some sort of curvature of the quantity F , now seen as a connection on a lower loop space L∂ Ω, made of the space of maps of the3 d − S d − to the border ∂ Ω of Ω. Therefore, there must be some sortof Poincar´e lemma playing a role here, implying that the curvature of a connection whichis already a curvature should vanish. So, in that sense J would play the role of the flatconnection in the approach proposed in [1, 2]. Note however that the integral form of theequations of motion (1.1) does not require the introduction of a connection in loop space toobtain conserved quantities for the theories we consider in this paper.The paper is organized as follows: in section 2 we implement our construction for integrablefield theories in 1 + 1 dimensions re-obtaining well known results in that research area usingthe integral form of the equations of motion (1.1). In section 3 we discuss the cases of theChern-Simons theory in the presence of a source as well as the Yang-Mills theories, both in2 + 1 dimensions. An important result of this section is the quantization of the charges in thecase where the two dimensional space sub-manifold has no border. In section 4 we discuss theinteresting case of non-abelian gauge theories in 3 + 1 dimensions, in the presence of mattercurrents. An important result here is an integral formulation of the Yang-Mills equations, andthe construction of gauge invariant conserved charges. In the appendices A and B we give theproofs of the non-abelian Stokes theorems used in our constructions. dimensions In a 1 + 1 dimensional space-time M we establish a dynamical equation relating a field g ( x ),element of a Lie group G , to another field C µ ( x ), a 1-form taking values in the Lie algebra G of G . The relation between those two fields is built as follows. Consider a path γ in M ,parametrized by σ , and define a quantity W through the differential equation dWdσ + C µ dx µ dσ W = 0 (2.1)where x µ , µ = 0 ,
1, are the Cartesian coordinates in M of the points of γ . Integration of (2.1)can be formally written as W = P e − (cid:82) γ dσC µ dxµdσ · W R , where P stands for the path-ordering,and W R is an integration constant corresponding to the value of W at the initial point x R ofthe path γ .Given any smooth path γ in M , with initial and final points denoted by x R and x f respectively, we impose the following equation for the fields g ( x ) and C µ ( x ) g ( x f ) · g ( x R ) − = P e − (cid:82) γ dσC µ ( x ) dxµdσ (2.2)with g ( x R ) and g ( x f ) corresponding to the values of g ( x ) at the end points x R and x f , and the4 R x f γ t γ L γ − L γ segunda-feira, 5 de setembro de 2011 Figure 1: The two paths γ = γ L · γ and γ = γ t · γ − L , connecting the points x R and x f ,which are used to construct the conserved charges as the eigenvalues of the operator (2.4).The horizontal paths are parallel to the space axis and the vertical ones to the time axis.r.h.s. of (2.2) is obtained by integrating (2.1) along the path γ , assuming that the integrationconstant is unit. Note that (2.2) has the form of (1.1) since the border of γ corresponds toits end points, i.e. ∂γ = { x R , x f } , and g ( x f ) · g ( x R ) − stands for the integration of F on ∂γ ,which in this case would be a zero-form.The first important consequence of (2.2) is that the path ordered integral of C µ ( x ) isindependent of the path. Indeed, if γ and γ are two paths in M with the same end points x R and x f , then (2.2) implies that P e − (cid:82) γ dσC µ ( x ) dxµdσ = P e − (cid:82) γ dσC µ ( x ) dxµdσ . That fact togetherwith some appropriate boundary conditions is a conservation law as we now explain. Considerthe space-time as being M = IR × IR, and let x ≡ t and x ≡ x be the time and spacecoordinates respectively. We choose the coordinates of x R as being ( t, x ) = (0 , − L ) and of x f as ( t, x ) = ( t, L ), with L being a length scale which will be taken to infinity at the end ofcalculations. We choose two paths joining x R and x f as shown in Figure 1, i.e. the first path is γ = γ L · γ and the second γ = γ t · γ − L . Note that γ t and γ are paths at constant time at t = t and t = 0 respectively. On the other hand γ L and γ − L are paths at constant space at x = L and x = − L respectively. If one assumes that the time component of the one-form satisfies theboundary condition C ( t, − L ) = C ( t, L ), for all values of t , then the path ordered integralsof C µ along γ L and γ − L are the same, i.e. P e − (cid:82) γ − L dσC µ ( x ) dxµdσ = P e − (cid:82) γL dσC µ ( x ) dxµdσ ≡ U ( t ).Therefore, the equality of the path ordered integrals of C µ along γ and γ leads to the iso-spectral evolution equation P e − (cid:82) γt dσC µ ( x ) dxµdσ = U ( t ) P e − (cid:82) γ dσC µ ( x ) dxµdσ U ( t ) − (2.3)Consequently the eigenvalues of the operator Q = P e − (cid:82) γt dσC µ ( x ) dxµdσ = g ( t, L ) g − ( t, − L ) (2.4)where in the last equality we have used (2.2), are constant in time. Similarly, one can express5hose constants of motion as Tr Q N , for any integer N . When we take the limit L → ∞ , oneobserves that the conserved charges are determined by the asymptotic values of the field g ( x ),which is a known result in soliton theory [3].Let us now consider the case where the space-time is of the form M = S ×
IR, where thespace submanifold S has no border, like for instance the circle S = S . It then follows from(2.2) that the path ordered integral of C µ on the whole space S must be unity, i.e. P e − (cid:82) S dσC µ ( x ) dxµdσ = 1l (2.5)since the initial and final points are the same and so g ( x R ) = g ( x f ). That is true for any valueof time, and consequently (2.5) can be interpreted as a conservation law, where the conservedcharges are in fact trivial. However, depending upon the theory under consideration, one gets(topological) quantization conditions for some quantities. A simple example would be that ofan abelian pure imaginary connection, C µ ≡ i J µ , where (2.5) leads to (cid:82) S dσJ µ ( x ) dx µ dσ = 2 π n ,with n integer.Note that (2.2) transforms covariantly under the gauge transformations C µ → h C µ h − − ∂ µ h h − g → h g (2.6)since (2.1) implies that P e − (cid:82) γ dσC µ ( x ) dxµdσ → h ( x f ) P e − (cid:82) γ dσC µ ( x ) dxµdσ h ( x R ) − , with x R and x f being the end points of γ . Therefore, the conserved charges given by the eigenvalues of (2.4)are invariant under those gauge transformations satisfying h ( t, − L ) = h ( t, L ).If γ is taken to be a path infinitesimally short, so that its end points approach each other,then g at x f can be written as an approximation of the value it has at x R by using a Taylorexpansion: g ( x f ) = g ( x R )+ ∂ µ g ( x R ) δx µ , with δx µ being the infinitesimal displacement between x f and x R . Therefore, the l.h.s. of (2.2), up to first order in δx µ , becomes g ( x f ) g ( x R ) − ∼
1l + ∂ µ g ( x R ) g ( x R ) − δx µ . In addition, the path-ordering effects in the integration of C µ areof higher order in δx µ , and therefore the r.h.s of (2.2), up to first order, becomes simply1l − C µ δx µ . Since that is valid for any infinitesimal path located anywhere in the space-time M , we get that (2.2) implies the following differential equation for the fields g ( x ) and C µ ( x ) C µ ( x ) = − ∂ µ g ( x ) g − ( x ) (2.7)Therefore C µ is of the form of a pure gauge field, and consequently its curvature vanishes, i.e. ∂ µ C ν − ∂ ν C µ + [ C µ , C ν ] = 0 (2.8)The relation (2.8) is the so-called Lax-Zakharov-Shabat equation [4] or the zero curvature6ondition, which is the basic structure used in the development of exact methods in solitontheory and two dimensional integrable field theories. The equation (2.2) is therefore an integralformulation of the Lax-Zakharov-Shabat equation. One can in fact obtain one from the other.However, there are some subtleties in the integral formulation, since it works with the twofields C µ and g ( x ) and a relation between them, namely (2.2). That approach leads in a quitenatural way, as shown in (2.4), to the fact that the conserved charges come from boundaryterms. Such result is known for a large class of soliton theories [3], but it is not so certainthat it holds for integrable field theories not possessing solitons. It would be interesting toinvestigate that issue further. In addition, the integral formulation leads to the triviality ofthe charges, or its topological quantization, in the case where the space sub-manifold has noborder. dimensions In the case of theories defined on a (2 + 1)-dimensional space-time M the basic ingredient ofour construction is the so-called non-abelian Stokes theorem for a one-form connection C µ .Let Σ be a two dimensional smooth surface on M , and let ∂ Σ be its border, i.e. a closed curveon M . The theorem states that the path-ordered integral of C µ around ∂ Σ is equal to thesurface ordered integral on Σ, of the curvature of C µ , i.e. P e − (cid:72) ∂ Σ dσ C µ dxµdσ · W R = W R · P e (cid:82) Σ dτ dσ W − G µν W dxµdσ dxνdτ (3.1)where G µν = ∂ µ C ν − ∂ ν C µ + [ C µ , C ν ] (3.2)A proof of (3.1) is given in the appendix A, but its meaning is the following. One chooses areference point x R on the border of Σ and scan it with closed loops starting and ending at x R . The loops are labelled by τ such that τ = 0 corresponds to the infinitesimal loop around x R , and τ = 2 π corresponds to the border ∂ Σ. Each closed loop is parametrized by σ suchthat σ = 0 and σ = 2 π corresponds to x R . The l.h.s. of (3.1) is obtained by integrating thedifferential equation (2.1) along ∂ Σ, and W R is the integration constant corresponding to thevalue of W at x R . The meaning of P is that such integration has to be path ordered. Asshown in the appendix A, the r.h.s. of (3.1) is obtained by integrating on Σ the differentialequation dVdτ − V J = 0 (3.3)7ith J ≡ (cid:90) π dσ W − G µν W dx µ dσ dx ν dτ (3.4)and the meaning of P is that such integration has to be surface ordered according to thescanning of Σ with loops as explained above. Again W R is the integration constant andcorresponds to the value of V on the infinitesimal loop around x R . That the two integrationconstants have to be the same can be understood by shrinking Σ to the reference point x R .We now show how to use the non-abelian Stokes theorem (3.1) to define an integral for-mulation of the Chern-Simons and Yang-Mills theories, both in the presence of sources, andon a space-time of 2 + 1 dimensions. We then show how to use (3.1) to construct conservedcharges for those theories. Consider a theory on a (2 + 1)-dimensional space-time M for a vector field A µ and a current J µ , with µ = 0 , ,
