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ANALYSIS OF OBSERVABLES IN

arXiv:hep-th/9110069v1 29 Oct 1991US-FT-10/91October, 1991ANALYSIS OF OBSERVABLES INCHERN-SIMONS PERTURBATION THEORYM. Alvarez and J.M.F.

Labastida⋆Departamento de F´ısica de Part´ıculasUniversidade de SantiagoE-15706 Santiago de Compostela, SpainABSTRACTChern-Simons Theory with gauge group SU(N) is analyzed from a perturba-tion theory point of view. The vacuum expectation value of the unknot is computedup to order g6 and it is shown that agreement with the exact result by Witten im-plies no quantum correction at two loops for the two-point function.

In addition, itis shown from a perturbation theory point of view that the framing dependence ofthe vacuum expectation value of an arbitrary knot factorizes in the form predictedby Witten.⋆E-mail: LABASTIDA@EUSCVX.DECNET

1. IntroductionChern-Simons gauge theory was solved exactly by Witten [1] using non-perturbative methods.This solution has been obtained subsequently by othergroups using both, the point of view of canonical quantization [2-10], and of cur-rent algebra [11,5] as originally proposed in [1].

The exact result for the vacuumexpectation value of the observables of the theory is analytic in the inverse ofthe Chern-Simons parameter k. Defining the Chern-Simons coupling constant asg =p4π/k the exact result suggests that the small coupling constant pertur-bation expansion should reproduce the exact result.One does not expect anynon-perturbative effect in Chern-Simons gauge theory. Perturbative approachesto the theory under consideration have been carried out during the last two years[12-25].

The main question which have been addressed in these works is which oneis the renormalization scheme which leads to the exact result obtained by Witten.In Chern-Simons gauge theory there are two problems which must be taken intoaccount. On the one hand, the loop expansion possesses divergences already at oneloop for two-point functions which must be regularized.

On the other hand, someof the observables of the theory, the Wilson lines, possess products of operators atcoincident points in their integration regions. The loop expansion divergences mustbe regulated in perturbation theory to obtain a finite answer to be compared tothe exact result.

The ambiguities present when considering products of operatorsat coincident points forces to make a choice in defining the observables of the the-ory. The main goal of this paper is to give a regularization procedure and a choiceto solve the problem of the ambiguity when considering products of operators atcoincident points whose perturbative expansion coincides with the exact result [1].The second aspect of the problem was solved successfully in [12] and we will followhere their approach.To handle the ambiguity associated to products of operators at coincidentpoints one must consider framed links instead of links [26,1,12], in other words onemust introduce a band instead of a knot and the corresponding integer number1

n which indicates the number of times that the band is twisted.In the non-perturbative approach leading to the exact result [1] the origin of the dependenceon the framing comes about because one must construct the observables on thesurface of a Riemann surface which then must be glued to another Riemann surfaceto build a three-dimensional manifold. The same knot can be obtained using thatprocedure in a variety of ways leading to different quantities for observables which,however, differ by a factor which is associated to the framing.As first foundout in [1] this factor is just exp(2πinh) where h is the conformal weight of therepresentation (for SU(N) in the fundamental representation, h =c2(R)/(k + N),c2(R) = (N2 −1)/2N) carried out by the Wilson line.The presence of ultraviolet divergences in the loop expansion of Chern-Simonstheory forces to regularize the theory and consequently to choose a renormaliza-tion scheme.

Certainly, from a perturbation theory point of view all schemes arephysically equivalent since they differ by a finite renormalization which can be ac-complished by adding finite counterterms to the action. However, one would liketo know if there exist a scheme which leads naturally to the exact result obtainedby Witten.

By naturalness we understand a scheme in which the intermediateregularized action leads after taking the limit in which the cutoffis removed tothe exact result obtained by Witten where the constant k we started with (barek) is the same constant k as the one appearing in the exact result. Certainly, thisconcept of naturalness has only meaning in a theory like Chern-Simons theory inwhich the beta function as well as the anomalous dimensions of the elementaryfields vanish at any order in perturbation theory [20,21,24,27].

This was the pointof view taken in [13] where a scheme based on Pauli-Villars regularization seemedto be natural in the sense discussed above. Indeed, the results obtained in [13]showed that a choice of scheme of that type seems to lead to the shift k→k + Nwhich appears in many of the equations corresponding to the exact result.

Thefact that the origin of the shift is a quantum effect was first pointed out by Witten[1] who showed its appearance using a gauge invariant regularization based on theeta function. There are other schemes which lead to results in agreement with2

[13,1] as the one used in [18]. All these schemes which seem to provide at oneloop an explanation of the origin of the shift share the common feature that theintermediate regularized action is gauge invariant.So far all calculations involving quantum corrections using gauge invariantregularized actions have concentrated on the effective action.

The fact that thequantum correction leads to an effective action whose constant k has been shiftedindicates that one would observe such effect when computing observables usingthose schemes. However, at present, only indirect calculations of observables havebeen carried out taking into account this quantum correction [25,12].

In this paperwe are going to present the computation of the Wilson line corresponding to theunknot in the fundamental representation of SU(N) up to order g6. This calcu-lation involves diagrams at two-loops for the two-point function whose calculationin some scheme whose regularized action is gauge invariant has to be carried out.We do not perform in this paper such a two-loop calculation but we will show thatagreement with the exact result implies that there is no correction a two-loops forthe two-point function.

The computation of the two-point function at two-loopsusing the Pauli-Villars regularization plus higher derivatives proposed in [13] isbeing carried out [28]. So far we have been able to prove that there is no need tointroduce higher derivative terms to regulate the theory at two loops and that asingle generation of Pauli-Villars is sufficient to render the two-loop graphs finite.However, we have not finished the calculation of the finite part that accordingto the results which we present in this work should be zero to have agreementwith the exact result, and, therefore, to be able to consider the scheme based onPauli-Villars as natural.To end with this introduction we will reproduce here the result obtained byWitten in [1] for the framed unknot in the fundamental representation of SU(N)lying on S3 with framing n. The corresponding vacuum expectation value is,⟨W⟩= qN2 −q−N2q12 −q−12 qn N2−12N ,(1.1)3

where,q = exp( 2πik + N ). (1.2)Expanding (1.1) in terms of 1/k up to terms of order 1/k3 one finds,⟨W⟩= N + 1kiπn(N2 −1)+ 1k2h−π26 N(N2 −1) −π2n22N (N2 −1)2 −iπnN(N2 −1)i+ 1k3h−iπ36 n(N2 −1)2 −iπ36N2n3(N2 −1)3 + π2n2(N2 −1)2+ π23 N2(N2 −1) + iπnN2(N2 −1)i+ O( 1k4).

(1.3)Notice that in this expansion all terms containing a power of π different that thepower of 1/k are originated by the fact that k appears shifted into k + N in (1.2).In our analysis we will show that those terms do indeed correspond to diagramswhich contain one-loop quantum corrections.Notice also that in the standardframing (n = 0) the series expansion has a simpler form. As a consequence ofour analysis we will be able to identify very simply all diagrams which provide theframing dependence.

Actually, we will derive from a perturbation theory point ofview the form of the framing dependence of the vacuum expectation value of anarbitrary knot. If, on the other hand, it turns to be correct the picture in whichthere are only one-loop corrections (which just account for the shift k→k +N) onecould extract all the effects due to framing and therefore one would be left witha series of diagrams which constitute the building blocks of the knot invariant.These building blocks lead to topological invariants which after considering themas the coefficients of a power series build the knot invariants leading to the Jonespolynomials [29] and its cousins [30,31].

We will discuss in more detail this pictureof the perturbation theory series expansion in our concluding remarks.In this paper we will consider the three dimensional manifold as R3 which allowsus to identify the corresponding observables to the ones in S3. We will not discuss4

the effect of the framing of the three-dimensional manifold from a perturbationtheory point of view. A good discussion of this point can be found in [25].The paper is organized as follows.

In sect. 2 we define the regularized Chern-Simons gauge theory using Pauli-Villars fields which we claim to correspond toa natural scheme in the sense discussed above.

In sect. 3 we compute (1.3) inperturbation theory the vacuum expectation value of the unknot carrying the fun-damental representation of SU(N) up to order g6.

In sect. 4 we will identify allthe framing dependence of the vacuum expectation value of a knot and we willshow its factorization in the form predicted by Witten.

Finally, in sect. 5 we stateour conclusions and make some final remarks .

Several appendices deal with ourconventions and with the proof of some results which are used in sects. 3 and 4.5

2. Perturbative Chern-Simons gauge theoryIn this section we will define Chern-Simons gauge theory from a perturbationtheory point of view.This is carried out in two steps.First a gauge fixing isperformed.

Second, after analyzing the ultraviolet behavior of the theory a reg-ularized action using Pauli-Villars fields is provided. Let us consider an SU(N)gauge connection Aµ on a boundaryless three-dimensional manifold M and thefollowing Chern-Simons action,S(Aµ) = k4πZMTr(A ∧dA + 23A ∧A ∧A),(2.1)where k is an arbitrary positive integer⋆and “Tr” denotes the trace in the fun-damental representation of SU(N) (normalized in such a way that Tr(T aT b) =−12δab).

