A General Families Index Theorem

James L. Heitsch ^{∗}

Department of Mathematics, Statistics and Computer Science University of Illinois at Chicago 851 S. Morgan Street (m/c 249) Chicago, Illinois 60607-7045 e-mail: heitsch@math.uic.edu and Connor Lazarov Department of Mathematics Herbert Lehman College, CUNY Bronx, New York 10468 e-mail: lazarov@lcvax.lehman.cuny.edu

Abstract

We consider the heat operator of a Bismut superconnection for a family of generalized Dirac operators defined along the leaves of a foliation with Hausdorff graph. We assume that the strong Novikov-Shubin invariants of the Dirac operators are greater than three times the codimension of the foliation. We compute the t → ∞ asymptotics associated to a rescaling of the metric by 1/t and show that the heat operator converges to the Chern character of the index bundle of the operator. Combined with previous results, this gives a general families index theorem for such operators.

∗Partially supported by NSF Grant DMS 9400676

1 Introduction

In this paper, we give for the first time general conditions which guarantee that the Bismut superconnection formalism extends in full generality to families of non-compact manifolds. This allows us to prove a families index theorem for generalized Dirac operators defined along such families. Special cases of our result give the Atiyah-Singer families index theorem, the Atiyah L^{2} index theorem [A76] and the Connes foliation cohomology index theorem [Con86]. It is also clear that the results here extend easily to the situation considered by Gong and Rothenberg in [GR]. In addition to known results, we get a new theorem for fiber bundles which is a combination of the first two above, namely an L^{2} families index theorem. Finally, our proof makes precise the dependence of the heat equation method on the small eigen value behavior of the Dirac operator under consideration.

The reader should note that Connes and Skandalis have proven a families index theorem for families of elliptic operators defined along the leaves of a foliation. See [CS84] and [Con87]. Their proof is in the spirit of the original proof of Atiyah-Singer, i.e. K-theory and operator algebras. It is a difficult and important problem to relate their results to the results presented here.

In [AS68] and [AS71], Atiyah and Singer proved their celebrated index theorems for elliptic operators and elliptic families respectively. Their proofs involved analysis of pseudodifferential operators and K-theory. A heat equation proof of the index theorem for Dirac operators was given by Atiyah, Bott and Patodi in [ABP73]. Their result extended previous results of Patodi [Pat71] and Gilkey [Gil73], [Gil79]. In [Q], Quillen conjectured that a heat equation proof of the families index theorem should be possible using superconnections. In [Bis86], Bismut constructed a remarkable superconnection for families of Dirac operators and gave two heat equation proofs of the families index theorem. See also [BV87] and [Don88].

In [Hei95] we showed how to define a Bismut superconnection for a family of generalized Dirac operators defined along the leaves of a foliation F of a compact manifold M with Hausdorff graph. Using the heat operator of the curvature of the superconnection, we defined a (non-normalized) Chern character for the family which lies in the Haefliger cohomology of the foliation. Rescaling the metric on M by 1/t and letting t → 0 gave the analog of the classical cohomological formula for the index of a family of Dirac operators.

In this paper we extend the results of [Hei95] by showing that the t → ∞ asymptotics give the Chern character of the index bundle of the family of generalized Dirac operators, i.e. we have a families index theorem for these operators. In order to do this, we need to assume that the spectral projections for small intervals about zero of the family of Dirac operators are reasonably well behaved. Our main assumption is that the strong Novikov-Shubin invariants of the Dirac operators are greater than three times the codimension of the foliation. This theorem is highly significant because it gives the first extension of results for compact families, where one must assume that zero is uniformly isolated in the spectrum, to a result for non-compact families where zero is not assumed to be isolated. We expect that further results are available. In particular, we conjecture that our families index theorem is still true provided one assumes only that the strong Novikov-Shubin invariants are positive and that the spectral projections for small intervals about zero are transversely smooth.

Some of the material in this paper is also in [HL]. We repeat it here so that this paper is self contained. In addition, things here are both simpler and more complicated than in the torsion case. The simplifications occur because we work with the pure exponential operator e^{−B_{t}^{2}} instead of the operator D_{t}e^{D_{t}^{2}}. The complications occur because B_{t} involves differentiation transverse to the foliation, while D_{t} is a pure fiberwise operator and so involves no transverse differentiation.

Finally, part of this work was done while the first author was visiting the Institut des Hautes Etudes Scientifique and the University of Paris VII. He would like to thank them both for their warm hospitality, as well as J.-M. Bismut for helpful conversations.

2 Bismut Superconnections for Foliations

In this section we recall the relevant results of [Hei95].

Let M be a smooth n dimensional Riemannian manifold and F a smooth p = 2k dimensional foliation on M of codimension q. Assume that the tangent bundle T F of F is oriented and spin with a fixed spin structure. Denote the normal and dual normal bundles of F by ν and ν^{∗} respectively. Let E be a Hermitian vector bundle over M with Hermitian connection.

The holonomy graph G of F consists of equivalence classes of paths γ : [0, 1] → M such that the image of γ is contained in a leaf of F. Two such paths are equivalent if they start and end at the same points and the holonomy along them is the same. There are two natural maps s, r : G → M given by s([γ]) = γ(0), r([γ]) = γ(1). G has the natural foliation F_{s} whose leaves are L̃_{x} ≡ s^{−1}(x), for x ∈ M. The normal bundle ν_{s} of F_{s} is canonically equivalent to s^{∗}T M. The metric g_{0} on M induces a canonical metric g_{0} on G as follows. T G = T F_{s} ⊕ s^{∗}T M and these bundles are mutually orthogonal. On s^{∗}T M define g_{0} to be s^{∗}g_{0} and on T F_{s} define g_{0} to be r^{∗}(g_{0}|T F). Note that r : L̃_{x} → L_{x} is the holonomy cover of L_{x}, the leaf of F containing x, and is a local isometry.

The dual normal bundle ν_{s}^{∗} of F_{s} is s^{∗}T^{∗}M.

Using the map r, we have the Hermitian bundle E ≡ r^{∗}E over G with Hermitian connection. Likewise, under r, the spin structure on T F induces one on T F_{s}.

As all the structures considered on G come from ones on the compact manifold M, they all have uniformly bounded geometry.

The homotopy graph H of F is the covering of G which consists of equivalence classes of leafwise paths where two such paths are equivalent if they are homotopy equivalent in their leaf, relative to their end points. The homotopy graph has essentially the same properties as the holonomy graph.

In order to use the results of [Hei95] we will assume that G or H is Hausdorff. Riemannian foliations and foliations constructed from the suspension of analytic group actions always have Hausdorff holonomy graphs. Note that if G is Hausdorff, then H is also, but the converse is not true. For example the suspension of a non identity diffeomorphism of S^{1} which is the identity on an open set has non-Hausdorff holonomy graph, but Hausdorff homotopy graph. Note also that the results of [Hei95] are still true if G is replaced by H. We will work here with a Hausdorff G with the understanding that all results are also true for any Hausdorff H.

Denote the bundle of spinors along the leaves of F_{s} by S_{0}. The tensor product connection on S_{0} ⊗ E|L̃_{x} determined by the Levi-Civita connection on S_{0} and the Hermitian connection on E determines a smooth family D_{E} = {D_{x} ⊗ E} of Dirac operators, where D_{x} ⊗ E acts on sections of S_{0} ⊗ E|L̃_{x}. Thus D_{E} is a smooth family of Dirac operators on G.

Recall that a Bismut superconnection B for F and E is a Dirac type operator on Γ(∧ν_{s}^{∗} ⊗ S_{0} ⊗ E), (see [Hei95] for the details of its construction), and that B satisfies the usual properties (suitably interpreted) for a Bismut superconnection. In particular, if ω ∈ Γ(∧T^{∗}M) and ϕ ∈ Γ(∧ν_{s}^{∗} ⊗ S_{0} ⊗ E), then

B(s^{∗}ω · ϕ) = d(s^{∗}ω) · ϕ + (−1)^{deg ω}s^{∗}ω · B(ϕ). (2.1)

We may use the Z grading on ∧ν_{s}^{∗} to grade B writing

B = B_{0} + B_{1} + B_{2} + ⋯

where B_{i} : Γ(∧^{k}ν_{s}^{∗} ⊗ S_{0} ⊗ E) ⟶ Γ(∧^{k+i}ν_{s}^{∗} ⊗ S_{0} ⊗ E). Note that B_{0} = D_{E}. For i ≠ 1, B_{i} commutes with the left action of ∧ν_{s}^{∗} = s^{∗}(∧T^{∗}M), so we may regard it as a smooth section of ∧^{i}T^{∗}M ⊗ End(S_{0} ⊗ E) considered as a ‘bundle’ over M. The reader should note that in general, ∧^{i}T^{∗}M ⊗ End(S_{0} ⊗ E) is not a bundle over M as G is in general not a bundle over M. When we say that an element φ is a smooth section of ∧^{i}T^{∗}M ⊗ End(S_{0} ⊗ E), we mean that for any smooth section X of ∧^{i}T^{∗}M, the family of operators φ(X) varies smoothly as a function of x ∈ M. This abuse of language will occur again in Section 4

Note that if F is the foliation given by the fibers of a fiber bundle, G is also a fiber bundle (it is the pull back of the bundle over itself) and B is a Bismut superconnection as in [Bis86],

[BV87] and [Don88].

