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Chern–Simons Perturbation Theory

arXiv:hep-th/9110056v1 20 Oct 1991Chern–Simons Perturbation TheoryScott AxelrodMath Department, Harvard University and MITI.M. SingerMath Department, MITAbstract.

We study the perturbation theory for three dimensional Chern–Simons quan-tum field theory on a general compact three manifold without boundary. We show thatafter a simple change of variables, the action obtained by BRS gauge fixing in the Lorentzgauge has a superspace formulation.

The basic properties of the propagator and the Feyn-man rules are written in a precise manner in the language of differential forms. Using theexplicit description of the propagator singularities, we prove that the theory is finite.

Fi-nally the anomalous metric dependence of the 2-loop partition function on the Riemannianmetric (which was introduced to define the gauge fixing) can be cancelled by a local coun-terterm as in the 1-loop case [28]. In fact, the counterterm is equal to the Chern–Simonsaction of the metric connection, normalized precisely as one would expect based on theframing dependence of Witten’s exact solution.This work was supported in part by the Divisions of Applied Mathematics of the U. S. Departmentof Energy under contracts DE-FG02-88ER25065 and DE-FG02-88ER25066.1

1. IntroductionOne of the most successful topological quantum field theories considered to date hasbeen the three dimensional Chern–Simons theory with the inclusion of Wilson loops.

Inhis seminal paper [28] and in subsequent work [29], E. Witten described the exact solutionfor this theory and showed that the observables lead to a broad class of invariants ofcompact three manifolds with imbedded knots (also with choices of orientations, framingsand labelings), generalizing the knot invariants defined by V. Jones [21]. Witten’s startingpoint was a Feynman path integral formulation of the observables.Of course from amathematical point of view this starting point is purely formal.

What Witten did, however,was perform formal manipulations of the path integral, based upon physical insight andexperience, to arrive at an ansatz for the solution. Subsequent work by several authors[10], [22], [26], [11] have verified and made rigorous Witten’s main results.

Also a theoryessentially equivalent to Witten’s solution of Chern–Simons was described in [24].When no links are present, the invariant for a compact oriented 3-manifold M (takenhere to have no boundary) is given by the partition functionZk(M, G) =ZDA eikCS(A),(1.1)The basic field A in Eq. 1.1 is a connection (gauge field) with compact gauge group G,and the Chern–Simons action is given byCS(A) = 14πZMTr(A ∧dA + 23A3).

(1.2)Here Tr is the basic trace on the Lie algebra g of G normalized so that the pairing (A, B) =−Tr(AB) on g is the basic inner product1. For notational simplicity, we have taken theunderlying principal bundle to be trivialized and identified the connection A with a gvalued one form on M.Witten’s solution follows not by “evaluating” the path integrals directly but by ex-ploiting deep connections with rational conformal field theory.

However, the motivationbehind and the intuition for the construction is that there is some a priori definition ofthe path integral which behaves as expected physically. Unfortunately, a direct defini-tion of the full non-perturbative path integral seems beyond the reach of our present-daytechniques.1Normalized so that it corresponds, under the Chern-Weil homomorphism, to a generator ofH4(B ˜G, ZZ), where ˜G is the simply-connected cover of G.2

Here we study the perturbative formulation. Chern–Simons perturbation theory onflat IR3 has been looked at previously by several groups of physicists.

In [17], the theoryup to 2-loops was found to be finite and to give knot invariants. In [7], [8], [12] a superspaceformulation of the gauged fixed action was given.

Two different arguments for finitenessto all orders are presented in [8] and [13], both assuming a nice regularization scheme andemploying special symmetries of the gauge fixed action to conclude that the β functionvanishes. In [4], Dror Bar-Natan gave a rigorous treatment of the perturbative definitionof knot invariants in IR3 up to 2 loops, and showed it agreed with the results expectedfrom Witten’s exact solution.In this paper, we will allow M to be an arbitrary compact, oriented, 3-manifoldwithout boundary and will obtain a succinct description of the l-loop contribution in thelanguage of differential forms which we show is finite directly.

Specifically, we perturbabout a solution, A(0), of the equations of motion (i.e.a flat connection).We shallassume that A(0) is isolated up to gauge transformations and that the group of gaugetransformations fixing A(0) is discrete. Equivalently, we assume that the cohomology ofd(0) vanishes, where d(0) is the exterior covariant derivative twisted by A(0) and acting onΩ∗(M; g), the space of forms with values in the associated adjoint bundle.

The differentialforms viewpoint instructs us to sum over all particle types before integrating and providesus with a natural point splitting of the propagator on the diagonal. One could say this isthe regularization scheme we use.To define the perturbative expansion, it is necessary to make a choice of gauge fixing.We choose BRS gauge fixing using Lorentz gauge, which depends on a choice of Riemannianmetric, g, on M. The perturbative expansion has the formZk(M, A(0), g) = Zsck (M, A(0), g)Zhlk+h(M, A(0), g),(1.3)where Zsck is the “semi-classical approximation” and Zhlk+h is the sum of the higher ordercorrections.

We have included an ad hoc shift in k here necessary for agreement withWitten’s exact solution. We will explain one reason this shift is needed in §6, although wecan offer, at present, no derivation of it.

The role of the shift in k in perturbation theoryhas been the subject of much discussion in the physics literature. An explanation of theproblem is given in [1].The expansion of Zhlk in inverse powers of k is given byZhlk =XV =0,2,4,...(−ik2π )−12 V IdiscV(M, A(0), g),(1.4)3

where IdiscVis the sum of the contributions of all Feynman diagrams with V vertices.Equivalently, the expansion can be written asZhlk = exp ∞Xl=2(−ik2π )1−lIconnl(M, A(0), g)!,(1.5)where Iconnlis the sum of the contribution of the connected Feynman diagrams with lloops.Our results are best summarized by outlining the paper.In §2, we give a form of the gauge fixed action which has a simple superspace formu-lation Eq. 2.17, and describe the superspace Feynman rules.In §3, we rewrite the superspace Feynman rules in a succinct way using the languageof differential forms.

We describe the basic propagator as a two form on M × M withvalues in g ⊗g and state its salient properties (PL1)-(PL6). IdiscVwill be written as anintegral over M V of a top form obtained from the propagator.In §4, we sketch the proof of finiteness, in particular that the multiple integral overM V defining IdiscVconverges despite the singularity of the propagator on the diagonal.The logic of the proof is as follows.

By a fairly general argument, the proof reduces toshowing convergence of diagrams for the flat space theory which may also have insertionsof edges with a propagator given by a subleading term in the singularity of the curvedspace propagator. By the convergence theorem, it thus suffices to show that any such flatspace diagrams which are superficially divergent vanish.

The latter follows by a simplesymmetry and power counting argument.In §5, we give a formal proof that IdiscV(M, A(0), g) is independent of the metric g. Bythat we mean, a proof that uses integration by parts ignoring the singularities. For thecase of two loops, we give a careful treatment using Stoke’s theorem.

We find an explicitanomaly given as a local integral over M of the form one would expect from power countingand symmetry considerations. The overall coefficient of the anomaly agrees with whatone would predict from Witten’s exact solution.

This means that we obtain a manifoldinvariant by subtracting a concrete counterterm from Idisc2= Iconn2.In §6, we make some comments about the probable relation of the results here toWitten’s exact solution and about possible extensions of our results.A more detailed exposition is in preparation [3]. In it will be found the derivationof the singular behaviour of the propagator near the diagonal and a careful discussion ofsigns and symmetry factors.4

2. Superpace Form of Gauge Fixed Action and Feynman RulesIn this section we derive a form of the gauge fixed action which has a superspaceformulation.

In fact, the gauge fixed action will be seen to have the same form as theoriginal Chern–Simons action, but applied to a superfield with certain constraints. Wewill first perform standard BRS gauge fixing in the Lorentz gauge, and then will changevariables and integrate out the field multiplying the gauge fixing condition.

We arrive ata form of the gauge fixed action which may be written in superspace.Expanding around A(0), the Chern–Simons action takes the formCS(A(0) + A) = CS(A(0)) + 14πZMTr(A ∧d(0)A + 13A ∧[A, A]),(2.6)where A is a Lie algebra valued one form.For the standard gauge fixing, we introduce Fermionic fields c and ¯c and a Bosonicfield b, all valued in the Lie algebra of the group of gauge transformations, Ω0(M; g). TheBRS operator Q is given byQA = −d(0)c −[A, c],Qc = 12[c, c]Q¯c = b,Qb = 0.

(2.7)The operator d(0) is the direct sum of operators d(0)q: Ωq(M; g) →Ωq+1(M; g). To definethe Lorentz gauge condition we choose a Riemannian metric g on M. This allows us todefine the Hodge ∗operator ∗: Ωq(M; g) →Ω3−q(M; g) which satisfies ∗2 = 1.