A summary of our group-theoretical conventions is contained in AppendixA. In defining the theory from a perturbation theory point of view we must give ameaning to vacuum expectation values of operators, i.e., to quantities of the form,⟨O⟩= 1ZZ[DAµ]O expiS(Aµ),(2.2)where Z is the partition function,Z =Z[DAµ] expiS(Aµ).

(2.3)The operators entering (2.2) are gauge invariant operators which do not depend onthe three-dimensional metric. These operators are knots, links and graphs [1,32].The first issue in defining (2.2) is to take care of the gauge fixing.

Indeed, the⋆A negative k will change the ǫ prescription in the perturbative series expansion leading toa shift of k at one loop with the opposite sign. With a negative k one makes connectionwith the exact result (1.1) after replacing q→q∗.6

exponential in (2.2) is invariant under gauge transformations of the form,Aµ→h−1Aµh + h−1∂µh,(2.4)where h is an arbitrary continuous map h : M→SU(N). Before carrying out thegauge fixing let us redefine the constant k and the field Aµ in such a way thatthe action (2.1) becomes standard from a perturbation theory point of view.

Wedefine,g =r4πk . (2.5)Then, after rescaling the gauge connection,Aµ→gAµ,(2.6)one obtains the following Chern-Simons action,S′(Aµ) =ZMTr(A ∧dA + 23gA ∧A ∧A).

(2.7)This form of the action contains the standard 1/2 factor for the kinetic termafter using (A3).From now on we will restrict ourselves to the case in whichthe three-dimensional manifold M is R3 which is the simplest case to treat from aperturbation theory point of view. Though (2.7) is metric independent, we will beforced to introduce a metric in carrying out the gauge fixing.

We will assume thatthis metric has signature (1, −1, −1).Our gauge choice will be the same as the one taken in [13]. The Lorentz-likegauge condition ∂µAµ = 0 is imposed using the standard Fadeev-Popov construc-tion which leads to the following action to be added to (2.7),Sgf(Aµ, c, ¯c, φ) =ZTr2¯c∂µDµc −2φ∂µAµ −λφ2,(2.8)where φ is the Lagrange multiplier which imposes the gauge condition, c and ¯care the Fadeev-Popov ghost, and λ is a gauge fixing parameter.

In (2.8), Dµ is7

the covariant derivative, Dµc = ∂µc + g[Aµ, c]. The action (2.7) as well as thegauge-fixing action (2.8) are invariant under the following BRST transformations,sAµ = Dµc,sc = −cc,s¯c = φ,sφ = 0.

(2.9)The field φ can be integrated out easily providing the following functional integralfor vacuum expectation values as the ones in (2.2):⟨O⟩= 1ZZ[DAµDcD¯c]O expiI(Aµ, c, ¯c),(2.10)where,I(Aµ, c, ¯c) =ZTrǫµνρ(Aµ∂νAρ + 23AµAνAρ) −λ−1Aµ∂µ∂νAν + 2¯c∂µDµc. (2.11)Of course, Z in (2.10) is appropriately defined taking into account the gauge fixing.The quantities obtained in (2.10) are independent of the value of λ.

In order toavoid the presence of infrared divergences we will work in the Landau gauge inwhich λ = 0The perturbative series expansion which one obtains from (2.10) and (2.11)possesses some divergences which need to be regularized. The analysis of the na-ture of these divergences was carried out in [13] by performing the correspondingpower counting.

There are many ways to regularize these divergences giving phys-ically equivalent results. In this work we will follow the regularization procedureintroduced in [13], i.e., we will use a gauge invariant regularization based on theintroduction of Pauli-Villars fields and, if needed, higher-derivative terms.

Thisseems to provide a scheme which is natural in the sense explained in sect.1.Further work have shown [28] that there is no need to introduce higher-derivativeterms. The Pauli-Villars fields which one introduces to regulate at one loop seemto be sufficient to render the theory finite to any order.

Of course, after the gaugefixing has been performed, when talking about a gauge invariant regularization wemean a regularization which preserves the BRST symmetry (2.9).8

Following [13] we introduce Pauli-Villars fields A(j)µ , c(i) and ¯c(i), j = 1, ..., Jand i = 1, ..., I. The regularized functional integral takes the form,⟨O⟩= 1ZZ[DAµDcD¯c]O expiI(Aµ, c, ¯c) JYj=1det−bj/2AjIYi=1detciCi,(2.12)where, of course, the same type of regularization is used for Z, and all the depen-dence on the Pauli-Villars fields is contained in the determinants,det−1/2Aj =Z[DA(j)µ ] expiZTr(ǫµνρA(j)µ DνA(j)ρ+ MjA(j)µ A(j)µ)detCi =Z[D¯c(i)Dc(i)] exp2iZTr(¯c(i)DµDµc(i) −m2i ¯c(i)c(i)).

(2.13)The masses entering into the determinants in (2.13) as well as the integers bj,j = 1, ..., J and ci, i = 1, ..., I are the regulating parameters. The relative valuesof these masses and these integers are fixed to make the theory finite in the limitin which the common scale of the masses Λ becomes large.

In [13] was shown thatthe following choice makes the theory finite at one loop,JXj=1bj = 1,JXj=1bjMj= 0,JXj=1bjM2j= 0,I = J,cj = 12bj,mj = Mj. (2.14)We conjecture that the limit Λ→∞of (2.12) with the choice (2.14) generates thesame values for the observables of the theory (once the ambiguities originated at co-incidence points of products of operators are handled as shown in the next section)as the ones in the exact result obtained by Witten [1].

The results presented in thispaper and in [28] provide certain evidence towards the validity of this conjecture.As shown in [13] the regularized action entering (2.12) is BRST invariant.Indeed, defining the BRST transformations of the Pauli-Villars fields as just gauge9

transformations of fields transforming in the adjoint representation whose gaugeparameter is the ghost field c,sA(j)µ= [A(j)µ , c],s¯c(i) = {¯c(i), c},sc(i) = {c(i), c},(2.15)it is simple to prove that the determinants entering (2.12) are BRST invariant.To end this section let us summarize the Feynman rules of the theory as wellas the one-loop results obtained in [13]. They will become very useful in the nextsection where the Wilson line corresponding to the unknot will be computed toorder g6.

We will work in space-time space. The two basic Feynman rules enteringour calculations are summarized in Fig.

1. In particular, the propagator associatedto the gauge field takes the form,Σµνab (x, y) = i4πδabǫµρν (x −y)ρ|x −y|3 .

(2.16)We do not give the Feynman rules corresponding to ghost and Pauli-Villars fieldssince these fields only enter in loops and we will take the results obtained in [13]for one-loop Green functions. These results are summarized in Fig.

2.10

3. Unknot to order g6.In this section we will compute the vacuum expectation value of the Wilson linecorresponding to the unknot in the fundamental representation of SU(N) using thefunctional integral defined in (2.12).

This calculation will provide the tools andmethods to analyze general features of the perturbative series expansion of vacuumexpectation values of knots as the one considered in the next section. Taking intoaccount the rescaling (2.6), the operator O entering in (2.12) has the form,W = TrPeg HA,(3.1)where the trace is taken over the fundamental representation and P denotes path-ordered product.

This choice of sign in the exponential leads to the convention(A1). The contour integral in (3.1) corresponds to any path diffeomorphic to theunknot.

To compute the vacuum expectation value of this operator in perturbationtheory we have to consider all diagrams which are not vacuum diagrams since, asshown in (2.12), we consider normalized vacuum expectation values, i.e., in (2.12),the functional integration where the operator is inserted is divided by the parti-tion function Z. The expansion of the path-ordered exponential in (3.1) reducesthe calculation to certain integrals of n-point functions.

These n-point functionsneed to be computed perturbatively up to certain order. We will use the standardFeynman diagrams to denote these n-point functions.

To denote the contour inte-gral we will attach their n-points by a circle. For convenience, let us express theperturbative series corresponding to the vacuum expectation value of (3.1) as,⟨W⟩=∞Xi=0w2ig2i.

(3.2)Clearly, to order g0 the computation of the vacuum expectation value of (3.1)reduces to the trace of the unit operator in the fundamental representation which11

is just N,w0 = N,(3.3)in agreement with (1.3). Higher orders up to g6 will be computed in the followingsubsections.3.1.

Order g2To this order, since there is a factor g in the exponential (3.1) there is only onediagram which just involves the propagator (2.16). This diagram is shown in Fig.

3. Its contribution to the perturbative series (3.2) is just,w2 =Tr(T aT b)IdxµxZdyν i4πδabǫµρν (x −y)ρ|x −y|3=12Tr(T aT b)IdxµIdyν i4πδabǫµρν (x −y)ρ|x −y|3 .

(3.4)To perform the step carried out in obtaining the second expression for w2 onemust first realize that the integration is well defined and finite, and symmetricunder the interchange xµ ↔yν.Notice that although it seems that there aresingularities at coincident points, a careful analysis of the integral shows that thisis not the case [26,12]. However, from a quantum field theory point of view thequantity entering (3.4) is not well defined.

The reason is that among the points ofintegration there are points where one is using quantities like ⟨Aµ(x)Aν(x)⟩whichare not well defined from a field theory point of view. One could add a finite part atthose coincident points making the integration ambiguous.

As shown in [12] thereis way to solve this ambiguity providing a procedure which is metric independentas it would be desirable from the point of view of topological field theory. The ideais to introduce an unit vector nµ normal to the path of integration and consider thepath corresponding to yν as the one constructed by yν = xν + εnν.

