STOCHASTIC COHOMOLOGY OF THE FRAME BUNDLE OF THE LOOP SPACE

R. Léandre

INTRODUCTION

Let L_{x}(M) be the based loop space of smooth applications γ_{s} from the circle into M such that γ_{0} = γ_{1} = x. Let Q → M be a principal bundle over M with structure group G. L_{e}(Q) is the set of based loop in Q over the based loop space of M. It is a based loop group bundle whose the structure group is L_{e}(G), the based loop group of G. If Q → M is the frame bundle, L_{e}(Q) is the frame bundle of L_{x}(M): the structure of L_{e}(Q) is of the main importance to study string structures (or spin structures) over the loop space ([C.M], [C.P], [M.L]), and has a deep place in the understanding the Dirac operator over the loop space ([M.L]).

Let us suppose that the loop space is simply connected, in order to avoid all torsion phenomenon. If the loop space is the space of smooth loop, there is an equivalence between the cohomology with values in Z and S^{1} bundles over the loop space. Let us now endow the loop space with the Brownian bridge measure, if the manifold is supposed riemannian. The equivalence is not at all clear in the stochastic context: let us clarify what it means. In the stochastic context, the loop are only continuous. A stochastic cohomology of L_{x}(M) is defined in [L_{5}], [L_{7}] and [L_{8}] ithe values in C or R: since L_{x}(M) is supposed simply connected, we can neglect all torsion phenomenon in order to construct a S^{1} bundle from a Z closed 2 form over the loop space of finite energy loops. But we have to choose distinguished paths in L_{x}(M) in order to shrink a loop in a constant loop: let l_{t}(γ)_{s} such a distinguished path. The law of l_{t}(γ)_{.} is not absolutely continuous with respect of the law of γ. So we have to consider special type of forms in order to overcome the problem: this avoids to use a Z stochastic cohomology of the brownian bridge, by considering only examples.

The goal of this paper is to do a stochastic cohomology of the frame bundle of L_{x}(M), to construct the stochastic forms which allow to consider a string structure over L_{x}(M). Namely, we have already constructed stochastic bundles over L_{e}(Q) by starting from a given deterministic form over this set, and the goal of this paper is to give a stochastic meaning to this form ([L_{12}]).

As in [L_{12}], we define a measure over L_{e}(Q), by putching together measures in the fiber: the fiber is a continuous loop group. We start with the equation in the fiber

(0.1)  dg_{s} = dB_{s}g_{s}

In [L_{12}], we have studied the equation

(0.2)  dg_{s} = B_{s}dsg_{s}

We choose this equation in order to reflect the fiber structure of L_{e}(Q), the only obstacle to the trivialization beeing the holonomy over a loop in the basical manifold. Namely, we can consider the Albeverio-Hoegh-Krohn quasi invariance formulas under the right translation g_{.} → g_{.}K_{.}. If K_{.} is deterministic in C^{1}, the quasi-invariance density belongs to all the L^{p} in the first case, while it belongs only to L^{1} in the second case, if K_{.} is C^{2}.

This allows us to define a tangent space of L_{e}(Q) by using an infinite dimensional connection and to get horizontal vector fields and vertical vector fields. We meet the following paradoxe: the big difference between the Sobolev Calculus over the loop group and the Sobolev Calculus over the loop space of a riemannian manifold is the following: in the first case, the tangent vector fields are stable by Lie Bracket, in the second case no. Apparently, if we follow this remark, we have to separate the treatment of the horizontal component and of the vertical component of a form, in order to define a stochastic exterior derivative over L_{e}(Q). Let us recall namely that, in order to define some cohomology groups over the loop spaces, we have imposed in [L_{5}] some regularity assumptions over the kernels of the associated forms, in order to simplify the treatment of the anticipative Stratonovitch integrals which appears in the definition of the exterior stochastic differential. These conditions lead to needless complications in the case of loop groups ([F.F]). But in our situation, we cannot neglect the curvature phenomena which appear: we are obliged to treat the horizontal and the vertical components in the same manner, in order to define some stochastic cohomology groups of L_{e}(Q). The Carey-Murray ([C.M]) form is closed for this stochastic cohomology (If the first Pontryaguin class of Q vanishes), because it is a mixture between a basical iterated integral and the canonical 2 form over a loop group: this gives the second aspect of the construction of the string bundle in our stochastic situation.

Moreover, this Calculus depends apparently of the connection over the frame bundle L_{e}(Q) → L_{x}(M). But we show that the functional spaces which are got with some regularity assumptions over the kernels are independant of this connection.

We thank the warm hospitality of the Max Planck Institut for Mathematic of Bonn where this work was done.

STOCHASTIC COHOMOLOGY OF THE LOOP SPACE OF THE BUNDLE

Let Q → M be a principal bundle with a compact connected structural Lie goup G. We suppose that M is endowed with a Riemannian metric: there exists a heat semi-group over M and a brownian bridge measure dP_{1,x} associated to the riemannian metric.It is a measure over the based continuous loop space.

Over G, we consider the following stochastic differential equation:

(1.1)  dg_{s} = dB_{s}g_{s} ; g_{0} = e

where B_{s} is a brownian motion independant of the law of the loop γ over M over Lie G.

We get a law Q which can be desintegrated over the pinned path space of path in the group joining e to g ([Gr_{2}][A.M]). We get a space of continuous paths in G L_{g}(G) endowed with a law Q_{g}. The non pinned based path group is denoted P (G).

We put over the bundle Q → M a connection ∇^{Q}: τ_{s}^{Q} the parrallel transport for a loop γ_{s} is therefore almost surley defined for the connection ∇^{Q}. We denote by L_{e}(Q) the space of loop q_{.} in Q such that q_{s} = τ_{s}^{Q}g_{s}, g_{1} = (τ_{1}^{Q})^{−1}. We get the following commutative diagramm ([C.M]):

(1.2)  \begin{aligned}L_{e}(Q) & → & P (G) \\ ↓ & & ↓ \\ L_{x}(M) & → & G \end{aligned}

The map from P(G) to G is the map which to g_{.} associates g_{1}. The map from L_{e}(Q) to L_{x}(M) is the projection map. The map f from L_{x}(M) to G is the map which to a stochastic loop γ_{.} associates (τ_{1}^{Q})^{−1}. The map from L_{e}(Q) to P (G) is the map which to q_{.} associates g_{.}. It is nothing else than f^{∗}.