We choosethe sign of ∗so that the inner product< χ, ψ >≡−ZMTr(∗χ ∧ψ),for χ, ψ ∈Ωq(M; g)(2.8)is positive. Relative to this inner product, the adjoint of d(0)qis (d(0)q )† = (−1)q+1 ∗d(0)2−q∗.d(0)† will also be denoted by δ(0).The Lorentz gauge condition is δ(0)A = 0.

This condition is implemented by the gaugefixed actionSgf(A, c, ¯c, b) = k CS(A(0) + A) + QVV = α < ¯c, δ(0)A >,(2.9)where α is a constant which we are free to select.Choosing α = k/2π, and definingC = ∗d(0)¯c and B = ∗d(0)b, we findSgf(A,c, ¯c, b) −kCS(A(0))= k2πZMTr12A ∧d(0)A + 16A[A, A] −C ∧d(0)c −C ∧[A, c] −B ∧A. (2.10)5

By acyclicity of A(0) and elementary Hodge theory, the change of variable from (b, ¯c) to(B, C) is an invertible map from pairs of elements of Ω0(M; g) to pairs of elements ofKer(δ(0)1 ). Note that the Jacobian for this change of variable is 1 because the Fermionicdeterminant for the change from ¯c to C cancels the Bosonic determinant for the changefrom b to B.

Also note that the integral over the field B just imposes the Lorentz gaugeconstraint δ(0)A = 0. Thus the gauge fixed path integral can be writtenZk(M, A(0), g) =ZDADcD¯cDbeiSgf(A,c,¯c,b) = eikCS(A(0))ZDADcDC e−S(A,c,C), (2.11)where the last integral is over the Bosonic field A ∈Ker(δ(0)0 ) and the Fermionic fieldsc ∈Ω0(M; g) and C ∈Ker(δ(0)1 ), and the (imaginary) action is given byS(A, c, C) = −ik2πZM12Aa∧d(0)Aa−Ca∧d(0)ca+ 16fabc(Aa∧Ab∧Ac−6Ca∧Ab∧cc).

(2.12)Here we have chosen coordinates on g corresponding to an orthonormal basis Ta (relativeto the basic inner product). The (totally antisymmetric) structure constants fabc of g aredefined by[Tb, Tc] = fabcTa.

(2.13)To give Eq. 2.12 a superspace interpretation, we need some supermanifold notation.For V a vector space, we let V−denote a Fermionic copy of V .

More generally, for E avector bundle over a base manifold N, we let E−denote E but with the fibers consideredFermionic. So functions on E−are sections over N of Λ∗(E)2.Our base supermanifold is TM−, that is the tangent bundle to M, but with the fibersconsidered to be Fermionic (and the base still bosonic).

A local coordinate system xµ onthe base M determine Fermionic coordinates θµ on the Fibers of TM−. The θµ essentiallybehave like the one forms dxµ.In fact, there is a correspondence between differentialforms, ˜a, on M and functions, a, on TM−(i.e.

“superfields”) given by˜a(x) =Xiaµ1...µidxµ1...dxµi ↔a(x, θ) =Xiθµ1...θµiaµ1...µi. (2.14)Under this correspondence, integration of top forms corresponds to integration of super-fields in the natural supervolume form on TM−, the exterior differential operator d onforms corresponds to the operator θµ∂∂xu on superfields, and wedge product of differential2More precisely, Bosonic (Fermionic) functions on E−correspond to a choice of a Bosonic (Fermionic)section of Λeven(E) together with a Fermionic (Bosonic) section of Λodd(E).6

forms corresponds to multiplication of superfields. Care must be taken with the corre-spondence of products because in the forms language one takes the dxµ to commute withFermions, whereas in the superspace language the θµ anti-commute with Fermions3.Now let A be a Fermionic g valued superfield on TM−.

The operators d(0) and δ(0)on g valued forms correspond to operators d(0) and δ(0) on the superfield A. So that wecan make contact with Eq.

2.11, we name the component fields of A as follows,Aa(x, θ) = ca(x) + θµAaµ(x) + θµθνCaµν(x) + θµθνθρBaµνρ(x). (2.16)Since A is Fermionic, c and C are Fermionic and A and B are Bosonic.

Observe thatthe condition δ(0)A = 0 means precisely that A is in Ker(δ(0)0 ), C is in Ker(δ(0)1 ), and Bequals zero. (The last fact follows from acyclicity of A(0).) So the set of A for which δ(0)Avanishes is equal to the set of triples (c, C, A) which are integrated over in Eq.

2.11. By asimple calculation, the action defined in Eq.

2.12 has a superspace formulation:S(A) = S(c, C, A) = λZdX12Aa(X)(d(0)A)a(X) + 16fabcAa(X)Ab(X)Ac(X). (2.17)For convenience in writing down the Feynman rules, we have introduced the symbol X forthe coordinates (x, θ), the abbreviation dX for the volume form d3xd3θ on TM−, and theconstantλ = −ik/2π.

(2.18)In summary, the path integral is given byZk(M) =Z{A;δ(0)A=0}DA e−S(A),(2.19)where A is a g valued Fermionic superfield on TM−and the action S is given by Eq. 2.17.To write the Feynman rules in the superspace formulation, we must take into accounttwo complications not usually present in the derivation of super-Feynman rules [16], [27].3Let a(i) = θµ1 ...θµi aµ1...µi denote the piece of a of degree i in the θµ, ˜a(i) denote the piece of ˜awhich has form degree i, and |a| = ±1 denote the statistics of a.

Then |aµ1...µi| = (−1)|a|+i. Theproduct of a with another superfield b isa(x, θ)b(x, θ) =Xi,j−1(i+|a|)jθµ1...θµiθν1...θνjaµ1...µi(x)bν1...νj(x).

(2.15)Thus the superfield ab obtained by superfield multiplication corresponds to the form Pi,j(−1)ij+|a|j ˜a(i)∧˜b(j).7

The first complication is to deal properly with the constraint δ(0)A = 0.The secondcomplication is to keep careful track of the overall sign in front of each graph.Caremust be taken here because the basic superfield is Fermionic and because the operation ofintegration over the base supermanifold TM−and the operator d(0) are Fermionic.LetZfree[J ] =ZA∈Ker(δ(0))DA eRdX [−λ2 Aa(X)(d(0)A)a(X)+J a(X)Aa(X)](2.20)be the partition function for the free theory coupled to a source J (X). J is taken to be aBosonic g valued superfield, so that the source term in Eq.

2.20 is Bosonic. To keep trackof the constraints we introduce the operators ˆπd and ˆπδ on g valued superfields whichorthogonally project onto the image of d(0) and δ(0), respectively.

By acyclicity of A(0)and Hodge theory, we haveˆπd + ˆπδ = ˆδg,(2.21)where ˆδg is the identity operator acting on g valued superfields. (ˆδ will denote the identityoperator on ordinary superfields.) In order to complete the square in the exponent in Eq.2.20 and evaluate Zfree[J] in the standard way, we introduce the Fermionic operator ˆLwhich is the “Hodge theory” inverse of d(0).

To define it, first let ˆL1 : Im(d(0)) →Ker(d(0))†be the inverse of d(0) as a map from the orthocomplement of its kernel to its image. ThenˆL is the operator on g valued superfields obtained by first using ˆπd to orthogonally projectonto Im(d(0)) and then applying ˆL1.

This definition is equivalent to the equationsKer(ˆL) = Im(d(0))† = Ker(δ(0))Im(ˆL) = Ker(d(0))† = Im(δ(0))d(0) ◦ˆL = ˆπdˆL ◦d(0) = ˆπδ. (2.22)Note that this definition of Hodge theory inverse makes sense even if A(0) is not acyclic,useful in generalizing the results here to allow for zero modes.

ˆL can also be defined byintroducing the Laplacian∇= δ(0)d(0) + d(0)δ(0)(2.23)which is invertible by acyclicity of A(0). The definition Eq.

2.22 is equivalent toˆL = δ(0) ◦∇−1. (2.24)8

Having introduced ˆL, we may complete the square,ZdX−λ2 Aa(X)(d(0)A)a(X)+J a(X)Aa(X)=ZdX−λ2 (A −(λ−1 ˆLJ ))a(X)d(0)(A −(λ−1 ˆLJ ))a(X)+ 12(λ−1 ˆLJ )a(X)J a(X),(2.25)and shift the variable A in the usual way to obtainZfree[J ] = Zfree[0]e12RdX (λ−1 ˆLJ )a(X)J a(X). (2.26)Now we want to write ˆL as an integral operator with an integral kernel L(X, Y ).For any operator ˆK on g-valued superfields (either Bosonic or Fermionic), we define thecorresponding integral kernel K to be the superfunction on TM−× TM−with values ing × g so that( ˆKΨ)a(X) =ZdY Kab(X, Y )Ψb(Y ).

