Analytic Torsion Forms and Torsion-Signature

Invariants

Donggeng Gong^{∗}

Abstract We define a torsion-signature invariant on the total space of a compact fiber bundle. The analytic torsion form of the fiber bundle is used to find an adiabatic limit formula of the torion-signature.

Mathematics Subject Classification (1991): 58G10, 53C07, and 57R20 Key words: Fiber bundle, analytic torsion form, mixed trace, and torsion-signature

1 Introduction

The analytic torsion form of Bismut-Lott[BL] arises naturally from a transgression formula of the smooth version of the Riemann-Roch-Grothendieck theorem. This torsion form is further extended to certain non-compact fiber bundles in [GR]. It is natural to study the contribution of the analytic torsion form of degree ≥ 2 to adiabatic limits of some invariants on the total space of the fiber bundle. Such a nontrivial contribution does not exist for the analytic torsion invariant simply because of the top dimensionality of the Euler form on the base space of the fiber bundle. Hence, we need to consider the adiabatic limits of other invariants.

The purpose of this paper is to introduce a new invariant, the torsion-signature on the total space, and to prove an adiabatic limit formula of this invariant, which involves both the analytic torsion form and a modified torsion form. This torsion-signature invariant is motivated by the following question: for a compact fiber bundle Z → M →^{π} B and its analytic torsion form T (T^{H}M, g^{Z}, g^{E}) associated with a flat complex vector bundle E over M, what homotopy invariances does the pairing < L(B)T (T^{H}M, g^{Z}, g^{E}), [B] > have, provided T (T^{H}M, g^{Z}, g^{E}) is closed? We refer to Section 2 for the detail of this question. We are led to consider an operator on the total space M which combines the signature operator on the base space B and the Euler-de Rham operator on the fiber. Instead of explicitly producing such an operator, we apply a new mixed super trace to the Euler-de Rham operator on M and get our invariant. The idea is based on the fact that applying two super traces, which correspond to the two gradings given by the degree of forms and by the Hodge star operator on a closed Riemannian manifold, to the Euler-de Rham operator, we obtain the Euler characteristic number and the signature of the manifold (cf. [BGV], Chapter 4), respectively. The mixed super trace is just the tensor product of these two super traces on the exterior cotangent bundle ∧(T^{∗}M) ≃ π^{∗}(∧(T^{∗}B)) ˆ⊗ ∧ (T^{∗}Z). The advantage of this trace approach is that we can directly use many existing formulas. For instance, we can quickly get a local mixed index formula. Because of these new features, we have rised more questions than what we have solved in this paper.

∗Research supported in part by NSF grant and MPIM, Bonn.

This paper is arranged as follows. In Section 2 we recall the analytic torsion form of Bismut-Lott. Section 3 is devoted to a modified torsion form that we need in the adiabatic limit formula. We introduce in Section 4 the mixed super trace and prove a local mixed index formula. We define the torsion-signature invariant in Section 5. This invariant reduces to the analytic torsion invariant when B = {pt}. Finally, we use in Section 6 the analytic and modified torsion forms to prove an adiabatic limit formula of the torsion-signature invariant. I am grateful to Mel Rothenberg for helpful comments.

2 Analytic Torsion Forms

Let B and Z be closed Riemannian manifolds and Z → M →^{π} B a smooth compact fiber bundle over B with the fiber Z. Suppose E is a flat complex vector bundle over M with a flat connection ∇^{E} and a Hermitian metric g^{E}. Denote by T Z the vertical tangent bundle of M consisting of tangent vectors of M to the fibers. We choose a connection on the fiber bundle M. Then T M = T Z ⊕ T^{H}M, where T^{H}M is the bundle of horizontal vectors, T^{H}M ≃ π^{∗}(T B). From this we can write the exterior cotangent bundle as ∧^{∗}(T^{∗}M) ≃ π^{∗}(∧^{∗}(T^{∗}B)) ˆ⊗ ∧^{∗} (T^{∗}Z).

Let E be the infinite dimensional ℤ-graded vector bundle over B whose fiber over x ∈ B is isomorphic to the space of E|_{M_{x}}-valued forms Ω(M_{x}, E|_{M_{x}}) on M_{x}, where M_{x} = π^{−1}(x) ≃ Z. There is a connection ∇^{E} on E given by

∇_{U}^{E}ϕ = L_{U^{H}}ϕ,

where ϕ ∈ C^{∞}(B, E), U^{H} ∈ C^{∞}(M, T^{H}M) is the horizontal lift of a vector field U on B, π_{∗}(U^{H}) = U, and L_{U^{H}} is the Lie differentiation on C^{∞}(B, E). We have Ω(M, E) ≃ Ω(B, E). Let T be the curvature of the fiber bundle M given by