The curvature B^{2} of B is a purely leafwise operator, that is for any ϕ ∈ Γ(∧ν_{s}^{∗} ⊗ S_{0} ⊗ E),

(B^{2}ϕ)|L̃_{x} = B_{x}^{2}(ϕ|L̃_{x})

where B_{x}^{2} is a second order differential operator on Γ(∧ν_{s}^{∗} ⊗ S_{0} ⊗ E|L̃_{x}) and the formulas for B_{x}^{2} in [Bis86], [BV87] and [Don88] are valid in our situation. It follows from the construction in [Don79] of the heat kernel on non-compact manifolds of bounded geometry that we may form the operator e^{−B_{x}^{2}} acting on Γ(∧ν_{s}^{∗} ⊗ S_{0} ⊗ E|L̃_{x}). Denote by K(γ, ξ) the Schwartz kernel of the family of operators e^{−B^{2}}. It is a result of [Hei95] that K(γ, ξ) is smooth in all its variables. The action of e^{−B^{2}} also commutes with the left action of ∧ν_{s}^{∗}, so for each [γ] ∈ L̃_{x} we may consider

K(γ, γ) ∈ ∧T^{∗}M_{x} ⊗ End((S_{0} ⊗ E)_{[γ]}),

and its pointwise trace, tr(K(γ, γ)) ∈ ∧T^{∗}M_{x}. As the dimension of F_{s} is even, the bundle S_{0} splits as S_{0} = S_{0}^{+} ⊕ S_{0}^{−}, and taking the pointwise supertrace associated to the splitting yields tr_{s}(K(γ, γ)) ∈ ∧T^{∗}M_{x}. In fact, both tr(K(γ, γ)) and tr_{s}(K(γ, γ)) are in ∧ν_{x}^{∗}. Exactly why this is so will be quite important for us.

Now scale the metric g_{0} by 1/t for t ∈ R^{+} and form the operators B_{t} and e^{−B_{t}^{2}} with Schwartz kernel K^{t}(γ, ξ) and supertrace tr_{s}(K^{t}(γ, γ)) ∈ ∧ν_{x}^{∗}. Denote by Ω the curvature of the bundle T F_{s} and by L the curvature of E. Let Â(z) and ch(z) be respectively the Â and Chern character polynomials and consider the differential form Â(Ω)ch(L) on G. As ∧T^{∗}G = s^{∗}(∧T^{∗}M) ∧ (∧T^{∗}F_{s}), we may consider the part of (Â(Ω)ch(L))([γ]) lying in s^{∗}(∧T^{∗}M_{x})∧(∧^{p}T^{∗}F_{s})_{[γ]}, and we write it as [Â(Ω)ch(L)]_{p,γ} ∧dx where dx is the volume form on L̃_{x}. Then [Â(Ω)ch(L)]_{p,γ} ∈ s^{∗}(∧T^{∗}M_{x}) and the proof of Theorem 3.4 of [Don88] gives the following generalization of a theorem of Bismut [Bis86] (also proven in [BV87]).

Theorem 2.2 For all [γ] ∈ G,

lim_{t→0} tr_{s}(K^{t}(γ, γ)) = (2πi)^{− \frac{p}{2}} [Â(Ω)ch(L)]_{p,γ}.

If F is a fiber bundle foliation of M where M is compact then Theorem 2.2 is a strong local form of the index theorem for families of elliptic operators. See [Bis86], [BV87]] and [Don88].

3 Chern character for foliations.

In this section we recall the definition of the Chern character for a family of generalized Dirac operators defined along the leaves of a foliation F of a compact Riemannian manifold M.

We begin by recalling the Haefliger cohomology of F, see [Hae80]. Let U = {U_{1}, …, U_{r}} be a good cover of M by foliation charts, [HL90]. Let T_{i} ⊂ U_{i} be a transversal and set T = ∪ T_{i}. We may assume that the T_{i} are disjoint. Let H be the holonomy pseudogroup induced by F on T. Let Ω_{c}^{k}(T) be the space of smooth k forms on T with compact support and denote by Ω_{c}^{k}(T/H) the quotient of Ω_{c}^{k}(T) by the vector subspace generated by elements of the form α−h^{∗}α where h ∈ H and α ∈ Ω_{c}^{k}(T) has support contained in the range of h. Give Ω_{c}^{k}(T/H) the quotient topology of the usual C^{∞} topology on Ω_{c}^{k}(T). The exterior derivative d : Ω_{c}^{k}(T) → Ω_{c}^{k+1}(T) induces a continuous differential d_{F} : Ω_{c}^{k}(T/H) → Ω_{c}^{k+1}(T/H). Note that Ω_{c}^{k}(T/H) and d_{F} are independent of the choice of cover U. We denote the cohomology of this complex by H_{c}^{∗}(T r F).

If F is the foliation given by a fiber bundle M → N, Ω_{c}^{∗}(T r F) is the usual de Rham complex of N and H_{c}^{∗}(T r F) is just the usual de Rham cohomology of N with compact supports.

As the bundle T F is oriented, there is a continuous open surjective linear map, called integration over the fiber,

∫_{F} : Ω^{p+k}(M) → Ω_{c}^{k}(F)

which commutes with d. It is given as follows. Choose a partition of unity {ϕ_{i}} subordinate to U. Given ω ∈ Ω^{p+k}(M), ϕ_{i} ω ∈ Ω_{c}^{p+k}(U_{i}), and we may integrate ϕ_{i} ω along the fibers of the submersion π_{i} : U_{i} → T_{i} to obtain ∫_{U_{i}} ϕ_{i} ω ∈ Ω_{c}^{k}(T_{i}). Define ∫_{F} ω to be the class of

∑_{i} ∫_{U_{i}} ϕ_{i} ω in Ω_{c}^{k}(F). It is independent of the choice of the ϕ_{i} and of the cover U.

For x ∈ M, we denote also by x ∈ L̃_{x} the class of the constant path at x. Denote the volume form along the leaves of F by dx. As tr(K(x, x)) and tr_{s}(K(x, x)) are in ∧ν_{x}^{∗}, tr(K(x, x))dx and tr_{s}(K(x, x))dx are smooth differential forms on M. We define

T r(e^{−B^{2}}) = ∫_{F} tr(K(x, x))dx and T r_{s}(e^{−B^{2}}) = ∫_{F} tr_{s}(K(x, x))dx.

The definition of ∫_{F} extends to measurable forms, which allows us to extend the notion of T r and T r_{s} to operators whose Schwartz kernels are only measurable.

The main result of [Hei95] is the following.

Theorem 3.1 The element T r_{s}(e^{−B^{2}}) in Ω_{c}^{∗}(T r F) is closed and its cohomology class in H_{c}^{∗}(T r F) is independent of the metric on M.

Definition 3.2 The non-normalized Chern character \overline{ch} (D_{E}) of D_{E} is the cohomology class of T r_{s}(e^{−B^{2}}) in H_{c}^{∗}(T r F).

Corollary 3.3 As Haefliger cohomology classes

\overline{ch} (D_{E}) = ∫_{F} (2πi)^{−p/2}Â(Ω_{F}) ch(L_{E}),

where Ω_{F} is the curvature of T F and L_{E} the curvature form on E over M.

This is immediate from Theorem 2.2 as the convergence there is uniform since the estimates are based on the local geometry of the leaves L̃_{x}, which is uniformly bounded. See [Bis86], [BV87] and [Don88].

4 The t → ∞ asymptotics.

In this section we justify Definition 3.2 by showing that \overline{ch} (D_{E}) is the non-normalized Chern character of the index bundle associated to the family D_{E}, i.e. we prove a families index theorem for these generalized Dirac operators. We do this by computing the t → ∞ asymptotics of T r_{s}(e^{−B_{t}^{2}}). This requires some regularity assumption on D_{E} just as in the classical fiber bundle case. In that case, one assumes that there is a uniform gap about zero in the spectrum of D_{E}. This implies that the Novikov-Shubin invariants of the family are infinite. We assume much less. In particular, our main assumption, 4.1 below, is that the strong Novikov-Shubin invariants of D_{E} are greater than three times the codimension of F. In addition, we will assume that the spectral projections of D_{E} and their transverse derivatives are well behaved for small intervals about zero.

Let B be a Bismut super connection for F and E as above. Denote by P the projection onto ker D_{E} = ker D_{E}^{2}, and for ϵ > 0, denote by P_{ϵ} and Q_{ϵ} the spectral projections of D_{E}^{2} associated to the intervals (0, ϵ) and [ϵ, ∞) respectively. We require that P and P_{ϵ} satisfy conditions 4.1, 4.2 and 4.3 below. When this occurs, we say that D_{E} is regular near zero.

Recalling that q = codim(F), we assume that

T r(P_{ϵ}) is O(ϵ^{β}) for β > 3q as ϵ → 0. (4.1)

When we say a Haefliger form Φ depending on ϵ is O(ϵ^{β}) as ϵ → 0 we mean that there is a constant C > 0 so that the function on T, ∥Φ∥_{T} ≤ Cϵ^{β} as ϵ → 0. Here ∥ ∥_{T} is the pointwise norm on forms on the transversal T induced from the metric on M. The fact that for any two good covers of M by foliation charts there is a integer N so that any placque of the first cover intersects at most N placques of the second cover implies easily that this condition does not depend on our choice of good cover.