(2.27)Note that since the operatorRdY is Fermionic, the integral kernel K will have the oppositestatistics to the operator ˆK.The kernel δ for the identity operator ˆδ on superfields, which one might call the“super-delta function”, is Fermionic and satisfiesZdXdY δ(X, Y )Ψ(Y )Φ(X) =ZdX Ψ(X)Φ(X). (2.28)This implies more generally thatZdXdY δ(X, Y )Ψ(X, Y, Z) =ZdX Ψ(X, X, Z)(2.29)for Ψ a function of three variable in TM−.

Since dY dX equals −dXdY , the super-deltafunction is antisymmetric under exchange of X and Y . Similarly, δgab(X, Y ) = δabδ(X, Y )is antisymmetric under simultaneous exchange of (X, a) and (Y, b).

One can show likewisethat the kernel for ˆL is antisymmetric,Lab(X, Y ) = −Lba(Y, X),(2.30)as one would expect for a Fermionic propagator.9

Next we write the partition function with interaction in terms of the free partitionfunction with source in the usual way,Zk = eiS(A(0)) exp−λ3!ZdX fabc∂∂J a(X)∂∂J b(X)∂∂J c(X)Zfree[J ]|J =0. (2.31)The expression∂∂J a(X) appearing here is defined as follows.

Given a variation δJ of J , welet δδJ be the operator of differentiation in the direction of δJ . This acts on functionalsΩ(J ) of J .

The operator∂∂J a(X) is then defined byδδJ Ω(J ) =ZdX δJ a∂∂J a(X)Ω(J ). (2.32)So, for example, we have∂∂J a(X)ZdY J a(Y )Aa(Y ) = A(X).

(2.33)Since J is Bosonic,∂∂J a(X) is Fermionic, and so∂∂J a(X)RdY equals −RdY∂∂J a(X). Hence∂∂J a(X)J b(Y ) = −δabδ(X, Y ) = δbaδ(Y, X).

(2.34)Expanding out the exponentials in Eq. 2.31 and Eq.

2.26, and writing ˆL in terms ofthe kernel L, we findZk = Zsck∞XI,V =0(−λ/3! )VV !λ−II!

(ΠVi=1Fi)LI|J =0,(2.35)whereZsck = eikCS(A(0))Zfree[0]Fi =ZdX faibici∂∂J ai(Xi)∂∂J bi(Xi)∂∂J ci(Xi),L = 12ZdY dZ J a(Y )Lab(Y, Z)J b(Z). (2.36)We may restrict the sum in Eq.

2.35 to V and I satisfying 3V = 2I since all other termsvanish.To write(ΠVi=1Fi)LI|J =0 in a form not involving functional derivatives, we considersources J of the formJ a(Y ) =VXi=1ja(i)δ(Y, Xj). (2.37)10

Here j(i) is a Fermionic g valued source at the ith position. Using∂∂J a(X)J b(Y ) = δabδ(Y, Xi) =∂∂ja(i)J b(Y ),(2.38)we find that the correction to the semiclassical approximation is given byZhlk ≡ZkZsck=X3V =2IλV −I(3!

)V (2! )IV !I!ΠVi=1"ZdXi faibici∂∂jai(i)∂∂jbi(i)∂∂jci(i)#j(i)=0LItot,(2.39)where, for given V ,Ltot =VXi,j=0Ls(Xi, j(i), Xj, j(j))Ls(Xi, j(i), Xj, j(j)) ≡Lab(Xi, Xj)ja(i)jb(j).

(2.40)For the meaning of this when i equals j see Eq. 3.52.

Ls is the propagator L, but rewrittenas a Bosonic superfunction on (TM ⊕g)−× (TM ⊕g)−.At the level of diagrams, Eq. 2.39 means the superspace Feynman rules assign a factorZdXfabc∂∂ja∂∂jb∂∂jcj=0,(2.41)to a vertex labeled by the point (X, j) ∈[TM ⊕g]−, and a propagator Ls(X, j, X′, j′)on an edge between vertices labeled by (X, j) and (X′, j′).

These Feynman rules have asuperspace formulation on [TM ⊕g]−even though the original action only had TM−asthe base supermanifold –the Lie algebra directions were not supercoordinates there. Infact, the superfield A can be viewed as a bosonic superfield A(x, θ, j) = Aa(x, θ)ja on[TM ⊕g]−satisfying constraints, δ(0)A = 0 and A depends linearly on j.The cubic“potential” term in the action Eq.

2.17 can be written as an integral over [TM ⊕g]−, inwhich Tr becomes a constant superfield. However, the quadratic “kinetic” term cannotbe naturally described this way.

One sees similar phenomena in other superspace fieldtheories; we hope our techniques will illuminate the special properties of those theories.11

3. A Closed Form for l-loop InvariantsIn this section we describe the results obtained in the previous section in the languageof differential forms rather than superspace.We also summarize in differential formslanguage the basic properties of the propagator.

Since the translation of results from theprevious section into the form language here is straight forward, we will not spell it out.Readers not familiar with superspace and gauge fixing can take the statements in thissection as a starting point (although we assume some notation from above). Although wedo not need it later in the paper, we will use the properties to write the perturbation seriesin a rather elegant form Eq.

3.57.We emphasize that the formulation of the Feynman rules given below could have beenderived by direct formal manipulations of the path integral (i.e. manipulations which haverigorous analogues for integrals over finite dimensional spaces) without having introducedFermionic coordinates or even BRS gauge fixing.

Such a direct derivation, however, wouldnot explain the simple form of the final answer we derive naturally here.Properties of the PropagatorFirst we describe the basic properties of the propagator L. It is the kernel for theoperator ˆL = δ(0)∇−1 acting on Ω∗(M; g), where ∇is Laplacian associated to d(0). Thismeans that L is a section of Ω∗(M × M; g ⊗g), which satisfies4(ˆLψ)a(x) ≡ZMyLab(x, y) ∧ψb(y).

(3.42)Note that this definition of L uses wedge products and integration of forms rather thaninner products and integration of functions with respect to the Riemannian volume. Thiswill enable us to make metric independence as manifest as possible.

The expression Myon the right hand side of Eq. 3.42 is simply an abbreviation to say that we are to integrateover the copy of M paramaterized by the y variable.

Since M is 3 dimensional and ˆLdecreases form degree by 1, Eq. 3.42 implies L is a 3 −1 = 2 form.When dealing with products of several copies of M throughout the rest of the paper,we will often adopt the notation used above of distinguishing a particular copy of M, andobjects associated with it, by adding the name of a variable parameterizing that copy asa subscript.

So, for example, we writeL ∈Ω2(Mx × My; gx ⊗gy). (3.43)4The unusual sign conventions used in this equation are discussed below.12

L is equivalently defined by(PL0)δ(0)x L(x, y) = 0,δ(0)y L(x, y) = 0together with(PL1)d(0)M×MLab(x, y) = (d(0)x+ d(0)y )Lab(x, y) = −δgab(x, y) ≡−δabδ(x, y).Here d(0)M×M is the exterior derivative operator on Ω∗(M × M; g × g) determined by theconnection (A(0), A(0)); and δ is the Poincare dual form to the diagonal, defined byZMx×Myδ(x, y)ψ(x, y) =ZMxψ(x, x)for ψ ∈Ω∗(M × M). (3.44)The fact that d(0)2 = 0 implies ˆL ◦ˆL = 0.

Written in terms of the kernel L, thisbecomes(PL2)ZMyLab(x, y)Lbc(y, z) = 0.Since it is a Fermionic propagator, L is antisymmetric under the involution of ad(P)×ad(P) gotten by exchanging the two copies of ad(P),(PL3)Lab(x, y) = −Lba(y, x).This can be shown directly from the fact that the involution reverses the orientation ofthe base M × M, which implies that δ(x, y) is antisymmetric under exchange of x andy. Then, since the involution leaves the operator d(0)M×M in (PL1) invariant, L must beantisymmetric.Now we wish to consider how L varies as the metric g changes by an infinitesimalvariation δg.

For K an object depending on g, we will denote the derivative of K in thedirection δg by either ˙K or δδgK. First notice that properties (PL1) -(PL3) can be statedwithout reference to the metric g. In particular, the right hand side of (PL1) does notvary as one changes the metric.

Therefore d(0)M×M ˙L vanishes. But acyclicity of d(0) impliesacyclicity of d(0)M×M.

