Asymptotic Expansion of the One-Loop Approximation of the Chern-Simons Integral in an Abstract Wiener Space Setting
aa r X i v : . [ m a t h . DG ] J un Asymptotic Expansion of the One-Loop Approximationof the Chern-Simons Integralin an Abstract Wiener Space Setting
Itaru Mitoma Seiki Nishikawa ∗ Abstract
In an abstract Wiener space setting, we construct a rigorous mathematical model ofthe one-loop approximation of the perturbative Chern-Simons integral, and derive itsexplicit asymptotic expansion for stochastic Wilson lines.
Contents
Since the pioneering work of Witten [21] in 1989, a multitude of people studied on therelationship between the
Chern-Simons integral , a formal path integration over an infinite-dimensional space of connections, and quantum invariants , new topological invariants ofthree-manifolds and knots (see, for instance, Atiyah [3] and Ohtsuki [20] for overviews ofrecent developments in this area). Amongst others, a rigorous mathematical model of theperturbative Chern-Simons integral was constructed by Albeverio and his colleagues; first
Mathematics Subject Classification.
Primary 57R56; Secondary 28C20, 57M27.
Key words and phrases.
Chern-Simons integral, one-loop approximation, asymptotic expansion, ab-stract Wiener space, stochastic holonomy, stochastic Wilson line. ∗ Partly supported by the Grant-in-Aid for Scientific Research (A) of the Japan Society for the Promotionof Science, No. 15204003. Itaru Mitoma and Seiki Nishikawa in the Abelian case as a Fresnel integral [1], and then for the non-Abelian case within theframework of white noise distribution [2].Recently, an explicit representation of stochastic oscillatory integrals with quadraticphase functions and the formula of changing variables, based on a method of computationof probability via “deformation of the contour integration”, have been established on abstract Wiener spaces by Malliavin and Taniguchi [17]. Motivated by these antecedentresults, the first-named author studied the Chern-Simons integral, in [18, 19], from thestandpoint of infinite dimensional stochastic analysis.The main objective of this paper is, based on the work of Bar-Natan and Witten [5]and the mathematical formulation of the Feynman integral due to Itˆo [15], to construct, inan abstract Wiener space setting, a rigorous mathematical model of the one-loop approx-imation of the perturbative Chern-Simons integral of Wilson lines, and derive its explicitasymptotic expansion.To state our result succinctly, let M be a compact oriented smooth three-manifold,and consider a (trivial) principal G -bundle P over M with a simply connected, connectedcompact simple gauge group G with Lie algebra g . We denote by Ω r ( M, g ) the spaceof g -valued smooth r -forms on M equipped with the canonical inner product ( , ), andidentify a connection on P with a g -valued 1-form A ∈ Ω ( M, g ). Let Q A = ( ∗ d A + d A ∗ ) J be a twisted Dirac operator acting on Ω r ( M, g ), where ∗ is the Hodge ∗ -operator definedby a Riemannian metric chosen on M , d A is the covariant exterior differentiation definedby a flat connection A on P , and J is an operator defined to be J ϕ = − ϕ if ϕ is a 0-formor a 3-form, and J ϕ = ϕ if ϕ is a 1-form or a 2-form. For a sufficiently large integer p ,we define the Hilbert subspace H p (Ω + ) of L (Ω + ) = L (cid:0) Ω ( M, g ) ⊕ Ω ( M, g ) (cid:1) with newinner product ( , ) p defined by (cid:0) ( A, φ ) , ( B, ϕ ) (cid:1) p = (cid:0) A, (cid:0) I + Q A (cid:1) p B (cid:1) + (cid:0) φ, (cid:0) I + Q A (cid:1) p ϕ (cid:1) , where I is the identity operator on L (Ω + ).Now, let H = H p (Ω + ) and ( B, H, µ ) be an abstract Wiener space (see Section 3 forthe precise definition). Let λ i and e i , i = 1 , , . . . , denote the eigenvalues and eigenvectorsof the self-adjoint elliptic operator Q A , and h i = (cid:0) λ i (cid:1) − p/ e i be the correspondingCONS of H , respectively. Choosing a sufficiently large p satisfying the condition ∞ X i =1 (cid:0) λ i (cid:1) − p | λ i | < ∞ , we define the normalized one-loop approximation of the Lorentz gauge-fixed Chern-Simonsintegral of the ǫ -regularized Wilson line F ǫA ( x ), defined in Section 4, to be(1.1) I CS (cid:0) F ǫA (cid:1) = lim sup n →∞ Z n Z B F ǫA (cid:0) √ nx (cid:1) e √− kCS ( √ nx ) µ ( dx ) , where Z n = Z B e √− kCS ( √ nx ) µ ( dx ) , CS ( x ) = ∞ X i =1 (cid:0) λ i (cid:1) − p λ i h x, h i i , symptotic Expansion of the Chern-Simons Integral h , i denotes the natural pairing of B and its dual space B ∗ .Then we obtain the following expansion theorem. Theorem .
For any fixed ǫ > and positive integer N , (1.2) I CS ( F ǫA ) = Z B F ǫA (cid:0) R k x (cid:1) µ ( dx ) = X m Here D ( A ) is the Feynman measure integrating over all gauge orbits, that is, over the space A / G of equivalence classes of connections modulo gauge transformations, Tr denotes thetrace in the adjoint representation of the Lie algebra g , that is, a multiple of the Killingform of g , normalized so that the pairing ( X, Y ) = − Tr XY on g is the basic inner product,and the parameter k is a positive integer called the level of charges .Among various integrands, the most typical example of gauge invariant observables isthe Wilson line defined by(2.3) F ( A ) = s Y j =1 Tr R j P exp Z γ j A, where P denotes the product integral (see [11], or equivalently [7]), γ j , j = 1 , , . . . , s ,are closed oriented loops, and the trace Tr is taken with respect to some irreduciblerepresentation R j of G assigned to each γ j . It should be noted that the term P exp R γ j A in (2.3) gives rise to the holonomy of A around γ j , which is defined to be a solutionof the parallel transport equation with respect to A along γ j . From the standpoint ofinfinite dimensional stochastic analysis, we need to regularize the Wilson line (2.3), in amanner similar to that in Albeverio and Sch¨afer [1], to obtain its ǫ -regularization F ǫ ( A )(see Section 3).We now recall the perturbative formulation of the Chern-Simons integral [4, 5] andadopt the method of superfields in the following manner. Let A be a critical point of theLagrangian L such that dA + A ∧ A = 0 , that is, A is a flat connection. For simplicity, we assume as in [4, 5] that A is isolatedup to gauge transformations and that the group of gauge transformations fixing A isdiscrete, or equivalently the cohomology H ∗ ( M, d A ) of d A vanishes, that is,(2.4) H ( M, d A ) = { } , H ( M, d A ) = { } , where d A is the covariant exterior differentialtion acting on Ω r ( M, g ), defined by d A = d + [ A , · ] . Here the bracket [ A, B ] of A = P A α ⊗ E α ∈ Ω r ( M, g ) and B = P B β ⊗ E β ∈ Ω r ( M, g )is defined to be [ A, B ] = X α, β A α ∧ B β ⊗ [ E α , E β ] ∈ Ω r + r ( M, g ) , where { E α } is a basis of the Lie algebra g .Then, for the standard gauge fixing, following [4, 5], we introduce a Bosonic 3-form φ ,a Fermionic 0-form c , a Fermionic 3-form ˆ c , which are g -valued smooth forms on M , andthe BRS operator δ . The BRS operator δ is defined by the laws δA = − D A c, δc = 12 [ c, c ] , δ ˆ c = √− φ, δφ = 0 , symptotic Expansion of the Chern-Simons Integral D A = d A + [ A, · ]. In order to define the Lorentz gauge condition, we now choosea Riemannian metric g on M and denote by ∗ : Ω r ( M, g ) → Ω − r ( M, g ) the Hodge ∗ -operator defined by g , which satisfies ∗ = id. Then the Lorentz gauge condition is givenby(2.5) ( d A ) ∗ A = 0 , where ( d A ) ∗ = ( − r ∗ d A ∗ denotes the adjoint operator of d A . We set V ( A ) = k π Z M Tr (ˆ c ∗ d A ∗ A ) , and define the gauge-fixed Lagrangian of (2.2) by L ( A + A ) − δV ( A ) , where δV ( A ) is given by δV ( A ) = k π Z M Tr (cid:0) √− φ ∗ d A ∗ A − ˆ c ∗ d A ∗ D A c (cid:1) . Noting that around the critical point A of L , L ( A + A ) is expanded as L ( A + A ) = L ( A ) − √− k π Z M Tr n A ∧ d A A + 23 A ∧ A ∧ A o , this leads to the gauge-fixed Chern-Simons integral written as Z A Z Φ Z b C Z C D ( A ) D ( φ ) D (ˆ c ) D ( c ) F ( A + A ) × exp (cid:20) L ( A ) − √− k π Z M Tr n A ∧ d A A + 23 A ∧ A ∧ A + 2 φ ∗ d A ∗ A + 2 √− c ∗ d A ∗ D A c o(cid:21) . (2.6)Geometrically, one can derive (2.6) in the following way. First recall that the tangentspace T A A ∼ = Ω ( M, g ) of the space of connections A at A is decomposed as T A A = Im d A ⊕ Ker ( d A ) ∗ , since for each c ∈ Ω ( M, g ) we have ( d/dt ) | t =0 (exp tc ) ∗ A = d A c . Thus the Lorentz gaugecondition (2.5) corresponds to the choice of the orthogonal complement of the tangentspace to the gauge orbit through A . Under the assumption (2.4) we may think that theLorentz gauge condition ( d A ) ∗ A = 0 has a unique solution on each gauge orbit of G . Then,denoting by det J ( A ) the Jacobian of the transformation G ∋ g ( d A ) ∗ ( g ∗ ( A + A )) ∈ Ω ( M, g ) at the identity element of G , we obtain the following basic identity for the Chern-Simons integral (2.1):(2.7) Z A / G F ( A ) e L ( A ) D ( A ) = Z A D ( A ) F ( A ) e L ( A ) δ (( d A ) ∗ A ) det J ( A ) , Itaru Mitoma and Seiki Nishikawa where δ denotes the Dirac delta function. Here it should be noted that the term δ (( d A ) ∗ A )can be read into the Lagrangian in the form Z Φ D ( φ ) exp (cid:20) −√− Z M Tr { ( d A ) ∗ A · φ } (cid:21) , and the term det J ( A ) in the form Z b C Z C D (ˆ c ) D ( c ) exp (cid:20)Z M Tr { ˆ c · ( d A ) ∗ D A c } (cid:21) , where ˆ c and c should be understood as Grassmann (anti-commuting) variables (cf. [22]).Encoding these contributions into (2.7), and taking account of the fact that, when derivingthe identity (2.7), the Lorentz gauge condition (2.5) may be replaced by κ ( d A ) ∗ A = 0for any non-zero constant κ ∈ C , we obtain (2.6), by choosing κ = − k/ π .Now, noticing that likewise one may simply substitute δ ( κ ( d A ) ∗ A ) for δ (( d A ) ∗ A ) in(2.7), we set A ′ = p / πA, φ ′ = p / πφ and c ′ = p k/ πc, ˆ c ′ = ∗ p k/ π ˆ c in (2.6), and collect the terms that are at most second order in A ′ , φ ′ , c ′ and ˆ c ′ . In theresult, we obtain the following Lorentz gauge-fixed path integral form of the one-loopapproximation of the Chern-Simons integral, written in variables c ′ , ˆ c ′ and ( A ′ , φ ′ ): Z A ′ Z Φ ′ Z b C ′ Z C ′ D ( A ′ ) D ( φ ′ ) D (ˆ c ′ ) D ( c ′ ) F ( A + A ′ ) × exp h L ( A ) + √− k (cid:0) ( A ′ , φ ′ ) , Q A ( A ′ , φ ′ ) (cid:1) + + (ˆ c ′ , ∆ c ′ ) i (2.8)(see [5, 18] for details). Here we denote by ( , ) + the inner product of the Hilbert space L (Ω + ) = L (cid:0) Ω ( M, g ) ⊕ Ω ( M, g ) (cid:1) given by(( A, φ ) , ( B, ϕ )) + = ( A, B ) + ( φ, ϕ ) , where the inner product and the norm on Ω r ( M, g ) are defined by(2.9) ( ω, η ) = − Z M Tr ω ∧ ∗ η, | · | = p ( · , · ) . Furthermore, Q A is a twisted Dirac operator defined by(2.10) Q A = ( ∗ d A + d A ∗ ) J, where J ϕ = − ϕ if ϕ is a 0-form or a 3-form, and J ϕ = ϕ if ϕ is a 1-form or a 2-form.It should be noted that Q A is a self-adjoint elliptic operator, and ∆ = ( d A ) ∗ d A is theLaplacian acting on Ω ( M, g ). symptotic Expansion of the Chern-Simons Integral L ( A ) as well as the Fermiintegral Z b C ′ Z C ′ D (ˆ c ′ ) D ( c ′ ) e (ˆ c ′ , ∆ c ′ ) , we arrive at, from (2.8), the normalized one-loop approximation of the Lorentz gauge-fixedChern-Simons integral:(2.11) 1 Z Z A Z Φ F ( A + A ) exp (cid:2) √− k (( A, φ ) , Q A ( A, φ )) + (cid:3) D ( A ) D ( φ ) , where Z = Z A Z Φ exp (cid:2) √− k (( A, φ ) , Q A ( A, φ )) + (cid:3) D ( A ) D ( φ ) . Our primary objective is to give a rigorous mathematical meaning to this normalizedone-loop approximation of the perturbative Chern-Simons integral (2.11). To handle the integral (2.11) in an abstract Wiener space setting, we need to extend theholonomy of a smooth connection A around a closed oriented loop γ , P exp Z γ A, to a rough connection A . To this end we regularize the Wilson line in a manner similarto that in [1], which is suitable for our abstract Wiener space setting.As in the previous section, let M be a compact oriented smooth three-manifold, G asimply connected, connected compact simple Lie group with Lie algebra g , and P → M aprincipal G -bundle over M . Let A be the space of connections on P , which is identifiedwith Ω ( M, g ), the space of g -valued smooth 1-forms on M , and denote by { E α } , 1 ≤ α ≤ d , a given basis of g . Let γ : [0 , ∋ τ γ ( τ ) ∈ M be a closed smooth curve in M , andset γ [ s, t ] = { γ ( τ ) | s ≤ τ ≤ t } . We regard γ [ s, t ] as a linear functional( γ [ s, t ])[ A ] = Z γ [ s,t ] A = Z ts A ( ˙ γ ( τ )) dτ, A ∈ A defined on the vector space A . Then γ [ s, t ] is continuous in the sense of distribution andhence defines a ( g -valued) de Rham current of degree two.To recall the regularization of currents, we first consider the case where γ is a closedsmooth curve in R and A is a g -valued smooth 1-form with compact support definedon R . Let φ be a non-negative smooth function on R such that the support of φ iscontained in the unit ball B with center 0 ∈ R and Z R φ ( x ) dx = 1 . Then define φ ǫ ( x ) = ǫ − φ ( x/ǫ ) for each ǫ > 0. If we write A = X α A α ⊗ E α = X i, α A iα dx i ⊗ E α , ˙ γ ( τ ) = X i ˙ γ i ( τ ) (cid:18) ∂∂x i (cid:19) γ ( τ ) Itaru Mitoma and Seiki Nishikawa for given A and γ , then we have(3.1) lim ǫ → sup s ≤ τ ≤ t (cid:12)(cid:12)(cid:12)(cid:12)Z R A iα ( x ) φ ǫ ( x − γ ( τ )) dx − A iα ( γ ( τ )) (cid:12)(cid:12)(cid:12)(cid:12) = 0 , and(3.2) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X i =1 Z ts (cid:18)Z R A iα ( x ) φ ǫ ( x − γ ( τ )) dx (cid:19) ˙ γ i ( τ ) dτ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ c ( ǫ ) k A α k L ( R ) | t − s | . Here and in what follows, we denote by c k ( ⋆ ) a constant depending on the quantity ⋆ andsimply write c k whenever no confusion may occur.Now, according to de Rham [10], the regulator of the current γ [ s, t ] is defined by( R ǫ γ [ s, t ])[ A ] = ( γ [ s, t ])[ R ∗ ǫ A ]= X i =1 Z ts (cid:18)Z R A iα ( γ ( τ ) + y ) φ ǫ ( y ) dy (cid:19) ˙ γ i ( τ ) dτ ⊗ E α = X i =1 Z ts (cid:18)Z R A iα ( x ) φ ǫ ( x − γ ( τ )) dx (cid:19) ˙ γ i ( τ ) dτ ⊗ E α , to which is associated an operator defined by( A ǫ γ [ s, t ])[ B ] = ( γ [ s, t ])[ A ∗ ǫ B ]= X i,j =1 Z ts (cid:26)Z R (cid:18)Z y i B ij α ( γ ( τ ) + ty ) dt (cid:19) φ ǫ ( y ) dy (cid:27) ˙ γ j ( τ ) dτ ⊗ E α , where B = P B ijα dx i ∧ dx j ⊗ E α is a g -valued smooth 2-form with compact supporton R . Then we have the following relation between the operators R ǫ and A ǫ , which isknown as the homotopy formula (see [10, § 15] for details). Proposition 1. For each ǫ > , R ǫ γ [ s, t ] and A ǫ γ [ s, t ] are currents whose supports arecontained in the ǫ -tubular neighborhood of γ [ s, t ] , and satisfy R ǫ γ [ s, t ] − γ [ s, t ] = ∂ A ǫ γ [ s, t ] + A ǫ ∂γ [ s, t ] , where ∂ is the boundary operator of currents. As in [10], the above construction of regularization generalizes to our case in thefollowing manner. First take a diffeomorphism h of R onto the unit ball B with center0 which coincides with the identity on the ball of radius 1 / s y the translation s y ( x ) = x + y and let s y be the map of R onto itself which coincides with h ◦ s y ◦ h − on B and with the identity at all other points, that is, s y ( x ) = ( h ◦ s y ◦ h − ( x ) if x ∈ B ,x if x B . symptotic Expansion of the Chern-Simons Integral h we may make s y to be a diffeomorphism. Thendefine R ǫ γ [ s, t ] and A ǫ γ [ s, t ] by the same equations above, but now replacing γ ( τ ) + y and γ ( τ ) + ty with s y ( γ ( τ )) and s ty ( γ ( τ )), respectively.Now, let { U i } be a finite open covering of M such that each U i is diffeomorphic to theunit ball B via a diffeomorphism h i , which can be extended to some neighborhoods ofthe closures of U i and of B . Using these diffeomorphisms, we transport the transformedoperators R ǫ and A ǫ defined on R to M . Indeed, let f be a cutoff function whichhas its support in the neighborhood of the closure of U i and is equal to 1 on U i . Set T = γ [ s, t ] for simplicity. Then T ′ = f T is a current which has its support contained inthe neighborhood of the closure of U i , and h i T ′ is a current which has its support containedin the neighborhood of the closure of B . Note that