Riemann-Roch for real varieties

Paul Bressler^{1}, Mikhail Kapranov^{2}, Boris Tsygan^{3}, and Eric Vasserot^{4}

1 I.A.S., Princeton bressler@math.ias.edu
2 Yale University mikhail.kapranov@yale.edu
3 Northwestern University tsygan@math.northwestern.edu
4 Université de Cergy-Pontoise eric.vasserot@@math.u-cergy.fr

1 Introduction

1.1

Let Σ be an oriented real analytic manifold of dimension d and let X be a complex envelope of Σ, i.e. a complex manifold of the same dimension containing Σ as a totally real submanifold. Then, (real) geometric objects on Σ can be viewed as (complex) geometric objects on X involving cohomology classes of degree d. For example, a C^{∞}-function f on Σ can be considered as a section of B_{Σ}, the sheaf of hyperfunctions on Σ which, according to Sato, can be defined as

B_{Σ} = \underline{H}_{Σ}^{d}(O_{X} ⊗ or_{Σ/X}) ,  (1)

where or_{Σ/X} is the relative orientation sheaf. So f can be viewed as a class in dth local cohomology.

More generally, the equality (1) suggests that various results of holomorphic geometry on X should have consequences for the purely real geometry on Σ, consequences that involve raising the cohomological degree by d. The goal of this paper is to investigate the consequences of one such result, the Grothendieck-Riemann-Roch theorem (GRR).

1.2

Let p : X → B be a smooth proper morphism of complex algebraic manifolds. We denote the fibers of p by X_{b} = p^{−1}(b) and assume them to be of dimension d. If E is an algebraic vector bundle on X, the GRR theorem says that

ch_{m}(Rp_{∗}(E)) = ∫_{X/B}[ch(E) · Td(T_{X/B})]_{2m+2d} ∈ H^{2m}(B, ℂ) . (2)

Here ∫_{X/B} : H^{2m+2d}(X, ℂ) → H^{2m}(B, ℂ) is the cohomological direct image (integration over the fibers of p).

In the case m = 1 the class on the left comes from the class, in the Picard group of B, of the determinantal line bundle det(Rp_{∗}E) whose fiber, at a generic point b ∈ B, is

det H^{•}(X_{b}, E) = ⨂_{i} (Λ^{max} H^{i}(X_{b}, E))^{⊗ (−1)^{i}} .  (3)

Deligne [9] posed the problem of describing det(Rp_{∗}E) in a functorial way as a refinement of GRR for m = 1. This problem makes sense already for the case B = pt when we have to describe the 1-dimensional vector space (3) as a functor of E. Deligne solved this problem for a family of curves and further results have been obtained in [11].

1.3

To understand the real counterpart of (2), assume first that B = pt, so X = X_{pt} and let Σ ⊂ X be as in 1.1. Denote by E the restriction of E to Σ and by C_{Σ}^{∞}(E) the sheaf of its C^{∞} sections. Then, similarly to (1), we have the embedding

C_{Σ}^{∞}(E) ⊂ \underline{H}_{Σ}^{d}(E ⊗ Ω_{X}^{d}) .

Assume further that d = 1, so X is an algebraic curve, and that Σ is a small circle in X cutting it into two pieces: X_{+} (a small disk) and X_{−}. Let E_{±} = E|_{X_{±}}. We are then in the situation of the Krichever correspondence [26]. Namely, the space Γ (E) of L^{2}-sections has a canonical polarization in the sense of Pressley and Segal [26] and therefore possesses a determinantal gerbe Det Γ (E). The latter is a category with every Hom-set made into a ℂ^{∗}- torsor (a 1-dimensional vector space with zero deleted). The extensions E_{±} of E to X_{±} define two objects [E_{±}] of this gerbe, and

det H^{•}(X, E) = Hom_{Det Γ (E)}([E_{+}], [E_{−}]) .

The real counterpart of the problem of describing the ℂ^{∗}-torsor det H^{•}(X, E) is the problem of describing the gerbe Det Γ (E). If we now have a family p : X → B as before (with d = 1), equipped with a subfamily of circles q : Σ → B, Σ ⊂ X, then we have an O_{B}^{∗}-gerbe Det q_{∗}(E) which, according to the the classification of gerbes [5], has a class in H^{2}(B, O_{B}^{∗}). The latter group maps naturally to H^{3}(B, ℤ) and in fact can be identified with the Deligne cohomology group H^{3}(B, ℤ_{D}(1)), see [6]. The Real Riemann-Roch for a circle fibration describes the above class (modulo 2-torsion) as

[Det q_{∗}(E)] = ∫_{Σ/B} ch_{2}(E) ∈ H^{3}(B, ℤ_{D}(2)) ⊗ ℤ [\frac{1}{2}] .  (4)

Here ∫_{Σ/B} : H^{4}(Σ, ℤ_{D}(2)) → H^{3}(B, ℤ_{D}(1)) is the direct image in Deligne cohomology. Note the absense of the characteristic classes of T_{Σ/B} (they are 2-torsion for a real rank one bundle). If one is interested in the image of the determinantal class in H^{3}(B, ℤ), then one can understand the RHS of the above formula in the purely topological sense.

Both sides of (4) do not involve anything other than q : Σ → B and a vector bundle E on Σ (equipped with CR-structures coming from the embeddings into X, E). One has a similar result for any C^{∞} circle fibration (no CR structure) and any C^{∞} complex bundle E on Σ. In this case we get a gerbe with lien C_{B}^{∞∗}, the sheaf of invertible complex valued C^{∞}-functions on B and its class lies in H^{2}(B, C_{B}^{∞∗}) = H^{3}(B, ℤ). It is this, purely C^{∞} setting, that we adopt and generalize in the present paper.

1.4

Let Σ be a compact oriented C^{∞}-manifold of arbitrary dimension d and E a C^{∞} complex vector bundle on Σ. One expects that the space Γ (E) should have some kind of d-fold polarization, giving rise to a “determinantal d-gerbe”, Det Γ (E). This structure is rather clear when Σ is a 2-torus but in general the theory of higher gerbes is not fully developed. In any case one expects that a C^{∞} family of such gerbes over a base B gives a class in H^{d+1}(B, C_{B}^{∞∗}) = H^{d+2}(B, ℤ). In this paper we consider a C^{∞} family q : Σ → B of relative dimension d and a C^{∞} bundle E on Σ. We then define by means of the Chern-Weil approach, what should be the characteristic class of the would-be d-gerbe Det(q_{∗}(E)):

C_{1}(q_{∗}(E)) ∈ H^{d+2}(B, ℂ) .

We denote it by C_{1} since it is a kind of d-fold delooping of the usual first Chern (determinantal) class. We then show the compatibility of this class with the gerbe approach whenever the latter can be carried out rigorously. Our main result is the Real Riemann-Roch theorem (RRR):

C_{1}(q_{∗}E) = ∫_{Σ/B}[ch(E) · Td(T_{Σ/B})]_{2d+2} ∈ H^{d+2}(B, ℂ) .

Here, T_{Σ/B} is the complexified relative tangent bundle and ∫_{Σ/B}, the integration along the fibers of q, lowers the degree by d.

Note that the above theorem is a statement of purely real geometry and is quite different from the “Riemann-Roch theorem for differentiable manifolds” proved by Atiyah and Hirzebruch [1]. The latter expresses properties of a Dirac operator on a real manifold Σ, while our RRR deals with the \overline{∂}-operator on a complex envelope X of Σ. The d = 1 case above can be deduced from a result of Lott [22] on “higher” index forms for Dirac operators (because the polarization in the circle case can be described in terms of the signs of eigenvalues of the Dirac operator). In general, however, our results proceed in a different direction.

1.5

Our definition of C_{1}(q_{∗}E) uses the description of the cyclic homology of differential operators [7] [29] which provides a construction of a natural Lie algebra cohomology class γ of the Atiyah algebra, i.e., of the Lie algebra of infinitesimal automorphisms of a pair (Σ, E) where Σ is a compact oriented d-dimensional C^{∞}-manifold and E is a vector bundle on Σ. The intuition with higher gerbes suggests that this class comes in fact from a group cohomology class of the infinite-dimensional group of all the automorphisms of (Σ, E), see Proposition 40 and, moreover, that there are similar classes coming from the higher Chern classes (39). This provides a new point of view on the rather classical subject of “cocycles on gauge groups and Lie algebras” i.e., on groups of diffeomorphisms of manifolds and automorphisms of vector bundles as well as their Lie algebra analogs.

There have been two spurs of interest in this subject. The first one was the study of the cohomology of the Lie algebras of vector fields following the work of Gelfand-Fuks, see [13] for a systematic account. In particular, Bott [3] produced a series of cohomology classes of the Lie algebra of vector fields on a compact manifold and integrated them to group cohomology classes of the group of diffeomorphisms. Later, group cocycles have been studied with connections with various anomalies in physics, see [27].

¿From our point of view, the approach of [27] can be seen as producing “integrals of products of Chern classes” in families over a base B, (cf. [9] [11]), in other words, as producing the ingredients for the right hand side of a group-theoretical RRR. This is the same approach that leads to the construction of the Morita-Miller characteristic classes for surface fibrations [24]. The anomalies themselves, however, should be seen as the hypothetical classes from Conjectures 39, 41 and whose description through integrals of products of Chern classes constitutes the RRR.

1.6

As far as the proof of the RRR goes, we use two types of techniques. The first is that of differential graded Lie algebroids (which can be seen as infinitesimal analogs of higher groupoids appearing in the heuristic discussion above). The second technique is that of “formal geometry” of Gelfand and Kazhdan, i.e., reduction of global problems in geometry of manifolds and vector bundles to problems related to cohomology of Lie algebras of formal vector fields and currents. The first work relating Riemann-Roch to Lie algera cohomology was [12] and this approach was further developed in [4]. To prove the RRR we use results of [25] and [4] on the Lie algebra cohomology of formal Atiyah algebras.

1.7

The second author would like to acknowledge support from NSF, Université Paris-7 and Max-Planck Institut für Mathematik. The fourth author is partially supported by the NSF.

2 Background on Lie algebroids, groupoids and gerbes.

2.1 Conventions

All manifolds will be understood to be C^{∞} unless otherwise specified. For a manifold Σ we denote by C_{Σ}^{∞} the sheaf of ℂ-valued C^{∞}-functions. By a vector bundle over Σ we mean a locally trivial, C^{∞} complex vector bundle, possibly infinite-dimensional. For such a bundle E we denote by C^{∞}(E) = C_{Σ}^{∞}(E) the sheaf of smooth sections, which is a locally free sheaf of C_{Σ}^{∞}-modules. By T_{Σ} we denote the complexified tangent bundle of Σ, so its sections are derivations of C_{Σ}^{∞}. We denote by D_{Σ} the sheaf of differential operators acting on C_{Σ}^{∞} and by D_{Σ,E} the sheaf of differential operators acting from sections of E to sections of E. The notations D(Σ) and D(Σ, E) will be used for the spaces of global sections of D_{Σ} and D_{Σ,E}.

2.2 Lie algebroids

Recall [23] that a Lie algebroid on Σ consists of a vector bundle G, a morphism of vector bundles α : G → T_{Σ} (the anchor map) and a Lie algebra structure in C^{∞}(G) satisfying the properties:

1. α takes the Lie bracket on sections of G to the standard Lie bracket on vector fields.

2. For any smooth function f on Σ and sections x, y of G we have

[fx, y] − f · [x, y] = Lie_{α(y)}(f) · x .

A Lie algebroid is called transitive if α is surjective.

Example 1. When Σ = pt, a Lie algebroid is the same as a Lie algebra.

Example 2. T_{X} with the standard Lie bracket and α = id is a Lie algebroid.

Example 3. If α = 0, then the bracket in G is C_{Σ}^{∞}-linear. In this case we say that G is a bundle of Lie algebras: every fiber of G is a Lie algebra.

Morphisms of Lie algebroids are defined in an obvious way. Note that for any transitive Lie algebroid G the kernel Ker(α) ⊂ G is a bundle of Lie algebras, i.e., a Lie algebroid with trivial anchor map, and the maps in the short exact sequence

0 → Ker(α) → G →^{α} T_{X} → 0

are morphisms of Lie algebroids.

2.3 The de Rham complex of a Lie algebroid

Let G be a Lie algebroid on Σ. Let

DR^{i}(G) := Hom(Λ^{i}G, C_{Σ}^{∞}) .

The differential d : DR^{i}(G) → DR^{i+1}(G) is defined by the standard formula of Cartan: for an antisymmetric i-linear function l : G^{i} → C_{Σ}^{∞} we set

dl(x_{1}, ..., x_{i+1}) = ∑_{j=1}^{i+1}(−1)^{j}Lie_{α(x_{j})}l(x_{1}, ..., x̂_{j}, ..., x_{i+1})

+ ∑_{j<k}(−1)^{j+k}l([x_{j}, x_{k}], x_{1}, ..., x̂_{j}, ..., x̂_{k}, ..., x_{i+1}) . (5)

We get a complex DR^{•}(G) called the de Rham complex of G. A morphism of Lie algebroids ϕ : G → H gives rise to the morphism of de Rham complexes ϕ^{∗} : DR^{•}(H) → DR^{•}(G).

