Asymptotic Expansion of Perturbative Chern-Simons Theory via Wiener Space

Sergio Albeverio, Itaru Mitoma

no. 322

Diese Arbeit ist mit Unterstützung des von der Deutschen Forschungs- gemeinschaft getragenen Sonderforschungsbereiches 611 an der Univer- sität Bonn entstanden und als Manuskript vervielfältigt worden. Bonn, Februar 2007 Asymptotic Expansion of Perturbative Chern-Simons Theory via Wiener Space

Sergio Albeverio Itaru Mitoma

Abstract The Chern-Simons integral is divided into a sum of finitely many resp: infinitely many contributions. A mathematical meaning is given to the “ finite part ” and an asymptotic estimate of the other part is given, using the abstract Wiener space setting. The latter takes the form of an asymptotic expansion in powers of a charge, using the infinite dimensional Malliavin-Taniguchi formula for a change of variables.

Contents

1 Introduction 1

2 Definitions and Results 3 2.1 Regularized Determinant ...... 6 2.2 Holonomy and the Poincar´eDual ...... 7 2.3 Definitions of “finite part” and “asymptotic estimate” ...... 12 2.4 Integrabilities and Theorems ...... 19

3 Proofs of Lemmas and Theorems 23 3.1 Proofs of Lemmas ...... 24 3.2 Proof of Theorem 1 ...... 31 3.3 Proof of Theorem 2 ...... 33

1 Introduction

After seminal work by Schwarz [39] and Witten [42], various papers have been published concerning the mathematical establishment of relations between Chern-Simons integrals, topological invariants of 3-manifolds and knot invariants ( [1, 2, 3, 6, 7, 9, 10, 11, 12, 13, 14, 20, 23, 28]).

2000 Mathematics Subject Classification. Primary 57R56; Secondary 28C70, 57M27. Key words and phrases. Chern-Simons integral,asymptotic expansion, abstract Winer space, Malliavin- Taniguchi formula, linking number. ∗ Partly supported by the Grant-in-Aid for Scientific Research (C) of the Japan Society for the Promotion of Science, No. 17540124.

1 Perturbative Expansion of Chern-Simons Theory 2

The first author and Sch¨aferhave constructed the integral and the corresponding invariants (linking numbers) for the abelian case by the technique of infinite dimensional Fresnel integrals [1]. A similar representation was given subsequently by Leukert and Sch¨afer,using methods of white noise analysis. The extension to the representation to the non-abelian case for R3 (resp. S2 × R) has been provided by using again methods of white noise analysis by the first author and Sengupta [2], ( who used a setting similar to Fr¨ohlich-King [20]). Recently, along this line, Hahn has defined rigorously and computed the Wilson loop variables for Non-Abelian Chern- Simons theory[24, 25]. See also the survey papers [?] and [5]. Hahn subsequently managed to extend the results to the case of S2 × S1, using “torus gauge” [26]. In the abelian case, some alternative methods for defining rigorously the integrals, via determinants and limits of simplicial approximation [1, 37] have been developed. Guadagnini, Martellini, Mintchev and Bar-Natan [23, 11, 13] (see also, e.g., [6, 9, 10, 12, 14] for further development) developed heuristically an asymptotic expansion of the Chern- Simons integral in powers of the relevant charge parameter, getting the relations with other topological invariants (Vassiliev invariants). First rigorous mathematical results are in [33, 34], who used methods of stochastic analysis, in particular a formula by Malliavin- Taniguchi [32] on the change of variables in infinite dimensional analysis. The present paper also uses such methods and an idea of Itˆoconcerning Feynman path integrals [27]. Let M be a compact oriented 3-dimensional manifold , G a skew symmetric matrix algebra, which is a linear space with the inner product (X,Y ) = Tr XY ∗ = − Tr XY , where Tr denotes the trace. Let A be the collection of all G-valued 1-forms. Let F be a complex-valued map on A. Then the Chern-Simons integral of F (A) is heuristically equal to Z F (A)eL(A)D(A), A where Z ik n ^ 2 ^ ^ o L(A) = − Tr A dA + A A A . 4π M 3 The parameter k, called charge, is a positive integer. D(A) is a heuristic “flat measure”. 2 Let us expand heuristically in powers of the “ 3 -part” under above integral, separating it in a “finite part” which consists of the terms up to order N in k and the rest :

Z 1 F (A)eL(A)D(A) Z A Z Z Z 1 XN h ik ^ in ik 2 ^ ^ o = F (A) exp − Tr A dA (− Tr A A A)r/r! D(A) Z 4π 4π 3 r=0 A M M Z Z Z 1 h ik ^ in X∞ ik 2 ^ ^ o + F (A) exp − Tr A dA (− Tr A A A)r/r! D(A). Z 4π 4π 3 A M r=N+1 M

Z denotes the heuristic normalization constant i.e. Z Z h ik ^ i Z = exp − Tr A dA D(A). A 4π M Perturbative Expansion of Chern-Simons Theory 3

Based on the perturbative formulation of the Chern-Simons integral [9, 13, 23] and the idea of Itˆo[27] for the Feynman path integral, we will give a meaning to the finite part and an asymptotic estimate of the “infinite part”, letting k tend to +∞, in an abstract Wiener space setting by using a formula for a change of variables in such a space(Theorems 1 and 2). We shall give the concrete calculation of the coefficients of the expansion in a further paper.

Throughout this paper, as integrand we take the typical exampleR of gauge invariant ob- Q A s γj servables (holonomy operators) such that F (A) = j=1 Tr P e , where γj, j = 1, 2, ...s are closed oriented loops, (these are called Wilson loop variables in the physics literature). In Section 2, we shall give the definitions and results. To derive rigorously the invariants first derived heuristically by Witten, we consider the ²-regularization of the Wilson line, following [42]. In doing this, we also need a regularization of a relevant determinant and this uses methods of the theory of super fields. In Section 3, we give the proofs of the results stated in Section 2. In these proofs an intermediate S-operator enters, following a technique by [27]. √ −π √ π In the sequel z denotes the branch of the root of z²C where 2 < arg z < 2 .

2 Definitions and Results

We consider the perturbative formulation of the Chern-Simons integral [9, 13, 23] and follow the method of super fields. Let A0 be a critical point of the “Chern-Simons action” L(A), i.e.

dA0 + A0 ∧ A0 = 0.

Let us recall the relations between external fields and the BRS operator δ (see e.g. [9, 13, 23] for background concepts). Let φ be a bosonic 3-form, c a fermionic 0-form and bc a fermionic 3-form. The BRS operator laws are 1 δA = −D c, δc = [c, c], A 2 δbc = iφ, δφ = 0,

where DA = dA0 + [A, ·] and dA0 is the covariant exterior derivative such that dA0 = Pd u d + [A0, ·]. Let Eu, u = 1, 2, ..., d be the orthonormal basis of G,A = u=1 A · Eu and Pd u P i V j B = u=1 B · Eu. We set [A, B] = i,j[EiEj − EjEi]A B . For simplicity, we assume

Assumption 1. A0 is isolated and irreducible, i.e. the whole de Rham cohomology vanishes, i.e.

∗ (2.1) H (M, dA0 ) = 0.

(see [9, 13]). Perturbative Expansion of Chern-Simons Theory 4

For the rest of this paper we will fix an auxiliary Riemannian metric g on M. By ∗ ∗ we will denote the corresponding Hodge ∗-operator and by (dA0 ) the adjoint operator of dA0 (with respect to the scalar product on A which is induced by g). ∗ Then we fix the Lorentz gauge such that (dA0 ) A = 0. Define Z k V (A) = Tr bc ∗ dA0 ∗ A, 2π M p p Let c0 = k/2πc and bc0 = ∗ k/2πbc. (see [23, p. 578]). Since Z ik n ^ 2 ^ ^ o L(A0 + A) = L(A0) − Tr A dA0 A + A A A 4π M 3 and Z k n o δV (A) = Tr iφ ∗ dA0 ∗ A − bc ∗ dA0 ∗ DAc , 2π M the Lorentz gauge fixed path integral form of the heuristic Chern-Simons integral is given by

Z Z Z Z h i 1 0 0 D(A)D(φ)D(bc )D(c )F (A0 + A) exp L(A0 + A) − δV (A) Zk A Φ Cb0 C0 Z Z Z Z h i (2.2) 1 0 0 = D(A)D(φ)D(bc )D(c )F (A0 + A) exp L(A0) Zk A Φ Cb0 C0 h Z ^ ^ Z i ik 2 0 0 × exp ik((A, φ),QA0 (A, φ))+ − Tr A A A + Tr bc dA0 ∗ DAc , 4π M 3 M where heuristically

Z Z Z Z 0 0 Zk = D(A)D(φ)D(bc )D(c ) A Φ Cb0 C0 h Z i 0 0 exp L(A0) + ik((A, φ),QA0 (A, φ))+ + Tr bc dA0 ∗ dA0 c . M Here Ωn is the space of G-valued n-forms with the inner product Z ^ (ω1, ω2)0 = − Tr ω1 ∗ω2 M and ((A, φ), (B, ϕ))+ = (A, B)0 + (φ, ϕ)0 2 2 1 3 denotes the inner product of the Hilbert space L (Ω+) = L (Ω ⊕ Ω ). We shall set

p p (2.3) k · k0 = (·, ·)0 and k · k+ = (·, ·)+. Perturbative Expansion of Chern-Simons Theory 5

1 Further QA0 = 4π (∗dA0 + dA0 ∗)J, where Jφ = −φ if φ is a zero or a three form and Jφ = φ if φ is a one or a two form. We note that the benefit of the above formulation is that the Lorentz gauge constraint ∗ (dA0 ) A = 0 is released in the domain of the path integral representation. 2 0 Set ∆0 = ∗dA0 ∗ dA0 , i.e. ∆0 is the Laplacian in L (Ω ). Let {νj, ξj, j = 1, 2, ···} be 2 0 the eigensystem of ∆0 in L (Ω ), X∞ X∞ 0 0 0 0 bc = ξjbcj and c = ξjcj. j=1 j=1 Since

Z 0 0 0 0 Tr bc dA0 ∗ DAc = −(bc , ∗dA0 ∗ DAc )0 M 0 0 0 0 = −(bc , ∗dA0 ∗ dA0 c )0 − (dA0 bc , [A, c ])0, we have heuristically Z Z h Z i 0 0 0 0 D(bc )D(c ) exp Tr bc dA0 ∗ DAc = det ∗dA0 ∗ DA b0 0 C C ¯ M ¯ ¯ ¯ ¯ a11 a12 · a1n · ¯ ¯ ¯ Y∞ ¯ a21 a22 · a2n · ¯ ¯ ¯ = νj · ¯ ····· ¯ ¯ ¯ j=1 ¯ an1 an2 · ann · ¯ ¯ ····· ¯

= det ∆0 detR ∗dA0 ∗ DA, where Z 1 ^ aii = 1 − Tr dA0 ξi ∗[A, ξi] , i = 1, 2, ··· νi M and Z 1 ^ aij = − Tr dA0 ξi ∗[A, ξj] , i, j = 1, 2, ··· . νi M Then considering Z Z h Z i 0 0 0 0 D(bc )D(c ) exp Tr bc dA0 ∗ dA0 c = det ∆0, Cb0 C0 M we arrive at the perturbative heuristic formulation of the normalized Chern-Simons inte- gral (2.2) such that

Z Z 1 D(A)D(φ)F (A + A) Z 0 (2.4) 1,k A Φ Z h ik 2 ^ ^ i × detR ∗dA0 ∗ DA exp ik((A, φ),QA0 (A, φ))+ − Tr A A A , 4π M 3 Perturbative Expansion of Chern-Simons Theory 6 where heuristically

Z Z h i

(2.5) Z1,k = D(A)D(φ) exp ik((A, φ),QA0 (A, φ))+ . A Φ

2.1 Regularized Determinant

To provide mathematical meaning to this, we have first of all to regularize the formal determinant detR ∗dA0 ∗ DA.