We also assume that for sufficiently small ϵ

(4.2) P and P_{ϵ} are smooth

(4.3) B_{1}(P) and B_{1}(P_{ϵ}) ∈ N and are bounded independently of ϵ.

See below for the precise meaning of 4.3. In essence it means that the transverse derivatives of both P and P_{ϵ} are bounded smoothing operators along the leaves of the graph. As for 4.2, note that leafwise smoothness of P and P_{ϵ} is automatic because of the bounded geometry of the leaves, so we are assuming that they are transversely smooth, and this implies that their Schwartz kernels are smooth.

In the classical fiber bundle case, the assumption that there is a uniform gap about zero implies regularity near zero since P_{ϵ} ≡ 0 for small ϵ and the Schwartz kernel K_{P} of P is a smooth function on the graph of the foliation given by the fibers. As the graph is compact, any derivative of K_{P} defines a bounded smoothing operator on the fibers. In the general foliation case, it is not difficult to construct examples where D_{E} is regular near zero even though a gap does not exist. In particular, any constant slope foliation of T^{8} of codimension 1 will do. The real obstruction to constructing more such examples is the paucity of information about the Novikov-Shubin invariants of non compact manifolds.

Denote by ∇ the operator P B_{1}P. Since P is smooth, ∇ is well defined. As P commutes with the action of ∧T^{∗}M on s^{∗}(∧T^{∗}M) ⊗ S_{0} ⊗ E, it is easy to see, using Equation 2.1, that ∇ is a connection on ker D_{E}. That is, for ω ∈ γ(∧T^{∗}M) and φ ∈ Γ(ker D_{E}),

∇(s^{∗}ω · φ) = d(s^{∗}ω) · φ + (−1)^{deg ω}s^{∗}ω · ∇φ.

It is easy to show that for arbitrary φ,

∇^{2}(s^{∗}ω · φ) = s^{∗}ω · ∇^{2}φ (4.4) and as ∇^{2} is nilpotent,

e^{−∇^{2}} = ∑_{j=0}^{[n/2]}(−1)^{j} \frac{∇^{2j}}{j!} .

Both ∇^{2} and e^{−∇^{2}} are smooth sections of ∧T^{∗}M ⊗ End(S_{0} ⊗ E), (in fact they are bounded elements of N, see below), so in particular T r(e^{−∇^{2}}) and T r_{s}(e^{−∇^{2}}) are well defined Haefliger forms.

The following is our main theorem.

Theorem 4.5 Let F be a foliation of a compact manifold M with Hausdorff graph and let E be a Hermitian bundle over the M. Assume that the dimension of F is even, and that it is oriented and spin with a fixed spin structure. Assume further that the family of generalized Dirac operators D_{E} is regular near zero. Then

lim_{t→∞} T r_{s}(e^{−B_{t}^{2}}) = T r_{s}(e^{−∇^{2}}).

Thus, as Haefliger classes,

∫_{F} (2πi)^{−p/2}Â(Ω_{F})ch(L_{E}) = \overline{ch} (D_{E}) = ∫_{F} tr_{s}(e^{−∇^{2}})dx.

Proof: Consider the ‘fiber bundle’ W = C_{c}^{∞}(S_{0} ⊗ E) over M whose ‘fiber’ over x ∈ M is the space W_{x} = C_{c}^{∞}(L̃_{x}; S_{0} ⊗ E). Filter the space M of all sections of ∧T^{∗}M ⊗ End(W, W) over M by the subspaces M_{i} of sections of ∑_{j≥i} ∧^{j}T^{∗}M ⊗ End(W, W). Filter the space N of all sections of ∧T^{∗}M ⊗ End_{S}(W, W) similarly, where End_{S}(W, W) is the space of smoothing operators. An element φ ∈ M assigns to each k vector X ∈ ∧^{k}T M_{x} an operator φ(X) on the space W_{x}, or more properly its L^{2} completion. We say φ is bounded if

∥φ∥ ≡ sup{∥φ(X)∥ | X ∈ ∧^{∗}T M, ∥X∥ = 1} < ∞.

We say φ is smooth (resp. measurable) if it is a smooth (resp. measurable) section of ∧T^{∗}M ⊗ End(W, W). Note that Q_{ϵ} is a bounded smooth element of M.

Denote by ∥ ∥_{r,s} the Sobolev r, s norm on End(W_{x}, W_{x}). We say φ ∈ N is bounded if for each r and s,

∥φ∥_{r,s} ≡ sup{∥φ(X)∥_{r,s} | X ∈ ∧^{∗}T M, ∥X∥ = 1} < ∞.

Note that P, and for sufficiently small ϵ, P_{ϵ} are bounded smooth elements of N. Because of the uniformly bounded geometry of the L̃_{x}, it follows from standard estimates that any bounded φ ∈ N has a Schwartz kernel K_{φ} which is leafwise smooth and pointwise uniformly bounded, as are all its leafwise derivatives. In particular, for sufficiently large ℓ, and for any x ∈ M and γ, ξ ∈ L̃_{x}, ∥K_{φ}(γ, ξ)∥ ≤ C_{ℓ}∥φ∥_{−ℓ,ℓ} where C_{ℓ} is a constant depending only on ℓ. Thus estimates on ∥φ∥_{−ℓ,ℓ} translate directly into estimates on T r(φ) and T r_{s}(φ). In particular, if φ_{t} ∈ N satisfies ∥φ_{t}∥_{−ℓ,ℓ} = O(t^{α}) as t → ∞, this implies T r(φ_{t}) = O(t^{α}) also.

Recall that B_{t} is the Bismut superconnection associated to F and E for the metric obtained by scaling the original metric by 1/t. For all t > 0, let ψ_{t} be the map of M which multiplies a section of ∧^{k}T^{∗}M ⊗ End(W, W) by t^{−k/2}. Then by [BV87],

B_{t} = \sqrt{t}ψ_{t}Bψ_{t}^{−1} = \sqrt{t}B_{0} + B_{1} + \frac{1}{\sqrt{t}}B_{2} + ⋯ (4.6)

and B_{t}^{2} = tψ_{t}B^{2}ψ_{t}^{−1}, so

B_{t}^{2} = tB_{0}^{2} + \sqrt{t}(B_{0}B_{1} + B_{1}B_{0}) + ⋯ = tD_{E}^{2} + \sqrt{t}[D_{E}, B_{1}] + ⋯ (4.7)

Set

B̄_{ϵ,t} = (P + Q_{ϵ})B_{t}(P + Q_{ϵ}) + P_{ϵ}B_{t}P_{ϵ}

and

A_{ϵ,t} = (P + Q_{ϵ})(B_{t} − \sqrt{t}B_{0})P_{ϵ} + P_{ϵ}(B_{t} − \sqrt{t}B_{0})(P + Q_{ϵ}).

As B_{0} = D_{E} commutes with the spectral projections of D_{E}^{2} and (P +Q_{ϵ})P_{ϵ} = 0 = P_{ϵ}(P +Q_{ϵ}), B_{t} = B̄_{ϵ,t} + A_{ϵ,t}. Note that 4.2 along with the fact that Q_{ϵ} = I − (P + P_{ϵ}) imply that B̄_{ϵ,t}, A_{ϵ,t}, and the operators B̂_{ϵ} and T_{ϵ,t} below are well defined.

As in [HL], see also [GR], we extend the results in Chapter 9 of [BGV92] to show that

lim_{t→∞} T r_{s}(e^{−B̄_{ϵ,t}^{2}}) = T r_{s}(e^{−∇^{2}}),

and then show that

lim_{t→∞} T r_{s}(e^{−B_{t}^{2}}) = lim_{t→∞} T r_{s}(e^{−B̄_{ϵ,t}^{2}}).

Set B̂_{ϵ} = (P + Q_{ϵ})B(P + Q_{ϵ}). Then B̄_{ϵ} ≡ B̄_{ϵ,1} = B̂_{ϵ} + P_{ϵ}BP_{ϵ}. Set T_{ϵ,t} = Q_{ϵ}B_{t}Q_{ϵ}, and T_{ϵ} = T_{ϵ,1}.

Proposition 4.8 There is a measurable section g_{ϵ} in M with g_{ϵ} − I and g_{ϵ}^{−1} − I ∈ N_{1} bounded so that under the decomposition W = P W ⊕ Q_{ϵ}W ⊕ P_{ϵ}W,

g_{ϵ}B̄_{ϵ}^{2}g_{ϵ}^{−1} = | \begin{matrix}∇^{2} & 0 & 0 \\ 0 & T_{ϵ}^{2} & 0 \\ 0 & 0 & (P_{ϵ}BP_{ϵ})^{2} \end{matrix} | mod | \begin{matrix}N_{3} & 0 & 0 \\ 0 & N_{2} & 0 \\ 0 & 0 & 0 \end{matrix} | .

Proof: Denote by G_{ϵ} the Green’s operator of Q_{ϵ}D_{E}^{2}Q_{ϵ}, i.e. G_{ϵ} is zero on P W ⊕ P_{ϵ}W and is the inverse of Q_{ϵ}D_{E}^{2}Q_{ϵ} on Q_{ϵ}W . It is a bounded operator with norm ||G_{ϵ}||_{s,s} ≤ ϵ^{−1} for all s.