The resultingintegral depends on the choice of nµ and it corresponds to the the Gauss integral12

which can be normalized such that its value is an integer n,n = 14πIdxµI ′dyνǫµνρ(x −y)ρ|x −y|3 . (3.5)In this equation the prime denotes that the second path is slightly separated fromthe first path as dictated by the unit vector nµ.

Often we will refer to this situationas saying that x runs over the knot and y over its frame. The integer value n isthe linking number of the two non-intersecting paths.

In general, the perturbativeexpansion of the Wilson line will possess terms containing the ambiguity discussedhere.From a field theory point of view, one may detect the presence of thisambiguity just observing if in the integrations of products of operators one isintegrating over coincident points. Fortunately, this seems to happen only whenthe two end points of a propagator may get together (“collapsible” propagator).It turns out that the three-point function possesses milder singularities than thepropagator at coincident points and it does not introduce any ambiguity.

We willdiscuss in more detail this feature in the next section.Using (3.5) and (A3) one finds for w2 in (3.4),w2 = i4n(N2 −1),(3.6)which is in agreement with (1.3) after taking into account that g2 = 4π/k.3.2. Order g4The diagrams contributing to this order are depicted in Fig.

4. It is at thisorder where the first appearance of a diagram involving quantum corrections ispresent.

Namely, diagram a of Fig. 4 contains the full two-point function at oneloop.

This two-point function was computed in [13] in the scheme adopted in thispaper. The result obtained there has been summarized in Fig.

2. Taking intoaccount that result and the previous calculation leading to w2 we can write very13

simply the contribution of diagram a of Fig. 4 to this order,w(a)4= −N4πw2 = −i16πnN(N2 −1).

(3.7)Notice that this contribution corresponds to the last one at order 1/k2 in (1.3).This term in (1.3) is such that the power of π and the power of 1/k are differentand therefore corresponds to the type of terms which are in the expansion of ⟨W⟩because k appears shifted into k + N in the exact result (1.1). This is the firstcase in which we will observe that a diagram present because of the existence ofquantum corrections gives a contribution which corresponds to the one originatedby the shift present in the exact result.The contribution of diagrams b, and c1, c2 and c3 of Fig.

4 has been analyzedin detail in [12,25]. We will use here their results and we will make a series ofremarks which will be useful in computations at higher order.

The contributionfrom b is,w(b)4=Tr(T aT bT c)IdxµxZdyνyZdzρZd3ω(−i)fabcǫν1ν2ν3i4πǫµρ1ν1 (x −w)ρ1|x −w|3i4πǫνρ2ν2 (y −w)ρ2|y −w|3i4πǫρρ3ν3 (z −w)ρ3|z −w|3=18N(N2 −1)ρ1(C),(3.8)where we have used (A1) and (A3) and,ρ1(C) =132π3IdxµxZdyνyZdzρZd3ωǫµν1ρ1ǫνν2ρ2ǫρν3ρ3ǫν1ν2ν3(x −w)ρ1(y −w)ρ2(z −w)ρ3|x −w|3|y −w|3|z −w|3. (3.9)This quantity has a special significance which we will discuss after analyzing thecontribution from the rest of diagrams at this order.

The argument of ρ1(C), C, isthe integration path. Notice that the integration entering ρ1(C) does not possess14

any ambiguity due to the presence of products of operators at coincident points andtherefore it is framing independent. The reason why ambiguities are not present isthat coincident points occur pairwise, i.e., the three endpoints never get together inthe integration, and singularities associated to this case are too mild to introduceambiguities.

Of course, this assertion needs a careful proof which indeed has beencarry out indirectly in [12,25]. Form a quantum field theory point of view, it seemsplausible and we will think about it as a general feature of the perturbative seriesexpansion.

For the unknot the quantity ρ1(C) was computed in [12] obtaining theresult,ρ1|unknot = −112. (3.10)Taking into account this value, the contribution from diagram b of Fig.

4 has theform,w(b)4= −196N(N2 −1),(3.11)which is just the first term of order 1/k2 in (1.3) after taking into considerationthat g2 = 4π/k.We are left with the contributions from diagrams c1, c2 and c3 of Fig.4.Diagrams c1 and c2 give the same contribution.However, diagram c3 has anentirely different nature. On the one hand, notice that diagram c3 does not possessambiguities.

The endpoints of a propagator never get together since they alwaysenclose an endpoint of another propagator.This means in particular that thecontribution from such a diagram is framing independent. In addition, the groupfactor from this diagram is different than the one from the other two diagrams.Non-planar diagrams as c3 possess different group factors than the correspondingplanar ones.

In general, using (A1) the group factor of a non-planar diagram canbe decomposed in a part containing the same structure as the planar one plusanother contribution. Namely using (A1) one finds,Tr(T aT bT aT b) =Tr(T aT aT bT b) + fabcTr(T aT cT b)=(N2 −1)24N−14N(N2 −1).

(3.12)15

the first group factor has the same form as the group factors of diagrams c1 and c2and we will consider all three contributions together. Actually it is simple to realizethat the resulting expression once the three contributions are taken into accountpossesses an integrand that is symmetric.

This allows to enlarge the integrationregion symmetrically and divide by a factor 4!. On the other hand the contributiondue to the second group factor in (3.12) is proportional to,ρ2(C) =18π2IdxµxZdyνyZdzρzZdωτǫµσ1ρǫνσ2τ (x −z)σ1|x −z|3(y −w)σ2|y −w|3 ,(3.13)which vanishes for the case in which the contour C can be contained in a plane asit is the case for the unknot.

Therefore, the contribution from diagrams c1, c2 andc3 of Fig. 4 takes the form,w(c)4= −(N2 −1)24N34!IdxµIdyνIdzρIdwτǫµσ1ν4π(x −y)σ1|x −y|3ǫρσ1τ4π(z −w)σ2|z −w|3= −n2(N2 −1)232N,(3.14)where in the last step we have used (3.5).

This contribution is just the remainingone at order 1/k2 in the expansion (1.3). Therefore, to this order we have fullagreement between the exact result and the perturbative calculation.

Notice thatto achieve this we have defined products of operators at coincident points in avery precise manner. We have argued that the ambiguity in those products onlyproduces a relevant effect when the points of coincidence are joined by a propagator.Coincidence of end-points which belong to a connected part of an n-point functionwith n > 2 does not introduce any ambiguity.

One may verify that the singularitiesappearing when n > 2 are milder than in the case n = 2 to justify in certain sensethat assertion. However, a complete proof of it would be desirable.

For the caseof ρ1(C) and ρ2(C), it has been shown [12,25] that both are framing independent,in agreement with our statement. Their sum must therefore correspond to a knot16

invariant. In fact, it was shown in [12] thatρ(C) = ρ1(C) + ρ2(C)(3.15)can be identified with the second coefficient of the Alexander-Conway polynomial.In general, the picture that emerges from the perturbative calculation is that theconnected n-point functions, n > 2, constitute the main building blocks of the knotinvariant (1.1).

This building blocks are knot invariants and build the perturbativeseries leading to (1.1). The two-point function takes care of the framing (planarcontribution) and of some corrections to the connected n-point functions, n > 2,as ρ2(C) above (non-planar contribution).

We will see how these facts are realizedat next order in perturbation theory. Their general features will be discussed insect.

4.3.3. Order g6This is the first order where a two-loop diagram takes place.

The diagramscontributing to this order are represented in Fig. 5 and Fig.

6. Diagram a1 involvesthe full two-loop one particle irreducible two-point function.

This quantity has notbeen computed yet in the regularization scheme considered in this paper. Oneof the aims of this work is to demonstrate that it must vanish in order to haveagreement with the exact result (1.1).

We will compute in this section all othercontributions at this order and we will prove that they generate all the terms atorder 1/k3 in (1.3).The contribution from diagram a2 is straightforward after using the expressionin Fig. 2.

It turns out,w(a2)6=i64π2nN2(N2 −1). (3.16)This contribution corresponds to the last term at order 1/k3 in (1.3).

Notice thatthis is one of the terms where the power of π is different than the power of 1/k and17

therefore is shift related. The other diagrams containing one-loop corrections areb, c1, c2 and c3, and d1, ..., d6 of Fig.

5. The contribution from these diagrams aresimple to compute using the form of the one-particle irreducible diagrams in Fig.2, and the results of the previous order.

From diagram b one finds,w(b)6=132πN2(N2 −1)ρ1(3.17)while, similarly, from diagrams c1, c2 and c3, which all give the same contribution,w(c)6= −332πN2(N2 −1)ρ1. (3.18)Finally, after rearranging the group factors as in (3.12), the contribution fromdiagrams d1, ..., d6 is,w(d)6=164πn2(N2 −1)2 −116πN2(N2 −1)ρ2.

(3.19)Collecting all the contributions and using (3.10) and the fact that for the unknotρ2 = 0 one finds,w(b)6+ w(c)6+ w(d)6= −116πN2(N2 −1)(ρ1 + ρ2) +164πn2(N2 −1)2=1192πN2(N2 −1) +164πn2(N2 −1)2,(3.20)which correspond to the other two contributions in (1.3) (all except the last one)whose power of π does not coincide with the power of 1/k.The rest of the diagrams contributing at this order do not contain loop cor-rections and are depicted in Fig. 6.