Since QA0 is self-adjoint and elliptic, QA0 has pure point spectrum [8]. Let µi, ei = A Pd u,A φ Pd u,φ (ei = u=1 ei · Eu, ei = u=1 ei · Eu), i = 1, 2, ··· be the eigenvalues resp. eigenvec- 2 tors of QA0 in L (Ω+). By the Assumption 1, the eigenvectors form a CONS(Complete 2 Orthonormal System) of L (Ω+). Define Z ^ ` 1 A aR,ij = − Tr ridA0 ξi ∗[e` , rjξj]. νi M

Since M is compact, we can choose appropriate real numbers rj > 0, j = 1, 2, ... such that

X ` (2.6) | aR,ij |< +∞, i,j and

X∞ | rj |< +∞. j=1

Then for smooth A²A, we consider instead of detR ∗dA0 ∗ DA, the regularized deter- minant defined by

detReg(A) = lim detm,R ∗dA ∗ DA, m→∞ 0 where ¯ ¯ ¯ R R R ¯ ¯ a11(A) a12(A) · a1m(A) ¯ ¯ R R R ¯ ¯ a21(A) a22(A) · a2m(A) ¯ detm,R ∗dA0 ∗ DA = ¯ ¯ , ¯ ···· ¯ ¯ R R R ¯ am1(A) am2(A) · amm(A) X∞ R ` aii (A) = 1 + ((A, φ), e`)+aR,ii `=1 and Perturbative Expansion of Chern-Simons Theory 7

X∞ R ` aij(A) = ((A, φ), e`)+aR,ij. `=1 We call (A, φ) smooth if for every natural number b,

2 2b k(A, φ)kb = ((A, φ),QA0 (A, φ))+ < +∞. Then for smooth (A, φ), these series converge and the regularized determinant is well defined since X X∞ ` |((A, φ), e`)+aR,ij| i,j `=1 X∞ X 1 ` ≤ k(A, φ)kN0 N |aR,ij| < +∞, | µ` | 0 `=1 i,j if we take some natural number N0 such that X∞ X 1 ` N | aR,ij |< +∞, | µ` | 0 `=1 i,j

which is guaranteed by the increasig rates of eigenvalues of QA0 ( (c) of Lemma 1.6.3 in [22] and [30]).

2.2 Holonomy and the Poincar´eDual

To handle the Chern-Simons integral in an abstract Wiener space setting, we need to extend the holonomy R A P e γ , ¿from smooth A to the rough A, which arises in the abstract Wiener space setting. We now regularize the holonomy like in Albeverio and Sch¨afer[1], (see also the relations to Witten’s considerations [42]). First we begin with introducing briefly Section 2 of Mitoma-Nishikawa [35]. As in the previous section, let M be a compact smooth oriented three-manifold and A the space of G-valued smooth 1-forms on M. Let γ : t ∈ [0, 1] 7→ γ(t) ∈ M be a smooth closed curve in M and set γ[s, t] = {γ(τ) | s ≤ τ ≤ t}. We regard γ[s, t] as a linear functional

Z Z t (γ[s, t])[A] = A = A(γ ˙ (τ))dτ, A ∈ A γ[s,t] s defined on the vector space A. Then γ[s, t] is continuous in the sense of distributions and hence 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 smooth closed curve in R3 and A is a G-valued smooth 1-form with compact support defined on Perturbative Expansion of Chern-Simons Theory 8

R3. Let φ be a nonnegative smooth function on R3 such that the support of φ is contained in the unit ball B3 with center 0 ∈ R3 and Z φ(x)dx = 1, R3

−3 and define φ²(x) = ² φ(x/²) for each ² > 0. If we write µ ¶ X X X ∂ (2.7) A = Au · E = A u dxi · E , γ˙ (τ) = γ˙ i(τ) u i u ∂xi u i,u i γ(τ) for given A and γ, then we have ¯Z ¯ ¯ ¯ ¯ u u ¯ (2.8) lim sup ¯ Ai (x)φ²(x − γ(τ))dx − Ai (γ(τ))¯ = 0, ²→0 s≤τ≤t R3 and ¯ ¯ ¯ Z µZ ¶ ¯ ¯X t ¯ (2.9) ¯ 3 A u(x)φ (x − γ(τ))dx γ˙ i(τ)dτ¯ ≤ c (²)kAuk |t − s|. ¯ i ² ¯ 1 L2(R3) i=1 s R3

Here and in what follows, we denote by ck(?) a constant depending on the quantity ? and simply write ck whenever no confusion may occur. Now, according to de Rham [18], the regulator of the current γ[s, t] is defined by

∗ (R²γ[s, t])[A] = (γ[s, t])[R² A] µ ¶ X3 Z t Z u = Ai (γ(τ) + y)φ²(y)dy γ˙ (τ)dτ · Eu 3 i=1 s R µ ¶ X3 Z t Z u i = Ai (x)φ²(x − γ(τ))dx γ˙ (τ)dτ · Eu. 3 i=1 s R to which there is associated an operator defined by

∗ (A²γ[s, t])[B] = (γ[s, t])[A² B] ½ µ ¶ ¾ X Z t Z Z 1 i u j = 3 y Bij (γ(τ) + ty)dt φ²(y)dy γ˙ (τ)dτ · Eu, i,j=1 s R3 0

P u i j where B = Bij dx ∧ dx · Eu is a G-valued smooth two-form with compact support on R3. Then we have the following relation which shows that A²γ[s, t] gives a chain homotopy between R²γ[s, t] and γ[s, t] (see [18, p. 65] for the proof).

Proposition 1. For each ² > 0, R²γ[s, t] and A²γ[s, t] are smooth currents whose supports are contained in the ² tubular neighborhood of γ[s, t], and satisfy

R²γ[s, t] − γ[s, t] = ∂A²γ[s, t] + A²∂γ[s, t], where ∂ is the boundary operator of currents. Perturbative Expansion of Chern-Simons Theory 9

As in [18], the above construction of regularization generalizes to our case in the following manner. First take a diffeomorphism h of R3 onto the unit ball B3 with center 0 which coincides with the identity on the ball of radius 1/3 with center 0. Denote by sy the translation sy(x) = x + y and let sy be the map of R3 onto itself which coincides with −1 h ◦ sy ◦ h on B3 and with the identity at all other points, that is, ( −1 h ◦ sy ◦ h (x) if x ∈ B3, sy(x) = x if x 6∈ B3.

Note that with a suitable choice of h we make sy to be a diffeomorphism. Then define R²γ[s, t] and A²γ[s, t] by the same equations above, but now by using sy and sty. Now, let {Ui} be a finite covering of M such that each Ui is diffeomorphic to the unit ball B3 via a diffeomorphism hi which can be extended to some neighborhoods of their closures. Using these diffeomorphisms, we transport the transformed operators R² and A² defined on R3 to M. In fact, let f be a cutoff function which has its support in the neighborhood of the closure of Ui and is equal to 1 on Ui. Set T = γ[s, t] for simplicity. Then T 0 = fT is a current which has its support contained in a neighborhood of the 0 closure of Ui, and hiT is a current which has its support contained in the neighborhood of the closure of B3. Note that the support of T 00 = T − T 0 does not meet the closure of Ui. We define i −1 0 00 i −1 0 R²T = hi ◦ R² ◦ hiT + T , A²T = hi ◦ A² ◦ hiT and set inductively (k) k (k) k−1 k R² T = R²1 ◦ R²2 ◦ · · · ◦ R² T, A² T = R²1 ◦ R²2 ◦ · · · R² ◦ A² T.

Then R²T and A²T are obtained to be XK (K) (k) R²T = R² T, A²T = A² T, k=1 where K is the number of coverings {Ui}. The construction of the operators R² and A² are easily generalized to any current T defined on a compact smooth manifold of arbitrary dimension, and we have the following properties for regularizations of currents. Proposition 2 ([18]). Let M be a compact smooth manifold. Then for each ² > 0 there exist linear operators R² and A² acting on the space of de Rham currents with the following properties: (1) If T is a current, then R²T and A²T are also currents and satisfy

R²T − T = ∂A²T + A²∂T.

(2) The supports of R²T and A²T are contained in an arbitrary given neighborhood of the support of T provided that ² is sufficiently small. (3) R²T is a smooth form. (4) For all smooth forms ϕ we have

R²T [ϕ] → T [ϕ] and A²T [ϕ] → 0 as ² → 0. Perturbative Expansion of Chern-Simons Theory 10

Given a smooth closed curve γ : [0, 1] → M in M, let Uγ be a tubular neighborhood of γ[0, 1] in M and j : Uγ → M denote the inclusion. For each t ∈ [0, 1] and sufficiently small ² > 0 we set ² ∗ Cγ(t) = j (R²γ[0, t]), which we know is a smooth two-form on Uγ and has a compact support in Uγ from Proposition 2. In particular, for t = 1, [35] showed the following observation, since

² ∗ ∗ ∗ (2.10) dCγ(1) = dj (R²γ[0, 1]) = j d(R²γ[0, 1]) = −j R²∂(γ[0, 1]) = 0. ² ∗ Proposition 3 ([1]). [Cγ(1)] = [j (R²γ[0, 1])] ∈ H2c(Uγ) is the compact Poincar´edual of γ in Uγ, where H2c(Uγ) denotes the second de Rham cohomology of Uγ with compact supports. Let A ∈ A be a G-valued one-form on M and express A as in (2.7). Denote the ² u-component of ∗Cγ(t) by

² ² ² u (2.11) Cu(t) = Cu(t)γ = ∗Cγ(t) · Eu. Then, recalling the construction of regulators of currents and noting (2.8) and (2.9), it is not hard to see that we have ¯ Z ¯ ¯ ² u¯ lim sup ¯(A, Cu(t))0 − A ¯ = 0, ²→0 0≤t≤1 γ[0,t]

¯ Z Z ¯ ¯ u u¯ (2.12) ¯ A − A ¯ ≤ c2(A)|t − s| γ[0,t] γ[0,s] and

² ² (2.13) kCu(t) − Cu(s)k0 ≤ c1(²)|t − s|, where k · k0 is the norm defined in (2.3). Let us define

Xd ² ² Aγ(t) = (A, Cu(t))0 · Eu u=1 and Z A(t) = A. γ[0,t] By Chen’s iterated integrals (Theorem 4.3 in § 1.4 of [19], p31, see also [16, 40]), we have R A +A P e γ 0 ∞ Z Z Z (2.14) X t1 tr−1 = I + 1 ··· d(A0 + A)(t1)d(A0 + A)(t2) ··· d(A0 + A)(tr). r=1 0 0 0 Perturbative Expansion of Chern-Simons Theory 11

Then, as [1], we define the ²-regularizations of the holonomy and the Wilson line as follows: X∞ ² ²,r (2.15) Wγ [A] = I + Wγ [A], r=1 where Z Z Z t1 tr−1 ²,r ² ² ² Wγ [A] = 1 ··· d(A0 + Aγ)(t1)d(A0 + Aγ)(t2) ··· d(A0 + Aγ)(tr) 0 0 0 and Ys ² ² (2.16) F (A0 + A) = Tr Wγj [A]. j=1

Setting

e² ² FA0 (A) = F (A0 + A) detReg(A) and x = (A, φ) = (xA, xφ), we arrive at a regularized form of (2.4) :

Z h Z ^ ^ i 1 e² ik 2 (2.17) FA0 (xA) exp ik(x, QA0 x)+ − Tr xA xA xA D(x), Z2,k X 4π M 3 where X is the space of all x and Z h i

Z2,k = exp ik(x, QA0 x)+ D(x). X By expanding heuristically, for any natural number N, we have that (2.17) is equal to I(k) + II(k), with

I(k) = Z Z 1 XN n ik 2 ^ ^ o h i Fe² (x ) (− Tr x x x )r/r! exp ik(x, Q x) D(x) Z A0 A 4π 3 A A A A0 + 2,k r=0 X M and

II(k) = Z n X∞ Z ^ ^ o h i 1 e² ik 2 r FA0 (xA) (− Tr xA xA xA) /r! exp ik(x, QA0 x)+ D(x). Z2,k 4π 3 X r=N+1 M Perturbative Expansion of Chern-Simons Theory 12

2.3 Definitions of “finite part” and “asymptotic estimate”

Now we proceed to give a meaning to I(k) and to derive an asymptotic estimate for II(k) in an abstract Wiener space setting, with k tending to ∞. We will first define the real Hilbert space H of the abstract Wiener space (X, H, µ) that will be used later. 2 We recall that L (Ω+) was the Hilbert space with the inner product

(2.18) ((A, φ), (B, ϕ))+ = (A, B)0 + (φ, ϕ)0 and the norm

2 2 2 (2.19) k(A, φ)k+ = (A, A)0 + (φ, φ)0 = kAk0 + kφk0.