The proof proceeds by induction on i and we need only prove the analogous result for B̂_{ϵ}^{2} restricted to P W ⊕ Q_{ϵ}W. For i = 1, we set g_{1} = I and as P B_{0} = B_{0}P = 0, we have

Lemma 4.9

B̂_{ϵ}^{2} = | \begin{matrix}P (B − B_{0})P & P (B − B_{0})Q_{ϵ} \\ Q_{ϵ}(B − B_{0})P & T_{ϵ} \end{matrix} |^{2} = | \begin{matrix}X_{1} & Y_{1} \\ Z_{1} & T_{1} \end{matrix} | ∈ | \begin{matrix}N_{2} & N_{1} \\ N_{1} & M_{0} \end{matrix} | ,

X_{1} − Y_{1}G_{ϵ}Z_{1} = ∇^{2} mod N_{3} and T_{1} = T_{ϵ}^{2} mod N_{2}. Furthermore, X_{1}, Y_{1}, Z_{1} and T_{1} − T_{ϵ}^{2} are bounded elements of N.

Proof: Given any x ∈ M, there are open neighborhoods U of x and Ũ ⊂ s^{−1}(U) of L̃_{x}, and a canonical identification, which respects the projections to U, of Ũ with an open neighborhood of L̃_{x} × {x} in L̃_{x} × U. With these coordinates on Ũ, we may write B_{1} = d_{U} + Θ where d_{U} is exterior differentiation in the base direction and Θ ∈ M_{1} is an order zero differential operator with uniformly bounded coefficients. See the proof of Proposition 2.4 of [HL].

We now explain Equation 4.3. By B_{1}(P) we mean the operator which is given in these coordinates by d_{U}(P) + Θ ∘ P, where d_{U}(P) is the operator whose Schwartz kernel is ∑_{i}(∂K_{x}(y, z)/∂x_{i})dx_{i} and K_{x}(y, z) is the Schwartz kernel of P on L̃_{x}. Similarly for B_{1}(P_{ϵ}).

The operators X_{1}, Y_{1}, Z_{1}, T_{1}, and T_{ϵ}^{2} are purely leafwise operators. In particular, they all satisfy Equation 4.4 with ∇^{2} replaced by the operator in question. Now T_{1} = T_{ϵ}^{2} + Q_{ϵ}(B − B_{0})P (B − B_{0})Q_{ϵ}, so Q_{ϵ}(B − B_{0})P (B − B_{0})Q_{ϵ} is also a leafwise operator, and we may write it in the above coordinates as a sum of compositions of operators where d_{U} does not appear as an operator, although it does appear as having acted on another operator. That is

Q_{ϵ}(B − B_{0})P (B − B_{0})Q_{ϵ} = Q_{ϵ}(d_{U} + Θ + B_{2} + ⋯)P (d_{U} + Θ + B_{2} + ⋯)(I − P − P_{ϵ})

may be written as

Q_{ϵ}(Θ + B_{2} + ⋯)P (Θ + B_{2} + ⋯)(I − P − P_{ϵ}) − Q_{ϵ}(Θ + B_{2} + ⋯)P_{ϵ}d_{U}(P + P_{ϵ})

+ Q_{ϵ}d_{U}(P)(Θ + B_{2} + ⋯)(I − P − P_{ϵ}) + Q_{ϵ}P d_{U}(Θ + B_{2} + ⋯)(I − P − P_{ϵ})

+ Q_{ϵ}P (Θ − B_{2} + ⋯)d_{U}(P + P_{ϵ}) − Q_{ϵ}d_{U}(P)d_{U}(P + P_{ϵ}).

Here d_{U}(Θ + B_{2} + ⋯) is the exterior derivative applied to the coefficients of the operator. Θ and the B_{j}, j ≥ 2, are at least in N_{1}, and are leafwise differential operators of order zero with uniformly bounded coefficients. Their derivatives are also uniformly bounded, so d_{U}(Θ) and the d_{U}(B_{j}) satisfy the same properties. In particular, their composition with a bounded element of N is a bounded element of N. P, P_{ϵ} are bounded, and by assumption are smooth elements of N. Also by assumption, d_{U}(P) and d_{U}(P_{ϵ}) are smooth elements of N_{1}. It follows that T_{1} − T_{ϵ}^{2} ∈ N_{2} and that it is bounded. It is easy to show that T_{ϵ}^{2} ∈ M_{0}, so we have T_{1} = T_{ϵ}^{2} mod N_{2} and T_{1} ∈ M_{0}.

Similar arguments show that X_{1} ∈ N_{2}, Y_{1}, Z_{1} ∈ N_{1}, and are bounded, and that X_{1} − Y_{1}G_{ϵ}Z_{1} = ∇^{2} mod N_{3}, (so ∇^{2} ∈ N_{2} and is bounded as claimed above).

Now we inductively define g_{i} ∈ M with g_{i} − I, g_{i}^{−1} − I ∈ N_{1} bounded and

g_{i}B̂_{ϵ}^{2}g_{i}^{−1} = | \begin{matrix}X_{i} & Y_{i} \\ Z_{i} & T_{i} \end{matrix} | ∈ | \begin{matrix}N_{2} & N_{i} \\ N_{i} & M_{0} \end{matrix} |

with X_{i}, Y_{i}, Z_{i} bounded and T_{i} − T_{ϵ}^{2} ∈ N_{2} bounded. To do this, set

h_{i} = | \begin{matrix}1 & −Y_{i}G_{ϵ} \\ G_{ϵ}Z_{i} & 1 \end{matrix} | , and | \begin{matrix}X_{i+1} & Y_{i+1} \\ Z_{i+1} & T_{i+1} \end{matrix} | = h_{i} | \begin{matrix}X_{i} & Y_{i} \\ Z_{i} & T_{i} \end{matrix} | h_{i}^{−1}.

Since | \begin{matrix}0 & −Y_{i}G_{ϵ} \\ G_{ϵ}Z_{i} & 0 \end{matrix} | is a nilpotent operator, h_{i}^{−1} = ∑_{k=0}^{∞}(−1)^{k} | \begin{matrix}0 & −Y_{i}G_{ϵ} \\ G_{ϵ}Z_{i} & 0 \end{matrix} |^{k} is a finite

sum, and

h_{i}^{−1} = | \begin{matrix}1 & Y_{i}G_{ϵ} \\ −G_{ϵ}Z_{i} & 1 \end{matrix} | mod N_{2i}.

Thus

| \begin{matrix}X_{i+1} & Y_{i+1} \\ Z_{i+1} & T_{i+1} \end{matrix} | = | \begin{matrix}1 & −Y_{i}G_{ϵ} \\ G_{ϵ}Z_{i} & 1 \end{matrix} | | \begin{matrix}X_{i} & Y_{i} \\ Z_{i} & T_{i} \end{matrix} | | \begin{matrix}1 & Y_{i}G_{ϵ} \\ −G_{ϵ}Z_{i} & 1 \end{matrix} | mod N_{2i}.

As T_{ϵ}^{2} − Q_{ϵ}D_{E}^{2}Q_{ϵ} ∈ M_{1}, we have

| \begin{matrix}0 & 0 \\ 0 & 1 − G_{ϵ}T_{i} \end{matrix} | and | \begin{matrix}0 & 0 \\ 0 & 1 − T_{i}G_{ϵ} \end{matrix} | ∈ M_{1}.

Using this, a direct computation, see [BGV92] p. 274, shows that X_{i+1} ∈ N_{2}, Y_{i+1} and Z_{i+1} ∈ N_{i+1}, and T_{i+1} − T_{ϵ}^{2} ∈ N_{2}, and are all bounded. Setting g_{i+1} = h_{i}g_{i}, we may continue the induction to obtain g_{ϵ} = g_{n+1} ⊕ 1 as N_{n+1} = 0. As in [BGV92], we have that X_{n+1} ≡ X_{1} − Y_{1}G_{ϵ}Z_{1} mod N_{3}, so g_{ϵ}B̄_{ϵ}^{2}g_{ϵ}^{−1} has the required form.

As all the Y_{i} and Z_{i} are in N_{1} and are bounded, it is obvious that g_{ϵ} −I, and g_{ϵ}^{−1} −I ∈ N_{1} and are bounded (by bounds which depend on ϵ).

□

An important fact to be gleaned from a careful analysis of the construction of g_{ϵ} is the following.

The number of G_{ϵ} s appearing in any monomial ocurring in g_{ϵ} or any monomial which is modded out in Proposition 4.8 is less than or equal to the number of B_{i} s, i ≥ 1 ocurring in the same monomial.

(4.10)

To see this, use the fact that Q_{ϵ}D_{E}^{2}Q_{ϵ}G_{ϵ} = (Q_{ϵ}D_{E}Q_{ϵ})^{2}G_{ϵ} = (Q_{ϵ}D_{E}Q_{ϵ})G_{ϵ}(Q_{ϵ}D_{E}Q_{ϵ}) = Q_{ϵ} to reduce the number of G_{ϵ} s.

Recall that β of Condition 4.1 satisfies β > 3q and choose a real number a ∈ (6, 2β/q). Set

ϵ = t^{− \frac{1}{a}} .