T (U_{1}, U_{2}) = −P^{Z}[U_{1}^{H}, U_{2}^{H}] ∈ C^{∞}(M, T Z),

where P^{Z} : T M = T Z ⊕ T^{H}M → T Z is the projection. T is a T Z-valued horizontal 2-form on M. Denote by i_{T} the interior multiplication by T. Then the exterior differential d_{M} on M defines a flat super connection on E of total degree 1,

d_{M} = d_{Z} + ∇^{E} + i_{T} .  (1)

The terms of (1) are degree-0, 1, and 2 parts of d_{M}, respectively. Let N be the number operator of E given by multiplication by j on C^{∞}(M, ∧^{j}(T Z) ⊗ E). Let g^{B} and g^{Z} be the Riemannian metrics on B and Z, respectively. Using these two metrics, we can define an L^{2}-inner product g^{E} on E, and hence an adjoint d_{M}^{∗} of d_{M} as a super connection on E. For t > 0, let

D_{t} = \frac{1}{2}(t^{−N/2}d_{M}^{∗}t^{N/2} − t^{N/2}d_{M}t^{−N/2})
= \frac{\sqrt{t}}{2}(d_{Z}^{∗} − d_{Z}) + \frac{1}{2}w(E, g^{E}) − \frac{1}{2\sqrt{t}}(T ∧ +i_{T}),  (2)

where w(E, g^{E}) = (∇^{E})^{∗}−∇^{E} = (g^{E})^{−1}(∇^{E}g^{E}). Take an odd function f(z) = z exp(z^{2}), z ∈ ℂ, and define φ(ϕ) = (2πi)^{− \frac{k}{2}} ϕ for ϕ ∈ Ω^{k}(B). We can use the usual super trace T r^{s} on E to define real odd and even differential forms on B, respectively,

f(D_{t}, g^{E}) = (2πi)^{\frac{1}{2}} φT r^{s}(f(D_{t})),  (3)

and

f′(D_{t}, g^{E}) = φT r^{s}(\frac{N}{2} f′(D_{t})).  (4)

Let H(Z, E|_{Z}) be the ℤ-graded complex vector bundle over B whose fiber over x ∈ B is isomorphic to the cohomology H^{∗}(M_{x}, E|_{M_{x}}). By Hodge theory, the metric g^{Z} on Z induces a metric g^{H(Z,E|_{Z})} on H(Z, E|_{Z}). Furthermore, d_{M} induces a canonical flat connection ∇^{H(Z,E|_{Z})} on H(Z, E|_{Z}). Let o(T Z) be the orientation line bundle of T Z, and e(T Z) be the Euler class of Z represented by an o(T Z)-valued closed form e(T Z, ∇^{Z}) on M,

e(T Z, ∇^{Z}) = \{ \begin{matrix}P f(\frac{R^{Z}}{2π}) ≡ T (exp \frac{−R^{Z}}{2π}), & dim(Z) even \\ 0, & dim(Z) odd, \end{matrix}

where ∇^{Z} = P^{Z}∇^{M} with ∇^{M} the Levi-Civita connection on T M associated with metric g^{M} = π^{∗}(g^{B}) ⊕ g^{Z}, T is the Berezin integral from ∧(T^{∗}Z) to ℝ [BGV], and R^{Z} is the curvature of ∇^{Z}. Define

f(∇^{E}, g^{E}) = (2πi)^{\frac{1}{2}} φT r^{s}(f(\frac{1}{2}((∇^{E})^{∗} − ∇^{E}))),

where w(E, g^{E}) = (∇^{E})^{∗}−∇^{E} = (g^{E})^{−1}(∇^{E}g^{E}). Similarly, we can define f(∇^{H(Z,E|_{Z})}, g^{H(Z,E|_{Z})}). Set

χ′(Z, E) = ∑_{j=0}^{dim(Z)}(−1)^{j}j dim(H^{j}(Z, E|_{Z})).

Let χ(Z) and rk(E) be the Euler characteristic number of Z and the rank of E, respectively. The properties of the two forms in (2) − (3) are summarized in the following lemma [BL].

Lemma 1 1) \frac{∂}{∂t}f(D_{t}, g^{E}) = \frac{1}{t}df′(D_{t}, g^{E}), t > 0.

2) As t → 0,

f(D_{t}, g^{E}) = \{ \begin{matrix}∫_{Z} e(T Z, ∇^{Z})f(∇^{E}, g^{E}) + O(t), & if dim(Z) even, \\ O(\sqrt{t}), & if dim(Z) odd, \end{matrix}

f′(D_{t}, g^{E}) = \{ \begin{matrix}\frac{1}{4} dim(Z)rk(E)χ(Z) + O(t), & if dim(Z) even, \\ O(\sqrt{t}), & if dim(Z) odd. \end{matrix}

3) As t → ∞,

f(D_{t}, g^{E}) = f(∇^{H(Z,E|_{Z})}, g^{H(Z,E|_{Z})}) + O(\frac{1}{\sqrt{t}}),
f′(D_{t}, g^{E}) = \frac{1}{2}χ′(Z, E) + O(\frac{1}{\sqrt{t}}).