Let Ωbn be the set of all real valued n-forms and for any α, β ∈ Ωbn, Z ^ R (α, β)0 = α ∗β. M Pd u Pd u 1 2 d u Noticing that xA = u=1 xA·Eu and xφ = u=1 xφ·Eu, we define forx ˆ = (x , x , ··· , x ), x = u u 1 2 d u u u (xA, xφ) andy ˆ = (y , y , ··· , y ), y = (yB, yϕ), Xd (ˆx, yˆ) = (xu, yu) u=1 and

u u u u R u u R (x , y ) = (xA, yB)0 + (xφ, yϕ)0 . 2 Then the real Hilbert space ofx ˆ corresponding to L (Ω+) is denoted by H with the inner product

X∞ 2 (ˆx, xˆ) = (ˆx, eˆj) j=1 and the corresponding norm such that

k · k2 = (·, ·), 1,A 1,φ 2,A 2,φ d,A d,φ wheree ˆj = ((ej , ej ), (ej , ej ), ··· , (ej , ej )). 2 Then H is isometric to L (Ω+) such that

(2.20) (ˆx, yˆ) = (x, y)+ and kxˆk = kxk+.

For smooth A, by (2.18) and (2.20), we have

² ² \² (2.21) (A, Cu(t))0 = ((A, φ), (Cu(t), 0))+ = (ˆx, (Cu(t), 0)), Perturbative Expansion of Chern-Simons Theory 13 where

\² ² u (Cu(t), 0) = ((0, 0), (0, 0), ··· , (∗Cγ(t) , 0), ··· , (0, 0)). Set

Xd ² \² xˆγ(t) = (ˆx, (Cu(t), 0)) · Eu. u=1 ² ² Then replacing Aγ(t) byx ˆγ(t) in the definitions (2.15) and (2.16), we have X∞ ² ²,r Wγ (ˆx) = I + Wγ (ˆx). r=1 and

Ys ² ² FA0 (ˆx) = Tr Wγj (ˆx). j=1 ² The convergence in H and the well definedness of Wγ (ˆx) is guaranteed by (2.13). We have

X∞ X∞ 2 A (x, QA0 x)+ = µj(ˆx, eˆj) , xA = (ˆx, eˆj)ej , j=1 j=1 Ys ² ² ² F (A0 + xA) = FA0 (ˆx) = Tr Wγj (ˆx), j=1 X∞ X∞ R ` R ` aii (xA) = 1 + (ˆx, eˆ`)aR,ii, aij(xA) = (ˆx, eˆ`)aR,ij `=1 `=1 and

Z −1 2 ^ ^ X∞ (2.22) Tr x x x = (ˆx, eˆ )(ˆx, eˆ )(ˆx, eˆ )βpj`, 4π 3 A A A p j ` M p,j,`=1 where Z ^ ^ pj` 1 A A A β = − Tr ep ej e` . M 6π Itˆotechnique [27] to handle oscillatory integrals is to introduce a Gaussian kernel with a suitable covariance operator. Thus for the m-dimensional approximation of I(k), we take the covariance operator Sm with eigenvalues {λj > 0, j = 1, 2, ··· , m} and consider Perturbative Expansion of Chern-Simons Theory 14

Z V λ XN n Xm o m,n e² m pj` r lim lim FA0 (x ) (ik xpxjx`β ) /r! (2.23) n→∞ m→∞ Zm,n m r=0 R p,j,`=1 h −1 m m i m m (Sm x , x )+ µm(dx) × exp ik(x ,QA x )+ − √ p , 0 m 2n ( 2π) det(nSm) where µm is the m-dimensional Lebesgue measure,

Xm Ym p m λ x = xjeˆj,Vm,n = 1 + nλj, j=1 j=1 Xm −1 m m −1 2 (Sm x , x )+ = λj xj j=1 and

Z h −1 m m i λ m m (Sm x , x )+ µm(dx) Zm,n = V exp ik(x ,QA x )+ − √ p . m,n 0 m Rm 2n ( 2π) det(nSm) In the sequel we will show that it is possible to find a realization of the expression (2.23) using a suitable abstract Wiener space (X, H, µ). The real Hilbert space H was already defined above. Now we will choose X and a Gaussian measure µ such that H is densely and continuously embedded into X and

Z 2 i − kξk e dµ(x) = e 2 , X holds, where < x, ξ > denotes the canonical bilinear form on X × X∗. We denote (x1, x2, ··· , xd) ∈ X by x and (ξ1, ξ2, ··· , ξd) ∈ X∗ by ξ. In the usual manner, we can extend the bilinear form to any element h ∈ H, then we denote it again by the same symbol < x, h >, h ∈ H such that for any x, h ∈ H, < x, h >= (x, h). Since (e ˆi, i = 1, 2, ··· ) forms a complete orthonormal system of the Hilbert space H,QA0 induces naturally a linear operator defined in H having the same spectrum as QA0 . We denote this linear operator in H again by the same symbol QA0 . Using the above CONS of H, we can construct a positive definite and self-adjoint operator S with eigenvalues {1 ≥ λj > 0, j = 1, 2, ···} on the Hilbert space H such that

X∞ p 4 3 (S.1) λj | µj |< +∞, j=1

X∞ p 4 (S.2) λj < +∞ j=1 Perturbative Expansion of Chern-Simons Theory 15 and

p X 4 ` (S.3) sup λ` | aR,ij |< +∞. ` i,j

( [38, 43]). For any ² > 0 and x ∈ X, we define

Xd ² \² xγ(t) = < x, (Cu(t), 0) >> ·Eu. u=1 Briefly we denote \² ²,u < x, (Cu(t), 0) > by xγ (t). ²,u Then by (2.19) and (2.20), we remark that xγ (t) is a Gaussian random variable such that

h i ²,u 2 \² 2 ² 2 (2.24) E xγ (t) = k(Cu(t), 0)k = kCu(t)k0.

Since (2.13) yields h i ²,u ²,u 2 2 2 (2.25) E |xγ (t) − xγ (s)| ≤ c1(²) |t − s| ,

²,u xγ (t) has a continuous version in t, by the Kolmogorov-Totoki criterion [41]. Henceforth ²,u we denote this continuous version, again by the same symbol xγ (t). For any natural number n, set

n X2 j j − 1 T = |x²,u( ) − x²,u( )|. n γ 2n γ 2n j=1

Since Tn ≤ Tn+1, we have

h i h i E lim Tn = lim E Tn n→∞ n→∞ n h X2 j j − 1 i = lim E |x²,u( ) − x²,u( )|. n→∞ γ 2n γ 2n j=1 n (2.26) X2 h j j − 1 i 1 ≤ lim E |x²,u( ) − x²,u( )|2. 2 n→∞ γ 2n γ 2n j=1 n X2 j j − 1 ≤ lim c1(²)| − | n→∞ 2n 2n j=1 ≤ 1, Perturbative Expansion of Chern-Simons Theory 16 which implies

(2.27) lim Tn < +∞ almost surely with respect to µ. n→∞

²,u Noticing that xγ (t) is continuous in t almost surely, by (2.27), we obtain, for almost all x ∈ X with respect to µ, that

²,u xγ (t) is of bounded variation . Therefore the Lebesgue-Stieltjes integral Z Z t t1 ² ² dxγ(t1) dxγ(t2) 0 0 is well defined and continuous in t almost surely with respect to µ and is equal to

d d Z Z X X t t1 ²,u ²,v dxγ (t1) dxγ (t2) · EuEv. u=1 v=1 0 0 Repeating such arguments, we arrive at the well definedness of the iterated integrals and in a similar manner as for the definitions of W ²(ˆx) and F ² (ˆx), we define γ A0 Ys ² ² FA0 (x) = Tr Wγj (x), j=1 where the well definiteness is guaranteed by (1) in Lemma 1 which will be stated below. We call, for ² > 0,F ² (x) the ²-regularization of the Wilson loop variable [1]. A0 R In the definitions of detn,R ∗dA0 ∗ DA and detReg(A), we replace aii (A) by √ X∞ p R ` aii ( nSx) = 1 + nλ` < x, eˆ` > aR,ii `=1 R and aij(A) by √ X∞ p R ` aij( nSx) = nλ` < x, eˆ` > aR,ij, `=1 √ √ which are detn,R ∗dA0 ∗ D nSx and detReg( nSx). The convergence of the series is guar- anteed by (S.2) and (S.3), since

X∞ p ` | nλ` < x, eˆ` > aR,ij| `=1 X∞ p p √ 4 4 ` ≤ n λ`| < x, eˆ` > | sup λ`|aR,ij|. `=1 ` Let us set Perturbative Expansion of Chern-Simons Theory 17

√ √ √ e² ² FA0 ( nSx) = FA0 ( nSx) detReg( nSx). We shall provide the definition of I(k) in the abstract Wiener space by setting it equal to

Definition 1.

XN Z √ h √ i 1 e² lim sup n FA0 ( nSx) exp ikQ( nSx) n→∞ Z (2.28) 2,k r=0 X n X∞ √ √ √ o pj` r × (ik < nSx, eˆp >< nSx, eˆj >< nSx, eˆ` > β ) /r! µ(dx), p,j,`=1

where Z h √ i n Z2,k = exp ikQ( nSx) µ(dx). X

We use the notations for real xn, yn :

lim sup(xn + iyn) = lim sup xn + i lim sup yn n→∞ n→∞ n→∞ and by above convergence assumptions (S.1) and (S.2), we get

√ X∞ 2 Q( nSx) = nλjµj < x, eˆj > < +∞ j=1 (cf. [32]). To derive an asymptotic estimate for II(k) as k −→ ∞, let us exploit compensations in the numerator and denominator. The m-dimensional approximation of II(k) is given by

Z n X∞ Xm o 1 e² m pj` r FA0 (x ) (ik xpxjx`β ) /r! (2.29) mZ2,k m R r=N+1 p,j,`=1 h i m m µm(dx) exp ik(x ,QA x )+ √ , 0 ( 2π)m where Z h i µ (dx) Z = exp ik(xm,Q xm) √m . m 2,k A0 + m Rm ( 2π) √ Set kxi = yi, then by the formula of changing variables of up and down integral, we obtain (2.29) Perturbative Expansion of Chern-Simons Theory 18

Z 1 1 n X∞ i Xm o h i µ (dy) e² √ m √ pj` r m m √m = m FA0 ( y ) ( ypyjy`β ) /r! exp i(y ,QA0 y )+ , Z m ( 2π)m 3,k R k r=N+1 k p,j,`=1 where Z h i µ (dy) Zm = exp i(ym,Q ym) √m . 3,k A0 + m Rm ( 2π) To discuss the asymptotic estimate of II(k), we may modify the Itˆotechnique by introducing the Gaussian kernel with parameter

1 (2.30) k2η, 0 < η < 12 and consider

√ N+1 lim k II(k) k→∞ Z n ∞ m o √ N+1 1 1 X i X (2.31) e² √ m √ pj` r = lim k lim m FA0 ( x ) ( xpxjx`β ) /r! k→∞ m→∞ Z m 4,k R k r=N+1 k p,j,`=1 h −1 m m i m m (Sm x , x )+ µm(dx) × exp i(x ,QA x )+ − √ p , 0 2η m 2η 2k ( 2π) det(k Sm) where

Z h −1 m m i m m m (Sm x , x )+ µm(dx) Z = exp i(x ,QA x )+ − √ p . 4,k 0 2η m 2η Rm 2k ( 2π) det(k Sm) We can find a realization of the expression (2.31) in the abstract Wiener space setting as equal to

Definition 2. (2.32)