Then ∥G_{ϵ}∥_{s,s} ≤ t^{\frac{1}{a}} for all s.

Lemma 4.11 Any element A ∈ N_{k} which is in g_{ϵ} − I, g_{ϵ}^{−1} − I or is modded out in Proposition 4.8, satisfies, for all r and s,

∥A_{t}∥_{r,s} is at least O(t^{− \frac{k}{2} + \frac{k}{a}}) as t → ∞,

where A_{t} = ψ_{t}Aψ_{t}^{−1}.

Proof: All such A are sums of words in the leafwise operators P, Q_{ϵ}, B_{1}(P), B_{1}(P_{ϵ}), G_{ϵ}, Θ, B_{1}(Θ), the B_{j} s and the B_{1}(B_{j}) s, j ≠ 1,. (Use the fact that B_{1}(Q_{ϵ}) = B_{1}(I − P − P_{ϵ})). Each word contains at least one P. (If necessary, use d_{U}(P) = d_{U}(P^{2}) = d_{U}(P)P +P d_{U}(P)). Any word in Θ, B_{1}(Θ), the B_{j} s and the B_{1}(B_{j}) s, j ≠ 1, is at worst a differential operator of finite order, say s_{0}, with uniformly bounded coefficients. As such, its s, s − s_{0} Sobolev norm is finite. In addition, ∥Q_{ϵ}∥_{s,s} ≤ 1 for all s, and by assumption P, B_{1}(P) and B_{1}(P_{ϵ}) are bounded smoothing operators. Now take any word W in A and assume that it is in N_{ℓ}, i.e. that it is purely of base degree ℓ ≥ k. Then ψ_{t}W ψ_{t}^{−1} = t^{− \frac{ℓ}{2}} W. Suppose for definiteness that

W = W_{1}G_{ϵ}W_{2}P W_{3}G_{ϵ}W_{4}

where the W_{i} are words which contain no G_{ϵ} s. At worst, W_{i} has finite order s_{i}, so that ∥W_{i}∥_{s,s−s_{i}} is finite for all s. Then we have

∥W ∥_{r,s} ≤ t^{− \frac{ℓ}{2}} ∥W_{1}∥_{k_{1},s}∥G_{ϵ}∥_{k_{1},k_{1}}∥W_{2}∥_{k_{2},k_{1}}∥P ∥_{k_{3},k_{2}}∥W_{3}∥_{k_{4},k_{3}}∥G_{ϵ}∥_{k_{4},k_{4}}∥W_{4}∥_{r,k_{4}}

where k_{1} = s + s_{1}, k_{2} = s + s_{1} + s_{2}, k_{3} = r − (s_{3} + s_{4}) and k_{4} = r − s_{4}. Now ∥P ∥_{k_{3},k_{2}} ≤ 1 for any k_{3}, k_{2} and ∥G_{ϵ}∥_{s,s} ≤ t^{\frac{1}{a}} for all s, so

∥W ∥_{r,s} ≤ t^{− \frac{ℓ}{2} + \frac{2}{a}} ∥W_{1}∥_{k_{1},s}∥W_{2}∥_{k_{2},k_{1}}∥W_{3}∥_{k_{4},k_{3}}∥W_{4}∥_{r,k_{4}} = Ct^{− \frac{ℓ}{2} + \frac{2}{a}}

where C is independent of t. By Remark 4.10 an arbitrary such W can contain at most ℓ G_{ϵ} s, so a trivial extension of the above argument gives the Lemma. □

The reader should note that the estimate in Lemma 4.11 can be strengthened. This requires a good deal of tedious bookkeeping and is achieved by showing that if A is modded out in Proposition 4.8, then the number of G_{ϵ} s appearing in any monomial occurring in A is actually less than the number of B_{i} s occurring in the same monomial. A second improvement comes from the fact that ∥Q_{ϵ}D_{E}Q_{ϵ}G_{ϵ}∥ ≤ \sqrt{∥G_{ϵ}∥}. These improvements translate into a weakening of the requirement that β > 3q. In particular, the constant 3 can be replaced by a smaller positive number.

All the A_{t} considered in Lemma 4.11 are nilpotent as they are at least in N_{1}. Since a > 6, if A ∈ N_{3}, then ∥A_{t}∥_{r,s} is at least O(t^{−(1+α)}) where α > 0. Thus we have

B̄_{ϵ,t}^{2} = tψ_{t}B̄_{ϵ}^{2}ψ_{t}^{−1} = ψ_{t}g_{ϵ}^{−1}ψ_{t}^{−1} | \begin{matrix}∇^{2} + O(tt^{−(1+α)}) & 0 & 0 \\ 0 & T_{ϵ,t}^{2} + O(tt^{−1+ \frac{2}{a}}) & 0 \\ 0 & 0 & (P_{ϵ}B_{t}P_{ϵ})^{2} \end{matrix} | ψ_{t}g_{ϵ}ψ_{t}^{−1}.

Lemma 4.12 We may assume without loss of generality that ψ_{t}g_{ϵ}ψ_{t}^{−1} = I = ψ_{t}g_{ϵ}^{−1}ψ_{t}^{−1}.

Proof: As g_{ϵ} = | \begin{matrix}g_{n+1} & 0 \\ 0 & 1 \end{matrix} |, we need only work with B̂_{ϵ,t}^{2} and g_{n+1}. Now

ψ_{t}g_{n+1}ψ_{t}^{−1} = I + | \begin{matrix}O(t^{− \frac{1}{2} + \frac{1}{a}}) & O(t^{− \frac{1}{2} + \frac{1}{a}}) \\ O(t^{− \frac{1}{2} + \frac{1}{a}}) & O(t^{− \frac{1}{2} + \frac{1}{a}}) \end{matrix} |

and similarly for ψ_{t}g_{n+1}^{−1}ψ_{t}^{−1}. Write B̂_{ϵ,t}^{2} = ψ_{t}g_{n+1}^{−1}ψ_{t}^{−1}Aψ_{t}g_{n+1}ψ_{t}^{−1}, where

A = | \begin{matrix}∇^{2} + O(tt^{−(1+α)}) & 0 \\ 0 & T_{ϵ,t}^{2} + O(tt^{−1+ \frac{2}{a}}) \end{matrix} | .

Denote by η the grading operator used in defining the supertrace, and note that ∥η∥_{s,s} = 1 for all s. Then we have

T r_{s}(e^{−B̂_{ϵ,t}^{2}}) = T r(ηψ_{t}g_{n+1}^{−1}ψ_{t}^{−1}e^{−A}ψ_{t}g_{n+1}ψ_{t}^{−1})
= T r(η[I + O(t^{− \frac{1}{2} + \frac{1}{a}})]e^{−A}[I + O(t^{− \frac{1}{2} + \frac{1}{a}})])
= T r(ηe^{−A}) + T r(ηO(t^{− \frac{1}{2} + \frac{1}{a}})e^{−A}) + T r(ηe^{−A}O(t^{− \frac{1}{2} + \frac{1}{a}})) +
T r(ηO(t^{− \frac{1}{2} + \frac{1}{a}})e^{−A}O(t^{− \frac{1}{2} + \frac{1}{a}}))
= T r(ηe^{−A}) + O(t^{− \frac{1}{2} + \frac{1}{a}})

provided we can show that ∥e^{−A}∥_{s,s} is finite for all s. The proof of Proposition 4.13 below

implies that ∥e^{−(T_{ϵ,t}^{2}+O(t^{\frac{2}{a}}))}∥_{s,s} is O(t^{−β}) for all s and any β > 0. To deal with e^{−(∇^{2}+O(t^{−α}))}, note that ∇^{2}, O(t^{−α}) and their sum are nilpotent, so this term is a finite sum of the form

∑ ∇^{2j_{1}}O(t^{−α})∇^{2j_{2}}O(t^{−α}) ⋯ O(t^{−α})∇^{2j_{k}}.

As both ∇^{2} and O(t^{−α}) are bounded elements of N, we are done. □

Note that we could just as well have written

e^{−(∇^{2}+O(t^{−α}))} = e^{−∇^{2}} + ∑ ∇^{2j_{1}}O(t^{−α})∇^{2j_{2}}O(t^{−α}) ⋯ O(t^{−α})∇^{2j_{k}},

so e^{−(∇^{2}+O(t^{−α}))} = e^{−∇^{2}} + O(t^{−α}), since every term in the sum now contains at least one O(t^{−α}). Thus

T r_{s}(e^{−B̄_{ϵ,t}^{2}}) = T r_{s}(e^{−(∇^{2}+O(t^{−α}))}) + T r_{s}(e^{−(T_{ϵ,t}^{2}+O(t^{\frac{2}{a}}))}) + T r_{s}(e^{−(P_{ϵ}B_{t}P_{ϵ})^{2}})
= O(t^{−α}) + T r_{s}(e^{−∇^{2}}) + T r_{s}(e^{−(T_{ϵ,t}^{2}+O(t^{\frac{2}{a}}))}) + T r_{s}(e^{−(P_{ϵ}B_{t}P_{ϵ})^{2}}).

Proposition 4.13 For any δ > 0, as t → ∞,

T r_{s}(e^{−(T_{ϵ,t}^{2}+O(t^{\frac{2}{a}}))}) = O(t^{−δ}).