Example 4. If Σ = pt, so G is a Lie algebra, then DR^{•}(G) = C^{•}(G) is the cochain complex of G with trivial coefficients.

Example 5. If G = T_{Σ}, then DR^{•}(G) = Ω_{Σ}^{•} is the C^{∞} de Rham complex of Σ.

2.4 The enveloping algebra of a Lie algebroid.

Let G be a Lie algebroid on Σ, as before. The enveloping algebra U(G) is the sheaf of associative algebras on Σ defined by generators x ∈ G (local sections) and f ∈ C_{Σ}^{∞} (local functions) subject to the relations:

xy − yx = [x, y]
f · x − x · f = Lie_{α(x)}(f) .

Example 6. If Σ = pt, so G is a Lie algebra, then U(G) is the usual enveloping algebra of G.

Example 7. If G = T_{Σ}, then U(G) = D_{Σ} is the sheaf of differential operators C_{Σ}^{∞} → C_{Σ}^{∞}.

Example 8. If G is any Lie algebroid, then the anchor map α induces a morphism

U(α) : U(G) → U(T_{Σ}) = D_{Σ}

of sheaves of associative algebras. In particular, C_{Σ}^{∞} is a left U(G)-module.

The sheaf U(G) has an increasing ring filtration {U^{m}(G)} with U^{m}(G) generated by products involving at most m sections of G. The following is then standard.

Proposition 9. The associated graded sheaf of algebras gr U(G) is identified with the symmetric algebra S^{•}(G).

2.5 The Koszul resolution

Let G be a Lie algebroid on Σ. We have then the complex

... → U(G) ⊗ Λ^{2}G → U(G) ⊗ G → U(G) → C_{Σ}^{∞} → 0 .  (6)

with the differential defined by:

d(u ⊗ (γ_{1} ∧ ... ∧ γ_{n})) = ∑_{j=1}^{n}(−1)^{j}(uγ_{j}) ⊗ (γ_{1} ∧ ... ∧ γ̂_{j} ∧ ... ∧ γ_{n})

+ ∑_{j<k}(−1)^{j+k}u ⊗ ([γ_{i}, γ_{j}] ∧ ... ∧ γ̂_{i} ∧ ... ∧ γ̂_{j} ∧ ... ∧ γ_{n}) .

Proposition 10. The complex (6) is exact. Thus, it provides a locally free resolution of C_{Σ}^{∞} as a U(G)-module.

Corollary 11. We have

DR^{•}(G) ≃ R\underline{Hom}_{U(G)}(C_{Σ}^{∞}, C_{Σ}^{∞}) .

2.6 The Atiyah algebra

Let G be a Lie group, 𝔤 be its Lie algebra, and ρ : P → Σ a principal G-bundle on Σ. The Atiyah algebra A_{P} is the sheaf of Lie algebras on Σ whose sections are G-invariant vector fields on P:

A_{P} = (ρ_{∗}T_{P})^{G} .

The map α = dρ makes A_{P} into a transitive Lie algebroid of the form

0 ⟶ Ad(P) ⟶ A_{P} ⟶^{α} T_{Σ} ⟶ 0.  (7)

Here, Ad(P) is the bundle of Lie algebras on Σ associated to P via the adjoint representation.

If Σ = ⋃ U_{i} is a covering in which P is trivialized: P |_{U_{i}} = U_{i} × G, and g_{ij} : U_{i} ∩ U_{j} → Aut(𝔤) are the transition functions, then A_{P} is glued out of A_{P} |_{U_{i}} = T_{U_{i}} × 𝔤 via the transition functions

(v, x) ↦ (v, i_{v}(dg_{ij} · g_{ij}^{−1}) + Ad_{g_{ij}}(x)) .  (8)

Example 12. Let G = GL_{r}(ℂ), so 𝔤 = 𝔤𝔩_{r}(ℂ). A principal G-bundle P corresponds then to a rank r vector bundle E on Σ. In this case A_{P} will also be denoted A_{E} and has a well known alternative description. It consists of differential operators L : E → E such that:

1. L has order ≤ 1.

2. The first order symbol of L (which is a priori a section of T_{Σ} ⊗ End(E)) lies in the subsheaf T_{Σ} = T_{Σ} ⊗ 1.

2.7 Modules over Lie algebroids

We follow [17] §3 but use a more geometric language. Let G be a Lie algebroid on Σ. A G-module is a vector bundle M on Σ equipped with a Lie algebra action (x, m) ↦ xm of G on the sections which satisfies

1. the Leibniz rule

x(f · m) − f · (xm) = (Lie_{α(x)}f) · m, f ∈ C_{Σ}^{∞}, x ∈ G, m ∈ M ;

in particular, the assignment x ↦ (m ↦ x · m) defines a map G → A_{M} which commutes with respective anchor maps

2. the map G → A_{M} is C_{Σ}^{∞}-linear.

Example 13. For any G the trivial bundle (whose sheaf of sections is) C_{Σ}^{∞} is a G-module with the G action given via the anchor map and the Lie derivations of functions.

Example 14. An ideal in G is a sub-Lie algebroid G′ such that [G, G′] ⊂ G′. Suppose that G′ is an ideal in G such that the restriction of the anchor map to G′ is trivial. Then, G′ is a G-module via the adjoint action.

Any G-module has a structure of a sheaf of modules over the sheaf of rings U(G).

2.8 Cohomology of Lie algebroids.

Let M be a G-module. The de Rham complex DR^{•}(G, M) with coefficients in M is defined by

DR^{i}(G, M) = \underline{Hom}(Λ^{i}G, M) .

with the differential of l : G^{i} → M defined by the modification of (5):

dl(x_{1}, ..., x_{i+1}) = ∑_{j=1}^{i+1}(−1)^{j}x_{j}(l(x_{1}, ..., x̂_{j}, ..., x_{i+1}))

+ ∑_{j<k}(−1)^{j+k}l([x_{j}, x_{k}], x_{1}, ..., x̂_{j}, ..., x̂_{k}, ..., x_{i+1}) .

Its cohomology sheaves will be denoted \underline{H}_{Lie}^{i}(G, M) and the corresponding cohomology groups of the complex of global smooth sections of DR^{•}(G, M) by H_{Lie}^{i}(G, M). See [23], §7.1. As before, it is easy to see that

DR^{•}(G, M) ≃ \underline{RHom}_{U(G)}(C_{Σ}^{∞}, M) .

Therefore,

\underline{H^{i}}_{Lie}(G, M) = \underline{Ext}_{U(G)}^{i}(C_{Σ}^{∞}, M), H_{Lie}^{i}(G, M) = Ext_{U(G)}^{i}(C_{Σ}^{∞}, M).

Example 15. The trivial bundle C_{Σ}^{∞} is always a G-module and for G = T_{Σ} we have H_{Lie}^{i}(T_{Σ}, C_{Σ}^{∞}) = H^{i}(Σ, ℂ).

2.9 The Hochschild-Serre spectral sequence and the transgression.

Let

0 → G′ → G → G′′ → 0  (9)

be an extension of Lie algebroids on Σ, so G′ is an ideal with zero anchor in G. Note that G′ is then a bundle of Lie algebras. Let M be a G-module. Then for every point x ∈ Σ the fiber M_{x} is a module over the Lie algebra G′_{x}. Assume that for any i ≥ 0 the sheaf of C_{Σ}^{∞}-modules Lie algebra cohomology spaces H_{Lie}^{i}(G′, M) is locally free. Then the sheaves \underline{H}_{Lie}^{i}(G′, M) are vector bundles on Σ with fiber H_{Lie}^{i}(G′_{x}, M_{x}) at x ∈ Σ. These vector bundles have natural structures of G′′-modules. In this case we have (a Lie algebroid generalization of) the Hochshild-Serre spectral sequence with

E_{2}^{pq} = H_{Lie}^{p}(G′′, \underline{H}_{Lie}^{q}(G′, M)) ⇒ H_{Lie}^{p+q}(G, M) .  (10)

The construction is parallel to the classical (Lie algebra) case as in [13]. One uses the short exact sequence (9) to produce, in a standard way, a filtration on DR^{•}(G, M). See [23], Section 7.4 for the treatment of the case G′′ = T_{Σ} which is the only case we will use in this paper.

Example 16. Similarly to the classical case, one can use (10) (or elementary considerations) to identify H_{Lie}^{2}(G, M) with the set of isomorphism classes of central extensions of Lie algebroids

0 → M → G̃ → G → 0.

Central extensions of this type with G = T_{Σ}, M = C_{Σ}^{∞}, and the G-action on M being the standard one (by Lie derivations), were called in [17] Picard Lie algebroids. The set of their isomorphism classes is thus identified with H_{Lie}^{2}(T_{Σ}, C_{Σ}^{∞}) which is the same as the topological (de Rham) cohomology H^{2}(Σ, ℂ).

Fix n > 0 and assume that

H^{j}(G′, M) = 0, 0 < j < n .  (11)

In this case E_{2}^{0,n} = E_{n+1}^{0,n} as well as E_{2}^{0,n+1} = E_{n+1}^{0,n+1}. We obtain therefore the transgression map

d_{n+1} : E_{n+1}^{0,n} = E_{2}^{0,n} = H_{Lie}^{n}(G′, M)^{G′′} ⟶

H_{Lie}^{n+1}(G′′, M^{G′}) = E_{2}^{n+1,0} = E_{n+1}^{n+1,0} . (12)

We will use this map later in the paper. Without the assumption (11) we have that E_{n+1}^{0,n} is a subspace of E_{2}^{0,n} = H_{Lie}^{n}(G′, M)^{G′′} namely the intersection of the kernels of d_{2}, ..., d_{n}. For convenience we will call elements of this space transgressive elements of E_{2}^{0,n}. Similarly, E_{n+1}^{n+1,0} is a quotient space of E_{2}^{n+1,0} = H_{Lie}^{n+1}(G′′, M^{G′}) by the union of images of d_{2}, ..., d_{n}.

Example 17. Suppose that n = 2 and Σ = pt, so (9) is a central extension of Lie algebras and M is a G-module in the usual sense. Let γ ∈ E_{2}^{0,2} = H_{Lie}^{2}(G′, M)^{G′′} be a G′′-invariant class in H^{2} and

0 → M → G̃′ → G′ → 0

be a central extension representing γ. The class γ is transgressive, (i.e., annihilated by d_{2}) if and only if G̃′ can be made into a G-equivariant central extension (as opposed to the fact that the class of the extension remains unchanged under the G-action or, what is the same, under G′′-action). Given such an equivariant extension, one obtains a crossed module of Lie algebras (i.e., a dg-Lie algebra situated in degrees (-1) and 0)

G̃′′ ⟶^{∂} G,

with Ker(∂) = M and Coker(∂) = G′′. As well known (see, e.g., [21], Example E.10.3), such a crossed module represents an element in H^{3}(G′′, M), and this element is the lifting of d_{3}(β). Different choices of equivariant structure on G̃′ correspond to the ambiguity of the values of d_{3} modulo the image of d_{2}. One can generalize this picture easily to the case of an arbitrary Σ.

2.10 Reminder on gerbes.

We follow the same conventions as in [19] and use [5] as the background reference.

If B is a topological space and and F is a sheaf of abelian groups on B, then we can speak of F-gerbes (= gerbes with lien F). Recall that such a gerbe 𝔊 is the following:

1. A category 𝔊(U) given for all open U ⊂ B, the restriction functors r_{UV} : 𝔊(U) → 𝔊(V) given for any morphism V ⊂ U and natural isomorphisms of functors s_{UV W} : r_{V W} ∘ r_{UV} ⇒ r_{UW} given for each W ⊂ V ⊂ U and satisfying the transitivity conditions.

2. The structure of F|_{U}-torsor (possibly empty) on each sheaf \underline{Hom}_{𝔊(U)}(x, y) compatible with the r_{UV} and such that the composition of morphisms is bi-additive.

These data have to satisfy the local uniqueness and gluing properties for which we refer to [5].

By a sheaf of F-groupoids we will mean a sheaf of categories ℭ on B (so both Ob ℭ and Mor ℭ are sheaves of sets) in which each sheaf \underline{Hom}_{ℭ(U)}(x, y) is either empty or is made into a sheaf of F|_{U}-torsors so that the composition is biadditive. A sheaf ℭ of F-groupoids is called locally connected if locally on B all the Ob ℭ(U) and Hom_{ℭ(U)}(x, y) are nonempty.

Each sheaf of F-groupoids can be seen as a fibered category over B, in fact it is a pre-stack, see, e.g., [20]. Recall (see, e.g., loc. cit. Lemma 2.2) that for any pre-stack ℭ there is an associated stack ℭ^{∼}. If ℭ is a locally connected sheaf of F-groupoids, then ℭ^{∼} is an F-gerbe.

