Stochastic Cohomology of the Frame Bundle of the Loop Space

We study the differential forms over the frame bundle of the based loop space. They are stochastics in the sense that we put over this frame bundle a probability measure. In order to understand the curvatures phenomena which appear when we look at the Lie bracket of two horizontal vector fields, we impose some regularity assumptions over the kernels of the differential forms. This allows us to define an exterior stochastic differential derivative over these forms. Introduction Let Lx(M) be the based loop space of smooth applications γs from the circle into M such that γ0 = γ1 = x. Let Q → M be a principal bundle over M with structure group G. Le(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 Le(G), the based loop group of G. If Q → M is the frame bundle, Le(Q) is the frame bundle of Lx(M): the structure of Le(Q) is of the main importance to study string structures (or spin structures) over the loop space ([8], [9], [38]), and has a deep place in the understanding the Dirac operator over the loop space ([38]). 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 S1 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 Lx(M) Copyright c ©1998 by R. Leandre 24 R. Leandre is defined in [27], [29] and [30] with values in C or R: since Lx(M) is supposed simply connected, we can neglect all torsion phenomenon in order to construct a S1 bundle from a Z closed 2 form over the loop space of finite energy loops. But we have to choose distinguished paths in Lx(M) in order to shrink a loop in a constant loop: let lt(γ)s such a distinguished path. The law of lt(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 Lx(M), to construct the stochastic forms which allow to consider a string structure over Lx(M). Namely, we have already constructed stochastic bundles over Le(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 [34]. As in [34], we define a measure over Le(Q), by putching together measures in the fiber: the fiber is a continuous loop group. We start with the equation in the fiber dgs = dBsgs. (0.1) In [34], we have studied the equation dgs = Bsdsgs. (0.2) We choose this equation in order to reflect the fiber structure of Le(Q), the only obstacle to the trivialization being 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 in the first case, while it belongs only to L1 in the second case, if K. is C 2. This allows us to define a tangent space of Le(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 Le(Q). Let us recall namely that, in order to define some cohomology groups over the loop spaces, we have imposed in [27] 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 [15]. 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 Le(Q). The Carey-Murray [38] 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 Le(Q) → Lx(M). But we show that the functional spaces which are got with some regularity assumptions over the kernels are independant of this connection. Stochastic Cohomology 25 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 dP1,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: dgs = dBsgs; g0 = e, (1.1) where Bs 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 paths in the group joining e to g ([17], [2], [3]). We get a space of continuous paths in G Lg(G) endowed with a law Qg. The non pinned based path group is denoted P (G). We put over the bundle Q → M a connection ∇: τ s the parrallel transport for a loop γs is therefore almost surley defined for the connection ∇ . We denote by Le(Q) the space of loop q. in Q such that qs = τ Q s gs, g1 = (τ Q 1 ) −1. We get the following commutative diagramm [38]: Le(Q) → P (G) ↓ ↓ Lx(M) → G . (1.2) The map from P(G) to G is the map which to g. associates g1. The map from Le(Q) to Lx(M) is the projection map. The map f from Lx(M) to G is the map which to a stochastic loop γ. associates (τ Q 1 ) −1. The map from Le(Q) to P (G) is the map which to q. associates g.. It is nothing else than f . Over Le(Q), we put the measure: dPtot = dP1,x ⊗ dQ(τ1)−1 . (1.3) Let us analyze a little bit more the Le(G) bundle P (G) → G. If g1 ∈ Gi is a small open neighborhood of G, we can choose a section gi,s(g1) of this bundle which is jointly smooth in s and in g1. It checks the following property: gi,0(g1) = e; gi,1(g1) = g1; gi,s(g1) ∈ 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 g1 ∈ Gi, the connection one form is a smooth path in the Lie algebra of G starting from 0 and arriving at 0 Ki,s(g1), which depends smoothly from g1 ∈ Gi and which is a one form in g1. The obstruction to trivialize Le(Q) over Lx(M) lies in (τ Q 1 ) −1: if (τ 1 ) −1 ∈ Gi, there is a local slice of Le(Q) which is gi,.((τ Q 1 ) −1). We look at the left transformation g. → (gi,.((τ Q 1 ) ))g.. Modulo this transformation, the bridge between e and (τ Q 1 ) −1 is transformed into the bridge between e and e. Let us recall namely the purpose of the quasiinvariance formula from Albeverio-Hoegh-Krohn [4]: if ks 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 and can be desintegrated along the appropriate bridge. We denote by Jr(k) and by Jl(k) the right quasi-invariance density and the left quasi-invariance density [4], [17]. 26 R. Leandre Therefore if (τ 1 ) −1 ∈ Gi: dPtot = dP1,x ⊗ Jl(gi,((τ Q 1 ) ))dQe. (1.4) Jl(gi,.((τ Q 1 ) −1) belongs to all the L and is bounded in L when (τ 1 ) −1 describes Gi. (1.4) produces a stochastic trivialization of our bundle. Let us recall that a vector field over Lx(M) is given by [7], [20] Xt = τtHt X0 = X1 = 0, (1.5) where τt is the parallel transport associated to the Levi-Civita connection and H. is a finite energy path in Tx. We choose as Hilbertian norm of X. the norm ‖X‖ = 1