Heat Kernel Measure on Central Extension of Current Groups in any Dimension

We define measures on central extension of current groups in any dimension by using infinite dimensional Brownian motion.


Introduction
If we consider a smooth loop group, the basical central extension associated to a suitable Kac-Moody cocycle plays a big role in mathematical physics [3,11,21,24]. Léandre has defined the space of L 2 functionals on a continuous Kac-Moody group, by using the Brownian bridge measure on the basis [16] and deduced the so-called energy representation of the smooth Kac-Moody group on it. This extends the very well known representation of a loop group of Albeverio-Hoegh-Krohn [2].
Etingof-Frenkel [13] and Frenkel-Khesin [14] extend these considerations to the case where the parameter space is two dimensional. They consider a compact Riemannian surface Σ and consider the set of smooth maps from Σ into a compact simply connected Lie group G. We call C r (Σ; G) the space of C r maps from Σ into G and C ∞ (Σ; G) the space of smooth maps from Σ into G. They consider the universal coverC ∞ (Σ; G) of it and construct a central extension by the Jacobian J of Σ of itĈ ∞ (Σ; G) (see [7,8,25] for related works).
We can repeat this construction if r > s big enough for C r (Σ; G). We get the universal cover C r (Σ; G) and the central extension by the Jacobian J of Σ ofC r (Σ; G) denoted byĈ r (Σ; G).
By using Airault-Malliavin construction of the Brownian motion on a loop group [1, 9], we have defined in [19] a probability measure onC r (Σ; J), and since the Jacobian is compact, we can define in [19] a probability measure onĈ r (Σ; G).
Maier-Neeb [20] have defined the universal central extension of a current group C ∞ (M ; G) where M is any compact manifold. The extension is done by a quotient of a certain space of differential form on M by a lattice. We remark that the Maier-Neeb procedure can be used if we replace this infinite dimensional space of forms by the de Rham cohomology groups H(M : Lie G) of M with values in Lie G. Doing this, we get a central extension by a finite dimensional Abelian groups instead of an infinite dimensional Abelian group. On the current group C r (M ; G) of C r maps from M into the considered compact connected Lie group G, we use heat-kernel measure deduced from the Airault-Malliavin equation, and since we get a central extensionĈ r (M ; G) by a finite dimensional group Z, we get a measure on the central extension of the current group. Let us recall that studies of the Brownian motion on infinite dimensional manifold have a long history (see works of Kuo [15], Belopolskaya-Daletskii [6,12], Baxendale [4,5], etc.).
Let us remark that this procedure of getting a random field by adding extra-time is very classical in theoretical physics, in the so called programme of stochastic-quantization of Parisi-Wu [23], which uses an infinite-dimensional Langevin equation. Instead to use here the Langevin equation, we use the more tractable Airault-Malliavin equation, that represents infinite dimensional Brownian motion on a current group.

A measure on the current group in any dimension
We consider C r (M ; G) endowed with its C r topology. The parameter space M is supposed compact and the Lie group G is supposed compact, simple and simply connected. We consider the set of continuous paths from [0, 1] into C r (M ; G) t → g t (·), where S ∈ M → g t (S) belongs to C r (M ; G) and g 0 (S) = e. We denote P (C r (M ; G)) this path space.
Let us consider the Hilbert space H of maps h from M into Lie G defined as follows: where ∆ is the Laplace Beltrami operator on M and dS the Riemannian element on M endowed with a Riemannian structure. We consider the Brownian motion B t (·) with values in H.
We consider the Airault-Malliavin equation (in Stratonovitch sense): Let us recall (see [17]): Theorem 1. If k is enough big, t → {S → g t (S)} defines a random element of P (C r (M ; G)).
We denote by µ the heat-kernel measure C r (M ; G): it is the law of the C r random field S → g 1 (S). It is in fact a probability law on the connected component of the identity C r (M ; G) e in the current group.

A brief review of Maier-Neeb theory
Let us consider Π 2 (C r (M ; G) e ) the second fundamental group of the identity in the current group for r > 1. The Lie algebra of this current group is C r (M ; Lie G) the space of C r maps from M into the Lie algebra Lie G of G [22]. We introduce the canonical Killing form k on Lie G. to (η, η 1 ), elements of the Lie algebra of the current group.
For that, let us recall that the Lie algebra of the current group is the set of C r maps η from the manifold into the Lie algebra of G. dη is a C r−1 1-form into the Lie algebra of G. Therefore k(η, dη 1 ) appears as a C r−1 1-form with values in the Lie algebra of G. Moreover dk(η, η 1 ) = k(dη, η 1 ) + k(η, dη 1 ).
This explains the introduction of the quotient in Y . Following the terminology of [20], we consider the period map P 1 which to σ belonging to Π 2 (C r (M ; G) e ) associates σ Ω. Apparently P 1 takes its values in Y , but in fact, the period map takes its values in a lattice L of H 1 (M ; Lie G).
It is defined on Π 2 (C r (M ; G) e ) since Ω is closed for the de Rham differential on the current group, as it is left-invariant and closed and it is a 2-cocycle in the Lie algebra of the current group [20]. We consider the Abelian group Z = H 1 (M ; Lie G)/L. Z is of finite dimension.
We would like to apply Theorem III.5 of [20]. We remark that the map P 2 considered as taking its values in Y /L is still equal to 0 when it is considered by taking its values in H 1 (M ; Lie G)/L.
We deduce the following theorem: We get a central extensionĈ r (M ; G) by Z of the current group C r (M ; G) e if r > 1.
Since Z is of finite dimension, we can consider the Haar measure on Z. We deduce from µ a measureμ onĈ r (M ; G).
Remark 1. Instead of considering C r (M ; Lie G), we can consider W θ,p (M ; Lie G), some convenient Sobolev-Slobodetsky spaces of maps from M into Lie G. We can deduce a central exten-sionĈ θ,p (M ; G) of the Sobolev-Slobodetsky current group C θ,p (M ; G) e . This will give us an example of Brzezniak-Elworthy theory, which works for the construction of diffusion processes on infinite-dimensional manifolds modelled on M-2 Banach spaces, since Sobolev-Slobodesty spaces are M-2 Banach spaces [9,10,18]. We consider a Brownian motion B 1 t with values in the finite dimensional Lie algebra of Z andB t = (B t (·), B 1 t ) where B t (·) is the Brownian motion in H considered in the Section 2. Then, following the ideas of Brzezniak-Elworthy, we can consider the stochastic differential equation onĈ θ,p (M ; G) (in Stratonovitch sense): dĝ t (·) =ĝ t (·)dB t .