ON THE INDEX FORMULA OF MELROSE AND NISTOR

ROBERT LAUTER AND SERGIU MOROIANU

Abstract. We improve two results of a paper by Melrose and Nistor [2]. We propose a simpler expression for the index of fully elliptic cusp pseudo-differential operators. We show that the local term in the index formula is given by a convergent indefinite integral.

The main goal of this note is to improve Proposition 16 from the paper [2] by Melrose and Nistor. Namely, let A, B be cusp operators. We claim that we obtain the index functional IF (A, B) defined in [2, (82)] as a sum of only two terms instead of the eight in [2, (90)], provided that A⊗B is a Hochschild 1-cycle in A. Moreover, we prove that the Atiyah-Singer-type term in the index formula from the Index Theorem of [2] is given by a convergent integral. This term is defined a priori as the regularization of a diverging integral. Our second result improves the Reduced Index Theorem from [2], where the integral is shown to be convergent under some extra assumptions.

To fix notations, recall the following definition of the “trace” functionals, equivalent to [2, (44)]:

zτT r(Ax^{z}Q^{−τ}) = T r_{∂,σ}(A) + τT̂ r_{∂}(A) + zT̂ r_{σ}(A) + O(z^{2}, zτ, τ^{2}). (1)

Proposition 1. Let A, B be cusp operators. Assume that A⊗B defines a Hochschild cycle in A (i.e. [A, B] belongs to the residual ideal I). Then

IF (A, B) = T̂r_{∂}(A[B, log x]) + T̂r_{σ}([log Q, B]A).

Proof: The cycle condition means that T r([A, B]) is well-defined, hence the function T r([A, B]x^{z}Q^{−τ}) is regular at z = 0, τ = 0. So

IF (A, B) = T r([A, B]x^{z}Q^{−τ})_{z=0,τ=0}
= T r [(A(B − x^{z}Bx^{−z})x^{z}Q^{−τ}) + (Q^{τ}BQ^{−τ} − B)Ax^{z}Q^{−τ})]_{z=0,τ=0}
= T r [zA([B, log x] + zH_{1}(z))x^{z}Q^{−τ}]_{z=0,τ=0}
+T r [τ([log Q, B] + τH_{2}(τ))Ax^{z}Q^{−τ}]_{z=0,τ=0} ,

where H_{1}(z), H_{2}(τ) are entire families of operators of fixed order.

Recall [2, Lemma 4] that for any holomorphic family C(z, τ) of cusp operators of fixed integral order, the function T r(Cx^{z}Q^{−τ}) has at most simple poles in each complex variable at real integers. It follows that the terms containing H_{1}(z), H_{2}(τ) do not contribute any constant term. We are left with the regularized values at z = 0, τ = 0 of T r(zA[B, log x]x^{z}Q^{−τ}) and T r(τ[log Q, B]Ax^{z}Q^{−τ}), which by (1) are just T̂r_{∂}(A([B, log x]), resp. T̂r_{σ}([log Q, B]A).

Let now A be a fully elliptic (or equivalently Fredholm) cusp operator. Then A has a parametrix B which inverts it up to remainders in the residual ideal I. Note that A ⊗ B is a Hochschild cycle in A. Let us make the following definitions: T̂r_{∂}(A([B, log x]) := −\frac{1}{2}η(A) and T̂r_{σ}([log Q, B]A) := \overline{AS}(A) (the second definition differs from ([2], (87)) by at least a minus sign). Of course, both the definitions of η(A) and \overline{AS}(A) are independent of the choice of the parametrix B. In fact, it suffices that B inverts A up to trace class remainders.

Theorem 2 (Variant of [2, Index Theorem]). Let A be a fully elliptic cusp operator. Then

Ind(A) = \overline{AS}(A) − \frac{1}{2}η(A).

Proof: Apply Proposition 1 to the cycle A ⊗ B, where B is a pseudo-inverse of A such that

(2) BA = I − P_{ker A}, AB = I − P_{cokerA}.

Notice now that IF (A, B) = Ind(A).

We claim now that the conclusion of [2, Lemma 15] holds provided only that A is fully elliptic, without any extra assumption. In other words:

Proposition 3. The integral for \overline{AS}(A) converges without regularization in x.