Over L_{e}(Q), we put the measure:

(1.3)  dP_{tot} = dP_{1,x} ⊗ dQ_{(τ_{1})^{−1}}

Let us analyze a little bit more the L_{e}(G) bundle P (G) → G. If g_{1} ∈ G_{i} a small open neighborhood of G, we can choose a section g_{i,s}(g_{1}) of this bundle which is jointly smooth in s and in g_{1}. It checks the following property: g_{i,0}(g_{1}) = e; g_{i,1}(g_{1}) = g_{1} ; g_{i,s}(g_{1}) ∈ G. This means that the transition functions of P (G) can be choosen to take their values in the smooth based loop space og G, L_{e}^{∞}(G). Since G is a compact manifold, we can choose a connection over the bundle P (G) → G whose the structural group is reduced to L_{e}^{∞}(G). Let us call ∇^{∞} this connection: if g_{1} ∈ G_{i}, the connection one form is a smooth path in the Lie algebra of G starting from 0 and arriving at 0 K_{i,s}(g_{1}), which depends smoothly from g_{1} ∈ G_{i} and which is a one form in g_{1}.

The obstruction to trivialize L_{e}(Q) over L_{x}(M) lies in (τ_{1}^{Q})^{−1}: if (τ_{1}^{Q})^{−1} ∈ G_{i}, there is a local slice of L_{e}(Q) which is g_{i,.}((τ_{1}^{Q})^{−1}). We look at the left transformation g_{.} → (g_{i,.}((τ_{1}^{Q})^{−1})^{−1})g_{.}. Modulo this transformation, the bridge between e and (τ_{1}^{Q})^{−1} is transformed into the bridge between e and e. Let us recall namely the purpose of the quasi-invariance formula from Albeverio-Hoegh-Krohn ([A.H-K]): if k_{s} is a deterministic C^{1} path in the group G, the law of g_{.}k_{.} and the law of k_{.}g_{.} are quasi-invariant with respect to the law of (1.1). Moreover the density of quasi-invariance belong to all the L^{p} and can be desintegrated along the appropriate bridge. We denote by J_{r}(k) and by J_{l}(k) the right quasi-invariance density and the left quasi-invariance density ([A.H-K], [Gr_{2}]).

Therefore if (τ_{1}^{Q})^{−1} ∈ G_{i}:

(1.4)  dP_{tot} = dP_{1,x} ⊗ J_{l}(g_{i,}((τ_{1}^{Q})^{−1}))dQ_{e}

J_{l}(g_{i,.}((τ_{1}^{Q})^{−1}) belongs to all the L^{p} and is bounded in L^{p} when (τ_{1}^{Q})^{−1} describes G_{i}. (1.4) produces a stochastic trivialization of our bundle.

Let us recall that a vector field over L_{x}(M) is given by [Bi] [J.L_{1}]

(1.5)  X_{t} = τ_{t}H_{t} ; X_{0} = X_{1} = 0

where τ_{t} is the parallel transport associated to the Levi-Civita connection and H_{.} is a finite energy path in T_{x}. We choose as Hilbertian norm of X_{.} the norm

(1.6)  ∥X∥^{2} = ∫_{0}^{1} ∥H′_{s}∥^{2}ds = ∥ H∥^{2}

Let us recall that a right vector field over L_{e}(G) is given by X_{t}^{r} = g_{t}K_{t} where K_{t} is a finite energy path with end points equal to 0 in the Lie algebra of G which checks ∫_{0}^{1} ∥K′_{s}∥^{2}ds = ∥K∥^{2} < ∞. A left vector field over L_{e}(G) is given by K_{s}g_{s} = X_{s}^{l} where K_{s} checks the same condition (See [L_{12}]).

We pullback the connection ∇^{∞} to be a connection over the stochastic bundle L_{e}(Q) → L_{x}(M). If s → K_{i,s}(g_{1})(dg_{1}) is the connection form for g_{1} ∈ G_{i}, the connection form of the pullback connection still denoted ∇^{∞} is s → K_{i,s}((τ_{1}^{Q})^{−1})(< d(τ_{1}^{Q})^{−1}, . >). For that, we recall that:

(1.7)  < dτ_{1}^{Q}, X >= τ_{1}^{Q} ∫_{0}^{1}(τ_{s}^{Q})^{−1}R^{Q}(dγ_{s}, X_{s})τ_{s}^{Q}

where R^{Q} is the curvature tensor of ∇^{Q}.

We define as tangent space of the total space L_{e}(Q) the orthonormal sum of the horizontal vector fields and vertical vector fields. In the trivialization given by (1.4), the horizontal vector fields are given by:

(1.8)  X^{H}(H)_{s} = τ_{s}H_{s} − K_{i,s}((τ_{1}^{Q})^{−1}) < d(τ_{1}^{Q})^{−1}, X > g_{i,s}

and the vertical vector fields are given by q_{s}K_{s} = X^{V} (K)_{s}. We choose as Hilbert norm of X^{H}(H) the quantity ∥H∥^{2} and of X^{V} (K) the quantity ∥K∥^{2}. These vector fields are consistently defined (See [L_{12}]).

Let us recall the following theorem ([L_{12}]):

Theorem I.1: Let us consider a cylindrical functional F (q_{s_{1}}, .., q_{s_{r}}) over L_{e}(Q). Then there exists a functional divX^{H}(H) and a functional divX^{V} (K) which belong to all the L^{p} such that for deterministic H and K:

(1.9)  E_{tot}[< dF, X^{H}(H) >] = E_{tot}[F divX^{H}(H)]

and such that

(1.10)  E_{tot}[< dF, X^{V} (K) >] = E_{tot}[F divX^{V} (K)]
Let us introduce over T^{H}(L_{e}(Q)) and T^{V} (L_{e}(Q)) a connection:
(1.11)  ∇X^{V} (K) = X^{V} (∇K)

and

(1.12)  ∇X^{H}(H) = X^{H}(∇H)

where ∇K_{s} is the H-derivative of K_{s} in the fixed Lie algebra of G and ∇H_{s} is the H-derivative in the fixed tangent space at x of M. The integration by parts (1.9) and (1.10) allow to define consistently these derivatives.