Diagrams e1, e2 and e3 of Fig. 6 involvethe tree-level four-point function.

Clearly, the first two diagrams are planar andidentical while the third one is non-planar. This third diagram, e3, possesses thegroup factor,Tr(T aT bT cT d)facefebd,(3.21)which, as shown in Appendix A, vanishes (see equation (A7)).The other twodiagrams, e1 and e2 of Fig.

6, which are the same, vanish for the case of theunknot as it is shown in Appendix B.18

Let us compute the contribution form the ten diagrams f1, ..., f10 of Fig. 6.These diagrams can be divided in planar and non-planar ones.

As in previouscases, non-planar diagrams possess group factors which decompose into the groupfactors of the planar ones plus an additional contribution. Indeed, from a diagramlike f6 the group factor is,Tr(T aT aT bT cT d)fbcd = −18(N2 −1)2,(3.22)while from a diagram like f2 the group factor is,Tr(T aT bT cT dT b)facd = −18(N2 −1)2 + 18N2(N2 −1).

(3.23)Non-planar diagrams of this type do not contribute for the case in which theWilson line corresponds to the unknot. This can be shown writing explicitly theintegration involved or using the lemma below.

The main idea behind the argumentbased on that lemma is that non-planar diagrams of the type under considerationare framing independent so one can choose any framing to compute it. For theunknot it is simple to realize that choosing a framing which is contained in the sameplane as the unknot the integrand vanishes trivially.

Before stating and provingthis lemma let us define “free” propagators as the ones that have both endpointson the knot.Lemma. Every framing independent diagram of the unknot containing a freepropagator is zero.Proof.Let us place the unknot C in a plane.Being the diagram framingindependent, choose a frame Cf coplanar to it.

The diagram contains the partcorresponding to the free propagator,I· · · dxµdyνǫµρν(x −y)ρ|x −y|3 · · ·(3.24)where dxµ ∈C and dyν ∈Cf. Due to the coplanarity, the previous term is a3 × 3 determinant whose rows are linearly dependent.

Then, it is zero and the19

lemma is proved. This result is very powerful once all the framing independentdiagrams of the perturbative series expansion are identified.

The theorem statedin the next section allows to characterize very simply all those diagrams. As wewill discuss there, it turns out that those diagrams are the ones not containingcollapsible propagators.

Thus, using the lemma above, we conclude that the onlynon-vanishing diagrams contributing to the perturbative series expansion of theunknot are the ones with no free propagators.We are left with planar diagrams of type f in Fig. 5.

Actually, it will be muchmore convenient to consider the whole set of the ten diagrams all with the samegroup factor (3.22). The reason for this is that then one can show the factorizationof the contribution into a product of contributions of the type appearing in Fig.3 times contributions of the type b in Fig.

4. This phenomena of factorizationis general for diagrams with disconnected one-particle irreducible subdiagrams.Indeed, in Appendix C we show the general form of the factorization theorem.The result of applying this theorem for diagrams f1, ..., f10 of Fig.

6 is explainedas an example in Appendix C. It turns out that it can be written as the followingproduct:w(f)6= −i8 (N2 −1)2g6 14π12Idxα11Idxα22 ǫα1α2α(x1 −x2)α|x1 −x2|3164π3×Idxα33x3Zdxα44x4Zdxα55Zd3zǫα3α6γǫα4α7δǫα5α8βǫα6α7α8(z −x3)γ(z −x4)δ(z −x5)β|z −x3|3|z −x4|3|z −x5|3 ,(3.25)i.e., a product of a linking number times ρ1,w(f)6= i32n(N2 −1)2ρ1 = −i384n(N2 −1)2. (3.26)In obtaining (3.26) we have used (3.6), (3.8) and (3.10).

This contribution is justthe first one at order 1/k3 in (1.3) after using the fact that g2 = 4π/k.Thisprocedure of using the lemma plus the factorization theorem of Appendix C is ageneral feature of the unknot. In general, for an arbitrary knot, the factorization20

theorem would force us to overcount diagrams giving additional contributions.However, for the unknot all those contributions vanish.To complete the perturbative computation at order g6 we are left with diagramsg1, ..., g15 of Fig. 6.

Again these diagrams can be divided i n planar and non-planarones. However, now the non-planar ones can be divided in three groups dependingon the number crossings.

The group factor decomposes differently in each group.A given diagram produces an additional group factor for each uncrossing neededto make it planar. If the group factor of the planar diagrams, g1 to g5 isTr(T aT aT bT bT cT c) = −18N2 (N2 −1)3,(3.27)the group factor of diagrams g6, ..., g11, which are of the first type, takes the form,Tr(T aT aT bT cT bT c) = −18N2(N2 −1)3 + 18(N2 −1)2,(3.28)where we have used simply (A1).

The diagrams of the second type are g12, g13 andg14, which similarly generate the following group factor,Tr(T aT bT aT cT bT c) = −18N2(N2 −1)3 + 18(N2 −1)2 + 18(N2 −1)(3.29)Finally, diagram g15 generates,Tr(T aT bT cT aT bT c) = −18N2(N2 −1)3 + 18(N2 −1)2 + 14(N2 −1)(3.30)Of all three types of group factors only the first one contributes in the case ofthe unknot. To the second group factor, 18(N2 −1)2, there are contributions fromthe last 10 diagrams.

Using the factorization theorem of Appendix C one finds thatthis contribution is proportional to ρ2 (diagram c3 in Fig. 4) and therefore vanishes.We have rearranged the group factors in order to get th e right weights which makeexplicit the factorization of ρ2.

To the third gr oup factor, 18(N2 −1), there are21

contributions from the last 4 diagrams wh ich can be shown explicitly to vanish forthe case of the unknot. We are left with t he first group factor, −18N 2 (N2 −1)3.There are contributions from all diagrams.

One can use the factorization theoremof Appendix C to write this contribution as a product of contributions of the typeshown in Fig. 3.

Us ing (3.6) one then finds,w(g)6= −i384N2n3(N2 −1)3,(3.31)which indeed corresponds to the second contribution at order 1/k3 in (1.3). Thecalculation is described in some detail as an example in Appendix C. This wasthe only contribution left to be obtained from the perturbative series expansion.The agreement found between the two results shows that the contribution fromdiagram a1 of Fig.

5 must be zero. This implies that the one-particle irreduciblediagram corresponding to the two-point function must vanish at two loops.22

4. Factorization of the framing dependenceIn this section we state a theorem about the framing independence of diagramswhich do not contain one-particle irreducible subdiagrams corresponding to two-point functions whose endpoints could get together.

This theorem refers to any kindof knot. Before stating it, some remarks are in order.

Let us consider an arbitrarydiagram whose n legs are attached to n points on the knot. The resulting integralruns over these points on the knot in a given order, i1 < i2 < · · · < in.

Suppose thatour diagram has a propagator with endpoints attaching two consecutive points, sayi1 and i2. Remember that the path ordered integration will make i1→i2, and thatthe propagator is singular in that case.

Albeit this singularity exists, the integralis finite but shape-dependent, as is well-known. The results for a circumferenceand for an ellipse are different and then it is not a topological invariant.

The wayout of this difficulty is the introduction of framings [26,1,12]. When the propagatorconnects the knot and the frame, the resulting integral is the linking number ofthe frame around the knot, and this is a topological invariant.

This suggests thatthe framing is relevant only when there are collapsible (free propagators whoseendpoints may get together upon integration) propagators. This is the idea behindthis theorem.

Its statement is:Theorem.A diagram gives a framing dependent contribution to the per-turbative expansion of the knot if and only if it contains at least one collapsiblepropagator. Moreover, the order of n in its contribution, the linking number, equalsthe number of collapsible propagators.Diagrams b and c3 of Fig.

4, e1, e2 and e3, f1 to f5, and g12 to g15 of Fig. 6are examples of framing independent diagrams.

Diagrams a, c1 and c2 of Fig. 4,and f6 to f10, and g1 to g11 of Fig.

6 are examples of framing dependent ones.Although we have no rigorous proof of this theorem, we do have results thatsuggest its validity. Two of them are the framing independence of ρ1(C) and ρ2(C)23

separately. The framing independence of these objects has been rigorously proven[12,25].

As argued in the previous section, from a quantum field theory point ofview one would expect that the ambiguity present in n-point functions at coincidentpoints would play a role when all n points get together. Since a Wilson line consistsof a path-ordered integration, such coincident points may occur only for the case oftwo-point functions (n = 2), in particular when they are collapsible.

By no meansthis argument provides a proof of the theorem but it makes its validity plausible.In rigorous terms one should think of the theorem above as a conjecture. In therest of this section we will find further evidence regarding its validity.

Assumingthat the theorem holds the following corollary follows.Corollary.If all the contribution to the self-energy comes from one loopdiagrams, thenW(C)= F(C; N) qn(N 2−1)/2N where F(C; N) is framing inde-pendent but knot dependent, and the exponential is manifestly framing dependentbut knot independent.Proof. Let us prove this corollary first forgetting about the shift k→k + N,or in other words, not including loops.

Let us recall that free propagators are theones with both endpoints on the knot. For example, diagram b of Fig.