Z √ h i √ N+1 1 k2ηS √ e² √ 2η lim k FA0 ( x) exp ik Q( Sx) k→∞ Zˆ2,k X k n X∞ X∞ √ √ √ o i 2η 2η 2η pj` r × (√ < k Sx, eˆp >< k Sx, eˆj >< k Sx, eˆ` > β ) /r! r=N+1 k p,j,`=1 µ(dx),

where Z h √ i 2η Zˆ2,k = exp ik Q( Sx) µ(dx). X Perturbative Expansion of Chern-Simons Theory 19

Here we add a further assumption to the operator S such that for some δ < η and a natural number M such that Mδ > 2η,

X∞ q M (S.4) λj | µj | = C ≤ c3 < +∞. j=1

2.4 Integrabilities and Theorems

Before stating Theorems, we shall discuss the integrability of the Wilson lines operator and of the determinant arising below. Set

1,0 Yγ (0) = I, Z Z Z Z X tl −1 tl −1 tr−1 1,r 1 ² i ² Yγ (i) = 1dA0(t1) ··· dxγ(tl1 ) ··· dxγ(tli ) ··· dA0(tr) 0 0 0 0 1≤l1

X Ys (2.33) Y 1,m(x) = Tr Y 1 (i ). A0 γj j i1+i2+···+is=m j=1

R R Furthermore, let us definea ˆii (x) = aii (x) − 1 and denote by Sm the collection of all permutations of {1, 2, ··· , m} and by Sm,t the collection of all σ ∈ Sm such that σ(ij) 6= ij, j = 1, 2, ··· t and otherwise σ(i) = i. Set Perturbative Expansion of Chern-Simons Theory 20

Xr X Y 2,r = lim ( signσaR (x)aR (x) ··· aR (x) m→∞ i1σ(i1) i2σ(i2) itσ(it) t=1 σ∈Sm,t X R R R × { aˆj1j1 (x)ˆaj2j2 (x) ··· aˆjr−tjr−t (x)} 1≤j1

Y 2,0 = 1. Set

X 0 Fe (x) = Y 1,r(x)Y 2,r (x). m A0 r+r0=m Define √ nS Rn,k = q √ √ . 1 − 2ik nSQA0 nS

For an n × n matrix E = (aij), define Xn |kE|k = | aij | . i,j=1

We define a regularization of QA0 such that the eigenvalues µj of QA0 satisfy the “renormalization condition” :

X∞ X 1 ` Renormalization. p (1 + | aR,ij |) < +∞. `=1 |µ`| i,j

We also assume

Assumption 2.

| βpj` |≤ β < +∞.

Remark 1. We expect that in the case where M is compact, this assumption is automat- ically satisfied.

The following Lemma will be proved later in Section 3. Perturbative Expansion of Chern-Simons Theory 21

Lemma 1. Let us denote any natural number by b and let E[·] the integral with respect to µ on X. (1) For any ² > 0, h i ² 2b E |kWγ (x)|k < +∞.

(2) For any fixed ² > 0, there exists a constant c4(²) independent of n such that h i ² 2b E |kWγ (Rn,kx)|k ≤ c4(²) < +∞. (3) h X∞ i 1 E Y 1,m(R x) = o((√ )N ), A0 n,k m=N k where o(( √1 )N ) means k √ N 1 lim k | o((√ )N ) |< +∞. k→∞ k (4) Under the Assumption 2,

h X∞ √ √ √ i pj` 2b E | ik < nSx, eˆp >< nSx, eˆj >< nSx, eˆ` > β | < +∞. p,j,`=1

(5) Under the Renormalization assumption on QA0 and the Assumption 2, there exists a constant c5(b) independent of n such that

h X∞ i X∞ pj` 2b 1 2b 1 6b E | ik < Rn,kx, eˆp >< Rn,kx, eˆj >< Rn,kx, eˆ` > β | ≤ c5(b)(√ ) ( p ) . |µ | p,j,`=1 k j=1 j

(6) Under the Assumption 2, h √ i 2b E | detReg( nSx) | < +∞.

(7) Under the same assumptions as in (5), there exists a constant c6(b) independent of n such that h i 2b E | detReg(Rn,kx) | ≤ c6(b) < +∞. (8) Under the same assumptions as in (5),

h X∞ i 2,r 1 N E Y (Rn,kx) = o((√ ) ). r=N k By Lemma 1, the Cauchy integral theorem in finite dimensions and a limiting argu- ment, we have Perturbative Expansion of Chern-Simons Theory 22

Theorem 1. Under the Assumption 2,

XN Z e² I(k) = lim sup FA0 (Rn,kx) n→∞ r=0 X n X∞ o pj` r × (ik < Rn,kx, eˆp >< Rn,kx, eˆj >< Rn,kx, eˆ` > β ) /r! µ(dx) p,j,`=1 is well-defined.

Under the Renormalization assumption on QA0 and the Assumption 2, the right hand side of the above equality is equal to :

X Z Fem(Rkx) m+m0≤N X n X∞ o pj` m0 0 × (ik < Rkx, eˆp >< Rkx, eˆj >< Rkx, eˆ` > β ) /m ! µ(dx) p,j,`=1 1 + o((√ )N+1) k X 1 m+m0 ²,m+m0 1 N+1 = (√ ) JCS + o((√ ) ), m+m0≤N k k where

Z 1 m+m0 ²,m+m0 e (√ ) JCS = Fm(Rkx) k X n X∞ o pj` m0 0 × (ik < Rkx, eˆp >< Rkx, eˆj >< Rkx, eˆ` > β ) /m ! µ(dx) p,j,`=1 and 1 Rk = p . −2ikQA0 Now we will proceed to the asymptotic estimate of II(k), as k → ∞. For 0 < δ < η, set √ n X∞ 4 o Sx δ D = x ∈ X; |< q √ √ , eˆj >|≤ 3k 2η j=1 (1 − 2ik SQA0 S) and √ k2ηS Vk = q √ √ . 2η 2η 1 − 2i k SQA0 k S) Perturbative Expansion of Chern-Simons Theory 23

Theorem 2. Under the Assumption 2,

(2.34)

Z √ 1 k2ηS h √ i e² √ 2η FA0 ( x) exp ik Q( Sx) Zˆ2,k X k n X∞ X∞ √ √ √ o i 2η 2η 2η pj` r × (√ < k Sx, eˆp >< k Sx, eˆj >< k Sx, eˆ` > β ) /r! µ(dx) r=N+1 k p,j,`=1 is well-defined and equal to Z V e² √k FA0 ( x) D k n X∞ X∞ o i pj` r × (√ < Vkx, eˆp >< Vkx, eˆj >< Vkx, eˆ` > β ) /r! µ(dx) r=N+1 k p,j,`=1 + R(k) where R(k) satisfies

δ − k (2.35) R(k) ≤ c7e 2 for sufficiently large k, so that under the Renormalization assumption on QA0 and the Assumption 2,

√ N+1 lim k | II(k) |< +∞. k→∞

Remark 2. If we make a stronger renormalization on QA0 such that

X∞ (βpj`)2 < +∞ | µp || µj || µ` | p,j,`=1 and impose appropriate conditions on the eigenvalues of S, we can release Assumption 2. As remarked before, Assumption 2 is expected to be automatically satisfied when M is compact.

3 Proofs of Lemmas and Theorems

First we remark that µ-almost surely,

Xd \² < Rn,kx, (Cu(t), 0) >> Eu u=1 √ Xd X∞ nλ = < x, eˆ > ((C\² (t), 0), eˆ )√ r E . r u r 1 − 2inkλ µ u u=1 r=1 r r Perturbative Expansion of Chern-Simons Theory 24

Since √ √ nλ nλ √ r = p r eiθ 4 1 − 2inkλrµr 1 + 4(nkλrµr)2 −π π = αn,k + iβn,k, where < θ < , r r 2 2 set P ² d \² Rn,kxγ(t) = u=1 < Rn,kx, (Cu(t), 0) >> Eu,

² u \² Rn,kxγ(t) =< Rn,kx, (Cu(t), 0) >>, P 1 ² u ∞ n,k \² Rn,kxγ(t) = r=1 αr < x, eˆr > ((Cu(t), 0), eˆr), and P 2 ² u ∞ n,k \² Rn,kxγ(t) = r=1 βr < x, eˆr > ((Cu(t), 0), eˆr).

Define

d cE = max |kEu|k. u=1

3.1 Proofs of Lemmas

Before we proceed to the proof of Lemma 1, we remark the following lemma used several times below.

Lemma 2. Let b be a natural number and Xi,j, i, j = 1, 2, ... be real numbers. Then X X X X 2b 2b 1 2b | Xi,j| ≤ ( [ |Xi,j| ] 2b ) i j j i

Especially for any random variables Xi, i = 1, 2, ... on a probability space, we have

h X i ³ X h i 1 ´2b 2b 2b 2b E | Xj| ≤ E |Xj| . j j Proof. Since X X 2b ( |Xi,j|) = |Xi,j1 ||Xi,j2 | · · · |Xi,j2b | j j1,j2,··· ,j2b and by the H¨older’sinequality recursively Perturbative Expansion of Chern-Simons Theory 25

X

|Xi,j1 ||Xi,j2 | · · · |Xi,j2b | i X 1 X 2b 2b−1 2b 2b 2b−1 2b ≤ [ |Xi,j1 | ] [ (|Xi,j2 | · · · |Xi,j2b |) ] i i X 1 X 1 2b 2b 2b 2b ≤ [ |Xi,j1 | ] [ |Xi,j2 | ] i i X 2b 2b−2 2b−2 2b [ (|Xi,j3 | · · · |Xi,j2b |) ] i and so on, we have

X X X X 2b | Xi,j| ≤ [ |Xi,j1 ||Xi,j2 | · · · |Xi,j2b |] i j i j ,j ,··· ,j X X 1 2 2b

= [ |Xi,j1 ||Xi,j2 | · · · |Xi,j2b |] j1,j2,··· ,j2b i X X 1 X 1 X 1 2b 2b 2b 2b 2b 2b ≤ [ |Xi,j1 | ] [ |Xi,j2 | ] ··· [ |Xi,j2b | ] , j1,j2,··· ,j2b i i i which is equal to the right hand side of the inequality in Lemma 2. This completes the proof.

Now we proceed to the proof of Lemma 1. Define Z u u A0 (t) = A0 . γ[0,t] Since

Z Z Z t1 tr−1 ² ² ² 1 ··· d(A0 + xγ)(t1)d(A0 + xγ)(t2) ··· d(A0 + xγ)(tr) 0 0 0 r Z Z X X tl −1 1 ² = 1dA0(t1) ··· dxγ(tl1 ) 0 0 m=0 1≤l1

(3.1) h i ² 2b E |kWγ (x)|k ∞ r d Z Z h X X X X tl1−1 j1 r ² jl1 ≤ E ( cE | 1dA0 (t1) ··· dxγ(tl1 ) 0 0 r=0 m=0 1≤l1

h i ² 2b E |kWγ (Rn,kx)|k h X∞ Xr X Xd Z r j1 ≤ E ( cE | 1dA0 (t1) ··· 0 r=0 m=0 1≤l1

Now we begin by recalling the following well known lemma [17] :

Lemma 3. Let Xi, i = 1, 2, ..., 2` be a mean-zero Gaussian system. Then

h i X h i h i h i

E X1X2 ··· X2` = E Xi1 Xj1 E Xi2 Xj2 ··· E Xi` Xj` comb(`) (3.3) 1 X h i h i h i = E X X E X X ··· E X X , 2``! σ(1) σ(2) σ(3) σ(4) σ(2`−1) σ(2`) σ∈S2` X where S2` denotes the collection of all permutations of {1, 2, ··· , 2`} and is the comb(`) summation taken over all pairs of {1, 2, ..., 2`} such that i1 < i2 < ··· < i`, ik ≤ jk for k = 1, 2, ..., `.

s0 On the other hand, let si, i = 0, 1, ··· , r be non-negative integers. Then setting t0 = 1 and ( 0 if si = 0, si s ti = t i−1 tsi−1 + i−1 if s ≥ 1 i ni i and

ji ji si+1 ji si A[si|0 ] = A0 (ti ) − A0 (ti ),

²,ji ² si+1 ji ² si ji x[si|γ ] = xγ(ti ) − xγ(ti ) , by (2.12) and (2.13), we have

