Connes-Chern Character for Manifolds with Boundary and ETA ...

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6 MATTHIAS LESCH, HENRI MOSCOVICI, AND MARKUS J. PFLAUM Comparing this expression with the local form of the pairing (0.13) leads to a generalization of the A-P-S odd-index formula [APS76, Prop. 6.2, Eq. (6.3)], from trivialized flat bundles to pairs of equivalent vector bundles in K-theory. Precisely (cf. Corollary 7.9), if E ′ , F ′ are two such bundles on a closed odd-dimensional spin manifold N, and h is the homotopy implementing the equivalence of E ′ with F ′ , then ∫ ξ(D F ′ g ) − ′ ξ(DE′ g ) = Â(∇ 2 ′ g ′) ∧ T/ch • (h) + SF(h, D g ′) , (0.19) N where D g ′ denotes the Dirac operator associated to a Riemannian metric g ′ on N; equivalently, ∫ 1 0 1 d ( η(ph(t) D g ′ p h(t) ) ) ∫ dt = 2 dt N Â(∇ 2 g ′) ∧ T/ch •(h) , (0.20) where p h(t) is the path of projections joining E ′ and F ′ , and the left hand side is the natural extension of the real-valued index in [APS76, Eq. (6.1)]. Let us briefly comment on the main analytical challenges encountered in the course of proving the results outlined above. In order to compute the limit as t ↘ 0 of the Chern character, one needs to understand the asymptotic behavior of expressions of the form b 〈A 0 , A 1 , . . . , A k 〉 √ tD ∫ := b Tr ( ) A 0 e −σ 0tD 2 A 1 e −σ 1tD 2 . . . A k e −σ (0.21) ktD 2 dσ, ∆ k where ∆ k denotes the standard simplex {σ 0 + . . . + σ k = 1, σ j ≥ 0} and A 0 , . . . , A k are b–differential operators. The difficulty here is twofold. Firstly, the b–trace is a regularized extension of the trace to b–pseudodifferential operators on the non-compact manifold M \∂M (recall that the b-metric degenerates at ∂M). Secondly, the expression inside the b-trace involves a product of operators. The Schwartz kernel of the product A 0 e −σ 0tD 2 A 1 e −σ 1tD 2 · . . . · A k e −σ ktD 2 does admit a pointwise asymptotic expansion (see [Wid79], [CoMo90], [BlFo90]), namely ( A 0 e −σ 0tD 2 A 1 e −σ 1tD 2 · . . . · A k e −σ ktD 2) (p, p) n∑ j−dim M−a (0.22) =: a j (A 0 , . . . , A k , D)(p) t 2 + O p (t (n+1−a−dim M)/2 ), where a = k ∑ j=0 j=0 ord A j . However, this asymptotic expansion is only locally uniform in p; it is not globally uniform on the non-compact manifold M \ ∂M. A further complication arises from the fact that the function a j (A 0 , . . . , A k , D) is not necessarily integrable. Nevertheless, a partie finie-type regularized integral, which we denote by
∫ b M CONNES-CHERN CHARACTER AND ETA COCHAINS 7 a j (A 0 , . . . , A k , D)d vol, does exist and we prove (cf. Theorem 5.3) that the corresponding b–trace admits an asymptotic expansion of the form b Tr (A 0 e −σ 0tD 2 A 1 e −σ 1tD 2 . . . A k e −σ ktD 2) = n∑ ∫ j=0 b M j−dim M−a a j (A 0 , . . . , A k , D)d vol t 2 + ( ( k∏ + O σ −a ) j/2 j t (n+1−a−dim M)/2) . j=1 (0.23) When D ∂ is invertible and hence D is a Fredholm operator, we can also prove the following estimate (cf. (3.43)) | b 〈A 0 (I − H), ..., A k (I − H)〉 √ tD | ≤ ˜C δ,ε t −a/2−(dim M)/2−ε e −tδ , for all 0 < t < ∞, (0.24) for any ε > 0 and any 0 < δ < inf spec ess D 2 . Here, H is the orthogonal projection onto Ker D. This estimate allows us to compute the limit as t ↗ ∞ and thus derive the formulæ (0.8) and (0.10). A few words about the organization of the paper are now in order. We start by recalling, in Section 1, some basic material on relative cyclic cohomology [LMP09], b- calculus [Mel93] and Dirac operators. As an illustration of the usefulness of the relative cohomology point of view for manifolds with boundary, in Subsection 1.5 we establish a cohomological analogue of the well-known McKean-Singer formula in the framework of relative cyclic cohomology for the pseudodifferential b-calculus, and employ it to recast in these terms Melrose’s approach to the proof of the Atiyah-Patodi-Singer index theorem (cf. [Mel93, Introduction]). In Subsection 1.6, refining an observation due to Loya [Loy05], we give an effective formula for the b-trace, which will turn out to be quite a useful technical device. Gerzler’s version of the relative entire Connes-Chern character in the setting of b- calculus [Get93a] is discussed in Section 2. Additional material on b-geometry can be found in Appendix A. In Section 3 we prove some crucial estimates for the heat kernel of a b-Dirac operator, which are then applied in Section 4 to analyze the short and long time behavior of the components of the b-analogue of the entire Chern character. More standard resolvent and heat kernel estimates are relegated to Appendix B. Section 5 is devoted to asymptotic expansions for the b-analogues of the Jaffe- Lesniewski-Osterwalder components. The retracted relative cocycle representing the Connes-Chern character in relative cyclic cohomology is constructed in Section 6, where we also compute the expressions of its small and large scale limits. Finally, Section 7 derives the ensuing pairing formulæ with the K-theory, establishes the connection with the Atiyah-Patodi-Singer index theorem, and discusses the geometric consequences. The paper concludes with an explanatory note (Subsection 7.1) elucidating the relationship between the results presented here and the prior work in this
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lim t→0+ CONNES-CHERN CHARACTER A
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CONNES-CHERN CHARACTER AND ETA COCH
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