So the fact that ˙L is closed implies it is exact,(PL4)˙L ≡δδgL = d(0)M×MB.Here B is a g × g valued one form on M × M which is also a one form on the space ofmetrics (i.e. it depends linearly on δg).13

The next important property of the propagator L is the explicit description of itssingularities and discontinuities near the diagonal, which we now describe. For (x, y) in aneighborhood of the diagonal, using the Hadamard parametrix method [18] we find5(PL5)L(x, y) = Lhad(x, y) + Lcont(x, y)Lhad(x, y) = Lsing(x, y) + Lbd(x, y)4πLsingab(x, y) = −12 det(g)12 ǫµνρuµ||u||3 fduνfduρδab4πLbdab(x, y) = −gµνuµ||u||ˆRνδabˆRν = 12 det(g)−12 ǫνρσRαβρσdzαdzβ,where Lcont is smooth away from the diagonal and continuous across the diagonal.

Herez ∈M and u ∈TzM are related to x and y by the exponential map,(x, y) = exp (z,z)((u, −u)). (3.45)The “horizontal” one forms fdui are defined byfduµ = duµ + Γµνρuνdxρ.

(3.46)The quantity ||u|| is the Riemannian norm of u. In the third and fourth lines of (PL5), wehave identified the Lie algebra at x with the Lie algebra at y using the parallel transportoperator determined by A(0) along the short geodesic from x to y (of length 2||u||).

In [3],we derive (PL5) from the Hadamard construction of the kernel for ∇−1, and also give analternate derivation using equivariant differential forms.Note that Lsing diverges quadratically as one approaches the diagonal and Lbd isdiscontinuous but bounded.For the proof of finiteness in §4, it will be also be useful to describe the propagatorsingularities in the following way (which, with some care, can be derived from the above).For z(0) ∈M and small y1, y2 ∈Tz(0)M, we let xi = expz(0)(yi), i = 1, 2. ThenLab(x1, x2) = −δab [ǫµνρˆuµdˆuνdˆuρ α + ǫµνρˆuµdˆuνγρ] + (bounded)ab,(3.47)where u = y1 −y2, ˆu = u/||u||, α is a bounded function of y1 and y2, γρ is a bounded1-form for each ρ, and (bounded)ab is a bounded 2-form for each a and b.5In our conventions, Rαβρσ = (Rαβ)τ σgρτ , where (Rαβ)τ σdzαdzβ is the curvature tensorthought of as an End(TM) valued 2-form.14

One can also derive an explicit description of the singularities of B near the diagonal6,(PL6)B(x, y) = Bhad(x, y) + Bcont(x, y)Bhad(x, y) = Bsing(x, y) + Bbd(x, y)4πBsingab(x, y) = 12 det(g)12 ǫµνρuµ||u||3 fduν(gρσδgστuτ)δab4πBbdab(x, y) = gµνuµ||u||ˆOνδabˆOν = 12 det(g)−12 ǫνρσOαρσdzαOαρσ = (δΓαρσ −12∇αδgρσ).Note that Bsing diverges linearly as one approaches the diagonal, and Bbd is bounded anddiscontinuous.Form of the Perturbative ExpansionHaving stated the important properties of the propagator in terms of forms, we willstate the precise transcription of Eq. 2.39 and Eq.

2.40 into that language.LetΓx1...xV = Γ(Mx1 × ... × MxV , Λ∗(⊕Vi=1([T ∗Mxi ⊕gxi])))(3.48)denote the space of sections of the bundle over Mx1 × ... × MxV whose fiber at (x1, ..., xV )is the graded Grassmann algebra generated by one forms dxµi and jaxi, i = 1, ..., V , corre-sponding to bases elements for the cotangent space of M and the adjoint bundle of P atthe points xi. Multiplication on Γx1...xV is pointwise wedge product.

The one forms jaxiwill also sometimes be denoted by ja(i). The operation of interior product with ja(i) will bedenoted∂∂ja(i)7.Given an element A ∈Ω∗(Mx × My; gx ⊗gy) (e.g.

the propagator L), we defineAs ∈Γxy to be the image of A under the natural injection from Ω∗(Mx × My; gx ⊗gy) toΓxy = Γ(Mx × My; Λ∗([TMx ⊕gx] ⊕[TMy ⊕gy])). (3.49)Also define Atot ∈Γx1,...,xV byAtot(x1, ..., xV ) =VXi,j=1As(xi, xj).

(3.50)6Here Γαρσ = gστ(Γα)τ σ, where (Γα)τ σdzα is the connection one-form for the metric connection.7 The definition of this requires a trivialization of g, and so can only be defined locally when g isnontrivial. But the equations below are always valid globally.15

Explicitly, the As(xi, xj) appearing here are given byAs(xi, xj) = Aab(xi, xj)ja(i)jb(j). (3.51)Although the propagator L is singular along the diagonal, the singularity is sym-metric in the group theory indices, because it is proportional to δab.

Hence Ls extendscontinuously across the diagonal,Ls(xi, xi) ≡Lcontab(xi, xi)ja(i)jb(i). (3.52)(This regularization can be stated in any number of equivalent ways as a point splittingregularization.

)Before going on, we make a rather technical comment about our sign conventions inEq. 3.42 which is a translation into forms language of conventions built into the superspacelanguage.

We adopt the unusual sign convention that the expressionRMy in Eq. 3.42 isdefined byZMy[ψ(y)χ(x)] ≡[ZMyψ(y)]χ(x)for χ ∈Ω∗(Mx), ψ ∈Ω∗(My).

(3.53)This convention is opposite to the usual mathematical definition (see e.g. [9], p.61])and means that the exterior derivative operator anticommutes withRMy rather thancommuting.

This corresponds, in the notation of the previous section, to the fact thatdRdXF(X, Y ) = −RdX dF(Y, X).More generally, we define the operatorRMxi onΓx1...xV by stating that for ψ(xi) in Γi and χ(x1, ..., xi−1, xi+1, ..., xV ) in Γx1...xi−1,xi+1,...,xV ,ZMxi1[ψ(xi)χ(x1, ..., xi−1, xi+1, ..., xV )]equals[ZMxiψ(xi)]χ(x1, ..., xi−1, xi+1, ..., xV )if ψ sits inside Ω∗(M) and equals zero if ψ is of degree one or more in the ja(i). Notethat, because we chose the nonstandard convention in Eq.

3.53, in order for the operatorsRMx1×...×MxV andRMx1 ...RMxV to agree, we must equip M V = Mx1 × ... × MxV with thenonstandard orientation (for which µV ...µ1 is a positive volume form on the product if theµi are positive volume forms on the factors).16

Now Eq. 2.39 becomesZhlk ≡ZkZsck=X3V =2I(−ik/2π)V −I(3!

)V (2! )IV !I!ΠVi=1"ZMxifaibici∂∂jai(i)∂∂jbi(i)∂∂jci(i)#Ltot(x1, .., xv)I.

(3.54)Let Tr(i) be the operator faibici∂∂jai(i)∂∂jbi(i)∂∂jci(i), and let TRTR : Γx1,...,xV →Ω∗(Mx1, ..., MxV )be the composition of the operator Tr(V )...Tr(1) followed by the restriction of an elementin Γx1...xV to the piece of degree 0 in the ja(i)8. Then we can rewrite Eq.

3.54 asZhlk =X3V =2I(−ik/2π)V −I(3! )V (2!

)IV !I!ZMV TR(Ltot(x1, .., xv)I). (3.55)Although we shall not use it anywhere else in the paper, we rewrite Zhlk in one moreway that may be useful in trying to sum the perturbation series.

LeteM =∞[V =0M V /SV ,(3.56)where SV is the permutation group of order V acting on M V by exchanging the differentcopies of M. (eM can be identified with the set of finite subsets of M.) ThenZhlk =ZeM TReγLtot,(3.57)where γ = 12(−ik/2π)−13 (3! )−23 .Diagramatic DescriptionWe now describe Eq.

3.54 in the language of Feynman diagrams. For the remainderof the paper, a Feynman diagram will mean a graph G, all of whose vertices have valency1, 2, or 3.

For a diagram G, Vr(G) will denote the number of r-valent vertices, V (G) =V1(G) + V2(G) + V3(G) the total number of vertices, I(G) the number of edges, C(G)8The map TR encodes the proper sign for the interaction vertices. It is possible to interpret it as ageneralized trace.17

the number of connected components, and l(G) the number of loops (the dimension of thefirst homology group). So we have,2I(G) = 3V3(G) + 2V2(G) + V1(G), andV (G) −I(G) = C(G) −l(G).

(3.58)When the diagram is clear, we will simply write V for V (G), I for I(G), etc.The diagrams we look at should be thought of as diagrams for truncated Greensfunctions. The number of external legs is defined to beE(G) = 3V (G) −2V (G) = V2(G) + 2V1(G).

(3.59)For example, in Figure 1 the external legs are numbered from 1 to 5.We now write down a generalization of the amplitude for a truncated graph G coupledto sources at the external legs. To do so, we order the vertices and let xi be the nameof a variable in M labeling the i’th vertex.