This lemma enables us to define an analytic torsion form T (T^{H}M, g^{Z}, g^{E}) of the fiber bundle M as

T (T^{H}M, g^{Z}, g^{E}) = ∫_{0}^{∞}[f′(D_{t}, g^{E}) − \frac{1}{2}χ′(Z, E) − (\frac{1}{4} dim(Z)rk(E)χ(Z) −
− \frac{1}{2}χ′(Z, E))f′(\frac{i\sqrt{t}}{2})]\frac{dt}{t}.  (5)

Note that this torsion form differs from that of [BZ] by a negative sign. Since the closed form f(D_{t}, g^{E}) is independent of t, we get from Lemma 1 the following transgression formula [BL].

Lemma 2

−dT (T^{H}M, g^{Z}, g^{E}) = ∫_{Z} e(T Z, ∇^{Z})f(∇^{E}, g^{E}) − f(∇^{H(Z,E|_{Z})}, g^{H(Z,E|_{Z})}).  (6)

The Riemann-Roch-Grothendieck theorem asserts that the two closed forms on the right hand side of (6) represent the same cohomology class. Lemma 2 gives explicitly the difference of these two forms. Formula (6) is also important in determining when the analytic torsion form is a closed even form. Indeed, we get by (6) the first part of the following lemma [BL].

Lemma 3 1) If dim(Z) is odd and H(Z, E|_{Z}) = 0, then the analytic torsion form is closed.

2) If g^{E} is covariantly constant with respect to ∇^{E} and dim(Z) is even, then T (T^{H}M, g^{Z}, g^{E}) = 0.

The property that T (T^{H}M, g^{Z}, g^{E}) is closed in certain cases is important in the application. So we want to discuss another case where the analytic torsion form is closed. To this aim, let {e_{i}} and {f_{α}} be local orthonormal bases of the vertical tangent bundle T Z and T B, respectively, with the corresponding dual bases {e^{i}} and {f^{α}}. We also use the same symbols {f_{α}} and {f^{α}} to denote the liftings of {f_{α}} and {f^{α}} to π^{∗}(T B) and π^{∗}(T^{∗}B), respectively. Denote by m^{i} (resp. m^{α}) the exterior multiplication by e^{i} (resp. f^{α}). Let c(X) = (X∧) − i_{X} and c_{v}(X) = (X∧) + i_{X} for X a tangent vector or 1-form. In particular, let c^{i} = c(e_{i}), c^{α} = c(f_{α}), c_{v}^{i} = c_{v}(e_{i}), c_{v}^{α} = c_{v}(f_{α}). Then

D_{t} = \frac{\sqrt{t}}{2}(−c_{v}^{j}∇_{e_{j}}^{Z⊗E} + \frac{1}{2}i_{e_{j}}ψ_{j}) + \frac{1}{2}w(E, g^{E}) − \frac{1}{2\sqrt{t}}c_{v}(T),  (7)

where ψ_{j} = (g^{E})^{−1}(∇_{e_{j}}^{E}g^{E}), ψ_{α} = (g^{E})^{−1}(∇_{f_{α}}^{E}g^{E}), and w(E, g^{E}) = m^{α}(w_{αjk}c^{j}c_{v}^{k} + ψ_{α}) with w_{αjk} given by

w_{abc} = X^{a}(∇_{X_{c}}X_{b})

for X_{a} = e_{j} or f_{α}.

Let Γ = π_{1}(B) be the fundamental group of B, ρ_{1} : Γ → Diff(Z) be a homomorphism, and B̃ be the universal classifying space of B, where Diff(Z) is the group of diffeomorphisms on Z. Let M = B̃ ×_{Γ} Z →^{π} B be a fiber bundle over B with the fiber Z. Suppose ρ_{2} : Γ → GL(ℂ^{k}) is a representation of Γ on ℂ^{k}. Then E = B̃ ×_{Γ} (Z × ℂ^{k}) is a flat vector bundle over M and its restriction to each fiber is a trivial bundle Ẽ = Z × ℂ^{k} over Z. Let g^{E} be a Hermitian metric on E such that g^{E} is locally independent of Z. One can construct such a metric as follows. Choose a finite open covering {U_{i}}_{i=1}^{l} of B together with smooth crosssections β_{i} : U_{i} → B̃ for the projection B̃ →^{π} B. For x ∈ β_{i}(U_{i}) we define a Hermitian metric g_{i} on ℂ_{x}^{k} = ℂ^{k}. The g′_{i}s may be different for i ≠ j. Then for νx ∈ νβ_{i}(U_{i}), ν ∈ Γ, we define a metric ρ_{2}(ν^{−1})g_{i} on ℂ_{νx}^{k}. We thus have a Γ-invariant Hermitian metric g̃_{i} on Γβ_{i}(U_{i}) × ℂ^{k}. Let {σ_{i}} be a partition of unity subordinate to {U_{i}} and {σ̃_{i}} be the pull back of {σ_{i}} to B̃. Then g̃ = ∑_{i} σ̃_{i}g̃_{i} is a Γ-invariant metric on the trivial vector bundle B̃ × ℂ^{k}. We consider g̃ as a Γ-invariant metric on the vector bundle B̃ × Z × ℂ^{k} which induces a required metric g on E. Let the flat connection ∇^{E} on E be locally given by the usual exterior differentiation. Since the fiber bundle M is locally a product, the curvature tensor T of M is zero. The operator D_{t} in (7) is now given by