the support of T ′′ = T − T ′ does notmeet the closure of U i . We define R iǫ T = h − i ◦ R ǫ ◦ h i T ′ + T ′′ , A iǫ T = h − i ◦ A ǫ ◦ h i T ′ and set inductively R ( k ) ǫ T = R ǫ ◦ R ǫ ◦ · · · ◦ R kǫ T, A ( k ) ǫ T = R ǫ ◦ R ǫ ◦ · · · ◦ R k − ǫ ◦ A kǫ T. Then R ǫ T and A ǫ T are obtained to be R ǫ T = R ( N ) ǫ T, A ǫ T = N X k =1 A ( k ) ǫ T, where N is the number of open sets in { U i } .The construction of the operators R ǫ and A ǫ are easily generalized to any current T defined on a compact smooth manifold of arbitrary dimension. We remark that thefollowing properties hold for regularization of currents. Proposition 2 ([10]) . Let M be a compact smooth manifold. Then for each ǫ > thereexist linear operators R ǫ and A ǫ acting on the space of de Rham currents with the followingproperties :(1) If T is a current, then R ǫ T and A ǫ T are also currents and satisfy R ǫ T − T = ∂ A ǫ T + A ǫ ∂T. (2) The supports of R ǫ T and A ǫ T are contained in an arbitrary given neighborhood ofthe support of T provided that ǫ is sufficiently small. (3) R ǫ T is a smooth form. (4) For all smooth forms ϕ we have R ǫ T [ ϕ ] → T [ ϕ ] and A ǫ T [ ϕ ] → as ǫ → . Given a closed smooth curve γ : [0 , → M in M , for each t ∈ [0 , 1] and sufficientlysmall ǫ > γ [0 , t ] defined by C ǫγ ( t ) = ∗R ǫ γ [0 , t ] , Itaru Mitoma and Seiki Nishikawa where ∗ is the Hodge ∗ -operator defined by a Riemannian metric chosen on M , and write C ǫγ ( t ) = P C ǫγ ( t ) α ⊗ E α . Let U γ be a tubular neighborhood of γ [0 , 1] in M and j : U γ → M denote the inclusion. Then j ∗ ( ∗ C ǫγ ( t )) = j ∗ ( R ǫ γ [0 , t ])is a g -valued smooth 2-form on U γ and has a compact support in U γ from Proposition 2.In particular, for t = 1 we see that dj ∗ ( ∗ C ǫγ (1)) = dj ∗ ( R ǫ γ [0 , j ∗ d ( R ǫ γ [0 , − j ∗ R ǫ ∂ ( γ [0 , , since ∂ ( γ [0 , ∅ .As a result, each j ∗ ( ∗ C ǫγ (1) α ) determines a cohomology class [ j ∗ ( ∗ C ǫγ (1) α )] ∈ H c ( U γ )in the second de Rham cohomology of U γ with compact support. Indeed, by virtue ofProposition 2 (1), it is not hard to see that Z U γ ω ∧ j ∗ ( ∗ C ǫγ (1) α ) = Z γ i ∗ ω holds for any [ ω ] ∈ H c ( U γ ), where i : γ [0 , → U γ denotes the inclusion. Namely, we have Proposition 3 ([1]) . [ j ∗ ( ∗ C ǫγ (1) α )] ∈ H c ( U γ ) is the compact Poincar´e dual of γ in U γ for each α = 1 , , . Recalling the construction of regulators of currents and noting (3.1) and (3.2), it is nothard to see that we havelim ǫ → sup ≤ t ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X i =1 Z t (cid:18)Z M A iα ( x ) φ ǫ ( x − γ ( τ )) dx − A iα ( γ ( τ )) (cid:19) ˙ γ i ( τ ) dτ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 , (3.3) (cid:12)(cid:12)(cid:12)(cid:12) Z γ [0 ,t ] A α − Z γ [0 ,s ] A α (cid:12)(cid:12)(cid:12)(cid:12) ≤ c ( A ) | t − s | , and(3.4) (cid:12)(cid:12) C ǫγ ( t ) − C ǫγ ( s ) (cid:12)(cid:12) ≤ c ( ǫ ) | t − s | , where | · | on the left side of (3.4) is the norm defined in (2.9).Now, in order to extend the holonomy to a rough connection A , for a non-negativeinteger p , let H p (Ω + ) denote the Hilbert subspace of L (Ω + ) = L (cid:0) Ω ( M, g ) ⊕ Ω ( M, g ) (cid:1) with new inner product ( , ) p defined by(3.5) (cid:0) ( A, φ ) , ( B, ϕ ) (cid:1) p = (cid:16) ( A, φ ) , (cid:0) I + Q A (cid:1) p ( B, ϕ ) (cid:17) + = (cid:0) A, (cid:0) I + Q A (cid:1) p B (cid:1) + (cid:0) φ, (cid:0) I + Q A (cid:1) p ϕ (cid:1) . Here I is the identity operator on L p (Ω + ), and the p -norm on H p (Ω + ) is defined as usualby k · k p = p ( · , · ) p . Henceforth we denote H p (Ω + ) briefly by H p whenever no confusionmay occur. symptotic Expansion of the Chern-Simons Integral A is extended to the stochastic holonomy of ( A, φ ) ∈ H p in the following manner. Since (cid:0) A, C ǫγ ( t ) (cid:1) = (cid:16) ( A, φ ) , (cid:0) I + Q A (cid:1) − p ( C ǫγ ( t ) , (cid:17) p , by setting(3.6) ˜ C ǫγ ( t ) = (cid:0) I + Q A (cid:1) − p ( C ǫγ ( t ) , , we obtain from (3.4) that(3.7) (cid:13)(cid:13) ˜ C ǫγ ( t ) − ˜ C ǫγ ( s ) (cid:13)(cid:13) p ≤ c ( ǫ ) | t − s | . Given ( A, φ ) ∈ H p , we now write(3.8) A ǫγ ( t ) = d X α =1 (cid:16) ( A, φ ) , ˜ C ǫγ ( t ) α ⊗ E α (cid:17) p E α , where ˜ C ǫγ ( t ) = P ˜ C ǫγ ( t ) α ⊗ E α , and define¯ A ( t ) = Z γ [0 ,t ] A. With these understood, recall that for the holonomy for a smooth connection A around A , it follows from (3.3) that, in terms of the product integral or Chen’s iterated integral(see Theorem 4.3 of [11, p. 31] and also [7]), it is given by(3.9) P exp Z γ A + A = I + ∞ X r =1 Z Z t · · · Z t r − d ( ¯ A + ¯ A )( t ) d ( ¯ A + ¯ A )( t ) · · · d ( ¯ A + ¯ A )( t r ) , where 0 ≤ t r − ≤ · · · ≤ t ≤ t = 1. Then, noting (3.7), for each ( A, φ ) ∈ H p we definethe ǫ -regularization of the holonomy by(3.10) W ǫγ ( A ) = I + ∞ X r =1 W ǫ,rγ ( A ) , where W ǫ,rγ ( A ) = Z Z t · · · Z t r − d ( ¯ A + A ǫγ )( t ) d ( ¯ A + A ǫγ )( t ) · · · d ( ¯ A + A ǫγ )( t r ) , and the ǫ -regularized Wilson line by(3.11) F ǫA ( A ) = s Y j =1 Tr R j W ǫγ j ( A ) , where the trace Tr is taken in the irreducible representation R j of G assigned to each γ j .2 Itaru Mitoma and Seiki Nishikawa We now proceed to extend the ǫ -regularized Wilson line F ǫA ( A ) in (3.11) even to anabstract Wiener space setting. To this end, let M and G be as in Section 3, and denote by H p (Ω + ) the Hilbert subspace of L (Ω + ) = L (cid:0) Ω ( M, g ) ⊕ Ω ( M, g ) (cid:1) with inner product( , ) p defined by (3.5). Then set H = H p (Ω + ) and let ( B, H, µ ) be an abstract Wienerspace such that µ is a Gaussian measure satisfying Z B e √− h x, ξ i dµ ( x ) = e −k ξ k p / for each ξ ∈ B ∗ . Here B is a real separable Banach space in which the separable Hilbertspace H is continuously and densely imbedded, h , i denotes the natural pairing of B andits dual space B ∗ , and B ∗ is considered as B ∗ ⊂ H under the usual identification of H with H ∗ (cf. [17]).We first note that the twisted Dirac operator Q A of (2.10) has pure point spectrum,since Q A is a self-adjoint elliptic operator (cf. [13]). Thus let λ i , e i = ( e Ai , e φi ) , i = 1 , , . . . , be the eigenvalues and eigenvectors of Q A . Recall that by our assumption (2.4) theeigenvectors { e i } form a CONS (complete orthonormal system) of L (Ω + ). If we define h j = (cid:0) λ j (cid:1) − p/ e j , j = 1 , , . . . , then the set { h j } gives rise to a CONS of H p , so that the increasing rate of the eigenvaluesof Q A guarantees the nuclearity of the system of semi-norms k · k q , q = 1 , , . . . (see, forinstance, Lemma 1.6.3 (c) in [13]). Hence there exists some integer p independent of p such that B is realized as H − p − p (cf. [12]), where H − q is the dual space of H q . If wechoose a sufficiently large p such that p > p and ∞ X i =1 (cid:0) λ i (cid:1) − p | λ i | < ∞ , if neccesary, then we see from (3.6) that˜ C ǫγ ( t ) ∈ H p + p = B ∗ . In what follows we take this suitable space as B throughout the paper.According to (3.8), for each ǫ > x ∈ B , we define x ǫγ ( t ) = d X α =1 h x, ˜ C ǫγ ( t ) α ⊗ E α i E α , where { E α } , 1 ≤ α ≤ d , is a basis of the Lie algebra g , and briefly denote x ǫ,αγ ( t ) = h x, ˜ C ǫγ ( t ) α ⊗ E α i , symptotic Expansion of the Chern-Simons Integral E (cid:2) x ǫ,αγ ( t ) (cid:3) = (cid:13)(cid:13) ˜ C ǫγ ( t ) α ⊗ E α (cid:13)(cid:13) p . Since it follows from (3.7) that(4.2) (cid:12)(cid:12) x ǫ,αγ ( t ) − x ǫ,αγ ( s ) (cid:12)(cid:12) ≤ c ( ǫ ) k x k B | t − s | , the Lebesgue-Stieltjes integral Z t dx ǫγ ( τ ) = d X α =1 Z t dx ǫ,αγ ( τ ) · E α is well-defined. Hence, according to (3.10), for each ǫ > ǫ -regularizedstochastic holonomy for x ∈ B by(4.3) W ǫγ ( x ) = I + ∞ X r =1 W ǫ,rγ ( x ) , where W ǫ,rγ ( x ) = Z Z t · · · Z t r − d ( ¯ A + x ǫγ )( t ) d ( ¯ A + x ǫγ )( t ) · · · d ( ¯ A + x ǫγ )( t r ) . Then the ǫ -regularized Wilson line for x ∈ B (cf. [1]) is given by(4.4) F ǫA ( x ) = s Y j =1 Tr R j W ǫγ j ( x ) . Now, we will see the well-definedness, the smoothness in H -Fr´echet differentiation andthe integrability of the ǫ -regularized Wilson line F ǫA ( x ) as an analytic function in