$L^2$-Betti Numbers of Infinite Configuration Spaces

Abstract

The space Γ X of all locally finite configurations in a infinite covering X of a compact Riemannian manifold is considered. The de Rham complex of square-integrable differential forms over Γ X , equipped with the Poisson measure, and the corresponding de Rham cohomology and the spaces of harmonic forms are studied. A natural von Neumann algebra containing the projection onto the space of harmonic forms is constructed. Explicit formulae for the corresponding trace are obtained. A regularized index of the Dirac operator associated with the de Rham differential on the configuration space of an infinite covering is considered.

L2 dimensions of spaces of braid-invariant harmonic forms

“…and STR P = e −χ(K) . The latter expression coincides with the formula derived in [1] for the case the de Rham complex over the configuration space equipped with the Poisson measure. Remark 6.1.…”

“…The right-hand side of formula (45) can be understood as a regularized index of the Dirac operator D + D * , where D := m N d m N , see [11] and e.g. [30] for the discussion of von Neumann supertraces in geometry and topology of Riemannian manifolds and their relation to L 2 -index theorems, and [1], [2], [3] for the extension of these notions to the framework of infinite configuration spaces.…”

“…In many interesting cases, X is a non-compact Riemannian manifold of infinite volume, which makes it natural to discuss the above questions in the framework of the theory of L 2 invariants. This programme has been started in [15], [13], [14], [1].…”

Traces of semigroups associated with interacting particle systems

“…Laplace operators with potentials q given by homogeneous random fields (see [10,21,29]). That is, q is supposed to be G-invariant in the sense that q(gx, ω) = q(x, U g ω), (2) where G g → U g is a representation of G by measure preserving transformations of the probability space. For such operators, the -function is defined by the formula…”

“…Let us remark that the interest in the analysis on infinite configuration spaces has risen in recent years, because of new approaches and rich applications in statistical mechanics and quantum field theory (see [3,4] and the review [30]). L 2 -Betti numbers of configuration spaces with Poisson and Lebesgue-Poisson measures, were computed in [2,15], respectively (see also [1,14,16]). …”