2, and let its classical equations of motion be defined as follows. On anytwo dimensional smooth surface Σ on M , with border ∂ Σ, the fields must satisfy the integralequations P e − ie (cid:72) ∂ Σ dσ A µ dxµdσ = P e ieκ (cid:82) Σ dτ dσ W − (cid:101) J µν W dxµdσ dxνdτ . (3.5)where (cid:101) J µν stands for the Hodge dual of the matter current i.e., (cid:101) J µν ≡ (cid:15) µνρ J ρ , and where e and κ are coupling constants of the theory. The meaning of the path ordered ( P ) and surfaceordered ( P ) integrals in (3.5) is the same as those in (3.1), i.e. the l.h.s. of (3.5) is obtainedby integrating (2.1) with C µ = i e A µ , and its r.h.s. by integrating (3.3) with G µν = i eκ (cid:101) J µν .Since (3.5) is valid on any Σ, then it has to hold true when Σ is taken to be an infinitesimalsurface. It then follows that (3.5) implies local differential equations for the fields as we nowexplain. Indeed, take Σ to be a planar surface of rectangular shape on the plane defined bytwo axis of the Cartesian coordinates, let us say x µ and x ν , with µ and ν fixed. The border ∂ Σis then a rectangle of infinitesimal sides δx µ and δx ν . We evaluate both sides of (3.5) by Taylorexpanding the integrands around one given corner of the rectangle, and keeping things at thelowest non-trivial order. One can check that the l.h.s. of (3.5) becomes 1l + i e F µν δx µ δx ν ,with no sum in µ and ν , and where F µν = ∂ µ A ν − ∂ ν A µ + i e [ A µ , A ν ] (3.6)The r.h.s. of (3.5) in lowest order is given by 1l + i eκ (cid:101) J µν δx µ δx ν (no sum in µ and ν ). Therefore,8or an infinitesimal surface, (3.5) implies the local differential equations for A µ F µν = 1 κ (cid:101) J µν = 1 κ (cid:15) µνρ J ρ (3.7)which are the equations of motion of the Chern-Simons theory in the presence of an externalcurrent J µ .On the other hand one observes that if one takes C µ in (3.1) as C µ = i e A µ , and therefore G µν = i e F µν , and uses (3.7), then one obtains (3.5). In other words, (3.5) is a direct con-sequence of the non-abelian Stokes theorem (3.1) and the Chern-Simons equations of motion(3.7). Since (3.5) implies (3.7), we see that (3.5) is indeed an integral formulation of theChern-Simons theory.Note that in obtaining (3.5) from the non-abelian Stokes theorem (3.1) we have dropped theintegration constant W R . That has to do with the covariance of (3.5) under gauge transforma-tions, as we now explain. The Chern-Simons equation (3.7) transforms covariantly under thegauge transformations A µ → g A µ g − + ie ∂ µ g g − , since F µν → g F µν g − , and (cid:101) J µν → g (cid:101) J µν g − .From (2.1) we have that under a gauge transformation W → g f W g − i , where g i and g f arethe values of g at the initial and final points of the curve where W is defined. Consequently,on a closed curve one has that W c → g R W c g − R , where g R is the value of g at the referencepoint x R , where the curve starts and ends. In addition, J defined in (2.1), with G µν re-placed by i eκ (cid:101) J µν , transforms as J → g R J g − R , and so from (3.3) we have that V → g R V g − R .However, if W c and V are solutions of (2.1) and (3.3) respectively, so are W c = W c k and V = h V , with k and h constant group elements. Under a gauge transformation one wouldthen have W ci → g R W ci g − R , and V i → g R V i g − R , with i = 1 ,
2. But since k and h are arbitrarygroup elements one should not expect them to depend upon A µ , and so be insensitive to itsgauge transformations. Therefore, one could as well conclude that W c → g R W c g − R k , and V → h g R V g − R . The only way to establish a compatibility is to assume that k and h shouldbelong to the center of the gauge group G , since g R can be any element of G . Since the inte-gration constants W R in (3.1) have the same status in this discussion, as k and h , we have totake them to lie in the center of G to have the gauge covariance of the integral Chern-Simonsequation (3.5). However, when that is done they drop out from (3.5) since they commute withthe path and surface ordered integrals. So, when integrating (2.1) and (3.3) to construct thel.h.s. and r.h.s. respectively of (3.5) one should keep in mind that those operators can carryan integration constant lying in the center of G without destroying the gauge covariance of(3.5). That fact may be important in some applications.Note that both sides of (3.5) depend upon the choice of the reference point x R and alsoon the choice of the scanning of Σ with loops. However, when one changes the scanningand the reference point, the non-abelian Stokes theorem (3.1) guarantees that both sides of93.5) change in a way that they remain equal to each other. In that sense one can say that(3.5) transforms “covariantly” under the change of scanning and reference point. In fact, eventhough we have defined the equation (3.5) on any surface Σ in M , it is formally defined onthe loop space L Σ = { γ : S → Σ | north pole → x R ∈ ∂ Σ } , consisting of maps from thecircle S into Σ, such that the north pole of S is mapped into x R . The images of such mapsare closed loops in Σ, starting and ending at x R . Since Σ is scanned by a collection of suchloops, and the loops are points in L Σ, one can see Σ as a path in L Σ. Therefore, a change inthe scanning of Σ corresponds to a change of path in L Σ representing the same physical Σ.Despite the fact that (3.5) is defined on loop space, it leads to very physical consequences,like conservation laws as we now explain. Consider the case where the surface Σ is a closedsurface Σ c , i.e. with no border. Then, the l.h.s. of (3.5) is trivial and we are lead to P e ieκ (cid:72) Σc dτ dσ W − (cid:101) J µν W dxµdσ dxνdτ = 1l (3.8)Being a closed surface, Σ c corresponds to a closed path in the loop space L Σ c , starting andending at the reference point x R . Consider now a point on that path corresponding to a loop γ in Σ c . It divide the path in two parts corresponding to two surfaces, i.e. we have Σ c = Σ +Σ .From the ordering defined by (3.3) we have that P e ieκ (cid:82) Σ1 dτ dσ W − (cid:101) J µν W dxµdσ dxνdτ P e ieκ (cid:82) Σ2 dτ dσ W − (cid:101) J µν W dxµdσ dxνdτ = 1l (3.9)By reverting the sense of the integration along the path, one obtains the inverse operator whenintegrating (3.3). Then, Σ and Σ − are two surfaces corresponding to two paths starting andending at the same points, namely the reference point x R and the loop γ , which is theircommon border. Therefore, (3.9) implies that the integration along two paths with the sameend points gives the same operator. Since that is valid for any closed surface Σ c and anypartition of it into two surfaces, we conclude that the quantity P e ieκ (cid:82) Σ dτ dσ W − (cid:101) J µν W dxµdσ dxνdτ is independent of the surface Σ. It depends only on the initial and final points of the pathcorresponding to Σ in loop space, i.e. the border ∂ Σ and the reference point x R on it. Notethat such independency corresponds to the change of the reparameterization (scanning) of thesurface, as well as to the change of the surface itself, but keeping the border and referencepoint fixed. Such surface independency leads to conservation laws as we now explain.First we consider the topology of space-time to be M = S ×
IR, with time being the realline IR, and the space being the closed two dimensional surface with no boundary S . A simplecase is when S is the two-sphere, i.e. S = S . Then if we evaluate (3.8) in space ( i.e. , Σ c = S )we get that the quantity Q S ≡ P e ieκ (cid:72) S dτ dσ W − (cid:101) J µν W dxµdσ dxνdτ = 1l, is conserved in time and equalto unity. That is an interesting relation since it may imply that the net charge on the whole10pace vanishes, or then that the charge must satisfy some quantization condition.For simplicity, let us consider the case of an abelian theory, with gauge group G = U (1),where the path and surface orderings are irrelevant. In such a case one has Q S = e ieκ q = 1,where q is the total charge in space, i.e. q = (cid:72) S dτ dσ (cid:101) J µν dx µ dσ dx ν dτ . Those equations establishesa quantization condition involving the total amount of charge in space and the Chern-Simonscoupling constants, i.e. q eκ = 2 π n with n integer (3.10)Such result has two important consequences for Chern-Simons theory on the space-time M = S ×