Z Z Z Z h 1 tl1−1 tlm−1 tr−1 i j1 jr ² jl1 ² jlm 2b E | dA0 (t1) ··· dxγ(tl1 ) ··· dxγ(tlm ) ··· dA0 (tr) | 0 0 0 0 n −1 h nX1−1 Xl1 j1 ²,jl1 = E | lim A[s1|0 ] ··· lim x[sl1 |γ ] n1→∞ nl1 →∞ s1=0 sl1 =0 n −1 Xlm nXr−1 i ²,jlm jr 2b ··· lim x[slm |γ ] ··· lim A[sr|0 ] | nlm →∞ nr→∞ slm =0 sr=0 n −1 n −1 h nX1−1 Xl1 Xlm nXr−1 ≤ E ( lim ··· lim ··· lim ··· lim ··· ··· ··· n1→∞ nl1 →∞ nlm →∞ nr→∞ s1=0 sl =0 sl =0 sr=0 i 1 m j1 ²,jl1 ²,jlm jr 2b |A[s1|0 ] ··· x[sl1 |γ ] ··· x[slm |γ ] ··· A[sr|0 ]|) Perturbative Expansion of Chern-Simons Theory 28

n −1 n −1 h nX1−1 Xl1 Xlm 2b(r−m) ≤ c2(A0) lim ··· lim ··· lim ··· lim E ( ··· ··· ··· n1→∞ nl1 →∞ nlm →∞ nr→∞ s1=0 sl1 =0 slm =0 nr−1 i X ²,j s1+1 s1 l1 ²,jlm sr+1 sr 2b |t1 − t1 | · · · |x[sl1 |γ ]| · · · |x[slm |γ ]| · · · |tr − tr |) sr=0 n −1 n −1 nX1−1 Xl1 Xlm 2b(r−m) ≤ c2(A0) lim ··· lim ··· lim ··· lim ( ··· ··· ··· n1→∞ nl1 →∞ nlm →∞ nr→∞ s1=0 sl1 =0 slm =0 nr−1 h i 1 X ²,j s1+1 s1 l1 ²,jlm sr+1 sr 2b 2b 2b E (|t1 − t1 | · · · |x[sl1 |γ ]| · · · |x[slm |γ ]| · · · |tr − tr |) ) . sr=0 By Lemma 3, we have

h i ²,jl1 ²,jlm 2b E (|x[sl1 |γ ]| · · · |x[slm |γ ]|) 2bm (2bm)!c (²) s +1 s s +1 s ≤ 1 |t l1 − t l1 |2b · · · |t lm − t lm |2b. 2bm(bm)! l1 l1 lm lm

Therefore we have

Z Z Z Z 1 h tl1−1 tlm−1 tr−1 i j1 jr 2b ² jl1 ² jlm 2b E | 1dA0 (t1) ··· dxγ(tl1 ) ··· dxγ(tlm ) ··· dA0 (tr) | 0 0 0 0 n −1 n −1 nX1−1 Xl1 Xlm r−m ≤ c2(A0) lim ··· lim ··· lim ··· lim ··· ··· ··· n1→∞ nl1 →∞ nlm →∞ nr→∞ s1=0 sl1 =0 slm =0 nr−1 2bm X 1 (2bm)!c1(²) s +1 s sl +1 sl sl +1 sl s +1 s { } 2b |t 1 − t 1 | · · · |t 1 − t 1 | · · · |t m − t m | · · · |t r − t r |. 2bm(bm)! 1 1 l1 l1 lm lm r r sr=0 Z Z t Z t (2bm)! 1 1 r−1 r−m m 2b ≤ c2(A0) c1(²) { bm } 1 ··· dt1dt2 ··· dtr. 2 (bm)! 0 0 0

Since (bm)! ≤ (m!bm)b, the last term is less or equal

n(m!)b(1 · 2 ··· (2bm − 2) · (2bm − 1) · 2bm)2bmbbm o 1 c (A )r 2b 8 0 (r!)2b((2 · 1) · (2 · 2) ··· 2(bm − 1) · 2bm)2 √ √ m m! ≤ c (A )r 2b , 8 0 r! where c8(A0) = max{c2(A0), c1(²)}. Thus we obtain that (3.1) is dominated by Perturbative Expansion of Chern-Simons Theory 29

∞ r √ X X √ m m! ( (dc c (A ))r C 2b )2b E 8 0 r m r! r=0 m=0 X∞ √ r r 1 2b ≤ ( (dcEc8(A0)) (1 + 2b) √ ) < +∞, r=0 r! which yields the proof of (1) of Lemma 1. By (2.13), we have for ζ = 1 or 2,

h i 1 ζ ² u ζ ² u 2 2 E |Rn,kxγ(t) − Rn,kxγ(s) | X∞ 1 −1 2 1 ≤ ( | µ | ((C\² (t), 0) − (C\² (s), 0), eˆ ) ) 2 2k r u u r r=1 (3.4) X∞ 1 −1 ² ² 2 1 = ( | µ | ((C (t) − C (s), 0), e ) ) 2 2k r u u r + r=1 1 ≤ √ kC² (t) − C² (s)k 2kρ u u 0 1 ≤ c (²)√ |t − s|, 1 2kρ where

(3.5) min | µr |= ρ > 0. r=1,2,···

In a manner similar to the proof of the former part, (3.2) is dominated by

√ X∞ Xr 2b 1 ( (dc c (A ))r C (2√ )m √ )2b E 8 0 r m 2kρ r=0 m=0 r! √ X∞ b 1 = ( (dc c (A )(1 + 2√ ))r √ )2b < +∞, E 8 0 kρ r=0 r! which provides the proofs of (2) and (3) of Lemma 1. Now we proceed to the proof of the rest of Lemma 1. Since

√ nλr | < Rn,kx, eˆr > | = |√ | < x, eˆr > | 1 − 2inkλrµr 1 ≤ p | < x, eˆr > |, 2k|µr| Perturbative Expansion of Chern-Simons Theory 30 by Lemmas 2 and 3 and the Assumption 2, we have under the Renormalization assumption,

h X∞ i pj` 2b E | < Rn,kx, eˆp >< Rn,kx, eˆj >< Rn,kx, eˆ` > β | p,j,`=1 h X∞ i β 2b ≤ E ( p p p | < x, eˆp > || < x, eˆj > || < x, eˆ` > |) p,j,`=1 2k|µp| 2k|µj| 2k|µ`| X∞ h i 1 β 2b 2b 2b ≤ ( p p p E (| < x, eˆp > || < x, eˆj > || < x, eˆ` > |) ) p,j,`=1 2k|µp| 2k|µj| 2k|µ`| X∞ 1 2b ≤ c5(b)( p p p ) p,j,`=1 k|µp| k|µj| k|µ`| X∞ 1 6b 1 6b = c5(b)(√ ) ( p ) < +∞. k i=1 |µi| √ The proofs of (4) and (5) of Lemma 1 are obtained by replacing Rn,k by nS, in effect the latter estimate is easier than the former. Since

√ X∞ nλ |aR(R x)| ≤ |a (0)| + | < x, eˆ > ||√ ` a` | ij n,k ij ` 1 − 2inkλ µ R,ij `=1 ` ` X∞ 1 ` ≤ 1 + | < x, eˆ` > |p |aR,ij|, `=1 2k|µ`| where aij(0) = δij, i, j = 1, 2, ··· , m, noticing the Renormalization assumption, we have

h i h X i 2b R R R 2b E | detReg(Rn,kx) | ≤ lim E ( | a (Rn,kx)a (Rn,kx) ··· a (Rn,kx) |) m→∞ 1σ(1) 2σ(2) mσ(m) σ∈Sm h Y X∞ i 1 ` 2b ≤ lim E ( (1 + | < x, eˆ` > |p |a |)) m→∞ R,ij i,j `=1 2k|µ`| h h X X∞ ii 1 ` , ≤ lim E exp 2b { | < x, eˆ` > |p |a |)} m→∞ R,ij i,j `=1 2k|µ`| h h X∞ X∞ ii 1 ` ≤ E exp 2b | < x, eˆ` > |p { |aR,ij|} < +∞, (see [29], `=1 2k|µ`| i,j=1 which yields the proofs of (6),(7) and (8) of Lemma 1. Perturbative Expansion of Chern-Simons Theory 31

3.2 Proof of Theorem 1

Now we proceed to the proof of Theorem 1. Set

Xd n XL o ² \² xL,γ(t) = < x, eˆi > ((Cu(t), 0), eˆi) · Eu u=1 i=1 ² ² ²,r and replace xγ(t) by xL,γ(t) in the definition of Wγ (x). Then the object to be obtained is denoted by

²,r WL,γ(x). Define

XJ ² ²,r Wγ,J,L(x) = I + WL,γ(x) r=1 and

Ys F ²,J (x) = T rW ² (x). L,A0 γj ,J,L j=1 √ √ R In the definition of detJ,R ∗dA0 ∗ D nSx, we replace aii ( nSx) by XL p ` 1 + nλ` < x, eˆ` > aR,ii `=1 √ R and aij( nSx) by

XL p ` nλ` < x, eˆ` > aR,ij, `=1 which is denoted by detJ,R,L(x). Setting

Fe²,J (x) = F ²,J (x) det (x), L,A0 L,A0 J,R,L

GL,J (< x, eˆ1 >, < x, eˆ2 >, ··· , < x, eˆL >) XN n XL √ √ √ o = Fe²,J (x) (ik < nSx, eˆ >< nSx, eˆ >< nSx, eˆ > βpj`)r/r! L,A0 p j ` r=0 p,j,`=1 and

XL 2 QL(x) = µj < x, eˆj > , j=1 Perturbative Expansion of Chern-Simons Theory 32 then by (1),(4) and (6) of Lemma 1, we have

XN Z √ h √ i e² FA0 ( nSx) exp ikQ( nSx) r=0 X n X∞ √ √ √ o pj` r × (ik < nSx, eˆp >< nSx, eˆj >< nSx, eˆ` > β ) /r! p,j,`=1 µ(dx) Z (3.6) = lim lim GL,J (< x, eˆ1 >, < x, eˆ2 >, ··· , < x, eˆL >) L→∞ J→∞ X h √ i × exp ikQL( nSx) µ(dx) ZZ Z

= lim lim ··· GL,J (x1, x2, ··· , xL) L→∞ J→∞ RL L L 2 h X i Y x 2 1 − j × exp ik[ nλjµjxj ] √ e 2 dxj. j=1 j=1 2π Setting π z = 1 − 2iknλ µ =| z | e−iαj , 0 < (sign µ )α < , j j j j j j 2 we get by using recursively the Cauchy integral theorem, that (3.6) is equal to

  ZZ Z XL α lim lim ··· exp i j  L→∞ J→∞ L 2 (3.7) R j=1 " # YL 2 i α1 i α2 i αL 1 | zj | xj × G (e 2 x , e 2 x , ··· , e 2 x ) √ exp − dx . L,J 1 2 L 2 j j=1 2π Using (2),(5) and (7) of Lemma 1, we see that (3.7) is equal to

Z 1 lim lim qQ GL,J (< Rn,kx, eˆ1 >, < Rn,kx, eˆ2 >, ··· , < Rn,kx, eˆL >)µ(dx) L→∞ J→∞ L X j=1(1 − 2iknλjµj) Z 1 XN = q Fe² (R x) √ √ A0 n,k X det(1 − 2ik nSQA0 nS) r=0 n X∞ o pj` r × (ik < Rn,kx, eˆp >< Rn,kx, eˆj >< Rn,kx, eˆ` > β ) /r! µ(dx). p,j,`=1 Therefore Perturbative Expansion of Chern-Simons Theory 33

Z h √ i n 1 Z2,k = exp ikQ( nSx) µ(dx) = q √ √ , X det(1 − 2ik nSQA0 nS) which, together with (3),(5) and (8) of Lemma 1, completes the proof of Theorem 1.