If k_{s} = ∑ k_{s}^{i}k_{i} where k_{i} is an orthonormal basis of the Lie algebra of G, we get:

(1.13)  ∇_{X}K_{s} = ∑ < dk_{s}^{i}, X > k_{i}

The same holds for H = ∑ h_{s}^{i}e_{i} where e_{i} is a fixed basis of the tangent space of M at x.

Let us consider a n cotensor ω over L_{e}(Q). Let us recall that ∇ω is defined as follows:

(1.14)  ∇ω(X_{1}, .., X_{n+1}) =< d(ω(X_{1}, .., X_{n})), X_{n+1} > − ∑_{i=1}^{n} ω(X_{1}, .., ∇_{X_{n+1}}X_{i}, .., X_{n})

This allows us to define iteratively the k covariant derivative of a n form σ. Let us describe a bit the situation: a n form is a n antisymmetric tensor over the tangent Hilbert space of q_{.}, which has a priori two types of behaviour:

-)The horizontal contribution. -)The vertical contribution. These contributions have two different behaviour:

(1.15)  \begin{matrix}σ(X^{H}(H_{1}), .., X^{H}(H_{n}) & , X^{V} (K_{1}), .., X^{V} (K_{m})) = \\ ∫ ∫ σ^{n,m}(s_{1}, .., s_{n}; t_{1}, .., t_{m})H′_{1,s_{1}}... & H′_{n,s_{n}}K′_{1,t_{1}}...K′_{m,t_{m}}ds_{1}..ds_{n}dt_{1}..dt_{m} \end{matrix}

where σ^{n,m} is a kernel which checks the antisymmetric conditions due to the antisymmetric conditions over σ. The covariant derivatives of σ have too two different contributions which are due to the vertical and horizontal vector fields. In order to simplify the exposure, we won’t do in the formulas the difference between the two type of contributions: a form σ is given by kernels σ(s_{1}, .., s_{n}) whose the covariant derivatives with respect to the connection ∇ are given by kernels σ(s_{1}, .., s_{n}; t_{1}, .., t_{k}). Moreover ∫ σ(s_{1}, .., s_{n}; t_{1}, .., t_{k})ds_{i} = 0 and ∫ σ(s_{1}, .., s_{n}; t_{1}, .., t_{k})dt_{j} = 0 since we work over the loop space.

Let us define the Nualart-Pardoux constants of σ. Let K be a connected component of [0, 1]^{n} × [0, 1]^{k} where we had removed the diagonals. We define the first Nualart-Pardoux constant as C(p, n, k)(Q) by the smallest constant such that:

(1.16) ∥σ(s_{1}, .., s_{n}; t_{1}, ..t_{k}) − σ(s′_{1}, .., s′_{n}; t′_{1}, .., t′_{k})∥_{L^{p}} ≤ C(p, n, k)(Q)(∑ \sqrt{| s_{i} − s′_{i} |} + ∑ \sqrt{| t_{i} − t′_{i} |}

over any K.

The second Nualart-Pardoux constan C′(p, n, k)(Q) is the smallest one such that for all s_{i} and all t_{j}:

(1.17)  ∥σ(s_{1}, .., s_{n}; t_{1}, .., t_{k})∥_{L^{p}} ≤ C′(p, n, k)(Q)

Definition I.2: A n form is said smooth in the Nualart-Pardoux sense if the collection of C(p, n, k)(Q) and C′(p, n, k)(Q) is finite.

We have a theorem whose the proof is the analoguous of the proof of the theorem I.2 of [L_{5}].

Theorem I.3: if σ is a n form which is smooth in the Nualart-Pardoux sense and if σ′ is a n′ form which is smooth in the Nualart-Pardoux sense, σ ∧ σ′ is a n + n′ form which is still smooth in the Nualart-Pardoux sense.

Over P (G), we can consider the brownian motion measure: g_{1} is free. The tangent space of a path g_{s} is given by the set of vector of the shape g_{s}K_{s} = X_{s} where K_{0} is equal to 0 and K_{1} is free. It is endowed with the Hilbert structure ∫_{0}^{1} ∥K′_{s}∥^{2}ds. We can repeat the previous considerations and give the definition of a form which is smooth in the Nualart-Pardoux sense over P (G): its Nualart-Pardoux constants are called C(p, n, k)(G) and C′(p, n, k)(G). We choose the same connection than in Q for the definition of iterated covariant derivatives of a form over P (G).

In the same way, over L_{x}(M), we can consider the brownian bridge measure. The tangent space of a loop is the space of τ_{s}H_{s}, H_{0} = H_{1} = 0 and we choose the Hilbert structure ∫_{0}^{1} ∥H′∥^{2}ds. We choose the same connection as before in order to iterate the covariant derivatives of a form. If σ is a n form, we can define its Nualart-Pardoux constant C(p, n, k)(M) and C′(p, n, k)(M).

Theorem I.4: Let σ(M) be a n form over L_{x}(M) which belongs to all the Nualart-Pardoux spaces. THen π^{∗}σ(M) = σ(Q) is a n form over L_{e}(Q) which belongs to all the Nualart-Pardoux spaces.

The proof is clear: the Nualart-Pardoux constants are the same. It is not the same for the next theorem: Theorem I.5: Let σ(G) be a n form over P (G) which belongs to all the Nualart-Pardoux spaces. Then

(f^{∗})^{∗}σ(G) = σ(Q) is a n form which belongs to all the Nualart-Pardoux spaces over L_{e}(Q).

Proof: Since the functional γ_{.} → h(τ_{1}^{Q}) belongs to all the Nualart-Pardoux spaces over L_{x}(M) if h is smooth, because the covariant derivatives of τ_{1}^{Q} are given by iterated integrals (See (1.7)), we can work in a region where L_{e}(Q) is trivial, by using a partition of unity over G associated to the G_{i}. L_{e}(Q) is locally a product, and we can speak of basical and (left or right) vertical vector fields. By the lemma A.2. of the appendix, the Nualart-Pardoux norms in terms of basical and right vertical vector fields are equivalent to the Nualart-Pardoux norms in term of the right vertical vector fields and the horizontal vector fields.