The generalization of a collection of externalsources will be an element Ψ of Γx1,...,xn. We let I(G) be the product of the propagatorsfor each of the edges of G. To write this down explicitly, we choose an orientation of thegraph and an ordering of the edges.

Let in(e) (out(e)) be the incoming (outgoing) vertexof the e’th edge. Then I(G) is given byI(G)(x1, ..., xV ) = ΠIe=1Ls(xin(e), xout(e)).

(3.60)The amplitude for the graph G coupled to the source Ψ isZMx1×...×MxVTR(I(G)(x1, ..., xV )Ψ(x1, ..., xV ));(3.61)or, more succinctly,RMV TR(I(G)Ψ).As an example, to get the amplitude for Figure 1 with a source Ji flowing in at thei’th external leg, we takeΨ(x1, x2, x3, x4, x5) = J1(x1)J2(x1)J3(x4)J4(x4)J5(x5). (3.62)The Feynman amplitude for a diagram G with no external legs (i.e.

a trivalent graph),is the amplitude when Ψ is equal to 1,I(G) ≡ZMV TR(I(G)). (3.63)For the special case of the empty graph, we set I(G) = 1.18

In order to write I(G) in a way making the group theory indices explicit, we makethe following definition.A labeling of a graph G is a choice of (i) an ordering of thevertices from 1 to V , (ii) an ordering of the edges from 1 to I, (iii) an orientation, and(iv) an ordering of the three edges incident on any given vertex. Graphs with labels willbe denoted ¯G, where G is the underlying unlabeled graph.

Given a labeled graph ¯G, onecan define an injectionF : {1, ..., I} × {1, 2} →{1, ..., 3V }(3.64)by setting F(e, 1) = 3(in(e)−1)+jin(e) and F(e, 2) = 3(out(e)−1)+jout(e) when the e’thedge points from the vertex in(e) to the vertex out(e) and is ordered as the jin(e)’th edgeincident on in(e) and the jout(e)’th edge incident on out(e). Similarly, such an injection Fdetermines a labeled graph ¯G.

The map F is onto precisely when the underlying graph Gis closed (has no external legs). It will be convenient to abbreviate F(e, 1) by e(1), F(e, 2)by e(2),The Feynman amplitude for a closed, labeled diagram ¯G isI( ¯G) =Z(x1,...,xV )∈MV σ( ¯G)fa1a2a3...fa3V −2a3V −1aV ∧Ie=1 Lae(1)ae(2)(xin(e), xout(e)), (3.65)where σ( ¯G) is equal to ±1.The overall sign σ( ¯G) is such that I( ¯G) is equal to theFeynman amplitude I(G) defined above.

This overall sign is irrelevant for the proof givenin the next section that the integral in Eq. 3.65 is convergent despite the singularitiesnear the various diagonals.

For the proof of metric independence up to local anomalies,however, the relative signs between graphs are crucial.The higher loop perturbation series Zhlk which is the focus of our study is given by aweighted sum of the Feynman amplitudes of the labeled trivalent graphs,Zhlk =X¯G(−ik/2π)V −I(3! )V (2!

)IV !I!I( ¯G). (3.66)Zhlk can be rewrittenZhlk =∞XV =0,2,...(−ik/2π)−12 V IdiscV,(3.67)where IdiscVis the contribution of all graphs (connected or not) with V vertices,IdiscV=X¯GV (G)=V1(3!

)V (2! )IV !I!I( ¯G).

(3.68)19

This can also be written as a sum over unlabeled graphs,IdiscV=XGV (G)=V1S(G)I(G),(3.69)where S(G) is the symmetry factor of the graph G9.Letting Iconnlbe the contribution of l-loop connected graphs, we haveZhlk = exp ∞Xl=0(−ik2π )1−lIconnl!.(3.70)4. Proof of FinitenessIn this section we sketch the proof of finiteness of Chern–Simons perturbation theoryfor all M, A(0), and g as above.

Further details will appear in [3]. The following strongdefinition of finiteness implies that the Feynman integrals computing all correlation func-tions (including the partition function itself) are absolutely convergent.

We say that thetheory is finite if, for every graph G with vertices labeled by points x1, ..., xV and everybounded section Ψ ∈Γx1,x2,...,xV , the integralRMV TR(I(G)Ψ) is convergent.Although we have been considering theories formulated on compact M, it also makessense to formulate perturbation theory for noncompact manifolds. To do so, one shouldimpose conditions at infinity to arrive at an appropriate definition of the propagator, andthen use the same formulas as above for the perturbation theory.

In particular, for flat IR3with the trivial connection, the propagator is the free propagator,4πLfreeab(x, y) = −12δabǫµνρuµ||u||3 duνduρ,(4.71)where x, y ∈IR3 and u equals x−y. We say a theory on a noncompact manifold is ultravioletfinite if the integralRMV TR(I(G)Ψ) is always locally integrable.

(For a compact manifoldultraviolet finiteness is the same as finiteness. )As a warmup to the general proof of finiteness, we first prove:9To define S(G), we let PV,I be the group of order (3!

)V (2! )IV !I!

generated by changes oforientation on any of the I edges, and permutations of the set {1, ...V } of vertex labels, the set {1, ..., I}of edge labels, and of the orderings of incident vertices on any of the V edges. PV,I acts on the set oflabeled graphs with V vertices and I edges.

The orbits of the action being sets of labeled graphs with thesame underlying unlabeled graph. S(G) is the number of elements of PV,I which fix a labeled graph ¯Gwith underlying graph G.20

Theorem 4.1 . The theory is ultra-violet finite in flat IR3.Then we shall prove:Theorem 4.2 .

The theory is finite for a general oriented compact 3-manifold M withoutboundary.Sketch of Proof of Theorem 4.1.For a general flat space theory, the superficial degree of divergence, ∆(G), for adiagram G is the degree by which one would expect its flat space Feynman integral todiverge due to the singularities when all the vertices approach one another. For the theorywe are considering, the superficial degree of divergence of G is∆(G) = 2I −3(V −C) = 3C −E.

(4.72)That is, one counts plus 2 for each edge because of the 1/||u||2 divergence of the propagator,minus 3 for the integration at each vertex, and plus 3 since the overall translation collectivecoordinate for each connected component does not help with convergence. The right handside of Eq.

4.72 also equals 3l −I as one one would expect from momentum space powercounting: plus 3 for the 3 momentum integrated over for each loop and minus 1 becausethe momentum space propagator falls offas one power of the momentum (since it is thekernel of a differential operator of order 1).We call a connected diagram superficially divergent if its superficial degree of diver-gence is non-negative and if it has at least one loop (tree graphs should not be considereddivergent).A general diagram is called superficially divergent if any of its connectedsubdiagrams are.Recall the convergence theorem for Feynman integrals which says that if a Feynmandiagram has no superficially divergent connected subdiagrams then it is absolutely conver-gent locally10.Thus, to prove Theorem 4.1, it suffices to show that I(G) vanishes whenever G is aconnected superficially divergent diagram. So let G be such a diagram, with the verticeslabeled by x1, ..., xV .10See [19], §8.1.4 and literature cited therein.

We use a version of the theorem easily proved by someslight modifications of the proof described in [19]. The proof there is for four dimensional scalar theories,but easily generalizes to any number of dimensions and any type of particles.

The proof in [19] also assumesa massive theory, but that is only to avoid infrared divergences, which do not concern us here (we onlyneed convergence locally). Finally, the theorem in [19] only refers to one-particle irreducible diagrams, butthat restriction is easily removed once one defines superficially divergent graphs in general as above.21

Plugging C = 1 into Eq. 4.72,∆(G) = 3 −E.

(4.73)Combining this with l > 0 and some algebra (or pictures), we find that either V = 1 andE = 1 or V > 1 and E < 4. The case V = 1, E = 1 is trivial because I(G) is Lfrees(x1, x1),which is zero in flat space.

So we may assume V > 1 and E < 4.Now letv(0) =VXi=1xµi∂∂xµi,(4.74)andv(µ) =VXi=1∂∂xµifor µ = 1, 2, 3(4.75)be the vector fields on (IR3)V generating an overall dilation and overall translations. Notethat, the v(α), α = 0, 1, 2, 3 are linearly independent as long as the xi are not all equal.A direct computation shows that interior product with any of these vector fieldsannihilates the propagator,i(v(α))Lfrees(xi, xj) = 0for α = 0, 1, 2, 3.

(4.76)Since I(G) is a product of propagators between different points, it is also annihilated byinterior product with any of the v(α). However, the form I(G) has degree 2I = 3V −E.Thus I(G) is a form on (IR3)V of codimension E less than 4 which is annihilated by interiorproduct with four vector fields on (IR3)V , linearly dependent almost everywhere.