D_{t} = \frac{\sqrt{t}}{2}(−c_{v}^{j}∇_{e_{j}}^{Z⊗E}) + \frac{1}{2}w(E, g^{E}).  (8)

Since ψ_{j} = (g^{E})^{−1}(∇_{e_{j}}^{E}g^{E}) = 0, w(E, g^{E}) = m^{α}ψ_{α} and w(E, g^{E}) = w(E, g^{E}) + m^{α}w_{αjk}c̃^{j}c̃_{v}^{k}.

Lemma 4 If w_{αjk} = 0 and g^{E} is independent of Z, then the analytic torsion form T (T^{H}M, g^{Z}, g^{E}) is closed on B.

Proof. By (6), it suffices to prove that

∫_{Z} e(T Z, ∇^{Z})f(∇^{E}, g^{E}) − f(∇^{H(Z,E|_{Z})}, g^{H(Z,E|_{Z})}) = 0.

Recall that f(∇^{E}, g^{E}) = (2πi)^{\frac{1}{2}} φT r^{s}(f(\frac{1}{2}w(E, g^{E}))) and

f(∇^{H(Z,E|_{Z})}, g^{H(Z,E|_{Z})}) = (2πi)^{\frac{1}{2}} φT r^{s}(f(\frac{1}{2}w(H, g^{H})))
= (2πi)^{\frac{1}{2}} φT r^{s}(f(\frac{1}{2}P_{Z}w(E, g^{E})P_{Z})),

where P_{Z} is the orthogonal projection of E onto the kernel of (d_{Z}^{∗} − d_{Z})^{2}. Since g^{E} is independent of Z, w(E, g^{E}) = m^{α}(g^{E})^{−1}(∇_{e_{α}}^{E}g^{E}) is a form on B which is independent of Z. Note that E|_{Z} is trivial. We get

(2πi)^{\frac{1}{2}} φT r^{s}(f(\frac{1}{2}P_{Z}w(E, g^{E})P_{Z})) = χ(Z)(2πi)^{\frac{1}{2}} φtr(f(\frac{1}{2}w(E, g^{E}))),

and

∫_{Z} e(T Z, ∇^{Z})f(∇^{E}, g^{E}) = χ(Z)(2πi)^{\frac{1}{2}} φtr(f(\frac{1}{2}w(E, g^{E}))).

These two identities prove the result. Q.E.D.

The condition that w_{αjk} = 0 can be satisfied in the following case. let q : B̃ × Z → M = B̃ ×_{Γ} Z be the natural projection. We have that q_{∗}(T B̃ ⊕ T (Z)) = T M. Here T (Z) denotes the tangent bundle of Z in order to distinguish the vertical bundle T Z. In particular, the vertical bundle T Z is equal to q_{∗}(T (Z)). Assume that there is a Γ-equivariant Riemannian metric on T (Z). Let the Riemannian metric g^{Z} on T Z be the image under q_{∗} of the Γinvariant metric on T (Z). Thus g^{Z} is locally independent of B. Let T^{H}M = q_{∗}(T B̃). It follows that a horizontal vector Ũ ∈ T^{H}M is locally independent of fibers. This implies that LŨg^{Z} = 0. Hence w_{αjk} =< S(ẽ_{j})ẽ_{k}, f_{α} >= \frac{1}{2}(LŨg^{Z})(e_{j}, e_{k}) = 0 (cf. [BC]). Thus w(E, g^{E}) = m^{α}ψ_{α} = w(E, g^{E}).

We now return to the general case. Let Z → M →^{π} B be a fiber bundle over B with a flat vector bundle E. When the analytic torsion form is closed, we define

T s(B, M, E) =< L(B)T (T^{H}M, g^{Z}, g^{E}), [B] >,  (9)

with the Hirzebruch L-class L(B) of B and the fundamental class [B] of B.

