Abstract
The goal of this note is to outline a proof that, for any l $\geq 0$, the JLO bivariant cocycle associated with a family of Dirac type operators along a smooth fibration $M\to B$ over the pair of algebras $(C^\infty (M), C^\infty(B))$, is entire when we endow $C^\infty(M)$ with the $C^{l+1}$ topology and $C^\infty(B)$ with the $C^{l}$ topology. As a corollary, we deduce that this cocycle is analytic when we consider the Frechet smooth topologies on $C^\infty(M)$ and $C^\infty(B)$.
Highlights
In this note we define a bivariant JLO cocycle in terms of which we can reformulate the local families index theorem [6, 5]
D can be regarded as a family of elliptic operators along the fibers parametrized by the elements of the base manifold B, i.e. D = (Db)b∈B where Db : C∞(Mb, E|Mb) → C∞(Mb, E|Mb) is an essentially self adjoint operator [5]
Even for a trivial fibration and under the assumption that ∇2 = 0, the bivariant JLO cocycle is not analytic in general with respect to the bornology associated with the Frechet C∞-topology on Ω∗(B)
Summary
In this note we define a bivariant JLO cocycle in terms of which we can reformulate the local families index theorem [6, 5]. The space C∞(M, E ⊗ Λπ∗T ∗B) is clearly a module over the algebra Ω∗B of differential forms on the base manifold B. The local coefficients of such operators are smooth in the base variables and the space Ψh(M |B; E) is a module over the algebra C∞(B) of smooth functions on B. We set ψh(M |B, E; Λ∗B) = Ψh(M |B, E)⊗Ω∗B, for the space of order h classical fiberwise pseudodifferential operators with coefficients in differential forms on the base B. The curvature operator ∇2 of ∇ is a fiberwise first order differential operator with coefficients in Ω2B, so ∇2 ∈ ψ1(M |B, E; Λ2B) This is a classical lemma that we proved in [2]. It is well known that e−tD2 belongs to ψ−∞(M |B, E), e−uB2σ belongs to ψ−∞(M |B, E; Λ∗B)
Sign-in/Register to view highlights & summary Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callMore From: Electronic Research Announcements in Mathematical Sciences
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