As well known (see, e.g., [5]), the set formed by F-gerbes up to equivalence is identified with H^{2}(B, F). The identification of the set of isomorphism classes of Picard Lie algebroids in Example 1.9.3 can be seen as an infinitesimal analog of this fact. Given an F-gerbe 𝔊, we denote by [𝔊] ∈ H^{2}(B, F) its class. Given a sheaf ℭ of F-groupoids, we denote by [ℭ] the class of the corresponding gerbe.

Let B be a C^{∞}-manifold. We will be particularly interested in C_{B}^{∞∗}-gerbes on B. Recall that we have the exponential sequence of sheaves on B:

0 → \underline{ℤ}_{B} → C_{B}^{∞} ⟶^{e^{2πix}} C_{B}^{∞∗} → 0 .  (13)

The corresponding coboundary map

δ_{n} : H^{n}(B, C_{B}^{∞∗}) → H^{n+1}(B, ℤ)  (14)

is an isomoprhism for n ≥ 1 since C_{B}^{∞} is a soft sheaf. Thus [𝔊] give rise to a class in H^{3}(B, ℤ).

Let 𝔊 be a C_{B}^{∞∗}-gerbe. Recall [6], that a connective structure Δ on 𝔊 is a set of data that associates to each open U ⊂ B and each object x ∈ Ob 𝔊(U) an Ω_{U}^{1}-torsor Δ(x) (whose sections can be thought of as “formal connections” on x) and for any local (iso)morphism g : x → y over U an identification of torsors g_{∗} : Δ(x) → Δ(y), satisfying the compatibility property plus the following gauge condition: if x = y so g ∈ C^{∞∗}(U) is an invertible function, then g_{∗}(∇) = ∇ − g^{−1}d(g).

A curving of a connective structure Δ is a rule K associating to any x as above and any global object ∇ ∈ Δ(x) a 2-form K(∇) ∈ Ω^{2}(U) satisfying the compatibility with pullbacks, invariance under isomorphisms as well as the gauge condition: K(∇ + α) = K(∇) + dα, α ∈ Ω^{1}(U). In this situation Brylinski defined the 3-curvature of the connective structure and curving, which is a closed 3-form S = S_{Δ,K} ∈ Ω^{3}(B).

Example 18. Let G be a Lie group and

1 → ℂ^{∗} → G̃ → G → 1

be a central extension of Lie groups. Let ρ : P → B be a principal G-bundle. We have then the C_{B}^{∞∗}-gerbe LiftG̃ ; G(P) whose objects over U ⊂ B are liftings of P |_{U} to a principal G̃-bundle over U, compare [2]. Let ∇_{P} be a connection on P. Then LiftG̃ ; G(P) has a connective structure Δ which to every lifting P̃ of P to a G̃-bundle associates the space of all connections on P̃ extending ∇_{P}. Further, let R_{∇} ∈ Ω^{2}(B) ⊗ Ad(P) be the curvature of ∇. A choice of a lifting of R_{∇} to a form R̃_{∇} ∈ Ω^{2}(B) ⊗ Ad(P̃) gives a curving K on Δ. This curving associates to any section ˜∇ of Δ(P̃), i.e., to a connection on P̃ extending ∇, the 2-form R ˜∇ − R̃_{∇}, where R ˜∇ is the curvature of ˜∇.

We will need the following result ([6], Thm. 5.3.12).

Theorem 19. If 𝔊 is a C_{B}^{∞∗}-gerbe with a connective structure Δ and a curving K, then the class of S_{Δ,K} in H^{3}(B, ℂ) is integral and is equal to the image of δ_{2}[𝔊] under the natural map from H^{3}(B, ℤ) to H^{3}(B, ℂ).

3 Background on homology of differential operators

3.1 Conventions

Let A be an associative algebra over ℂ. We denote by Hoch_{•}(A) the Hochschild complex of A with coefficients in A:

... → A ⊗ A ⊗ A → A ⊗ A → A

with the differential given by the formula

b(a_{0} ⊗ ... ⊗ a_{p}) =

∑_{i=0}^{p−1}(−1)^{i}a_{0} ⊗ ... ⊗ a_{i}a_{i+1} ⊗ ... ⊗ a_{p} + (−1)^{p}a_{p}a_{0} ⊗ a_{1} ⊗ ... ⊗ a_{p−1}.

By HH_{•}(A) we denote the homology of Hoch_{•}(A). As well known,

HH_{•}(A) = Tor_{•}^{A⊗A^{op}}(A, A) .  (15)

Put

τ(a_{0} ⊗ ... ⊗ a_{p}) = (−1)^{p}a_{1} ⊗ ... ⊗ a_{p} ⊗ a_{0} .

Let N = 1 + τ + τ^{2} + ... + τ^{n} on Hoch_{n}(A). The cyclic complex of A is defined as the total complex

CC_{•}(A) =

Tot_{•}{⋯ Hoch_{•}(A) →^{1−τ} Hoch_{•}(A) →^{N} Hoch_{•}(A) →^{1−τ} Hoch_{•}(A)} . (16)

The cyclic homology HC_{•}(A) is the homology of the complex CC_{•}(A). We recall the theorem relating the cyclic homology with the Lie algebra homology of the algebra of matrices, see [21].

Theorem 20. H_{•}^{Lie}(𝔤𝔩(A)) = S^{•}(HC_{•−1}(A))

Corollary 21. If HC_{j}(A) = 0 for j = 0, ..., p − 1, then H_{j}^{Lie}(𝔤l(A)) = 0 for j = 1, ..., p, and H_{p+1}^{Lie}(𝔤l(A)) = HC_{p}(A).

3.2 Homology of differential operators: algebro-geometric version.

Let X be a smooth affine algebraic variety over ℂ of dimension d and E be an algebraic vector bundle on X. Then the Hochschild-Kostant-Rosenberg theorem (together with Morita invariance of HH_{•}) gives an identification:

HH_{m}(End(E)) = Ω^{m}(X) ,

where on the right we have the space of global regular m-forms on X. Furthermore,

HC_{m}(End(E)) =

Ω^{m}(X)/dΩ^{m−1}(X) ⊕ H^{m−2}(X, ℂ) ⊕ H^{m−4}(X, ℂ) ⊕ ….

where on the right we have the usual topological (de Rham) cohomology, see [21] Th. 3.4.12. Let D(E) be the ring of global differential operators from E to E. Then, the results of [7], [29] imply:

HH_{m}(D(E)) = H^{2d−m}(X, ℂ) .

Furthermore,

HC_{m}(D(E)) = ⨁_{i≥0} H^{2d−m+2i}(X, ℂ).

We recall that the approach of loc. cit is to use the filtration by the order of differential operators and realize the E_{1}-term of the corresponding spectral sequence for HH as the complex of forms on the cotangent bundle with the differential adjoint to the de Rham differential by means of the symplectic form. The spectral sequence is then seen to degenerate at E_{2}.

Let us note the particular case when X = 𝔸^{d} and E = O_{𝔸^{d}} is the trivial bundle of rank 1. Then D(E) = W_{d} is the Weyl algebra with generators x_{i}, ∂_{i}, i = 1, ..., d, and relations

[x_{i}, x_{j}] = [∂_{i}, ∂_{j}] = 0, [∂_{i}, x_{j}] = δ_{ij} · 1 .

The above results imply that

HH_{i}(W_{d}) = 0 if i ≠ 2d,  HH_{2d}(W_{d}) = ℂ  (17)

and

HC_{i}(W_{d}) = ℂ, i − 2d ∈ 2ℤ_{≥0}, HC_{i}(W_{d}) = 0, i − 2d ∉ 2ℤ_{≥0} .

3.3 The C^{∞} version.

Let Σ be an oriented C^{∞}-manifold of dimension d and E be a smooth complex vector bundle on Σ. We have then the algebras End(E), D(E) of smooth endomorphisms and differential operators on E. Following [29] we present the analogs of the results cited in 3.2 for these algebras. These rings have natural Fréchet topologies. As pointed out in loc. cit., to get reasonable results, all tensor products occuring in the Hochschild and cyclic complexes of the above algebras should be taken in the category of topological vector spaces, i.e., be completed. In plain terms, this means that the End(E)^{⊗p} should be understood as the ring of endomorphisms of the vector bundle E^{⊠p} on the p-fold Cartesian product Σ^{p} and similarly for differential operators. Under these conventions, we have:

HH_{m}(D(E)) = H^{2d−m}(Σ, ℂ),  (18)
HC_{m}(D(E)) = ⨁_{i≥0} H^{2d−m+2i}(Σ, ℂ),  (19)

where on the right we have the topological cohomology.

Remark 22. The Lie algebra cochain complexes of D(E) and of 𝔤𝔩_{N}D(E) = D(E⊗ℂ^{r}) involve exterior products of these algebras over ℂ. If we understand these products in the completed sense as above (compare also with Fuks [13]), then the analog of Theorem 20 holds, and we have the following.

Corollary 23. Let Σ be a compact, oriented C^{∞} manifold of dimension d. Then, for N ≫ 0 we have:

H_{i}^{Lie}𝔤𝔩_{N}D(E) = 0, 0 < i < d + 1
H_{d+1}^{Lie}𝔤𝔩_{N}D(E) = ℂ

3.4 The formal series version.

Let

Ŵ_{d} = W_{d} ⊗_{ℂ[x_{1},...,x_{d}]} ℂ[[x_{1}...., x_{d}]]

be the algebra of differential operators whose coefficients are formal power series. Similarly to the above, we consider the Hochschild and cyclic complexes of Ŵ_{d} using the adic topology on ℂ[[x_{1}, ..., x_{d}]] and taking completions. Thus Ŵ_{d}^{⊗p} is understood as the ring of differential operators whose coefficients are power series in p groups of d variables. With this understanding, we have the analog of (17):

HH_{2d}(Ŵ_{d}) = ℂ, HH_{i}(Ŵ_{d}) = 0, i ≠ 2d .

For the proof, see [12]. One can also apply the spectral sequence argument of [7] and [29] and then use the Poincare lemma on the (contangent bundle to the) formal disk.

Our next step is to consider such formal completions simultaneously at all points of a given C^{∞}-manifold Σ. So, let Σ, E be as above. Let { \atop Hoch}_{p}(D(E)) be the completion of D(E^{⊠(p+1)}) (differential operators in the bundle E^{⊠(p+1)} on Σ^{p+1}) along the diagonal Σ ⊂ Σ^{p+1}. This is a sheaf on Σ.

Then the Hochschild differential extends to { \atop Hoch}_{•}(D(E)), making it into a complex, and we denote by ĤH_{•}(D(E)) its homology. Similarly, we define the completed cyclic complex ĈC_{•}(D(E)) by the procedure identical to (16) and denote its homology by ĤC_{•}(D(E)). Thus, ĤH_{•}(D(E)) and ĈC_{•}(D(E)) are sheaves on Σ.

Proposition 24. We have ĤH_{p}(D(E)) = \underline{ℂ}_{Σ} (constant sheaf) for p = 2d and ĤH_{p}(D(E)) = 0 for p ≠ 2d.

Proof. Consider the case when Σ is an open contractible domain in ℝ^{d} and E is trivial. Let us prove that in this case the complex of global sections of ĤH_{•}(D(E)) is exact everywhere except degree 2d where the cohomology is isomoprhic to ℂ. (This is the standard Hochschild-Kostant-Rosenberg theorem in the context of completed Hochschild complexes).

We start with the case of { \atop Hoch}_{•}(C_{Σ}^{∞}) defined, as before, using the completion of the functions on Σ^{•+1} along the diagonal. Recall the interpretation of HH as Tor, see (15). Assume for a moment that Σ is the affine space viewed as an affine algebraic variety. Choose the standard Koszul resolution of ℂ[Σ] over ℂ[Σ × Σ]. We see that

HH_{•}(ℂ[Σ]) = Ω^{•}(Σ),

and the same will hold if we replace ℂ[Σ] by a matrix algebra (i.e., take E of higher rank).

Now let us get back to the C^{∞} case. There is a small difference, namely that we are using completed tensor products and therefore the standard argument of comparing two projective resolutions is not quite applicable. But if we follow this standard argument in the algebraic case, we see that it gives the embedding of complexes i : Ω^{•}(Σ) → Hoch_{•}(ℂ[Σ]), a projection j : Hoch_{•}(ℂ[Σ]) → Ω^{•}(Σ), and a homotopy s : Hoch_{•}(ℂ[Σ]) → Hoch_{•+1}(ℂ[Σ]) such that ji = 1, ij − 1 = sd + ds. It is easy to see that the maps i, j, and s extend from ℂ[Σ] to C^{∞}(Σ) and from the algebraic Hochschild complex to the completed one. We conclude that

ĤH_{•}(C^{∞}(Σ)) = Ω^{•}(Σ),

and the same will hold if we replace ℂ[Σ] by a matrix algebra.

Next, we replace C_{Σ}^{∞} by the sheaf of commutative algebras

A = S^{•}(T_{Σ})

(polynomial functions on the contangent bundle) and define { \atop Hoch}_{•}(A) using the completions of sheaves of sections of A^{⊠(p+1)} on Σ^{p+1} along the diagonals. The same argument will apply, so we conclude that

ĤH_{•}(A) = p_{∗}(Ω_{T^{∗}Σ}^{•}),  (20)

where p : T^{∗}Σ → Σ is the projection. Again, a similar statement will hold for matrices.