IR. The equations of motion (3.7) imply that the time component J of the current,namely the charge density ρ , is proportional to the space components of the field tensor,which is the pseudo-scalar magnetic field B , i.e. ρ = κ B . So, the effect of the Chern-Simonsequation of motion is to attach magnetic flux to the electric charge [5]. For a point particlewe have ρ = e δ ( (cid:126)x − (cid:126)x ), with (cid:126)x being the position vector of the particle. Therefore, themagnetic flux associated to such a particle is Φ = eκ , and (3.10) implies it is quantized asΦ = π nq . The second consequence of (3.8) and the topology M = S ×
IR, is that the phasegained by a non-relativistic particle that moves around another, due to a Aharonov-Bohmtype interaction, is no longer dependent on the Chern-Simons coupling constant κ . For N such particles the charge density reads ρ = e (cid:80) Na δ ( (cid:126)x − (cid:126)x a ) and the attached magnetic field B = eκ (cid:80) Na δ ( (cid:126)x − (cid:126)x a ). After a double interchange of two particles, their phase exchange isgiven by ∆ θ = e πκ [5]. Due to the quantization condition (3.10) and using that q = N e weget ∆ θ = n N , which is a rational number, and not any number as would be the case if (3.8),and so (3.10), was not used.Let us now consider the case where space-time is IR , and discuss how conserved chargescan be constructed using (3.8). As we have seen, as a consequence of (3.8), the quantity V (Σ) ≡ P e ieκ (cid:82) Σ dτ dσ W − (cid:101) J µν W dxµdσ dxνdτ (3.11)is independent of the surface Σ, as long as its border ∂ Σ is kept fixed and also the referencepoint x R on it. In addition it is independent of the scanning (parameterization) of Σ withloops. We shall consider two surfaces, Σ and Σ , with the same borders as follows. The firstone, shown in figure 2, is made of two parts. The first part is a disk D (0) ∞ on the plane x x attime x = 0, and of a radius which will be taken to be infinite. The second part is a cylinder S ∞ × I , where I is a segment of the x -axis going from x = 0 to x = t , and S ∞ is a circleof infinite radius parallel to the plane x x . We take the reference point x R to be on theborder of the disk D (0) ∞ , as shown in 2, and scan Σ = D (0) ∞ ∪ ( S ∞ × I ), with loops starting and11 urface Σ = D (0) ∞ ∪ S ∞ × I Surface Σ = S × I ∪ D ( t ) ∞ x R D (0) ∞ S ∞ × I segunda-feira, 5 de setembro de 2011 x R D ( t ) ∞ S × I segunda-feira, 5 de setembro de 2011 x ( t ) R segunda-feira, 12 de setembro de 2011 Figure 2: The surfaces Σ and Σ , with the same border S , ( t ) ∞ , and reference point x R , usedin the construction of conserved charges.ending at x R , labeled by τ , such that for τ ∈ [0 , π ] we scan D (0) ∞ , with τ = 0 correspondingto the infinitesimal loop around x R , and τ = π corresponding to S ∞ , the border of D (0) ∞ . For τ ∈ [ π, π ] we scan S ∞ × I with loops, which start at x R , go up in the x direction upon to x = t (cid:48) ∈ I , go round S ∞ , and come down to x R again. By varying t (cid:48) within I we scan thecylinder S ∞ × I . Following the notation of (3.11), and the ordering defined by (3.3), one gets V (Σ ) = V (cid:0) D (0) ∞ (cid:1) V (cid:0) S ∞ × I (cid:1) (3.12)The second surface Σ , as shown in figure 2, is also made of two parts. The first part is acylinder S × I , with I being the same time interval as above, and S a circle parallel to the x x plane with infinitesimal radius. The reference point x R is on the border of the base ofsuch cylinder at x = 0. The second part is a disk D ( t ) ∞ on the plane x x at time x = t , andof a radius which will be taken to be infinite. The surface Σ = ( S × I ) ∪ D ( t ) ∞ is scannedwith loops starting and ending at x R , labelled by τ , such that for τ ∈ [0 , π ], we scan S × I with loops, which start at x R , go up in the x direction upon to x = t (cid:48) ∈ I , go round S , andcome down to x R again. By varying t (cid:48) within I we scan the cylinder S × I . For τ ∈ [ π, π ]we scan D ( t ) ∞ with loops which start at x R , go up to x = t , go round a closed loop on D ( t ) ∞ ,and come down to x R again. By keeping the two legs going up and down fixed and varyingthe closed loops on D ( t ) ∞ , we scan it entirely. Again, following the notation of (3.11), and theordering defined by (3.3), one gets V (Σ ) = V (cid:0) S × I (cid:1) V (cid:0) D ( t ) ∞ (cid:1) (3.13)Since Σ and Σ have the same reference point and the same border, namely S ∞ at x = t ,we have from the surface independency of (3.11) that V (Σ ) = V (Σ ). We now impose the12ollowing boundary condition on our system˜ J = J ∼ r δ T (ˆ r ) for r → ∞ (3.14)with δ > r = ( x ) + ( x ) , and T (ˆ r ) being an element of the Lie algebra of G , dependingon the spatial direction defined by ˆ r = (cid:126)rr . That condition implies that the quantity J ≡ i eκ (cid:82) π dσ W − ˜ J µν W dx µ dσ dx ν dτ , vanishes on loops at spatial infinity, and therefore from (3.3)one gets that V ( S ∞ × I ) = 1l. In addition, since the circle S has vanishing radius wealso get that V ( S × I ) = 1l. Therefore, from the equality of (3.12) and (3.13) we get that V (cid:16) D ( t ) ∞ (cid:17) = V (cid:16) D (0) ∞ (cid:17) . Those two operators are calculated using the same reference point x R , which lies at the border of D (0) ∞ . Consider now a reference point x ( t ) R , which have thesame space coordinates as x R , but at a time x = t , i.e. it lies on the border of D ( t ) ∞ ,just above x R (see figure 2). By changing the reference point the quantity J changes as J → W − ( x ( t ) R , x R ) J W ( x ( t ) R , x R ), where W ( x ( t ) R , x R ) is obtained by integrating (2 .
1) on thepath joining x R to x ( t ) R . Therefore, if one now integrates (3.3) on D ( t ) ∞ with this new referencepoint, one gets that V x R ( D ( t ) ∞ ) = W − ( x ( t ) R , x R ) V x ( t ) R ( D ( t ) ∞ ) W ( x ( t ) R , x R ). Therefore, one gets that V x ( t ) R (cid:0) D ( t ) ∞ (cid:1) = W (cid:16) x ( t ) R , x R (cid:17) V x R (cid:0) D (0) ∞ (cid:1) W − (cid:16) x ( t ) R , x R (cid:17) (3.15)where the subindices indicate which reference point is being used in the integration of (3.3).Note that in this way V x ( t ) R (cid:16) D ( t ) ∞ (cid:17) and V x R (cid:16) D (0) ∞ (cid:17) correspond to surface ordered integrals overthe entire space at times x = t and x = 0 respectively, and with reference points at spatialinfinity (border of the infinite disks) and at the same times. Consequently (3.15) constitutean iso-spectral time evolution for the operator V x ( t ) R (cid:0) D ( t ) ∞ (cid:1) = P e ieκ (cid:82) D ( t ) ∞ dτ dσ W − (cid:101) J µν W dxµdσ dxνdτ = P e − ie (cid:72) S ∞ dσ A µ dxµdσ (3.16)where in the last equality we have used the Chern-Simons integral equation (3.5), and wherethe spatial circle with infinite radius S ∞ stands for the border of D ( t ) ∞ . Therefore, its eigenval-ues, or equivalently Tr (cid:104) V x ( t ) R (cid:16) D ( t ) ∞ (cid:17)(cid:105) N , are constant in time. Those are the conserved quan-tities for the Chern-Simons theory. Note that such conserved quantities are gauge invariant,since under a gauge transformation we have that V x ( t ) R (cid:16) D ( t ) ∞ (cid:17) → g (cid:16) x ( t ) R (cid:17) V x ( t ) R (cid:16) D ( t ) ∞ (cid:17) g (cid:16) x ( t ) R (cid:17) − ,where g (cid:16) x ( t ) R (cid:17) is the element of the gauge group, performing the gauge transformation, eval-uated at the reference point x ( t ) R . The conserved quantities are also independent of the waywe scan D ( t ) ∞ , since we have already shown above that the operators of the type (3.11) arescanning independent. In fact, it was that property that lead to the conservation laws. Inaddition, the conserved quantities are independent of the choice of the reference point on the13order of D ( t ) ∞ . That has to with the fact that by changing the reference point, V x ( t ) R (cid:16) D ( t ) ∞ (cid:17) changes by conjugation by an element W obtained by integrating (2.1) along a path on theborder joining the two reference points. Consequently, its eigenvalues are unchanged. We now consider another theory on a (2 + 1)-dimensional space-time M , with the same fieldcontent, i.e. a vector field A µ and a current J µ , with µ = 0 , ,