Proof: Using T_{ϵ,t}^{2} = tψ_{t}T_{ϵ}^{2}ψ_{t}^{−1}, write

e^{−(T_{ϵ,t}^{2}+O(t^{\frac{2}{a}}))} = e^{−Q_{ϵ}tD_{E}^{2}Q_{ϵ}+C_{t}}

where C_{t} is an element of M_{1}. Note that we are again abusing notation and should be considering T r_{s}(Q_{ϵ}e^{−Q_{ϵ}tD_{E}^{2}Q_{ϵ}+C_{t}}Q_{ϵ}). The Volterra series for the exponential of a perturbed operator, [BGV92] p. 78, gives

e^{−Q_{ϵ}tD_{E}^{2}Q_{ϵ}+C_{t}} = ∑_{k=0}^{∞} ∫_{Δ_{k}} e^{−x_{1}Q_{ϵ}tD_{E}^{2}Q_{ϵ}}C_{t}e^{−x_{2}Q_{ϵ}tD_{E}^{2}Q_{ϵ}}C_{t} … C_{t}e^{−x_{k+1}Q_{ϵ}tD_{E}^{2}Q_{ϵ}}dx_{k} … dx_{1},

where Δ_{k} is the standard k simplex Δ_{k} = {(x_{1}, …, x_{k+1}) | x_{i} ≥ 0, ∑ x_{i} = 1}. As the operator C_{t} ∈ M_{1}, the sum is finite, in particular k ≤ n = dim M. (Below, we shall improve this bound to k ≤ q = codim F). Fix k. At each point in Δ_{k} there is at least one x_{i} ≥ 1/(k + 1). We may assume that i = k + 1, since the general case is handled in the same way. We may also assume without loss of generality that all x_{i} are positive.

Write the integrand as

e^{−x_{1}Q_{ϵ}tD_{E}^{2}Q_{ϵ}}t^{− \frac{1}{2}} C_{t}e^{−x_{2}Q_{ϵ}tD_{E}^{2}Q_{ϵ}}t^{− \frac{1}{2}} C_{t} … t^{− \frac{1}{2}} C_{t}t^{\frac{k}{2}} e^{−x_{k+1}Q_{ϵ}tD_{E}^{2}Q_{ϵ}}.

Since T_{ϵ,t}^{2} is a leafwise operator of order two in terms of differentiation, and Q_{ϵ}tD_{E}^{2}Q_{ϵ} is its order two part, at worst t^{− \frac{1}{2}} C_{t} is a leafwise operator of order one with uniformly bounded coefficients (independently of t for say t ≥ 1/2). Thus ∥t^{− \frac{1}{2}} C_{t}∥_{ℓ,ℓ−1} is bounded independently of t. The Spectral Mapping Theorem implies that the operator e^{−xQ_{ϵ}tD_{E}^{2}Q_{ϵ}} satisfies

∥e^{−xQ_{ϵ}tD_{E}^{2}Q_{ϵ}}∥_{ℓ−1,ℓ} ≤ max_{z≥0} e^{−xtz^{2}} \sqrt{1 + z^{2}} = \{ \begin{matrix}e^{xt}/\sqrt{2ext} & if 2xt ≤ 1 \\ 1 & if 2xt ≥ 1. \end{matrix}

In particular, if t ≥ 1/2,

∥e^{−xQ_{ϵ}tD_{E}^{2}Q_{ϵ}}∥_{ℓ−1,ℓ} ≤ x^{− \frac{1}{2}}

since e^{xt} ≤ \sqrt{e} for 2xt ≤ 1, and 1 ≤ x^{− \frac{1}{2}} as 0 < x ≤ 1. Thus, for t ≥ 1/2,

∥e^{−x_{j}Q_{ϵ}tD_{E}^{2}Q_{ϵ}}t^{− \frac{1}{2}} C_{t}∥_{ℓ,ℓ} ≤ ∥e^{−x_{j}Q_{ϵ}tD_{E}^{2}Q_{ϵ}}∥_{ℓ−1,ℓ} ∥t^{− \frac{1}{2}} C_{t}∥_{ℓ,ℓ−1} ≤ x_{j}^{− \frac{1}{2}} ∥t^{− \frac{1}{2}} C_{t}∥_{ℓ,ℓ−1},

so for t ≥ 1/2,

∥e^{−x_{1}Q_{ϵ}tD_{E}^{2}Q_{ϵ}}t^{− \frac{1}{2}} C_{t} … e^{−x_{k}Q_{ϵ}tD_{E}^{2}Q_{ϵ}}t^{− \frac{1}{2}} C_{t}∥_{ℓ,ℓ} ≤ C(x_{1} … x_{k})^{− \frac{1}{2}} ,

where the constant C is independent of t.

Now write

t^{\frac{k}{2}} e^{−x_{k+1}Q_{ϵ}tD_{E}^{2}Q_{ϵ}} = t^{(−m+ \frac{m}{a} + \frac{k}{2a})}(t^{− \frac{1}{a}} G_{ϵ})^{m+ \frac{k}{2}} (Q_{ϵ}tD_{E}^{2}Q_{ϵ})^{m+ \frac{k}{2}} e^{−x_{k+1}Q_{ϵ}tD_{E}^{2}Q_{ϵ}}

where m is a positive integer as large as we please. ∥t^{− \frac{1}{a}} G_{ϵ}∥_{ℓ,ℓ} is bounded independently of t. The Spectral Mapping Theorem gives that (Q_{ϵ}tD_{E}^{2}Q_{ϵ})^{m+ \frac{k}{2}} e^{−x_{k+1}Q_{ϵ}tD_{E}^{2}Q_{ϵ}} is a bounded smoothing operator whose −ℓ, ℓ norm satisfies

∥(Q_{ϵ}tD_{E}^{2}Q_{ϵ})^{m+ \frac{k}{2}} e^{−x_{k+1}Q_{ϵ}tD_{E}^{2}Q_{ϵ}}∥_{−ℓ,ℓ} ≤ max_{z≥0} (1 + z^{2})^{ℓ}(tz^{2})^{m+ \frac{k}{2}} e^{−x_{k+1}tz^{2}}

which has a bound depending only on ℓ, m and n since x_{k+1} ≥ 1/(k + 1) and k ≤ n. As ∫_{Δ_{k}}(x_{1} … x_{k})^{− \frac{1}{2}} dx_{k} … dx_{1} is finite, e^{−T_{ϵ,t}^{2}+O(t^{\frac{2}{a}})} is at worst t^{−m+ \frac{m}{a} + \frac{n}{2a}} times a bounded

smoothing operator whose −ℓ, ℓ norm is bounded independently of t and so at worst

T r_{s}(e^{−T_{ϵ,t}^{2}+O(t^{\frac{2}{a}})}) = O(t^{−m+ \frac{m}{a} + \frac{n}{2a}}).

Since n is fixed, a > 1 and m is arbitrarily large, we are done. □

This completes the proof of Theorem 4.5 in the case P_{ϵ} = 0 for some positive ϵ. In this case, B̄_{ϵ,t} = B_{t} and (P_{ϵ}B_{t}P_{ϵ}) = 0 for sufficiently large t. Note also that in this case, we only need that P is smooth and B_{1}(P) ∈ N and is bounded.

We now consider the case where P_{ϵ} ≠ 0 for all positive ϵ.

Proposition 4.14 lim_{t→∞} T r_{s}(e^{−(P_{ϵ}B_{t}P_{ϵ})^{2}}) = 0.

Proof: Note that we are abusing notation here as we are really working on the subspace P_{ϵ}W of W. Thus T r_{s}(e^{−(P_{ϵ}B_{t}P_{ϵ})^{2}}) is really T r_{s}(P_{ϵ}e^{−(P_{ϵ}B_{t}P_{ϵ})^{2}}P_{ϵ}). This sort of abuse of notation will occur again. Setting A = ηP_{ϵ}e^{−(P_{ϵ}B_{t}P_{ϵ})^{2}}, we have T r_{s}(e^{−(P_{ϵ}B_{t}P_{ϵ})^{2}}) = T r(AP_{ϵ}).

Lemma 4.15 t^{− \frac{q}{2}} ∥A∥ is bounded independently of t for t large.