T s(B, M, E) is a secondary invariant in the sense that the analytic torsion form appears in the Riemann-Roch-Grothendieck theorem as a secondary class form. Since T (T^{H}M, g^{W} , g^{E}) is an even form, T s(B, M, E) is zero for the odd dimensional B. Obviously, if M = B × Z, E = B × (Z̃ ×_{ρ} ℂ^{k}) with E|_{Z} = Z̃ ×_{ρ} ℂ^{k} and ρ : π_{1}(Z) → GL(ℂ^{k}). Then

T s(B, M, E) = sig(B)T (Z, E|_{Z}),

where sig(B) is the signature of B, and T (Z, E|_{Z}) is the analytic torsion of Z associated with the bundle E|_{Z}. For convention, for Z = {pt}, M = B, we understand T s(B, M, E) = sig(B). But for B = {pt}, M is equal to Z. We interpret T s(B, M, E) = T (Z, E|_{Z}).

One open question is whether T s(B, M, E) has certain homotopy property. Namely, if h : B_{1} → B is certain homotopy equivalence, does the following equality hold

< L(B_{1})h^{∗}(T (T^{H}M, g^{Z}, g^{E})), [B_{1}] >=< L(B)T (T^{H}M, g^{Z}, g^{E}), [B] >?

If M is the trivial fiber bundle B ×Z over B, and E|_{Z} is independent of B, then T s(B, M, Z) is clearly a homotopy invariant. In view of the Kahn theorem [Ka], T s(B, M, Z) is unlikely a homotopy invariant in general. Now we list some elementary properties of the invariant T s(B, M, E).

Proposition 1 (i) T s(B, M, E) is a diffeomorphism invariant. (ii) Let Z_{1} → M_{i} →^{π_{i}} B_{i} be a fiber bundle and E_{i} a flat bundle over M_{i} such that the analytic torsion form T (T^{H}M_{i}, g^{Z_{i}}, g^{E_{i}}) is closed, i = 1, 2. Let Z_{1} × Z_{2} → M_{1} × M_{2} → B_{1} × B_{2} be the product of the fiber bundles M_{1} and M_{2}, and P_{i} : M_{1} × M_{2} → M_{i} be the projection. Then

T s(B_{1} × B_{2}, M_{1} × M_{2}, P_{1}^{∗}(E_{1}) ⊗ P_{2}^{∗}(E_{2})) = rk(E_{1})χ(Z_{1})sig(B_{1})T s(B_{2}, M_{2}, E_{2})
+ rk(E_{2})χ(Z_{2})sig(B_{2})T s(B_{1}, M_{1}, E_{1}).

Proof. Since L(B) and [B] are diffeomorphism invariants, it suffices to prove that the analytic torsion form T (T^{H}M, g^{Z}, g^{E}) is independent of the metric g^{B}. But this is obvious from the definition of the analytic torsion form.

Part (ii) follows from the product formula of the analytic torsion form (see Theorem 3.28 [BL] for the case B_{1} = B_{2}) . Q.E.D.

3 Modified Analytic Torsion Forms

In this section we introduce a modified analytic torsion form of the compact fiber bundle M. This torsion form will be used in the adiabatic limit in Section 6.

We use the notation of Section 2. Let for t > 0

C_{t} ≡ \frac{1}{2}(t^{− \frac{N}{2}} (d_{M})^{∗}t^{\frac{N}{2}} + t^{\frac{N}{2}} d_{M}t^{− \frac{N}{2}} )
= \frac{\sqrt{t}}{2}(d_{Z}^{∗} + d_{Z}) + ∇^{E} + \frac{1}{2}w(E, g^{E}) − \frac{1}{2\sqrt{t}}c(T).  (10)

Let δ_{t} be the automorphism of Ω(B, E) given by δ_{t}(ξ) = t^{− \frac{j}{2}} ξ, ξ ∈ Ω^{j}(B, E). Then C_{t} = t^{\frac{1}{2}} δ_{t}((d_{M})^{∗} +d_{M})δ_{t}^{−1} and D_{t} = t^{\frac{1}{2}} δ_{t}((d_{M})^{∗} −d_{M})δ_{t}^{−1}. Hence, C_{t}^{2} = −D_{t}^{2}. As shown in [BGV], T r^{s}(e^{−C_{t}^{2}}) is a closed form on B.

Lemma 5 1) As t → ∞,

e^{−C_{t}^{2}} = e^{( \frac{1}{2} w(H(Z,E|_{Z}),g^{H(Z,E|_{Z})}))^{2}} + O(\frac{1}{\sqrt{t}}),

in the sense of all C^{l}-norms.

2) As t → 0, the heat kernel e^{−C_{t}^{2}}(x, x) satisfies

e^{−C_{t}^{2}}(x, x) ∼ (4πt)^{− \frac{dim(Z)}{2}} ∑_{j=0}^{∞} t^{j}K_{j}(x),  (11)

where K_{j}(x) ∈ ∑_{2l≤2j} Ω^{2l}(M, End_{C(T^{∗}Z)}(∧(T^{∗}Z) ⊗ E|_{Z})).

Proof. See Theorems 9.19 and 10.21 in [BGV]. Q.E.D.