Finally, we use the approach of [7] [29] and consider the spectral sequence for ĤH_{•}(D(E)) associated to the filtration by degree of operators. We get the E_{1}-term to be (20) with the differential being the adjoint of the de Rham differential on T^{∗}Σ. Since we assumed Σ to be a contractible domain in the flat space, we conclude that the E_{2}-term reduces to one space ℂ. Moreover, we see that the class of the cycle

1 ⊗ Alt_{S_{2d}}(∂_{x_{1}} ⊗ … ⊗ ∂_{x_{d}} ⊗ x_{1} ⊗ … ⊗ x_{d})

is a generator of ĤH_{2d}(D(E)). We will call it the canonical generator. Note also that the above argument works not only for the ring of algebraic or smooth (or holomorphic) differential operators but also for formal differential operators, i.e. differential operators whose coefficients are formal power series.

Now consider a diffeomorphism from one contractible domain in the flat space to another. It induces an isomorphism of the rings of differential operators. It is enough to show that this isomorphism sends the canonical generator to the canonical generator. Take a point of Σ. We have seen that the homomorphism which associates to a function its jet at this point induces an isomorphism on the Hochschild homology. Furthermore, any shift in the affine space sends the canonical generator to itself. We are reduced to proving that any formal coordinate change induces an automorphism of the ring of formal differential operators that sends the canonical generator to itself.Since a reflection preserves the canonical generator, we may assume that our formal coordinate change is oriented. Therefore it may be included into a one-parameter group of formal coordinate changes. We are reduced to proving that if X is a formal vector field then the corresponding derivation of the ring of formal differential operators is trivial on the Hochschild homology. But such derivation is inner, and any inner derivation acts on the Hochschild homology trivially (the operator ι_{X} from (41) is a contracting homotopy).

More generally, any change of the trivialization of the vector bundle E induces an automorphism of ĤH^{•}(D_{(}E)) that sends the fundamental generator to itself.

We have proven that the only sheaf of cohomology of ĤH_{•}(D(E)) in the case when Σ is a contractible domain in a flat space (and thus in the general case) is \underline{ℂ}_{Σ}. ⊓ ⊔

Furthermore, we need a relative version of the above statements. Let

q : Σ → B

be a submersion (smooth fibration) of C^{∞}-manifolds, whose fibers are of dimension d and are oriented. Let E be C^{∞}-bundle on Σ, as above. We have then the subring

D_{Σ/B}(E) ⊂ D(E),

consisting of differential operators that are q^{−1}C_{B}^{∞}-linear, i.e., act along the fibers only.

Let Σ_{B}^{p+1} ⊂ Σ^{p+1} be the (p+1)-fold fiber product of Σ over B. We denote by E_{B}^{⊠(p+1)} the restriction of E^{⊠(p+1)} to Σ_{B}^{p+1}.

Let { \atop Hoch}_{p}(D_{Σ/B}(E)) denote the completion of D_{Σ_{B}^{p+1}/B}(E_{B}^{⊠(p+1)}) along

the diagonal. Then the Hochschild differential extends to { \atop Hoch}_{p}(D_{Σ/B}(E)). We also define the completed cyclic complex ĈC_{•}(D_{Σ/B}(E)) by implementing (16).

Theorem 25.

1. The complex { \atop Hoch}_{p}(D_{Σ/B}(E)) is acyclic in degrees other than 2d, and its 2dth cohomology sheaf is isomorphic to q^{−1}C_{B}^{∞}. In other words, we have an isomorphism in the derived category of sheaves of q^{−1}C_{B}^{∞}-modules on Σ:

μ_{D} : { \atop Hoch}_{p}(D_{Σ/B}(E)) → q^{−1}C_{B}^{∞}[2d].

2. We have H^{i}(ĈC_{•}(D_{Σ/B}(E)) = 0 unless i = −2d + k, k ∈ ℤ_{+}, and

H^{−2d+k}(ĈC_{•}(D_{Σ/B}(E)) = q^{−1}C_{B}^{∞}.

Proof. Similar to 24. ⊓ ⊔

Corollary 26. We have a morphism (no longer an isomorphism) in the derived category

ν_{D} : ĈC_{•}(D_{Σ/B}(E)) → q^{−1}C_{B}^{∞}[2d] .

4 Characteristic classes from Lie algebra cohomology.

4.1 The finite-dimensional case

Let G be a Lie group with Lie algebra 𝔤. We denote by C^{•}(𝔤) the cochain complex of 𝔤 with trivial coefficients ℂ and by H^{n}(𝔤) its nth cohomology space.

Let γ ∈ H^{n}(𝔤) be a cohomology class. We want to associate (under certain conditions) to γ a characteristic class of principal G-bundles. In other words, we want to produce, for each C^{∞}-manifold B and each smooth principal Gbundle P on B, a topological (de Rham) cohomology class

c_{γ}(P) ∈ H^{n+1}(B) = H^{n+1}(B, ℂ)

(note the shift of degree by 1).

Indeed, let a principal G-bundle ρ : P → B be given and let A_{P} be its Atiyah algebra. We have then the extension of Lie algebroids (7) on B and the corresponding Hochschild-Serre spectral sequence (10) which in our case has the form:

E_{2}^{pq} = H_{Lie}^{p}(T_{B}, \underline{H}_{Lie}^{q}(Ad(P), C_{B}^{∞})) ⇒ H_{Lie}^{p+q}(A_{P} , C_{B}^{∞}).  (21)

This sequence was considered in [23], Thm. 7.4.19. Note that \underline{H}_{Lie}^{q}(Ad(P), C_{B}^{∞}) is the cohomology of the cochain complex of Ad(P) as a Lie algebra over C_{B}^{∞}, i.e., of the complex of bundles formed by the duals of the fiberwise exterior products of fibers of Ad(P). We will also use the notation C^{•}(Ad(P)_{/B}) for this complex.

Lemma 27. For any q ≥ 0 the bundle H_{Lie}^{q}(Ad(P), C_{B}^{∞}) = H^{q}(Ad(P)_{/B}) on B formed by the Lie algebra cohomology spaces of the fibers of Ad(P) is canonically identified with the trivial bundle with fiber H^{q}(𝔤).

Proof. This follows from the fact the the adjoint action of G on 𝔤 induces the trivial action on H^{q}(𝔤). ⊓ ⊔

Corollary 28. The E_{2}-term of the spectral sequence (21) is given by E_{2}^{pq} = H^{p}(B) ⊗ H^{q}(𝔤) In particular, the assignment γ ↦ 1 ⊗ γ defines a map our class H^{n}(𝔤) → E_{2}^{0n}.

Assume now that there exists n > 0 such that the Lie algebra 𝔤 satisfies the acyclicity condition:

H^{i}(𝔤) = 0, 0 < i < n .  (22)

Then we are in the situation of (11), so we have the transgression map (12) which in our case has the form

d_{n+1} : H^{n}(𝔤) → H^{n+1}(B),  (23)

and we define

c_{γ}(P) = d_{n+1}(1 ⊗ γ).  (24)

Without the assumption (22) we have that c_{γ}(P) is defined only if 1 ⊗ γ is transgressive (i.e., annihilated by d_{2}, ..., d_{n} and takes value not in H^{n+1}(B) but in the quotient of H^{n+1}(B) by the images of d_{2}, ..., d_{n}).

If the latter is true for a cohomology class γ, we say that γ is transgressive.

Example 29. Let n = 1. Then the condition (22) is trivially satisfied. A class γ is just a trace functional γ : 𝔤 → ℂ. The class c_{γ}(P) ∈ H^{2}(B) can be obtained by choosing a connection ∇ in P with curvature R ∈ Ω_{B}^{2} ⊗ 𝔤 and taking the class of the closed 2-form γ(R) ∈ Ω_{B}^{2}. Alternatively, one can use γ to produce a trace functional γ_{P} : Ad(P) → C_{B}^{∞} and then use γ_{P} to push forward the extension (7) to a central extension of Lie algebroids

0 → C_{B}^{∞} → G → T_{B} → 0.

As well known (1.7) the set of isomorphism classes of such central extensions is identified with H_{Lie}^{2}(T_{B}, C_{B}^{∞}) = H^{2}(B, ℂ).

Example 30. Let n = 2, so γ is represented by a central extension

0 → ℂ → 𝔤̃ → 𝔤 → 0.  (25)

A sufficient condition for γ to be basic for any P is that 𝔤̃ can be made into a G-equivariant central extension, compare Example 17. Suppose that such an equivariant structure has been chosen. Then the class c_{γ}(P) ∈ H^{3}(B, ℂ) can be constructed as follows. We have the representation Ãd of G on 𝔤̃, and therefore an extension of associated vector bundles on B:

0 → C_{B}^{∞} → Ãd(P) → Ad(P) → 0.

Choose a connection ∇ in P. Then we have associated linear connections ∇_{Ad} in Ad(P) and ∇Ãd in Ãd(P). We also have the curvature R_{∇} ∈ Ω^{2}(B) ⊗ Ad(P). Choose a lifting R̃_{∇} of R_{∇} to Ω^{2}(B) ⊗ Ãd(P), and take

S = ∇Ãd(R̃_{∇}) ∈ Ω^{3}(B) ⊗ Ãd(P).

By the Bianchi identity ∇(R_{∇}) = 0 and so S lies in the tensor product of Ω^{3}(B) and the subbundle C_{B}^{∞} ⊂ Ãd(P)), i.e., it is a scalar differential form S ∈ Ω^{3}(B). Furthermore, it is clear that S is a closed 3-form. The class c_{γ}(P) is then the class of the form S. A different choice of an equivariant structure on 𝔤̃ leads to change of the class of S by an element from the image of d_{2}.

Example 31. Let G = GL_{N}(ℂ), so 𝔤 = 𝔤𝔩_{N}(ℂ). Then H^{•}(𝔤) is the exterior algebra on generators γ_{1}, ..., γ_{N} with γ_{i} ∈ H^{2i−1}(𝔤). A principal G-bundle P on B is the same as a rank N vector bundle E. In this case each 1 ⊗ γ_{i} is transgressive, and c_{γ_{i}}(P) is the image of c_{i}(E) ∈ H^{2i}(B) under the natural projection H^{2i}(B) → E_{n+1}^{0,n+1}. Here c_{i}(E) is the usual ith Chern class of E.

4.2 Other interpretations

Here we collect, for future use, some more or less straightforward reformulations of the construction of c_{γ}(P).

4.2.1 The Chern-Weil picture If we choose a connection ∇ in P, then the sequence (7) splits (such splitting is in fact the definition of a connection following Atiyah). So we can identify

Ω^{•}(P)^{G} = DR^{•}(A_{P}) = Ω_{B}^{•} ⊗ C^{•}(Ad(P)_{/B}) .  (26)

Let R be the curvature of ∇. Then the differential in the RHS of (26) has the form ∂ + ∇ + i_{R}, where ∂ is the differential in C^{•}(𝔤) and

i_{R} : Ω_{B}^{•} ⊗ C^{•}(𝔤) → Ω_{B}^{•+2} ⊗ C^{•−1}(𝔤)

is the contraction with R. This leads to a definition of c_{γ}(P) in terms of differential forms. Namely, we have an injective map of complexes followed by a surjective one:

Ω_{B}^{•} = Ω_{B}^{•} ⊗ C^{0}(𝔤) ↪^{ϕ} Ω_{B}^{•} ⊗ C^{•}(𝔤) ⟶^{ψ} Ω_{B}^{0} ⊗ C^{•}(𝔤).

Here, ψ is identified with the projection to gr_{F}^{0} , where F is the filtration from (21). If our class γ is basic, then it lifts uniquely to a class in H^{n}(Coker(ϕ)), so c_{γ}(P) is the image of that lifted class under the coboundary map corresponding to the short exact sequence

0 → Ω_{B}^{•} ↪^{ϕ} Ω_{B}^{•} ⊗ C^{•}(𝔤) → Coker(ϕ) → 0.

4.2.2 The differential graded picture Let 𝔄 denote the cone of the map i : Ad(P) → A_{P} viewed as a differential graded Lie algebroid. Thus A_{P} is put in degree 0, and Ad(P) in degree (−1). The anchor map α induces the quasi-isomorphism of Lie algebroids 𝔄 → T_{B}, hence the map of respective universal enveloping (differential graded) algebras U(𝔄) → U(T_{B}) = D_{B} (the latter concentrated in degree zero) which is a quasi-isomorphism. Define the map

DR^{•}(A_{P})/DR^{•}(T_{B}) ⟶^{δ} DR^{•+1}(𝔄)

as follows. For X ∈ Ad(P), denote by \underline{X} the element (X, 0) in the cone 𝔄 of i; for Y ∈ A_{P}, denote the element (0, Y) simply by Y. Given a p-cochain ω from DR^{•}(A_{P}), define the cochain δω by

δω(\underline{X}_{1}, …, \underline{X}_{q}, Y_{1}, …, Y_{r}) = ω(\underline{X}_{1}, Y_{1}, …, Y_{r})

for q = 1 and zero for q ≠ 1.

It is easy to see that the sequence

DR^{•}(A_{P})/DR^{•}(T_{B}) ⟶^{δ} DR^{•+1}(𝔄) ← DR^{•+1}(T_{B}) = Ω_{B}^{•+1}

represents the boundary map

H^{•}(DR^{•}(A_{P})/DR^{•}(T_{B})) → H^{•+1}(DR^{•}(T_{B})) = H^{•+1}(B).  (27)

A basic class γ as above defines an n-dimensional cohomology class γ̃ of DR^{•}(A_{P})/DR^{•}(T_{B}), and c_{γ}(P) is the image of γ̃ under (27).

4.2.3 The D-module picture Consider the short exact sequence

0 → C^{≥1}(Ad(P)_{/B}) → C^{•}(Ad(P)_{/B}) → C_{B}^{∞} → 0  (28)

coming from the fact that C_{B}^{∞} = C^{0}(Ad(P)_{/B}) is the 0th term of the relative cochain complex. If 𝔄 is as in (b), then all three complexes in (28) are graded U(𝔄)-modules in the following way. Elements Y = (0, Y), Y ∈ A, act via the adjoint action. Element \underline{X} = (X, 0), X ∈ Ad(P),acts by contraction, i.e. by substitution of X into a cochain. The action of U(𝔄) on C_{B}^{∞} is via the quasiisomorphism with D_{B}.

Note that (28) splits as a short exact sequence of complexes of vector bundles but not of U(𝔄)-modules. We will use the corresponding connecting morphism

δ : C_{B}^{∞} → C^{≥1}(Ad(P)_{/B})[1]

in D(U(𝔄)), the derived category of differential graded U(𝔄)-modules.

As 𝔄 is quasiisomorphic to T_{B}, the DG algebra U(𝔄) is quasiisomorphic to D_{B}, and the category D(U(𝔄)) is equivalent to D(D_{B}). Now recall (Corollary 11) that

H^{m}(B; ℂ) = Hom_{D(D_{B})}(C_{B}^{∞}, C_{B}^{∞}[m]) .

On the other hand, suppose that 𝔤 is such that H^{i}(𝔤) = 0 for 0 < i < n, see (3.1.11). Then H^{i}(Ad(P)_{/B}) = H^{i}(𝔤)⊗C_{B}^{∞} = 0 for 0 < i < n as well. In other words, the complex C^{≥1}(Ad(P)_{/B}) is acyclic in degrees < n and therefore each class ξ in its nth cohomology (which is isomorphic to H^{n}(𝔤) ⊗ C_{B}^{∞}) defines a morphism in the derived category of complexes of vector bundles

ξ̃: C^{≥1}(Ad(P)_{/B}) → C_{B}^{∞}[n] .

Furthermore, “constant” class ξ, i.e., a class of the form γ ⊗ 1, γ ∈ H^{n}(𝔤), defines in fact a morphism in the category D(U(𝔄)) ∼ D(D_{B}). Composing { \atop γ ⊗ 1} with δ, we get a morphism

C_{B}^{∞} → C_{B}^{∞}[n + 1] ,  (29)

i.e., a class in H^{n+1}(B; ℂ).

Proposition 32. The class in H^{n+1}(B; ℂ) corresponding to (29) is equal to c_{γ}(P).

Proof. This follows directly from the definitions (in fact, we could take (29) as the definition of c_{γ}(P)). Indeed, the morphism in the derived category from the cohomology of a quotient complex such as C_{B}^{∞} to the homology of a subcomplex such as C^{≥1}(Ad(P)_{/B}) acyclic up to degree n, is precisely the differential d_{n+1} in the corresponding spectral sequence. ⊓ ⊔

4.3 Infinite-dimensional groups

Slightly reformulating the approach of K.-T. Chen [8], we introduce the following definition.

Definition 33. A differentiable space is an ind-object in the category of C^{∞}- manifolds.

For background on ind-objects, see [10]. Thus a differentiable space M is a formal limit “ lim_{⟶}′′_{α∈A}M_{α} of (finite-dimensional) C^{∞}-manifolds. In particular, M defines a functor

S ↦ M(S) = C^{∞}(S, M) = lim_{⟶} C^{∞}(S, M_{α})  (30)

on such manifolds and can in fact be identified with this functor. In practice, however, we will identify M with the set M(pt) = lim_{⟶} M_{α} with (30) providing an additional structure on this set (description of what it means for an element of this set to vary in a smooth family).

For a differential space M we define (compare [8]) the space of p-forms (in particular, of C^{∞}-functions) on M by

Ω^{p}(M) = lim_{⟵} Ω^{p}(M_{α}) .

For a point m ∈ M(pt) the tangent space T_{m}M is defined by

T_{m}M = lim_{⟶} T_{s}S ,

where the limit is taken over C^{∞}-maps (S, s) → (M, m).

A differentiable group G is a group object in the category of differentiable spaces. For such a group the space 𝔤 = T_{e}G is a Lie algebra in the standard way.

Example 34 (Groups of diffeomorphisms). Let Σ_{0} be a compact oriented C^{∞}-manifold of dimension d. Then we have a differentiable group G = Diffeo(Σ_{0}) of orientation preserving diffeomorphisms. The corresponding functor (30) is as follows. A smooth map S → Diffeo(Σ_{0}) is a diffeomorphism of S × Σ_{0} preserving the projection to S. The Lie algebra of this group is Vect(Σ_{0}), the algebra of C^{∞} vector fields.

Example 35 (Gauge groups). Let Σ_{0} be as before and E_{0} be a C^{∞} complex vector bundle on Σ_{0} Then we have the differentiable group Aut(E_{0}) of C^{∞}- automorphisms of E_{0} (the differentiable structure defined as in Example 34). Its Lie algebra is End(E_{0}).

Example 36 (Atiyah groups). Let Σ_{0}, E_{0} be as before. The Atiyah group AT (Σ_{0}, E_{0}) consists of pairs (ϕ, f), where ϕ is an orientation preserving diffeomorphism of Σ_{0}, and f : ϕ^{∗}E_{0} → E_{0} is an isomorphism of vector bundles. Thus we have an extension of differentiable groups:

1 → Aut(E_{0}) → AT (Σ_{0}, E_{0}) → Diffeo(Σ_{0}) → 1.

The Lie algebra of AT (Σ_{0}, E_{0}) is A_{E_{0}}(Σ_{0}), the algebra of global C^{∞}-sections of the Atiyah Lie algebroid.

More generally, one can replace the vector bundle in Examples 35, 36 by a principal bundle with an arbitrary structure Lie group. In this paper we will be interested in the vector bundle case and will concentrate on the Example 36 as the most general.

Let us now describe a class of principal bundles with structure groups as in Example 36. Suppose that q : Σ → B is a smooth fibration with compact oriented fibers of dimension d. Suppose that B is connected. Then all the fibers Σ_{b} = q^{−1}(b), b ∈ B, are diffeomorphic to each other. Let Σ_{0} be one such fiber. Futher, let E be a smooth ℂ-vector bundle on Σ and E_{b} = E|_{Σ_{b}}. Then, for different b the pairs (Σ_{b}, E_{b}) are isomorphic, in particular, isomorphic to (Σ_{0}, E_{0}). Let G = AT (Σ_{0}, E_{0}). We have the principal G-bundle

ρ : P = P (Σ/B, E) → B  (31)

whose fiber P_{b} = ρ^{−1}(b), b ∈ B, consists of isomoprhisms of pairs (Σ_{0}, E_{0}) → (Σ_{b}, E_{b}).

For any differentiable G-bundle P over a finite-dimensional base B the Atiyah algebra A_{P} can be defined by (8). In the example where G = AT (Σ_{0}, E_{0}) and P = P (Σ/B, E), this gives

A_{P (Σ/B,E)} = q_{∗}A_{E}

(the sheaf-theoretic direct image of the Atiyah algebra of E).

4.4 The first Chern class

Let q : Σ → B and E be as before, so that we have a principal bundle P = P (Σ/B, E) → B with structure group G = AT (Σ_{0}, E_{0}). As the corresponding Lie algebra 𝔤 = A_{E_{0}}(Σ_{0}) consists of global sections of the Atiyah Lie algebroid of Σ_{0}, we have the embeddings

𝔤 ↪ D(E_{0}) ↪ 𝔤𝔩(D(E_{0})) .

By Corollary 23, 𝔤𝔩(D(E_{0})) has a unique continuous (in the Fréchet topology) cohomology class c in degree d + 1. We denote by γ the restriction of c to 𝔤.

Proposition 37.

1. There exists a Lie algebroid

0 → q_{∗}(𝔤𝔩(D_{Σ/B}(E))) → A_{Σ/B,E} ⟶^{α} T_{B} → 0

and a morphism (embedding) of Lie algebroids A_{P} → A_{Σ/B,E} which restricts to the embedding 𝔤 ↪ 𝔤𝔩(D(E_{0})).

2. The class 1 ⊗ γ is transgressive, so d_{d+2}(1 ⊗ γ) is defined.

Proof. The construction of A_{Σ/B,E} is given in 4.5 below.

The fibers of Ker(α) are Lie algebras isomorphic to 𝔤𝔩(D_{Σ_{0},E_{0}}) via an isomorphism defined uniquely up to an inner automorphism and thus satisfy the acyclicity condition (22) with n = d + 1. Therefore the class c is transgressive.

The Hochschild-Serre spectral sequence for A_{Σ/B,E} maps into the analogous spectral sequence for A_{P}. Since γ is the restriction of c, the naturality of the Hochschild-Serre spectral sequence implies that γ is transgressive. ⊓ ⊔

Definition 38. The first Chern class C_{1}(q_{∗}E) is defined by

C_{1}(q_{∗}E) := d_{d+2}(1 ⊗ γ) ∈ H^{d+2}(B, ℂ) ,

The class C_{1}(q_{∗}E) will be the main object of study in the rest of the paper.

4.5 Construction of A_{Σ/B,E}.

We start with the Atiyah Lie algebroid on Σ:

0 → End(E) →^{i} A_{E} ⟶^{α} T_{Σ} → 0 .

Let U(A_{E})_{/B} denote the centralizer of q^{−1}C_{B}^{∞} in U(A_{E}). Let F_{1}U(A_{E}) = {a|[a, q^{−1}C_{B}^{∞}] ⊆ q^{−1}C_{B}^{∞}}. Then, F_{1}U(A_{E}) is a Lie algebra under the commutator, U(A_{E})_{/B} is a Lie ideal in F_{1}U(A_{E}), and there is an exact sequence

0 → U(A_{E})_{/B} → F_{1}U(A_{E}) → q^{−1}T_{B} → 0  (32)

exhibiting F_{1}U(A_{E}) as a transitive q^{−1}C_{B}^{∞}-algebroid.

The inclusion A_{E} → D_{Σ}(E) induces the surjective map U(A_{E})_{/B} → D_{Σ/B,E} with kernel being the ideal generated by the relation which identifies 1 ∈ C_{Σ}^{∞} ⊂ U(A_{E})_{/B} with 1 ∈ \underline{End}_{C_{Σ}^{∞}}(E) ⊂ U(A_{E})_{/B}. The pushout of the exact sequence (32) by the map U(A_{E}) → D_{Σ}(E) gives the transitive Lie algebroid (the middle term in the exact sequence)

0 → D_{Σ/B,E} → F_{1}D_{Σ,E} → q^{−1}T_{B} → 0.  (33)

Replacing E by its tensor product by the trivial bundle of rank r in the above example, (33) can be rewritten as

0 → 𝔤𝔩_{r}(D_{Σ/B,E}) → F_{1}𝔤𝔩_{r}(D_{Σ,E}) → q^{−1}T_{B} → 0 .

Taking the limit over inclusions 𝔤𝔩_{r} → 𝔤𝔩_{r+1} we obtain a q^{−1}C_{B}^{∞}-algebroid

0 → 𝔤𝔩(D_{Σ/B,E}) → A_{q,E} → q^{−1}T_{B} → 0 .  (34)

Let 𝔄_{q,E} denote the cone of the inclusion 𝔤𝔩(D_{Σ/B,E}) → A_{q,E}. There are quasi-isomorphisms

𝔄_{q,E} → q^{−1}T_{B}, U_{q^{−1}C_{B}^{∞}}(𝔄_{q,E}) → q^{−1}D_{B} .

Taking the direct image of (34) under q and pulling back by the canonical map T_{B} → q_{∗}q^{−1}T_{B} we obtain the transitive (because R^{1}π_{∗}𝔤l_{r}(D_{Σ/B,E}) = 0) Lie algebroid on B:

0 → G → A_{Σ/B,E} → T_{B} → 0 ,

where G = q_{∗}𝔤𝔩(D_{Σ/B,E}), as we wanted. Let 𝔄_{Σ/B,E} denote the differential graded Lie algebroid on B equal to the cone of the inclusion G → A_{Σ/B,E}.