It remains to show that (f^{∗})^{∗}σ has locally Nualart-Pardoux constants for the basical and the right vertical vector fields which are finite.

The vertical derivative are given by the vertical derivatives over a pinned path group: the basical one are view by using the derivatives of g_{i,.}((τ_{1}^{Q})^{−1}) which check the Nualart-Pardoux conditions, because (τ_{1}^{Q}) checks the Nualart-Pardoux conditions and the g_{i,s} are smooth in τ_{1}^{Q} and s together. It remains to solve the problem that we don’t consider the form σ over P (G) but the form over L_{(τ_{1}^{Q})^{−1}}(G) isomorphic to

L_{e}(G) by the map g_{.} → g_{i,.}((τ_{1}^{Q})^{−1})g_{.} This leads to the vector field < dg_{i,.}((τ_{1}^{Q})^{−1}, X > g_{.} which is a left vector field over P (G). But a map which checks the Nualart-Pardoux conditions over P (G) for right vector fields checks still the Nualart-Pardoux conditions for left vector fields (See lemma A.3): if we had consider as trivialization the couple of L_{x}(M) × P (G) with the trivialization map (γ_{.}, g_{.}) → (γ_{.}, g_{i,.}((τ_{1}^{Q})^{−1})g_{.}), the proof would be finished. But a map over P (G) can be reduced into a map over L_{e}(G) if it satisfies the Nualart-Pardoux conditions (See [L_{5}] beginning of the chapter II for the Riemannian case which is more complicated). So if σ(γ_{.}, g_{.}) checks the Nualart-Pardoux conditions over L_{x}(M) × P (G), it checks still the Nualart-Pardoux conditions over L_{x}(M) × L_{e}(G): the map which is associated is the map (γ_{.}, g_{.}) → g_{1}) which gives L_{x}(M) × L_{e}(G) as a finite codimensional manifold of L_{x}(M) × P (G).

♢

Example: Let Q → M a principal bundle with structure group G which is supposed simply connected simply laced ([P.S]). Over P (G) we consider the form which at the level of the Lie algebra of P (G) is equal for right vector fields to:

(1.18)  c(X, Y) = \frac{1}{8π^{2}} ∫_{0}^{1} < X_{s}, dY_{s} > − < Y_{s}, dX_{s} >

The form (f^{∗})^{∗}c satisfies over L_{E}(Q) the Nualart-Pardoux conditions by the theorem I.5.

Let μ be the form over L_{x}(M):

(1.19)  μ = \frac{1}{8π^{2}} ∫_{0<u<s<1} < (τ_{s}^{Q})^{−1}R^{Q}(dγ_{s}, .)τ_{s}^{Q} ∧ (τ_{u}^{Q})^{−1}RQ(dγ_{u}, .)τ_{u}^{Q} >

It satisfies the Nualart-Pardoux conditions over L_{x}(M). Therefore π^{∗}μ satisfies the Nualart-Pardoux conditions over L_{e}(Q). Let ν be a form such that dν = p_{1}^{Q}, the first Pontryaguin class of the bundle Q which is supposed to be zero in class. Let τ(ν) be the two form over L_{x}(M)

(1.20)  τ(ν) = ∫_{0}^{1} ν(dγ_{s}, ., .)

It satisfies to the Nualart-Pardoux conditions. Therefore the Carey-Murray two form ([C.M]) F_{Q} = (f^{∗})^{∗}c − π^{∗}(μ + τ(ν)) satisfies to the Nualart-Pardoux conditions over the big space L_{e}(Q).

Theorem I.6: The space of n form which are smooth in the Nualart-Pardoux sense is independent of the connection ∇^{∞}.

Proof: Let us consider two connections ∇^{∞} and ∇^{′∞}. After using a partition of unity associated to the open neighborhoods G_{i}, since τ_{1}^{Q} satisfies to the Nualart-Pardoux conditions, we come back to a situation where the bundle L_{e}(Q) is trivialized. The Nualart-Pardoux constants of σ with respect of the connection ∇^{∞} can be estimated in terms of the basical Nualart-Pardoux constants, which can be estimated themselves in terms of the Nualart-Pardoux constants with respect of the connection ∇^{′∞} (See lemma A.2). Therefore the result . ♢

Let us recall that the exterior derivative of a n − 1 form σ is defined as follows:

(1.21)  \begin{matrix}dσ(X_{1}, .., X_{n}) = ∑(−1)^{i−1} & < dσ(X_{1}, .., X_{i−1}, X_{i+1}..X_{n}), X_{i} > \\ + ∑_{i<j}(−1)^{i+j}σ([X_{i}, X_{j}], X_{1} & , ..X_{i−1}, X_{i+1}, .., X_{j−1}, X_{j+1}, ..X_{n}) \end{matrix}

Our goal is to define an exterior derivative over l_{e}(Q). (1.20) shows that we need to compute some Lie brackets.

-)Let us compute the Lie bracket of two (right) vertical vector fields. It is nothing else than g_{t}[K_{t}^{1}, K_{t}^{2}] if X_{t}^{V} (K^{i}) = g_{t}K_{t}^{i} for deterministic process K_{t}^{i} in the Lie algebra of G.