thesense of Malliavin and Taniguchi [17]. Indeed, in the representation R j of G assigned toeach loop γ j , if we define for a given basis { E α } of g and an n × n matrix A = ( a ij ), c E = max ≤ α ≤ d k E α k , k A k = n X i,j =1 | a ij | , then we have the following Lemma 1. For ǫ > and x ∈ B , define the ǫ -regularizations W ǫγ ( x ) and F ǫA ( x ) by (4.3) and (4.4) , respectively. Then the following hold. (1) W ǫγ ( x ) is well defined and C ∞ in H-Fr´echet differentiation. (2) For any positive integer q we have E h(cid:13)(cid:13) W ǫγ ( x ) (cid:13)(cid:13) q i < ∞ . (3) For any positive integer q and positive number s we have ∞ X k =0 s k k ! E " X i ,i ,...,i k (cid:13)(cid:13) D k W ǫγ ( x )( h i , h i , . . . , h i k ) (cid:13)(cid:13) ! q / q < ∞ Itaru Mitoma and Seiki Nishikawa and ∞ X k =0 s k k ! E " X i ,i ,...,i k (cid:12)(cid:12) D k F ǫA ( x )( h i , h i , . . . , h i k ) (cid:12)(cid:12) ! q / q < ∞ . Proof. First we prove (1). It follows from (3.3) and (4.2) that for any t ≥ (cid:13)(cid:13)(cid:13)(cid:13)Z t d ¯ A (cid:13)(cid:13)(cid:13)(cid:13) ≤ σc ( A ) t, (cid:13)(cid:13)(cid:13)(cid:13)Z t dx ǫγ ( τ ) (cid:13)(cid:13)(cid:13)(cid:13) ≤ σc ( ǫ ) k x k B t, where σ = d · c E . Then it is not hard to see that for x ∈ B (4.5) (cid:13)(cid:13) W ǫγ ( x ) (cid:13)(cid:13) ≤ ∞ X r =0 (cid:13)(cid:13)(cid:13)(cid:13)Z Z t · · · Z t r − d ( ¯ A + x ǫγ )( t ) d ( ¯ A + x ǫγ )( t ) · · · d ( ¯ A + x ǫγ )( t r ) (cid:13)(cid:13)(cid:13)(cid:13) ≤ ∞ X r =0 ( σ ( c ( A ) + c ( ǫ ) k x k B )) r Z Z t · · · Z t r − dt dt · · · dt r ≤ ∞ X r =0 ( σ ( c ( A ) + c ( ǫ ) k x k B )) r /r != e σ ( c ( A )+ c ( ǫ ) k x k B ) , which implies the well-definedness of W ǫγ ( x ).To see the smoothness of W ǫγ ( x ) in H -Fr´echet differentiation, we first note that for h ∈ HDW ǫγ ( x )( h ) = lim s → (cid:8) W ǫγ ( x + sh ) − W ǫγ ( x ) (cid:9) /s = lim s → s ∞ X r =1 (cid:26)Z Z t · · · Z t r − d ( ¯ A + ( x + sh ) ǫγ )( t ) · · · d ( ¯ A + ( x + sh ) ǫγ )( t r ) − Z Z t · · · Z t r − d ( ¯ A + x ǫγ )( t ) · · · d ( ¯ A + x ǫγ )( t r ) (cid:27) . Then, in a manner similar to the previous estimate, we have for | s | ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) s ∞ X r =1 (cid:26)Z Z t · · · Z t r − d ( ¯ A + ( x + sh ) ǫγ )( t ) · · · d ( ¯ A + ( x + sh ) ǫγ )( t r ) − Z Z t · · · Z t r − d ( ¯ A + x ǫγ )( t ) · · · d ( ¯ A + x ǫγ )( t r ) (cid:27) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X r =1 r X m =1 Z Z t · · · Z t r − d ( ¯ A + x ǫγ )( t ) · · · d ( ¯ A + x ǫγ )( t m − ) · dh ǫγ ( t m ) d ( ¯ A + ( x + sh ) ǫγ )( t m +1 ) · · · d ( ¯ A + ( x + sh ) ǫγ )( t r ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) symptotic Expansion of the Chern-Simons Integral ≤ ∞ X r =1 r X m =1 σ r ( c ( A ) + c ( ǫ ) k x k B ) m − × c ( ǫ ) k h k B ( c ( A ) + c ( ǫ ) {k x k B + k h k B } ) r − m /r ! ≤ ∞ X r =1 σ r ( c ( A ) + c ( ǫ ) {k x k B + k h k B } ) r − c ( ǫ ) k h k B / ( r − σc ( ǫ ) k h k B e σ ( c ( A )+ c ( ǫ )( k x k B + k h k B )) < ∞ . This, together with Lebesgue’s convergence theorem, implies that W ǫγ ( x ) is H -Fr´echetdifferentiable. Repeating this argument, we then obtain that W ǫγ ( x ) is C ∞ in H -Fr´echetdifferentiation.For the proof of (2) we recall the following lemma due to Fernique (see [16]). Lemma 2. There exists δ > such that Z B e δ k x k B µ ( dx ) < ∞ . Then it follows from (4.5) that E h k W γ ( x ) k q i ≤ E h e qσ ( c ( A )+ c ( ǫ ) k x k B ) i , which together with Lemma 2 shows (2) of Lemma 1.Before proceeding to the proof of (3), we remark the following Lemma 3. Let q be a positive integer and X i,j , i, j = 1 , , . . . , be real numbers. Then X i (cid:12)(cid:12)(cid:12) X j X i,j (cid:12)(cid:12)(cid:12) q ≤ X j (cid:16) X i (cid:12)(cid:12) X i,j (cid:12)(cid:12) q (cid:17) / q ! q . Proof of Lemma 3. Note that (cid:16) X j (cid:12)(cid:12) X i,j (cid:12)(cid:12)(cid:17) q = X j ,j ,...,j q (cid:12)(cid:12) X i,j (cid:12)(cid:12)(cid:12)(cid:12) X i,j (cid:12)(cid:12) · · · (cid:12)(cid:12) X i,j q (cid:12)(cid:12) , and by using H¨older’s inequality recursively we have X i (cid:12)(cid:12) X i,j (cid:12)(cid:12)(cid:12)(cid:12) X i,j (cid:12)(cid:12) · · · (cid:12)(cid:12) X i,j q (cid:12)(cid:12) ≤ (cid:16) X i (cid:12)(cid:12) X i,j (cid:12)(cid:12) q (cid:17) / q (cid:16) X i (cid:16)(cid:12)(cid:12) X i,j (cid:12)(cid:12) · · · (cid:12)(cid:12) X i,j q (cid:12)(cid:12)(cid:17) q/ (2 q − (cid:17) (2 q − / q ≤ (cid:16) X i (cid:12)(cid:12) X i,j (cid:12)(cid:12) q (cid:17) / q (cid:16) X i (cid:12)(cid:12) X i,j (cid:12)(cid:12) q (cid:17) / q × (cid:16) X i (cid:16)(cid:12)(cid:12) X i,j (cid:12)(cid:12) · · · (cid:12)(cid:12) X i,j q (cid:12)(cid:12)(cid:17) q/ (2 q − (cid:17) (2 q − / q and so on.6 Itaru Mitoma and Seiki Nishikawa Hence we obtain X i (cid:12)(cid:12)(cid:12) X j X i,j (cid:12)(cid:12)(cid:12) q ≤ X i (cid:16) X j ,j ,...,j q (cid:12)(cid:12) X i,j (cid:12)(cid:12)(cid:12)(cid:12) X i,j (cid:12)(cid:12) · · · (cid:12)(cid:12) X i,j q (cid:12)(cid:12)(cid:17) = X j ,j ,...,j q (cid:16) X i (cid:12)(cid:12) X i,j (cid:12)(cid:12)(cid:12)(cid:12) X i,j (cid:12)(cid:12) · · · (cid:12)(cid:12) X i,j q (cid:12)(cid:12)(cid:17) ≤ X j ,j ,...,j q (cid:16) X i (cid:12)(cid:12) X i,j (cid:12)(cid:12) q (cid:17) / q (cid:16) X i (cid:12)(cid:12) X i,j (cid:12)(cid:12) q (cid:17) / q · · · (cid:16) X i (cid:12)(cid:12) X i,j q (cid:12)(cid:12) q (cid:17) / q = X j (cid:16) X i (cid:12)(cid:12) X i,j (cid:12)(cid:12) q (cid:17) / q ! q , which completes the proof of Lemma 3.Now we proceed to proving (3) of Lemma 1. Noting that X i ,i ,...,i k (cid:13)(cid:13) D k W ǫγ ( x )( h i , h i , . . . , h i k ) (cid:13)(cid:13) ≤ X i ,i ,...,i k (cid:16) ∞ X r = k (cid:13)(cid:13) D k W ǫ,rγ ( x )( h i , h i , . . . , h i k ) (cid:13)(cid:13)(cid:17) , and by making use of Lemma 3 recursively, it is immediate to see that the right side ofthe above inequality is dominated by ∞ X r = k (cid:16) X i ,i ,...,i k (cid:13)(cid:13) D k W ǫ,rγ ( x )( h i , h i , . . . , h i k ) (cid:13)(cid:13) (cid:17) / ! . Let us denote for simplicity X ≤ l Hence, noting that ∞ X r = k σ r r − k )! ( c ( A ) + c ( ǫ ) k x k B ) r − k c ( ǫ ) k = ∞ X r =0 σ r + k r ! ( c ( A ) + c ( ǫ ) k x k B ) r c ( ǫ ) k = ( σc ( ǫ )) k e σ ( c ( A )+ c ( ǫ ) k x k B ) , we see with Lemma 2 that ∞ X k =0 s k k ! E X i ,i ,...,i k (cid:13)(cid:13) D k W ǫγ ( x )( h i , h i , . . . , h i k ) (cid:13)(cid:13) ! q / q ≤ ∞ X k =0 s k k ! ( σc ( ǫ )) k E h e qσ ( c ( A )+ c ( ǫ ) k x k B ) i / q < ∞ , which verifies the first part of (3).By a similar argument we can also obtain the second half of (3), so is omitted thedetail. The aim of this section is to give a rigorous mathematical meaning, in an abstract Wienerspace setting, to the normalized one-loop approximation of the Lorentz gauge-fixed Chern-Simons integral (2.11). We keep the notation in Section 4.First, recall that for each x = ( A, φ ) ∈ L (Ω + ) = L (Ω ⊕ Ω ) we have( x, Q A x ) + = ∞ X i =1 λ i ( x, e i ) = ∞ X j =1 (cid:0) λ j (cid:1) − p λ j ( x, h j ) p . Then, adopting an idea due to Itˆo [15], we implement convergent factorsexp (cid:20) − ( x, x )2 n (cid:21) with n > L (Ω + ). This leads us to the following m -dimensional approximation of (2.11) written aslim n →∞ Z m,n Z R m F ǫA ( x m ) exp (cid:20) √− k ( x, Qx ) m, + − ( x, x ) m n (cid:21) µ m ( dx ) (cid:0) √ π (cid:1) m , where µ m is the m -dimensional Lebesgue measure, x m = m X j =1 x j h j , ( x, Qx ) m, + = m X j =1 (cid:0) λ j (cid:1) − p λ j x j , ( x, x ) m = m X j =1 x j symptotic Expansion of the Chern-Simons Integral Z m,n = Z R m exp (cid:20) √− k ( x, Qx ) m, + − ( x, x ) m n (cid:21) µ m ( dx ) (cid:0) √ π (cid:1) m . Note that, by setting x = √ ny , this can be rewritten in the formlim n →∞ Z m,n Z R m F ǫA ( √ ny m ) exp (cid:2) √− k ( √ ny, Q √ ny ) m, + (cid:3) × (cid:0) √ π (cid:1) m exp (cid:20) − ( y, y ) m (cid:21) µ m ( dy ) , where Z m,n = Z R m exp (cid:2) √− k ( √ ny, Q √ ny ) m, + (cid:3) (cid:0) √ π (cid:1) m exp (cid:20) − ( y, y ) m (cid:21) µ m ( dy ) . We then consider(5.1) lim n →∞ lim m →∞ Z m,n Z R m F ǫA ( √ ny m ) exp (cid:2) √− k ( √ ny, Q √ ny ) m, + (cid:3) × (cid:0) √ π (cid:1) m exp (cid:20) − ( y, y ) m (cid:21) µ m ( dy ) . However, the canonical Gaussian measure cannot be defined on the Hilbert space L (Ω + ),so that we are going to achieve a realization of (5.1) in an abstract Wiener space setting.Thus, let H = H p and ( B, H, µ ) the abstract Wiener space described in Section 4.Then, within this framework, we now define the normalized one-loop approximation of theperturbative Chern-Simons integral of the ǫ -regularized Wilson line to be(5.2) I CS (cid:0) F ǫA (cid:1) = lim sup n →∞ Z n Z B F ǫA (cid:0) √ nx (cid:1) e √− kCS ( √ nx ) µ ( dx ) , where Z n = Z B e √− kCS ( √ nx ) µ ( dx ) ,CS ( x ) = (cid:10) x, (cid:0) I + Q A (cid:1) − p Q A x (cid:11) = ∞ X j =1 (cid:0) λ j (cid:1) − p λ j h x, h j i , and lim sup n →∞ ( x n + √− y n ) = lim sup n →∞ x n + √− n →∞ y n for real numbers x n and y n .Given ǫ > 0, we also set Z ǫ, γ (0) = I,Z ǫ,rγ ( i ) = X ≤ l Theorem 1. For any fixed ǫ > and positive integer N , I CS ( F ǫA ) = lim sup n →∞ Z B F ǫA (cid:0) R n,k x (cid:1) µ ( dx ) = Z B F ǫA (cid:0) R k x (cid:1) µ ( dx )= X m 1. By making use of the so-called Fresnel integral formula1 √ π Z + ∞−∞ exp (cid:20) − zx (cid:21) dx = 1 √ z , z ∈ C symptotic Expansion of the Chern-Simons Integral Z n = h det n I − √− nk (cid:0) I + Q A (cid:1) − p Q A oi − / . Also, it follows from (3.7) that (cid:13)(cid:13) √ n (cid:0) ˜ C ǫγ ( t ) − ˜ C ǫγ ( s ) (cid:1)(cid:13)(cid:13) p ≤ c ( ǫ ) | t − s | . Hence, by mimicing the proof of (3) of Lemma 1, we see that for any sufficiently small fixed ǫ > 0, the same inequalities in the course of the proof hold with W ǫγ ( x ) being replaced by W ǫγ ( √ nx ). This, together with (1) of Lemma 1, then yields that ∞ X k =0 s k k ! E " X i ,i ,...,i k (cid:12)(cid:12) D k F ǫA (cid:0) √ nx (cid:1) ( h i , h i , ..., h i k ) (cid:12)(cid:12) ! q / q < ∞ for any positive number s , implying the analyticity of F ǫA ( √ nx ).Therefore, we can apply the formula of Malliavin-Taniguchi [17, Theorem 7.8] to theright side of (5.2) to obtain, for any sufficiently small fixed ǫ > 0, that(5.6) I CS (cid:0) F ǫA (cid:1) = lim sup n →∞ Z B F ǫA (cid:0) R n,k x (cid:1) µ ( dx ) . Step 2. In order to determine the limit in (5.6), we first note that for any positiveinteger q we have(5.7) E h(cid:13)(cid:13) W ǫγ ( R n,k x ) (cid:13)(cid:13) q i < ∞ . To see this and for later use as well, we now carry out a more precise estimate than thatof proving (2) of Lemma 1 in the following way.For the twisted Dirac operator Q A , we define a jn,k , b jn,k ∈ R by a jn,k + √− b jn,k = √ n q − √− nk (cid:0) λ j (cid:1) − p λ j , where λ j are eigenvalues of Q A as above. Then we set R n,k ˜ C ǫγ ( t ) α ⊗ E α = ∞ X j =1 a jn,k (cid:0) ˜ C ǫγ ( t ) α ⊗ E α , h j (cid:1) p h j ,R n,k ˜ C ǫγ ( t ) α ⊗ E α = ∞ X j =1 b jn,k (cid:0) ˜ C ǫγ ( t ) α ⊗ E α , h j (cid:1) p h j . Note that, for each x ∈ B and t ∈ [0 , R n,k defined by (5.4) gives rise toan element(5.8) R n,k x ǫγ ( t ) = d X α =1 h x, R n,k ˜ C ǫγ ( t ) α ⊗ E α i E α Itaru Mitoma and Seiki Nishikawa in the complexification of g , where R n,k ˜ C ǫγ ( t ) α ⊗ E α is defined by R n,k ˜ C ǫγ ( t ) α ⊗ E α = R n,k ˜ C ǫγ ( t ) α ⊗ E α + √− R n,k ˜ C ǫγ ( t ) α ⊗ E α . For convenience we denote the accompanying Gaussian random variables by(5.9) R n,k x ǫ,αγ ( t ) = h x, R n,k ˜ C ǫγ ( t ) α ⊗ E α i , R n,k x ǫ,αγ ( t ) = h x, R n,k ˜ C ǫγ ( t ) α ⊗ E α i and set R n,k x ǫ,αγ ( t ) = R n,k x ǫ,αγ ( t ) + √− R n,k x ǫ,αγ ( t ) . Now, noting that Z Z t · · · Z t r − d ( ¯ A + R n,k x ǫγ )( t ) d ( ¯ A + R n,k x ǫγ )( t ) · · · d ( ¯ A + R n,k x ǫγ )( t r )= r X m =0 X ≤ l Lemma 4. Let X i , i = 1 , , . . . , l , be a mean-zero Gaussian system. Then E (cid:2) X X · · · X l (cid:3) = 12 l l ! X σ ∈ S l E (cid:2) X σ (1) X σ (2) (cid:3) E (cid:2) X σ (3) X σ (4) (cid:3) · · · E (cid:2) X σ (2 l − X σ (2 l ) (cid:3) , where S l denotes the group of permutations of { , , . . . , l } . Then it follows from (5.12) together with Lemma 4 that E (cid:20)(cid:16)(cid:12)(cid:12) R ν n,k x ǫ,α l γ [ s l ] (cid:12)(cid:12) · · · (cid:12)(cid:12) R ν m n,k x ǫ,α lm γ [ s l m ] (cid:12)(cid:12)(cid:17) q (cid:21) ≤ (2 qm )! (cid:0) c ( ǫ ) / √ kρ (cid:1) qm qm ( qm )! (cid:12)(cid:12) t s l +1 l − t s l l (cid:12)(cid:12) q · · · (cid:12)(cid:12) t s lm +1 l m − t s lm l m (cid:12)(cid:12) q , from which we see that (5.11) is then dominated by(5.13) c ( A ) r − m lim n ,...,n r →∞ n − X s =0 · · · nr − X s r =0 ( (2 qm )! (cid:0) c ( ǫ ) / √ kρ (cid:1) qm qm ( qm )! ) / q · (cid:12)(cid:12) t s +11 − t s (cid:12)(cid:12) · · · (cid:12)(cid:12) t s r +1 r − t s r r (cid:12)(cid:12) ≤ c ( A ) r − m (cid:18) c ( ǫ ) √ kρ (cid:19) m (cid:26) (2 qm )!2 qm ( qm )! (cid:27) / q Z Z t · · · Z t r − dt dt · · · dt r ≤ c ( A ) r (cid:18) √ q √ kρ (cid:19) m √ m ! r ! , since ( qm )! ≤ ( m ! q m ) q , where c ( A ) = max { c ( A ) , c ( ǫ ) } .Consequently, summing up these estimates and denoting σ = d · c E , we obtain(5.14) E h(cid:13)(cid:13) W ǫγ ( R n,k x ) (cid:13)(cid:13) q i ≤ ∞ X r =0 (cid:0) σc ( A ) (cid:1) r r X m =0 r C m (cid:18) r qkρ (cid:19) m √ r ! ! q = ∞ X r =0 (cid:26) σc ( A ) (cid:18) r qkρ (cid:19)(cid:27) r √ r ! ! q < ∞ with the bound being independent of n . Step 3. Since B ∗ is dense in H , for each h ∈ H , there is a sequence { ξ n } ∞ n =1 of elementsin B ∗ such that lim n →∞ k h − ξ n k p = 0. As is well-known, h · , ξ n i then converges to h · , h i symptotic Expansion of the Chern-Simons Integral L ( B, R ; µ ) as n → ∞ . Hence, taking a subsequence if necessary, we may assume that h x, ξ n i converges to h x, h i for µ -almost every x ∈ B . Then we define for x ∈ B and h ∈ H (5.15) h x, h i = ( lim n →∞ h x, ξ n i if it exists,0 otherwise , as usual.It should be noted that, given ξ ∈ B ∗ , the operator R k defined by (5.5) takes ξ into H ; not into B ∗ in general. This leads us to define, by virtue of (5.15), elements in thecomplexification of g , associated with x ∈ B and ˜ C ǫγ ( t ) ∈ B ∗ , by R k x ǫγ ( t ) = d X α =1 h x, R k ˜ C ǫγ ( t ) α ⊗ E α i E α ,R k ˜ C ǫγ ( t ) α ⊗ E α = R k ˜ C ǫγ ( t ) α ⊗ E α + √− R k ˜ C ǫγ ( t ) α ⊗ E α , and the accompanying Gaussian random variables R k x ǫ,αγ ( t ) = h x, R k ˜ C ǫγ ( t ) α ⊗ E α i , R k x ǫ,αγ ( t ) = h x, R k ˜ C ǫγ ( t ) α ⊗ E α i in a manner similar to that in defining R n,k x ǫγ ( t ) and R n,k x ǫ,αγ ( t ), R n,k x ǫ,αγ ( t ) in (5.8) and(5.9), respectively. Then it is immediate from (5.12) that we have(5.16) E h(cid:12)(cid:12) R k x ǫ,αγ ( t ) − R k x ǫ,αγ ( s ) (cid:12)(cid:12) i ≤ c ( ǫ ) | t − s | . Hence, by virtue of the Kolmogorov-Delporte criterion [9], R k x ǫ,αγ ( t ) has a continuousmodification in t . Henceforth we denote such continuous modification by the same symbol R k x ǫ,αγ ( t ).Now, for any positive integer n , set T n = n X j =1 (cid:12)(cid:12)(cid:12)(cid:12) R k x ǫ,αγ (cid:16) j n (cid:17) − R k x ǫ,αγ (cid:16) j − n (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) . Then, since T n ≤ T n +1 , it is easy to see from (5.16) that E h lim n →∞ T n i = lim n →∞ E " n X j =1 (cid:12)(cid:12)(cid:12)(cid:12) R k x ǫ,αγ (cid:16) j n (cid:17) − R k x ǫ,αγ (cid:16) j − n (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) ≤ lim n →∞ n X j =1 E "(cid:12)(cid:12)(cid:12)(cid:12) R k x ǫ,αγ (cid:16) j n (cid:17) − R k x ǫ,αγ (cid:16) j − n (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) / ≤ lim n →∞ n X j =1 c ( ǫ ) (cid:12)(cid:12)(cid:12)(cid:12) j n − j − n (cid:12)(cid:12)(cid:12)(cid:12) ≤ c ( ǫ ) , Itaru Mitoma and Seiki Nishikawa which implies that lim n →∞ T n < ∞ µ -almost everywhere . Since R k x ǫ,αγ ( t ) is continuous in t almost surely, this implies that R k x ǫ,αγ ( t ) is of boundedvariation for all x ∈ B ′ ⊂ B with µ ( B ′ ) = 1. Therefore the Lebesgue-Stieltjes integral(5.17) Z Z t · · · Z t r − d ( ¯ A + R k x ǫγ )( t ) d ( ¯ A + R k x ǫγ )( t ) · · · d ( ¯ A + R k x