4 doesnot contain free propagators while diagrams c2 and c3 of Fig. 4 contain two freepropagators.

In diagram c2 of Fig. 4 these two free propagators are collapsible.To prove the corollary we will organize the perturbative series expansion of theWilson line in the following way.First, select all the diagrams which do notcontain free propagators.

Let us denote by M the set of these diagrams. Thesimplest diagram of this set is the one with no internal line at all, which is thezeroth order diagram.

Diagram b of Fig. 4 is the order g4 diagram in the set M.In virtue of the theorem above the contribution of each of the diagrams in thisset is framing independent.

Now take each of the diagrams of this set and dressit with free propagators in all possible ways. Certainly, this organization exhauststhe perturbative series.

The proof will consist in demonstrating that the effect ofdressing by free propagators each diagram in M is such that the contribution tothe perturbative series expansion factorizes as stated in the corollary. To be more24

specific, we will show that the form of the contribution of a diagram in M plusall the diagrams resulting of its dressing by free propagators factorizes in a partcontaining all the framing dependence which has the form qn(N 2−1)/2N times a partwhich is framing independent.Let us consider a diagram A ∈M and let us denote by {DpA} the set of dia-grams resulting after dressing the diagram A with p free propagators. This set ofdiagrams in {DpA} has been schematically drawn in Fig.

7. Given a diagram in{DpA}, one can work out its group trace and notice that after commuting appropri-ately the generators of SU(N) entering into this trace in such a way that generatorswith the same index get together one generates a series of terms, being the last ofthem the group factor of the diagram in {DpA} with p collapsible propagators.

Forexample, the representative of {DpA} shown in Fig. 7 would provide a group struc-ture whose last (or leading) term is as the one of the diagram pictured in Fig.

8.This procedure is the one which we have followed, for example, in the derivation ofthe group factors (3.28), (3.29) and (3.30). To gain a better understanding aboutthe types of group factors which appear we will consider several subsets of {DpA}.At first sight one could think that diagrams with p free propagators with the samenumber of crossings lead to the same group structure.

This is not entirely true.It holds for diagrams without three-vertices with one crossing of free propagatorsbut it is not true in general. Given a diagram in {DpA} with c crossings one findsdifferent group factors.

For example one can check explicitly that the group factorof the diagram in Fig. 9 is different than the one of diagram g12 of Fig.

6 withone more (collapsible) propagator. Let us denote by {Dp,c,jA} the set of diagramsin {DpA} with c crossings and group factor j.

Certainly,{Dp,c,jA} ⊂{DpA}. (4.1)Given a diagram Dp,c,jAone finds after working out the group factor that italways contains one which corresponds to the power of order p of the quadraticCasimir, [(N2 −1)/2N]p times the group factor corresponding to diagram A. In25

the process one finds other group structures with lower powers. Let us concentratefirst on the group structure with the highest power.

Certainly, all diagrams inDp,c,jAfor a fixed value of p contribute to this group structure.To apply thefactorization theorem of Appendix C we need to have as many diagrams as domains.As shown at the end of Appendix C, if diagram A is connected the differencebetween the number of domains and the number of diagrams comes about becausewhile diagrams with ni identical subdiagrams count as one, from the point of viewof domains they should count as ni! to have the adequate relabelings to be in thehypothesis of the factorization theorem.

Thus, for the case in which A is connectedone just has to repeat p! times the diagrams and make the adequate relabelingsto be in hypothesis of the factorization theorem.

Of course, this implies that onemust divide the result of the theorem by p!. For A connected the contributioncorresponding to the group structure [(N2 −1)/2N]p from all diagrams in {Dp,c,jA}is,1p!np2p (ig2)pN2 −12NpDA = 1p!in2πkN2 −12NpDA,(4.2)where the factor 2p appears after enlarging the integral of each propagator to thewhol e knot (which provides the factor np, where n is the winding number (3.5)).Notice that in (4.2) DA represents the contribution from diagram A, which isframing independent, and that we have used (2.5).

Notice also that after summingin p (4.2) gives the form stated for the Wilson line in the corollary. However, this isnot the only framing dependent contribution.

One certainly has more contributionswith other group structures. Also, one has to discuss the situation in which A isnot connected.

We will consider that situation later.To the next group structure (next to leading) not all the diagrams in Dp,c,jAcontribute. Indeed, only diagrams with c > 0 do.

There are, however, diagramswhich contribute and are framing dependent so we have to work out this depen-dence. For example, diagrams g6 to g11 of Fig.

6 ar e framing dependent. Onewould like to have enough diagrams to be able to use the factorization theorem ofAppendix C and factorize the contribution as the one from diagram A with one26

crossing of free propagators, which is framing independent, times the contributiondue to p −2 collapsible propagators which i s framing dependent and proportionalto n(p−2). To explain how one must arrange the perturbative series to extract theeffect of the framing we will consider first in some detail the case correspondingto diagrams g6 to g11 of Fig.

6. This is a particular case in which p = 3 and A istrivial but it possesses the essential features in which we will focus our attentionin the general proof.Diagrams g1 to g15 of Fig.

6 contribute to the group structures worked out in(3.27), (3.28), (3.29) and (3.30). All fifteen diagrams g1 to g15 shown in Fig.

6contribute to the group structure −(N2 −1)3/8N2. It corresponds to the generalcase which led to (4.2).

We will describe it in this example for completeness. Thecontribution to this group structure can be written as follows:I3 =Ii

As explained, not allpossible domains are represented in (4.3). One possesses 90 domains while thereare only 15 diagrams.

The ratio between these two numbers is just 3!, the numberof possible orderings of the three free propagators which from the point of view ofdomains should be different. Introducing all the relabelings needed to apply thefactorization theorem of Appendix C one must therefore divide by 3!.

The resultis,I3 = 16Ii

endpoints i and j attached to the knot. The linking number n appears when welet the endpoints of each propagator go freely over the knot and the frame, andmultiply by 1/2 per propagator.

This provides the additional factor (1/2)3 = 1/8.The result, in agreement with (4.2) is,I3 = n33!23. (4.5)The diagrams contributing to the next group structure, are g6 to g15 of Fig.

6,and in order to apply the factorization theorem we need to have 15 diagrams sincethat is the number of domains. Therefore, we have to overcount some of them.Notice that now one of the subdiagrams is a free propagator while the other is c3of Fig.

4, which is not connected. We can not apply the simple strategy describedat the end of Appendix C. The number of domains is different than the numberof diagrams because in diagrams like g12, g13 and g14 one has two possible choicesof domains while in diagrams g15 one has three.

Thus let us add what we need,i.e., let us consider two diagrams of types g12, g13 and g14 and make the apropiaterelabelings in one of each pair, and 3 diagrams of type g15 and make relabelings intwo of them. Certainly, we must subtract what we have added.

Now one can notjust simply divide by a factor. We need therefore to subtract once diagrams g12,g13 and g14 and twice diagram g15.

These diagrams are framing independent andtherefore they do not contribute to the framing dependent part. In the generalproof, at this stage, the diagrams which one must subtract contribute to a lowerpower of n. Therefore, we may use the algorithm safely power by power.

Usingour previous notation, the contribution to the next group structure, 18[N2 −1]2 is,28

I1 =Ii

˜I1 =Ii

(4.9)In (4.8), the quantity f2 is the integrand corresponding to diagram c3 of Fig. 4.After integration, it results the quantity ρ2 in (3.13).

The contributions remainingto the next group structures plus the left over represented by (4.9) are framingindependent. This means that we have extracted all the framing dependence fromthe set of diagrams g1 to g15 of Fig.

6. Using (4.5) and (4.8) we can write such acontribution as−18N2(N2 −1)3 n33!23 + 18(N2 −1)2n2= −N813!

(N2 −1)n2N3+ N8 (N2 −1) 11!N2 −12N1ρ2(4.10)In the second line of this equation we have rewritten the contribution to show thegeneral structure which we will find out now.30

Let us discuss which one is the algorithm to arrange the perturbative seriesexpansion to extract the framing dependence for the group structure next to theleading one. In the example considered we have seen that diagrams contributing tothis type of group structure were g6 to g15 of Fig.

6. Certainly, these were not allthe 15 diagrams which were needed to use the factorization theorem and factorizethe contribution to the group factor (N2 −1) as a product of a diagram like theone in Fig.

3 times the one originating ρ2 (diagram c3 of Fig. 4).

However, wewere able to add pieces of diagrams with the adequate group structure with twoor more crossings (which one must subtract when considering the situation leadingto a factorization of diagrams with two crossings) to have the 15 needed and applythe factorization theorem of Appendix C. Clearly, this is the procedure which onemust carry out when considering the next to the leading group structure. One addspieces of diagrams to complete the set in such a way that the factorization theoremof Appendix C can be applied, and, on the other hand one subtracts them.

Theimportant point is that all diagrams which are involved in this operation contain alower power of n, the linking number, and therefore, in the process one extracts allthe framing dependence for the next to leading group structure. The correspondingcontribution from all the diagrams in Dp,c,jAfor a fixed value of p is,1(p −2)!n(p−2)2(p−2) (ig2)(p−2)N2 −12N(p−2)D(1,i)A=1(p −2)!in2πkN2 −12N(p−2)D(1,i)A,(4.11)where the origin of each factor is similar to the case of (4.2) and D(1,i)Ais one ofthe possible types of diagrams resulting after dressing diagram A with two freepropagators with one crossing.