For any Lie algebra 𝔥, we denote by C_{+}(𝔥) the positive part of the Chevalley-Eilenberg complex, i.e. ⊕_{p>0}Λ^{p}𝔥 with the Chevalley-Eilenberg differential. There is an exact sequence of complexes

0 → C_{+}(𝔥) → C_{•}(𝔥) → C_{0}(𝔥) → 0

The exact sequence

0 → C_{+}(G) → C_{•}(G) → C_{0}(G) → 0

is, in fact, an exact sequence of differential graded U(𝔄_{Σ/B})-modules (this is a construction dual to (28)). Note that C_{0}(G) = C_{B}^{∞}. Let

δ_{Σ/B} : C_{B}^{∞} → C_{+}(G)[1]  (35)

denote the correponding morphism in the derived category of differential graded modules over the universal enveloping (differential graded) algebra U(𝔄_{Σ/B}).

4.6 Smooth cohomology and characteristic classes

A more traditional way of getting characteristic classes of principal G-bundle is by using group cohomology classes of G. Let us present a framework which we will then compare with the Lie algebra framework above.

Let S be a topological space and F be a sheaf of abelian groups on S. We denote by Φ^{•}(F) the standard Godement resolution of F by flabby sheaves. Thus Φ^{0}(F) = DS(F) is the sheaf of (possibly discontinuous) sections of the (étale space associated to) F, and Φ^{n+1}(F) = DS(Φ^{n}(F)). In this and the next sections we write RΓ (S, F) for the complex of global sections Γ (S, Φ^{•}(F)).

Let G be a differentiable group and B_{•}G be its classifying space. Thus B_{•}G = (B_{n}G)_{n≥0} is a simplicial object in the category of differentiable spaces with B_{n}G = G^{n}, and the face and degeneracy maps given by the standard formulas. We define the smooth cohomology of G with coefficients in ℂ^{∗} to be

H_{sm}^{n}(G, ℂ^{∗}) = ℍ^{n}(B_{•}G, C^{∞∗}) .

Here the hypercohomology on the right is defined as the cohomology of the double complex whose rows are the complexes RΓ (B_{n}G, C_{B_{n}G}^{∞∗}) and the differential between the neighboring slices coming from the simplicial structure on B_{•}G. This is a version of the Segal cohomology theory for topological groups ([13], p. 305). In particular, we have a spectral sequence

H^{i}(B_{n}G, ℂ^{∗}) ⇒ H_{sm}^{i+n}(G, ℂ^{∗}) .

We will use some other natural (complexes of) sheaves on B_{•}G to get natural cohomology theories for G. For example, the Deligne cohomology

H_{sm}^{n}(G, ℤ_{D}(p)) = ℍ^{n}(B_{•}G, ℤ_{D}(p)) ,

where for any differentiable space M we set

ℤ_{D}(p) = \{\underline{ℤ}_{M} → Ω_{M}^{0} → Ω_{m}^{1} → ... → Ω_{M}^{p−1}\} ,

with \underline{ℤ}_{M} placed in degree zero, compare [6].

Let B be a C^{∞}-manifold and U = {U_{i}}_{i∈I} be an open covering of B. We denote by N_{•}U the simplicial nerve of U, i.e., the simplicial manifold with

N_{n}U = ∐_{i_{0},...,i_{n}} U_{i_{0}} ∩ ... ∩ U_{i_{n}} .

For any sheaf F on B there is a natural isomorphism

ℍ^{i}(N_{•}U, F_{•}) = H^{i}(B, F) ,

where F_{•} is the natural sheaf on N_{•}U whose nth component is the sheaf on N_{n}U formed by the restrictions of F.

Let ρ : P → B be a principal G-bundle and suppose that P is trivial on each U_{i}. Then a collection of trivializations (i.e., sections) τ = (τ_{i} : U_{i} → P) gives a morphism of simplicial differentiable spaces

u_{τ} : N_{•}U → B_{•}G .

Given a class β ∈ H_{sm}^{n}(G, ℂ^{∗}), we define the characteristic class

𝔠_{β}(P) = u_{ϕ}^{∗}(β) ∈ H^{n}(B, C_{B}^{∞∗}) .  (36)

Similarly one can define characteristic classes corresponding to group cohomology classes with values in the Deligne cohomology.

4.7 Integrality and integrability

Let G be as in 4.6, and 𝔤 be the Lie algebra of G. We construct the “derivative” map

∂ : H_{sm}^{n}(G, ℂ^{∗}) → H_{Lie}^{n}(𝔤, ℂ) .  (37)

To do this, we first remark that for any topological space S, any sheaf of abelain groups F on S and any point s_{0} ∈ S we have a natural morphism of complexes

ϵ_{s_{0}} : RΓ (S, F) → F_{s_{0}} ,

where F_{s_{0}} is the stalk of F at s_{0}. To construct ϵ_{s_{0}}, we first project RΓ (S, F) = Γ (S, Φ^{•}(F)) to its 0th term Γ (S, Φ^{0}(F)) which, by definition, is the space of all sections ϕ = (s ↦ ϕ_{s}) of the étale space of F. Thus any such ϕ is a rule which to any point s ∈ S associates an element of F_{s}. We define ϵ_{s_{0}} by further mapping any ϕ as above to ϕ_{s_{0}} ∈ F_{s_{0}}.

We now specialize to S = B_{m}G = G^{m}, to s_{0} = e_{m} := (1, ..., 1) and to F = C_{S}^{∞∗}. We get a morphism from the double complex

{RΓ (B_{m}G, C_{B_{m}G}^{∞∗})}_{m≥0}  (38)

to the complex of stalks

ℂ^{∗} → C_{G,e_{1}}^{∞∗} → C_{G×G,e_{2}}^{∞∗} → …

Thus, an n-cocycle in (38) gives a germ of a smooth function

ξ = ξ(g_{1}, ..., g_{n}) : G^{n} → ℂ^{∗}

satisfying the group cocycle equation (on a neighborhood of e_{n+1} in G^{n+1}). Similarly to [13], p. 293, one associates to ξ a Lie algebra cocycle ∂(ξ) ∈ C^{n}(𝔤) by

∂(ξ)(x_{1}, ..., x_{n}) = \frac{d}{dt}Alt log ξ(exp(tx_{1}), ..., exp(tx_{n}))|_{t=0} .

A Lie algebra cohomology class γ ∈ H^{n}(𝔤, ℂ) will be called integrable if it lies in the image of the map ∂ from (37). Consider the exponential exact sequence (13) of sheaves on B and ts coboundary map δ_{n} from (14). The intuition with determinantal d-gerbes (1.4) suggests the following.

Conjecture 39.

1. The class γ ∈ H^{d+1}(A_{E_{0}}(Σ_{0})) constructed in 4.4 is integrable and comes from a natural class β ∈ H_{sm}^{d+1}(AT (Σ_{0}, E_{0}), ℂ^{∗}) (the “higher determinantal class”).

2. Furthermore, for any q : Σ → B and E as above, the class C_{1}(q_{∗}E) = c_{γ}(P) ∈ H^{d+2}(B, ℂ) is integral and is the image of the following class in the integral cohomology:

δ_{d+1}(𝔠_{β}(P)) ∈ H^{d+2}(B, ℤ) .

This conjecture holds for d = 1 (i.e., for the case of a circle fibration). We will verify this in Section 6. In general, the second statement seems to follow from the first by virtue of some compatibility result between group cohomology classes with coefficients in ℂ^{∗} and Lie algebra cohomology classes with coefficients in ℂ. Here we present a d = 1 version of such a result.

Let G be a differentiable group with Lie algebra 𝔤. Let β ∈ H_{sm}^{2}(G, ℂ^{∗}) and γ = ∂(β) ∈ H_{Lie}^{2}(𝔤, ℂ) be the derivative of β. Suppose β is represented by an extension of differentiable groups

1 → ℂ^{∗} → G̃ → G → 1 ,

whose Lie algebra is the extension (25) representing γ. Let ρ : P → B be a principal G-bundle over a C^{∞}-manifold B. Then we have the characteristic class c_{γ}(P) ∈ H^{3}(B, ℂ) (lifting to H^{3} well defined because 𝔤̃ is a G-module via the adjoint representation of G̃, see Example 30). On the other hand, β gives rise to a class 𝔠_{β}(P) ∈ H^{2}(B, C_{B}^{∞∗}), see (36).

Proposition 40. In the above situation c_{γ}(P) ∈ H^{3}(B, ℂ) is the image of δ_{2}(𝔠_{β}(P)) ∈ H^{3}(B, ℤ) under the natural homomorphism from the integral to the complex cohomology.

Proof. This follows from Theorem 19 using Example 30 and an obvious generalization of Example 18 to differentiable groups. ⊓ ⊔

Conjecture 41. We further conjecture the existence of the natural “deloopings” of the higher Chern classes as well, i.e., the existence of classes

β_{m} ∈ H_{sm}^{d+2m}(AT (Σ_{0}, E_{0}), ℤ_{D}(m)), m ≥ 1,  (39)

which then give characteristic classes in families:

C_{m}(q_{∗}E) ∈ H^{d+2m}(B, ℤ_{D}(m)).  (40)

5 The Real Riemann-Roch

Here is the main result of the present paper.

Theorem 42. Let q : Σ → B be a C^{∞} fibration with compact oriented fibers of dimension d. Let E be a complex C^{∞} vector bundle on Σ. Then:

C_{1}(q_{∗}E) = ∫_{Σ/B}[ch(E) · Td(T_{Σ/B})]_{2d+2} ∈ H^{d+2}(B, ℂ).

The proof consists of several steps.

5.1 A D-module interpretation of C_{1} using A_{Σ/B,E}.

We use the notation of 4.4 and introduce the following abbreviations:

G = q_{∗}(𝔤𝔩(D_{Σ/B}(E))) .

This is a bundle of infinite-dimensional Lie algebras on B.

𝔄 = 𝔄_{q,E} .

This is a DG Lie algebroid on Σ quasiisomorphic to q^{−1}T_{B}.

U𝔄 = U_{q^{−1}C_{B}^{∞}}(𝔄_{q,E}) .

This is a sheaf of DG-algebras on Σ quasiisomorphic to q^{−1}D_{B}.

Now, U𝔄 acts on C_{+}(𝔤𝔩(D_{Σ/B,E}))_{B}. Furthermore, a similar algebra acts on the Hochschild and cyclic complexes of D_{Σ/B,E}. Let 𝔄_{0} be the Lie algebroid defined exactly in the same way as 𝔄 but without tensoring by 𝔤𝔩. In the same spirit as in 4.2.3, elements Y = (0, Y), Y ∈ A_{q,E}, act via the adjoint action. Elements of the form \underline{X} = (X, 0) act via the shuffle multiplication

ι_{X}(a_{0} ⊗ ... ⊗ a_{p}) = ∑_{i=0}^{p}(−1)^{i}a_{0} ⊗ ... ⊗ a_{i} ⊗ X ⊗ a_{i+1} ⊗ ... ⊗ a_{p} .  (41)

Denoting by b, B the standard operators on Hoschshild chains, see [21], we have

[b, ι_{X}] = ad(X), [B, ι_{X}] = 0.

Therefore U𝔄_{0} acts on both the Hochschild and the cyclic complexes. This action extends to the completions described in 3.4. Furthermore, the morphisms μ_{D}, ν_{D} from Theorem 25 and Corollary 26 are in fact morphisms in D(U𝔄). Indeed, there is a spectral sequence

E_{2}^{pq} = Ext_{q^{−1}D_{B}}^{p}(\underline{H}^{q}({ \atop Hoch}_{•}(D_{Σ/B}(E))), C_{B}^{∞}) ⇒

Ext_{U𝔄_{0}}^{p+q}(({ \atop Hoch}_{•}(D_{Σ/B}(E)), C_{B}^{∞}) , (42)

and similarly for the cyclic complex. The action of q^{−1}D_{B} on \underline{H}^{q}({ \atop Hoch}_{•}(D_{Σ/B}(E))) is induced on the cohomology by the action of U𝔄 on { \atop Hoch}_{•}(D_{Σ/B}(E)). The map μ_{D} defines an element of E_{2}^{0d}, and E_{2}^{pq} = 0 for q < d, so μ_{D} gives rise to a well defined class in Ext^{d} on the RHS of (42). Similarly for ν_{D}.

We would like to compare the Lie algebra chain complex to the cyclic complex as modules over the algebras above. Roughly speaking, this comparison involves the embedding of 𝔄_{0} into 𝔄 induced by the embedding of differential operators into matrix-valued differential operators as diagonal matrices all of whose diagonal entries are the same. Unfortunately, these operators are not finite and therefore do not lie in 𝔤𝔩. This causes a minor technical difficulty that we are going to address next.