ǫγ )( t r )is well-defined for all x ∈ B ′ ⊂ B with µ ( B ′ ) = 1. According to (4.3) and (4.4), we thendefine the stochastic holonomy given by R k x to be W ǫ,rγ ( R k x ) = ( (5.17) for x ∈ B ′ , x ∈ B \ B ′ ,W ǫγ ( R k x ) = I + ∞ X r =1 W ǫ,rγ ( R k x ) , and the associated Wilson line by F ǫA ( R k x ) = s Y j =1 Tr R j W ǫγ j ( R k x ) . The well-definedness of W ǫγ ( R k x ) can be seen as follows. First we note that(5.18) E "(cid:12)(cid:12)(cid:12)(cid:12) Z Z t · · · Z t r − d ( ¯ A α + R k x ǫ,α γ )( t ) · · · d ( ¯ A α r + R k x ǫ,α r γ )( t r ) (cid:12)(cid:12)(cid:12)(cid:12) q ≤ E " lim n ,...,n r →∞ (cid:12)(cid:12)(cid:12)(cid:12) n − X s =0 (cid:12)(cid:12) A α [ s ] + R k x ǫ,α γ [ s ] (cid:12)(cid:12) · · · nr − X s r =0 (cid:12)(cid:12) A α r [ s r ] + R k x ǫ,α r γ [ s r ] (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q ≤ lim n ,...,n r →∞ n − X s =0 · · · nr − X s r =0 E (cid:20)(cid:16)(cid:12)(cid:12) A α [ s ] + R k x ǫ,α γ [ s ] (cid:12)(cid:12) · · · · (cid:12)(cid:12) A α r [ s r ] + R k x ǫ,α r γ [ s r ] (cid:12)(cid:12)(cid:17) q (cid:21) / q ! q On the other hand, it is easy to see from (5.16) together with Lemma 4 that E h(cid:12)(cid:12) A α i [ s i ] + R k x ǫ,α i γ [ s i | ] (cid:12)(cid:12) m i ≤ c ( A , m ) (cid:12)(cid:12) t s i +1 i − t s i i (cid:12)(cid:12) m for any positive integer m , so that (5.18) is dominated by c ( ǫ ) (cid:18)Z Z t · · · Z t r − dt dt · · · dt r (cid:19) q . symptotic Expansion of the Chern-Simons Integral E "(cid:12)(cid:12)(cid:12)(cid:12) Z Z t · · · Z t r − d ( ¯ A α + R k x ǫ,α γ )( t ) · · · d ( ¯ A α r + R k x ǫ,α r γ )( t r ) (cid:12)(cid:12)(cid:12)(cid:12) q = lim n ,...,n r →∞ E "(cid:12)(cid:12)(cid:12)(cid:12) n − X s =0 (cid:0) A α [ s ] + R k x ǫ,α γ [ s ] (cid:1) · · · nr − X s r =0 (cid:0) A α r [ s r ] + R k x ǫ,α r γ [ s r ] (cid:1)(cid:12)(cid:12)(cid:12)(cid:12) q , which assures that the above estimates obtained for W ǫγ ( R n,k x ) in (5.10) through (5.14)also hold for W ǫγ ( R k x ) without essential change. In consequence, we obtain(5.20) E h(cid:13)(cid:13) W ǫγ ( R k x ) (cid:13)(cid:13) q i < ∞ , showing that W ǫγ ( R k x ) is well-defined for each x ∈ B . Step 4. Furthermore, since R νn,k ˜ C ǫγ ( t ) α ⊗ E α converges to R νk ˜ C ǫγ ( t ) α ⊗ E α in H as n → ∞ for ν = 1 , 2, it also follows from Lebesgue’s convergence theorem that(5.21) lim n →∞ E h(cid:13)(cid:13) W ǫγ ( R n,k x ) − W ǫγ ( R k x ) (cid:13)(cid:13) q i = 0 . Indeed, as in the estimation in (5.10) it holds that E h(cid:13)(cid:13) W ǫγ ( R n,k x ) − W ǫγ ( R k x ) (cid:13)(cid:13) q i ≤ ∞ X r =0 r X m =0 X ≤ l On the other hand, by an argument similar to that in obtaining (5.11), we see thateach term of the right side of (5.22) is dominated by c ( A ) r − m lim n ,...,n r →∞ n − X s =0 · · · nr − X s r =0 E (cid:20)(cid:16)(cid:12)(cid:12) t s +11 − t s (cid:12)(cid:12) · · · (cid:12)(cid:12) R ν k x ǫ,α l γ [ s l ] (cid:12)(cid:12) · · · · (cid:12)(cid:12) R ν j n,k x ǫ,α lj γ [ s l j ] − R ν j k x ǫ,α lj γ [ s l j ] (cid:12)(cid:12) · · · (cid:12)(cid:12) R ν m n,k x ǫ,α lm γ [ s l m ] (cid:12)(cid:12) · · · (cid:12)(cid:12) t s r +1 r − t s r r (cid:12)(cid:12)(cid:17) q (cid:21) / q , where it also holds as in (5.12) that(5.23) E h(cid:12)(cid:12) R ν j n,k x ǫ,α lj γ [ s l j ] − R ν j k x ǫ,α lj γ [ s l j ] (cid:12)(cid:12) i = (cid:13)(cid:13)(cid:13)(cid:0) R ν j n,k − R ν j k (cid:1) ˜ C ǫγ (cid:0) t s lj +1 l j (cid:1) α ⊗ E α − (cid:0) R ν j n,k − R ν j k (cid:1) ˜ C ǫγ (cid:0) t s lj l j (cid:1) α ⊗ E α (cid:13)(cid:13)(cid:13) p ≤ kρ c ( ǫ ) (cid:12)(cid:12) t s lj +1 l j − t s lj l j (cid:12)(cid:12) . Hence, by the same reasoning as in (5.13), we obtain that(5.24) E h(cid:12)(cid:12) B j (cid:12)(cid:12) q i / q ≤ c ( A ) r − m lim n ,...,n r →∞ n − X s =0 · · · nr − X s r =0 (cid:26) (2 qm )!( √ c ( ǫ ) / √ kρ ) qm qm ( qm )! (cid:27) / q · (cid:12)(cid:12) t s +11 − t s (cid:12)(cid:12) · · · (cid:12)(cid:12) t s r +1 r − t s r r (cid:12)(cid:12) ≤ c ( A ) r (cid:18) r qkρ (cid:19) m √ m ! r ! . Since each R νn,k ˜ C ǫγ ( t ) α ⊗ E α converges to R νk ˜ C ǫγ ( t ) α ⊗ E α in H as n → ∞ , it followsfrom the first identities in (5.12) and (5.23) combined with Lemma 4 thatlim n →∞ E (cid:20)(cid:16)(cid:12)(cid:12) t s +11 − t s (cid:12)(cid:12) · · · (cid:12)(cid:12) R ν k x ǫ,α l γ [ s l ] (cid:12)(cid:12) · · · · (cid:12)(cid:12) R ν j n,k x ǫ,α lj γ [ s l j ] − R ν j k x ǫ,α lj γ [ s l j ] (cid:12)(cid:12) · · · (cid:12)(cid:12) R ν m n,k x ǫ,α lm γ [ s l m ] (cid:12)(cid:12) · · · (cid:12)(cid:12) t s r +1 r − t s r r (cid:12)(cid:12)(cid:17) q (cid:21) = 0 . This, together with the estimates (5.23) and (5.24) with the bound independent of n , thenyields by Lebesgue’s convergence theorem that c ( A ) r − m lim n ,...,n r →∞ n − X s =0 · · · nr − X s r =0 E (cid:20)(cid:16)(cid:12)(cid:12) t s +11 − t s (cid:12)(cid:12) · · · (cid:12)(cid:12) R ν k x ǫ,α l γ [ s l ] (cid:12)(cid:12) · · · · (cid:12)(cid:12) R ν j n,k x ǫ,α lj γ [ s l j ] − R ν j k x ǫ,α lj γ [ s l j ] (cid:12)(cid:12) · · · (cid:12)(cid:12) R ν m n,k x ǫ,α lm γ [ s l m ] (cid:12)(cid:12) · · · (cid:12)(cid:12) t s r +1 r − t s r r (cid:12)(cid:12)(cid:17) q (cid:21) / q = 0 , so that lim n →∞ E (cid:20)(cid:12)(cid:12)(cid:12) D r,m (cid:2) R νn,k x, R νk x (cid:3)(cid:12)(cid:12)(cid:12) q (cid:21) / q = 0 . symptotic Expansion of the Chern-Simons Integral u + v ) m ≤ m ( u m + v m )for u, v ≥ 0, we have(5.25) E (cid:20)(cid:12)(cid:12)(cid:12) D r,m (cid:2) R νn,k x, R νk x (cid:3)(cid:12)(cid:12)(cid:12) q (cid:21) / q ≤ E "(cid:12)(cid:12)(cid:12)(cid:12) Z d ¯ A α ( t ) · · · Z t l − dR ν n,k x ǫ,α l γ ( t l ) · · · · Z t lm − dR ν m n,k x ǫ,α lm γ ( t l m ) · · · Z t r − d ¯ A α r ( t r ) (cid:12)(cid:12)(cid:12)(cid:12) q / q + E "(cid:12)(cid:12)(cid:12)(cid:12) Z d ¯ A α ( t ) · · · Z t l − dR ν k x ǫ,α l γ ( t l ) · · · · Z t lm − dR ν m k x ǫ,α lm γ ( t l m ) · · · Z t r − d ¯ A α r ( t r ) (cid:12)(cid:12)(cid:12)(cid:12) q / q ! . Recalling that the estimates in (5.10) through (5.14) are valid for both R n,k x and R k x ,and the bounds in the estimates (5.12) and (5.14) are independent of n , it follows from(5.25) and Lebesgue’s convergence theorem thatlim n →∞ ∞ X r =0 r X m =0 X ≤ l 5. Finally, taking into account of (5.3), we note that the following integrabilitycan be proved in a manner similar to that in obtaining the estimates described above.Namely, we have Lemma 5. For any positive integer N , E " ∞ X m = N F ǫ,mA ( R n,k x ) = O (cid:0) k − N/ (cid:1) , where O (cid:0) k − N/ (cid:1) means lim k →∞ k N/ (cid:12)(cid:12)(cid:12) O (cid:0) k − N/ (cid:1)(cid:12)(cid:12)(cid:12) < ∞ . Itaru Mitoma and Seiki Nishikawa Then Lemma 5 and the fact that Z B F A ( R k x ) µ ( dx ) = X m 3, an orthonormal basis of the Lie algebra g = su (2) with respect tothe inner product ( X, Y ) = − Tr XY for X, Y ∈ g . For simplicity, we also assume for the ǫ -regularized Wilson line (4.4) that A = 0, and write F ǫ ( x ) = Y j =1 Tr R W ǫγ j ( x ) . Step 1. Recalling (4.3), we begin with the evaluation of(6.1) E " Y j =1 Tr R W ǫ, γ j ( R k x ) . Writing briefly (cid:10) R k x, ˜ C ǫγ ( t ) α ⊗ E α (cid:11) by (cid:0) R k x αγ (cid:1) ( t ) , we see that (6.1) is equal to(6.2) E (cid:2) Tr R W ǫ, γ ( R k x ) ⊗ W ǫ, γ ( R k x ) (cid:3) = X α ,α ,β ,β =1 Tr E α E α ⊗ E β E β · E (cid:20)Z Z t d (cid:0) R k x α γ (cid:1) ( t ) d (cid:0) R k x α γ (cid:1) ( t ) Z Z τ d (cid:0) R k x β γ (cid:1) ( τ ) d (cid:0) R k x β γ (cid:1) ( τ ) (cid:21) . Then, by changing the order of taking sum and expectation, in a similar manner as in theproof of (5.19), we obtain(6.3) E (cid:20)Z Z t d (cid:0) R k x α γ (cid:1) ( t ) d (cid:0) R k x α γ (cid:1) ( t ) Z Z τ d (cid:0) R k x β γ (cid:1) ( τ ) d (cid:0) R k x β γ (cid:1) ( τ ) (cid:21) = lim n ,n →∞ m ,m →∞ n − X s =0 n − X s ( s )=0 m − X s =0 m − X s ( s )=0 E h(cid:16)(cid:0) R k x α γ (cid:1)(cid:0) t s +11 (cid:1) − (cid:0) R k x α γ (cid:1)(cid:0) t s (cid:1)(cid:17)(cid:16)(cid:0) R k x α γ (cid:1)(cid:0) t s ( s )+12 (cid:1) − (cid:0) R k x α γ (cid:1)(cid:0) t s ( s )2 (cid:1)(cid:17) · (cid:16)(cid:0) R k x β γ (cid:1)(cid:0) τ s +11 (cid:1) − (cid:0) R k x β γ (cid:1)(cid:0) τ s (cid:1)(cid:17)(cid:16)(cid:0) R k x β γ (cid:1)(cid:0) τ s ( s )+12 (cid:1) − (cid:0) R k x β γ (cid:1)(cid:0) τ s ( s )2 (cid:1)(cid:17)i . symptotic Expansion of the Chern-Simons Integral i = 1 , t s i ( s i − ) i = ( s i ( s i − ) = 0 ,t s i ( s i − ) − i + t s i − ( s i − ) i − / n i if s i ( s i − ) ≥ , and τ s i ( s i − ) i = ( s i ( s i − ) = 0 ,τ s i ( s i − ) − i + τ s i − ( s i − ) i − / m i if s i ( s i − ) ≥ , where s i ( s i − ) are non-negative integers and we use the convention such that s ( s ) = s , s ( s − ) = 1 and t = τ = 1.Writing for brevity j i = (cid:0) R k x α i γ (cid:1)(cid:0) t s i ( s i − )+1 i (cid:1) − (cid:0) R k x α i γ (cid:1)(cid:0) t s i ( s i − ) i (cid:1) if i ≤ , (cid:0) R k x β i − γ (cid:1)(cid:0) τ s i − ( s i − )+1 i − (cid:1) − (cid:0) R k x β i − γ (cid:1)(cid:0) τ s i − ( s i − ) i − (cid:1) if i > , we see from Lemma 4 that the right side of (6.3) is equal tolim n ,n →∞ m ,m →∞ n − X s =0 2 n − X s ( s )=0 2 m − X s =0 2 m − X s ( s )=0 X σ ∈ S E (cid:2) j σ (1) j σ (2) (cid:3) E (cid:2) j σ (3) j σ (4) (cid:3) = lim n ,n →∞ m ,m →∞ n − X s =0 2 n − X s ( s )=0 2 m − X s =0 2 m − X s ( s )=0 X σ ∈ S E (cid:2) j j σ (1)+2 (cid:3) E (cid:2) j j σ (2)+2 (cid:3) + T self = lim n ,n →∞ m ,m →∞ n − X s =0 2 n − X s ( s )=0 2 m − X s =0 2 m − X s ( s )=0 X σ ∈ S E h(cid:16)(cid:0) R k x α γ (cid:1)(cid:0) t s +11 (cid:1) − (cid:0) R k x α γ (cid:1)(cid:0) t s (cid:1)(cid:17) · (cid:16)(cid:0) R k x β σ (1) γ (cid:1)(cid:0) τ s σ (1) ( s σ (1) − )+1 σ (1) (cid:1) − (cid:0) R k x β σ (1) γ (cid:1)(cid:0) τ s σ (1) ( s σ (1) − ) σ (1) (cid:1)(cid:17)i × E h(cid:16)(cid:0) R k x α γ (cid:1)(cid:0) t s ( s )+12 (cid:1) − (cid:0) R k x α γ (cid:1)(cid:0) t s ( s )2 (cid:1)(cid:17) · (cid:16)(cid:0) R k x β σ (2) γ (cid:1)(cid:0) τ s σ (2) ( s σ (2) − )+1 σ (2) (cid:1) − (cid:0) R k x β σ (2) γ (cid:1)(cid:0) τ s σ (2) ( s σ (2) − ) σ (2) (cid:1)(cid:17)i + T self , where T self stands for the collection of self-linking terms containing E h(cid:16)(cid:0) R k x α γ (cid:1) ( t l +11 ) − (cid:0) R k x α γ (cid:1) ( t l ) (cid:17)(cid:16)(cid:0) R k x α γ (cid:1) ( t l +12 ) − (cid:0) R k x α γ (cid:1) ( t l ) (cid:17)i or E h(cid:16)(cid:0) R k x β γ (cid:1) ( τ l +11 ) − (cid:0) R k x β γ (cid:1) ( τ l ) (cid:17)(cid:16)(cid:0) R k x β γ (cid:1) ( τ l +12 ) − (cid:0) R k x β γ (cid:1) ( τ l ) (cid:17)i . Since R k x αγ i ( t ) and R k x βγ j ( t ) are independent if α = β , we then have E h(cid:16)(cid:0) R k x α γ (cid:1)(cid:0) t s +11 (cid:1) − (cid:0) R k x α γ (cid:1)(cid:0) t s (cid:1)(cid:17) Itaru Mitoma and Seiki Nishikawa · (cid:16)(cid:0) R k x β σ (1) γ (cid:1)(cid:0) τ s σ (1) ( s σ (1) − )+1 σ (1) (cid:1) − (cid:0) R k x β σ (1) γ (cid:1)(cid:0) τ s σ (1) ( s σ (1) − ) σ (1) (cid:1)(cid:17)i × E h(cid:16)(cid:0) R k x α γ (cid:1)(cid:0) t s ( s )+12 (cid:1) − (cid:0) R k x α γ (cid:1)(cid:0) t s ( s )2 (cid:1)(cid:17) · (cid:16)(cid:0) R k x β σ (2) γ (cid:1)(cid:0) τ s σ (2) ( s σ (2) − )+1 σ (2) (cid:1) − (cid:0) R k x β σ (2) γ (cid:1)(cid:0) τ s σ (2) ( s σ (2) − ) σ (2) (cid:1)(cid:17)i = δ α β σ (1) E h(cid:16)(cid:0) R k x α γ (cid:1)(cid:0) t s +11 (cid:1) − (cid:0) R k x α γ (cid:1)(cid:0) t s (cid:1)(cid:17) · (cid:16)(cid:0) R k x β σ (1) γ (cid:1)(cid:0) τ s σ (1) ( s σ (1) − )+1 σ (1) (cid:1) − (cid:0) R k x β σ (1) γ (cid:1)(cid:0) τ s σ (1) ( s σ (1) − ) σ (1) (cid:1)(cid:17)i × δ α β σ (2) E h(cid:16)(cid:0) R k x α γ (cid:1)(cid:0) t s ( s )+12 (cid:1) − (cid:0) R k x α γ (cid:1)(cid:0) t s ( s )2 (cid:1)(cid:17) · (cid:16)(cid:0) R k x β σ (2) γ (cid:1)(cid:0) τ s σ (2) ( s σ (2) − )+1 σ (2) (cid:1) − (cid:0) R k x β σ (2) γ (cid:1)(cid:0) τ s σ (2) ( s σ (2) − ) σ (2) (cid:1)i = E h(cid:16)(cid:0) R k x α γ (cid:1)(cid:0) t s +11 (cid:1) − (cid:0) R k x α γ (cid:1)(cid:0) t s (cid:1)(cid:17) · (cid:16)(cid:0) R k x α γ (cid:1)(cid:0) τ s σ (1) ( s σ (1) − )+1 σ (1) (cid:1) − (cid:0) R k x α γ (cid:1)(cid:0) τ s σ (1) ( s σ (1) − ) σ (1) (cid:1)(cid:17)i × E h(cid:16)(cid:0) R k x α γ (cid:1)(cid:0) t s ( s )+12 (cid:1) − (cid:0) R k x α γ (cid:1)(cid:0) t s ( s )2 (cid:1)(cid:17) · (cid:16)(cid:0) R k x α γ (cid:1)(cid:0) τ s σ (2) ( s σ (2) − )+1 σ (2) (cid:1) − (cid:0) R k x α γ (cid:1)(cid:0) τ s σ (2) ( s σ (2) − ) σ (2) (cid:1)(cid:17)i . Furthermore, since R k x αγ i ( t ) and R k x βγ i ( t ) are identically distributed if α = β , we obtain(6.4) (6.3) = Z Z t Z Z τ X σ ∈ S dE (cid:2)(cid:0) R k x α γ (cid:1) ( t ) (cid:0) R k x α γ (cid:1) ( τ σ (1) ) (cid:3) · dE (cid:2)(cid:0) R k x α γ (cid:1) ( t ) (cid:0) R k x α γ (cid:1) ( τ σ (2) ) (cid:3) + T self = Z Z t Z Z τ X σ ∈ S dE (cid:2)(cid:0) R k x αγ (cid:1) ( t ) (cid:0) R k x αγ (cid:1) ( τ σ (1) ) (cid:3) · dE (cid:2)(cid:0) R k x αγ (cid:1) ( t ) (cid:0) R k x αγ (cid:1) ( τ σ (2) ) (cid:3) + T self . Consequently, (6.2), (6.3) and (6.4) yield for each α = 1 , , E " Y j =1 Tr R W ǫ, γ j ( R k x ) = Tr X α ,α =1 E α E α ⊗ E α E α × Z Z t Z Z τ X σ ∈ S dE (cid:2)(cid:0) R k x αγ (cid:1) ( t ) (cid:0) R k x αγ (cid:1) ( τ σ (1) ) (cid:3) dE (cid:2)(cid:0) R k x αγ (cid:1) ( t ) (cid:0) R k x αγ )( τ σ (2) ) (cid:3) + T self . symptotic Expansion of the Chern-Simons Integral Z Z τ X σ ∈ S dE (cid:2)(cid:0) R k x αγ (cid:1) ( t ) (cid:0) R k x αγ (cid:1) ( τ σ (1) ) (cid:3) dE (cid:2)(cid:0) R k x αγ (cid:1) ( t ) (cid:0) R k x αγ (cid:1) ( τ σ (2) ) (cid:3) = Z Z dE (cid:2)(cid:0) R k x αγ (cid:1) ( t ) (cid:0) R k x αγ (cid:1) ( τ ) (cid:3) dE (cid:2)(cid:0) R k x αγ (cid:1) ( t ) (cid:0) R k x αγ (cid:1) ( τ ) (cid:3) and Z Z t Z Z dE (cid:2)(cid:0) R k x αγ (cid:1) ( t ) (cid:0) R k x αγ (cid:1) ( τ ) (cid:3) dE (cid:2)(cid:0) R k x αγ (cid:1) ( t ) (cid:0) R k x αγ (cid:1) ( τ ) (cid:3) = Z Z t Z Z dE (cid:2)(cid:0) R k x αγ (cid:1) ( t ) (cid:0) R k x αγ (cid:1) ( τ ) (cid:3) dE (cid:2)(cid:0) R k x αγ (cid:1) ( t ) (cid:0) R k x αγ (cid:1) ( τ ) (cid:3) , we see from (6.5) that E " Y j =1 Tr R W ǫ, γ j ( R k x ) = Tr X α =1 E α ⊗ E α ! × Z Z Z Z dE (cid:2)(cid:0) R k x αγ (cid:1) ( t ) (cid:0) R k x αγ (cid:1) ( τ ) (cid:3) dE (cid:2)(cid:0) R k x αγ (cid:1) ( t ) (cid:0) R k x αγ (cid:1) ( τ ) (cid:3) + T self = Tr X α =1 E α ⊗ E α ! E (cid:2)(cid:0) R k x αγ (cid:1) (1) (cid:0) R k x αγ (cid:1) (1) (cid:3) + T self . On the other hand, it follows from (3.5), (4.1) and (5.5) that E (cid:2)(cid:0) R k x αγ (cid:1) (1) (cid:0) R k x αγ (cid:1) (1) (cid:3) = E (cid:2)(cid:10) x, R k ˜ C ǫγ (1) α ⊗ E α (cid:11)(cid:10) x, R k ˜ C ǫγ (1) α ⊗ E α (cid:11)(cid:3) = (cid:16) R k ˜ C ǫγ (1) α ⊗ E α , R k ˜ C ǫγ (1) α ⊗ E α (cid:17) p = (cid:16) R k ( ˜ C ǫγ (1) α ⊗ E α , , (cid:0) Q (cid:1) p R k ( ˜ C ǫγ (1) α ⊗ E α , (cid:17) + = − √− k (cid:16) ( C ǫγ (1) α ⊗ E α , , Q − ( C ǫγ (1) α ⊗ E α , (cid:17) + = − √− k (cid:0) C ǫγ (1) α ⊗ E α , ω α ⊗ E α (cid:1) , where ω = 1-form part of Q − ( C ǫγ (1) , . Recall that, as seen in Proposition 3, ∗ C ǫγ (1) α is a representative of the compactPoincar´e dual of γ extended by zero to all of S , and the second de Rham cohomology H DR ( S ) = { } , so that we have dω α = ∗ C ǫγ (1) α , since ∗ C ǫγ (1) α is closed and exact.Hence, for each α = 1 , , (cid:0) C ǫγ (1) α , ω α (cid:1) Itaru Mitoma and Seiki Nishikawa yields the linking number L ( γ , γ ) of loops γ and γ , provided that ǫ is sufficiently smallso that the ǫ -tubular neighborhoods of γ j are not intersected (see [6] for details). Also, byinvestigating deformed Wilson loops, it has been proved by Hahn [14] that T self = 0 fornon-self-intersected links. Step 2. We proceed to evaluate m -th order coefficients of the expansion, that is,(6.6) E (cid:2) Tr R W ǫ,m γ ( R k x ) Tr R W ǫ,m γ ( R k x ) (cid:3) , where m = m + m . Note that if m is odd, then (6.6) is equal to zero. Even if m iseven, when m = m , the term (6.6) belongs to T self , where T self denotes the collection ofself-linking terms containing the limits of E h · · · (cid:16)(cid:0) R k x α γ (cid:1) ( t l +11 ) − (cid:0) R k x α γ (cid:1) ( t l ) (cid:17)(cid:16)(cid:0) R k x α γ (cid:1) ( t l ′ +12 ) − (cid:0) R k x α γ (cid:1) ( t l ′ ) (cid:17)i or E h · · · (cid:16)(cid:0) R k x β γ (cid:1) ( τ l +11 ) (cid:17) − (cid:0) R k x β γ (cid:1) ( τ l ) (cid:17)(cid:16)(cid:0) R k x β γ (cid:1) ( τ l ′ +12 ) − (cid:0) R k x β γ (cid:1) ( τ l ′ ) (cid:17)i as (cid:12)(cid:12) t l +1 j − t lj (cid:12)(cid:12) , (cid:12)(cid:12) τ l ′ +1 j ′ − τ l ′ j ′ (cid:12)(cid:12) → 0. Hence it suffices to evaluate the case with m = m .Consequently, (6.6) is equal to(6.7) E (cid:2) Tr R W ǫ,m γ ( R k x ) ⊗ W ǫ,m γ ( R k x ) (cid:3) = X α ,α ,...,α m =1 3 X β ,β ,...,β m =1 Tr E α E α · · · E α m ⊗ E β E β · · · E β m × E (cid:20)Z Z t · · · Z t m − Z Z τ · · · Z τ m − d (cid:0) R k x α γ (cid:1) ( t ) d (cid:0) R k x α γ (cid:1) ( t ) · · ·· d (cid:0) R k x α m γ (cid:1) ( t m ) d (cid:0) R k x β γ (cid:1) ( τ ) d (cid:0) R k x β γ (cid:1) ( τ ) · · · d (cid:0) R k x β m γ (cid:1) ( τ m ) (cid:21) + T self . Then writing for brevity j i = (cid:0) R k x α i γ (cid:1) ( t i ) if i ≤ m , (cid:0) R k x β i − m γ (cid:1) ( τ i − m ) if i > m , we obtain, in a manner similar to the derivation of (6.3), that(6.8) E (cid:20)Z Z t · · · Z t m − Z Z τ · · · Z τ m − d (cid:0) R k x α γ (cid:1) ( t ) d (cid:0) R k x α γ (cid:1) ( t ) · · ·· d (cid:0) R k x α m γ (cid:1) ( t m ) d (cid:0) R k x β γ (cid:1) ( τ ) d (cid:0) R k x β γ (cid:1) ( τ ) · · · d (cid:0) R k x β m γ (cid:1) ( τ m ) (cid:21) = Z Z t · · · Z t m − Z Z τ · · · Z τ m − m ! 2 m X σ ∈ S m dE (cid:2) j σ (1) j σ (2) (cid:3) · dE (cid:2) j σ (3) j σ (4) (cid:3) · · · dE (cid:2) j σ ( m − j σ ( m ) (cid:3) . symptotic Expansion of the Chern-Simons Integral σ ( i − 1) and σ ( i ) both in { , , . . . , m } or { m + 1 , m + 2 , . . . , m } belong to T self , it follows that(6.8) = Z Z t · · · Z t m − Z Z τ · · · Z τ m − X σ ∈ S m dE (cid:2) j j m + σ (1) (cid:3) dE (cid:2) j j m + σ (2) (cid:3) · · · dE (cid:2) j m j m + σ ( m ) (cid:3) + T self = Z Z t · · · Z t m − Z Z τ · · · Z τ m − X σ ∈ S m dE (cid:2)(cid:0) R k x α γ (cid:1) ( t ) (cid:0) R k x β σ (1) γ (cid:1) ( τ σ (1) ) (cid:3) · dE (cid:2)(cid:0) R k x α γ (cid:1) ( t ) (cid:0) R k x β σ (2) γ (cid:1) ( τ σ (2) ) (cid:3) · · · dE (cid:2)(cid:0) R k x α m γ (cid:1) ( t m ) (cid:0) R k x β σ ( m γ (cid:1) ( τ σ ( m ) ) (cid:3) + T self . Again, since (cid:0) R k x αγ (cid:1) ( t ) and (cid:0) R k x βγ (cid:1) ( t ) are independent and identically distributed if α = β , we have E (cid:2)(cid:0) R k x α j γ (cid:1) ( t j ) (cid:0) R k x β σ ( j ) γ (cid:1) ( τ σ ( j ) ) (cid:3) = δ α j β σ ( j ) E (cid:2)(cid:0) R k x α j γ (cid:1) ( t j ) (cid:0) R k x α j γ (cid:1) ( τ σ ( j ) ) (cid:3) = δ α j β σ ( j ) E (cid:2)(cid:0) R k x αγ (cid:1) ( t j ) (cid:0) R k x αγ (cid:1) ( τ σ ( j ) ) (cid:3) from which we see that the right side of (6.8) is equal to(6.9) Z Z t · · · Z t m − Z Z τ · · · Z τ m − X σ ∈ S m m Y j =1 δ α j β σ ( j ) dE (cid:2)(cid:0) R k x αγ (cid:1) ( t ) (cid:0) R k x αγ (cid:1) ( τ σ (1) ) (cid:3) · dE (cid:2)(cid:0) R k x αγ (cid:1) ( t ) (cid:0) R k x αγ (cid:1) ( τ σ (2) ) (cid:3) · · · dE (cid:2)(cid:0) R k x αγ (cid:1) ( t m ) (cid:0) R k x αγ (cid:1) ( τ σ ( m ) ) (cid:3) + T self . It then follows from (6.7), (6.8) and (6.9) that(6.10) E (cid:2) Tr R W ǫ,m γ ( R k x ) Tr R W ǫ,m γ ( R k x ) (cid:3) = X α ,α ,...,α m =1 Tr E α E α · · · E α m ⊗ E α E α · · · E α m × Z Z t · · · Z t m − Z Z τ · · · Z τ m − X σ ∈ S m dE (cid:2)(cid:0) R k x αγ (cid:1) ( t ) (cid:0) R k x αγ (cid:1) ( τ σ (1) ) (cid:3) · dE (cid:2)(cid:0) R k x αγ (cid:1) ( t ) (cid:0) R k x αγ (cid:1) ( τ σ (2) ) (cid:3) · · · dE (cid:2)(cid:0) R k x αγ (cid:1) ( t m ) (cid:0) R k x αγ (cid:1) ( τ σ ( m ) ) (cid:3) + T self . Now, noting that Z Z τ · · · Z τ m − X σ ∈ S m dE (cid:2)(cid:0) R k x αγ (cid:1) ( t ) (cid:0) R k x αγ (cid:1) ( τ σ (1) ) (cid:3) Itaru Mitoma and Seiki Nishikawa · dE (cid:2) R k x αγ (cid:1) ( t ) (cid:0) R k x αγ (cid:1) ( τ σ (2) ) (cid:3) · · · dE (cid:2)(cid:0) R k x αγ (cid:1) ( t m ) (cid:0) R k x αγ (cid:1) ( τ σ ( m ) ) (cid:3) = Z Z · · · Z dE (cid:2)(cid:0) R k x αγ (cid:1) ( t ) (cid:0) R k x αγ (cid:1) ( τ ) (cid:3) dE (cid:2)(cid:0) R k x αγ (cid:1) ( t ) (cid:0) R k x αγ (cid:1) ( τ ) (cid:3) · · · dE (cid:2)(cid:0) R k x αγ (cid:1) ( t m ) (cid:0) R k x αγ (cid:1) ( τ m ) (cid:3) , and for any σ ∈ S m Z Z t · · · Z t m − Z Z · · · Z dE (cid:2) R k x αγ (cid:1) ( t ) (cid:0) R k x αγ (cid:1) ( τ ) (cid:3) · dE (cid:2)(cid:0) R k x αγ (cid:1) ( t ) (cid:0) R k x αγ (cid:1) ( τ ) (cid:3) · · · dE (cid:2) R k x αγ (cid:1) ( t m ) (cid:0) R k x αγ (cid:1) ( τ m ) (cid:3) = Z Z t · · · Z t m − Z Z · · · Z dE (cid:2) R k x αγ (cid:1) ( t σ (1) ) (cid:0) R k x αγ (cid:1) ( τ ) (cid:3) · dE (cid:2)(cid:0) R k x αγ (cid:1) ( t σ (2) ) (cid:0) R k x αγ (cid:1) ( τ ) (cid:3) · · · dE (cid:2)(cid:0) R k x αγ (cid:1) ( t σ ( m ) ) (cid:0) R k x αγ (cid:1) ( τ m ) (cid:3) , we find from (6 . 10) that for each α = 1 , , E (cid:2) Tr R W ǫ,m γ ( R k x ) Tr R W ǫ,m γ ( R k x ) (cid:3) = Tr X α =1 E α ⊗ E α ! m m ! × Z Z t · · · Z t m − Z Z · · · Z X σ ∈ S m dE (cid:2)(cid:0) R k x αγ (cid:1) ( t σ (1) ) (cid:0) R k x αγ (cid:1) ( τ ) (cid:3) · dE (cid:2)(cid:0) R k x αγ (cid:1) ( t σ (2) ) (cid:0) R k x αγ (cid:1) ( τ ) (cid:3) · · · dE (cid:2)(cid:0) R k x αγ (cid:1) ( t σ ( m ) ) (cid:0) R k x αγ (cid:1) ( τ m ) (cid:3) + T self = Tr X α =1 E α ⊗ E α ! m m ! × Z Z · · · Z Z Z · · · Z dE (cid:2)(cid:0) R k x αγ (cid:1) ( t ) (cid:0) R k x αγ (cid:1) ( τ ) (cid:3) · dE (cid:2)(cid:0) R k x αγ (cid:1) ( t ) (cid:0) R k x αγ (cid:1) ( τ ) (cid:3) · · · dE (cid:2)(cid:0) R k x αγ (cid:1) ( t m ) (cid:0) R k x αγ (cid:1) ( τ m ) (cid:3) + T self = Tr X α =1 E α ⊗ E α ! m m ! E (cid:2)(cid:0) R k x αγ (cid:1) (1) (cid:0) R k x αγ (cid:1) (1) (cid:3) m + T self . Summing up the above argument together with Lebesgue’s convergence theorem guar-anteed by an estimate similar to that in the proof of (2) of Lemma 1, we finally obtain I CS ( F ǫ ) = E (cid:2) F ǫ ( R k x ) (cid:3) = E " Y j =1 Tr R W ǫγ j ( R k x ) symptotic Expansion of the Chern-Simons Integral 37= (Tr I ) + ∞ X n =1 Tr X α =1 E α ⊗ E α ! n n ! E h(cid:0) R k x αγ (cid:1) (1) (cid:0) R k x αγ (cid:1) (1) i n + T self . Step 3. Now, noting that an orthonormal basis of su (2) is given by E = 1 √ (cid:20) √− −√− (cid:21) , E = 1 √ (cid:20) − 11 0 (cid:21) , E = 1 √ (cid:20) √− √− (cid:21) , so that E ⊗ E = 12 − − , E ⊗ E = 12 − − ,E ⊗ E = 12 − 10 0 − − − , we have X α =1 E α ⊗ E α = 12 − − − − . Since the eigenvalues of 2 P E α ⊗ E α are − , − , − , 3, we obtainTr X α =1 E α ⊗ E α ! n = ( − n + ( − n + ( − n + 3 n n . Consequently, we have I CS ( F ǫ ) = E (cid:2) F ǫ ( R k x ) (cid:3) = (Tr I ) + ∞ X n =1 Tr X α =1 E α ⊗ E α ! n n ! E h(cid:0) R k x αγ (cid:1) (1) (cid:0) R k x αγ (cid:1) (1) i n + T self = 4 + ∞ X n =1 ( − n + ( − n + ( − n + 3 n n n ! (cid:18) − √− k L ( γ , γ ) (cid:19) n + T self = 4 + ∞ X n =1 √− n { ( − n + ( − n + ( − n + 3 n } (4 k ) n n ! L ( γ , γ ) n + T self = 3 e −√− L ( γ ,γ ) / k + e √− L ( γ ,γ ) / k + T self , where L ( γ , γ ) = the linking number of loops γ and γ . Itaru Mitoma and Seiki Nishikawa Acknowledgment . During the preparation of the paper the second-named author stayed atUniversity of Brest, Technical University of Berlin and Chinese University of Hong Kong. Hewould like to thank, in particular, Professors Paul Baird, Udo Simon, Tom Wan and Thomas Aufor their hospitalities. References [1] S. Albeverio and J. Sch¨afer, Abelian Chern-Simons theory and linking numbers via oscillatoryintegrals, J. Math. Phys. 36 (1995), 2157–2169.[2] S. Albeverio and A. 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Phys.77, Oxford Science Publications, Oxford University Press, New York, 1989.Itaru MitomaDepartment of MathematicsSaga University840-8502 SagaJapan E-mail address : [email protected] Seiki NishikawaMathematical InstituteTohoku University980-8578 SendaiJapan E-mail address ::