By Lemma 5, we can define the modified analytic torsion form of M as follows. Definition 1. The modified analytic torsion form T_{1}(T^{H}M, g^{Z}, g^{E}) of M associated with the flat complex vector bundle E is

T_{1}(T^{H}M, g^{Z}, g^{E}) = ∫_{τ}^{∞} φT r^{s}(\frac{N}{2} (e^{−C_{4t}^{2}} − e^{( \frac{1}{2} w(H(Z,E|_{Z}),g^{H(Z,E|_{Z})}))^{2}})\frac{dt}{t}
+ \frac{d}{dr}(\frac{1}{Γ(r)} ∫_{0}^{τ} t^{r−1}φT r^{s}(\frac{N}{2} (e^{−C_{4t}^{2}} − e^{( \frac{1}{2} w(H(Z,E|_{Z}),g^{H(Z,E|_{Z})}))^{2}})dt)_{r=0},(12)

where τ is any small positive number.

T_{1}(T^{H}M, g^{Z}, g^{E}) is an even form on B. It shares some properties of the analytic torsion form T (T^{H}M, g^{Z}, g^{E}). We list two properties in the following.

Proposition 2 Let B and Z_{i} be closed Riemannian manifolds. Suppose Z_{i} → M_{i} →^{π_{i}} B is a compact fiber bundle, and E_{i} is a flat complex vector bundle over M_{i} with Hermitian metric g^{E_{i}}, i = 1, 2. Let Z = Z_{1}×Z_{2} with the product metric g^{Z} = g^{Z_{1}}×g^{Z_{2}}, and Z → M → B be the product fiber bundle of M_{1} and M_{2}. Form E = P_{1}^{∗}(E_{1})⊗P_{2}^{∗}(E_{2}) and g^{E} = P_{1}^{∗}(g^{E_{1}})⊗P_{2}^{∗}(g^{E_{2}}), where P_{i} : M → M_{i} is the natural projection. Then

T_{1}(T^{H}M, g^{Z}, g^{E}) = rk(E_{1})χ(Z_{1})T_{1}(T^{H}M_{2}, g^{Z_{2}}, g^{E_{2}}) + rk(E_{2})χ(Z_{2})T_{1}(T^{H}M_{1}, g^{Z_{1}}, g^{E_{1}}). (13)

Proof. Note that

Ω(M, E) ≃ Ω(B) ˆ⊗C^{∞}(M, E)
≃ Ω(B) ˆ⊗C^{∞}(B, (∧(T^{∗}Z_{1}) ⊗ E_{1}|_{Z_{1}}) ˆ⊗(∧(T^{∗}Z_{2}) ⊗ E_{2}|_{Z_{2}})),
N = N_{1} ˆ⊗I + I ˆ⊗N_{2},

and

D_{t}^{2} = D_{1,t}^{2} ˆ⊗I + I ˆ⊗D_{2,t}^{2}.

Hence,

T r^{s}(\frac{N}{2} e^{D_{t}^{2}}) = T r^{s}(\frac{N}{2} e^{D_{1,t}^{2}}e^{D_{2,t}^{2}})
= T r^{s}(\frac{N_{1}}{2} e^{D_{1,t}^{2}})T r^{s}(e^{D_{2,t}^{2}}) + T r^{s}(\frac{N_{2}}{2} e^{D_{2,t}^{2}})T r^{s}(e^{D_{1,t}^{2}}).

By local index theory techniques, one can check (see [BL]) that

T r^{s}(e^{D_{i,t}^{2}}) = rk(E_{i})χ(Z_{i}).

Now (13) follows clearly from (12). Q.E.D.

Recall that the analytic torsion invariant T (M_{x}, E_{x}) of each fiber M_{x} associated with vector bundle E_{x} = E|_{M_{x}}, x ∈ B, is defined to be

T (M_{x}, E_{x}) = e^{\frac{1}{2} \frac{∂}{∂u} T r^{s}(N(−P_{Vx}^{⊥}(V_{x})^{2}P_{Vx}^{⊥}))^{−u}|_{u=0}},  (14)

where V_{x} = (d_{Z}^{∗} − d_{Z})|_{M_{x}}, P_{V_{x}}^{⊥} = 1 − P_{V_{x}}, and P_{V_{x}} is the orthogonal projection onto the kernel of V_{x}. The following lemma asserts that the zero-form component T_{1}(T^{H}M, g^{Z}, g^{E})_{[0]} of T_{1}(T^{H}M, g^{Z}, g^{E}) at x ∈ B is equal to log T (M_{x}, E_{x}).

Proposition 3 T_{1}(T^{H}M, g^{Z}, g^{E})_{[0]}(x) = log T (M_{x}, E_{x}).