Proof: As η and P_{ϵ} are bounded, we may ignore them. Write

(P_{ϵ}B_{t}P_{ϵ})^{2} = P_{ϵ}tD_{E}^{2}P_{ϵ} + C_{t}

where t^{− \frac{1}{2}} C_{t} ∈ M_{1} is a leafwise differential operator of at most order one with uniformly bounded coefficients (independently of t for t large). Thus t^{− \frac{1}{2}} ∥C_{t}∥_{ℓ,ℓ−1} is bounded independently of t for t large. As in Proposition 4.13, consider the Volterra series for A. The Spectral Mapping Theorem gives that for all x_{j}, ∥e^{−x_{j}P_{ϵ}tD_{E}^{2}P_{ϵ}}∥ ≤ 1 and ∥e^{−x_{j}P_{ϵ}tD_{E}^{2}P_{ϵ}}∥_{ℓ−1,ℓ} ≤ \sqrt{1 + ϵ^{2}}. Thus the lemma will follow provided we show that at most q C_{t} s occur in any element of the Volterra series. Consider again the situation in Lemma 4.9. We may choose U so that it is a foliation chart U ≃ R^{p} ×R^{q}. The proof of the Lemma on page 512 of [Hei95] implies that we may write B = B_{0} +d_{U} +Θ+B_{2} +⋯ where the B_{j}, j = 0, 2, 3, … and Θ are in fact sections of ∧ν^{∗} ⊗ End(W, W) which do not depend on the leaf direction (i.e. on R^{p}). As P_{ϵ} is independent of the leaf direction, it follows immediately that (P_{ϵ}B_{t}P_{ϵ})^{2} ∈ Γ(∧ν^{∗} ⊗ End(W, W)), so C_{t} ∈ Γ(∧ν^{∗} ⊗ End(W, W)) as well. As ∧^{q+1}ν^{∗} = 0, we have the lemma. □

Thus we have

T r_{s}(e^{−(P_{ϵ}B_{t}P_{ϵ})^{2}}) = T r(AP_{ϵ}) = T r(P_{ϵ}AP_{ϵ}),

as P_{ϵ} = P_{ϵ}^{2} and P_{ϵ} commutes with η. Denote by A_{x} the action of A on W_{x} and let ω_{1}, ω_{2}, … be a basis of ∧ν_{x}^{∗}. As in [HL], we may write A_{x} = ∑_{j} ω_{j} ⊗ A_{j} where A_{j} ∈ End(W_{x}, W_{x}). Let v_{1}, …, v_{k} be an orthonormal basis of (S_{0} ⊗ Ẽ)x̄, where x̄ ∈ L̃_{x} is the class of the constant path at x ∈ M. Denoting the Schwartz kernel of P_{ϵ}AP_{ϵ} by H_{ϵ}, we have

tr(H_{ϵ}^{x}(x, x)) = ∑_{i,j} ω_{j}⊗ < P_{ϵ}A_{j}P_{ϵ}(δ_{v_{i}}), δ_{v_{i}} > .

Now

| < P_{ϵ}A_{j}P_{ϵ}(δ_{v_{i}}), δ_{v_{i}} > | = | < A_{j}P_{ϵ}(δ_{v_{i}}), P_{ϵ}(δ_{v_{i}}) > | ≤
∥A_{j}P_{ϵ}(δ_{v_{i}})∥ ∥P_{ϵ}(δ_{v_{i}})∥ ≤ ∥A_{j}∥ ∥P_{ϵ}(δ_{v_{i}})∥^{2} ≤ ∥A∥ ∥P_{ϵ}(δ_{v_{i}})∥^{2}.

Denote the Schwartz kernel of P_{ϵ} by K_{ϵ}. Then

∑_{i} ∥P_{ϵ}(δ_{v_{i}})∥^{2} = ∑_{i} < P_{ϵ}(δ_{v_{i}}), P_{ϵ}(δ_{v_{i}}) > = ∑_{i} < P_{ϵ}^{2}(δ_{v_{i}}), δ_{v_{i}} > =

∑_{i} < P_{ϵ}(δ_{v_{i}}), δ_{v_{i}} > = ∑_{i} K_{ϵ}^{x}(v_{i}, v_{i}) = tr(K_{ϵ}^{x}(x, x)).

Let U and {ϕ_{U} | U ∈ U} be a good foliation cover and partition of unity subordinate to it. Consider a chart U ∈ U with transversal T_{U}. Let ω_{1}^{T} , ω_{2}^{T} , … be an orthonormal basis of ∧T^{∗}T_{U}. We may assume that ω_{j} = π^{∗}(ω_{j}^{T}), where π : U → T_{U} is the projection. On T_{U}, the Haefliger form T r(P_{ϵ}AP_{ϵ}) is given by

∫_{U} ϕ_{U} tr(H_{ϵ}^{x}(x, x))dx = ∫_{U} ϕ_{U} ∑_{i,j} ω_{j}⊗ < P_{ϵ}A_{j}P_{ϵ}(δ_{v_{i}}), δ_{v_{i}} > dx =

∑_{j} ω_{j}^{T} ∫_{U} ϕ_{U} ∑_{i} < P_{ϵ}A_{j}P_{ϵ}(δ_{v_{i}}), δ_{v_{i}} > dx.

Now

| ∫_{U} ϕ_{U} ∑_{i} < P_{ϵ}A_{j}P_{ϵ}(δ_{v_{i}}), δ_{v_{i}} > dx| ≤ ∫_{U} ϕ_{U} ∑_{i} | < P_{ϵ}A_{j}P_{ϵ}(δ_{v_{i}}), δ_{v_{i}} > |dx ≤

∫_{U} ϕ_{U} ∑_{i} ∥A∥ ∥P_{ϵ}(δ_{v_{i}})∥^{2} = ∥A∥ ∫_{U} ϕ_{U} tr(K_{ϵ}^{x}(x, x))dx ≤ C∥A∥ϵ^{β}

for some constant C since ∫_{U} ϕ_{U} tr(K_{ϵ}^{x}(x, x))dx is the form T r(P_{ϵ}) on T_{U}. Now ϵ = t^{− \frac{1}{a}}, so

∥A∥ϵ^{β} = t^{− \frac{q}{2}} ∥A∥t^{\frac{q}{2} − \frac{β}{a}} .

The term t^{− \frac{q}{2}} ∥A∥ is bounded as t → ∞ and 6 < a < 2β/q so the result follows. □

Thus,

lim_{t→∞} T r_{s}(e^{−B̄_{ϵ,t}^{2}}) = T r_{s}(e^{−∇^{2}}),

and to finish the proof of Theorem 4.5, we have

Proposition 4.16

lim_{t→∞} T r_{s}(e^{−B_{t}^{2}}) = lim_{t→∞} T r_{s}(e^{−B̄_{ϵ,t}^{2}}).

Proof: For 0 ≤ z ≤ 1, set B_{t}(z) = B̄_{ϵ,t} + zA_{ϵ,t}, i.e.

B_{t}(z) = \sqrt{t}D_{E} + (P + Q_{ϵ})(B_{t} − \sqrt{t}D_{E})(P + Q_{ϵ}) + P_{ϵ}(B_{t} − \sqrt{t}D_{E})P_{ϵ}+
z[(P + Q_{ϵ})(B_{t} − \sqrt{t}D_{E})P_{ϵ} + P_{ϵ}(B_{t} − \sqrt{t}D_{E})(P + Q_{ϵ})].

By Equation 3.10 of [Hei95], we have

T r_{s}(e^{−B̄_{ϵ,t}^{2}}) − T r_{s}(e^{−B_{t}^{2}}) = ∫_{0}^{1} ∫_{0}^{1} T r(νe^{(s−1)B_{t}^{2}(z)} \frac{d(B_{t}^{2}(z))}{dz}e^{−sB_{t}^{2}(z)})dsdz. (4.17)

Note that B_{t}^{2}(z) is a leafwise operator of order two whose order two part, namely tD_{E}^{2}, does not depend on z, so d(B_{t}^{2}(z))/dz is a leafwise operator of order one. Just as in Lemma 4.9, we may write

d(B_{t}^{2}(z))/dz = B_{t}(z)A_{ϵ,t} + A_{ϵ,t}B_{t}(z) = \sqrt{t}D_{E}A_{1} + A_{2}\sqrt{t}D_{E} + A_{3}

where

A_{i} = C_{i,1}P_{ϵ}C_{i,2} = (C_{i,1}P_{ϵ})P_{ϵ}(P_{ϵ}C_{i,2})

with the C_{i,j} ∈ M_{1} being sums of words in Θ, B_{1}(Θ), the t^{(1−j)/2}B_{j} s and t^{(1−j)/2}B_{1}(B_{j}) s, j ≥ 2, (which are differential operators of order zero with uniformly bounded coefficients, the bounds being independent of t and s for t large), and P, P_{ϵ}, Q_{ϵ}, B_{1}(P), and B_{1}(P_{ϵ}), which are bounded operators with bounds independent to t. (Use the fact that d_{U}(P_{ϵ}) = d_{U}(P_{ϵ}^{2}) = d_{U}(P_{ϵ})P_{ϵ} + P_{ϵ}d_{U}(P_{ϵ}) to see that each term must contain at least one P_{ϵ}. Use (P + Q_{ϵ})P_{ϵ} = P_{ϵ}(P + Q_{ϵ}) = 0 to see that the order one differential operator B_{1}(D_{E}) cannot occur). Thus the C_{i,1}P_{ϵ} and P_{ϵ}C_{i,2} are bounded smoothing operators with bounds independent of t.

Thus T r_{s}(e^{−B̄_{ϵ,t}^{2}}) − T r_{s}(e^{−B_{t}^{2}}) can be written in the form T r(AP_{ϵ}) where

A = ∫_{0}^{1} ∫_{0}^{1} C_{1,2}e^{−sB_{t}^{2}(z)}νe^{(s−1)B_{t}^{2}(z)}\sqrt{t}D_{E}C_{1,1}+
C_{2,2}\sqrt{t}D_{E}e^{−sB_{t}^{2}(z)}νe^{(s−1)B_{t}^{2}(z)}C_{2,1} + C_{3,2}e^{−sB_{t}^{2}(z)}νe^{(s−1)B_{t}^{2}(z)}C_{3,1}dsdz.