-)In order to compute the Lie bracket of two horizontal vector fields, we work in a local trivialization of L_{e}(Q). The first horizontal vector fild is given by

(1.22)  X^{H}(H^{1})_{t} = τ_{t}H_{t}^{1} − K_{i,t}(< d(τ_{1}^{Q})^{−1}, τ_{.}H_{.}^{1} >)g_{t}

and the second one is given by

(1.23)  X^{H}(H^{2})_{t} = τ_{t}H_{t}^{2} − K_{i,t}(< d(τ_{1}^{Q})^{−1}, τ_{.}H_{.}^{2} >)g_{t}

for deterministic H_{t}^{1} and H_{t}^{2}. We have:

(1.24) \begin{matrix}[X^{H}(H^{1}), X^{H}(H^{2})]_{t} = τ_{t} & ∫_{0}^{t} τ_{s}^{−1}R(dγ_{s}, τ_{s}H_{s}^{2})τ_{s}H_{t}^{1} \\ − < d(K_{i,t}(< d(τ_{1}^{Q})^{−1}, τ_{.}H_{.}^{1} >), τ_{.}H^{2} > g_{t} & + < K_{i,t}(< d(τ_{1}^{Q})^{−1}, τ_{.}H_{.}^{1} >)K_{i,t}(< d(τ_{1}^{Q})^{−1}, τ_{.}H_{.}^{2} >)g_{t} \\ +antisymmetry = X^{H}[τ_{.} & H_{.}^{1}, τ_{.}H_{.}^{2}] + R^{∞}(H_{.}^{1}, H_{.}^{2})g_{.} \end{matrix}

In other terms, the Lie bracket of horizontal vector fields is not an horizontal vector field associated to the generalized vector field [τ_{.}H_{.}^{1}, τ_{.}H_{.}^{2}]: some curvature phenomenon appears, which leads to some extra (left) vertical fields over the fiber (and not some (right) vector fields).

-)The Lie bracket of an horizontal vector field X^{H}(H) (H deterministic) and of a vertical vector field X^{V} (K) (K deterministic) is equal to zero.

We are ready to state the following theorem:

Theorem I.7: Let σ be an n form which is smooth in the Nualart-Pardoux sense over L_{e}(Q). Then dσ is a n + 1 form which is smooth in the Nualart-Pardoux sense over L_{e}(Q) and its Nualart-Pardoux constants can be estimated in terms of the Nualart-Pardoux constants of σ.

Proof: Only the contribution of the Lie bracket in (1.20) gives any problem. Since the Lie bracket of two (right) vertical vector fields is still a (right) vertical field, only the contribution of the Lie bracket of two horizontal vector fields put any problem.

We treat first the contribution of X^{H}[τ_{.}H_{.}^{1}, τ_{.}H_{.}^{2}]: this leads to a Stratonovitch integral in dγ_{s}: the lemma A.2 of [L_{5}] and more precisely the lemma A.1 of this work allow to show that this contribution satisfies to the Nualart-Pardoux conditions.

We consider now the contribution of R_{t}^{∞}(H_{.}^{1}, H_{.}^{2})g_{t} where R_{t} is a process with finite energy in the Lie algebra, which satisfies to the Nualart-Pardoux conditions because K_{i,t} is smooth in t and because τ_{1}^{Q} and τ_{s} satisfi to the Nualart-Pardoux conditions simultaneously. But

(1.25)  R_{t}^{∞}(H_{.}^{1}, H_{.}^{2})g_{t} = g_{t}(g_{t}^{−1}R_{t}^{∞}(H^{1}, H^{2})g_{t})

which is a generalized (right) vector field over L_{e}(G). The proposition A.4 allows to show that this contribution satisfies still the Nualart-Pardoux conditions: the only stochastic integral which appears does not occur from the derivative in time t of R_{t}^{∞} but of the time differential element of g_{t}^{−1} and of g_{t}. ♢

Example: let σ(G) be a n form over P (G) which belongs to all the Nualart-Pardoux spaces over P (G). dσ(G) is a n + 1 form over P (G) which satisfies to the Nualart-Pardoux conditions. We get

(1.26)  d(f^{∗})^{∗}σ(G) = (f^{∗})^{∗}dσ(G)

Namely this property is true if we consider finite energy loop over Q, and reflects some algebraic identities between iterated integrals; these algebraic identities remain true in the stochastic context.

If we consider a n form σ(M) over L_{x}(M) which satisfies to the Nualart-Pardoux conditions, dσ(M) is a n + 1 form over L_{x}(M) which satisfies to the Nualart-Pardoux conditions, and we have clearly:

(1.27)  π^{∗}dσ(M) = dπ^{∗}σ(M)

In particular if the first Pontryaguin class of the bundle Q is equal to zero, we can use the result of [C.M]

(1.28)  dF_{Q} = 0

because dF_{Q} is equal to zero over the finite energy loop space of Q: dF_{Q} is given by iterated integrals: these formulas remain true in the stochastic context.

APPENDIX: ANTICIPATIVE STRATONOVITCH INTEGRALS

Let us recall the following fact: if X_{s} = τ_{s}H_{s}; H_{0} = H_{1} = 0 is a deterministic vector field (This means, it corresponds to the deterministic vector field H_{s}), we get the following integration by parts formula, for a cylindrical functional F:

(a.1)  E[< dF, X >] = E[F divX]

divX is defined by the formula:

(a.2)  divX = ∫_{0}^{1} < τ_{s}H′_{s}, δγ_{s} > +\frac{1}{2} ∫_{0}^{1} < S_{X_{s}}, δγ_{s} >

S is the Ricci tensor and δ the Ito integral with respect to the Levi-Civita connection.

If K_{s}g_{s} is a left vector field (K_{0} = K_{1} = 0; K_{s} deterministic) over the basical loop group (or any pinned path space in the group), we get over the loop group an integration by parts formula analoguous to (a.1), but this time

(a.3)  div(K_{.}g_{.}) = ∫_{0}^{1} < K′_{s}, δB_{s} >

if g_{.} is given by the equation (1.1).

If g_{s}K_{s} is a right vector field over the basical loop group (K_{.} deterministic; K_{0} = K_{1} = 0), we get:

(a.4)  div(g_{.}K_{;}) = ∫_{0}^{1} < g_{s}K′_{s}g_{s}^{−1}, δB_{s} >

We have:

Proposition A.1: Let u(s, t; u, s̃) a random variable with value in T_{x}(M) which satisfies to the Nualart-Pardoux conditions over L_{e}(Q), the both type of derivatives included and s, t, u, s̃ included. Then the anticipative Stratonovitch integral:

(a.5)  ∫_{s}^{t} < τ_{u}u(s, t; us̃), dγ_{u} >= I(s, t; s̃)

satisfies to the Nualart-Pardoux conditions, the both type of derivatives included, and s, t, s̃ included.