3.3 Proof of Theorem 2

Now we proceed to the proof of Theorem 2. Since Z h √ i 2η 1 exp ik Q( Sx) µ(dx) = q √ √ , X 2η det(1 − 2ik SQA0 S) we have that (2.34) is equal to

q Z √ √ √ k2ηS h √ i 2η e² √ 2η det(1 − 2ik SQA0 S) FA0 ( x) exp ik Q( Sx) X k (3.8) n X∞ X∞ √ √ √ o i 2η 2η 2η pj` r (√ < k Sx, eˆp >< k Sx, eˆj >< k Sx, eˆ` > β ) /r! r=N+1 k p,j,`=1 µ(dx).

By the convergence assumption (S,2), we have

h h X∞ p ii 4 E exp λi |< x, eˆi >| < +∞, i=1 so that by the Chebyshev inequality, we get

h X∞ p i 4 δ −3kδ (3.9) µ x ∈ X; λi |< x, eˆi >|> 3k ≤ c9e . i=1 Let

n X∞ p o 4 δ Xk = x ∈ X; λi |< x, eˆi >|≤ 3k . i=1 Then we prove Perturbative Expansion of Chern-Simons Theory 34

Lemma 4. q Z √ √ √ k2ηS h √ i 2η e² √ 2η det(1 − 2ik SQA0 S) FA0 ( x) exp ik Q( Sx) c Xk k n X∞ X∞ √ √ √ o i 2η 2η 2η pj` r × (√ < k Sx, eˆp >< k Sx, eˆj >< k Sx, eˆ` > β ) /r! µ(dx) r=N+1 k p,j,`=1 δ = o(e−k ).

Proof. First we show that for sufficiently large k,

q √ √ kδ 2η − (3.10) lim det(1 − 2ik SQA S)e 2 = 0. k→∞ 0 By the convergence assumption (S,4),we have for some natural number M such that Mδ > 2η and

X∞ q M λj | µj | = C ≤ c3 < +∞. j=1 Hence we have

P p q h ∞ M i √ √ λj | µj | 2η δ j=1 | det(1 − 2ik SQA0 S) exp − k | p 2C Y∞ q h M λ | µ |i = | 1 − 2ik2ηλ µ | exp − kδ j j j j 2C j=1 p Y∞ | 1 − 2ik2ηλ µ | ≤ q j j , Mδ k λj |µj | j=1 1 + M!(2C)M 2 which gives (3.10). 3 1 Since 3η < 12 < 2 and

√ k2ηS n XN e² √ FA0 ( x) k r=1 X∞ √ √ √ o i 2η 2η 2η pj` r (√ < k Sx, eˆp >< k Sx, eˆj >< k Sx, eˆ` > β ) /r! (3.11) k p,j,`=1 kη √ n XN e² √ = FA0 ( Sx) k r=1 3η X∞ √ √ √ o k pj` r (i√ < Sx, eˆi >< Sx, eˆj >< Sx, eˆ` > β ) /r! , k p,j,`=1 Perturbative Expansion of Chern-Simons Theory 35 by the estimations similar to the proof of Lemma 1, we get under Assumption 2,

Z η √ e k | FA0 (√ Sx) X k n XN X∞ √ √ √ o i 2η 2η 2η pj` r 2 × (√ < k Sx, eˆp >< k Sx, eˆj >< k Sx, eˆ` > β ) /r! | µ(dx) r=1 k p,j,`=1

≤ c10(N) < +∞, so that by (3.9) and (3.10) we have

(3.12)

q Z √ √ √ k2ηS h √ i 2η e² √ 2η det(1 − 2ik SQA0 S) FA0 ( x) exp ik Q( Sx) c Xk k n XN X∞ √ √ √ o i 2η 2η 2η pj` r (√ < k Sx, eˆi >< k Sx, eˆj >< k Sx, eˆ` > β ) /r! µ(dx) r=1 k p,j,`=1 δ = o(e−k ).

Similarly we get Z √ k2ηS e² √ 2 | FA0 ( x) | µ(dx) ≤ c11 < +∞, X k so that using that

√ 3η X∞ √ √ √ 2η k pj` Q( k Sx) + √ < Sx, eˆp >< Sx, eˆj >< Sx, eˆ` > β k p,j,`=1 is real, by (3.9) and (3.10) we have

q Z √ √ √ k2ηS h √ i 2η e² √ 2η det(1 − 2ik SQA0 S) FA0 ( x) exp ik Q( Sx) c Xk k (3.13) h X∞ √ √ √ i i 2η 2η 2η pj` × exp √ < k Sx, eˆp >< k Sx, eˆj >< k Sx, eˆ` > β µ(dx) k p,j,`=1 δ = o(e−k ).

Then Perturbative Expansion of Chern-Simons Theory 36

Z √ k2ηS h √ i e² √ 2η FA0 ( x) exp ik Q( Sx) c Xk k n X∞ X∞ √ √ √ o i 2η 2η 2η pj` r (√ < k Sx, eˆp >< k Sx, eˆj >< k Sx, eˆ` > β ) /r! µ(dx) r=N+1 k p,j,`=1 Z √ k2ηS h √ i e² √ 2η = FA0 ( x) exp ik Q( Sx) c Xk k ³ h X∞ √ √ √ i i 2η 2η 2η pj` × exp √ < k Sx, eˆp >< k Sx, eˆj >< k Sx, eˆ` > β k p,j,`=1 n XN X∞ √ √ √ o´ i 2η 2η 2η pj` r − (√ < k Sx, eˆp >< k Sx, eˆj >< k Sx, eˆ` > β ) /r! µ(dx), r=1 k p,j,`=1 which , together with (3.12) and (3.13), completes the proof of Lemma 4.

By Lemma 4, for sufficiently large k we have

(3.14)

q Z √ √ √ k2ηS h √ i 2η e² √ 2η det(1 − 2ik SQA0 S) FA0 ( x) exp ik Q( Sx) X k n X∞ X∞ √ √ √ o i 2η 2η 2η pj` r × (√ < k Sx, eˆp >< k Sx, eˆj >< k Sx, eˆ` > β ) /r! r=N+1 k p,j,`=1 µ(dx) q Z √ √ √ k2ηS h √ i 2η e² √ 2η = det(1 − 2ik SQA0 S) FA0 ( x) exp ik Q( Sx) Xk k n X∞ X∞ √ √ √ o i 2η 2η 2η pj` r × (√ < k Sx, eˆp >< k Sx, eˆj >< k Sx, eˆ` > β ) /r! r=N+1 k p,j,`=1 δ µ(dx) + o(e−k ). Set

HL,J (< x, eˆ1 >, < x, eˆ2 >, ··· , < x, eˆL >) √ k2ηS n X∞ i XL √ √ √ o = Fe²,J ( √ x) (√ < k2ηSx, eˆ >< k2ηSx, eˆ >< k2ηSx, eˆ > βpj`)r/r! L,A0 p j ` k r=N+1 k p,j,`=1 and

n XL p o L 4 δ Xk = x ∈ X; λj |< x, eˆj >|≤ 3k . j=1 Perturbative Expansion of Chern-Simons Theory 37

Then by the estimations similar to (1),(4) and (6) of Lemma 1, we have

Z √ k2ηS h √ i e² √ 2η FA0 ( x) exp ik Q( Sx) Xk k n X∞ X∞ √ √ √ o i 2η 2η 2η pj` r × (√ < k Sx, eˆp >< k Sx, eˆj >< k Sx, eˆ` > β ) /r! µ(dx) r=N+1 k p,j,`=1 Z h √ i 2η = lim lim exp ik QL( Sx) HL,J (< x, eˆ1 >, < x, eˆ2 >, ··· , < x, eˆL >)µ(dx). L→∞ J→∞ L Xk Since

Z h √ i 2η exp ik QL( Sx) HL,J (< x, eˆ1 >, < x, eˆ2 >, ··· , < x, eˆL >)µ(dx) L Xk Z Z h L i L 2 X Y 1 xj 2η 2 − 2 = ··· P √ exp ik λjµjxj HL,J (x1, x2, ··· , xL) √ e dxj L 4 δ 2π { j=1 λj |xj |≤3k } j=1 j=1 Z Z Z

= √ ··· P √ 3kδ 1 δ 4 1 δ L−1 4 |x1|≤ √4 |x2|≤ √4 (3k − λ1|x1|) |xL|≤ √4 (3k − j=1 λj |xj |) λ1 λ2 λL YL 1 h 1 i × H (x , x , ··· , x ) √ exp − (1 − 2ik2ηλ µ )x2 dx , L,J 1 2 L 2 j j j j j=1 2π setting π z = 1 − 2ik2ηλ µ =| z | e−iαj , 0 < sign (µ )α < , j j j j j j 2 we get by the Cauchy integral theorem,

Z h i 1 1 2η 2 P √ HL,J (x1, x2, ··· , xL)√ exp − (1 − 2ik λLµL)xL dxL 1 δ L−1 4 2 |xL|≤ √4 (3k − j=1 λj |xj |) 2π Z λL αL αL 1 |zL| 2 i 2 i 2 − 2 xL = P √ e HL,J (x1, ··· , e xL)√ e dxL 1 δ L−1 4 |xL|≤ √4 (3k − j=1 λj |xj |) 2π λL Z αL Z αL L−1 2 π+ 2 h X p i 1 zL 1 δ 4 iθ 2 + ( + )√ exp − [ √ (3k − λj | xj |)e ] 2 4 λ 0 π 2π L j=1 LX−1 p LX−1 p 1 δ 4 iθ 1 δ 4 iθ × HL,J (x1, ··· , √ (3k − λj | xj |)e ) √ (3k − λj | xj |)ie dθ. 4 λ 4 λ L j=1 L j=1

Further Perturbative Expansion of Chern-Simons Theory 38

Z Z Z Z αL Z αL 2 π+ 2 | √ ··· P √ ( + ) 3kδ 1 δ 4 1 δ L−2 4 |x1|≤ √4 |x2|≤ √4 (3k − λ1|x1|) |xL−1|≤ √4 (3k − j=1 λj |xj |) 0 π λ1 λ2 λL−1 h LX−1 p i LX−1 p 1 zL 1 δ 4 iθ 2 1 δ 4 iθ √ exp − [ √ (3k − λj | xj |)e ] HL,J (x1, ··· , √ (3k − λj | xj |)e ) 2 4 λ 4 λ 2π L j=1 L j=1 LX−1 p LY−1 £ i 1 δ 4 iθ 1 1 2 × √ (3k − λj | xj |)ie √ exp − zjx dxjdθ | 4 λ 2 j L j=1 j=1 2π

Z Z Z Z |αL| 2 ≤ √ ··· P √ 3kδ 1 δ 4 √ 1 δ L−2 4 |x1|≤ √4 |x2|≤ √4 (3k − λ1|x1|) |xL−1|≤ 4 (3k − j=1 λj |xj |) 0 λ1 λ2 λL−1 h 2iθ LX−1 p i X1 ν LX−1 p 1 Re(zLe ) δ 4 2 (−1) δ 4 √ exp − √ (3k − λj | xj |) | HL,J (x1, ··· , √ (3k − λj | xj |) 2 λ 4 λ 2π L j=1 ν=0 L j=1

L−1 2 Y x iθ 1 δ 1 − j × e ) | √ 3k √ e 2 dxjdθ. 4 λ L j=1 2π

Since

Z Z Z αL αL i 2 i 2 √ ··· P √ e HL,J (x1, ··· , e xL) 3kδ 1 δ 4 1 δ L−1 4 |x1|≤ √4 |x2|≤ √4 (3k − λ1|x1|) |xL|≤ √4 (3k − j=1 λj |xj |) λ1 λ2 λL L−1 2 Y z x 1 − j j 1 − |zL| x2 √ e 2 dxj √ e 2 L dxL j=1 2π 2π Z Z L−1 z x2 αL αL Y 1 j j 1 |zL| 2 i 2 i 2 − 2 − 2 xL = ··· P √ e HL,J (x1, ··· , e xL) √ e dxj √ e dxL, L 4 δ 2π 2π { j=1 λj |xj |≤3k } j=1 the right hand side of the above equality is equal to

Z Z Z

√ ··· P √ 3kδ 1 δ 4 1 δ L−2 4 |x1|≤ √4 |x2|≤ √4 (3k − λ1|x1|) |xL|≤ √4 (3k − j=1 λj |xj |) Z λ1 λ2 λL αL αL i 2 i 2 P √ e HL,J (x1, ··· , e xL) 1 δ L 4 |xL−1|≤ √4 (3k − j6=L−1,j=1 λj |xj |) λL−1 L−2 2 Y z x z x2 1 − j j 1 − |zL| x2 1 − L−1 L−1 √ e 2 dxi √ e 2 L dxL √ e 2 dxL−1. j=1 2π 2π 2π

Using the Cauchy integral theorem, we have Perturbative Expansion of Chern-Simons Theory 39

Z z x2 αL 1 L−1 L−1 i 2 − 2 P √ HL,J (x1, ··· , e xL)√ e dxL−1 |x |≤ √ 1 (3kδ− L 4 λ |x |) 2π L−1 4 λ j6=L−1,j=1 j j Z L−1

= P √ 1 δ L 4 |xL−1|≤ √4 (3k − j6=L−1,j=1 λj |xj |) λL−1 2 α α |z |x i L−1 i L−1 i αL 1 − L−1 L−1 e 2 HL,J (x1, ··· , e 2 xL−1, e 2 xL)√ e 2 dxL−1 + R. 2π R is a remainder, which is estimated as follows.

