Asymptotic expansion of perturbative Chern–Simons theory via Wiener space

Bull. Sci. math. 133 (2009) 272–314 www.elsevier.com/locate/bulsci

Asymptotic expansion of perturbative Chern–Simons theory via Wiener space ✩ Sergio Albeverio a,∗ , Itaru Mitoma b a Institut für Angewandte Mathematik, Universität Bonn, Wegeler Strasse 6, 53115, Bonn, Germany b Department of Mathematics, Saga University, 840-8502 Saga, Japan

Received 9 July 2007; accepted 9 July 2007 Available online 7 November 2007

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. © 2007 Published by Elsevier Masson SAS. MSC: primary 57R56; secondary 28C70, 57M27 Keywords: Chern–Simons integral; Asymptotic expansion; Abstract Winer space; Malliavin–Taniguchi formula; Linking number

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 [1–4,6–14,20,23,28]. The first author and Schäfer have 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 Schäfer, using methods ✩

Partly supported by the Grant-in-Aid for Scientific Research (C) of the Japan Society for the Promotion of Science, No. 17540124. * Corresponding author. E-mail addresses: [email protected] (S. Albeverio), [email protected] (I. Mitoma). 0007-4497/$ – see front matter © 2007 Published by Elsevier Masson SAS. doi:10.1016/j.bulsci.2007.07.003

S. Albeverio, I. Mitoma / Bull. Sci. math. 133 (2009) 272–314

273

of white noise analysis. The extension of the representation to the non-Abelian case for R 3 (resp. S 2 × 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 Fröhlich–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 [3] and [5]. Hahn subsequently managed to extend the results to the case of S 2 × S 1 , using the “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 [11,13,23] (see also, e.g., [6,9,10,12,14] for further developments) 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 Itô concerning Feynman path integrals [27] (there are also some relations with methods in, e.g., [31,36]). 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  F (A)eL(A) D(A), A

where L(A) = −

ik 4π



  2 Tr A ∧ dA + A ∧ A ∧ A . 3

M

The parameter k, called charge, is a positive integer. D(A) is a heuristic “flat measure”. Let us expand heuristically in powers of the “ 32 -part” under above integral, separating it in a “finite part” which consists of the terms up to order N in k and the rest:  1 F (A)eL(A) D(A) Z A

=

r      N  ik ik 2 1 F (A) exp − Tr A ∧ dA − Tr A ∧ A ∧ A /r! D(A) Z 4π 4π 3 r=0 A M M     1 ik + F (A) exp − Tr A ∧ dA Z 4π M A

∞  r    ik 2 − Tr A ∧ A ∧ A /r! D(A). × 4π 3 r=N +1

M

Z denotes the heuristic normalization constant i.e.     ik Tr A ∧ dA D(A). Z = exp − 4π A

M

274

S. Albeverio, I. Mitoma / Bull. Sci. math. 133 (2009) 272–314

Based on the perturbative formulation of the Chern–Simons integral [9,13,23] and the idea of Itô [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 example of gauge invariant observ A ables (holonomy operators) such that F (A) = sj =1 Tr P e γj , 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 -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 introduced in [27]. √ √ π In the sequel z denotes the branch of the root of z ∈ C where −π 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 cˆ a fermionic 3-form. The BRS operator laws are δA = −DA c, δ cˆ = iφ,

1 δc = [c, c], 2 δφ = 0,

where DA = dA0 + [A, ·] and dA0 is the covariant exterior derivative such that dA0 = d + [A0 , ·].   Let Eu , u = 1, 2, . . . , d, be the orthonormal basis of G, A = du=1 Au · Eu and B = du=1 B u · Eu . We set [A, B] = i,j [Ei Ej − Ej Ei ]Ai ∧ B j . For simplicity, we assume Assumption 1. A0 is isolated and irreducible, i.e. the whole de Rham cohomology vanishes, i.e. H ∗ (M, dA0 ) = 0

(2.1)

(see [9,13]). 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  k V (A) = Tr cˆ ∗ dA0 ∗ A. 2π M

S. Albeverio, I. Mitoma / Bull. Sci. math. 133 (2009) 272–314

275

√ √ k/2π c and cˆ = ∗ k/2π cˆ (see [23, p. 578]). Since    ik 2 L(A0 + A) = L(A0 ) − Tr A ∧ dA0 A + A ∧ A ∧ A 4π 3

Let c =

M

and δV (A) =



k 2π

Tr{iφ ∗ dA0 ∗ A − cˆ ∗ dA0 ∗ DA c}, M

the Lorentz gauge fixed path integral form of the heuristic Chern–Simons integral is given by       1 D(A)D(φ)D(cˆ )D(c )F (A0 + A) exp L(A0 + A) − δV (A) Zk  C  A Φ C

=

1 Zk

   

  D(A)D(φ)D(cˆ )D(c )F (A0 + A) exp L(A0 )

 C  A Φ C

      ik 2 × exp ik (A, φ), QA0 (A, φ) + − Tr A ∧ A ∧ A + Tr cˆ dA0 ∗ DA c , 4π 3 M

M

(2.2)

where heuristically     Zk = D(A)D(φ)D(cˆ )D(c )  C  A Φ C

     × exp L(A0 ) + ik (A, φ), QA0 (A, φ) + + Tr cˆ dA0 ∗ dA0 c . M

Here Ω n is the space of G-valued n-forms with the inner product  (ω1 , ω2 )0 = − Tr ω1 ∧ ∗ω2 M

and



 (A, φ), (B, ϕ) + = (A, B)0 + (φ, ϕ)0

denotes the inner product of the Hilbert space L2 (Ω+ ) = L2 (Ω 1 ⊕ Ω 3 ). We shall set   · 0 = (·, ·)0 and · + = (·, ·)+ .

(2.3)

+ dA0 ∗)J , where J φ = −φ if φ is a zero or a three form and J φ = φ if Further QA0 = 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. Set 0 = ∗dA0 ∗ dA0 , i.e. 0 is the Laplacian in L2 (Ω 0 ). Let {νj , ξj , j = 1, 2, . . .} be the eigensystem of 0 in L2 (Ω 0 ), 1 4π (∗dA0

cˆ =

∞  j =1

ξj cˆj

and c =

∞  j =1

ξj cj .

276

S. Albeverio, I. Mitoma / Bull. Sci. math. 133 (2009) 272–314

Since 

Tr cˆ dA0 ∗ DA c = −(cˆ , ∗dA0 ∗ DA c )0

M

  = −(cˆ , ∗dA0 ∗ dA0 c )0 − dA0 cˆ , [A, c ] 0 ,

we have heuristically  

    D(cˆ )D(c ) exp Tr cˆ dA0 ∗ DA c = det ∗dA0 ∗ DA 



 C  C

M

  a11 a12 · a1n  ∞  a21 a22 · a2n   = νj ·  · · · ·  j =1  an1 an2 · ann  · · · · = det 0 detR ∗dA0 ∗ DA ,

where aii = 1 −

1 νi

 ·  ·  ·  ·  ·

 Tr dA0 ξi ∧ ∗[A, ξi ],

i = 1, 2, . . .

M

and aij = −

1 νi

 Tr dA0 ξi ∧ ∗[A, ξj ],

i, j = 1, 2, . . . .

M

Then considering     D(cˆ )D(c ) exp Tr cˆ dA0 ∗ dA0 c = det 0 ,  C  C

M

we arrive at the perturbative heuristic formulation of the normalized Chern–Simons integral (2.2) such that   1 D(A)D(φ)F (A0 + A) Z1,k A Φ

     ik 2 Tr A ∧ A ∧ A , × detR ∗dA0 ∗ DA exp ik (A, φ), QA0 (A, φ) + − 4π 3

(2.4)

M

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

(2.5)

S. Albeverio, I. Mitoma / Bull. Sci. math. 133 (2009) 272–314

277

2.1. Regularized determinant To provide mathematical meaning to the above objects, we have first of all to regularize the formal determinant detR ∗dA0 ∗ DA . Since QA0 is self-adjoint and elliptic, and M is compact, QA0 has pure point spectrum [22].   φ u,φ Let μi , ei = (eiA = du=1 eiu,A · Eu , ei = du=1 ei · Eu ), i = 1, 2, . . . , be the eigenvalues resp. 2 eigenvectors of QA0 in L (Ω+ ). By Assumption 1, the eigenvectors form a CONS (Complete Orthonormal System) of L2 (Ω+ ). Define  1  aR,ij = − Tr ri dA0 ξi ∧ ∗[eA , rj ξj ]. νi M

Since M is compact, we can choose appropriate real numbers rj > 0, j = 1, 2, . . . , such that   |aR,ij | < +∞, (2.6) i,j

and ∞ 

|rj | < +∞.

j =1

Then for smooth A ∈ A, we consider instead of detR ∗dA0 ∗ DA , the regularized determinant defined by detReg (A) = lim detm,R ∗dA0 ∗ DA , m→∞

where

 a R (A)  11  R  a (A) detm,R ∗dA0 ∗ DA =  21  ·  R am1 (A)

aiiR (A) = 1 +

R (A) a12 R (A) a22 · R (A) am2

R (A)  · a1m  R (A)  · a2m ,  · · R (A)  · amm

∞     (A, φ), e + aR,ii =1

and aijR (A) =

∞     (A, φ), e + aR,ij . =1

We call (A, φ) smooth if for every natural number b,     (A, φ)2 = (A, φ), Q2b (A, φ) < +∞. A0 b + Then for smooth (A, φ), these series converge and the regularized determinant is well defined since ∞ ∞         (A, φ), e a    (A, φ) R,ij + N i,j =1

=1

1   |aR,ij | < +∞, 0 |μ |N0  i,j

278

S. Albeverio, I. Mitoma / Bull. Sci. math. 133 (2009) 272–314

if we take some natural number N0 such that ∞ 

1   |aR,ij | < +∞, |μ |N0 i,j

=1

which is guaranteed by the increasing rates of eigenvalues of QA0 ((c) of Lemma 1.6.3 in [22] and [30]). 2.2. Holonomy and the Poincaré dual To handle the Chern–Simons integral in an abstract Wiener space setting, we need to extend the holonomy

Pe

γ

A

,

from smooth A to the rough A which arise in the abstract Wiener space setting. We now regularize the holonomy like in Albeverio and Schäfer [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] → 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   γ [s, t] [A] =

t

 A=

γ [s,t]