Proof: We integrate by part in order to compute E[(I(s, t; s̃)^{p}] for som even integer p. u(s, t; u, s̃) is a vector field over the loop space if ∫_{s}^{t} u(s, t; u, s̃)du = 0. If the previous inequality is not checked, we can remove the average of u in order to recognize a vector field over the based loop space.

Let us put:

(a.6)  X_{s,t,s̃}(u) = τ_{u} ∫_{0}^{u} u(s, t; v, s̃)dv

In the definition of the divergence, we have to add the counterterm \frac{1}{2} ∫_{0}^{1} < S_{X_{s,t,s̃}(u)}, δγ_{u} > which is only apparently an anticipative integral, by integrating by part, in order to recognize the beginning of a curved Skorohod integral (See [L_{2}], [L_{3}]). The second counterterm we have to add is the integral of the kernels of some H-derivative of u(s, t; u, s̃) in order to recognize a complete Skorohod integral. For that, we begin by studying E[(I(s, t; s̃)^{p}] for the discrete approximation of the anticipative integral by Riemann sum of \frac{1}{t_{i+1}−t_{1}} ∫_{t_{i}}^{t_{i+1}} u(s, t; v, s̃)dv for a suitable subdivision of [0,1]. We find a sum of integral over [s, t]^{k} of polynomial expression in the derivatives of u and of the derivatives of τ_{.} and of dγ_{u}, with possible contraction over the diagonals. Let us recall that:

(a.7)  ∇_{X}τ_{s} = τ_{s} ∫_{0}^{s} τ_{u}^{−1}R(dγ_{u}, X_{u})τ_{u}

We work in a small trivialization of the bundle such that we can speak of basical derivatives (associated to basical vector fields) instead of horizontal vector derivatives (associated to horizontal vector fields): the derivatives which apper after integrating by parts in E[(I(s, t; s̃)^{p}] are basical derivatives, and not the horizontal derivatives which are given by the Nualart-Pardoux norms. This leads apparently to a problem, which is solved by the Lemma.A.2: we postpone the proof of this lemma later.

Lemma.A.2: Let be a local trivialization of the bundle L_{e}(Q)), such that we can speak of a basical vector field and of a (right) vertical vector field. We can speak of the Nualart-Pardoux constants of a form σ for the basical vector fields and the (right) vertical vector field. They can be estimated in terms of the Nualart-Pardoux constants in terms of horizontal and (right) vertical vector fields. The converse is true.

In the previous discussion, we did not speak of the fact we have removed to u(s, t; u, s̃) the quantity \frac{1}{t−s} ∫_{s}^{t} u(s, t; u, s̃)du as well for the higher derivatives, because some auxiliary terms which arise from the derivative of the parallel transport can appear. We find after this remark a finite sum of integrals over [s, t]^{k} of polynomial expressions in the basical derivatives of u with possible contraction over the diagonals, and some expressions in the basical derivatives of the parallel transport and of the curvature tensor. It is possible to divide this integral by a power k′ of t − s, but we have always k′ + p/2 ≤ k: namely the division by t − s appears by an operation of averaging in order to recognize a vector field over the loop space and not from an integration by parts, which leads to at most p/2 contractions.

We deduce that the discrete approximation of the integral converges in all the L^{p} to I(s, t; s̃), this from the regularity assumption over the kernels of u(s, t; u, s̃). Moreover ∥I(s, t; s̃∥_{L^{p}}^{p} is a sum of iterated integrals of the basical kernels of u(s, t; u, s̃) over [s, t]^{k} with contraction of the basical kernels and half-limits over the diagonals. Therefore ∥I(s, t; s̃)∥_{L^{p}} is bounded by the Nualart-Pardoux Sobolev norms of the vector fields by the lemma A.2..

In order to check the regularity assumption in s, t, s̃, we suppose in order to simplify that s_{1} < s̃_{1,1} < .. < s̃_{n,1} < t_{1} and that s_{2} < s̃_{1,2}.. < s̃_{n,2} < t_{2}. We splitt the integral between s_{1} and t_{1} into smaller integrals over the intervals defined by the contiguous time of the subdivision, as it was already done in the proof of the lemma A.2 of [L_{5}].

We consider the inegral of u(s_{1}, t_{1}; u, s̃_{1}) and of u(s_{2}, t_{2}; u, s̃_{2}) over [s̃_{i,1}, s̃_{i+1,1}] ∩ [s̃_{i,2}, s̃_{i+1,2}]. We substract the necessary counterterm in order to get an anticipative integral over the based loop space, and we get the expectation of an integral power of it as before. The main remark is that we remein in the same connected component of the parameter set outside the diagonals in the integrals which are got. We get an estimate of the L^{p} norm in term of \sqrt{| s̃_{i,1} − s̃_{i,2} |} + \sqrt{| s̃_{i+1,1} − s̃_{i+1,2} |}. If we integrate outside the intersections, the distance between the extremities of the considered intervals is smaller than the sum of the distance between s̃_{i,1} and s̃_{i,2}. We get an estimate in terms of the Nualart-Pardoux constants of the second type (1.17) and \sqrt{| s̃_{i,1} − s̃_{i,2} |} + \sqrt{| s̃_{i+1,1} − s̃_{i+1,2} |}.

In order to finish the proof, let us precise the effect of the operation u(s, t; u; s̃) → u(s, t; u, s̃) − \frac{1}{t−s} ∫_{s}^{t} u(s, t; u, s̃)du over I in order to get a tangent vector over the based loop space. It has only the consequence to substract to the initial anticipative integral the non-anticipative Stratonobitsh integral ∫_{s}^{t} < τ_{u}C, dγ_{u} > for a suitable C.

For the derivative of I, we deduce from the previous discussion, that we can take derivative under the sign ∫, for the vertical and horizontal vector fields. We conclude by using the fact that:

(a.8)  ∇_{X}dγ_{s} = τ_{s}H′_{s}ds

if X_{s} = τ_{s}H_{s}. ♢

Proof of the lemma A.2: the lemma A.2. will be proved if, after choosing a trivialization which does not affect the Nualart-Pardoux conditions as we will see later, we prove the following fact: a functional which belongs to all the Nualart-Pardoux spaces for the vertical derivatives along right vector fields g_{.}K_{.} belongs to all the Nualart-Pardoux spaces for the vertical derivatives along (left) vector fields K_{.}g_{.}, and its Nualart-Pardoux constants can be estimated in term of the Nualart-Pardoux constants for the first type of vector fields. Namely K_{s} depends in the definition of an horizontal vector field only on (τ_{1}^{Q})^{−1} and its derivatives, which satisfy to the Nualart-Pardoux conditions if we consider horizontal vecto fields.