  Z |αL−1| L 2 2iθ X p 1 Re(zL−1e ) δ 4 2 | R |≤ √ exp − p (3k − λj | xj |)  2π 0 2 λL−1 j6=L−1,j=1 1 L X ν X p α (−1) δ 4 iθ i L 1 δ × | H (x , ··· , p (3k − λ | x |)e , e 2 x ) | p 3k dθ. L,J 1 4 j j L 4 ν=0 λL−1 j6=L−1,j=1 λL−1

By repeating this argument recursively, we have

ZZ Z h XL i 2η 2 ··· P √ exp ik λjµjxj HL,J (x1, x2, ··· , xL) L 4 δ { j=1 λj |xj |≤3k } j=1

L 2 Y x 1 − j √ e 2 dx (3.15) j j=1 2π ZZ Z P α i L j α1 α2 αL j=1 2 i 2 i 2 i 2 = ··· P √ e HL,J (e x1, e x2, ··· , e xL) L 4 δ { j=1 λj |xj |≤3k } L 2 Y |z |x 1 − j j √ e 2 dxj + RL, j=1 2π where Perturbative Expansion of Chern-Simons Theory 40

| RL | XL Z Z Z ≤ √ ··· P √ √3kδ √1 δ 4 √1 δ L−1 4 |x1|≤ 4 |x2|≤ 4 (3k − λ1|x1|) |xL|≤ 4 (3k − j6=m,j=1 λj |xj |) m=1 λ1 λ2 λL Z |αm| h 2iθ L i 2 1 Re(z e ) X p 1 √ √m δ 4 2 √ δ (3.16) exp − (3k − λj | xj |) 4 3k 2π 2 λ λ 0 m j6=m,j=1 m 1 ν L X X p αm+1 (−1) δ 4 iθ i | HL,J (x1, ··· , √ (3k − λj | xj |)e , e 2 xm+1, ··· 4 λ ν=0 m j6=m,j=1

m−1 2 L 2 Y x Y |z |x i αL 1 − j 1 − j j ··· , e 2 xL) | √ e 2 dxj √ e 2 dxjdθ. j=1 2π j=m+1 2π

Now, we prove that for sufficiently large k,

q √ √ kδ 2η − 2 (3.17) | det(1 − 2ik SQA0 S) | supL | RL |= o(e ).

We begin by remarking that if

n 3kδ 1 p L √ √ δ 4 x ∈ x ∈ R ; | x1 |≤ 4 , | x2 |≤ 4 (3k − λ1 | x1 |), ··· , λ1 λ2 1 mX−2 p 1 mX−1 p | x |≤ p (3kδ − 4 λ | x |), | x |≤ p (3kδ − 4 λ | x |), ··· , m−1 4 j j m+1 4 j j λm−1 j=1 λm+1 j=1 LX−1 p o 1 δ 4 | xL |≤ √ (3k − λj | xj |) , 4 λ L j6=m,j=1 the following inequalities hold. We have

P √ m−1 iθ δ L 4 L X p p X p α ` e (3k − `6=m,`=1 λ` | x` |) m i ` ` | λ`x`a + λm √ a + λ`e 2 x`a | R,ij 4 λ R,ij R,ij `=1 m `=m+1 P √ n mX−1 p p eiθ(3kδ − L 4 λ | x |) XL p o 4 4 `6=m,`=1 ` ` 4 ≤ λ` | x` | + | λm √ | + λ` | x` | 4 λ `=1 m `=m+1 p p 4 ` δ 4 ` × sup λ` | aR,ij |≤ 3k sup λ` | aR,ij | . ` `

2 1 Then by (S.3) and η + δ ≤ 2η < 12 < 2 , Perturbative Expansion of Chern-Simons Theory 41

P | det ∗d ∗ D n iθ δ L √4 α o | J,R A0 η P √ √ e (3k − λj |xj |) P √ j √k m−1 j6=m,j=1 L i 2 j=1 λj xj eˆj + λm √4 eˆm+ j=m+1 λj e xj eˆj k λm h kη X∞ p i δ 4 ` c12 ≤ exp √ 3k sup λ` | aR,ij | ≤ e . k i,j=1 `

Since

m−1 iθ δ PL √ X p p e (3k − λ` | x` |) \² `6=m,`=1 \² | λ`x`((C (t), 0), eˆ`) + λm √ ((C (t), 0), eˆm) u λ u `=1 m L X p α i ` \² + λ`e 2 x`((Cu(t), 0), eˆ`) | `=m+1 P √ n mX−1 p p iθ δ L XL p o e (3k − `6=m,`=1 λ` | x` |) ² ≤ λ` | x` | + | λm √ | + λ` | x` | kC (t)k0 λ u `=1 m `=m+1 δ ² ≤ 3k kCu(t)k0 and

u | A0 (t) |≤ c2(A0)t, similarly we have for sufficiently large k,

ν L X p αm+1 α ²,J (−1) δ 4 iθ i i L | F (x1, ··· , √ (3k − λj | xj |)e , e 2 xm+1, ··· , e 2 xL) | L,A0 λ m j6=m,j=1

≤ c13(A0).

1 By Assumption 2 and since we take δ < η < 12 ,

P p ∞ m−1 iθ δ L 4 L X 3η X p p X p α k e (3k − p6=m,p=1 λp | xp |) i p | i√ { λpxp + λm √ + λpe 2 xp} 4 λ r=N+1 k p=1 m p=m+1 p m−1 iθ δ PL L n X p p 4 X p α o e (3k − j6=m,j=1 λj | xj |) i j × λjxj + λm √ + λje 2 xj 4 λ j=1 m j=m+1 P √ m−1 iθ δ L 4 L n X p p X p α o e (3k − `6=m,`=1 λ` | x` |) i ` pj` r × λ`x` + λm √ + λ`e 2 x` β | /r! 4 λ `=1 m `=m+1 3η k√ (3kδ)3β ≤ e k = c14(β).

Then for sufficiently large k, Perturbative Expansion of Chern-Simons Theory 42

PL p iθ δ 4 α ν e (3k − j6=m,j=1 λj | xj |) i m+1 i αL √ 2 2 | HL,J (··· , (−1) 4 , e xm+1, ··· , e xL) | λm c12 ≤ e c13(A0)c14(β) = c15(A0, β) < +∞.

Now first we consider the case where

XL p 3kδ 3kδ − 4 λ | x |> . j j 2 j6=m,j=1 Since

2iθ i(−αm+2θ) zme = |zm|e and | − αm + 2θ| ≤ |αm|, 2iθ we get Re(zme ) ≥ 1 and further noticing that

√ 2η 2η | αm |≤ 2k λm | µm |, | zm |=| 1 − 2 −1k λmµm |≥ 1 and noticing (S.1), we have for sufficiently large k,

L Z Z Z Z |αm| X 2 | √ ··· P √ 3kδ 1 δ 4 1 δ L−1 4 |x1|≤ √ |x2|≤ √ (3k − λ1|x1|) |x |≤ √ (3k − λj |xj |) 0 m=1 4 4 L 4 λ j6=m,j=1  λ1 λ2 L 2iθ XL p 1 Re(zme ) δ 4 2 1 δ √ exp − √ (3k − λj | xj |)  √ 3k 2π 2 λ 4 λ m j6=m,j=1 m

1 ν L X X p αm+1 α (−1) δ 4 iθ i i L × | HL,J (x1, ··· , √ (3k − λj | xj |)e , e 2 xm+1, ··· , e 2 xL) | 4 λ ν=0 m j6=m,j=1

m−1 2 L 2 Y x Y |z |x 1 − j 1 − j j P √ 2 2 × χ δ L 4 3kδ √ e dxj √ e dxjdθ | {3k − j6=m,j=1 λj |xj |> 2 } j=1 2π j=m+1 2π XL 1 h (3kδ)2 i ≤ 2c (A , β)√ √ 3kδ | α | exp − p 15 0 4 m m=1 2π λm 8 max λm m=1,2,··· ,L XL p h 2δ i 12c15(A0, β) 4 3 2η+δ 9k ≤ √ λm | µm | k exp − p m=1 2π 8 max λm m=1,2,··· ,L −kδ ≤ c16e .

Secondly we look at the complementary case :

XL p 3kδ (3.18) 3kδ − 4 λ | x |≤ . j j 2 j6=m,j=1 Perturbative Expansion of Chern-Simons Theory 43

In this case, dominating

h 2iθ XL p i Re(zme ) δ 4 2 exp − √ (3k − λj | xj |) 2 λ m j6=m,j=1 by 1 in the above estimation, we have that the estimation of the right hand side of (3.13) for this case is reduced to

3kδ XL p µˆ( ≤ 4 λ | x |), 2 j j j6=m,j=1

x2 |z |x2 Qm−1 1 − j QL 1 − j j whereµ ˆ = √ e 2 dx √ e 2 dx . j=1 2π j j=m+1 2π j Since

Z Z L m−1 2 L 2 h X p i Y xj Y |zj |xj 4 1 − 1 − ··· exp λj | xj | √ e 2 dxj √ e 2 dxj L−1 2π 2π R j6=m,j=1 j=1 j=m+1 h h X∞ p ii 4 ≤ E exp λj |< x, eˆj >| ≤ c17, j=1 so that

δ L X p 3kδ 3k 4 − µˆ( ≤ λ | x |) ≤ c e 2 . 2 j j 17 j6=m,j=1 Hence in the case of (3.18), we have

L Z Z Z Z |αm| X 2 | √ ··· P √ √3kδ √1 δ 4 √1 δ L−1 4 |x1|≤ 4 |x2|≤ 4 (3k − λ1|x1|) |xL|≤ 4 (3k − j6=m,j=1 λj |xj |) 0 m=1 λ1 λ2 λL h 2iθ XL p i 1 Re(zme ) δ 4 2 1 δ √ exp − √ (3k − λj | xj |) √ 3k 2π 2 λ 4 λ m j6=m,j=1 m 1 ν L X X p αm+1 α (−1) δ 4 iθ i i L × | HL,J (x1, ··· , √ (3k − λj | xj |)e , e 2 xm+1, ··· , e 2 xL) | 4 λ ν=0 m j6=m,j=1

m−1 2 L 2 Y x Y |z |x 1 − j 1 − j j P √ 2 2 χ δ L 4 3kδ √ e dxj √ e dxjdθ | {3k − j6=m,j=1 λj |xj |≤ 2 } j=1 2π j=m+1 2π L X p δ 12c17c15(A0, β) 4 3 2η+δ − 3k −kδ ≤ √ λm | µm | k e 2 ≤ c18e . m=1 2π This, together with (3.10), yields the completion of the proof of (3.17). Since we take δ < η, we have for x ∈ D, Perturbative Expansion of Chern-Simons Theory 44

√ X∞ 4 Sx η |< q √ √ , eˆj >|≤ 3k . 2η j=1 1 − 2ik SQA0 S Hence by the Assumption 2,