  A γ˙ (τ ) dτ,

A∈A

s

defined on the vector space A. Then γ [s, t] is continuous in the sense of distributions (see, e.g., [21]) 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 R 3 and A is a G-valued smooth 1-form with compact support defined on R 3 . Let φ be a non-negative smooth function on R 3 such that the support of φ is contained in the unit ball B 3 with center 0 ∈ R 3 and  φ(x) dx = 1, R3

and define φ (x) =  −3 φ(x/) for each  > 0. If we write     ∂ u u i i A= A · Eu = Ai dx · Eu , γ˙ (τ ) = γ˙ (τ ) ∂x i γ (τ ) u i,u

for given A and γ , then we have        lim sup  Ai u (x)φ x − γ (τ ) dx − Ai u γ (τ )  = 0, →0 sτ t

and

(2.8)

R3

 3 t          Ai u (x)φ x − γ (τ ) dx γ˙ i (τ ) dτ   c1 () Au L2 (R 3 ) |t − s|.    i=1 s

(2.7)

i

R3

(2.9)

S. Albeverio, I. Mitoma / Bull. Sci. math. 133 (2009) 272–314

279

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     R γ [s, t] [A] = γ [s, t] [R∗ A] =

3 t    i=1 s

=

R3

3 t    i=1 s

  Ai u γ (τ ) + y φ (y) dy γ˙ (τ ) dτ · Eu   Aui (x)φ x − γ (τ ) dx γ˙ i (τ ) dτ · Eu

R3

to which there is associated an operator defined by     A γ [s, t] [B] = γ [s, t] [A∗ B]  

  1 3 t    i u = y Bij γ (τ ) + ty dt φ (y) dy γ˙ j (τ ) dτ · Eu , i,j =1 s



R3

0

where B = Biju dx i ∧ dx j · Eu is a G-valued smooth two-form with compact support on R 3 . Then we have the following relation which shows that A γ [s, t] gives a chain homotopy between R γ [s, t] and γ [s, t] (see [18, p. 65] for the proof). Proposition 1. For each  > 0, R γ [s, t] and A γ [s, t] are smooth currents whose supports are contained in the  tubular neighborhood of γ [s, t], and satisfy R γ [s, t] − γ [s, t] = ∂A γ [s, t] + A ∂γ [s, t], where ∂ is the boundary operator of currents. As in [18] (see also, e.g., [15]), the above construction of regularization generalizes to our case in the following manner. First take a diffeomorphism h of R 3 onto the unit ball B 3 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 s y be the map of R 3 onto itself which coincides with h ◦ sy ◦ h−1 on B 3 and with the identity at all other points, that is,  3 −1 s y (x) = h ◦ sy ◦ h (x) if x ∈ B 3 , x if x ∈ /B . Note that with a suitable choice of h we make s y to be a diffeomorphism. Then define R γ [s, t] and A γ [s, t] by the same equations above, but now by using s y and s ty . Now, let {Ui } be a finite covering of M such that each Ui is diffeomorphic to the unit ball B 3 via a diffeomorphism hi which can be extended to some neighborhoods of their closures. Using these diffeomorphisms, we transport the transformed operators R and A defined on R 3 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  = f T is a current which has its support contained in a neighborhood of the closure of Ui , and hi T  is a current which has its

280

S. Albeverio, I. Mitoma / Bull. Sci. math. 133 (2009) 272–314

support contained in the neighborhood of the closure of B 3 . Note that the support of T  = T − T  does not meet the closure of Ui . We define   Ri T = h−1 i ◦ R ◦ hi T + T ,

 Ai T = h−1 i ◦ A ◦ hi T

and set inductively 1 2 k R(k)  T = R ◦ R ◦ · · · ◦ R T ,

1 2 k−1 A(k) ◦ Ak T .  T = R ◦ R ◦ · · · ◦ R

Then R T and A T are obtained to be R T = R(K)  T,

A T =

K 

A(k)  T,

k=1

where K is the number of coverings {Ui }. The construction of the operators R and A are easily generalized to any current T defined on a compact smooth manifold of arbitrary dimension, and we have the following properties for the regularizations of currents. Proposition 2. (See [18].) Let M be a compact smooth manifold. Then for each  > 0 there exist linear operators R and A acting on the space of de Rham currents with the following properties: (1) If T is a current, then R T and A T are also currents and satisfy R T − T = ∂A T + A ∂T . (2) The supports of R T and A T are contained in an arbitrary given neighborhood of the support of T provided that  is sufficiently small. (3) R T is a smooth form. (4) For all smooth forms ϕ we have R T [ϕ] → T [ϕ] and A T [ϕ] → 0 as  → 0. Given a smooth closed curve γ : [0, 1] → M in M, let Uγ be a tubular neighborhood of γ [0, 1] in M and let j : Uγ → M denote the inclusion. For each t ∈ [0, 1] and sufficiently small  > 0 we set   Cγ (t) = j ∗ R γ [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, using       (2.10) dCγ (1) = dj ∗ R γ [0, 1] = j ∗ d R γ [0, 1] = −j ∗ R ∂ γ [0, 1] = 0. Proposition 3. (See [1].) [Cγ (1)] = [j ∗ (R γ [0, 1])] ∈ Hc2 (Uγ ) is the compact Poincaré dual of γ in Uγ , where Hc2 (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 u-component of ∗Cγ (t) by Cu (t) = Cu (t)γ = ∗Cγ (t)u · Eu .

(2.11)

S. Albeverio, I. Mitoma / Bull. Sci. math. 133 (2009) 272–314

281

Then, recalling the construction of regulators of currents and noting (2.8) and (2.9), it is not hard to see that we have        u  A  = 0, lim sup  A, Cu (t) 0 − →0 0t1

    

Au −

γ [0,t]

and

γ [0,t]

  u A   c2 (A)|t − s|



(2.12)

γ [0,s]

   C (t) − C  (s)  c1 ()|t − s|, u u 0

(2.13)

where · 0 is the norm defined in (2.3). Let us define Aγ (t) =

d    A, Cu (t) 0 · Eu u=1

and

 A(t) =

A.

γ [0,t]

By Chen’s iterated integrals (Theorem 4.3 in §1.4 of [19], p. 31, see also [16,40]), we have

Pe

γ

A0 +A

=I +

∞ 1t1 

tr−1 ··· d(A0 + A)(t1 ) d(A0 + A)(t2 ) · · · d(A0 + A)(tr ).

r=1 0 0

(2.14)

0

Then, as [1], we define the -regularizations of the holonomy and the Wilson line as follows: Wγ [A] = I

+

∞ 

Wγ,r [A],

(2.15)

r=1

where 1t1 Wγ,r [A] =

tr−1 ··· d(A0 + Aγ )(t1 ) d(A0 + Aγ )(t2 ) · · · d(A0 + Aγ )(tr )

0 0

0

and F  (A0 + A) =

s 

Tr Wγj [A].

(2.16)

j =1

Setting A (A) = F  (A0 + A) detReg (A) F 0

and x = (A, φ) = (xA , xφ ), we arrive at a regularized form of (2.4):     1 2 A (xA ) exp ik(x, QA0 x)+ − ik x Tr ∧ x ∧ x F A A A D(x), 0 Z2,k 4π 3 X

M

(2.17)

282

S. Albeverio, I. Mitoma / Bull. Sci. math. 133 (2009) 272–314

where X is the space of all x and    Z2,k = exp ik(x, QA0 x)+ D(x). X

By expanding heuristically in powers of k, for any natural number N , we have that (2.17) is equal to I (k) + II(k), with r    N  1   ik 2 I (k) = Tr xA ∧ xA ∧ xA /r! FA0 (xA ) − Z2,k 4π 3 r=0 X M   × exp ik(x, QA0 x)+ D(x) and

∞  r    ik 2   − Tr xA ∧ xA ∧ xA /r! FA0 (xA ) 4π 3 r=N +1 M X   × exp ik(x, QA0 x)+ D(x).

1 II(k) = Z2,k



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. We recall that L2 (Ω+ ) was the Hilbert space with the inner product   (A, φ), (B, ϕ) + = (A, B)0 + (φ, ϕ)0 (2.18) and the norm   (A, φ)2 = (A, A)0 + (φ, φ)0 = A 2 + φ 2 . 0 0 +

(2.19)

n be the set of all real valued n-forms and for any α, β ∈ Ω n , Let Ω  (α, β)R α ∧ ∗β. 0 = M

  Noticing that xA = du=1 xAu · Eu and xφ = du=1 xφu · Eu , we define for xˆ = (x 1 , x 2 , . . . , x d ), x u = (xAu , xφu ) and yˆ = (y 1 , y 2 , . . . , y d ), y u = (yBu , yϕu ), (x, ˆ y) ˆ =

d  (x u , y u ) u=1

and u u R (x u , y u ) = (xAu , yBu )R 0 + (xφ , yϕ )0 .

Then the real Hilbert space of xˆ corresponding to L2 (Ω+ ) is denoted by H with the inner product (x, ˆ x) ˆ =

∞  (x, ˆ eˆj )2 j =1

S. Albeverio, I. Mitoma / Bull. Sci. math. 133 (2009) 272–314

283

and the corresponding norm such that · 2 = (·,·), 1,φ

2,φ

d,φ

where eˆj = ((ej1,A , ej ), (ej2,A , ej ), . . . , (ejd,A , ej )). Then H is isometric to L2 (Ω+ ) such that (x, ˆ y) ˆ = (x, y)+

and x ˆ = x + .

(2.20)

For smooth A, by (2.18) and (2.20), we have          A, Cu (t) 0 = (A, φ), Cu (t), 0 + = x, ˆ Cu (t), 0 ,

(2.21)

where        Cu (t), 0 = (0, 0), (0, 0), . . . , ∗Cγ (t)u , 0 , . . . , (0, 0) . Set xˆγ (t) =

d      x, ˆ Cu (t), 0 · Eu . u=1

Then replacing Aγ (t) by xˆγ (t) in the definitions (2.15) and (2.16), we have ˆ =I + Wγ (x)

∞ 

Wγ,r (x) ˆ

r=1

and FA 0 (x) ˆ =

s 

Tr Wγj (x). ˆ

j =1

The convergence in H and the well definedness of Wγ (x) ˆ is guaranteed by (2.13). We have (x, QA0 x)+ =

∞ 

μj (x, ˆ eˆj )2 ,

xA =

j =1

F  (A0 + xA ) = FA 0 (x) ˆ =

j =1 s 

Tr Wγj (x), ˆ

j =1

∞  R  (x, ˆ eˆ )aR,ii , aii (xA ) = 1 + =1

and −1 4π



∞  (x, ˆ eˆj )ejA ,

aijR (xA ) =

∞ 

 (x, ˆ eˆ )aR,ij

=1

∞  2 Tr xA ∧ xA ∧ xA = (x, ˆ eˆp )(x, ˆ eˆj )(x, ˆ eˆ )β pj  , 3 p,j,=1

M

where



β pj  = −

Tr M

1 A e ∧ ejA ∧ eA . 6π p

(2.22)