Proof. This is obvious, since w(H(Z, E|_{Z}), g^{H(Z,E|_{Z})}) is a one-form on B, by (12) and (14),

T_{1}(T^{H}M, g^{Z}, g^{E})_{[0]}(x) = ∫_{τ}^{∞} φT r^{s}(\frac{N}{2} (e^{V_{x}^{2}} − P_{V_{x}}))\frac{dt}{t}
+ \frac{d}{dr}(\frac{1}{Γ(r)} ∫_{0}^{τ} t^{r−1}φT r^{s}(\frac{N}{2} (e^{V_{x}^{2}} − P_{V_{x}}))dt)_{r=0}
= log T (M_{x}, E_{x}).

Q.E.D.

The modified torsion form T_{1}(T^{H}M, g^{Z}, g^{E}) is still quite mystery. For instance, we do not know how this form depends on its arguments and when it is a closed form on B. The difficulty is that we do not have at moment (2) and (3) of Lemma 1 for the complementary error function erfc(x). From this point of view, the analytic torsion form of Bismut-Lott has advantage over the modified torsion form. However, it is much more convenient to use the modified torsion form in the adiabatic limit. This is the main reason that we discuss the modified torsion form here.

4 Local Mixed Index Theorem

The main purpose of this section is to introduce a mixed super trace on ∧(T^{∗}M) ≃ π^{∗}(∧(T^{∗}B)) ˆ⊗∧(T^{∗}Z), which combines the super trace on π^{∗}(∧(T^{∗}B)) corresponding to the grading given by the Hodge star operator and the super trace on ∧(T^{∗}Z) associated with the grading given by the degree of forms. We apply this mixed super trace to the Euler-de Rham operator on the total space M and get a local mixed index theorem.

Let Z → M →^{π} B be the compact fiber bundle as in Section 2. Recall that the Hodge operator ∗_{M} on M is given by the chirality operator on ∧^{∗}(T^{∗}M), namely, ∗_{M}(ξ) = i^{p}c(x^{1}) ⋯ c(x^{dim(M)})ξ, where {x^{i}} is a local orthonormal basis of T^{∗}M, c(x) = (x∧) − i_{x}, p = \frac{dim(M)}{2} for dim(M) even, and p = \frac{dim(M)+1}{2} for dim(M) odd. The definition of ∗_{M} is slightly different from the usual one since ∗_{M}^{2} = 1. Let C(T^{∗}M) be the bundle of Clifford algebras over M whose fiber over x ∈ M is the Clifford algebra C(T_{x}^{∗}M) of the Euclidean space T_{x}^{∗}M. Then ∧^{∗}(T^{∗}M) is a Z_{2}-graded Clifford-bundle over M with a C(T^{∗}M)-action given by vξ = c(v)ξ, v ∈ T_{x}^{∗}M and ξ ∈ ∧^{∗}(T_{x}^{∗}M), and with a grading given by the degree of forms. We have a natural Z_{2}-grading on End(∧^{∗}(T^{∗}M)) and a super trace tr^{s} on C^{∞}(M, End(∧^{∗}(T^{∗}M))) given by

tr^{s}(a) = tr(a_{11}) − tr(a_{22})

for a = (a_{ij})_{i,j=1}^{2} on ∧^{∗}(T^{∗}M) = ∧^{ev}(T^{∗}M) ⊕ ∧^{odd}(T^{∗}M). There is a super trace T r^{s} on L^{2}(M, ∧^{∗}(T^{∗}M)) which is given for an operator A with a smooth kernel K_{A}(x, y) by

T r^{s}(A) = ∫_{M} tr^{s}(K_{A}(x, x)).

We assume for the moment that M is even dimensional. Then End(∧^{∗}(T^{∗}M)) = C(T^{∗}M) ⊗ End_{C(T^{∗}M)}(∧^{∗}(T^{∗}M)), and for a(x) ∈ C(T^{∗}M), b ∈ End_{C(T^{∗}M)}(∧^{∗}(T^{∗}M)),

tr^{s}(a(x) ⊗ b(x)) = (−2i)^{\frac{dim(M)}{2}} σ_{dim(M)}(a(x))tr_{r}^{s}(b(x)),

where σ_{dim(M)} : C(T^{∗}M) → ∧^{dim(M)}(T^{∗}M) is the top dimensional piece of the symbol map, (−2i)^{\frac{dim(M)}{2}} σ_{dim(M)} is the super trace on C(T^{∗}M) and tr_{r}^{s} is the relative trace given by

tr_{r}^{s}(b) = 2^{− \frac{dim(M)}{2}} tr^{s}(∗_{M}b).

tr_{r}^{s} extends to a linear map from Ω(M, End_{C(T^{∗}M)}(∧^{∗}(T^{∗}M))) to Ω(M). Thus,

tr^{s}(K_{A}(x, x)) = (−2i)^{\frac{dim(M)}{2}} tr_{r}^{s}(σ_{dim(M)}(K_{A}(x, x))).  (15)