In Fig. 10 a particular situation of (4.11) has beendepicted.One has now to analyze the next group structure.

In this case one has to takeinto consideration all the left-overs from the previous one. Certainly, this is goingto change the numerical factor in front of this contribution but once this is takeninto account one may proceed similarly as the previous case.

It is clear now thatone may proceed performing this construction for a fixed value of p with all types31

which appear at each value of c. The result that one obtains in this way is,1p!in2πkN2 −12NpDA +(p2)Xc=1ncXi=11(p −c −1)!in2πkN2 −12N(p−c−1)D(c,i)A, (4.12)where nc is the number of types which one finds at c crossings, and we have usedthe fact that for the case of p free propagators the maximum number of possiblecrossings isp2. It is important to remark that the coefficients DA, D(c,i)A, whichappear in (4.12) are framing independent.

In (4.12) we have singled out all theframing dependence at a given order of free propagators for a diagram A ∈M.Summing over p one obtains the anticipated exponential behavior:(DA +∞Xc=1ncXi=1D(c,i)A) exp12ing2N2 −12N= (DA +∞Xc=1ncXi=1D(c,i)A)qn N2−12N , (4.13)where we have used (2.5) and (1.2). Therefore the corollary is proven.As this exponential is a common factor, the sum of the perturbative seriesfactorizes as the sum of all diagrams without collapsible propagators (framingindependent, but knot dependent) multiplied by some adequate coefficients timesthe preceding exponential (manifestly framing dependent).

Note that this factordoes not depend on the kind of knot because the Gauss integral only sees thelinking number.Also is interesting to note that the framing independent andknot dependent factor is intrinsic to the knot and then includes all the diagramsthat give the building blocks of its topological invariants. This is schematicallyrepresented in Fig.

11.All along our discussion regarding the proof of the corollary we have assumedthat A was a connected diagram. Let us remove now that fact.

If A is not connectedit is clear that one can use the technic of adding and removing pieces of diagrams toapply the factorization theorem at each stage of the proof described for the case inwhich A was connected. This will introduce some numerical factors in (4.13) whichare important in what regards the building blocks of the knot, but are irrelevant32

for the framing dependence since the exponential behavior has been shown for eachterm.Finally we have to include the shift k→k + N. Assuming that diagrams withmore than one loop do not contribute to the self-energy, we have to factorize prop-agators with and without self-energy insertions, and then sum the resulting series.Remember again that the rest of the diagrams add up to a framing independentand knot dependent factor.This series is hard to manage because there can be any number of self-energyinsertions at each propagator, and any number of propagators. The best organi-zation is as follows: call {Dqp} the set of diagrams with p propagators in which wehave inserted q self-energies in all the possible ways, and f(n), the sum of all ofthem, which is the framing dependent and knot independent factor inW(C).Notice that,f(n) =∞Xp=0∞Xq=0{Dqp}.

(4.14)The important point to use here is that the distribution of self-energies is suchthat they are indistinguishable. The number of possible distributions of q identicalinsertions in p lines is a Bose-Einstein combinatory factor.

The lines in which weinsert self-energies are also indistinguishable. For example, insertions done in setsof diagrams as the ones in Fig.

7 introduces a factor 1/p!. Therefore, the prefactorof {Dqp} is1p!p + q −1q.

(4.15)Now, each insertion amounts to a factor −N/k (k = 4π/g2), and each line to afactor x/k (x = in2π(N2 −1)/2N). Hence,f(n) =∞Xp=0∞Xq=01p!p + q −1qxkp−Nkq.

(4.16)33

This series in q is the expansion of a factor that provides the shift:1 + Nk−p=∞Xq=0p + q −1q−Nkq,(4.17)and therefore, the final result isf(n) =∞Xp=01p!xkp1 + Nk−p= ein2π(N2−1)2N1k+N = qn N2−12N ,(4.18)where we have used (1.2). This proof also works in the other way around.

Supposethat two-loop diagrams also contribute. These are identical among themselves,but distinguishable from the one loop insertions, and so there must appear two“bosonic” combinatory factors.

The sum of this series has to be different from theshifted exponential, because expanding the exponential with shift we find just onebosonic combinatory factor. Then, the exact result implies that the only quantumcorrections relevant to the framing dependent part are the one-loop self-energies.This corollary shows from a perturbation theory point of view that all depen-dence on the framing in the vacuum expectation value of a knot factorizes in theform predicted by Witten [1].

Notice also that assuming that there are only one-loop quantum corrections, we have found for an arbitrary knot in the fundamentalrepresentation of SU(N) the shift in the framing dependent factor of the WilsonLine through a purely perturbative approach.We have proved that the effect of one-loop contributions on the framing depen-dent factorized part of the vacuum expectation value is just a shift in k. Certainly,this is going to hold also for the rest. This, together with the factorization of theframing dependence means that we can write from (1.1) the full contribution fromthe building blocks or diagrams in M with no loop insertions.

One just has to setn = 0 in (1.1) and remove the shift.34

5. Conclusion and final remarksIn this paper we have shown that agreement between the exact result foundby Witten [1] for the vacuum expectation value of the unknot and the Chern-Simons perturbative series expansion implies that the two-loop contribution tothe one-particle irreducible two-point function must vanishes.We have workedwithin a renormalization scheme which is gauge invariant and which provides aone loop correction to the two and three-point functions which, as shown here, isresponsible for the shift of k into k +N observed in [1].

Consistency with the exactresult implies that the two-loop contribution in renormalization schemes providingquantum corrections at one loop must vanish. Work is under completion regardingthis issue [28] for the renormalization scheme proposed in sect.

2 of this paper.Our analysis of the structure of the series expansion which appears in theperturbative calculation of the vacuum expectation value of Wilson lines showsthat in general one can disentangle the framing dependence from the rest of thecontribution. We have shown that under the assumption that the theorem statedin sect.

4 holds, the framing dependence of the vacuum expectation value of Wilsonlines factorizes in the form predicted in [1]. We have shown this for a Wilson linecarrying the fundamental representation of SU(N) but it is clear from the proofthat similar arguments hold for any other representation.Although a rigorousproof of the theorem (conjecture) stated in sect.4 would be very valuable tomake our discussion on the factorization of the framing dependence complete, thearguments based on physical grounds utilized in sect.

3 make the validity of thisconjecture rather plausible.It is important to remark that the factorization theorem proved in Appendix Chas played an essential role in the factorization of the framing dependence achievedin sect. 4, as well as in the explicit calculation of the vacuum expectation value ofthe unknot in the fundamental representation at order g6.

In general, this theoremdecreases th e number of integrations needed at a given order by one, reducing thecomputation to just the building blocks of the perturbative expansion.35

The study carried out in sect. 4 to extract the framing dependence out of theperturbative series expansions of the vacuum expectation value of a Wilson linehas also provided information about the building blocks of the perturbative seriesexpansion.

Presumably, these building blocks generate a whole series of topologicalinvariants whose integral form is easy to write down using the Feynman rules of thetheory. Certainly, these building blocks are framing independent since, accordingto the corollary of sect.

4, all framing dependence has been factorized out. Toprove their topological invariance is much harder and it may well happen that at agiven order in g not all the building blocks by themselves are topological invariantsbut adequate combinations of them.

It would be desirable to have some generalresult in this respect.In this paper we have carried out an explicit calculation of the vacuum ex-pectation value of the unknot in the fundamental representation of SU(N) up toorder g6. It is straightforward to generalize this calculation to any other represen-tation.

One would like, however, to analyze the case of a non-trivial knot to verifyif the same conclusion holds and to compute some of its building blocks. Chern-Simons theory provides a whole series of topological invariants whose integral formis simple (but tedious) to write down, which would be interesting to classify andcharacterize.

For example, one would like to know if the degree of complexity of aknot is related to the number of building blocks which are different from zero. Forthe case of the unknot we have that the lemma of sect.

3 plus the theorem of sect.4 imply that its building blocks are diagrams which do not contain free propaga-tors. The quantity ρ1 defined in (3.10) is the first non-trivial building block.

Itis represented by diagram b of Fig. 4.

The building blocks of the unknot at nextorder are represented by diagrams e1, e2 and e3 of Fig. 6.

As shown in sect. 3together with Appendix B the contribution from these diagrams vanishes.

There-fore, the next possibly non-vanishing building blocks for the unknot correspondsto diagrams containing two three-vertices with all their legs attached to the Wil-son line (a representative is diagram a of Fig. 12) and diagrams containing threethree-vertices (a representative is diagram b of Fig.

12). The contribution from36

these building blocks should be computed and compared to the exact result. Asargued at the end of Appendix B, all building blocks of the unknot correspondingto connected tree-level diagrams with an even number of vertices vanish.

This isin agreement with the full result (1.1). From (1.1), as explained at the end of sect.4, it is rather simple to obtain the contribution from the building blocks.

One hasjust to set n = 0 and remove the shift. The remaining series is clearly even in 1/kwhich implies that only terms at order g4m are different from zero, in agreementwith the observation made at the end of Appendix B.In this work we have shown how to extract from the perturbative series ex-pansion of knots in Chern-Simons theory their framing dependence as well as theeffect of quantum corrections.