A vector field g_{s}K_{s} corresponds to the vector field g_{s}K′_{s}g_{s}^{−1} over the leading brownian motion. Let us suppose we can get a prolongation over the path group of our functional which checks the (right) Nualart-Pardoux conditions and whose the (right) Nualart-Pardoux norms over the whole path group can be estimated into the Nualart-Pardoux norms over the pinned path group. We get therefore a functional over the leading brownian motion, which belongs to all the Sobolev spaces. The flat derivative of g_{s} checks s included the Nualart-Pardoux conditions: the flat Nualart-Pardoux norms can be estimated in terms of the Nualart-Pardoux norms over the pinned loop group. Moreover a (left) vector field K_{.}g_{.} gives the vector [K_{s}, dB_{s}] + K′_{s}ds, which is a generalized flat vector field. We can repeat therefore the proof of the theorem A.1. of [L_{5}] in order to conclude. Let us precise this statement.

Let us precise the prolongation: we do in order to simplify as we were working over the based loop group. We consider a path in G with the condition that g_{1} is closed from e. We associate to a path g_{s} the loop g_{s}exp[−sLogg_{1}]. The vector field g_{s}K_{s} is transformed into the vector field:

(a.9)  \begin{matrix}g_{s}exp[−sLogg_{1}] & (exp[sLogg_{1}]K_{s}exp[−sLogg_{1}]) \\ +g_{s}exp[−sLog_{1}] & (exp[sLogg_{1}]\frac{∂}{∂g_{1}} exp[−sLogg_{1}]K_{s}] \end{matrix}

Therefore a vector field g_{s}K_{s} is transformed into

(a.10)  g_{s}exp[sLogg_{1}](K_{s}exp[−sLogg_{1}] + \frac{∂}{∂g_{1}} exp[−sLogg_{1}]K_{s})

g_{1} satisfies to the Nualart-Pardoux conditions and the law of g_{s}exp[−sLogC] is absolutely continuous with a density which belongs to the L^{p} with respect to the law of g_{s}. It follows than the functional over the path group:

(a.11)  F_{tot}(g_{.}) = F (g_{.}exp[−.Logg_{1}])ϕ(g_{1})

where ϕ(g_{1}) is a cutoff functional destinated to ensure the existence of Logg_{1} belongs to all the (right) vertical Nualart-Pardoux Sobolev spaces, and its vertical Nualart-Pardoux constants can be estimated by the vertical Nualart-Pardoux constants over the pinned loop group.

Let us now repeat the scheme of the proof of the theorem A.1. of [L_{5}]. Let σ_{n} = σ(B_{u_{1}}, .., B_{u_{n}}), 0 = u_{1} < .. < u_{n} = 1, u_{i+1} − u_{i} = 1/n for a dyadic subdivision of length 2^{k}. Let F_{n} = E[F |σ_{n}]. It is a functional which depends only from a finite number of flat variables. Since dB_{s} = dg_{s}g_{s}^{−1}, F_{n} belongs to all the Sobolev spaces for g_{.} related to the (left) vector fields K_{.}g_{.}. The flat kernel of F_{n} are given by:

(a.12)  \frac{1}{∏(u_{k_{i+1}} − u_{k_{i}})} ∫ ∫_{∏[u_{k_{i}} ,u_{k_{i+1}} ]} E[k(t_{1}, .., t_{r})| σ_{n}]dt_{1}..dt_{n}

k denotes a flat kernel of F. F_{n} = G_{n}(g_{.}). We will show that G_{n} is a Cauchy sequence for the Sobolev spaces relatively to the left vector fields K_{.}g_{.} (We call them the left Sobolev spaces). The kernel associated to G_{n} are Stratonovitch integrals in dB_{s}. We use for this the formula:

(a.13)  ΔB_{u_{i}} = ∫_{u_{i}}^{u_{i+1}} dg_{s}g_{s}^{−1}

The derivative of dg_{s} is given by K′_{s}g_{s}ds + K_{s}dg_{s} and the derivative of g_{s}^{−1} is given by −g_{s}^{−1}K_{s}. The kernel of the derivative of G_{n} are iterated Stratonovitch integrals with frozen time: we integrate expressions in the flat derivatives of F_{n} and algebraic expressions in g_{s}, g_{s}^{−1} and dB_{s} which are non anticipative.

It remains to pass at the limit: we see that the half limits over the diagonals of the flat kernels of F appear when we go to the limit. In order to pass at the limit, there are two procedures as in the proof of the theorem A.1. of [L_{5}]. Π_{n} is the procedure of conditional expectation over the σ-algebra σ_{n} and χ_{n} the procedure of averaging. Let KerF be a flat kernel of F. The associated flat kernel of F_{n} is Π_{n}χ_{n}KerF. We get:

(a.14)  Π_{n}χ_{n}KerF − Π_{n′}χ_{n′}KerF = Π_{n}(χ_{n} − χ_{n′})KerF + (Π_{n} − Π_{n′})χ_{n′}KerF

Π_{n}KerF satisfies to the Nualart-Pardoux conditions with the same constants than KerF. We can apply the Kolmogorov lemma. Let us denote by U any connected component of the complement of the diagonals. We get:

(a.15)  Sup_{U×U} \frac{∥E[k(t_{1}, .., t_{r})|F_{n}] − E[k(t′_{1}, .., t′_{r})|F_{n}]∥_{L^{p}}}{∑ | t_{i} − t′_{i} |^{α}} < ∞

for a certain α < 1/2.