HenceI(G) vanishes.Sketch of Proof of Theorem 4.2.Since M V is compact, it suffices to show that every (x(0)1 , ..., x(0)V ) ∈M V has an openneighborhood U so that the integralIU ≡ZUTR(I(G)Ψ)(4.77)is convergent for every bounded Ψ ∈Γx1,...,xV . We will take U to be of the formU = {(x1, ..., xV ) ∈M V ; xi = expx(0)i (yi), yi ∈Tx(0)i M, ||yi|| < ǫ},where ǫ will be chosen sufficiently small.

For the rest of the proof, (x1, ..., xV ) will alwaysbe a point in U.22

If the x(0)iare all distinct, then IU converges because the propagators are bounded inU. Let z(0)1 , ..., z(0)Kbe the distinct points in the set {x(0)1 , ..., x(0)V }.

By choosing ǫ smallenough, there is some constant C so that the distance between xi and xj is greater thanC unless x(0)iequals x(0)j . For J between 1 and K, let IJ be the product of propagatorsfor edges e connecting vertices close to z(0)J , i.e.IJ = Πe∈SJL(xin(e), xout(e)),SJ = {e; x(0)in(e) = x(0)out(e) = z(0)J }.

(4.78)ThenIU =ZUTRΠKJ=1IJΨ′,(4.79)where Ψ′ ∈Γx1,...,xV is bounded on U. Now U = U1 × ... × UK, where UJ is the set ofpositions of the xi for i such that x(0)i= z(0)J .

Also Ψ′ can be uniformly approximated by asum of terms of the form ΠKJ=1ΨJ, where the ΨJ are bounded and (as forms) only dependon the variables in UJ. Consequently, to prove that IU converges, it suffices to show thatRUJ TR(I(GJ)ΨJ) is finite, where GJ is the subgraph of G consisting of the vertices closeto z(0)Jand the edges connecting two such vertices (so IJ above equals I(GJ)).

ReplacingG above by GJ, we can assume that K = 1.To recapitulate, it suffices to show convergence ofIU =Z(x1,...,xV )∈MV ;xi=expz(0) (yi),yi∈Tz(0) M,||yi||<ǫTRΨΠEe=1L(xin(e), xout(e)),(4.80)for any graph G and any bounded Ψ. Now use the exponential map to pull back the integralin Eq.

4.80 to an integral over the yj’s; choose a basis {eµ} of Tz(0)M; and substitute theexpression in Eq. 3.47.

We obtainIU =Z(y1,...,yV );yi∈Tz(0) M,||yi||<ǫTRΨΠIe=1Beµνd(ˆue)µd(ˆue)ν + Ceµd(ˆue)µ + De,(4.81)whereˆue =yin(e) −yout(e)||yin(e) −yout(e)||(4.82)and the B’s, C’s, and D’s are all bounded. The right hand side of Eq.

4.81 is a sum ofterms of the formRP ∧ω, where ω is a bounded form and P is a product of the d(ˆue)µ.We can think of P as the Feynman integrand, I(Γ), for a graph Γ in a theory withthree different types of propagators which can be attached to an edge e connecting yin(e) toyout(e), namely the d(ˆue)µ for µ = 1, 2, 3, The vertex interaction is wedge product followed23

by integration of top forms. The graph Γ may have vertices of valency greater than threeand is not allowed to have any edges connecting a vertex to itself.By the convergence theorem, it suffices to prove that if Γ is a superficially divergentconnected diagrams then its Feynman integrand I(Γ) vanishes.

Now, the degree of di-vergence of any of the propagators d(ˆue)µ is one, as is its form degree. So the degree ofdivergence of Γ is E(Γ)−3(V (Γ)−1).

Thus Γ is superficially divergent if 3V (Γ)−E(Γ) ≤3(and it has at least one loop). But E(Γ) is also the form degree of I(Γ).

If Γ is superfi-cially divergent, I(Γ) is a form of codimension less than 4. However, I(Γ) is annihilatedby interior product with the four vector fields generating overall dilations and translations(which are linearly independent when V (Γ) > 1 as is the case for a diagram with at leastone loop and no tadpoles).

Thus I(Γ) vanishes if Γ is superficially divergent.5. Formal Proof of Metric Independence and 2-loop AnomaliesIn this section we discuss the dependence of IdiscV(M, A(0), g) on the metric g. Wegive a formal proof that IdiscVis independent of g; i.e., we show that the derivativeδδgIdiscV(M, A(0), g) vanishes for an arbitrary variation δg of the metric g. Our argumentis formal; later in this section we compute δδgIdisc2rigorously and show that it is a localanomaly.Formal Metric IndependenceLet ˙IdiscVdenote δδgIdiscV(M, A(0), g).

Using integration by parts formally, we find˙IdiscV= δδgcVZMV TR(LItot)= cV IZMV TR( ˙LtotLI−1tot )= cV IZMV TR((dBtot)LI−1tot )= cV IZMV TR(Btotd(LI−1tot ))= −cV I(I −1)ZMV TR(BtotδgtotLI−2tot )= −cV I(I −1)ZMV TR(δgtotBtotLI−2tot ),(5.83)where cV = [(3! )V (2!

)IV !I!]−1. Here, we have useddLtot = −δtot, and˙Ltot = dBtot,(5.84)24

which follow from (PL1) and (PL4).In Eq. 5.84,δgtot =Xi̸=jδ(xi, xj)δab ja(i)jb(j).

(5.85)Note that the terms with i = j in Eq. 5.85 vanish because δab is symmetric and the j’s areFermionic.To show that the last expression in Eq.

5.83 vanishes, it is perhaps most expeditiousto describe the basic cancellations in terms of diagrams. The last expression in Eq.

5.83 isequal to a sum over labeled trivalent graphs with a δg propagator on the first edge, a Bpropagator on the second edge, and an L propagator on all the other edges;˙IdiscV= −X¯GcVZMV TRδgs (xin(1), xout(1))Bs(xin(2), xout(2))ΠVe=3Ls(xin(e), xout(e)). (5.86)This can be rewritten as a sum, with suitably combinatorial factors, of Feynman amplitudesfor trivalent graphs with two marked edges.

The amplitude is the same as that for anunmarked graph, except one is to make an insertion of a δg (rather than the propagatorL) on one marked edge and an insertion of a B on the other marked edge. Now, if theδg edge connects the points y and z, then by integrating out the δ function, we find anamplitude which corresponds to a graph with a four valent vertex inserted.

One obtains, inthis manner, all diagrams with one four valent vertex and one B edge (with the remainingvertices trivalent and the remaining edges unmarked). In fact, each diagram of this type isobtained in three different ways.

So, the Feynman rule associated to the four-valent vertexwill be a sum of three terms. Figure 3 illustrates the situation.

In the equations below, wechoose to write amplitudes with Lie algebra indices explicit, rather than imbedded withinform notation. So in Figure 3 we have written the Lie algebra indices that will be used inthe equations below as well as the names of the positions of the vertices.The shaded region in each of the diagrams depicted in Figure 3 is the same exceptfor the location of the external legs.

Let W cdef(x1, x2, x3, x4) be the amplitude for thisregion when the external legs are at arbitrary positions x1, ..., x4 as in Figure 4. Then theamplitude for the top left, top middle, and top right diagrams in Figure 3 are, respectively,ZMy×Mzfacdfbefδabδ(y, z)W cdef(y, y, z, z),(5.87.1)ZMy×Mzfacffbdeδabδ(y, z)W cdef(y, z, z, y), and(5.87.2)ZMy×Mzfacefbdfδabδ(y, z)W cdef(y, z, y, z)(5.87.3).25

By a more careful analysis, one can check that the overall combinatorial factors and signsare the same for each of these diagrams. Hence the amplitude for the four vertex diagram(IV) isRMx GcdefW cdef(x, x, x, x), where the effective Feynman rule at the four valentvertex is Gcdef = facdfaef + facffade + facefadf.

But this vanishes by the Jacobi identity!For the case V = 2, the argument above simplifies.We now provide the details,including the precise combinatorial factors and signs. There are only two, 2-loop diagrams,which are both connected.

We call them the dumbbell diagram and the sunset diagramand they are illustrated in Figure 2.The Feynman amplitudes, with the correct signs and symmetry factors, are:Idumbbell = −18ZMx1×Mx2fa1b1c1fa2b2c2La1c1(x1, x1)La2c2(x2, x2)Lb1b2(x1, x2)(5.88)for the dumbbell diagram; andIsunset = + 112ZMx1×Mx2fa1b1c1fa2b2c2La1a2(x1, x2)Lc1c2(x1, x2)Lb1b2(x1, x2)(5.89)for the sunset diagram. These are the two nonvanishing terms in c2RM2 TR(L3tot).