284

S. Albeverio, I. Mitoma / Bull. Sci. math. 133 (2009) 272–314

Itô technique [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

 r  N  m λ   Vm,n  m pj  A (x ) ik xp xj x β /r! F lim lim 0 n→∞ m→∞ Zm,n r=0R m

p,j,=1

  μm (dx) (S −1 x m , x m )+ × exp ik(x m , QA0 x m )+ − m , √ √ 2n ( 2π)m det(nSm )

(2.23)

where μm is the m-dimensional Lebesgue measure, xm =

m 

xj eˆj ,

λ Vm,n =

j =1

m   j =1

and

 λ Zm,n = Vm,n Rm

1 + nλj ,

−1 m m (Sm x , x )+ =

m 

2 λ−1 j xj

j =1

  μm (dx) (S −1 x m , x m )+ exp ik(x m , QA0 x m )+ − m . √ √ 2n ( 2π)m 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  ξ 2 ei x,ξ  dμ(x) = e− 2 , X

holds, where x, ξ  denotes the canonical bilinear form on X × X ∗ . We denote (x 1 , x 2 , . . . , x d ) ∈ X by x and (ξ 1 , ξ 2 , . . . , ξ d ) ∈ 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 (eˆi , 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 ∞   4

λj 3 |μj | < +∞,

(S.1)

λj < +∞

(S.2)

j =1 ∞   4

j =1

and

   sup 4 λ |aR,ij | < +∞ 

[38,43].

i,j

(S.3)

S. Albeverio, I. Mitoma / Bull. Sci. math. 133 (2009) 272–314

285

For any  > 0 and x ∈ X, we define xγ (t) =

d  

   x, C u (t), 0 · Eu .

u=1

Briefly we denote     x, Cu (t), 0 by xγ,u (t). Then by (2.19) and (2.20), we remark that xγ,u (t) is a Gaussian random variable such that   2   2    = C (t) . (2.24) E xγ,u (t)2 =  C u u (t), 0 0 Since (2.13) yields 2   E xγ,u (t) − xγ,u (s)  c1 ()2 |t − s|2 ,

(2.25)

xγ,u (t) has a continuous version in t, by the Kolmogorov–Totoki criterion [41]. Henceforth we denote this continuous version again by the same symbol xγ,u (t). For any natural number n, set    2n     ,u j ,u j − 1  x − x Tn = γ .  γ n n 2 2 j =1

Since Tn  Tn+1 , we have E[ lim Tn ] = lim E[Tn ] n→∞ n→∞  2n       ,u j ,u j − 1   xγ − xγ = lim E   n n n→∞ 2 2 j =1

  2  1    2 ,u j ,u j − 1    lim − x E x γ γ n→∞ 2n 2n  2n

j =1 n

 lim

n→∞

2  j =1

  j j − 1 c1 () n − n  2 2

 1,

(2.26)

which implies lim Tn < +∞

n→∞

almost surely with respect to μ.

(2.27)

Noticing that xγ,u (t) is continuous in t almost surely, by (2.27), we obtain, for almost all x ∈ X with respect to μ, that xγ,u (t) is of bounded variation. Therefore the Lebesgue–Stieltjes integral t

t1 dxγ (t1 )

0

dxγ (t2 ) 0

286

S. Albeverio, I. Mitoma / Bull. Sci. math. 133 (2009) 272–314

is well defined and continuous in t almost surely with respect to μ and is equal to d  d  

t1

t

dxγ,v (t2 ) · Eu Ev .

dxγ,u (t1 )

u=1 v=1 0

0

Repeating such arguments, we arrive at the well definedness of the iterated integrals and in a ˆ and FA 0 (x), ˆ we define similar manner as for the definitions of Wγ (x) FA 0 (x) =

s 

Tr Wγj (x),

j =1

where the well definiteness is guaranteed by (1) in Lemma 1 which will be stated below. We call, for  > 0, FA 0 (x) the -regularization of the Wilson loop variable [1]. In the definitions of detn,R ∗dA0 ∗ DA and detReg (A), we replace aiiR (A) by aiiR

∞   √   nSx = 1 + nλ x, eˆ aR,ii =1

and

aijR (A) aijR

by

∞  √    nSx = nλ x, eˆ aR,ij , =1

√ which are detn,R ∗dA0 ∗ D√nSx and detReg ( nSx). The convergence of the series is guaranteed by (S.2) and (S.3), since   ∞      nλ x, eˆ aR,ij     =1



∞     √   n 4 λ  x, eˆ  sup 4 λ |aR,ij |. 

=1

Let us set √ √ √     A nSx = F nSx det nSx . F Reg A 0 0 We shall provide the definition of I (k) in the abstract Wiener space by setting it equal to Definition 1. N  √    1   √ FA0 nSx exp ikQ( nSx) n n→∞ Z2,k r=0 X

 r  ∞  √ √ √  pj  × ik nSx, eˆp nSx, eˆj nSx, eˆ β /r! μ(dx),

lim sup

p,j,=1

where n Z2,k

 = X

 √  exp ikQ nSx μ(dx).

(2.28)

S. Albeverio, I. Mitoma / Bull. Sci. math. 133 (2009) 272–314

287

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 ∞   √ nλj μj x, eˆj 2 < +∞ Q nSx = 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

∞  r   m   1  m pj   xp xj x β /r! FA0 (x ) ik m Z2,k r=N +1

Rm

p,j,=1

  μm (dx) × exp ik(x m , QA0 x m )+ √ , ( 2π)m where



m Z2,k

Set

√

= Rm

(2.29)

 μm (dx)  exp ik(x m , QA0 x m )+ √ . ( 2π)m

kxi = yi , then by the formula of changing variables, we obtain (2.29)  r     ∞ m  1 1 i  m pj  A √ y = m yp yj y β /r! F √ 0 Z3,k k k p,j,=1 r=N +1 m R

  μm (dy) × exp i(y m , QA0 y m )+ √ , ( 2π)m where



m Z3,k = Rm

 μm (dy)  exp i(y m , QA0 y m )+ √ . ( 2π )m

To discuss the asymptotic estimate of II(k), we may modify the Itô technique by introducing the Gaussian kernel with parameter k 2η ,

0<η<

1 12

(2.30)

and consider √ N +1 lim k II(k) k→∞

 r   ∞ m i  pj  xp xj x β /r! √ k→∞ k r=N +1 p,j,=1 Rm   −1 m m μm (dx) (S x , x )+  × exp i(x m , QA0 x m )+ − m , (2.31) √ 2k 2η ( 2π)m det(k 2η Sm )

√ N +1 = lim k lim

1 m→∞ Z m 4,k



A F 0



1 √ xm k

288

S. Albeverio, I. Mitoma / Bull. Sci. math. 133 (2009) 272–314

where



m Z4,k = Rm

  μm (dx) (S −1 x m , x m )+  exp i(x m , QA0 x m )+ − m . √ 2η 2k ( 2π)m det(k 2η Sm )

We can find a realization of the expression (2.31) in the abstract Wiener space setting it as equal to Definition 2. lim

√

k

N +1

1



A F 0

 √ 2η √   k S √ x exp ik 2η Q Sx k

Zˆ 2,k X

∞  r  ∞     pj  i   2η 2η 2η × k Sx, eˆp k Sx, eˆj k Sx, eˆ β /r! μ(dx), √ k p,j,=1 r=N +1 (2.32)

k→∞

where Zˆ 2,k =



 √  exp ik 2η Q Sx μ(dx).

X

Here we add a further assumption to the operator S such that for some 0 < δ < η and a natural number M such that Mδ > 2η, ∞   M

λj |μj | = C  c3 < +∞.

(S.4)

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 Yγ1,0 (0) = I, Yγ1,r (i) =

tl1 −1

1





dA0 (t1 ) · · ·

1l1
tli −1



dxγ (tl1 ) · · · 0

dxγ (tli ) · · · 0

tr−1 dA0 (tr ) 0

and Yγ1 (i) =

∞ 

Yγ1,r (i).

r=i

Since 1 t1 Wγ,r (x) = 0 0

tr−1 r  ··· d(A0 + xγ )(t1 )d(A0 + xγ )(t2 ) · · · d(A0 + xγ )(tr ) = Yγ1,r (i), 0

i=0

S. Albeverio, I. Mitoma / Bull. Sci. math. 133 (2009) 272–314

289

we have Wγ (x) = I

+

∞ 

Wγ,r (x) =

r=1

∞ 

Yγ1 (i).

i=0

To proceed further, we need several notations. Let us set YA1,m (x) = 0



s 

i1 +i2 +···+is =m j =1

Tr Yγ1j (ij ).

(2.33)

Furthermore, let us define aˆ iiR (x) = aiiR (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 ) = ij , j = 1, 2, . . . , t, and otherwise σ (i) = i. Set  r   sign σ aiR1 σ (i1 ) (x)aiR2 σ (i2 ) (x) · · · aiRt σ (it ) (x) Y 2,r = lim m→∞

t=1 σ ∈Sm,t





×

1j1


+

1j1
 aˆ jR1 j1 (x)aˆ jR2 j2 (x) · · · aˆ jRr−t jr−t (x) 

aˆ jR1 j1 (x)aˆ jR2 j2 (x) · · · aˆ jRr jr (x)

and Y 2,0 = 1. Set m (x) = F

 r+r  =m



YA1,r (x)Y 2,r (x). 0

Define

√ nS Rn,k =  √ . √ 1 − 2ik nSQA0 nS

For an n × n matrix E = (aij ), define |||E||| =

n 

|aij |.

i,j =1

We define a regularization of QA0 such that the eigenvalues μj of QA0 satisfy the “renormalization condition”: Renormalization.  ∞   1  1+ |aR,ij | < +∞. √ |μ | i,j =1 We also assume

290

S. Albeverio, I. Mitoma / Bull. Sci. math. 133 (2009) 272–314

Assumption 2. |β pj  |  β < +∞. Remark 1. We expect that in the case where M is compact this assumption is automatically satisfied. The following lemma will be proved later in Section 3. Lemma 1. Let us denote any natural number by b and let E[·] the integral with respect to μ on X. (1) For any  > 0, 2b   E Wγ (x) < +∞. (2) For any fixed  > 0, there exists a constant c4 () independent of n such that 2b   E Wγ (Rn,k x)  c4 () < +∞.  ∞    1,m 1 N , (3) E YA0 (Rn,k x) = o √ k m=N where o(( √1 )N ) means k

lim

k→∞

√

   1 N  N k o √  < +∞. k

(4) Under Assumption 2, 2b   ∞   √ √ √  pj    E ik nSx, eˆp nSx, eˆj nSx, eˆ β  < +∞.   p,j,=1

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

Rn,k x, eˆp  Rn,k x, eˆj  Rn,k x, eˆ β    p,j,=1

6b  ∞  1 2b  1   c5 (b) √ . |μj | k j =1 (6) Under Assumption 2,  √ 2b  E detReg nSx  < +∞. (7) Under the same assumptions as in (5), there exists a constant c6 (b) independent of n such that 2b   E detReg (Rn,k x)  c6 (b) < +∞.