As in the proof of Proposition 4.13 we may write t^{− \frac{q}{2}} e^{−sB_{t}^{2}(z)}νe^{(s−1)B_{t}^{2}(z)} as

∑_{k,m=0}^{q} t^{\frac{k+m−q}{2}} ∫_{Δ_{k}} e^{−x_{1}stD_{E}^{2}}st^{− \frac{1}{2}} C_{t}e^{−x_{2}stD_{E}^{2}}st^{− \frac{1}{2}} C_{t} … st^{− \frac{1}{2}} C_{t}e^{−x_{k+1}stD_{E}^{2}}dx_{k} … dx_{1}

ν ∫_{Δ_{m}} e^{−y_{1}rtD_{E}^{2}}rt^{− \frac{1}{2}} C_{t}e^{−y_{2}rtD_{E}^{2}}rt^{− \frac{1}{2}} C_{t} … rt^{− \frac{1}{2}} C_{t}e^{−y_{m+1}rtD_{E}^{2}}dy_{m} … dy_{1},

where r = 1−s. Note that the proof of Lemma 4.15 gives that the sum stops when k+m = q, so t^{\frac{k+m−q}{2}} is bounded for t large. Just as in the proof of 4.13, we have ∥t^{− \frac{1}{2}} C_{t}∥_{ℓ,ℓ−1} is uniformly bounded for large t, and ∥e^{−atD_{E}^{2}}∥_{ℓ−1,ℓ} ≤ a^{− \frac{1}{2}} for t ≥ 1/2. Thus for t ≥ 1/2

∥st^{− \frac{1}{2}} C_{t}e^{−x_{j}stD_{E}^{2}}∥_{ℓ,ℓ} ≤ ∥t^{− \frac{1}{2}} C_{t}∥_{ℓ,ℓ−1} ∥se^{−x_{j}stD_{E}^{2}}∥_{ℓ−1,ℓ} ≤ (∥t^{− \frac{1}{2}} C_{t}∥_{ℓ,ℓ−1}) s^{\frac{1}{2}} x_{j}^{− \frac{1}{2}} ,

so for such t,

∥st^{− \frac{1}{2}} C_{t}e^{−x_{2}stD_{E}^{2}} … st^{− \frac{1}{2}} C_{t}e^{−x_{k+1}stD_{E}^{2}}∥_{ℓ,ℓ} ≤ C(x_{2} … x_{k+1})^{− \frac{1}{2}} .

Similarly,

∥e^{−y_{1}rtD_{E}^{2}}rt^{− \frac{1}{2}} C_{t} … e^{−y_{m}rtD_{E}^{2}}rt^{− \frac{1}{2}} C_{t}∥_{ℓ,ℓ} ≤ C(y_{1} … y_{m})^{− \frac{1}{2}} ,

where the constant C is independent of t for t ≥ 1/2. As ∥ν∥, ∥e^{−x_{1}stD_{E}^{2}}∥ and ∥e^{−y_{m+1}rtD_{E}^{2}}∥ are bounded by 1, and ∫_{Δ_{k}}(x_{2} … x_{k+1})^{− \frac{1}{2}} dx_{k} … dx_{1} and ∫_{Δ_{m}}(y_{1} … y_{m})^{− \frac{1}{2}} dy_{m} … dy_{1} are finite, we have that t^{− \frac{q}{2}} ∥e^{−sB_{t}^{2}(z)}νe^{(s−1)B_{t}^{2}(z)}∥ is bounded independently of t for t large.

Note that ∥\sqrt{t}D_{E}e^{−ztD_{E}^{2}}∥ = ∥e^{−ztD_{E}^{2}} \sqrt{t}D_{E}∥ is bounded by a multiple of z^{− \frac{1}{2}} , which combined with the argument above shows that t^{− \frac{q}{2}} ∥e^{−sB_{t}^{2}(z)}νe^{(s−1)B_{t}^{2}(z)}\sqrt{t}D_{E}∥ is bounded by a multiple of (1 − s)^{− \frac{1}{2}} , and t^{− \frac{q}{2}} ∥\sqrt{t}D_{E}e^{−sB_{t}^{2}(z)}νe^{(s−1)B_{t}^{2}(z)}∥ by a multiple of s^{− \frac{1}{2}} for t large. Thus A is a bounded smoothing operator with bound depending on t, and t^{− \frac{q}{2}} ∥A∥ is bounded independently of t for t large. Now, T r(AP_{ϵ}) = T r(AP_{ϵ}^{2}) = T r(P_{ϵ}AP_{ϵ}) as both A and P_{ϵ} are bounded smoothing operators. The method of proof of Proposition 4.14 gives the result and completes the proofs of the Proposition and Theorem 4.5. □

5 Applications

5.1 The Atiyah-Singer Index Theorem

In [AS68], Atiyah and Singer proved the index theorem for elliptic operators on a compact manifold M. For a generalized Dirac operator, the Atiyah-Singer theorem is a special case of our general families index theorem. Take the foliation of M with only one leaf, namely M itself. It has T F = T M and its holonomy graph is G = M × M. Regularity near zero is automatic in this case as there is a gap in the spectrum about zero and P is transversely constant as there is only one leaf to the foliation. As a transversal we take a single point in M, and the Haefliger cohomology of this foliation is just the usual cohomology of the point. Theorem 4.5 is then the Atiyah-Singer cohomology index theorem.

5.2 The Atiyah L^{2} Covering Index Theorem

In [A76], Atiyah proved an index theorem for a non-compact regular covering space M̃ of a compact manifold M, the L^{2} covering index theorem. For Dirac operators, this is also a special case of our theorem. Instead of using the holonomy graph as above, use the homotopy graph H = M × M̃. The supertrace T r_{s} in this situation is the Atiyah L^{2} covering space supertrace and Theorem 4.5 is the Atiyah L^{2} covering index theorem. As above, Conditions 4.2 and 4.3 are automatic as there is only one leaf, so P and P_{ϵ} are transversely constant. We need not be concerned with Condition 4.1 as the transverse space is zero dimensional, so the places where we need to invoke 4.1 do not occur.

5.3 The Atiyah-Singer Families Index Theorem

In [AS71], Atiyah and Singer proved the families index theorem for elliptic operators defined along the fibers of a fiber bundle M with base space B. In this case, the Haefliger cohomology of the foliation given by the fibers is just the usual deRham cohomology of B. Let D_{E} be a family of generalized Dirac operators defined along the fibers of M. In all proofs of the families index theorem using the heat equation method, one makes an assumption equivalent to the assumption that there is a uniform gap in the spectrum about zero. When such a gap exists, D_{E} is regular near zero. Theorem 4.5 is then the Atiyah-Singer cohomology index theorem for the family D_{E}. Having proven the theorem for such operators, one extends to arbitrary operators by K-theory methods.

5.4 The L^{2} Families Index Theorem

If we consider again the situation of the Atiyah-Singer families case, but use the homotopy graph H, (which is a fiber bundle over M whose fiber is the universal cover Z̃ of Z), Theorem 4.5 gives a families index theorem for families of covering spaces, i.e. a combination of the L^{2} covering case with the families case. In particular, the cohomology class on B, ∫_{Z} (2πi)^{−p/2}Â(Ω_{F})ch(L_{E}) is the Chern character of the original family of Dirac operators on

the bundle M and the Chern character of the pullback of the family to the fibers Z̃ of the graph H. In this case, we must require regularity near zero.

5.5 The Connes’ Foliation Index Theorem

In [Con86], see also [Con94], Connes proves an index theorem for elliptic operators defined along the leaves of a foliation. In its original form, this index is a real valued number constructed using a transverse invariant measure. As explained in [HL], the transverse measure is a red herring and Connes’ index actually lives in the zero dimensional Haefliger cohomology of the foliation. It is the order zero term of our families index. A careful reading of the proof of Theorem 4.5 shows that in order to obtain Connes’ theorem, we do not need to assume regularity near zero.

As noted in the introduction, Connes and Skandalis have proven a families index theorem for families of elliptic operators defined along the leaves of a foliation. See [CS84] and [Con87]. This index takes its values in the K-theory of the C^{∗} algebra of the foliation. It is a difficult and important problem to relate their results to those presented here.

5.6 Gong-Rothenberg Index Theorem

In [GR], Gong and Rothenberg consider a smooth fiber bundle Z̃ → M→B with fiber Z̃ a regular covering Z̃ →^{π} Z of a closed connected Riemannian manifold Z with covering group Γ. They assume that M admits a local action by Γ so that locally, their data is pulled back under the projection π^{−1}(U) ≃ U × Z̃ →^{π} U × Z. As our results here and in [Hei95] are local in this sense, it is clear that they extend immediately to the Gong-Rothenberg situation.

5.7 Lichnerowicz Regularity Theorem

The well know theorem of Lichnerowicz [L62] extends to the case of foliations to give a general condition for regularity near zero. In particular, we have (see Theorem 8.17 of [LM89], and [Con94].)

Theorem 5.1 Let E be a flat Hermitian bundle over the compact manifold M with foliation F with Hausdorff graph. Assume that the dimension of F is even, F spin and that T F admits a metric of positive scalar curvature. Then the generalized Dirac operator D_{E} is regular near zero.

Note that in the above case, ker D_{E} = 0.

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