2, and with its classical equationsof motion being defined as follows. On any two dimensional smooth surface Σ on M , withborder ∂ Σ, the fields must satisfy the integral equations P e − ie (cid:72) ∂ Σ dσ ( A µ + β (cid:101) F µ ) dxµdσ = P e ie (cid:82) Σ dτ dσ W − ( F µν − β (cid:101) J µν + ie β [ (cid:101) F µ , (cid:101) F ν ]) W dxµdσ dxνdτ . (3.17)where e is the coupling constant of the theory, β is a free parameter, (cid:101) J µν is the Hodge dual ofthe matter current i.e., (cid:101) J µν ≡ ε µνρ J ρ , (cid:101) F µ is the Hodge dual of the curvature of the connection,i.e. (cid:101) F µ = ε µνρ F νρ , and F µν = ∂ µ A ν − ∂ ν A µ + ie [ A µ , A ν ]. The meaning of the path ordered( P ) and surface ordered ( P ) integrals in (3.17) is the same as those in (3.1), i.e. the l.h.s. of(3.5) is obtained by integrating (2.1) with C µ = i e (cid:16) A µ + β (cid:101) F µ (cid:17) (3.18)and its r.h.s. by integrating (3.3) with G µν = i e (cid:16) F µν − β (cid:101) J µν + ie β (cid:104) (cid:101) F µ , (cid:101) F ν (cid:105)(cid:17) (3.19)In order to obtain the corresponding local equations of motion we consider the integralequation (3.17) on an infinitesimal surface Σ of a rectangular shape, on the plane defined bytwo axis of the Cartesian coordinates, let us say x µ and x ν , with µ and ν fixed. The border ∂ Σ is then the rectangle of infinitesimal sides δx µ and δx ν . Evaluating the r.h.s. of (3.17) inlowest order, and Taylor expanding the integrand around one given corner of the rectangle,we get 1l + i e (cid:16) F µν − β (cid:101) J µν + ie β (cid:104) (cid:101) F µ , (cid:101) F ν (cid:105)(cid:17) δx µ δx ν , with no sum in µ and ν . Analogously,evaluating the l.h.s. of (3.17) in lowest order, and Taylor expanding around the same corner,one gets 1l + ( ∂ µ C ν − ∂ ν C µ + [ C µ , C ν ]) δx µ δx ν , again with no sum in µ and ν , and with C µ given by (3.18), and so ∂ µ C ν − ∂ ν C µ + [ C µ , C ν ] = ie (cid:16) F µν + β (cid:16) D µ (cid:101) F ν − D ν (cid:101) F µ (cid:17) + ie β (cid:104) (cid:101) F µ , (cid:101) F ν (cid:105)(cid:17) (3.20)14herefore, equating both sides of (3.17), in lowest order, one gets D µ (cid:101) F ν − D ν (cid:101) F µ = − (cid:101) J µν (3.21)where D µ ∗ = ∂ µ ∗ + ie [ A µ , ∗ ]. Taking the Hodge dual one gets the Yang-Mills equations in(2 + 1) dimensions in the presence of mater currents D ν F νµ = J µ (3.22)Note that if one takes the non-abelian Stokes theorem (3.1) with the connection C µ givenby (3.18), and so its curvature G µν given by (3.20), one gets the integral equation (3.17) byusing the Yang-Mills equations (3.21) . Therefore, (3.17) is a direct consequence of the non-abelian Stokes theorem (3.1) and the Yang-Mills equations. In this sense, (3.17) is an integralformulation of the Yang-Mills theory in (2+1) dimensions in the presence of matter currents.We have put therefore the Yang-Mills theory in the same footing as the Chern-Simonstheory in the presence of matter currents, with its integral equation being given by (3.5). Con-sequently, most of the results we obtained for Chern-Simons are also valid for the Yang-Millsusing the same techniques. For instance, the integral equation (3.17) transform covariantlyunder the gauge transformations A µ → g A µ g − + ie ∂ µ g g − , and J µ → g J µ g − . In addition,it transforms covariantly under re-parameterization of the surface Σ with loops, and changeof the reference point, as explained in section 3.1. But the most important result followingfrom (3.17) is that if Σ c is a closed surface with no border, then its l.h.s. is trivial and so P e ie (cid:72) Σ c dτ dσ W − ( F µν − β (cid:101) J µν + ie β [ (cid:101) F µ , (cid:101) F ν ]) W dxµdσ dxνdτ = 1l (3.23)That is the equivalent for Yang-Mills of the equation (3.8) for Chern-Simons, and it leads toconservation laws. In particular, for a space-time of the form M = S ×
IR, with S being thespace sub-manifold, assumed closed with no border, like for instance the two-sphere S , onegets that Q S = P e ie (cid:72) S dτ dσ W − ( F µν − β (cid:101) J µν + ie β [ (cid:101) F µ , (cid:101) F ν ]) W dxµdσ dxνdτ = 1l, is constant in time andequal to unity. Again, it implies that the total charge in space vanishes, or then it may leadto quantization conditions. In the case of an abelian gauge group, for instance U (1), one getsthat Φ − β q = 2 π ne for n integer (3.24)where Φ = (cid:72) S dτ dσ F µν dx µ dσ dx ν dτ is the total magnetic flux (or magnetic charge), and q = (cid:72) S dτ dσ (cid:101) J µν dx µ dσ dx ν dτ , is the total electric charge in space.Again following the reasoning used in the case of the Chern-Simons theory in section 3.1,15e get that (3.23) implies that the quantity V (Σ) = P e ie (cid:82) Σ dτ dσ W − ( F µν − β (cid:101) J µν + ie β [ (cid:101) F µ , (cid:101) F ν ]) W dxµdσ dxνdτ (3.25)is invariant under smooth deformations of the surface Σ as long as its boundary and referencepoint x R are kept fixed. In addition, it is also invariant under the change of the scanningof Σ with loops based at x R . Those facts can be used to construct conserved charges forthe Yang-Mills theory. Let us consider space-time to be IR , and let us assume that thespace components of the field tensor and time component of the currents satisfy the boundaryconditions (cid:101) J = J ∼ r δ T (ˆ r ) F ∼ r δ (cid:48) T (cid:48) (ˆ r ) for r → ∞ (3.26)with δ , δ (cid:48) > r = ( x ) + ( x ) , T (ˆ r ) and T (cid:48) (ˆ r ) being elements of the Lie algebra of thegauge group G , depending on the spatial direction at infinity defined by ˆ r = (cid:126)rr . Consider nowa disk D ( t ) ∞ , of infinite radius on the plane x x , at a given time t , and let it be scanned withclosed loops starting and ending at a reference point x ( t ) R on its border. Consider the followingoperator obtained by integrating (3.3) on D ( t ) ∞ , with G µν given by (3.19) V x ( t ) R (cid:0) D ( t ) ∞ (cid:1) = P e ie (cid:82) D ( t ) ∞ dτ dσ W − ( F µν − β (cid:101) J µν + ie β [ (cid:101) F µ , (cid:101) F ν ]) W dxµdσ dxνdτ = P e − ie (cid:72) S ∞ dσ ( A µ + β (cid:101) F µ ) dxµdσ (3.27)where in the last equality we have used the integral equation (3.17), and where S ∞ is theborder of D ( t ) ∞ , i.e. a circle at spatial infinity. Then following the arguments used in section3.1, leading to (3.16), one concludes that the eigenvalues of the operator (3.27) are constantin time. Equivalently, one can write those conserved charges as Tr (cid:104) V x ( t ) R (cid:16) D ( t ) ∞ (cid:17)(cid:105) N . Againfollowing those same arguments one concludes that such conserved charges are gauge invariant,and independent of the scanning of D ( t ) ∞ with loops, and also on the choice of the referencepoint x ( t ) R on its border. We now comment on the connection of our integral formulation of Chern-Simons and Yang-Mills theories in (2 + 1) dimensions and the approach of [1, 2] using flat connections on loopspaces.Given the Lie algebra-valued 1-form C = C µ dx µ and the 2-form B = B µν dx µ ∧ dx ν on M ,we construct a 1-form connection A in the loop space LM = { γ : S → M | north pole → x R } ,16s [1, 2] A = (cid:90) π dσ W − B µν W dx µ dσ δx ν (3.28)where δ stands for the exterior derivative on the space of all parametrized loops with basepoint x R , with δ = 0 and δx µ ( σ ) ∧ δx ν ( σ (cid:48) ) = − δx ν ( σ (cid:48) ) ∧ δx µ ( σ ) . The curvature F = δ A + A∧A is given by F = − (cid:90) π dσ W ( σ ) − [ D λ B µν + D µ B νλ + D ν B λµ ] ( x ( σ )) W ( σ ) dx λ dσ δx µ ( σ ) ∧ δx ν ( σ )+ 12 (cid:90) π dσ (cid:90) σ dσ (cid:48) (cid:32) θ ( σ − σ (cid:48) ) (cid:2) B Wκµ ( x ( σ (cid:48) )) − G Wκµ ( x ( σ (cid:48) )) , B Wλν ( x ( σ )) (cid:3) − θ ( σ (cid:48) − σ ) (cid:2) B Wλν ( x ( σ )) − G Wλν ( x ( σ )) , B Wκµ ( x ( σ (cid:48) )) (cid:3) (cid:33) dx κ dσ (cid:48) dx λ dσ δx µ ( σ (cid:48) ) ∧ δx ν ( σ )where G µν is the curvature of C µ , i.e. G µν = ∂ µ C ν − ∂ ν C µ + [ C µ , C ν ], and W is constructedout of C µ by integration of (2.1).In the case of Chern-Simons theory we consider C µ = ieA µ and B µν = 1 κ (cid:101) J µν . Lemma 2 . B µν − G µν = 0then F = 0. Then, for our choice of C µ and B µν given above we see that F = 0 ⇔ Chern-Simons equation is satisfied . For Yang-Mills theory we take C µ = ie (cid:16) A µ + β (cid:101) F µ (cid:17) and B µν = i e (cid:16) F µν − β (cid:101) J µν + ie β (cid:104) (cid:101) F µ , (cid:101) F ν (cid:105)(cid:17) . The condition B µν − G µν = 0 leads to the Yang-Mills equation D ν F νµ = J µ , and thereforegives the zero curvature representation of this theory in loop space.Therefore, for the cases of Chern-Simons and Yang-Mills theories in (2 + 1) dimensions,the integral formulation approach and zero curvature on loop space lead to the same results,and also to the same conserved charges. 17 The case of volumes: theories in dimensions
For 3+1 dimensional theories we consider the generalization of the non-abelian Stokes theoremfor a 2-form connection proved in appendix B. Given a three dimensional volume Ω and itstwo dimensional border ∂ Ω, a closed surface, the theorem relates the surface-ordered integralof the connection W − B µν W along ∂ Ω with the volume-ordered integral of K = (cid:90) π dτ V (cid:40) (cid:90) π dσ W − ( D ρ B µν + D µ B νρ + D ν B ρµ ) W dx µ dσ dx ν dτ dx ρ dζ + − (cid:90) π dσ (cid:90) σ dσ (cid:48) (cid:2) B Wκλ ( σ (cid:48) ) − F µνκλ ( σ (cid:48) ) , B Wµν ( σ ) (cid:3) dx κ dσ (cid:48) ( σ (cid:48) ) dx µ dσ ( σ ) ×× (cid:18) dx λ dτ ( σ (cid:48) ) dx ν dζ ( σ ) − dx λ dζ ( σ (cid:48) ) dx ν dτ ( σ ) (cid:19) (cid:41) V − in Ω: V R P e (cid:82) ∂ Ω dτdσW − B µν W dxµdσ d xνd τ = P e (cid:82) Ω dζ K V R . (4.1)In the next few lines we summarize the construction of the above equation. A reference point x R is defined on the border of Ω, and around it we construct an infinitesimal volume, whoseborder is the infinitesimal closed surface Σ R . We then scan Ω with closed surfaces based atthe reference point x R . In doing so, we define the parameter ζ , such that ζ = 0 stands forthe infinitesimal surface Σ R and ζ = 2 π , for the boundary ∂ Ω. During this variation thequantities V are calculated for each surface, in each step, through equation dVdτ − V A = 0 (4.2)with A = (cid:82) π dσW − B µν W dx µ dσ dx ν dτ dσ . The surface Σ is scanned with loops parametrized by σ ∈ [0 , π ], starting and ending at the reference point, and the parameter τ labels these loops.The Wilson lines W are calculated on the loops using equation (A.1). The l.h.s of (4.1) istherefore obtained after integrating (4.2) over the surface ∂ Ω, with the ordering given by theway we scan it with loops. The quantity V R is the integration constant (the value of V forthe infinitesimal surface Σ R ).Once the surface Σ is closed, V can also be obtained from the equation (as we show in theappendix) dVdζ − K