Re-peating the derivation of Eq. 5.83 at the diagramatic level, we find˙Idumbbell = +18ZMx1×Mx2fa1b1c1fa2b2c2Ba1c1(x1, x1)La2c2(x2, x2)δb1b2δ(x1, x2)+ permutations,(5.90)and˙Isunset = −112ZMx1×Mx2fa1b1c1fa2b2c2Ba1a2(x1, x2)Lc1c2(x1, x2)δb1b2δ(x1, x2)+ permutations.

(5.91)The permutation terms mean that we should sum over permutations of B, L, and δg.The terms where the δ function is placed on the handle of the dumbbell vanish becauseδabfabc vanishes. Collecting equal terms, relabeling indices, and integrating out the deltafunctions, we find˙Idumbbell = +14ZMxfacdfaefBcd(x, x)Lef(x, x),(5.92)and˙Isunset = +12ZMxfacefafdBcd(x, x)Lef(x, x)= +14ZMx[facefafd −facffaed]Bcd(x, x)Lef(x, x).

(5.93)For the last equality, we used antisymmetry of Lef(x, x) under exchange of e and f. By theJacobi identity, facdfaef +facefafd+facffade = 0, we conclude that ˙I2 = ˙Idumbbell+ ˙Isunsetvanishes.26

2-loop AnomaliesTo evaluate ˙IV rigorously, we will replace the formal use of integration by parts in Eq.5.83 by a proper use of Stoke’s theorem.We make the following definitions:∆ij = {(x1, ..., xV ) ∈M V ; xi = xj} for i ̸= j,∆tot =V[i,j=1i̸=j∆ij, andNǫ = {(x1, ..., xV ) ∈M V ; d(xi, xj) < ǫ for some i and j}. (5.94)Note that when V = 2, ∆12 is ∆tot and equals the diagonal in Mx1 × Mx2 Moreover, Nǫis isomorphic (by the exponential map) to Nǫ, the ball bundle of radius ǫ in TM.

Itsboundary, ∂¯Nǫ in M × M is isomorphic to Sǫ, the sphere bundle of radius ǫ in TM.The finiteness result of §4 implies1cVIdiscV= limǫ→0ZMV −NǫTr(LItot). (5.95)Hence11,1cV˙IdiscV= limǫ→0δδgZMV −NǫTR(LItot)= limǫ→0IZMV −NǫTR( ˙LtotLI−1tot )= limǫ→0IZMV −NǫTR((dBtot)LI−1tot ).

(5.96)Now using Stoke’s theorem on the manifold M V −Nǫ, we find1cV˙IdiscV= limǫ→0"IZMV −NǫTR(Btotd(LI−1tot )) + IZ∂(MV −Nǫ)TR(BtotLI−1tot )#. (5.97)But dLtot vanishes away from ∆tot, so the first term on the right hand side of Eq.

5.97vanishes. Thus we find,1cV˙IdiscV= limǫ→0IZ∂(MV −Nǫ)TR(BtotLI−1tot ).

(5.98)11By generalizing the proof of finiteness in §4, one can prove that the integralRMV TR( ˙LtotLI−1tot )converges absolutely. This implies that the limits in the last two lines of Eq.

2.31 converge and justifiesour exchanging the order of integration and metric variation for the second equality of Eq. 2.31.27

When V = 2, we obtain˙I2 = 3c2 limǫ→0Z(z,u)∈SǫTR(Btot(x, y)L2tot(x, y)). (5.99)We remind the reader that the (z, u) coordinates are related to the x, y coordinates by theexponential map Eq.

3.45.For the next lemma, it is convenient to define bounded pieces of propagators:Lbded = Lbd + Lcont, and(5.100)Lbdedtot (x, y) = Ls(x, x) + Ls(y, y) + 2Lbdeds(x, y). (5.101)Note also thatLsingtot (x, y) = 2Lsings(x, y)Lbdtot(x, y) = 2Lbds (x, y).

(5.102)We make similar definitions and observations for B.We now proveLemma 5.3 .˙I2 = 3c2 limǫ→0ZSǫTR(Lsingtot Bbdedtot Lbdedtot ). (5.103)To do so, we substitute Btot = Bsingtot+ Bbdedtotand Ltot = Lsingtot+ Lbdedtotinto Eq.

5.99 andexpand the result into eight terms. The term involving only bounded pieces (i.e.

onlyLbdedtotand Bbdedtot ) vanishes because the integrand is bounded and the measure of the regionof integration shrinks to zero as ǫ →0. The terms involving at least two singular piecesvanishes because (Lsingtot )2 and Lsingtot Bsingtotboth vanish.

(Since they are forms of degreegreater than 2 which only depend on the components of fdu in the directions orthogonalto u.) Finally, the quantityRSǫ TR(Bsingtot Lbdedtot Lbdedtot ) vanishes in the limit as ǫ goes to 0because the volume of the area of the sphere Sǫ|x at each point x ∈M shrinks like ǫ2,while Bsingtotdiverges like 1/ǫ, and Lbdedtotremains bounded.Next, we haveLemma 5.4 .˙I2 = 3c2 limǫ→0ZSǫTR(Lsingtot BbdtotLbdtot).

(5.104)28

To prove the Lemma, we use Eq. 5.100 and expand Eq.

5.103 into four terms. The Lemmafollows from0 = limǫ→0ZSǫTR(Lsingtot BbdtotLconttot )(5.105.1)0 = limǫ→0ZSǫTR(Lsingtot Bconttot Lbdtot)(5.105.2)0 = limǫ→0ZSǫTR(Lsingtot Bconttot Lconttot ).

(5.105.3)Now the operation limǫ→0 ◦RSǫ TR equals the operation −RMz TR ◦limǫ→0 ◦Ru∈Sǫ|z. Theminus sign appears because we’ve taken the non-standard orientation on M × M. So theright hand side of Eq.

5.105.1 equals−ZMzTR 4 Lconttot (z, z)limǫ→0Zu∈Sǫ|zLsings(x, y)Bbds (x, y)!. (5.106)Substituting the explicit expressions for Lsing and Bbd given in (PL5) and (PL6)12, Eq.5.106 equals−ZMzTR Xi,j4[ja(1)ja(2)]2Lconttot (z, z) −12ǫνρσ(δΓαρσ(z) −12∇αδgρσ(z))dzαlimǫ→0Zu∈Sǫ|z(−12ǫβγδuβ||u||duγ||u||duδ||u||) uν||u||.

(5.107)The second line in Eq. 5.107 vanishes since it is the integral of a linear function over thetwo-sphere with its standard volume form.

This proves Eq. 5.105.1.

The proof of Eq.5.105.2 is similar.Define Lsing,0 so that Lsingab= δabLsing,0 and make similar definitions for Lbd,0 andBbd,0. Note that, when restricted to the fibers of Sǫ above any point in M, Lsing,0 isminus Ω, where Ωis the spherical volume form of unit area with respect to the standardorientation.To verify Eq.

5.105.3, we havelimǫ→0ZSǫTRLsingtot Bconttot Lconttot= −ZMzTRLconttot (z, z)Bconttot (z, z)[ja(1)ja(2)]limǫ→0"ZSǫ|z2Lsing,0s#= 2ZMzTRLconttot (z, z)Bconttot (z, z)[ja(1)ja(2)]. (5.108)Finally, after unraveling all the notation, the last term in Eq.

5.108 vanishes by the formalproof of metric independence.12We use orthonormal coordinates on TzM so that gij = δij.29

Theorem 5.5 .˙I2 = −h dim(G)2418π2ZMzδΓναρRβγρνdzαdzβdzγ,(5.109)where h equals the dual Coxeter number of G.In the basis of g we have chosen, h is given by 2hdim(G) = fabcfabc.Substituting Eq. 5.102 int Eq.

5.104 and performing the TR operation, we find˙I2 = −14fabcfabcZSǫLsing,0(x, y)Bbd,0(x, y)Lbd,0(x, y). (5.110)Now substitute in the explicit expressions for Lsing, Lbd, and Bbd and perform the integralover the sphere:˙I2 = h dim(G)8(2π)2ZMzlimǫ→0Zu∈Sǫ|zΩ(gµνuµ||u||ˆRν)(gρσuρ||u||)gρσ ˆOσ= h dim(G)24(2π)2ZMzˆRµ ˆOνgµν.

(5.111)The final expression equals is a sum of two terms: the first term is the right hand side ofEq. 5.109; the second term vanishes by integration by parts and the Bianchi identity.Let CSgrav(g, s) be the Chern–Simons action for the metric connection associatedto g defined using a framing s of the manifold M and normalized using minus half thetrace in the adjoint representation of SO(3).

(By framing we mean a homotopy class oftrivializations of the tangent bundle of M.) SinceδδgCSgrav(g, s) = −14πZMzδΓναρRβγρνdzαdzβdzγ,(5.112)we obtainCor. 5.6 .