S. Albeverio, I. Mitoma / Bull. Sci. math. 133 (2009) 272–314

291

(8) Under the same assumptions as in (5),  ∞    1 N 2,r E . Y (Rn,k x) = o √ k r=N By Lemma 1, the Cauchy integral theorem in finite dimensions and a limiting argument, we have Theorem 1. Under Assumption 2, I (k) = lim sup n→∞

 ×

N  

A (Rn,k x) F 0

r=0 X ∞ 

ik



r

Rn,k x, eˆp  Rn,k x, eˆj  Rn,k x, eˆ β

pj 

/r! μ(dx)

p,j,=1

is well-defined. Under the Renormalization assumption on QA0 and Assumption 2, the right-hand side of the above equality is equal to:   m (Rk x) F m+m N X



×

ik



m

∞ 

Rk x, eˆp  Rk x, eˆj  Rk x, eˆ β

pj 



/m ! μ(dx)

p,j,=1

1 N +1 √ k     1 m+m ,m+m 1 N +1 = , JCS +o √ √ k k m+m N 

+o

where 

1 √ k

m+m



m (Rk x) F

X

 ×



,m+m JCS =

ik

∞ 

Rk x, eˆp  Rk x, eˆj  Rk x, eˆ β



m pj 



/m ! μ(dx)

p,j,=1

and Rk = 

1 . −2ikQA0

Now we will proceed to the asymptotic estimate of II(k), as k → ∞.

292

S. Albeverio, I. Mitoma / Bull. Sci. math. 133 (2009) 272–314

For 0 < δ < η, set

 √ ! ∞  4    Sx δ   D = x ∈ X;   √ √ , eˆj   3k (1 − 2ik 2η SQA0 S) j =1 and

√ k 2η S . Vk =  √ √ 2η 2η 1 − 2i k SQA0 k S)

Theorem 2. Under Assumption 2,  √ 2η  √   k S 1  A F √ x exp ik 2η Q Sx 0 k Zˆ 2,k X

∞  r  ∞     pj  i   2η 2η 2η × k Sx, eˆp k Sx, eˆj k Sx, eˆ β /r! μ(dx) √ k p,j,=1 r=N +1 (2.34) is well-defined and equal to  r     ∞ ∞  V i k  pj  A √ x

Vk x, eˆp  Vk x, eˆj  Vk x, eˆ β /r! μ(dx) F √ 0 k k p,j,=1 r=N +1 D

+ R(k), where R(k) satisfies kδ

R(k)  c7 e− 2

(2.35)

for sufficiently large k, so that under the Renormalization assumption on QA0 and Assumption 2, √ N +1   II(k) < +∞. lim k k→∞

Remark 2. If we make a stronger renormalization on QA0 such that ∞  p,j,=1

(β pj  )2 < +∞ |μp ||μj ||μ |

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, √ d  ∞        nλr  (t), 0 E =  (t), 0 , eˆ √ Rn,k x, C

x, e ˆ  C Eu . u r r u u 1 − 2inkλr μr u=1 u=1 r=1

d  

Since

S. Albeverio, I. Mitoma / Bull. Sci. math. 133 (2009) 272–314

√ √ nλr nλr = eiθ √ 4 2 1 − 2inkλr μr 1 + 4(nkλr μr ) = αrn,k + iβrn,k ,

where

293

π −π <θ < , 2 2

set Rn,k xγ (t) =

d  

   Rn,k x, C u (t), 0 Eu ,

u=1

Rn,k xγ (t)u 1 Rn,k xγ (t)u

    = Rn,k x, C u (t), 0 , =

∞ 

    αrn,k x, eˆr  C u (t), 0 , eˆr ,

r=1

and 2 Rn,k xγ (t)u =

∞ 

    βrn,k x, eˆr  C u (t), 0 , eˆr .

r=1

Define d

cE = max | Eu | . u=1

3.1. Proofs of lemmas Before we proceed to the proof of Lemma 1, we prove 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  2b   1 2b     2b  2b   Xi,j   |Xi,j | .  i

j

j

i

Especially for any random variables Xi , i = 1, 2, . . . , on a probability space, we have   2b       1 2b   E  Xj  E |Xj |2b 2b . j

j

Proof. Since  2b |Xi,j | = j



|Xi,j1 ||Xi,j2 | · · · |Xi,j2b |

j1 ,j2 ,...,j2b

and by the Hölder’s inequality recursively  |Xi,j1 ||Xi,j2 | · · · |Xi,j2b | i



 i

 1   2b−1 2b 2b   2b 2b−1 |Xi,j2 | · · · |Xi,j2b | |Xi,j1 | 2b

i

294

S. Albeverio, I. Mitoma / Bull. Sci. math. 133 (2009) 272–314





|Xi,j1 |

2b

 1  2b

i

i

and so on, we have 2b       X i,j    i

j



=

j1 ,j2 ,...,j2b





 1   2b−2 2b 2b   2b 2b−2 |Xi,j3 | · · · |Xi,j2b | |Xi,j2 | 2b

j1 ,j2 ,...,j2b



i

i

 |Xi,j1 ||Xi,j2 | · · · |Xi,j2b |

 j1 ,j2 ,...,j2b

 |Xi,j1 ||Xi,j2 | · · · |Xi,j2b |

i



|Xi,j1 |2b

 1  2b

i

1

2b

|Xi,j2 |2b

···



i

1 |Xi,j2b |2b

2b

,

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  Au0 (t) = Au0 . γ [0,t]

Since 1 t1

tr−1 ··· d(A0 + xγ )(t1 ) d(A0 + xγ )(t2 ) · · · d(A0 + xγ )(tr )

0 0

=

0 r 

tl1 −1

1





dA0 (t1 ) · · ·

m=0 1l1
=

r 



t lm −1

dxγ (tl1 ) · · ·

dxγ (tlm ) · · ·

0

1

d 

0

tr−1 dA0 (tr ) 0

tl1 −1 j



dxγ (tl1 )jl1 · · ·

dA01 (t1 ) · · ·

m=0 1l1
0

t lm −1

tr−1 j  jlm dxγ (tlm ) · · · dA0r (tr )Ej1 Ej2 · · · Ejr

0

0

and 1 t1

tr−1 ··· d(A0 + Rn,k xγ )(t1 ) d(A0 + Rn,k xγ )(t2 ) · · · d(A0 + Rn,k xγ )(tr )

0 0

=

0 r 



1

d 

m=0 1l1
tr−1

dRn,k xγ (tlm )jlm · · · 0

j

tl1 −1 j dA01 (t1 ) · · ·



dRn,k xγ (tl1 )jl1 · · · 0

dA0r (tr )Ej1 Ej2 · · · Ejr , 0

2

S. Albeverio, I. Mitoma / Bull. Sci. math. 133 (2009) 272–314

295

we have, by Lemma 2, 2b   E Wγ (x)  E

∞  r 



 1 

d 

r=0 m=0 1l1


0

2b  tr−1   j dxγ (tlm )jlm · · · dA0r (tr ) 

0



j

dA01 (t1 ) · · ·

t lm −1

dxγ (tl1 )jl1 · · · 

r  cE  

0 ∞  r 

0



d 

r cE E

r=0 m=0 1l1
tl1 −1  1 t lm −1    ··· ···  ···  0

0

0

2b  1 2b 2b   j1 jr jl1   jlm dA0 (t1 ) · · · dxγ (tl1 ) · · · dxγ (tlm ) · · · dA0 (tr ) 

tr−1

(3.1)

0

and 2b   E Wγ (Rn,k x)  E

∞  r 



 1 

d 

r=0 m=0 1l1


t lm −1

dRn,k xγ (tl1 )jl1

···

0

dRn,k xγ (tlm )jlm

r  cE  

0

2b  tr−1   jr ··· dA0 (tr ) 

0

 E

∞  r 

0



d 

tl1 −1 ζ1  dRn,k xγ (tl1 )jl1

0

 

ζm  dRn,k xγ (tlm )jlm

∞  r 

0



tl1 −1

d 

ζ

t lm −1

1 dRn,k xγ (tl1 )jl1 · · ·

0

2b  tr−1   jr ··· dA0 (tr ) 

0

r=0 m=0 1l1


ζ1 ,ζ2 ,...,ζm =1,2 0

t lm −1

···

ζ



r cE

ζ1 ,ζ2 ,...,ζm =1,2

 1   j E  dA01 (t1 ) · · ·  0

2b  1 2b 2b   jr dA0 (tr ) . 

tr−1

m  dRn,k xγ (tlm )jlm · · ·

0

 1   j  dA01 (t1 ) · · · 



r cE

r=0 m=0 1l1


j

dA01 (t1 ) · · ·

0

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

(3.2)

296

S. Albeverio, I. Mitoma / Bull. Sci. math. 133 (2009) 272–314

Lemma 3. Let Xi , i = 1, 2, . . . , 2, be a mean-zero Gaussian system. Then  E[X1 X2 · · · X2 ] = E[Xi1 Xj1 ]E[Xi2 Xj2 ] · · · E[Xi Xj ] comb()

=

1 2 !



E[Xσ (1) Xσ (2) ]E[Xσ (3) Xσ (4) ] · · · E[Xσ (2−1) Xσ (2) ], (3.3)

σ ∈S2

 where S2 denotes the collection of all permutations of {1, 2, . . . , 2} and comb() is the summation taken over all pairs of {1, 2, . . . , 2} such that i1 < i2 < · · · < i , ik  jk for k = 1, 2, . . . , . Further we recall the integrability of bounded variation norm of xγ,u (·): Lemma 4. Let b be any natural number, then   E ( lim Tn )2b < +∞, n→∞

where    2n     ,u j ,u j − 1   Tn = − xγ . x γ n n 2 2 j =1

Proof. Since Tn  Tn+1 , by Lemmas 2 and 3 we have     E ( lim Tn )2b = lim E (Tn )2b n→∞ n→∞  2n    2b  1 2b    2b ,u j ,u j − 1   − xγ E x γ  lim  n n n→∞ 2 2   lim

n→∞

j =1

1 2n   (2b)! 2b j =1

2b b!

2b  j j − 1   c1 () n − n  2 2

< +∞, which completes the proof of this lemma.