V = 0 (4.3)and this result is expressed in the r.h.s of the theorem above.18e now proceed to show how to formulate an integral version of Yang-Mills theory throughthis non-abelian Stokes theorem, and also, how it leads to conserved charges. We consider the Yang-Mills theory in (3 + 1) dimensions for a gauge group G and in thepresence of matter currents J µ . The classical equations of motion are given by D ν (cid:101) F νµ = 0 D ν F νµ = J µ (4.4)where F µν = ∂ µ A ν − ∂ ν A µ + i e [ A µ , A ν ], and (cid:101) F µν is the Hodge dual of the field tensor, i.e. , F µν ≡ ε µνρλ (cid:101) F ρλ .One can obtain an integral equation for the Yang-Mills theory using the Stokes theorem(4.1) as follows (see [6] for more details). Take B µν = ie (cid:104) α F µν + β (cid:101) F µν (cid:105) , with α and β beingarbitrary constants, and using Yang-Mills differential equations (4.4) to replace D ρ B µν + D µ B νρ + D ν B ρµ by ( − ieβ (cid:101) J µνλ ). With that, the quantity K above is now given by K = (cid:82) π dτ V J V − , with J ≡ (cid:90) π dσ (cid:40) ieβW − (cid:101) J µνρ W dx µ dσ dx ν dτ dx λ dζ ++ e (cid:90) σ dσ (cid:48) (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) ×× dx κ dσ (cid:48) dx µ dσ (cid:18) dx ρ ( σ (cid:48) ) dτ dx ν ( σ ) dζ − dx ρ ( σ (cid:48) ) dζ dx ν ( σ ) dτ (cid:19) (cid:41) and V is constructed by integrating (4.2) with A ≡ 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 we have used the notation X W ≡ W − X W . Then we have [6]: P e ie (cid:82) ∂ Ω dτdσ [ αF Wµν + β (cid:101) F Wµν ] dxµdσ dxνdτ = P e (cid:82) Ω dζdτV J V − . (4.5)which is a direct consequence of the generalized non-abelian Stokes theorem (4.1) and theYang-Mills equations (4.4). On the other hand, the integral equation (4.5) implies the differ-ential equations (4.4), as we now explain.Equation (4.5) is defined for an arbitrary volume Ω, and in particular, for an infinitesimalone. Take Ω as an infinitesimal cube of sides δx µ , δy ν and δz λ , with the indices fixed (seeFigure 3). We choose the reference point x R to be at one of the vertices, and when oppositesurfaces are scanned with loops based on x R one has to pay special attention to the fact that19 R δz δxδy sexta-feira, 26 de agosto de 2011 Figure 3: The scanning of an infinitesimal cube.the signs of the velocities in τ -direction changes.Now, considering only first order contributions to equation (4.5), the integrand in its l.h.scan be evaluated at any point on the cube’s face, since the differences will be of higher order,and therefore, evaluating it on the face δz λ = 0 one gets − ie (cid:104) αF µν + β (cid:101) F µν (cid:105) x R δx µ δy ν . Forthe face at x R + δz the contribution comes from ie (cid:16) W − (cid:104) αF µν + β (cid:101) F µν (cid:105) W (cid:17) ( x R + δz λ ) δx µ δy ν .For this term we have to expand both the Wilson line and the field strength so that their valuesat x R + δz are approximated to their values in x R . The equation for the Wilson line gives foran infinitesimal variation σ → σ + δσ along the z direction, W ( x R + δz λ ) ∼ − ieA λ ( x R ) δx λ andTaylor expanding the field strength in this direction, F µν ( x R + δz ) = F µν ( x R ) + ∂ λ F µν ( x R ) δz λ ,we end up with ieD λ (cid:104) αF µν + β (cid:101) F µν (cid:105) x R δx µ δy ν δz λ . Doing the same for the other two pairs offaces one gets P e ie (cid:82) ∂ Ω dτdσ [ αF Wµν + β (cid:101) F Wµν ] dxµdσ dxνdτ ≈
1l + ie ( D λ [ αF µν + β (cid:101) F µν ] + cyclic perm.) x R δx µ δy ν δz λ . For the r.h.s of (4.5), considering only the first order contributions, we notice that the com-mutator term is of higher order with respect to the first term involving only the current, forit has one more integration along the loop. Then, up to lowest order P e (cid:82) Ω dζdτV J V − ≈
1l + ieβ (cid:101) J µνλ δx µ δy ν δz λ and clearly equating the previous result with this one, we get the set of Yang-Mills equations(4.4), as the coefficients of the parameters α and β .Comparing the integral equation (4.5) with (4.1) one notices that the integration constants V R are missing. One has to keep in mind that while (4.1) is a mathematical relation, (4.5) isa physical equation, and therefore gauge covariance is important, and in order to guaranteethat those integration constants must lie in the center of the gauge group, as we now explainin detail. Consider the gauge transformation of the Yang-Mills field A µ → gA µ g − + ie ∂ µ gg − , The minus sign is due to the choice of the direction of scanning. F µν → gF µν g − , and thematter current transforms in the same way, J µ → gJ µ g − . From (A.1), the Wilson linechanges as W → g f W g − i where g i and g f stand for the value the gauge group element takesat the points x i and x f respectively; for the reference point we denote as g R the value ofthe gauge group element there. Then it is direct to see (by replacing W , F µν and J µ by therespective gauge transformed quantities given before) that all the important quantities relatedto the integral Yang-Mills equation (4.5) (such as V , J , K and A ) transform as V → g R V g − R , J → g R J g − R , K → g R K g − R and A → g R A g − R .The l.h.s of (4.5) comes from the l.h.s of the non-abelian Stokes theorem previously pre-sented, which, in turn, is the result of the integration of equation (4.2). We notice that if V is a solution of this equation, then V (cid:48) = kV is also a solution, where k is a constant elementof the gauge group. Under a gauge transformation one gets V (cid:48) → g R V (cid:48) g − R = g R kV g − R . Onthe other hand, since k is an arbitrary constant, and so insensitive to transformations of thegauge field, one could have written it as V (cid:48) → kg R V g − R , and the only way to guarantee thecompatibility is to have g R k = kg R , i.e. , to have k in the center Z ( G ) of the group. The sameargument can be applied to the r.h.s of (4.5), which comes from the r.h.s of the non-abelianStokes theorem, which, in turn, is the solution of equation (4.3). In that case, if V is a solu-tion, so is V (cid:48) = V h , with h a constant element of G . Finally we conclude that (4.5) is gaugecovariant only if the integration constants are in the center of the gauge group. Then theycan be cancelled trivially, since they commute with the surface and volume integrals, and thatis the reason why they do not appear in the integral Yang-Mills equation (4.5).An important issue concerns the fact that equation (4.5) is formulated in a way that itdepends on the particular choice of the reference point x R and of the scanning of the volumewith surfaces. Although it is at first sight unwanted, the Stokes theorem (4.1) guarantees thatif one changes any of these things, each side of (4.5) will change in a way to remain equalto each other; in this sense, this equation transforms “covariantly” under reparametrization.This can be better understood once we realize that this equation is formulated in the loopspace L Ω = { γ : S → Ω | north pole → x R ∈ ∂ Ω } , formed by maps from S into Ω, such thatthe north pole of S is mapped into the reference point x R . The images of this map are closedsurfaces in Ω, starting and ending at x R . The volume Ω is scanned by a family of these closedsurfaces, which are points in L Ω, thus, Ω is a path in the loop space L Ω. Then, a change inthe scanning of Ω corresponds to a change on the parametrization of the path in L Ω, whichdoes not change the path, nor the physical results from it.Equation (4.5) does not only describes the Yang-Mills theory in loop space, but also leadsto a conservation laws as we now discuss. For a closed path in loop space, corresponding to aclosed volume Ω c in space-time, with no boundary, the l.h.s of equation (4.5) becomes trivial21nd we get P e (cid:82) Ωc dζdτV J V − = 1l . (4.6)Consider now a given point in that path, in space-time, the surface Σ, which divides it intotwo parts: Ω c = Ω + Ω . Then, we can split equation (4.6) as P e (cid:82) Ω2 dζdτV J V − · P e (cid:82) Ω1 dζdτV J V − = 1l . (4.7)Each term of this equation is obtained from integration of (4.3). It can be written as V (Ω ) · V (Ω ) = 1l. Now, reverting the order of integration for the second path Ω , onegets equivalently V − (Ω − ) · V (Ω ) = 1l, which implies that V (Ω ) = V (Ω − ). In other words,the integration of (4.3), for the volumes Ω and Ω − , with the same boundary Σ and referencepoint x R , leads to the same operator. In fact, since equation (4.7) holds for any closed volumeΩ c and for any partition of it into two other volumes, what we just saw is that not only theintegration of (4.3) along two volumes with same boundary gives the same operator but alsothat P e (cid:82) Ω dζdτV J V − is independent of the volume, for any volume Ω. This means that as longas the border is kept fixed, one can change the volume and there will be no consequences to V (Ω). The fact that this operator depends only on the border ∂ Ω and on the reference point x R on it, leads to conservation laws as we now explain.Consider first the case in which space-time M has the topology S ×