1 X∞ (√ |< V x, eˆ >< V x, eˆ >< V x, eˆ > βpj` |)r (3.19) k p k j k ` k p,j,`=1 1 ≤ (27β)r(√ )rk6ηr, k

6 1 so that 0 < 6η < 12 = 2 gives some natural number n(N + 1) such that

√ k6η (3.20) lim ( k)N+1(√ )n(N+1) = 0. k→∞ k By (3.19) and the estimations similar to (1) and (6) of Lemma 1, we have

Z 1 V q Fe² (√k x) √ √ A0 2η D k det(1 − 2ik SQA0 S) n X∞ X∞ o i pj` r × (√ < Vkx, eˆp >< Vkx, eˆj >< Vkx, eˆ` > β ) /r! µ(dx) r=N+1 k p,j,`=1 ZZ Z P α i L j α1 α2 αL j=1 2 i 2 i 2 i 2 = lim lim ··· P √ e HL,J (e x1, e x2, ··· , e xL) L→∞ J→∞ L 4 δ { j=1 λj |xj |≤3k } L 2 Y |z |x 1 − j j √ e 2 dxj, j=1 2π which ,together with (3.13), (3.14), (3.15),(3.16) and (3.17), completes the proofs of (2.34) and (2.35). In a manner similar to the proof of (5) of Lemma 1, under the Renormalization assumption on QA0 and the Assumption 2, we further have

h X∞ i i pj` 2b E |√ < Vkx, eˆp >< Vkx, eˆj >< Vkx, eˆ` > β | k p,j,`=1 X∞ 1 2b 1 6b ≤ c19(b)(√ ) ( p ) . k j=1 |µj|

This and estimations similar to (2) and (7) of Lemma 1, together with (3.19) and (3.20) , completes the proof of Theorem 2. Perturbative Expansion of Chern-Simons Theory 45

Acknowledgement. Both authors gratefully acknowledge a kind invitation of R. L´eandreto attend the workshop held in C.I.R.M. in Luminy in 2004. The second author would like to express his sincere thanks to Prof. A. Hahn for his valuable suggestions.

References

[1] Albeverio, S., Sch¨afer,J.: Abelian Chern-Simons theory and linking numbers via oscillatory integrals. J. Math. Phys. 36, 2157–2169 (1995)

[2] Albeverio, S., Sengupta, A.: A mathematical construction of the non-abelian Chern- Simons functional integral. Commun. Math. Phys. 186, 563–579 (1997)

[3] Albeverio, S., Hahn, A., Sengupta, A.: Rigorous Feynman path integrals with ap- plications to quantum theory, gauge fields and topological invariants. To appear in Proc. Lisbon Conf. Editors J.C. Zambrini et al., 2004

[4] Albeverio, S., Mazzucchi, S.: Some new developments in the theory of path integrals with applications to quantum theory. J. Stat. Phys. 115, 191–215 (2004)

[5] Albeverio, S., Hφegh-Krohn, R., Mazzucchi, S.: Mathematical Theory of Feynman Path Integrals- An Introduction, 2nd ed.. Berlin, Heidelberg, New York : Springer, 2006

[6] Altschuler, D., Freidel, L.: Vassiliev knot invariants and Chern-Simons perturbation theory to all orders. Comm. Math. Phys. 187, 261–287 (1997)

[7] Atiyah, M.: The geometry and Physics of knots.: Cambridge U.P.,1990

[8] Aref’eva, I. Ya.: Nonabelian Stokes formula, Theoret. and Math. Phys. 43, 353–356 (1980)

[9] Axelrod, S., Singer, I. M.: Chern-Simons perturbation theory. Proceedings of the XXth International Conference on Differential Geometric Methods in Theoretical Physics, Vol. 1 (New York, 1991), 3–45, World Sci. Publishing, River Edge, N. J., 1992

[10] Axelrod, S., Singer, I. M.: Chern-Simons perturbation theory. II. J. Differ. Geom. 39(1), 173–213 (1994)

[11] Bar-Natan, D.: Perturbative Chern-Simons theory. J. Knot Theory and its Ramifi- cations 4, 503–547 (1995)

[12] Bar-Natan, D.: On the Vassiliev knot invariants. Topology 34, 423–472 (1995)

[13] Bar-Natan, D., Witten, E.: Perturbation expansion of Chern-Simons theory with non-compact gauge group. Commun. Math. Phys. 141, 423–440 (1991)

[14] Bott, R., Taubes, C.: On the self-linking of knots. J. Math. Phys. 35(10), 5247–5287 (1994) Perturbative Expansion of Chern-Simons Theory 46

[15] Bott, R., Tu, L. W.: Differential Forms in Algebraic Topology . Grad. Texts in Math. 82. New York, Heidelberg, Berlin : Springer, 1982

[16] Chen, K. T.: Iterated path integrals. Bull. Amer. Soc. 83, 831–879 (1977)

[17] Dalecky, Yu.L., Fomin, S.V.: Measures and Differential Equations in Infinite- Dimensional Space. Dordrech, Boston, London : Kluwer Academic Publishers, 1991

[18] De Rham, G.: Differentiable Manifolds. Grundlehren Math. Wiss. 266. Berlin, Hei- delberg, New York, Tokyo : Springer, 1984

[19] Dollard, J. D., Friedman, C. N.: Product Integration with Applications to Differ- ential Equations. Encyclopedia of Math. Appl. 10. Massachusetts : Addison-Wesley Publishing Co.,1979

[20] Fr¨ohlich, J., King, C.: The Chern-Simons theory and knot polynomials. Commun. Math. Phys. 126, 167–199 (1989)

[21] Gelfand, I. M., Vilenkin, N. Ya.: Generalized Functions, Vol. 4, Applications of Harmonic Analysis. New York, London : Academic Press, 1964

[22] Gilkey, P. B.: Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem. Mathematics Lecture Series 11. Wilmington, Del.,: Publish or Perish Inc., 1984

[23] Guadagnini, E., Martellini, M., Mintchev, M.: Wilson lines in Chern-Simons theory and link invariants. Nuclear Phys. B 330,575–607 (1990)

[24] Hahn, A.: Chern-Simons theory on R3 in axial gauge. Dissertation, Rheinische Friedrich-Wilhelms-Universit¨atBonn, Bonn, 2000 : Bonner Math. Schriften 345, Univ. Bonn, Bonn, 2001

[25] Hahn, A.: The Wilson loop observables of Chern-Simons theory on R3 in axial gauge. Commun. Math. Phys. 248, 467–499 (2004)

[26] Hahn, A.: Chern-Simons models on S2 × S1, torus gauge fixing, and link invariants I. J. Geom. Phys. 53, 275–314 (2005)

[27] Itˆo,K.: Generalized uniform complex measures in the Hilbertian metric space with their application to the Feynman integral. Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66), Vol. II, Contributions to Probability Theory, Part 1, 145–161, Berkeley, Calif.: Univ. California Press, 1967

[28] Kaufmann, L. H. : Witten’s integral and Kontsevich integral. in ”Particles ,fields and gravitation”, Lect. Notes 308-381. 1998

[29] Kuo, H. H.: Gaussian Measures in Banach Space. Lecture Notes in Math. 463. Berlin, Heidelberg, New York : Springer, 1975

[30] Lawson, Jr. H. B., Michelsohn, M. L.:, Spin Geometry. Princeton Math. Ser. 38. Princeton, N. J.: Princeton Univ. Press, 1989. Perturbative Expansion of Chern-Simons Theory 47

[31] L´eandre,R.: Invariant Sobolev calculus on the free loop space. Acta Appl. Math. 46, 267–350 (1997)

[32] Malliavin, P., Taniguchi, S.: Analytic functions, Cauchy formula, and stationary phase on a real abstract Wiener space. J. Funct. Anal. 143, 470–528 (1997)

[33] Mitoma, I.: One loop approximation of the Chern-Simons integral. Dimensional Acta Appl. Math. 63, 253–273 (2000)

[34] Mitoma, I.: Wiener space approach to a perturbative Chern-Simons integral. Stochas- tic Processes, Physics and Geometry, New Interplays II: (Leipzig, 1999), 471–480, Providence, R. I.: CMS Conf. Proc. 29, Amer. Math. Soc., 2000

[35] Mitoma, I., Nishikawa, S.: Asymptotic expansion of the one loop approximation to the Chern-Simons integral. Preprint 2006

[36] Morrey, Jr., C. B.: Multiple Integrals in the Calculus of Variation. Grundlehlen Math. Wiss. 130, Berlin, Heidelberg, New York : Springer, 1966

[37] Niemann, M.: Das Chern-Simons Modell : Quantisierung der kopplungs konstante und simpliciale approximation. Diploma thesis, Univ. Bonn, Bonn. Sept. 2004

[38] Reed, M., Simon, B.: Methods of Modern Mathematical Physics I, Functional Anal- ysis. Second edition, New York : Academic Press Inc., 1980

[39] Schwarz, A.: The partition function of a degenerate functional . Commun. Math. Phys. 67, 1-16 (1973)

[40] Tavares, J. N.: Chen integrals, generalized loops and loop calculus. Internat. J. Modern Phys. A 9, 4511–4548 (1994)

[41] Totoki, H.: A method of construction for measures on function spaces and its applica- tions for stochastic processes. Mem.. Fac. Sci. Kyushu Univ. Ser. A, Math.15,178– 190 (1962)

[42] Witten, E.: Quantum field theory and the Jones polynomial. Commun. Math. Phys. 121, 351–399 (1989)

[43] Yosida, K.: Functional Analysis. Grundlehlen Math. Wiss. 123, Second edition, Berlin, Heidelberg, New York : Springer, 1968

Sergio Albeverio Institut f¨urAngewandte Mathematik Universit¨atBonn Wegeler Strasse 6, 53115, Bonn Germany E-mail address: [email protected]

Itaru Mitoma Department of Mathematics Perturbative Expansion of Chern-Simons Theory 48

Saga University 840-8502 Saga Japan E-mail address: [email protected] Bestellungen nimmt entgegen:

Institut für Angewandte Mathematik der Universität Bonn Sonderforschungsbereich 611 Wegelerstr. 6 D - 53115 Bonn

Telefon: 0228/73 4882 Telefax: 0228/73 7864 E-mail: [email protected] http://www.iam.uni-bonn.de/sfb611/

Verzeichnis der erschienenen Preprints ab No. 310

310. Eberle, Andreas; Marinelli, Carlo: Stability of Sequential Markov Chain Monte Carlo Methods

311. Eberle, Andreas; Marinelli, Carlo: Convergence of Sequential Markov Chain Monte Carlo Methods: I. Nonlinear Flow of Probability Measures

312. Albeverio, Sergio; Hryniv, Rostyslav; Mykytyuk, Yaroslav: Reconstruction of Radial Dirac and Schrödinger Operators from Two Spectra

313. Eppler, Karsten; Harbrecht, Helmut: Tracking Neumann Data for Stationary Free Boundary Problems

314. Albeverio, Sergio; Mandrekar, Vidyadhar; Rüdiger, Barbara: Existence of Mild Solutions for Stochastic Differential Equations and Semilinear Equations with Non-Gaussian Lévy Noise

315. Albeverio, Sergio; Baranovskyi, Oleksandr; Pratsiovytyi, Mykola; Torbin, Grygoriy: The Set of Incomplete Sums of the First Ostrogradsky Series and Anomalously Fractal Probability Distributions on it

316. Gottschalk, Hanno; Smii, Boubaker: How to Determine the Law of the Noise Driving a SPDE

317. Gottschalk, Hanno; Thaler, Horst: AdS/CFT Correspondence in the Euclidean Context

318. Gottschalk, Hanno; Hack, Thomas: On a Third S-Matrix in the Theory of Quantized Fields on Curved Spacetimes

319. Müller, Werner; Salomonsen, Gorm: Scattering Theory for the Laplacian on Manifolds with Bounded Curvature

320. Ignat, Radu; Otto, Felix: 2-d Compactness of the Néel Wall

321. Harbrecht, Helmut: A Newton Method for Bernoulli’s Free Boundary Problem in Three Dimensions

322. Albeverio, Sergio; Mitoma, Itaru: Asymptotic Expansion of Perturbative Chern-Simons Theory via Wiener Space