Let

C_{+}(𝔤𝔩(D_{Σ/B}(E)))_{B} ⟶^{β} CC_{•}(D_{Σ/B}(E))_{B}[1]

be the standard map from the Lie algebra chain complex to the cyclic complex, see [21], (10.2.3). Observe that this map factors into the composition

C_{+}(𝔤𝔩((D_{Σ/B}(E)))_{B} ⟶^{proj} (C_{+}(𝔤𝔩(D_{Σ/B}(E)))_{B})_{𝔤𝔩(ℂ)} → CC_{•}(D_{Σ/B}(E))_{B}[1]

(the complex in the middle is the complex of coinvariants). For each p the coinvariants stabilize: the projection

(C_{p}(𝔤𝔩_{N}(D_{Σ/B}(E)))_{B})_{𝔤𝔩_{N} (ℂ)} ⟶^{proj_{N}} (C_{p}(𝔤𝔩(D_{Σ/B}(E)))_{B})_{𝔤𝔩(ℂ)}

is an isomorphism for N > p. The DG Lie algebroid 𝔄_{0} acts on the complex of 𝔤𝔩_{N}-coinvariants via the diagonal embedding of D_{Σ/B}(E) into 𝔤𝔩_{N}(D_{Σ/B}(E)) for N big enough; this action is independent of N.

Let

α : (C_{+}(𝔤𝔩(D_{Σ/B}(E)))_{B})_{𝔤𝔩(ℂ)} → q^{−1}C_{B}^{∞}[2d]

denote the composition

(C_{+}(𝔤𝔩(D_{Σ/B}(E)))_{B})_{𝔤𝔩(ℂ)} ⟶^{β} CC_{•}(D_{Σ/B}(E))_{B}[1] →

ĈC_{•}(D_{Σ/B}(E))_{B}[1] ⟶^{ν_{D}[1]} q^{−1}C_{B}^{∞}[2d + 1] . (43)

It is checked directly that β commutes with the operators ι_{X}, so it is U𝔄_{0}-invariant. Therefore, all maps in (43) and the map α are morphisms in D(U𝔄_{0}).

Let us now take the direct image and define the morphism

∫_{Σ/B} α : (C_{+}(G)_{B})_{𝔤𝔩(ℂ)} → C_{B}^{∞}[d]

as the composition

(C_{+}(G)_{B})_{𝔤𝔩(ℂ)} → q_{∗}(C_{+}𝔤l(D_{Σ/B}(E))_{B})_{𝔤𝔩(ℂ)} ⟶^{∼}

Rq_{∗}(C_{+}𝔤l(D_{Σ/B}(E))_{B})_{𝔤𝔩(ℂ)} ⟶^{α} Rq_{∗}q^{−1}C_{B}^{∞}[2d + 1] \underrightarrow{_{∫_{Σ/B}}} C_{B}^{∞}[d + 1].

Here the last map is the integration over the relative (topological) fundamental class of Σ/B. Consider the composition

C_{B}^{∞} ⟶^{δ_{Σ/B}} (C_{+}(G)_{B})_{𝔤𝔩(ℂ)}[1] \underrightarrow{∫_{Σ/B} α} C_{B}^{∞}[d + 2] ,  (44)

where δ_{Σ/B} is as in (35). As both maps in (44) are morphisms in D(D_{B}), the composition (denote it C) is an element

C ∈ Ext_{D_{B}}^{d+2}(C_{B}^{∞}, C_{B}^{∞}) = H^{d+2}(B, ℂ).

Proposition 43. We have C = C_{1}(q_{∗}E).

Proof. This follows from the interpretation of C_{1}(q_{∗}E) = c_{γ}(P (Σ/B, E)) given in 4.2.2 and 4.2.3, and from the compatibility of the Atiyah algebroid of P (Σ/B) with A_{Σ/B,E}. ⊓ ⊔ 5.2 A local RRR in the total space

Proposition 43 reduces the RRR to the following “local” statement taking place in the total space Σ.

Theorem 44. Let ξ be the morphism in D(q^{−1}D_{B}) defined as the composition

q^{−1}C_{B}^{∞} → C_{+}(𝔤𝔩(D_{Σ/B,E}))_{𝔤𝔩(ℂ)}[1] → q^{−1}C_{B}^{∞}[2d + 2] .

Then the class in

Ext_{q^{−1}D_{B}}^{2d+2}(q^{−1}C_{B}^{∞}, q^{−1}C_{B}^{∞}) = H^{2d+2}(Σ, ℂ)

corresponding to ξ is equal to

[ch(E) · Td(T_{Σ/B})]_{2d+2} .

We now concentrate on the proof of Theorem 44. First, we remind the definition of periodic cyclic homology [21]. Let A be an associative algebra. The “negative” cyclic complex of A is defined, similarly to (16), as

CC_{•}^{−}(A) = Tot{Hoch_{•}(A) ⟶^{N} Hoch_{•}(A) ⟶^{1−τ} Hoch_{•}(A) → …}

Here, the grading of the copies of Hoch_{•}(A) in the horizontal direction goes in increasing integers 0,1,2 etc. So CC_{•}^{−}(A) s a module over the formal Taylor series ring ℂ[[u]] where u has degree (−2). The original cyclic complex is a module over the polynomial ring ℂ[u^{−1}]. Finally, the periodic cyclic complex CC_{•}^{per}(A) is obtained by merging together CC_{•}(A) and CC_{•}^{−}(A) into one double complex which is repeated 2-periodically both in the positive and negative horizontal directions. In other words,

CC_{•}^{per}(A) = CC_{•}^{−}(A) ⊗_{ℂ[[u]]} ℂ((u)).

We extend these construction to other situations (see Section 3) where the tensor products are understood in the sense of various completions. In particular, the morphism ν_{D} of 26 extends to morphisms

ν_{D}^{−} : CC_{•}^{−}(D_{Σ/B,E}) → q^{−1}C_{B}^{∞}[2d][[u]]
ν_{D}^{per} : CC_{•}^{per}(D_{Σ/B,E}) → q^{−1}C_{B}^{∞}[2d]((u))

These morphisms include into the commutative diagram

CC_{•}^{−}(D_{Σ/B,E}) ⟶ CC_{•}^{per}(D_{Σ/B,E}) ⟶ CC(D_{Σ/S,E})[2]
ν_{D}^{−}↓  ^{ν_{D}^{per}} ↓  ↓^{ν_{D}}
C_{B}^{∞}[2d][[u]] ⟶ C_{B}^{∞}[2d]((u)) ⟶^{Res_{u=0}} C_{B}^{∞}[2d + 2]

We now want to reduce Theorem 44 to the following statement.

Theorem 45. The composition

C_{B}^{∞} ⟶^{1} CC_{•}^{per}(D_{Σ/B,E}) ⟶^{ν_{D}^{per}} C_{B}^{∞}[2d]((u))

defines an element of Ext_{q^{−1}D_{B}}^{•}(q^{−1}C_{B}^{∞}, q^{−1}C_{B}^{∞}[2d])((u)) which is equal to

∑_{i=0}^{∞} u^{i} · [ch(E)Td(T_{Σ/B})]_{2(d−i)} .

Proof (Theorem 44). Assuming Theorem 45 it is sufficient to prove that the composition

q^{−1}C_{B}^{∞} → C_{+}(𝔤𝔩(D_{Σ/B,E}))_{𝔤𝔩(ℂ)}[1] → CC_{•}(D_{Σ/B,E})[2]

is equal to the composition

q^{−1}C_{B}^{∞} ⟶^{1} CC_{•}^{per}(D_{Σ/B,E}) → CC_{•}(D_{Σ/B,E})[2] ,

as the latter one is related to Chern and Todd via Theorem 45. In order to perform the comparison, let K be the cone of the inclusion C_{+}(𝔤𝔩(D_{Σ/B,E}) → C_{•}(𝔤𝔩(D_{Σ/B,E})), so that we have a quasi-isomorphism K → q^{−1}C_{B}^{∞} as well as an isomorphism of distinguished triangles

C_{•}(𝔤𝔩(D_{Σ/B,E}))_{𝔤𝔩(ℂ)} ⟶ K ⟶ C_{+}(𝔤𝔩(D_{Σ/B,E}))_{𝔤𝔩(ℂ)}[1]
↓  ↓  ↓
C(𝔤𝔩(D_{Σ/B,E}))_{𝔤𝔩(ℂ)} ⟶ q^{−1}C_{B}^{∞} ⟶ C_{+}(𝔤𝔩(D_{Σ/B,E}))[1]_{𝔤𝔩(ℂ)}

(with the top row a short exact sequence of complexes). Note, that there is a morphism of distinguished triangles

C_{•}(𝔤𝔩(D_{Σ/B,E}))_{𝔤𝔩(ℂ)} ⟶  K  ⟶ C_{+}(𝔤𝔩(D_{Σ/B,E}))_{𝔤𝔩(ℂ)}[1]
↓  ↓  ↓
CC_{•}^{−}(D_{Σ/B,E}) ⟶ CC_{•}^{per}(D_{Σ/B,E}) ⟶ CC_{•}(D_{Σ/B,E})[2]

It remains to notice further that the diagram

q^{−1}C_{B}^{∞} ⟵ K ⟶ CC_{•}^{per}(D_{Σ/B,E})

represents the morphism C_{B}^{∞} ⟶^{1} CC_{•}^{per}(D_{Σ/B,E}) in the derived category. ⊓ ⊔ 5.3 Proof of Theorem 45

This statement can be deduced from the results of [25] on the cohomology of the Lie algebras of formal vector fields and formal matrix functions. We recall the setting of [25] which extends that of the Chern-Weil definition of characteristic classes. Recall that the latter provides a map

S^{•}[[𝔥_{0}]]^{H_{0}} → H^{2•}(Σ, ℂ) ,  (45)

where H_{0} = GL_{d}(ℂ) × GL_{r}(ℂ) with r = rk(E), while 𝔥_{0} is the Lie algebra of H_{0}, i.e., 𝔤𝔩_{d}(ℂ) ⊕ 𝔤𝔩_{r}(ℂ). To be precise, the elementary symmetric functions of the two copies of 𝔤𝔩 are mapped to the Chern classes of T_{Σ/B} and E.

In [25], this construction was generalized in the following way. Let k = dim(B), and 𝔤̂ be the Lie algebra of formal differential operators of the form

∑_{i=1}^{k} P_{i}(y_{1}, …, y_{k})\frac{∂}{∂y_{i}} +

∑_{j=1}^{d} Q_{j}(x_{1}, …, x_{d}, y_{1}, …, y_{k})\frac{∂}{∂x_{i}} + R(x_{1}, …, x_{d}, y_{1}, …, y_{k})

where P_{i}, Q_{j} are formal power series, and R(x) is an r×r matrix whose entries are power series. Thus 𝔤̂ is the formal version of the relative Atiyah algebra. Consider the Lie subalgebra 𝔥 of fields such that all P_{i} and Q_{j} are of degree one and all entries of R are of degree zero. We can identify this subalgebra with

𝔥 = 𝔤𝔩_{d}(ℂ) ⊕ 𝔤𝔩_{k}(ℂ) ⊕ 𝔤𝔩_{r}(ℂ)

Let

H = GL_{d}(ℂ) × GL_{k}(ℂ) × GL_{r}(ℂ)

be the corresponding Lie group. Thus (𝔤̂, H) form a Harish-Chandra pair. Following the ideas of “formal geometry” (or “localization”) of Gelfand and Kazhdan, one sees that every (𝔤̂, H)-module L induces a sheaf L on Σ. Similarly, a complex L^{•} of modules gives rise to a complex of sheaves L^{•}. A complex L^{•} of modules is called homotopy constant if the action of 𝔤̂ extends to an action of the differential graded Lie algebra (𝔤̂[ϵ], \frac{∂}{∂ϵ}). Here ϵ is a formal variable of degree −1 and square zero. In this case, there is a generalization of the Chern-Weil map constructed in [25]:

CW : ℍ^{•}(𝔥_{0}[ϵ], 𝔥_{0}; L^{•}) → ℍ^{•}(Σ, L^{•})

which gives (45) when L = ℂ with the trivial action. Consider the following (𝔤̂, H)-modules:

D = {∑ P_{α}(x_{1}, …, x_{d}, y_{1}, …, y_{k})∂_{x}^{α}},

where P_{α} are r × r matrices whose entries are power series, and

Ω^{•} = {∑_{I} P_{I}(x_{1}, …, x_{d}, y_{1}, …, y_{k})d^{I}x},

which is the space of differential forms whose coefficients are formal power series. The latter is a complex with the (fiberwise) De Rham differential. Moreover, Ω^{•} is homotopy constant (ϵ𝔤̂ acts on it by exterior multiplication). The Hochschild, cyclic, etc. complexes of D inherit the (𝔤̂, H)-module structure; moreover, they also become homotopy constant (an element ϵX ∈ ϵ𝔤̂ act by the operator ι_{X} from (4.2.4)). One constructs ([4], pt. II, Lemma 3.2.4) a class

ν ∈ ℍ^{0}(𝔥_{0}[ϵ], 𝔥_{0}; \underline{Hom}(CC_{−•}^{per}(D), Ω^{2d+•}))

such that CW(ν) coincides with

ν_{D} ∈ ℍ^{0}(Σ; \underline{Hom}(CC_{−•}^{per}(D_{Σ/B}), Ω_{Σ/B}^{2d+•})) .

To be precise, the cited lemma concerns the Weyl algebra of power series in both coordinates and derivations with the Moyal product (clearly, differential operators of finite order form a subalgebra). Second, the construction there is for the relative cohomology of the pair (𝔤, 𝔥) but it extends to the case of the pair (𝔤[ϵ], 𝔥) of which (𝔥_{0}[ϵ], 𝔥_{0}) is a sub-pair.