IR, with S being aclosed unbounded spatial submanifold and IR the time. Then, equation (4.6) can be evaluatedfor Ω c = S , giving that the quantity Q S ≡ P e (cid:72) S dζdτV J V − = 1l . (4.8)is conserved in time and is equal to the unit. This has two main implications: the first isthat the net charge on the whole space vanishes, and the other is that it might lead to somequantization condition. In particular, for the case of the Maxwell theory, where the (abelian)gauge group is G = U (1), this equation for Q S is satisfied if q ≡ (cid:82) S dζdτ dσ (cid:101) J µνλ dx µ dσ dx ν dτ dx λ dζ , issuch that q = 2 πneβ with n integer. If for some reason β can be fixed at the quantum level, then (4.8) express thequantization of electric charge [7].Next we consider the space-time M to have the topology IR × IR. Lets take two pointsin the loop space, which correspond in space-time to the following two surfaces (see figure 4):the infinitesimal 2-sphere S around x R , at time x = 0 and the 2-sphere which is the borderof the entire space at time x = t ; and which we denote as S , ( t ) ∞ . Now, these two points22 R S , (0) ∞ S , ( t ) ∞ S ∞ × I Ω ∞ Ω t ∞ S × I segunda-feira, 29 de agosto de 2011 Figure 4: The schematic representation in loop space L Ωof the “path” used in the construction of the conserved charge.in L Ω are joined by two different paths (volumes in space-time) and each of these paths arecomposed by two parts as we now explain. The first part of the first path is the infinitesimalhyper-cylinder S × I , with I being the interval in IR from 0 to t and S the infinitesimalsphere around x R . The second part of this path is the volume inside the sphere S , ( t ) ∞ , whichwe make by “blowing up” the infinitesimal sphere S when we reach the point x = t ; wedenote this volume by Ω ( t ) ∞ . This first path is then the composition Ω = ( S × I ) ∪ Ω ( t ) ∞ . Thesecond path has a first part, which in space-time corresponds to the whole spatial volumeΩ (0) ∞ , inside S , (0) ∞ at x = 0, and its second part is the hyper-cylinder S ∞ × I . This path is thecomposition Ω = Ω (0) ∞ × ( S ∞ × I ). Basically we are dealing with different paths (volumes)which boundaries - S and S , ( t ) ∞ - are common, and thus, following our previous result, theoperator V (Ω) = P e (cid:82) Ω dζdτV J V − , is independent of the volume, and should depend only onthe boundaries of Ω. Therefore, it follows that V (Ω ) = V (Ω ), and this can be written as V ( S (2) ∞ × I ) V (Ω (0) ∞ ) = V (Ω ( t ) ∞ ) V ( S × I ) . (4.9)In order to obtain V (Ω) for each step it is necessary to evaluate K = (cid:82) π dτ V J V − on thesurfaces scanning each volume Ω. We shall scan a hyper-cylinder S × I with surfaces, basedat x R , of the form given in figure (5.b), with t (cid:48) denoting a time in the interval I . Each oneof such surfaces are scanned with loops, labelled by τ , in the following way. For 0 ≤ τ ≤ π ,we scan the infinitesimal cylinder as shown in figure (5.a), then for π ≤ τ ≤ π we scan thesphere S as shown in figure (5.b), and finally for π ≤ τ ≤ π we go back to x R with loopsas shown in figure (5.c). Then, we can split K in three parts: K a + K b + K c , each of themcorresponding to one of the three surfaces, defined by the τ intervals.From the physical point of view it is very reasonable to take the current and the field23trength to satisfy the boundary conditions J µ ∼ R δ F µν ∼ R + δ (cid:48) for R → ∞ , with δ and δ (cid:48) bigger than zero. With that, integration of J over S ∞ vanishesand we get K b = 0 for the path Ω . We notice that for the path Ω , the infinitesimal sphere S does not contribute to K b , and we have also K b = 0 in this case. Then we conclude that K calculated in both spheres S ∞ and S gives the same result and therefore V ( S (2) ∞ × I ) = V ( S × I ) . It is now possible to contract the cylinders into a line, so that the loops scanning them inthe intervals τ ∈ [0 , π ] and τ ∈ [ π , π ] become exactly the same. However, since one setof loops is “going up”, and the other is “going down” (the sign of the velocity dx µ dτ changes),the contributions K a and K c exactly cancel. Note that V inside K , obtained from (4.2) doesnot change sign since it does not see the direction the loops are going to but only takes intoaccount the profile of the loop for each τ . 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) Figure 5: Surfaces of type (b) scan a hyper-cylinder S × I .Finally what remains is exactly V calculated on Ω ( t ) ∞ , whose border is the sphere S , ( t ) ∞ ,where we need to evaluate K . In order to scan the sphere, we establish the point x ( t ) R , onits boundary. This is the point x R at time x = t , so that the path starts at x R , goes upto x ( t ) R , and from this point we scan the sphere with loops. Then, we go back to x R . So,we construct the Wilson line composed of the two parts: W = W ( x, x ( t ) R ) W ( x ( t ) R , x R ) theone corresponding to the “leg” that goes up from x R to x ( t ) R , denoted by W ( x ( t ) R , x R ), andthe one that corresponds to the path from x ( t ) R to the point x ( σ ), on the sphere. Using thisdecomposition we can re-write every needed quantity in terms of the new reference point x ( t ) R .In particular, A in equation (4.2) is decomposed as A x R = W ( x ( t ) R , x R ) − A x ( t ) R W ( x ( t ) R , x R ).24his leads to V x R = W − ( x ( t ) R , x R ) V x ( t ) R W ( x tR , x R ); In the same way the quantities K b and J transform as K b,x R = W − ( x ( t ) R , x R ) K b,x ( t ) R W ( x tR , x R ) and J x R = W − ( x ( t ) R , x R ) J x ( t ) R W ( x tR , x R ),so that V (Ω) obtained from (4.3) becomes V x R (Ω ( t ) ∞ ) = W − ( x ( t ) R , x R ) V x tR (Ω ( t ) ∞ ) W ( x ( t ) R , x R ) . Plugging this into the equation (4.9) we finally get V x tR (Ω ( t ) ∞ ) = U ( t ) · V (Ω (0) ∞ ) · U − ( t ) (4.10)with U ( t ) = W ( x ( t ) R , x R ) · V ( S × I ).Therefore, the operator V x ( t ) R (Ω ( t ) ∞ ) has an iso-spectral evolution in time, and thus, itseigenvalues, or equivalently Tr (cid:16) V x R (Ω ( t (cid:48) ) ∞ ) (cid:17) N , are conserved in time. Using Yang-Mills integralequation one can write V x tR (Ω ( t ) ∞ ) as a volume or a surface ordered integral: V x ( t ) R (Ω ( t ) ∞ ) = P e ie (cid:82) S , ( t ) ∞ dτ dσ ( αF Wµν + β (cid:101) F Wµν ) dxµdσ dxνdτ = P e (cid:82) Ω( t ) ∞ dζ dτ V J V − . (4.11)The fact that this operator is independent of the parameterization of the volume guaranteesthat the conserved charges constructed from it are also independent of this parametrization.One can, however, argue about the dependence of our results with respect to the choice ofthe reference point. As we saw above, by changing x ( t ) R to (cid:101) x tR , V x ( t ) R changes under conjugationwith respect to W ( (cid:101) x tR , x tR ), and so its eigenvalues (conserved charges) are invariant underchange of reference point. In addition, the reference point is placed at the border of Ω x tR (the spatial infinity), and the field strength is supposed to vanish there, which implies thatthe gauge field A µ is asymptotically flat, and therefore the Wilson line depends only on thepoints and not on the path joining them. The conserved charges are also gauge invariant,since under a gauge transformation we have seen that V x ( t ) R (Ω ( t ) ∞ ) → g R V x ( t ) R (Ω ( t ) ∞ ) g − R , with g R being the group element, performing the gauge transformation, at x ( t ) R . Note in addition thatif V x ( t ) R (Ω ( t ) ∞ ) has an iso-spectral evolution so does g c V x ( t ) R (Ω ( t ) ∞ ), with g c and element of thecentre Z ( G ) of the gauge group G . That fact has to do with the freedom we have to choosethe integration constants to lie in Z ( G ), without spoiling the gauge covariance of (4.5).Note that there are several integral and loop space formulations of Yang-Mills theories [8].Our approach shares some of the ideas of those formulations in the sense of using orderedintegrals of the gauge potential and of the field strength tensor. However, it differs in anessential way because it is based on the new eq. (4.5). In addition, it leads in a quite noveland direct way to gauge invariant conserved charges.25 .2 The conserved charges for Dyons Consider a theory where the gauge group G is spontaneously broken into a Higgs field in theadjoint representation. Consider a BPS dyon solution: E i = − F i = sin θ D i φ B i = − (cid:15) ijk F jk = cos θD i φ with θ an arbitrary constant angle.At spatial infinity ( r → ∞ , ˆ r = 1), we have D i φ → ˆ r i πr G (ˆ r ) , where G (ˆ r ) is an element of the Lie algebra of H , covariantly constant D µ G (ˆ r ) = 0[9]. Also,the field strength goes to zero there, which leads to a asymptotically flat gauge field A µ = ie ∂ µ W W − and therefore, at S , ( t ) ∞ we get G (ˆ r ) = W G R W − where G R stands for the value of G (ˆ r ) at x R .Then, the operator V (Ω) discussed previously becomes V x ( t ) R (Ω ( t ) ∞ ) = P e ie (cid:82) S , ( t ) ∞ dτdσ [ αF Wµν + β (cid:101) F Wµν ] dxµdσ dxνdτ = e [ − ie ( α cos θ + β sin θ ) G R ] and the conserved charges are given by the eigenvalues of G R , which contain the magneticand electric charges of the dyon solution. It is important to remark that since G R is at spatialinfinity, according to our observations a change of the reference point makes it change byconjugation with the Wilson line of the path between the new point and the old one; thischanges nothing in the eigenvalues of G R . Acknowledgements
The authors are grateful to fruitful discussions with O. Alvarez,E. Castellano, P. Klimas, M.A.C. Kneipp, R. Koberle, J. S´anchez-Guill´en, N. Sawado andW. Zakrzewski. LAF is partially supported by CNPq, and GL is supported by a CNPqscholarship. 26 ppendices
We give in the next two appendices the proofs, following the arguments of [1, 2], of thestandard non-abelian Stokes theorem for a 1-form connection and then its generalization fora 2-form connection.