This leaves the series with the essential ingredientswhich we have called building blocks and contain all t he topological information.Further work is needed to study the general features and the classification of thesebuilding blocks.37

APPENDIX AIn this appendix we present a summary of our group-theoretical conventions.We choose the generators of SU(N), T a, a = 1, ..., N2 −1, to be antihermitiansuch that[T a, T b] = −fabcT c,(A.1)and fabc are completely antisymmetric, satisfying,facdfbcd = Nδab. (A.2)The convention chosen in (A.1) seems unusual but it is the right one when theWilson line is defined as in (3.1).

If we had chosen ifabc instead of −fabc, theexponential of the Wilson line would have had ig instead of g. Our conventionalso introduces a −1 in the vertex (see Fig. 1).

The fundamental representationof SU(N) is normalized in such a way that,Tr(T aT b) = −12δab. (A.3)The quadratic Casimir in the fundamental representation has the form,N 2−1Xa=1T aT a = −N2 −12N.

(A.4)One of the group factors which appear in subsect. 3.3 of the paper is the following,Tr(T aT bT cT d)facefebd,(A.5)which can be shown to be zero.

In fact, using the invariance of the trace undercyclic permutations one finds, after relabeling,Tr(T aT bT cT d)facefebd = Tr(T bT cT dT a)facefebd = Tr(T aT bT cT d)fdbefeac,(A.6)which is just the same as (A.5) but with the opposite sign. Therefore,Tr(T aT bT cT d)facefebd = 0.

(A.7)38

APPENDIX BIn this appendix we show that the contribution from diagrams e1 or e2 of Fig.6 vanishes for the case of the unknot. These are the integrals that appear in thefour-point g6 contribution to the unknot, represented in diagrams e1, e2 and e3of Fig.

6. The idea of the calculation is as follows.

According to the framingindependence theorem of sect. 4, each diagram is framing independent, so we canthink that the four points are all in the unknot.

Also we assume that it correspondsto a topological invariant and therefore we choose the unknot to be a circumferenceon the x0 = 0 plane, centered at the origin. Call p and q the points of integrationover R3 ⊗R3.

Now observe that the integrand contains an odd number of ǫαβγcontracted in such a way that it is a pseudoscalar.Its sign is different in thex0 > 0 and x0 < 0 regions.In other words, for each (p, q) ∈R3 ⊗R3 thereare (p′, q′) ∈R3 ⊗R3 such that p′0 = −p0, q′0 = −q0 and all other componentsunchanged, for which the integrands are equal in magnitude but different in sign.Then, in the p0, q0 plane we have and odd integrand and so the integral vanishes.Let us verify this explicitly.The integrations entering this contribution are of the type,Zd3p d3q ǫµρ1ν1dxµ(x −p)ρ1|x −p|3 ǫνρ2ν2dyν (y −p)ρ2|y −p|3 ǫν1ν2τ1ǫρρ3ν3dzρ(z −q)ρ3|z −q|3ǫτρ4ν4dwτ (w −q)ρ4|w −q|3 ǫν3ν4τ2ǫτ1τ2ρ5(p −q)ρ5|p −q|3 ,(B.1)where x, y, z, w lie on the knot and hence x0 = y0 = z0 = w0 = 0. The denominatorof the integrand of this expression is invariant under p0, q0→−p0, −q0.

The struc-ture of the numerator is more complicated and we will consider its form separately.The first three factors of the numerator of (B.1) become,ǫ1ρ1ν1dx1(x −p)ρ1 + ǫ2ρ1ν1dx2(x −p)ρ1= ǫ10ν1dx1(x −p)0 + ǫ12ν1dx1(x −p)2 + ǫ20ν1dx2(x −p)0 + ǫ21ν1dx2(x −p)1. (B.2)39

The following three factors give an analogous contribution,ǫ1ρ1ν2dy1(y −p)ρ1 + ǫ2ρ1ν2dy2(y −p)ρ1= ǫ10ν2dy1(y −p)0 + ǫ12ν2dy1(y −p)2 + ǫ20ν2dy2(y −p)0 + ǫ21ν2dy2(y −p)1,(B.3)which becomes (B.2) after changing y→x and ν2→ν1. Contracting (B.2) and (B.3)with ǫν1ν2τ one obtains,ǫ20τ1dx1p0dy1(y −p)2 −dx1p0dy2(y −p)1 −dx1(x −p)2dy1p0 + dx2(x −p)1dy1p0−ǫ01τ1dx1p0dy2(x −p)2 −dx2p0dy1(y −p)2 + dx2(y −p)1dy2p0 −dx2(x −p)1dy2p0+ǫ21τ1−dx1(p0)2dy2 + dx2(p0)2dy1.

(B.4)The rest of the factors in (B.1) except the last two are treated similarly, obtain-ing an expression similar to (B.4) with x→z, y→w, p→q, and τ1→τ2. Finally,multiplying (B.4) by the remaining factor of (B.1), ǫτ1τ2ρ5(p −q)ρ5,one gets,−p0q20(p −q)2· · ·+ p0q0(p −q)0· · ·−q0p20(p −q)2· · ·−q0p20(p −q)1· · ·+ p0q0(p −q)0· · ·−p0q20(p −q)1· · ·,(B.5)where by· · ·it is meant a part that does not depend on p0 or q0.

As arguedabove, this expression is odd under p0, q0→−p0, −q0 and therefore the integrationover p0, q0 in (B.1) vanishes. This way of showing the vanishing of integrationsas (B.1) suggests that this property is a general feature of tree level connecteddiagrams with an even (and non zero) number of R3 points of integration.

As wediscuss in sect. 5, this assertion is substantiated by the full result (1.1).40

APPENDIX CIn this appendix we state and prove the factorization theorem. First, let usintroduce some notation.

We will be considering diagrams corresponding to a givenorder g2m in the perturbative expansion of a knot, and to a given number of pointsrunning over it, namely n. Note that n and m fix the number of vertices, nv, andpropagators, np, that are in each diagram: nv = 2m−n and np = 3m−n. We willdenote by {i1, i2, .

. .

, in} a domain of integration where the order of integration isi1 < i2 < ... < in, being i1, i2, . .

. , in the points on the knot (notice the condensednotation) where the internal lines of the diagram are attached.The integrandcorresponding to that diagram will be denoted as f(i1, i2, .

. .

, in). Diagrams are ingeneral composed of subdiagrams, which may be connected or non-connected.

Fora given diagram we will make specific choices of subdiagrams depending on thetype of factorization which is intended to achieve. For example, for a diagram likeg6 of Fig.

6 one may choose as subdiagrams the three free propagators, or one maychoose a subdiagram to be the collapsible propagator and other subdiagram to bethe one built by two crossed free propagators. We will consider a set of diagrams Ncorresponding to a given order g2m, to given number of points attached to the knot,n, and to a given kind.

By kind we mean all diagrams containing ni subdiagrams oftype i, i = 1, ..., T. By pi we will denote the number of points which a subdiagramof type i has attached to the knot. For example, if one considers diagrams at orderg6 with n = 6 points attached to the knot, with three subdiagrams which are justfree propagators, this set is made out of diagrams g1 to g15 of Fig.

6. However, ifone considers diagrams at order g6 with n = 6 with a subdiagram consisting of afree propagator and another subdiagram of the type c3 of Fig.

4, this set is madeout of diagrams g6 to g15. The contribution from all diagrams in N can be writtenas the following sum:Xσ∈ΠnIi1,i2,...,inf(iσ(1), iσ(2), .

. .

, iσ(n))(C.1)where σ ∈Πn, being Πn ⊂Pn a subset of the symmetric group of n elements.41

Notice that Πn reflects the different shapes of the diagrams in N . In (C.1) theintegration region has been left fixed for all the diagrams and one has introduceddifferent integrands.

One could have taken the opposite choice, namely, one couldhave left fixed the integrand and sum over the different domains associated to N .The first statement regarding the factorization theorem just refers to these twopossible choices. Let us define the domain resulting of permuting {i1, i2, .

. .

, in}by an element σ of the symmetric group Pn bydσ = { iσ(1), iσ(2), . .

. , iσ(n) },(C.2)then the following result immediately follows.Statement 1.

The contribution to the Wilson line of the sum of diagramswhose integrands are of the form:f(iσ(1), iσ(2), . .

. , iσ(n)),(C.3)where σ runs over a given subset Πn ∈Pn with a common domain of integrationis equal to the sum of the integral of f(i1, i2, .

. .

, in) over dσ where σ ∈Π−1n :Ii1,i2,...,inXσ∈Πnf(iσ(1), iσ(2), . .

. , iσ(n)) =Xσ∈Π−1nIdσf(i1, i2, .

. .

, in). (C.4)The idea behind the factorization theorem is to organize the diagrams in N insuch a way that one is summing over all possible permutations of domains.

Sum-ming over all domains implies that one can consider the integration over the pointscorresponding to each subdiagram as independent and therefore one can factorizethe contribution into a product given by the integrations of each subdiagram in-dependently. Our aim in the rest of this appendix will be to prove the followingstatement.42

Statement 2. (Factorization theorem) Let Π′n be the set of all possible per-mutations of the domains of integration of diagrams containing subdiagrams oftypes i = 1, ..., T. If Π−1n= Π′n, the sum of integrals over dσ, σ ∈Π−1n , is theproduct of the integrals of the subdiagrams over the knot, being the domains allindependent,Xσ∈Π−1nIdσf(i1, .