The previous L^{p} norms are smaller than the L^{p} norms which are got, when we don’t take any conditional expectation. When we take the L^{p} norm of the difference of the Stratonovitch integral of Π_{n}χ_{n}KerF and of Π_{n′}χ_{n′}KerF, we get iterated integrals without the stochastic term dB_{s} with some half limits of the kernels of F_{n} − F_{n′} over the diagonals. We splitt it into an expression polynomial in (Π_{n} − Π_{n′})χ_{n}KerF and a polynomial expression in (χ_{n} − χ_{n′})Π_{n′}KerF. The first type of expression goes uniformly to 0 in all the L^{p} by using the Kolmogorov lemma when n → ∞ and n′ → ∞. It is the same for the second type of expressions, by using the criterium of continuity of the kernels of F. G_{n} is a Cauchy sequence in the (left) Sobolev spaces.

Moreover, by the Kolmogorov lemma:

(a.16)  E[k(s_{1}, .., s_{r})|σ_{n}] − k(s_{1}, .., s_{r}) → 0

uniformly in (s_{1}, .., s_{r}) outside the diagonals in all the L^{p}. The derivatives of G_{n} tend to the Stratonovitch integral which are got formally when we replace the flat dH_{s} by [K_{s}, dB_{s}] + K′_{s}ds.

It remains to restricts the functional G over the based loop group as well as its kernels. It is the purpose of the quasi-sure analysis: we consider the measure

(a.17)  f → E[Gf(g_{1})]

which has a density. The (left) Nualart-Pardoux Sobolev norms for the pinned loop group are estimated in terms of the (left) Nualart-Pardoux Sobolev norms overthe path group. In order to show that , let us consider the vector fields X_{.}^{1} = K_{.}^{1}g_{.},.., X_{.}^{r} = K_{.}^{r}g_{.}. We get the following integration by parts formula for any integer p:

(a.18)  E[|G(s_{1}, .., s_{r}) − G̃(s̃_{1}, .., s̃_{r})|^{p} < d < .. < df(g_{1}), X^{r} > .. > X^{1} >] = E[ξf(g_{1})]

where ξ is a polynomial expression in G(s_{1}, .., s_{r}) − G̃(s_{1}, .., s_{r}) and its (left) kernels integrated and the divergence of the vector fields X and therir derivatives. We apply the lemme A.2 of [L_{5}] in order to conclude.

We have proved the lemma A.2. for functionals: for forms, we associate to a form over (right) vertical vector fields a form over flat vector fields, and after we do the transformation dH_{s} → [K, s, dB_{s}] + K′_{s}ds in order to get a form over the (left) vertical vector filds: we get Stratonovitch iterated integrals, and we apply the lemma A.2. of [L_{5}] in order to conclude.

The last point it remains to clarify is that the operation of trivialization in order to come to a product situation has no effect over the right vertical Nualart-Pardoux Sobolev norms. If (τ_{1}^{Q})^{−1} ∈ G_{i}, we can find a mollifer f(g_{1}) which belongs to all the Nualart-Pardoux spaces with compact support in a small neighborhood of G_{i}. We put over the path group

(a.19)  F_{tot}(g_{.}) = F (g_{.}exp[−sLog[τ_{1}^{Q}g_{1}])f(g_{1})

We enlarge by this the functional over the vertical pinned path going from e to (τ_{1}^{Q})^{−1} to a functional over the total space of the path group which checks still the Nualart-Pardoux conditions: its right Nualart-Pardoux norms can be estimated in term of the Nualart-Pardoux norms of the non extended functional, since (τ_{1}^{Q}) satisfies to the Nualart-Pardoux conditions. We perform after the transformation:

(a.20)  F_{tot}(g_{.}) → F_{tot}(g_{i,.}((τ_{1}^{Q})^{−1})g_{.})

The left Nualart-Pardoux constants of the new global functional can be estimated in term of the right Nualart-Pardoux constants of F_{tot}(g_{.}). Since the law og g_{i}.((τ_{1}^{Q})^{−1})g_{.} is equivalent to the law of g_{.} with a density which belongs to all the L^{p}, we deduce that the global Nualart-Pardoux constants of F_{tot}(g_{i.}((τ_{1}^{Q})^{−1}g_{.}) can be estimated in terms of the right Nualart-Pardoux constants of F (g_{.}). ♢

We had shown too the following lemma:

Lemma A.3.: A functional over a trivialization which checks the (right) Nualart-Pardoux conditions for (right) vertical vector fields over L_{e}(G) (orP (G)) checks still the (left) Nualart-Pardoux conditions for (left) vertical vector fields.

We get the proposition:

Proposition A.4.: Let u(s, t; us̃) a random application with values in LieG which belongs to the (right) Nualart-Pardoux spaces over the total space, s, t, s̃ included. Let I(s, t, s̃) the anticipative Stratonovitch integral:

(a.21)  I(s, t; s̃) = ∫_{s}^{t} < g_{u}u(s, t; u, s̃), dg_{u} >

I(s, t; s̃) checks the Nualart-Pardoux conditions over the total space L_{e}(Q).

Proof: we begin to write dg_{u} = dB_{u}g_{u}, such that we come back to a Stratonovitch integral

(a.22)  ∫_{s}^{t} < ũ(s, t; u, s̃), dB_{u} >

where ũ checks still the Nualart-Pardoux conditions. We extend ũ(s, t; u, s̃) over the path group, such that it checks still the Nualart-Pardoux conditions over the path group. We use the isometry given in the beginning of the proof of the lemma A.2.. We get ũ which depends on B and γ which checks the Nualart-Pardoux conditions in B and γ. We come back to the flat case and to a flat Stratonovitch integral. We can use the results of [L_{5}]: I(s, t; s̃) extended satisfies to the Nualart-Pardoux conditions in B and γ, for the basical vector field in γ? this from the lemma A.2. of [L_{5}]. Then I(s, t; s̃) extended satisfies to the Nualart-Pardoux conditions in g_{.} and γ_{.}, for basical vector fields in γ. By the lemma A.2., it satisfies to the Nualart-Pardoux conditions in g_{.} and γ_{.} for horizontal vector fields, which are intrisically defined. ♢

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département de Mathématiques Institut Elie Cartan Faculté des Sciences Université de Nancy I 54000. Vandoeuvre-les-Nancy FRANCE department of mathematics Max Planck Institut fur Mathematics D5300. Bonn GERMANY