A The “standard” non-abelian Stokes theorem
Consider a Lie algebra valued 1-form C = C µ ( x ) dx µ , defined in a d + 1 dimensional simplyconnected space-time M . Given a path γ parametrized by σ ∈ [0 , π ], such that x µ ( σ = 0) ≡ x R and x µ ( σ = 2 π ) ≡ x f , the Wilson line is constructed from dW γ [ σ, dσ + C µ ( x ) dx µ dσ W γ [ σ,
0] = 0 , (A.1)with initial condition W [0 ,
0] = W R .After integrated, equation (A.1) gives the path-ordered integral W γ [ σ,
0] = P exp (cid:18) − (cid:90) σ C µ dx µ dσ (cid:48) dσ (cid:48) (cid:19) · W R . (A.2)There are many paths one can choose to link x R to x f , and a natural question is whether theWilson line depends upon this choice. Being M simply connected, a different path γ (cid:48) withthe same end points x R and x f is related to γ by continuous transformations and the Wilsonline reads W γ (cid:48) [2 π, τ ∈ [0 , π ], labeling the path one chooses to go from x R to x f . For γ we set τ = 0, and for γ (cid:48) , τ = 2 π . All other values give intermediary pathsbetween these two, that arise due to the variation process γ → γ + δγ . With that, each pointis now characterized by two parameters, τ and σ : the first tells in which curve the point is,while the second, where in this curve.The variation is performed as follows. At a given point, immediately after x R we define avector T µ = dx µ dτ , joining γ and γ + δγ . Basically we are defining a way to map one point of thecurve at τ = 0 to a point of the curve at τ (cid:54) = 0. In order to answer if this holds for every otherpoint, we need to parallel transport the vector T µ along γ , i.e. , along the direction of S µ = dx µ dσ ;the Lie derivative of T µ in the direction of S µ gives L S T µ = S ν ∂ ν T µ − T ν ∂ ν S µ = d x µ dσdτ − d x µ dτdσ = 0and therefore if we define the variation to be orthogonal to the curve γ at a given point, this The notation W γ [ σ,
0] says that the quantity W is evaluated on the path γ from the point with σ = 0 tothe point x µ ( σ ). γγ + δγ T µ S µ x R sexta-feira, 17 de junho de 2011 Figure 6: The variation of a path with fixed end points.will be the case for every other point. A variation of the path implies a variation of theholonomy, which can be calculated taking the variation of the equation (A.1): δ (cid:18) dW γ dσ + C µ ( x ) dx µ dσ W γ (cid:19) = 0 . Computing this explicitly (omitting some symbols that we shall reintroduce at the end): W − ddσ ( δW ) + W − C µ dx µ dσ δW + W − δ (cid:18) C µ dx µ dσ (cid:19) W = 0 ddσ (cid:0) W − δW (cid:1) − (cid:18) ddσ W − (cid:19) δW + W − C µ dx µ dσ δW + W − δ (cid:18) C µ dx µ dσ (cid:19) W = 0 ddσ (cid:0) W − δW (cid:1) + W − δ (cid:18) C µ dx µ dσ (cid:19) W = 0where after the variation was performed we multiplied the equation by W − from the left(line 1), used the chain rule in order to rewrite the first term (line 2), and with the identity dW − dσ = − W − dWdσ W − and equation (A.1), two terms were mutually canceled (line 3). Thislast equation can be integrated from x µ ( σ = 0) to x µ ( σ ), a point of γ . Taking into accountthe fact that the holonomy does not change at the initial point ( δW = 0), we get δW γ [ σ,
0] = − W γ [ σ, (cid:90) σ dσ (cid:48) W − γ [ σ (cid:48) , δ (cid:18) C µ dx µ dσ (cid:48) (cid:19) W γ [ σ (cid:48) , . We introduced a new, but obvious, notation: δW γ [ τ ; σ, x µ ( σ ) in the curve τ . (cid:90) σ dσ (cid:48) W − δ (cid:18) C µ dx µ dσ (cid:48) (cid:19) W = W − ( σ ) C µ ( x ( σ )) W ( σ ) δx µ ( σ ) + (cid:90) σ dσ (cid:48) (cid:26) W − δC µ dx µ dσ (cid:48) W − ddσ (cid:48) (cid:0) W − C µ W (cid:1) δx µ (cid:27) = W − C µ W δx µ + (cid:90) σ dσ (cid:48) W − (cid:26) ∂ ν C µ δx ν dx µ dσ (cid:48) − dC µ dσ (cid:48) δx µ + (cid:20) C µ , dWdσ (cid:48) W − (cid:21) δx µ (cid:27) W = W − C µ W δx µ + (cid:90) σ dσ (cid:48) W − (cid:26) ∂ ν C µ δx ν dx µ dσ (cid:48) − ∂ ν C µ dx ν dσ (cid:48) δx µ − (cid:20) C µ , C ν dx ν dσ (cid:48) (cid:21) δx µ (cid:27) W = W − ( σ ) C µ ( x ( σ )) W ( σ ) δx µ ( σ ) − (cid:90) σ dσ (cid:48) W − F µν W dx µ dσ (cid:48) δx ν where F µν = ∂ µ C ν − ∂ ν C µ + [ C µ , C ν ] is the curvature of C µ .From line 1 to line 2 we performed an integration by parts and used the fact that δx µ (0) = 0.Then, since the variation in the connection is due to a variation in the space-time point( x → x + δx ), we took δC µ = ∂ ν C µ δx ν in line 3, and performed the derivative. After that,we used chain rule for the derivative of the connection in σ (cid:48) , and equation (A.1) inside thecommutator, which gives line 4. Then, relabeling the indices we get a curvature, showed inline 5.Finally, the variation of the Wilson line due to a variation of the path γ to another path,labeled by τ reads δW γ [ τ ; σ,
0] = − C µ W γ [ τ ; σ, δx µ ( σ ) + W γ [ τ ; σ, (cid:90) σ dσ (cid:48) W − γ [ τ ; σ (cid:48) , F µν W γ [ τ ; σ (cid:48) , dx µ dσ (cid:48) δx ν (A.3)Notice that this is a general result for the variation is not specified: one can do it in bothdirections, tangent to the path, or orthogonal to it. Of course, in the tangent direction, avariation is just a reparametrization σ → (cid:101) σ = f ( σ ) , which, from equation (A.1), changesnothing as long as f ( σ ) is monotonic. On the other hand, in the orthogonal direction onehas δx µ = T µ and δW [ τ ] = W [ τ + δτ ] − W [ τ ] = dWdτ δτ , which, in (A.3) (with σ = 2 π , sothat the end points are the same for both paths, and therefore δx µ (0) = δx µ (2 π ) = 0) gives adifferential equation for W : ddτ W γ [ τ ; 2 π, − W γ [ τ ; 2 π, (cid:90) π dσ W − γ [ τ ; σ, F µν W γ [ τ ; σ, dx µ dσ dx ν dτ = 0 . (A.4)29hat we conclude is that there are two possible ways to calculate the Wilson line of a givencurve. Consider, for instance, the path γ (cid:48) . The Wilson line W γ (cid:48) [ τ = 2 π ; x f , x R ] can beobtained, first, from equation (A.1), after integration over γ (cid:48) . The second possibility is tothink of γ (cid:48) as obtained from γ , using the variations we discussed above. Then the holonomythere is given after integrating (A.4) from τ = 0 to τ = 2 π : W γ (cid:48) [2 π ; 2 π,
0] = W γ [0; 2 π, · P exp (cid:18)(cid:90) π dτ (cid:90) π dσ W − γ (cid:48) F µν W γ (cid:48) dx µ dσ dx ν dτ (cid:19) where P stands for the “surface-ordering”, i.e. , the ordering due to the non-abelian characterof the fields, according to the way we vary τ .It is clear that the quantity appearing in the r.h.s of the above equation is the flux of F Wµν ≡ W − F µν W through the space-time surface Σ ⊂ M , whose boundary is ∂ Σ = Γ ≡ γ (cid:48)− · γ ;lets call it Φ( F W , Σ), and rewrite the equation as W γ (cid:48) = W γ · P e Φ( F W , Σ) . (A.5)Consider the case where the path discussed above is a loop Γ with no self intersections: x µ : σ ∈ [0 , π ] → M σ (cid:55)→ x µ ( σ ); x µ (0) = x µ (2 π ) . Then, integration of (A.1) for the whole loop gives W Γ = P e (cid:72) Γ C µ dx µ · W R . (A.6)Being Γ the boundary of a two dimensional submanifold Σ ⊂ M , we also have that this sameWilson line can be calculated taking a variation from the point loop P x R = x R ∀ σ ∈ [0 , π ](whose Wilson line is W R ) W Γ = W R · P e Φ( F W , Σ) . (A.7)The fact that it is possible to calculate W Γ integrating the connection C over Γ, or integratingthe curvature F ( C ) over the area bounded by Γ is exactly the statement of the Stokes theorem: P exp (cid:18)(cid:73) ∂ Σ C µ dx µ (cid:19) · W R = W R · P exp (cid:0) Φ( F W , Σ) (cid:1) . (A.8)30 A generalization of the non-abelian Stokes theoremfor a 2-form connection
