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1-MD Author: Ryszard Nest, Boris Tsygan Short title: Algebraic index theorem for families Control number: 1 337 107 Primary classification: 58G12 Secondary classification(s): 58G15 35S05 19D55 19K56 47G30 46L85 81S10 58H10 Short title: Algebraic index theorem Control number: 1 350 407 Primary classification: 58G12 Secondary classification(s): 58G15 35S05 19D55 19K56 47G30 46L85 81S10 58H10 Review text: These two papers present some new generalizations of the Atiyah-Singer index theorems in a more general algebraic context. The basic philosophy here comes viewing the (noncommutative) calculus of pseudodifferential operators as a quantized version of the (commutative) multiplication of symbols. As originally pointed out by J. Block and J. Fox in [Geometric and topological invariants for elliptic operators, 1–32, Contemp. Math., vol. 105, Amer. Math. Soc., Providence, RI, 1990; MR 91e:58186], it is useful for index theory to consider “asymptotic pseudodifferential operators” (AψDO’s), which roughly speaking are one-parameter families of operators. In the paper of Block and Fox, AψDO’s were required to be given by a family of symbols indexed by a parameter t (corresponding to the “time” variable in the usual heat equation proof of the index theorem), and to have an asymptotic expansion with respect to t. Here this approach is carried to one further level of abstraction via the notion of a “deformation quantization” of the algebra of functions on a symplectic manifold (M, ω), or still more generally, a Poisson manifold. (In the classical case corresponding to the index theorem on a compact manifold X, the symplectic manifold would be the cotangent bundle T^{∗}X of the compact manifold X. In the case of the index theorem for families, one needs to consider a Poisson manifold with a fibration to a parameter space B, where the fibers are the symplectic leaves of the Poisson structure.) For Nest and Tsygan, an AψDO is viewed merely as an operatorvalued formal power series in “Planck’s constant” ℏ, which replaces the time parameter t in the Block and Fox approach. This is reasonable in view of the fact that only the first few coefficients in the asymptotic expansions are needed for the proof of the index theorem, and is also in accord with physical intuition regarding the “classical limit” as ℏ → 0.

To be a little more precise, we briefly outline the statement and proof of the authors’ “algebraic index theorem.” The “algebraic index theorem for families” is similar, but slightly more involved technically. Fix a symplectic manifold (M^{2n}, ω), not necessarily compact, with Poisson bracket { , }. Consider a deformation quantization of C^{∞}(M). This is a formal deformation of the commutative multiplication on A to a non-commutative (but associative) ∗-product

f ∗ g = fg + ℏφ_{1}(f, g) + ℏ^{2}φ_{2}(f, g) + …, f, g ∈ A,

where φ_{1}, φ_{2}, … are bilinear differential operators, 1 ∗ f = f ∗ 1 = f for all f ∈ A, and

φ_{1}(f, g) − φ_{1}(g, f) = {f, g}.

The last of these properties ensures the “correspondence principle” between classical and quantum mechanics, that

\frac{f ∗ g − g ∗ f}{ℏ} |_{ℏ=0} = \{f, g\}.

Existence of a ∗-product with these properties has been proved by M. De Wilde and P. B. A. Lecomte [Lett. Math. Phys. 7 (1983), no. 6, 487–496; MR 85j:17021]—see also [Note Mat. 10 (1990), Suppl. 1, 205–216 (1992); MR 94g:58068]—and by B. V. Fedosov [J. Diff. Geom. 40 (1994), no. 2, 213–238; MR 95h:58062]. Such a ∗-product extends to an associative multiplication on 𝔸^{ℏ}(M) = C^{∞}(M)[[ℏ]] making this into an algebra over ℂ[[ℏ]]. Locally, this algebra is given by the Weyl quantization of C^{∞}(ℝ^{2n}). From the patching of copies of C^{∞}(ℝ^{2n}) corresponding to coordinate charts on M, the authors define a characteristic class for the deformation quantization,

θ ∈ H^{2}(M; ℂ[[ℏ]]) = [ω] + O(ℏ),

where [ω] is the de Rham class of ω. Let 𝔸_{0}^{ℏ}(M) = C_{c}^{∞}(M)[[ℏ]]. View 𝔸^{ℏ}(M) as the algebra of AψDO’s, 𝔸_{0}^{ℏ}(M) as the ideal of smoothing AψDO’s. We let 𝔸̃_{0}^{ℏ}(M) be the result of adjoining a unit to 𝔸_{0}^{ℏ}(M).