The cochain ν is actually independent of y. There is the canonical class 1 in HC_{0}^{per}(D); it is 𝔥_{0}-invariant, and it is shown in [25] how to extend it to a class in ℍ^{0}(𝔥_{0}[ϵ], 𝔥_{0}; CC_{−•}^{per}(D)). On the other hand,

ℍ^{0}(𝔥_{0}[ϵ], 𝔥_{0}; Ω^{•})

can be naturally identified with

ℍ^{0}(𝔥_{0}[ϵ], 𝔥_{0}; ℂ)

It remains to show that

ν(1) = ∑[ch · Td]_{2(d+i)} · u^{i}

where ch is the corresponding invariant power series in H^{•}(𝔤𝔩_{r}[ϵ], 𝔤𝔩_{r}; ℂ) and Td is the corresponding invariant power series in H^{•}(𝔤𝔩_{d}[ϵ], 𝔤𝔩_{d}; ℂ). This was carried out in [4], Lemma 5.3.2. ⊓ ⊔

6 Comparison with the gerbe picture

6.1 L^{2}-sections of a vector bundle on a circle.

Let Σ be an oriented C^{∞}-manifold diffeomorphic to the circle S^{1} with the standard orientation, and let E be a complex C^{∞}-vector bundle on Σ. Choose a smooth Riemannian metric g on Σ and a smooth Hermitian metric h on E. Let Γ (Σ, E) be the space of C^{∞}-sections of E. The choise of g, h defines a positive definite scalar product on this space and we denote by L_{g,h}^{2}(Σ, E) the Hilbert space obtained by completion with respect to this scalar product.

Lemma 46. For a different choice g′, h′ of metrics on Σ, E we have a canonical identification of topological vector spaces

L_{g,h}^{2}(Σ, E) → L_{g′,h′}^{2}(Σ, E).

Proof. The Hilbert norms on Γ (Σ, E) associated to (g, h) and (g′, h′) are equivalent, since Σ is compact. ⊓ ⊔

We will denote the completion simply by L^{2}(Σ, E).

Consider now the case when Σ = S^{1} is the standard circle and E = ℂ^{r} is the trivial bundle of rank r. In this case L^{2}(Σ, E) = L^{2}(S^{1})^{⊕r}. Let us denote this Hilbert space by H. It comes with a polarization in the sense of Pressley and Segal [26]. In other words, H is decomposed as H_{+} ⊕ H_{−} where H_{+}, H_{−} are infinite-dimensional ortogonal closed subspaces defined as follows.

H_{+} consists of vector functions extending holomorphically into the unit disk D_{+} = {|z| < 1}. The space H_{−} consists of vector functions extending holomorphically into the opposite annulus D_{−} = {|z| > 1} and vanishing at ∞.

The decomposition H = H_{+} ⊕ H_{−} yields the groups GL_{res}(H) ⊂ GL(H), see [26] (6.2.1), as well as the Sato Grassmannian Gr(H) on which GL_{res}(H) acts transitively. We recall that Gr(H) consists of closed subspaces W ⊂ H whose projection to H_{+} is a Fredholm operator and the projection to H_{−} is a Hilbert-Schmidt operator, see [26] (7.1.1).

Given arbitrary Σ, E as before, we can choose an orientation preserving diffeomorphism ϕ : S^{1} → Σ and a trivialization ψ : ϕ^{∗}E → ℂ^{r}. This gives an identification

u_{ϕ,ψ} : L^{2}(Σ, E) → H = L^{2}(S^{1})^{⊕r}.

In particular, we get a distinguished set of subspaces in L^{2}(Σ, E), namely

Gr_{ϕ,ψ}(Σ, E) = u_{ϕ,ψ}^{−1}(Gr(H)),

and a distinguished subgroup of its automorphisms, namely

GL_{res}^{ϕ,ψ}(L^{2}(Σ, E)) = u_{ϕ,ψ}^{−1}GL_{res}(H)u_{ϕ,ψ}.

Lemma 47. The subgroup GL_{res}^{ϕ,ψ}(L^{2}(Σ, E)) and the set Gr_{ϕ,ψ}(L^{2}(Σ, E)) are independent of the choice of ϕ and ψ.

Proof. Any two choices of ϕ, ψ differ by an element of the Atiyah group AT (S^{1}, ℂ^{r}), see Example 36. This group being a semidirect product of Diffeo(S^{1}) and GL_{r}C^{∞}(S^{1}), our statement follows from the known fact that both of these groups are subgroups of GL_{res}(H), see [26]. ⊓ ⊔ We will drop ϕ, ψ from the notation, writing Gr(L^{2}(Σ, E)) and GL_{res}(L^{2}(Σ, E)). Recall further that Gr(H) × Gr(H) is equipped with a line bundle Δ (the

relative determinantal bundle) which has the following additional structures:

1. Equivariance with respect to GL_{res}(H).

2. A multiplicative structure, i.e., an identification

p_{12}^{∗}Δ ⊗ p_{23}^{∗}Δ → p_{13}^{∗}Δ  (46)

of vector bundles on Gr(H)×Gr(H)×Gr(H), which is equivariant under GL_{res}(H) and satisfies the associatiivity, unit and inversion properties.

It follows from the above that we have a canonically defined line bundle (still denoted Δ) on Gr(L^{2}(Σ, E)) × Gr(L^{2}(Σ, E)) equivariant under GL_{res}(L^{2}(Σ, E)) and equipped with a multiplicative structure. For W, W′ ∈ Gr(L^{2}(Σ, E)) we denote by Δ_{W,W′} the fiber of Δ at (W, W′).

As is well known, the multiplicative bundle Δ gives rise to a category (ℂ^{∗}-gerbe) Det L^{2}(Σ, E) whose set of objects is Gr(L^{2}(Σ, E)), while

Hom_{Det L^{2}(Σ,E)}(W, W′) = Δ_{W,W′} − {0} .

The composition of morphisms comes from the identification

Δ_{W,W′} ⊗ Δ_{W′,W′′} → Δ_{W,W′′}

given by (46).

6.2 L^{2}-direct image in a circle fibration.

Let now q : Σ → B be a fibration in oriented circles and E be a vector bundle on Σ. We have then a bundle of Hilbert spaces q_{∗}^{L^{2}}(E) whose fiber at b ∈ B is L^{2}(Σ_{b}, E_{b}). Furthermore, by Lemma 47 this bundle has a GL_{res}(H)- structure, where H = L^{2}(S^{1})^{⊕r}. Therefore we have the associated bundle of Sato Grassmannians Gr(q_{∗}^{L^{2}}(E)) on B and the (fiberwise) multiplicative line bundle Δ on

Gr(q_{∗}^{L^{2}}(E)) ×_{B} Gr(q_{∗}^{L^{2}}(E)).

We define a sheaf of C_{B}^{∞∗}-groupoids on B whose local objects are local sections of Gr(q_{∗}^{L^{2}}(E)) and for any two such sections defined on U ⊂ B

\underline{Hom}(s_{1}, s_{2}) = (s_{1}, s_{2})^{∗}Δ − 0_{U},

where 0_{U} stands for the zero section of the induced line bundle. This sheaf of groupoids is locally connected and so gives rise to a C_{B}^{∞∗}-gerbe which we denote Det(q_{∗}E). So we have the class

[Det(q_{∗}e)] ∈ H^{2}(B, C_{B}^{∞∗}).

Alternatively, consider the Atiyah group G = AT (S^{1}, ℂ^{r}), see Example 36. By the above, G ⊂ GL_{res}(H). The determinantal ℂ^{∗}-gerbe Det(H) (over a point) with G-action gives a central extension G̃ of G by ℂ^{∗}. A circle fibration q : Σ → B gives a principal G-bundlle P (Σ/B), as in (31), and the following is clear.

Proposition 48. The gerbe Det(q_{∗}E) is equivalent to LiftG̃ ; G(P (Σ/B, E)), see Example 18.

Consider the exponential sequence (13) of sheaves on B and the corresponding coboundary map δ_{2}, see (14). Then we have the class

δ_{2}[Det(q_{∗}E)] ∈ H^{3}(B, ℤ) .

Theorem 49. The image of δ[Det(q_{∗}E)] in H^{3}(B, ℂ) coincides with negative of the class C_{1}(q_{∗}E) (see Definition 38).

Proof. We apply Proposition 40 to G = AT (S^{1}, ℂ^{r}) and β being the class of the central extension G̃. Then 𝔤 = A_{ℂ^{r}}(S^{1}) is the Atiyah algebra of the trivial bundle on S^{1} and γ is the class of the “trace” central extension induced from the Lie algebra 𝔤𝔩_{res}(H) of GL_{res}(H). We have the embeddings

𝔤 ⊂ 𝔤𝔩_{r}(D(S^{1})) ⊂ 𝔤𝔩_{res}(H),

and the trace central extension is represented by an explicit cocycle Ψ of 𝔤𝔩_{res}(H) (going back to [28]). Let z be the standard complex coordinate on S^{1} such that |z| = 1. Then the formula for the restriction of Ψ to 𝔤𝔩_{r}(D(S^{1})) was given in [15], see also [16], formula (1.5.2):

Ψ(f(z)∂_{z}^{m}, g(z)∂_{z}^{n}) = \frac{m!n!}{(m + n + 1)!}Res_{z=0}dz · Tr(f^{(n+1}(z)g^{(m)}(z)) ,

where f^{(n)} denotes the nth derivative with respect to z. Our statement now reduces to the following lemma. ⊓ ⊔

Lemma 50. The second Lie cohomology class of 𝔤𝔩_{r}D(S^{1}) given by the cocycle Ψ is equal to the negative of the class corresponding to the fundamental class of S^{1} via the identification of Corollary 23.

Proof. As the space of (continuous) Lie algebra homology in question is 1dimensional, it is enough to evaluate the cocycle Ψ on the Lie algebra 2homology class σ from 23 and to show that this value is precisely equal to 1. For this it is enough to consider r = 1. Let D = D(S^{1}) for simplicity.

We need to recall the explicit form of the identification (18) for the case n = 1 (first Hochschild homology maps to the second Lie algebra homology). In other words, we need to recall the definition of the map.

ϵ : HH_{1}(D) → H_{2}^{Lie}(𝔤𝔩(D)) → ℂ .

As explained in B̧G and [29], this map is defined via the order filtration F on the ring D and uses the corresponding spectral sequence. This means we need to start with a Hochschild 1-cycle σ = ∑ P_{i} ⊗ Q_{i} ∈ D ⊗ D and form its highest symbol cycle

Smbl(σ) = ∑ Smbl(P_{i}) ⊗ Smbl(Q_{i}) ∈ gr(D) ⊗ gr(D) ,

which gives an element in Hoch_{1}(gr(D)). As gr(D) is the ring of polynomial functions on T^{∗}S^{1}, Hochschild-Kostant-Rosenberg gives HH_{1}(gr(D)) = Ω^{1}(T^{∗}S^{1}), the space of 1-forms on T^{∗}S^{1} polynomial along the fibers. So the class of Smbl(σ) is a 1-form ω = ω(σ) on T^{∗}S^{1}. This is an element of the E_{1}-term of the spectral sequence for the Hochschild homology of the filtered ring D.

Furthermore, one denotes by ∗ the symplectic Hodge operator on forms on T^{∗}S^{1}. The results of loc. cit. imply the differential in the E_{1}-term is ∗d∗ where d is the de Rham differential on T^{∗}S^{1} while higher differentials vanish. This means that under our assumptions ∗ω(σ) is a closed 1-form and

ϵ(σ) = ∫_{S^{1}} ∗ω(σ) .

To finish the proof we need to exhibit just one σ as above such that

0 ≠ ϵ(σ) = Ψ(σ) := ∑ Ψ(P_{i}, Q_{i}) .

We take

σ = z^{2} ⊗ z^{−1}∂_{z} − 2z ⊗ ∂_{z} .

Then one sees that σ is a Hochschild 1-cycle and Ψ(σ) = 1. On the other hand, let θ be the real coordinate on S^{1} so that z = exp(2πiθ). Then the real coordinates on T^{∗}S^{1} are θ, ξ with ξ = Smbl(∂/∂θ), so the Poisson bracket {θ, ξ} is equal to 1. In terms of the coordinate z it means that ξ = Smbl(z∂/∂z) and {z, ξ} = z. Therefore

Smbl(σ) = z^{2} ⊗ z^{−2}ξ − 2z ⊗ z^{−1}ξ

and, hence,

ω(σ) = z^{2}d(z^{−2}ξ) = 2zd(z^{−1}ξ) = −dz − z^{−1}ξ ,

see [21] p.11. The symplectic (volume) form on T^{∗}S^{1} is (dz/z) ∧ dξ, so the symplectic Hodge operator is given by

∗dξ = dz/z, ∗dz/z = dξ, ∗^{2} = 1 .

Therefore,

∗ω(σ) = −dz/z − ξdξ, ∫_{S^{1}} ∗ω(σ) = −1

and we are done. ⊓ ⊔

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