Let M be an oriented compact 3-manifold without boundary.Let s be aframing and A(0) an acyclic flat G-connection. Then the quantity˜I2 = I2(M, A(0), g) −h dim(G)2412π CSgrav(g, s)(5.113)is independent of the metric g13.13We warn the reader that we are not confident of the minus sign in the Corollary because of thenumber of sign conventions used in its derivation.30

6. Concluding RemarksI.

Shift in k and Higher Loop Anomalies.Having described a closed form for the l-loop contribution, and having found a man-ifold invariant at 2 loops, we discuss the relation between our perturbative results so farwith Witten’s exact solution. For a manifold M with framing s, the exact solution will bedenoted by Zexactk(M, s).Recall Witten’s analysis of the semiclassical approximation, Zsc.

Expanding arounda flat acyclic connection A(0), he found thatZsc(M, A(0), g) = τ(M, [A(0)])12 eiπ|G|4ηgrav(M,g)ei(k+h)CS(A(0)),(6.114)where τ(M, [A(0)]) is the analytic torsion of the flat connection A(0) on M [23], [25], andηgrav(M, g) is the η invariant of the curl operator on M 14.As it stands, Zsc is not independent of the metric g. However, from [2], one findsδδgηgrav(M, g) = −16π δδgCSgrav(g, s),(6.115)for any framing s Thus, by adding a local counterterm |G|/24CSgrav(g, s) to the classicalaction, one obtains an invariant˜Zsc(M, A(0), s) = Zscei|G|24 CSgrav(g,s)(6.116)of framed manifolds with flat acyclic connection. Note that the counterterm added is, upto a constant, the counterterm added in Eq.

5.113 to get a 2-loop invariant.One expects that the leading asymptotics for large k of Witten’s exact solution will re-produce the semi-classical approximation once the counterterms above have been included.More specifically, one expects that, asymptotically for large k,Zexactk(M, s) ∼XA(0)˜Zsck (M, A(0), s). (6.117)This has been confirmed for many examples [14], [20], [15]15.

Also, as observed in [28], thebehaviour of Zsck and Zexactkunder a change of framing are consistent with Eq. 6.117:Zexactk(M, s + 1) = Zexactk(M, s)e2πi24kk+h |G|(6.118.1)˜Zsck = ˜Zsck e2πi24 |G|,(6.118.2)14We normalize η invariants as in [2].

The η invariants in [28] are smaller by a factor of two.15In these examples, some of the flat connections have non-zero betti numbers βi. In that case, thereis an extra factor (k +h)12 (β1−β0) in the definition of Zsck .

In order to reproduce the large k asymptoticsof the exact solution in these examples, it is also necessary to include a constant factor of one over theorder of the center of G whose origin is explained in [30].31

where we used CSgrav(g, s + 1) = CSgrav(g, s) + 2π.Eq. 6.117 describes the leading asymptotics of Zexactk.

Ignoring for the moment theintegration over the moduli space of flat connections, one expects for any nice gauge fixingand regularization procedures that the asymptotics with higher order terms will be givenbyZexactk∼˜Zperpkdef=XA(0)˜Zsck exp ∞Xl=2(−ik′2π )1−l ˜Iconnl!. (6.119)Here ˜Il is the regularized contribution of l loop graphs with a counterterm added to insuremetric independence, and k′ is a function of k whose form depends on the regularizationscheme.

(There has been much discussion in the physics literature on this point, seefor example [1]. )The only regularization implicit in this paper was (i) summing overall particle types (BRS ghosts as well as the original dynamical field) before integratingover M, and (ii) point splitting regularization of the propagator on the diagonal (i.e.

thedefinition of Ls(x, x)).We now argue that k′ = k + h for our regularization scheme. First, we observe thatIconnl(−M, A(0), g) = (−1)1−lIconnl(M, A(0), g)(6.120)where −M is the manifold M with the opposite orientation.

Next we note that the 2-loopcounterterm above changes sign under orientation reversal. We expect counterterms athigher loops to be local and respect the symmetry above, so that˜Iconnl(−M, A(0)) = (−1)1−l ˜Iconnl(M, A(0)).

(6.121)Now let’s focus our attention on the case M = S3 with s the framing invariant underorientation reversing diffeomorphisms. The exact solution in that case isZexactk= (2k + h)12 sin(πk + h).

(6.122)Although the only flat connection on S3, the trivial connection, is not acyclic, we can stillshow Eq. 6.121.

Thus ˜Il(S3, 0) vanishes for l even, so that (6.119) implies k′ = k + h.Next we discuss the specific form of the counterterms, assuming Eq. 6.119 with k′ =k + h. Let us assume that˜Il = Il + βlCSgrav(g, s) + al(g),(6.123)where al(g) is a local counterterm independent of any choice of framing and βl is a constant.We have already seen that this is the case for l = 1, 2, with a1 = a2 = 0.

Preliminary32

investigations based on naive power counting support this assumption and even suggestthat al(g) = 0.The framing dependence˜Zperpk(M, s + 1) = ˜Zperpk(M, s)e2πP(−i(k+h)2π)1−lβl(6.124)togeteher with Eq. 6.119 and Eq.

6.118.1 imply that (i) β1 = i|G|/24. (ii) β2 = −h|G|/24,and (iii) βl = 0 for l > 2.

Item (i) is the agreement of the framing dependence of thesemiclassical approximation with that obtained from the exact solution. Item (ii) saysthat the framing dependence of our 2-loop invariant given in Cor.

5.6 agrees with thatexpected from the exact solution. (The caveat to the footnote in Cor.

4.2 holds here aswell. Other authors seem to have had as much trouble with sign conventions as we do.

)Finally, item (iii) implies that βl = 0. So, we are hopeful that, for l > 2, Il is metricindependent.II.

Extensions.In the paragraphs above, we discussed the l-loop counterterm and the shift from k tok + h which needs to be understood better16. We now list other possible extensions of ourresults and comment on them.i.

One should remove the assumption of acyclicity of A(0). In fact, our proof of finitenessof Il(M, A(0), g) remains valid without the assumption of acyclicity.

(First, the propagatoris still well defined; see the comment after Eq. 2.22.

Second, Lhad has the same formeven when the cohomology doesn’t vanish.) However, one must show that the integral ofIdiscV(M, A(0), g) over the moduli space of flat connections is finite17.

One knows formallythat the variation of IdiscVis a total divergence, so that its integral is metric independent.This formal proof must be made explicit and rigorous up to local anomalies. Consideringthe complicated structure of the moduli space, it seems at present a formidable task.

Wecan at least say that Cor. 5.6 holds for M a rational homology sphere and A(0) the trivialconnection.ii.

Our results should be extended to include knot invariants. Actually, since we havealready shown finiteness of the integral of Greens functions against a bounded source,little needs to be added to show that the terms in the perturbative expansion of Wilson16A derivation was suggested in a talk by one of us (SA) in May, 1991 at a Conference on “Topologicaland Geometrical Methods in Field Theory” in Turku, Finland.

But evidence against this was subsequentlysupplied by E. Witten.17The integral is relative to Zsck , which still needs to be defined precisely as a measure on the modulispace of flat connections, as opposed to a section of a line bundle.33

loop expectation values is finite. The l-loop contribution to Wilson loop expectation valuescan be expressed in terms of the untrunctated Greens functions with E external legs.

Thelatter can be interpreted as elements of ΩE(M E; g⊗E). Formally, assuming acyclicity ofA(0), these Greens functions are closed, and their derivatives with respect to a metricvariation are exact.

This will lead to the formal proof of metric independence.iii.It is natural to try to extend our results to the case when the manifold M has aboundary. Now the exact solution produces a state vector in a Hilbert space associated tothe boundary in a nonperturbative way.

It is not clear, however, where the perturbativeterms belong.Since the the sewing rule is one of the fundamental properties of a topological fieldtheory reflecting the intuition of path integrals, one would hope to capture some ana-logue of it at the perturbative level. More modestly, perhaps one could get some feel forfactorization perturbatively by trying to prove directly the sewing formula for connectedsums:Zexactk(M1#M2) = Zexactk(M1)Zexactk(M2)Zexactk(S3).

(6.125)That might require computing the perturbative terms for Zexactk(S3) (or anything) directly.iv. To reproduce the full exact solution, rather than just its asymptotic behavior to allorders, it would be necessary to sum the perturbation series.

Explicit examples seem toindicate that the full partition function is analytic in1k+h near 0. Eq.

3.57 might be usefulfor showing this.v. Finally, it would be of great interest, for example in the application to 3-dimensionalgravity, to generalize the results contained here to non-compact gauge groups.

Hopefullyone could extend the 1-loop analysis of this problem given in [5] to a sensible regularizationto all orders in perturbation theory. The manifold invariants thus obtained would have, inour opinion, a rich geometric interpretation in their own right.Acknowledgements:We would like to thank Edward Witten and Dror Bar-Natan foruseful discussions.34

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