In his previous work on deformation quantization, Fedosov had shown that there is an essentially unique trace on 𝔸_{0}^{ℏ}(M)[ℏ^{−1}], with values in the field of fractions k = ℂ⟨⟨ℏ⟩⟩ of ℂ[[ℏ]]. This is a k-linear map Tr such that Tr(f ∗g) = Tr(g ∗ f). It is given by a local expression of the form

Tr(f) = \frac{1}{n! ℏ^{n}} ∫_{M} fω^{n} + O (\frac{1}{ℏ^{n−1}} ) .

According to the philosophy of the non-commutative geometry of Connes [Inst. Hautes Études Sci. Publ. Math. no. 62 (1985), 257–360; MR 87i:58162], this trace should be viewed a cyclic 0-cocycle on 𝔸_{0}^{ℏ}(M)[ℏ^{−1}]. The Nest-Tsy-gan index theorem depends on the calculation of the higher cyclic cohomology of 𝔸_{0}^{ℏ}(M)[ℏ^{−1}].

It is crucial to the approach here that the ∗-product is local on M. Thus everything associated to 𝔸^{ℏ} can be “sheafified,” and reduces the calculation of the cyclic cohomology to a local calculation. The result is that

HC_{odd}^{per} (𝔸^{ℏ}(M)[ℏ^{−1}]) ≅ k,

with a canonical generator Ω. This generator has a “Poincaré dual”

χ_{Tr}(Ω) ∈ HC_{per}^{even} (𝔸_{0}^{ℏ}(M)[ℏ^{−1}]) .

The main theorem comes from computing the result of pairing this cyclic cocycle with the cycle in periodic cyclic homology associated to a projection, and then taking the limit as ℏ → 0. The final index calculation then says the following. Let e be an idempotent matrix over 𝔸̃_{0}^{ℏ}(M), and let e(∞) be the limit of e(x) as x ∈ M tends to infinity, so that e − e(∞) has compact support. Let e_{0} and e_{0}(∞) be the values of e_{0} and e_{0}(∞) in the “classical limit” as ℏ → 0. Then

Tr (e − e(∞)) = ∫_{M} (e_{0} − e_{0}(∞)) · Td(ω) · e^{−c_{1}(ω)/2}e^{θ/ℏ}.

Here c_{1}(ω) and Td(ω) are the first Chern class and Todd class of an almost complex structure on M compatible with the symplectic structure. When e is the projection onto the kernel of an elliptic pseudodifferential operator, this basically recaptures the Atiyah-Singer index theorem. It should be pointed out that a similar result was obtained a number of years ago by Fedosov [Dokl. Akad. Nauk SSSR 305 (1989), no. 4, 835–838; English transl. in Soviet Phys. Dokl. 34 (1989), no. 4, 319–321; MR 90g:58123], though it is difficult comparing the authors’ approach to that of Fedosov since Fedosov never published detailed proofs.

In the case of the “Algebraic index theorem for families,” the authors prove a similar result for the case of a fibration π : M → B with symplectic fibers. One has to assume one has a “constant” family of deformation quantizations of the leaves, meaning that the classes θ_{b} ∈ H^{2}(π^{−1}(b); ℂ[[ℏ]]) come from a single class θ ∈ H^{2}(M; ℂ[[ℏ]]). This ensures the existence of a suitable connection D in the bundle b ↦ 𝔸^{ℏ}(π^{−1}(b)) and makes it possible to define a suitable “Chern character” ch_{D} with values in the de Rham cohomology of B. Then if p and q are idempotent matrices over 𝔸^{ℏ}(M) whose difference has compact support, and if p_{0}, q_{0} are their “classical limits” when ℏ → 0, the index formula becomes

ch_{D} (p − q) = ε ∫_{M/B} (ch p_{0} − ch q_{0}) · Â(ω)e^{θ(D)/ℏ}.

Here ε is a sign depending on the degree of the form modulo 4, and ∫_{M/B} denotes integration over the fibers of π.

These papers are not such easy reading, in part because it is assumed the reader is already familiar with the motivation for the work and with quite a bit of differential geometry and homological algebra, and in part because of a large number of misprints, fortunately not serious ones, and a few mislabeled or missing references. (For example, [R] is cited in both papers, but only appears in the bibliography of “Algebraic index theorem for families.”) However, the persistent reader will be rewarded by much deep and interesting mathematics.