Let us consider the antisymmetric field B µν ( x ), defined in the d + 1 dimensional space-time M . A family of (homotopically equivalent) loops with base point x R can be used to scan thetwo-dimensional hypersurface Σ, starting from the infinitesimal loop around x R and takingvariations along the T µ direction, as explained in the previous section.A new quantity V is introduced in analogy with the Wilson line in equation (A.4), definedby dV Σ [ τ, dτ − V Σ [ τ, (cid:90) π dσ W − [ τ ; σ, B µν W Γ [ τ ; σ, dx µ dσ dx ν dτ = 0 , (B.1)with the initial condition V [0 , ≡ V Σ R , being Σ R the infinitesimal surface around x R .The surface-holonomy V Σ [ τ,
0] is defined on the surface whose boundary is the loop labeledby τ . The quantity T π ( B, C, τ ) ≡ (cid:90) π dσ W − [ τ ; σ, B µν W Γ [ τ ; σ, dx µ dσ dx ν dτ plays the role of a “non-local connection” , defined on each loop. The Wilson lines appearinginside the integral are calculated from (A.1), running from x R to x µ ( σ ) on the loop τ .After integrating (B.1) one gets the surface-ordered integral V Σ [ τ,
0] = V R · P exp (cid:18)(cid:90) τ dτ (cid:48) T π ( B, C, τ (cid:48) ) (cid:19) . (B.2)In analogy with the fact that the Wilson line is defined over the path that links two (boundary)points, the surface-holonomy is defined over the surface that links two (boundary) loops. Takethese two loops to be those at τ = 0 and τ = 2 π , fixed. Clearly one can use different surfacesto link them. Consider Σ and Σ (cid:48) , two possibilities. Each of these choices might lead to adifferent solution of (B.1). Since Σ (cid:48) can be obtained from Σ after a variation in the Z µ ≡ dx µ dζ direction ( ζ ∈ [0 , π ] ) we can calculate the difference δV = V [ ζ + δζ ] − V [ ζ ] = dVdζ δζ inanalogy with what we did for the path-holonomy, taking first the variation of the definingequation (B.1): δ (cid:18) dV Σ [ τ, dτ − V Σ [ τ, T π ( B, C, τ ) (cid:19) = 0 The 2-form B = B µν ( x ) dx µ ∧ dx ν is not necessarily exact. For future reference we call it surface-holonomy. It was shown in [1, 2] that it is in fact a 1-form connection in the loop space. This direction is in the normal direction to the surface Σ. ddτ (cid:0) δV V − (cid:1) − δV (cid:18) dV − dτ + T π ( B, C, τ ) V − (cid:19) − V δT π ( B, C, τ ) V − = 0 ddτ (cid:0) δV V − (cid:1) − V δT π ( B, C, τ ) V − = 0We took the variation, and multiplied by V − from the right. After that the chain rule wasused to get the first two terms in line 1. Then, using the identity dV − dτ = − V − dVdτ V − andequation (B.1) the term inside the parenthesis in line 2 vanishes. The next step is to expandthe term V T π ( B, C, τ ) V − appearing above. V δT π ( B, C, τ ) V − = V (cid:90) π dσ (cid:0) δW − B µν W + W B µν δW − + W δB µν W − (cid:1) dx µ dσ dx ν dτ V − + V (cid:90) π dσ W − B µν W dδx µ dσ dx ν dτ V − + V (cid:90) π dσ W − B µν W dx µ dσ dδx ν dτ V − = V (cid:90) π dσ (cid:0)(cid:2) B Wµν , W − δW (cid:3) + W − ∂ ρ B µν W δx ρ (cid:1) dx µ dσ dx ν dτ V − − V (cid:90) π dσ (cid:18)(cid:20) B Wµν , W − dWdσ (cid:21) dx ν dτ + W − ∂ ρ B µν W dx ρ dσ dx ν dτ + B Wµν d x ν dσdτ (cid:19) δx µ V − + ddτ (cid:0) V T π ( B, C, δ ) V − (cid:1) − dVdτ T π ( B, C, δ ) V − − V T π ( B, C, δ ) dV − dτ − V (cid:90) π dσ ddτ (cid:18) W − B µν W dx µ dσ (cid:19) V − δx ν . When the integration by parts of the second term was performed we used the fact that δx µ ( σ )vanishes at the boundaries. For the commutators we use δW given by (A.3) and dWdσ given by(A.1). In line 5 we can use the identity dV − dτ = − V − dVdτ V − and we calculate the derivativein the last term using equation (B.1). 32lugging the result back into the equation we started with, and integrating gives δV Σ [ τ,
0] = (cid:0)
V T π ( B, C, δ ) V − (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) τ V Σ [ τ,
0] ++ (cid:90) τ dτ (cid:48) V (cid:40) (cid:90) π dσW − ( D ρ B µν + D µ B νρ + D ν B ρµ ) W dx µ dσ dx ν dτ (cid:48) δx ρ ++ (cid:90) π dσ (cid:2) B Wµν , T σ ( F, C, δ ) (cid:3) dx µ dσ dx ν dτ (cid:48) − (cid:90) π dσ (cid:2) B Wµν , T σ ( F, C, τ (cid:48) ) (cid:3) dx µ dσ δx ν ++ [ T π ( B, C, δ ) , T π ( B, C, τ (cid:48) )] (cid:41) V − V Σ [ τ, . The second term will be called K , so that we can write δV Σ [ τ,
0] = (cid:0)
V T π ( B, C, δ ) V − (cid:1) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) τ V Σ [ τ,
0] + K V Σ [ τ, . (B.3)Taking τ = 2 π , the first term on the RHS vanishes and we get δV Σ [2 π, − K V Σ [2 π,
0] = 0 . (B.4)Then, taking the variation to be along the Z µ direction so that δV = dVdζ δζ , we get thedifferential equation for V (now we drop some of the symbols that have being used so far): ddζ V − K V = 0 , (B.5)where all δx µ in K is replaced by dx µ dζ δζ , and we write it in the nicer form: K = (cid:90) π dτ V (cid:40) (cid:90) π dσ W − ( D ρ B µν + D µ B νρ + D ν B ρµ ) W dx µ dσ dx ν dτ dx ρ dζ + − (cid:90) π dσ (cid:90) σ dσ (cid:48) (cid:2) B Wκλ ( σ (cid:48) ) − F µνκλ ( σ (cid:48) ) , B Wµν ( σ ) (cid:3) dx κ dσ (cid:48) ( σ (cid:48) ) dx µ dσ ( σ ) ×× (cid:18) dx λ dτ ( σ (cid:48) ) dx ν dζ ( σ ) − dx λ dζ ( σ (cid:48) ) dx ν dτ ( σ ) (cid:19) (cid:41) V − . ◦ , so that the solution of the equation (B.1) reads V Σ ◦ = V R · P exp (cid:18)(cid:90) π dτ T π ( B, A, τ ) (cid:19) . (B.6)The surface Σ ◦ is the boundary of the three-dimensional submanifold (volume) Ω, so thesurface-holonomy V Σ ◦ can be calculated from variations in the Z µ direction starting at theinfinitesimal surface Σ R around the reference point x R , using (B.5): V Σ ◦ = P exp (cid:18)(cid:90) π dζ K (cid:19) V Σ R . (B.7)The fact that there are two ways to compute the surface-holonomy V Σ ◦ is the statement ofthe Stokes theorem: V Σ R P exp (cid:18)(cid:90) π dτ T π ( B, A, τ ) (cid:19) = P exp (cid:18)(cid:90) π dζ K (cid:19) V Σ R (B.8)being the l.h.s computed over the surface Σ which is the boundary of the volume Ω, where weintegrate the r.h.s. In order to make it more explicit we rewrite this as V Σ R P exp (cid:18)(cid:90) ∂ Ω dτ dσ W − B µν W dx µ dσ dx ν dτ (cid:19) = P exp (cid:18)(cid:90) π dζ K (cid:19) V Σ R . eferences [1] O. Alvarez, L. A. Ferreira and J. Sanchez Guillen, “A New approach to integrable theoriesin any dimension,” Nucl. Phys. B , 689 (1998) [arXiv:hep-th/9710147].[2] O. Alvarez, L. A. Ferreira and J. Sanchez-Guillen, “Integrable theories and loop spaces:Fundamentals, applications and new developments,” Int. J. Mod. Phys. A , 1825 (2009)[arXiv:0901.1654 [hep-th]].[3] L. A. Ferreira, W. J. Zakrzewski, “A Simple formula for the conserved charges of solitontheories,” JHEP , 015 (2007). [arXiv:0707.1603 [hep-th]].[4] P. Lax, Comm. Pure Appl. Math. (1968) 467-490.V.E. Zakharov and A.B. Shabat, Zh. Exp. Teor. Fiz. (1971) 118-134; english transl. Soviet Phys. JETP (1972) 62-69.[5] G. V. Dunne, “Aspects of Chern-Simons theory,” Lectures at the 1998 Les HouchesSummer School: Topological Aspects of Low Dimensional Systems , [hep-th/9902115].[6] L. A. Ferreira and G. Luchini;
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