2 s

On the other hand, let si , i = 0, 1, . . . , r, be non-negative integers. Then setting t00 = 1 and

0 if si = 0, si si−1 ti = si −1 ti−1 ti + ni if si  1 and A[si |0i ] = A0i (tisi +1 ) − A0i (tisi ), j

j

j

 si +1 ji i ) − xγ (tisi )ji , x[si |,j γ ] = xγ (ti

by (2.12), (2.13) and Lemma 4 which guarantee the well known condition for using the Lebesgue convergence theorem, we have

S. Albeverio, I. Mitoma / Bull. Sci. math. 133 (2009) 272–314

297

tl1 −1 2b   1 t lm −1  tr−1     j1 jr jl1   jlm E  dA0 (t1 ) · · · dxγ (tl1 ) · · · dxγ (tlm ) · · · dA0 (tr )   0

0

0

 nl1 −1 n 1 −1   ,jl  j1 A[s1 |0 ] · · · lim x[sl1 |γ 1 ] · · · = E  lim nl1 →∞ n1 →∞ s1 =0

nlm −1



lim

nlm →∞

,j x[slm |γ lm ] · · ·

slm =0

 E

sl1 =0

lim

nr →∞

n r −1 sr =0

nl1 →∞

nlm →∞

2b  

j  A[sr |0r ] 

lim · · · lim · · · lim · · · lim

n1 →∞

0

nr →∞

n 1 −1

nl1 −1

···

s1 =0



 ,j l1 j  A[s1 |j1 ] · · · x[sl |,j ] · · · x[slm |γ lm ] · · · A[sr |0r ] 1 γ 0

sr =0

 c2 (A0 ) n r −1

|t1s1 +1

lim · · · lim · · · lim · · · lim E

n1 →∞

nl1 →∞

nlm →∞

 c2 (A0 ) n r −1

 n −1 1 

lim · · · lim · · · lim · · · lim

n1 →∞

nl1 →∞

nlm →∞

nl1 −1

···

s1 =0

  ,jl  ,j  − t1s1 | · · · x[sl1 |γ 1 ] · · · x[slm |γ lm ] · · · |trsr +1

2b(r−m)

···

slm =0

2b 

nr →∞

sr =0



···

sl1 =0

n r −1

2b(r−m)

nlm −1

 n −1 1 

nr →∞



nlm −1

sl1 =0



2b 

− trsr  nl1 −1

···

s1 =0



nlm −1

···

sl1 =0



By Lemma 3, we have   ,jl  ,j 2b  E x[sl1 |γ 1 ] · · · x[slm |γ lm ] (2bm)!c1 ()2bm sl1 +1 sl s +1 s |tl1 − tl1 1 |2b · · · |tlmlm − tlmlm |2b . bm 2 (bm)!

Therefore we have tl1 −1 2b  1  1 t lm −1  tr−1 2b     j1 jr jl1   jlm E  dA0 (t1 ) · · · dxγ (tl1 ) · · · dxγ (tlm ) · · · dA0 (tr )   0

 c2 (A0 )

r−m

n r −1 sr =0

0

0

lim · · · lim · · · lim · · · lim

n1 →∞

nl1 →∞

(2bm)!c1 ()2bm 2bm (bm)!

1

2b

nlm →∞

nr →∞

n 1 −1

···

s1 =0

sl +1

|t1s1 +1 − t1s1 | · · · |tl1 1

nl1 −1



sl1 =0 sl

− t l1 1 | · · ·

nlm −1

···



slm =0

···

slm =0

   2b  1 ,jl  ,j  2b E |t1s1 +1 − t1s1 | · · · x[sl1 |γ 1 ] · · · x[slm |γ lm ] · · · |trsr +1 − trsr |

0

···

slm =0

sr =0





···

···

2b .

298

S. Albeverio, I. Mitoma / Bull. Sci. math. 133 (2009) 272–314 s +1

× |tlmlm

− tlmlm | · · · |trsr +1 − trsr | s

  c2 (A0 )

r−m

m

c1 ()

(2bm)! bm 2 (bm)!

 1 1 t1 2b

tr−1 ··· dt1 dt2 · · · dtr .

0 0

Since

(bm)!  (m!bm )b ,

0

the last term is less or equal



(m!)b (1 · 2 · · · (2bm − 2) · (2bm − 1) · 2bm)2bm bbm c8 (A0 ) (r!)2b ((2 · 1) · (2 · 2) · · · 2(bm − 1) · 2bm)2 √ √ m m! r  c8 (A0 ) 2b , r! where c8 (A0 ) = max{c2 (A0 ), c1 ()}. Thus we obtain that (3.1) is dominated by  ∞ √ 2b r  √ m m! r  dcE c8 (A0 ) r Cm 2b r! r=0 m=0  ∞ 2b  √ r 1 r dcE c8 (A0 ) (1 + 2b) √  < +∞, r! r=0 r

1

2b

which yields the proof of (1) of Lemma 1. By (2.13), we have for ζ = 1 or 2, 2  1  ζ ζ E Rn,k xγ (t)u − Rn,k xγ (s)u  2  1 ∞       2 2 1  −1   Cu (t), 0 − Cu (s), 0 , eˆr  |μr | 2k  =

r=1 ∞

  2 1  |μr |−1 Cu (t) − Cu (s), 0 , er + 2k

1 2

r=1

 1  C  (t) − C  (s)  c1 () √ 1 |t − s|, √ u u 0 2kρ 2kρ

(3.4)

where min |μr | = ρ > 0.

r=1,2,...

In a manner similar to the proof of the former part, (3.2) is dominated by  ∞ 2b  √ m r  r  2b 1 dcE c8 (A0 ) √ r Cm 2 √ 2kρ r! r=0 m=0  ∞ 2b √  r  b 1 dcE c8 (A0 ) 1 + 2 √ = < +∞, √ kρ r! r=0 which provides the proofs of (2) and (3) of Lemma 1. Now we proceed to the proof of the rest of Lemma 1.

(3.5)

S. Albeverio, I. Mitoma / Bull. Sci. math. 133 (2009) 272–314

299

Since

√       nλr   Rn,k x, eˆr  =  √  1 − 2inkλ μ x, eˆr  r r   1  x, eˆr , √ 2k|μr |

by Lemmas 2 and 3 and Assumption 2, we have under the Renormalization assumption, 2b   ∞     pj   E 

Rn,k x, eˆp  Rn,k x, eˆj  Rn,k x, eˆ β    p,j,=1



E 

∞ 

    β  x, eˆp  x, eˆj  x, eˆ    √ 2k|μp | 2k|μj | 2k|μ | p,j,=1

2b 

∞ 

  2b  1  β 2b    E  x, eˆp  x, eˆj  x, eˆ  √ 2k|μ | 2k|μ | 2k|μ | p j  p,j,=1  ∞ 2b  1    c5 (b) √ k|μp | k|μj | k|μ | p,j,=1 6b  ∞  1 6b  1 = c5 (b) √ < +∞. √ |μi | k i=1 The proofs of (4) and (5) of Lemma 1 are obtained by replacing Rn,k by estimate is easier than the former. Since √   ∞   R       nλ   a (Rn,k x)  aij (0) +  x, eˆ  √ a ij  1 − 2inkλ μ R,ij    =1 1+

√ nS, in effect the latter

∞      x, eˆ  √ 1 |, |aR,ij 2k|μ |  =1

where aij (0) = δij ,

i, j = 1, 2, . . . , m,

noticing the Renormalization assumption, we have 2b   E detReg (Rn,k x)     2b  R R R   a1σ (1) (Rn,k x)a2σ (2) (Rn,k x) · · · amσ (m) (Rn,k x)  lim E m→∞

  lim E m→∞

σ ∈Sm

2b   ∞     1   x, eˆ  √ | |aR,ij 1+ 2k|μ |  i,j =1

2b

300

S. Albeverio, I. Mitoma / Bull. Sci. math. 133 (2009) 272–314







∞     1   

x, eˆ  √  lim E exp 2b |a | m→∞ 2k|μ | R,ij i,j =1   ∞ 

∞    1   

x, eˆ  √  E exp 2b |a | < +∞ (see [29]), 2k|μ | i,j =1 R,ij =1 which yields the proofs of (6), (7) and (8) of Lemma 1. 3.2. Proof of Theorem 1 Now we proceed to the proof of Theorem 1. Set 

L d         x (t) = ·E

x, eˆ  C (t), 0 , eˆ i

L,γ

u=1

u

i

u

i=1

 (t) in the definition of W ,r (x). Then the object to be obtained is and replace xγ (t) by xL,γ γ denoted by ,r (x). WL,γ

Define Wγ,J,L (x) = I +

J 

,r WL,γ (x)

r=1

and ,J FL,A (x) = 0

s  j =1

Tr Wγj ,J,L (x).

√ In the definition of detJ,R ∗dA0 ∗ D√nSx , we replace aiiR ( nSx) by 1+

L  

 nλ x, eˆ aR,ii

=1

√ and aijR ( nSx) by L  

 nλ x, eˆ aR,ij ,

=1

which is denoted by detJ,R,L (x). Setting ,J (x) = F ,J (x) detJ,R,L (x), F L,A0 L,A0   GL,J x, eˆ1 , x, eˆ2 , . . . , x, eˆL 

 r  N L   √ √ √ ,J  (x) ik

nSx, eˆp  nSx, eˆj  nSx, eˆ β pj  /r! F = L,A0

r=0

p,j,=1

S. Albeverio, I. Mitoma / Bull. Sci. math. 133 (2009) 272–314

301

and QL (x) =

L 

μj x, eˆj 2 ,

j =1

then by (1), (4) and (6) of Lemma 1, we have N  

A F 0

r=0 X

√

  √  nSx exp ikQ nSx



∞  √ √ √  ik nSx, eˆp nSx, eˆj nSx, eˆ β pj 

×

r

 /r! μ(dx)

p,j,=1



= lim lim

L→∞ J →∞

 √    GL,J x, eˆ1 , x, eˆ2 , . . . , x, eˆL  exp ikQL nSx μ(dx)

X





= lim lim

···

L→∞ J →∞

  × exp ik

GL,J (x1 , x2 , . . . , xL )

RL L 



nλj μj xj2

j =1

L 

2

xj 1 √ e− 2 dxj . 2π j =1

(3.6)

Setting π , 2 we get by using recursively the Cauchy integral theorem, that (3.6) is equal to  L     αj lim lim · · · exp i L→∞ J →∞ 2 zj = 1 − 2iknλj μj = |zj |e−iαj ,

0 < (sign μj )αj <

j =1

RL

  L |zj |xj2 α α  i α1  1 i 22 i 2L 2 dxj . × GL,J e x1 , e x2 , . . . , e xL √ exp − 2 2π j =1 Using (2), (5) and (7) of Lemma 1, we see that (3.7) is equal to 1

lim lim  L→∞ J →∞ L  ×

j =1 (1 − 2iknλj μj )

  GL,J Rn,k x, eˆ1 , Rn,k x, eˆ2 , . . . , Rn,k x, eˆL  μ(dx)