Proof: Consider the function f(ϵ) = ∫_{^{c}S^{∗}X∩{x=ϵ}}([log Q, B]A)_{−n}ı_{∂_{x}}ı_{R}ω_{c}^{n}. By [2, (55)],

\overline{AS}(A) = ∫_{^{c}S^{∗}X∩\{x≥x_{0}\}}([log Q, B]A)_{−n}ı_{R}ω_{c}^{n} + LIM_{ϵ→0} ∫_{ϵ}^{x_{0}} f(x)dx. (3)

Here LIM is the Hadamard regularization of the limit, defined as in [1]. We remark that the volume form ı_{∂_{x}}ı_{R}ω_{c}^{n}, and hence also f(x), is a smooth multiple of x^{−2}. Replace f by its asymptotic expansion at x = 0. As far as convergence is concerned, only the coefficients of x^{−2} and x^{−1} matter. The product on the boundary algebra A_{∂} has as leading term the suspended product, and the other terms are multiples of x. So let A_{0}, B_{0}, Q_{0} be the indicial operators of A, B, Q in the suspended algebra. The term in (3) diverging like ϵ^{−1} is given by the leading component of f, i.e. f_{−2} = ∫_{^{c}S^{∗}X∩{x=0}}([log Q_{0}, B_{0}]A_{0})_{−n}ı_{∂_{x}}ı_{R}ω_{c}^{n}.

Lemma 4. f_{−2} = 0.

Proof: We can express f_{−2} as the residue of \overline{T r}(Q_{0}^{τ}[log Q_{0}, B_{0}]A_{0}) at τ = 0. The rest of the lemma is formally identical to the index formula [2, (3)]. We claim that f_{−2} is equal to \overline{T r}([B_{0}, A_{0}]), which equals zero (because \overline{T r} is a trace, and also because [B_{0}, A_{0}] = 0). Going backwards,

\overline{T r}([B_{0}, A_{0}]) = \overline{T r}(Q_{0}^{τ}[B_{0}, A_{0}])_{τ=0} = \overline{T r}([Q_{0}^{τ}, B_{0}]A_{0})_{τ=0}

= \overline{T r} (τQ_{0}^{τ} \frac{B_{0} − Q_{0}^{−τ}B_{0}Q_{0}^{τ}}{τ} A_{0})_{τ=0} .

Let F (τ) = (B_{0} − Q_{0}^{−τ}B_{0}Q_{0}^{τ})/τ. Then F (τ) is entire, of fixed order, and F (0) = [log Q_{0}, B_{0}]. Note that \overline{T r}(Q_{0}^{τ}F (τ)A_{0}) has at most simple poles at real integer τ, which implies the claim.

Let us now examine the other diverging term. It is given by the coefficient of x^{−1} in f(x), which by [2, (55)] is just Tr_{∂,σ}([log Q, B]A). By [2, Lemma 12], we have Tr_{∂,σ}([log Q, B]A) = T̂r_{∂}([A, B]). By assumption, [A, B] belongs to I and so T̂r_{∂}([A, B]) = 0.

Therefore, only integrable terms are left in the asymptotic expansion of the integrand in (3). This finishes the proof of the proposition.

As for the η term, this is not quite the η invariant from [1], even with the assumptions on Q at the boundary from [2, Proposition 17]. The difference arises from the fact that the regular value at z = 0 of \overline{T r}(Q^{z}A) and \overline{T r}(A) do not always agree. As shown by Nistor [3], they do agree for differential operators A, in particular for Dirac operators.

Acknowledgments: We have learnt about the cusp algebra in Richard Melrose’s seminar in MIT. We would like to thank him and Victor Nistor for providing us with preprints of their papers and for many useful discussions. This note arose in conjunction with our work on a related subject suggested by Richard Melrose.

References

[1] R. B. Melrose. The eta invariant and families of pseudodifferential operators. Math. Res. Letters, 2:541–561, 1995.

[2] R. B. Melrose and V. Nistor. Homology of pseudodifferential operators I. Manifolds with boundary. Version 3.0A, to appear in Amer. J. Math.

[3] V. Nistor. An index theorem for families of elliptic operators invariant with respect to a bundle of Lie groups. in preparation.

FB 17-Mathematik, Universität Mainz, D-55099 Mainz, Germany lauter@mathematik.uni-mainz.de

IMAR, PO. Box 1-764, RO-70700, Bucharest, Romania Sergiu.Moroianu@imar.ro