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

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58 MATTHIAS LESCH, HENRI MOSCOVICI, AND MARKUS J. PFLAUM Given a de Rham current C of degree j then C defines naturally a cochain ˜C ∈ C j ( b C ∞ (M)) by putting ˜C(a 0 , . . . , a j ) := 1 j! 〈C, a 0da 1 ∧ · · · ∧ da j 〉. One has b ˜C = 0 and B ˜C = ˜∂C, where ∂ is the codifferential. Because of this identification we will from now on omit the ∼ from the notation if no confusion is possible. Given a closed b-differential form ω on M of even degree. By C ω we denote the de Rham current ∫b ω ∧ −. There is a natural pullback M ι∗ ω at ∞ (cf. Definition and Propostion 1.5), which is a closed even degree form on ∂M. We now find ∫ ∫ ∫ 〈∂C ω , α〉 = ω ∧ dα = d(ω ∧ α) = ι ∗ (ω ∧ α) = b M b M ∂M ∫ (6.21) = ι ∗ (ω) ∧ ι ∗ (α) = 〈C ι ∗ ω, ι ∗ (α)〉. ∂M In view of Subsection 1.2 this means that the pair (C ω , C ι ∗ ω) is a relative de Rham cycle or via the above mentioned identification between de Rham currents and cochains that (˜C ω , ˜C ι ∗ ω) is a relative cocycle in the relative total complex Eq. (6.14). If ω = ∑ j≥0 ω 2j with closed b-differential forms of degree 2j then the pair (ω, ι ∗ ω) still gives rise to a relative cocycle of degree dim M in the relative total complex. These considerations certainly apply to the even degree forms b ω D , ω D∂ which satisfy ι ∗ ( b ω) = ω D∂ . The limit results (6.18), (6.19), (6.5) can then be summarized as lim t→0+ ( b ˜ch k t (D), ch k−1 t (D ∂ ) ) = ( ∫b M ∫ b ω D ∧ −, ∂M ) ω D∂ ∧ − . (6.22) The limit on left is understood pointwise for each component of pure degree. Finally we need to relate ( k b ˜ch t (D), ch k−1 t (D ∂ ) ) to the Chern character of [D] ∈ K m (M, ∂M). First recall from Eq. (1.11) that HP •( C ∞ (M), C ∞ (∂M) ) is naturally isomorphic to HP •( J ∞ (∂M, M) ) . Under this isomorphism, the class of the pair ( k b ˜ch t (D), ch k−1 t (D ∂ ) ) is mapped to b k ˜ch t (D) |J ∞ (∂M,M), just because elements of J ∞ (∂M, M) vanish on ∂M. We note in passing that by (A.5) a smooth function f on M ◦ lies in J ∞ (∂M, M) iff in cylindrical coordinates one has for all l, R and every differential operator P on ∂M ∂ l xDf(x, p) = O(e Rx ), x ↦→ −∞. In view of (6.17), the class of ( k b ˜ch t (D), ch k−1 t (D ∂ ) ) in HP •( J ∞ (∂M, M) ) equals that of ˜Cb ω D . As explained in Section 1.8, HP •( J ∞ (∂M, M) ) is naturally isomorphic to H• dR (M, ∂M; C). Under this isomorphism, ˜Cb ω D corresponds to the relative de Rham cycle ( ) Cb ω D , C ωD∂ . Finally, note that under the Poincaré duality isomorphism H• dR (M, ∂M; C) ∼ = H dR • (M \ ∂M; C), the relative de Rham cycle ( ) Cb ω D , C ωD∂ is mapped onto the closed form b ω D . This line of argument shows that the class of ( k b ˜ch t (D), ch k−1 t (D ∂ ) ) depends only on the absolute de Rham cohomology class of the closed form b ω D = c·Â(b ∇g) 2 on the open manifold M \∂M. The transgression formula in Chern–Weil theory shows that the (absolute) de Rham cohomology class of Â ( ) ∇ 2 g is independent of the metric g on M ◦ . Thus the de Rham class of b ω D equals that of
CONNES-CHERN CHARACTER AND ETA COCHAINS 59 ω D = c·Â( ) ∇ 2 g 0 for any smooth metric g0 . Choosing g 0 to be smooth up to the boundary we infer from Section 1.8 and Proposition 1.12 that the class of ( k b ˜ch t (D), ch k−1 t (D ∂ ) ) in HP •( C ∞ (M), C ∞ (∂M) ) ∼ ( = HP • J ∞ (∂M, M) ) equals that of the Connes–Chern character of [D]. □ 6.2. The large time limit and higher η–invariants. Let us now assume that the boundary Dirac D ∂ is invertible. In view of Proposition 4.4 and Proposition 4.1 we can now, for k ≥ dim M, form the transgressed cochain b T/ch k ∞ (D)(a 0, ..., a k ) = In view of Eq. (2.52) we arrive at ∫ ∞ 0 b /ch k (sD, D) ds. (6.23) B b T/ch k+1 ∞ (D)(a 0, ..., a k ) k∑ ∫ ∞ = (−1) j+1 s k+1 b 〈[D, a 0 ], . . . , [D, a j ], D, [D, a j+1 ], . . . , [D, a k ]〉 ds j=0 0 k∑ ∫ (6.24) ∞ ( = (−1) j+1 s k+1 j=0 0 ∫∆ b Str q [D, a0 ]e −σ 0s 2 D 2 . . . k+1 ) [D, a j ]e −σ js 2 D 2 De −σ j+1s 2 D 2 . . . [D, a k ]e −σ k+1s 2 D 2 dσ ds. Together with Proposition 4.5 we have proved the analogue of [CoMo93, Thm. 1] in the relative setting: Theorem 6.2. Let k ≥ dim M be of the same parity as q and assume that D ∂ is invertible. Then the pair of retracted cochains ( b ch k t (D), ch k+1 t (D ∂ ) ) , t > 0, has a limit as t → ∞. For k = 2l even we have l∑ b ch k ∞(D) = κ 2j (D) + B b T/ch k+1 ∞ (D) , (6.25) j=0 ch k+1 ∞ (D ∂ ) = B T/ch k+2 ∞ (D ∂) . If k = 2l + 1 is odd then b ch k ∞(D) = B b T/ch k+1 ∞ (D) , ch k+1 ∞ (D ∂ ) = B T/ch k+2 ∞ (D ∂) . 7. Relative pairing formulæ and geometric consequences (6.26) Let us briefly recall some facts from the theory of boundary value problems for Dirac operators [BBWo93]. Given a Dirac operator D acting on sections of the bundle W on a compact Riemannian manifold with boundary (M, g). In contrast to the rest of the paper g is a ”true” Riemannian metric, smooth and non-degenerate up to the boundary, and not a b-metric. We assume that all structures are product near the boundary, that is there is a collar U = [0, ε) × ∂M of the boundary such that g |U = dx 2 ⊕ g |∂M is a product metric. In particular the formulæ of Remark 1.10 hold.
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CONNES-CHERN CHARACTER FOR MANIFOLD
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CONNES-CHERN CHARACTER AND ETA COCH
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CONNES-CHERN CHARACTER AND ETA COCH
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Page 53 and 54: Applying (5.9) to A 1 we get e −
Page 57: lim t→0+ CONNES-CHERN CHARACTER A
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