X

1 √

N  

A (Rn,k x) F √ 0 det(1 − 2ik nSQA0 nS) r=0 X

 r  ∞ 

Rn,k x, eˆp  Rn,k x, eˆj  Rn,k x, eˆ β pj  /r! μ(dx). × ik

=

p,j,=1

(3.7)

302

S. Albeverio, I. Mitoma / Bull. Sci. math. 133 (2009) 272–314

Therefore n Z2,k



=

 √  exp ikQ nSx μ(dx) = 

X

1 , √ √ 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   √  1 exp ik 2η Q Sx μ(dx) =  √ √ , det(1 − 2ik 2η SQA0 S) X we have that (2.34) is equal to √  √ √   √   k 2η S 2η A ( √ det(1 − 2ik SQA0 S) F x) exp ik 2η Q Sx 0 k X

∞  r  ∞     pj  i   2η 2η 2η × k Sx, eˆp k Sx, eˆj k Sx, eˆ β /r! μ(dx). √ k p,j,=1 r=N +1

(3.8)

By the convergence assumption (S.2), we have   ∞     4   E exp λi x, eˆi  < +∞, i=1

so that by the Chebyshev inequality, we get   ∞     δ 4 λi  x, eˆi  > 3k δ  c9 e−3k . μ x ∈ X;

(3.9)

i=1

Let



 ∞     4 δ λi  x, eˆi   3k . Xk = x ∈ X; i=1

Then we prove Lemma 5.  √ 2η   √ √   √   k S 2η  det 1 − 2ik SQA0 S FA0 √ x exp ik 2η Q Sx k c

×

∞ 



r=N +1

= o(e−k ). δ

Xk

i √ k

∞ 



p,j,=1

k 2η Sx, eˆ

 p

k 2η Sx, eˆ

 j

k 2η Sx, eˆ

r

 

β

pj 

 /r! μ(dx)

S. Albeverio, I. Mitoma / Bull. Sci. math. 133 (2009) 272–314

303

Proof. First we show that for sufficiently large k,   √ √  kδ lim det 1 − 2ik 2η SQA0 S e− 2 = 0.

(3.10)

k→∞

By the convergence assumption (S.4), we have for some natural number M such that Mδ > 2η and ∞   M

λj |μj | = C  c3 < +∞.

j =1

Hence we have  ∞ M    √ √  λj |μj |      det 1 − 2ik 2η SQA S exp −k δ j =1 0   2C      ∞  ∞ | 1 − 2ik 2η λ μ | M λ |μ |  j j   j j δ 2η   =  1 − 2ik λj μj exp −k , " 2 2C k Mδ λj |μj | j =1 j =1 1 + M!(2C)M which gives (3.10). 3 Since 3η < 12 <

1 2

and 

 √ 2η  N  ∞      pj  r i k S  A k 2η Sx, eˆp k 2η Sx, eˆj k 2η Sx, eˆ β /r! F √ x √ 0 k k r=1 p,j,=1  r   η  N ∞ 3η  √ √ √ √  pj  k k  A √ Sx =F Sx, eˆi Sx, eˆj Sx, eˆ β /r! , i√ 0 k k p,j,=1 r=1 (3.11)

by the estimations similar to the proof of Lemma 1, we get under Assumption 2,  η   k √  F Sx √  A0  k X



 r 2 N ∞      pj  i   2η  × k Sx, eˆp k 2η Sx, eˆj k 2η Sx, eˆ β /r!  μ(dx) √  k p,j,=1 r=1  c10 (N ) < +∞,

so that by (3.9) and (3.10) we have 

√ √   det 1 − 2ik 2η SQA0 S

×

N 



r=1

= o(e−k ). δ



Xkc

A F 0

 √ 2η √   k S √ x exp ik 2η Q Sx k

∞    i   2η k Sx, eˆi k 2η Sx, eˆj k 2η Sx, eˆ β pj  √ k p,j,=1

r

 /r! μ(dx) (3.12)

304

S. Albeverio, I. Mitoma / Bull. Sci. math. 133 (2009) 272–314

Similarly we get √      k 2η S 2 F A ( √ x)  0  μ(dx)  c11 < +∞, k X

so that using that ∞  √ √  k 3η  √ Q( k 2η Sx) + √ Sx, eˆp Sx, eˆj Sx, eˆ β pj  k p,j,=1

is real, by (3.9) and (3.10) we have  √ 2η   √ √   √   k S 2η A det 1 − 2ik SQA0 S F √ x exp ik 2η Q Sx 0 k c Xk



     pj  i k 2η Sx, eˆp k 2η Sx, eˆj k 2η Sx, eˆ β × exp √ μ(dx) k p,j,=1 ∞ 

= o(e−k ). δ

Then  Xkc

A F 0

(3.13)

 √ 2η √   k S √ x exp ik 2η Q Sx k

×

∞ 



i √

∞  

k 2η Sx, eˆ

 p

k p,j,=1 √  2η  √   k S A = F √ x exp ik 2η Q Sx 0 k c

k 2η Sx, eˆ

 j

k 2η Sx, eˆ





r



β

pj 

/r! μ(dx)

r=N +1

Xk





i × exp √

−

N  r=1



∞  

k p,j,=1

k 2η Sx, eˆp



k 2η Sx, eˆj







k 2η Sx, eˆ β pj 

∞    i   2η k Sx, eˆp k 2η Sx, eˆj k 2η Sx, eˆ β pj  √ k p,j,=1

which, together with (3.12) and (3.13), completes the proof of Lemma 5.

r

 /r!

μ(dx),

2

By Lemma 5, for sufficiently large k we have  √ 2η   √ √   √   k S 2η  det 1 − 2ik SQA0 S FA0 √ x exp ik 2η Q Sx k X r 

∞  ∞     pj  i   2η k Sx, eˆp k 2η Sx, eˆj k 2η Sx, eˆ β /r! μ(dx) × √ k r=N +1 p,j,=1

S. Albeverio, I. Mitoma / Bull. Sci. math. 133 (2009) 272–314

=



√ √   det 1 − 2ik 2η SQA0 S



∞ 

×

r=N +1



A F 0

Xk

i √ k

∞  

305

 √ 2η √   k S √ x exp ik 2η Q Sx k

k 2η Sx, eˆp



  k 2η Sx, eˆj k 2η Sx, eˆ β pj 

r

 /r! μ(dx)

p,j,=1

+ o(e−k ). δ

(3.14)

Set   HL,J x, eˆ1 , x, eˆ2 , . . . , x, eˆL   √ 2η k S ,J  = FL,A0 √ x k

∞  r  L     pj  i   2η k Sx, eˆp k 2η Sx, eˆj k 2η Sx, eˆ β /r! × √ k r=N +1 p,j,=1 and

XkL

= x ∈ X;

L   4

   δ λj  x, eˆj   3k .

j =1

Then by the estimations similar to (1), (4) and (6) of Lemma 1, we have  √ 2η  √   k S  A F √ x exp ik 2η Q Sx 0 k Xk

∞  r  ∞     pj  i   2η 2η 2η × k Sx, eˆp k Sx, eˆj k Sx, eˆ β /r! μ(dx) √ k p,j,=1 r=N +1  √     = lim lim exp ik 2η QL Sx HL,J x, eˆ1 , x, eˆ2 , . . . , x, eˆL  μ(dx). L→∞ J →∞ XkL

Since 

√     exp ik 2η QL Sx HL,J x, eˆ1 , x, eˆ2 , . . . , x, eˆL  μ(dx)

XkL

 = {

L

j =1





··· √ 4

exp ik

λj |xj |3k δ }



2η

L 

 λj μj xj2

HL,J (x1 , x2 , . . . , xL )

j =1

··· δ

λ1

√ 4

1 |x2 | √ (3k δ − λ1 |x1 |) 4 λ2

1 |xL | √ 4

λL

 (3k δ − L−1 j =1

  1 1 2η 2 × √ exp − (1 − 2ik λj μj )xj dxj , 2 2π j =1 L 

2

xj 1 √ e− 2 dxj 2π j =1





= 3k |x1 | √ 4

L 

√ 4

λj |xj |)

HL,J (x1 , x2 , . . . , xL )

306

S. Albeverio, I. Mitoma / Bull. Sci. math. 133 (2009) 272–314

setting zj = 1 − 2ik 2η λj μj = |zj |e−iαj ,

0 < sign(μj )αj <

π , 2

we get by the Cauchy integral theorem,  HL,J (x1 , x2 , . . . , xL ) √  L−1 4 1 δ |xL | √ 4

λL

(3k −

λj |xj |)

j =1

  1 1 × √ exp − (1 − 2ik 2η λL μL )xL2 dxL 2 2π  αL αL |zL | 2 1 = ei 2 HL,J (x1 , . . . , ei 2 xL ) √ e− 2 xL dxL 2π L−1 √ 1 4 δ |xL | √ 4

λL

 α2L

(3k −

λj |xj |)

j =1

 2     L−1  1 1 zL δ 4 + + λj |xj | eiθ 3k − √ exp − √ 2 4 λL 2π j =1 π 0     L−1  1 δ 4 − λj |xj | eiθ × HL,J x1 , . . . , √ 3k 4 λL j =1   L−1  1 4 ×√ λj |xj | ieiθ dθ. 3k δ − 4 λL j =1 Further     



π+





αL 2

 α2L



 ···

3k |x1 | √ 4

δ

λ1

√ 4

1 |x2 | √ (3k δ − λ1 |x1 |) 4 λ2

|xL−1 | √ 4

1 λL−1

 (3k δ − L−2 j =1

√ 4



π+

+

λj |xj |)

0

 2     L−1  1 1 zL 4 λj |xj | eiθ 3k δ − √ exp − √ 2 4 λL 2π j =1     L−1  1 δ 4 λj |xj | eiθ 3k − × HL,J x1 , . . . , √ 4 λL j =1      L−1 L−1    1 1 1  δ iθ 2 4 dx z − λ |x | ie x dθ 3k ×√ exp − √  j j j j j 4  2 λL 2π j =1 j =1     ··· √ L−2 √ δ 1 4 1 δ 4 δ 3k |x1 | √ 4 |αL |

2 0

λ1

|x2 | √ 4

λ2

(3k − λ1 |x1 |)

|xL−1 | √ 4

λL−1

(3k −

j =1

2    L−1  1 Re(zL e2iθ ) δ 4 λj |xj | 3k − √ exp − √ 2 λL 2π j =1

λj |xj |)