. .

, in) =TYi=1 Ii1,...,ipifi(i1, . .

. , ipi)ni,(C.5)In (C.5) ni denotes the number of subdiagram of type i and pi its number of pointsattached to the knot.The proof of this statement is trivial since having all possible domains it isclear that one can write the integration considering subdiagram by subdiagram,the result being the product of all the partial integrations over subdiagrams.In sect.4 we considered situations where we were forced to add and sub-tract pieces of diagrams in such a way that the theorem above was utilized.

Forcompleteness, we will show now that for the case in which all subdiagrams areconnected the overcounting needed to apply the theorem is very simple and thatit just amount to divide by an adequate combinatory factor. Let us discuss firstan example to understand the strategy leading to the general situation.Let us consider the four point g4 contribution or, better to say, its part with(N2 −1)2/4N as SU(N) factor.

The diagrams are c1, c2 and c3 of Fig. 4, whosecontribution can be written according to Statement 1, is,Ii1<...

There are no more than three diagrams, and the number of domains is six.43

These are:i1 < i2 < i3 < i4i1 < i3 < i2 < i4i1 < i3 < i4 < i2i3 < i1 < i2 < i4i3 < i1 < i4 < i2i3 < i4 < i1 < i2(C.7)so we need three more diagrams. Now the overcounting consists of rewriting thediagrams after a relabeling.

In the course of the relabeling we will use the fact thatp(i1, i2) = p(i2, i1). The relabelings needed in the overcounting are the following:σ1 = 12343412!σ2 = 12343142!σ3 = 12343124!

(C.8)Here the subindex of each σ indicates the integrand over which it acts. Note alsothat each relabeling is in fact a repetition of the diagram, due to the symmetry ofthe propagator pointed out above.

Therefore, all we have to do is to multiply thesum of the six terms by 1/2. The new integrands are:σ1[f(i1, i2, i3, i4)] =f(i3, i4, i1, i2) = f(i1, i2, i3, i4),σ2[f(i1, i3, i2, i4)] =f(i3, i4, i1, i2) = f(i1, i2, i3, i4),σ3[f(i1, i4, i2, i3)] =f(i3, i4, i1, i2) = f(i1, i2, i3, i4),(C.9)and the result is:Ii1<...

(C.10)Now, in the language of the theorem, Π−14= Π′4, i.e., each propagator runs freely44

over the knot and therefore we obtain,Ii1<...

This is easily achieved in the integrals we have by simplyleaving i1 and i2 free, and multiplying by 1/2. The same should be done for thepair i3 and i4.

This is again an example of factorization. The final 1/8 is the factor3/4!

that appears for w(c)4in (3.14).The example suggests the idea of a general proof. We should count the numberof domains and the number of diagrams and observe their relation, as well as theorigin of their difference.

We are able to provide formulae for the number of do-mains for an arbitrary diagram. However, our formula for the number of diagramsin terms of the features of their subdiagrams only holds when all subdiagramsare connected.

Using these formulae we will show that we can make equivalentthe overcounting and the original contribution simply introducing a combinatoryfactor.First, let us compute the number of domains, d′, corresponding to a general setof diagrams at a given order in the perturbative expansion of the knot, to the samenumber of points on the knot, and to the same types of subdiagrams. Supposethat we construct the diagram adding its sudiagrams in a given order.

The firstone can put its pi points in n places. The second one has to distribute its points inthe remaining n −pi, and so on.

For example, if there are only a points attachedby propagators and b points attached by three-vertices (and so n = 2a + 3b),d′ =n2n −22n −42. .

.n −2a + 22n −2a3n −2a −33. .

.. . .n −2a −3b + 33=n!(2!)a(3!)b.

(C.12)In general the denominator will include the product of every pi!, where pi denotes45

the number of points attached to the knot corresponding to a subdiagram of type i.If we denote by d the number of diagrams of the set under consideration, it is clearthat d ≤d′ due to the possible identity of some subdiagrams. In those cases we haveto divide d′ by the number of permutations of all the identical subdiagrams.

If someof the subdiagrams were not connected we would have to consider additional factorswhich would imply to introduce more data about each subdiagram. Therefore, letus restrict ourselves to the case of connected subdiagrams.

The final form of theformulae is:d′ =n!Qki=1(pi! )ni ,d =n!Qki=1 ni!(pi!

)ni . (C.13)The relation between d and d′ is always an integer:d′d =kYi=1ni!

(C.14)This is the combinatory factor we were searching for. Therefore, when the subdi-agrams are connected one just has to overcount evenly each diagram to have asmany diagrams as domains and divide by (C.14).

For non-connected subdiagramsthe previous formula for d fails. An example is the factorization of ρ2(C) in the18(N2 −1)2 part of diagrams g6, .

. .

, g15 of Fig. 6.

The subdiagram that we wouldfactorize is the corresponding to diagram c3 of Fig. 4, which is not connected.

Theprevious formula gives d = d′ = 15, exactly the same result as if the subdiagramwere e1 or e2 of Fig. 6, but there are just 10 diagrams.

The number of domains,however, is correct.46

FIGURE CAPTIONS1) Basic Feynman rules of the theory.2) Two-point function and three-point function at one loop.3) Diagrams corresponding to g2. Thick lines represent the Wilson line whilethin lines refer to Feynman diagrams.4) Diagrams corresponding to g4.5) Part of the diagrams corresponding to g6.6) Rest of diagrams corresponding to g6.7) A general set of diagrams with p propagators in the knot8) The less crossed diagram of Fig.

7.9) A framing independent diagram.10) Factorization of ρ211) Factorization of the framing dependence.12) Diagrams at order g8 corresponding to building blocks47

REFERENCES1. E. Witten, Comm.

Math. Phys.

121(1989), 3512. M. Bos and V.P.

Nair, Phys. Lett.

B223(1989), 61, Int. J. Mod.Phys.

A5(1990), 9593. H. Murayama, “Explicit Quantization of the Chern-Simons Action”, Univer-sity of Tokyo preprint, UT-542, March 19894.

J.M.F. Labastida and A.V.

Ramallo, Phys. Lett.

B227(1989), 92, Phys.Lett. B228(1989), 2145.

S. Elitzur,G. Moore,A.SchwimmerandN.

Seiberg,Nucl.Phys. B326(1989), 1086.

J. Frohlich and C. King, Comm. Math.

Phys. 126(1989), 1677.

G.V. Dunne, R. Jackiw, and C.A.

Trugenberger, Ann. Phys.

194(1989), 1978. S. Axelrod, S. Della Pietra and E. Witten, J. Diff.

Geom. 33 (1991) 7879.

J.M.F. Labastida, P.M. Llatas and A.V.

Ramallo, Nucl. Phys.

B348(1991),65110. K. Gawedzki and A. Kupiainen, Comm.

Math. Phys.

135(1991), 53111. G. Moore and N. Seiberg, Phys.

Lett. B220(1989), 42212.

E. Guadagnini, M. Martellini and M. Mintchev, Phys. Lett.

B227(1989),111, Phys. Lett.

B228(1989), 489, Nucl. Phys.

B330(1990), 57513. L. Alvarez-Gaum´e,J.M.F.

Labastida,and A.V. Ramallo,Nucl.Phys.

B334(1990), 103, Nucl. Phys.

B (Proc. Suppl.) 18B (1990) 114.

R.D. Prisarski and S. Rao, Phys.

Rev. D22(1985), 20815.

D. Birmingham, M. Rakowski and G. Thompson, Nucl. Phys.

B234(1990),8316. F. Delduc, F. Gieres and S.P.

Sorella, Phys. Lett.

B225(1989), 36748

17. P.H.

Damgaard and V.O. Rivelles, Phys.

Lett. B245(1989), 4818.

C.P. Mart´ın, Phys.

Lett. B241(1990), 51319.

M. Asorey and F. Falceto, Phys. Lett.

B241(1990), 3120. A. Blasi and R. Collina, Nucl.

Phys. B345(1990), 47221.

F. Delduc, C. Lucchesi, O. Piguet and S.P. Sorella, Nucl.

Phys. B346(1990),51322.

D. Daniel and N. Dorey, Phys. Lett.

B246(1990), 8223. N. Dorey, Phys.

Lett. B246(1990), 8724.

C.P. Mart´ın, Phys.

Lett. B263(1991), 6925.

D. Bar-Natan, “Perturbative Chern-Simons Theory”, Princeton preprint,199026. A. M. Polyakov, Mod.

Phys. Lett.

A3(1988), 32527. S. Axelrod and I.M.

Singer, MIT preprint, October 199128. M. Alvarez, L. Alvarez-Gaum´e, J.M.F.

Labastida and A.V. Ramallo, CERNpreprint to appear29.

V.F.R. Jones, Bull.

Am. Math.

Soc. 12 (1985) 103, Ann.

Math. 126 (1987)33530.

P. Freyd, D. Yetter, J. Hoste, W.B.R. Lickorish, K. Millet and A. Ocneanu,Bull.

Am. Math.

Soc. 12 (1985) 23931.

L.H. Kauffman, Trans.

Amer. Math.

Soc. 318 (1990) 41732.

E. Witten, Nucl. Phys.

B322(1989), 62949

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