π



αL 2

S. Albeverio, I. Mitoma / Bull. Sci. math. 133 (2009) 272–314

307

     1  L−1    (−1)ν  δ iθ  1 4 × − λ |x | e 3k δ 3k √ HL,J x1 , . . . , √ j j 4  4 λL  λL j =1

ν=0

×

L−1  j =1

Since

2

xj 1 √ e− 2 dxj dθ. 2π





 ···

3k |x1 | √ 4

√ 4

1 |x2 | √ (3k δ − λ1 |x1 |) 4

δ

1 |xL | √ 4

λ2

λ1

L−1 

λL

 (3k δ − L−1 j =1

ei

√ 4

αL 2

HL,J (x1 , . . . , ei

λj |xj |)

2

zj xj |zL | 2 1 1 × √ e− 2 dxj √ e− 2 xL dxL 2π 2π j =1   αL αL = ··· ei 2 HL,J (x1 , . . . , ei 2 xL ) L √ 4 δ

{

j =1

×

L−1  j =1

λj |xj |3k }

2

zj xj |zL | 2 1 1 √ e− 2 dxj √ e− 2 xL dxL , 2π 2π

the right-hand side of the above equality is equal to    ··· 3k |x1 | √ 4

√ 1 |x2 | √ (3k δ − 4 λ1 |x1 |) 4

δ

λ2

λ1

|xL−1 | √ 4

×

1 λL−1

L−2  j =1

 1 |xL | √ (3k δ − L−2 j =1 4 λL



 (3k δ − L j =L−1,j =1

ei

√ 4

αL 2

√ 4

λj |xj |)

HL,J (x1 , . . . , ei

αL 2

xL )

λj |xj |)

2

2

zj xj zL−1 xL−1 |zL | 2 1 1 1 dxL−1 . √ e− 2 dxi √ e− 2 xL dxL √ e− 2 2π 2π 2π

Using the Cauchy integral theorem, we have  2 zL−1 xL−1 αL 1 HL,J (x1 , . . . , ei 2 xL ) √ e− 2 dxL−1 2π √  L 1 4 δ |xL−1 | √ 4

λL−1

(3k −

j =L−1,j =1

= |xL−1 | √ 4

λj |xj |)



√

 1 4 (3k δ − L j =L−1,j =1 λL−1

× HL,J (x1 , . . . , ei

ei

αL−1 2

αL−1 2

λj |xj |)

xL−1 , ei

αL 2

2

|zL−1 |xL−1 1 2 xL ) √ e − dxL−1 + R. 2π

R is a remainder, which is estimated as follows.

αL 2

xL )

308

S. Albeverio, I. Mitoma / Bull. Sci. math. 133 (2009) 272–314 |αL−1 | 2



|R|  0

2    L   1 Re(zL−1 e2iθ ) δ 4 λj |xj | 3k − √ exp − √ 2 λL−1 2π j =L−1,j =1

   1   (−1)ν  × 3k δ − HL,J x1 , . . . , √ 4  λL−1 ν=0

L 

 4



λj |xj | e , e iθ

i

j =L−1,j =1

αL 2

   xL  

1 ×√ 3k δ dθ. 4 λL−1 By repeating this argument recursively, we have  {

L

j =1





√ 4

···

exp ik

2η

= {

L

×

j =1

√ 4

 ei

··· λj |xj

 λj μj xj2

HL,J (x1 , x2 , . . . , xL )

j =1

λj |xj |3k δ }



L 

L

αj j =1 2

HL,J (ei

α1 2

x1 , e i

α2 2

x2 , . . . , e i

L 

2

xj 1 √ e− 2 dxj 2π j =1

αL 2

xL )

|3k δ }

L 

2

|zj |xj 1 √ e− 2 dxj + RL , 2π j =1

(3.15)

where |RL | 



L 

···

√ m=1 3k δ |x2 | √1 (3k δ − 4 λ1 |x1 |) |x1 | √ 4 λ 4 λ1

|αm |

2 0





2

 1 |xL | √ (3k δ − L−1 j =m,j =1 4 λL

√ 4

λj |xj |)

2    L   1 1 Re(zm e2iθ ) δ 4 λj |xj | 3k δ 3k − √ exp − √ √ 4 2 λm λ 2π m j =m,j =1

    1  L    (−1)ν  δ 4 × λj |xj | eiθ , 3k − HL,J x1 , . . . , √ 4  λ m ν=0 j =m,j =1   αm+1 αL  ei 2 xm+1 , . . . , ei 2 xL   ×

m−1  j =1

L xj2 |zj |xj2  1 1 √ e− 2 dxj √ e− 2 dxj dθ. 2π 2π j =m+1

Now, we prove that for sufficiently large k, δ √ √     det 1 − 2ik 2η SQA S  sup |RL | = o(e− k2 ). 0 L

We begin by remarking that if

(3.16)

(3.17)

S. Albeverio, I. Mitoma / Bull. Sci. math. 133 (2009) 272–314

309



 3k δ 1  δ  3k − 4 λ1 |x1 | , . . . , x ∈ x ∈ R L ; |x1 |  √ , |x2 |  √ 4 4 λ1 λ2   m−2  1 4 |xm−1 |  √ λj |xj | , 3k δ − 4 λm−1 j =1   m−1  1 δ 4 |xm+1 |  √ λj |xj | , . . . , 3k − 4 λm+1 j =1   L−1   1 δ 4 |xL |  √ λj |xj | , 3k − 4 λL j =m,j =1 the following inequalities hold. We have  m−1  √  L  4     eiθ (3k δ − L α =m,=1 λ |x |) m   i 2   λ x aR,ij + λm a + λ e x a √    R,ij R,ij  4  λm =1 =m+1

m−1  √    L 4    eiθ (3k δ − L λ |x |)    = m,=1 4 4 4 +  λ |x | +  λm λ |x | √  4 λm =1 =m+1     × sup 4 λ |aR,ij |  3k δ sup 4 λ |aR,ij |. 



Then by (S.3) and η + δ  2η < < 12 ,  detJ,R ∗dA ∗ D 0 √ √ k η m−1 2 12

√



{

k

j =1

λj xj eˆj +

∞ 

√

 4 λ |x |) eiθ (3k δ − L  j j j =m,j =1 λm eˆm + L √ j =m+1 4λ m



√

αj i λj e 2

  xj eˆj }

 kη   exp √ 3k δ sup 4 λ |aR,ij |  ec12 . k  i,j =1 Since  m−1 √ L       eiθ (3k δ − =m,=1 λ |x |)       Cu (t), 0 , eˆm λ x Cu (t), 0 , eˆ + λm √   λm =1  L   α     (t), 0 , eˆ  + λ ei 2 x C  u  =m+1

m−1  √    L    λ |x |)   eiθ (3k δ − L    = m,=1 +  λ |x | +  λm λ |x | Cu (t)0 √  λm =1 =m+1   δ    3k Cu (t) 0 and

 u  A (t)  c2 (A0 )t, 0

similarly we have for sufficiently large k,

310

S. Albeverio, I. Mitoma / Bull. Sci. math. 133 (2009) 272–314

    (−1)ν  ,J 3k δ − FL,A0 x1 , . . . , √  λm

L 

 4

 λj |xj | e , e iθ

i

αm+1 2

xm+1 , . . . , e

i

j =m,j =1

αL 2

   xL  

 c13 (A0 ). 1 , By Assumption 2 and since we take δ < η < 12  

  ∞  3η m−1 L 4     eiθ (3k δ − L α  p=m,p=1 λp |xp |)  k i 2p λp xp + λm + λp e xp √ i √ 4  k λm r=N +1 p=1 p=m+1

m−1    L 4    eiθ (3k δ − L  i αj j =m,j =1 λj |xj |) 2 × λj xj + λm + λj e xj √ 4 λm j =1 j =m+1

m−1 √  4   eiθ (3k δ − L =m,=1 λ |x |) × λ x + λm √ 4 λm =1   L  r  α i 2 pj   + λ e x β  /r!  =m+1

e

k√3η k

(3k δ )3 β

= c14 (β).

Then for sufficiently large k,      4 eiθ (3k δ − L   αL j =m,j =1 λj |xj |) i αm+1 HL,J . . . , (−1)ν 2 xm+1 , . . . , e i 2 xL  , e √   4 λ m

 ec12 c13 (A0 )c14 (β) = c15 (A0 , β) < +∞. Now first we consider the case where L 

3k δ −

j =m,j =1

 3k δ 4 . λj |xj | > 2

Since zm e2iθ = |zm |ei(−αm +2θ)

and |−αm + 2θ |  |αm |,

e2iθ )  1

and further noticing that √ |αm |  2k 2η λm |μm |, |zm | = |1 − 2 −1k 2η λm μm |  1

we get Re(zm

and noticing (S.1), we have for sufficiently large k,  L     ···   √ m=1 3k δ |x2 | √1 (3k δ − 4 λ1 |x1 |) |x1 | √ 4 λ 4 λ1

|αm |

2 0

2



 1 |xL | √ (3k δ − L−1 j =m,j =1 4 λL

√ 4

λj |xj |)

2    L   1 1 Re(zm e2iθ ) δ 4 λj |xj | 3k δ 3k − √ exp − √ √ 4 2 λm λ 2π m j =m,j =1

S. Albeverio, I. Mitoma / Bull. Sci. math. 133 (2009) 272–314

311

    1  L    (−1)ν  δ 4 × λj |xj | eiθ , 3k − HL,J x1 , . . . , √ 4  λ m ν=0 j =m,j =1   αm+1 αL  √ ei 2 xm+1 , . . . , ei 2 xL χ δ L δ  {3k − j =m,j =1 4 λj |xj |> 3k2 }  2 L m−1  x2  1  1 − |zj |xj  − 2j × dxj √ e √ e 2 dxj dθ   2π 2π j =1

j =m+1

  1 (3k δ )2 δ 3k |α | exp − √ √ m 8 maxm=1,2,...,L λm 2π 4 λm m=1   L  12c15 (A0 , β)  9k 2δ 4 2η+δ 3 λm |μm |k exp −  √ √ 8 maxm=1,2,...,L λm 2π m=1 

L 

2c15 (A0 , β) √

 c16 e−k . δ

Secondly we look at the complementary case: 3k δ −

L 

 4

λj |xj | 

j =m,j =1

3k δ . 2

(3.18)

In this case, dominating  2   L   Re(zm e2iθ ) δ 4 exp − λj |xj | 3k − √ 2 λm j =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   L   3k δ 4 μˆ  λj |xj | , 2 j =m,j =1

x |zj |x − 2j − 2j √1 √1 dxj L dxj . where μˆ = m−1 j =m+1 2π e j =1 2π e Since  L  m−1   L xj2 |zj |xj2    1  1 4 · · · exp λj |xj | √ e− 2 dxj √ e− 2 dxj 2π 2π j =m,j =1 j =1 j =m+1 R L−1  ∞     4  E exp λj  x, eˆj   c17 , 2

2

j =1

so that



3k δ  μˆ 2

L 

 4

 λj |xj |  c17 e−

j =m,j =1

Hence in the case of (3.18), we have

3k δ 2

.

312

S. Albeverio, I. Mitoma / Bull. Sci. math. 133 (2009) 272–314

 L    





 ···

√ m=1 3k δ |x2 | √1 (3k δ − 4 λ1 |x1 |) |x1 | √ 4 λ 4 λ1

2

 1 |xL | √ (3k δ − L−1 j =m,j =1 4 λL

√ 4

λj |xj |)

|αm |

2    L   1 1 Re(zm e2iθ ) 4 λj |xj | 3k δ 3k δ − √ exp − √ √ 4 2 λm λ 2π m j =m,j =1 0     1  L    (−1)ν  δ 4 × λj |xj | eiθ , 3k − HL,J x1 , . . . , √ 4  λ m ν=0 j =m,j =1   αm+1 αL  √ ei 2 xm+1 , . . . , ei 2 xL χ δ L δ  {3k − j =m,j =1 4 λj |xj | 3k2 }  2 L m−1  x2  1  1 − |zj |xj  − 2j × dxj √ e √ e 2 dxj dθ   2π 2π 2

j =1



j =m+1

L  3k δ 12c17 c15 (A0 , β)  δ 4 λm 3 |μm |k 2η+δ e− 2  c18 e−k . √ 2π m=1

This, together with (3.10), yields the completion of the proof of (3.17). Since we take δ < η, we have for x ∈ D, √ ! ∞  4    Sx η     √ √ , eˆj   3k . 2η 1 − 2ik SQA0 S j =1 Hence by Assumption 2, r   ∞  1 r 6ηr 1   pj  

Vk x, eˆp  Vk x, eˆj  Vk x, eˆ β  (27β)r √ k , √ k p,j,=1 k so that 0 < 6η <

6 12

= 

√ lim ( k)N +1

k→∞

gives some natural number n(N + 1) such that k 6η n(N +1) = 0. √ k

(3.19)

1 2

By (3.19) and the estimations similar to (1) and (6) of Lemma 1, we have   1 Vk A √  F x √ √ 0 k det(1 − 2ik 2η SQA0 S) D r 

∞  ∞  i  pj 

Vk x, eˆp  Vk x, eˆj  Vk x, eˆ β /r! μ(dx) × √ k r=N +1 p,j,=1   L αj α1 α2 αL = lim lim ei j =1 2 HL,J (ei 2 x1 , ei 2 x2 , . . . , ei 2 xL ) ··· L→∞ J →∞ L √ 4 δ {

j =1

λj |xj |3k }

(3.20)

S. Albeverio, I. Mitoma / Bull. Sci. math. 133 (2009) 272–314

×

L 

313

2

|zj |xj 1 √ e− 2 dxj , 2π j =1

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 Assumption 2, we further have 2b   ∞  i    pj   E √

Vk x, eˆp  Vk x, eˆj  Vk x, eˆ β   k  p,j,=1

6b  ∞  1 2b  1  .  c19 (b) √ |μj | k j =1 This and estimations similar to (2) and (7) of Lemma 1, together with (3.19) and (3.20), completes the proof of Theorem 2. Acknowledgement Both authors gratefully acknowledge a kind invitation of R. Léandre to 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 and to Prof. K. Funakubo for his kind guidance in Mathematical Physics. References [1] S. Albeverio, J. Schäfer, Abelian Chern–Simons theory and linking numbers via oscillatory integrals, J. Math. Phys. 36 (1995) 2157–2169. [2] S. Albeverio, A. Sengupta, A mathematical construction of the non-Abelian Chern–Simons functional integral, Comm. Math. Phys. 186 (1997) 563–579. [3] S. Albeverio, A. Hahn, A. Sengupta, Rigorous Feynman path integrals with applications to quantum theory, gauge fields and topological invariants, in: J.C. Zambrini et al. (Eds.), Proc. Lisbon Conf., 2004. [4] S. Albeverio, S. Mazzucchi, Some new developments in the theory of path integrals with applications to quantum theory, J. Stat. Phys. 115 (2004) 191–215. [5] S. Albeverio, R. Høegh-Krohn, S. Mazzucchi, Mathematical Theory of Feynman Path Integrals – An Introduction, second ed., Springer, Berlin, Heidelberg, New York, 2007. [6] D. Altschuler, L. Freidel, Vassiliev knot invariants and Chern–Simons perturbation theory to all orders, Comm. Math. Phys. 187 (1997) 261–287. [7] M. Atiyah, The Geometry and Physics of Knots, Cambridge U.P., 1990. [8] I.Ya. Aref’eva, Non-Abelian Stokes formula, Theoret. Phys. Math. 43 (1980) 353–356. [9] S. Axelrod, I.M. Singer, Chern–Simons perturbation theory, in: Proceedings of the XXth International Conference on Differential Geometric Methods in Theoretical Physics, vol. 1, New York, 1991, World Sci. Publishing, River Edge, NJ, 1992, pp. 3–45. [10] S. Axelrod, I.M. Singer, Chern–Simons perturbation theory. II, J. Differ. Geom. 39 (1) (1994) 173–213. [11] D. Bar-Natan, Perturbative Chern–Simons theory, J. Knot Theory Ramifications 4 (1995) 503–547. [12] D. Bar-Natan, On the Vassiliev knot invariants, Topology 34 (1995) 423–472. [13] D. Bar-Natan, E. Witten, Perturbation expansion of Chern–Simons theory with non-compact gauge group, Comm. Math. Phys. 141 (1991) 423–440. [14] R. Bott, C. Taubes, On the self-linking of knots, J. Math. Phys. 35 (10) (1994) 5247–5287. [15] R. Bott, L.W. Tu, Differential Forms in Algebraic Topology, Grad. Texts in Math., vol. 82, Springer, New York, Heidelberg, Berlin, 1982.

314

S. Albeverio, I. Mitoma / Bull. Sci. math. 133 (2009) 272–314

[16] K.T. Chen, Iterated path integrals, Bull. Amer. Soc. 83 (1977) 831–879. [17] Yu.L. Dalecky, S.V. Fomin, Measures and Differential Equations in Infinite-Dimensional Space, Kluwer Academic Publishers, Dordrecht, Boston, London, 1991. [18] G. De Rham, Differentiable Manifolds, Grundl. Math. Wiss., vol. 266, Springer, Berlin, Heidelberg, New York, Tokyo, 1984. [19] J.D. Dollard, C.N. Friedman, Product Integration with Applications to Differential Equations, Encyclopedia of Math. Appl., vol. 10, Addison–Wesley Publishing Co., MA, 1979. [20] J. Fröhlich, C. King, The Chern–Simons theory and knot polynomials, Comm. Math. Phys. 126 (1989) 167–199. [21] I.M. Gelfand, N.Ya. Vilenkin, Generalized Functions, vol. 4, Applications of Harmonic Analysis, Academic Press, New York, London, 1964. [22] P.B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah–Singer Index Theorem, Mathematics Lecture Series, vol. 11, Publish or Perish Ind., Wilmington, DE, 1984. [23] E. Guadagnini, M. Martellini, M. Mintchev, Wilson lines in Chern–Simons theory and link invariants, Nucl. Phys. B 330 (1990) 575–607. [24] A. Hahn, Chern–Simons theory on R 3 in axial gauge, Dissertation, Rheinische Friedrich-Wilhelms-Universität Bonn, Bonn, 2000, Bonner Math. Schriften 345, Univ. Bonn, Bonn, 2001. [25] A. Hahn, The Wilson loop observables of Chern–Simons theory on R3 in axial gauge, Comm. Math. Phys. 248 (2004) 467–499. [26] A. Hahn, Chern–Simons models on S 2 × S 1 , torus gauge fixing, and link invariants I, J. Geom. Phys. 53 (2005) 275–314. [27] K. Itô, Generalized uniform complex measures in the Hilbertian metric space with their application to the Feynman integral, in: Proc. Fifth Berkeley Sympos. Math. Statist. and Probability, vol. II, Contributions to Probability Theory, Part 1, Berkeley, CA, 1965/66, Univ. California Press, Berkeley, CA, 2000, pp. 145–161. [28] L.H. Kaufmann, Witten’s integral and Kontsevich integral, in: Particles, Fields and Gravitation, Lect. Notes, 1998, pp. 308–381. [29] H.H. Kuo, Gaussian Measures in Banach Space, Lecture Notes in Math., vol. 463, Springer, Berlin, Heidelberg, New York, 1975. [30] H.B. Lawson Jr., M.L. Michelsohn, Spin Geometry, Princeton Math. Ser., vol. 38, Princeton Univ. Press, Princeton, NJ, 1989. [31] R. Léandre, Invariant Sobolev calculus on the free loop space, Acta Appl. Math. 46 (1997) 267–350. [32] P. Malliavin, S. Taniguchi, Analytic functions, Cauchy formula, and stationary phase on a real abstract Wiener space, J. Funct. Anal. 143 (1997) 470–528. [33] I. Mitoma, One loop approximation of the Chern–Simons integral, Acta Appl. Math. 63 (2000) 253–273. [34] I. Mitoma, Wiener space approach to a perturbative Chern–Simons integral, in: Stochastic Processes, Physics and Geometry, New Interplays II, Leipzig, 1999, in: CMS Conf. Proc., vol. 29, Amer. Math. Soc., Providence, RI, 2000, pp. 471–480. [35] I. Mitoma, S. Nishikawa, Asymptotic expansion of the one loop approximation of the Chern–Simons integral in an abstract Wiener space setting, J. Funct. Anal. 253 (2007) 729–771. [36] C.B. Morrey Jr., Multiple Integrals in the Calculus of Variation, Grundl. Math. Wiss., vol. 130, Springer, Berlin, Heidelberg, New York, 1966. [37] M. Niemann, Das Chern–Simons Modell: Quantisierung der Kopplungskonstante und simpliciale Approximation, Diploma thesis, Univ. Bonn, Bonn, Sept. 2004. [38] M. Reed, B. Simon, Methods of Modern Mathematical Physics I, Functional Analysis, second ed., Academic Press, New York, 1980. [39] A. Schwarz, The partition function of a degenerate functional, Comm. Math. Phys. 67 (1973) 1–16. [40] J.N. Tavares, Chen integrals, generalized loops and loop calculus, Internat. J. Modern Phys. A 9 (1994) 4511–4548. [41] H. Totoki, A method of construction for measures on function spaces and its applications for stochastic processes, Mem. Fac. Sci. Kyushu Univ. Ser. A Math. 15 (1962) 178–190. [42] E. Witten, Quantum field theory and the Jones polynomial, Comm. Math. Phys. 121 (1989) 351–399. [43] K. Yosida, Functional Analysis, second ed., Grundl. Math. Wiss., vol. 123, Springer, Berlin, Heidelberg, New York, 1968.