On the other hand, we can also use the Hodge star operator ∗_{M} to grade ∧^{∗}(T^{∗}M) and define a usual trace tr_{∗} = 2^{− \frac{dim(M)}{2}} tr_{∧^{∗}(T^{∗}M)} on End_{C(T^{∗}M)}(∧^{∗}(T^{∗}M)), which can be extended to a linear map from Ω(M, End_{C(T^{∗}M)}(∧^{∗}(T^{∗}M))) to Ω(M). We have thus another pointwise super trace

tr_{∗}^{s}(K_{A}(x, x)) = (−2i)^{\frac{dim(M)}{2}} tr_{∗}(σ_{dim(M)}(K_{A}(x, x)))  (16)

and then another super trace

T r_{∗}^{s}(A) = ∫_{M} tr_{∗}^{s}(K_{A}(x, x)).  (17)

The difference between the two traces T r^{s} and T r_{∗}^{s} is that we use a relative super trace in (15) and a usual trace in (16). Let D_{M} = d̄_{M} + d_{M}, where d̄_{M} is the adjoint of d_{M} with respect to the Hermitian structure on M. Applying these two traces to the operator A = e^{−tD_{M}^{2}} and letting t → 0, we obtain the index formulas for the Euler characteristic number and the signature of M, respectively. See ([BGV] Chapter 4) for details.

Now for the total space of the fiber bundle M, we have T^{∗}M = (T^{H}M)^{∗} ⊕ T^{∗}Z and ∧^{∗}(T^{∗}M) = ∧^{∗}((T^{H}M)^{∗}) ˆ⊗ ∧^{∗} (T^{∗}Z). We may use a tensor product of the above two traces tr^{s} and tr_{∗}^{s} to get a new trace. More specifically, we assume throughout this section that the dimension of B is even. Let tr_{M}^{s} be the tensor product of the trace tr_{∗}^{s} on ∧^{∗}((T^{H}M)^{∗}) and the super trace tr^{s} on ∧^{∗}(T^{∗}Z). For an even dimensional fiber Z, tr_{M}^{s} acts on the smooth kernel K_{A}(x, y) of an operator A as tr_{M}^{s}(K_{A}(x, x)) = (−2i)^{\frac{dim(M)}{2}} tr_{∗,r}^{s}(σ_{dim(M)}(K_{A}(x, x))), where tr_{∗,r}^{s} is the tensor product of the pointwise traces tr_{r}^{s} and tr_{∗}. The trace tr_{M}^{s} together with an integral on M produces a super trace T r_{M}^{s} on L^{2}(M, ∧^{∗}(T^{∗}M)). If we tensor ∧^{∗}(T^{∗}M) with the flat vector bundle E over M and use the usual trace on E, we get a trace as above which is still denoted by T r_{M}^{s}. Thus if A is of T r_{M}^{s}-trace class operator on L^{2}(M, ∧^{∗}(T^{∗}M) ⊗ E) with a smooth kernel K_{A}(x, y), then

T r_{M}^{s}(A) = ∫_{M} tr_{M}^{s}(K_{A}(x, x)).

We first apply the trace T r_{M}^{s} to the operator D_{M} = d̄_{M} + d_{M} and consider the following invariant for an even dimensional Z,

ind_{w}(D_{M}) = lim_{t→0} T r_{M}^{s}(e^{−tD_{M}^{2}} ).  (18)

ind_{w}(D_{M}) is well defined by Theorem 4.1 [BGV]. The geometric meaning of ind_{w}(D_{M}) is that it represents a mixture of the Euler characteristic number and the signature of M.

Theorem 1 Let Z → M →^{π} B be a fiber bundle with Z and B closed Riemannian manifolds. Let B be even dimensional and E be a flat vector bundle over M with a flat connection ∇^{E}.

(i) If Z is even dimensional, then

ind_{w}(D_{M}) = (2πi)^{− \frac{dim(M)}{2}} rk(E) ∫_{B} Â(B) ∫_{Z} Â(Z)tr_{∗,r}^{s}(e^{−R̂^{M}}),  (19)

where Â(B) is the Â-class of B, R̂^{M} = \frac{1}{4}R_{jk}^{M}c_{v}^{j}c_{v}^{k}, R_{jk}^{M} =< e_{j}, R^{M}e_{k} > with the curvature R^{M} of the Levi-Civita connection ∇^{M}. Let N̄ be the number operator of the total space M.

As t → 0,

(ii) T r_{M}^{s}(e^{−tD_{M}^{2}} ) = \{ \begin{matrix}ind_{w}(D_{M}) + O(t), & dim(Z) even, \\ O(\sqrt{t}), & dim(Z) odd. \end{matrix}
(iii) T r_{M}^{s}(N̄e^{−tD_{M}^{2}} ) = \{ \begin{matrix}\frac{dim M}{2} ind_{w}(D_{M}) + O(t), & dim(Z) even, \\ \frac{1}{\sqrt{t}}rk(E) ∫_{M} a_{M} + O(\sqrt{t}), & dim(Z) odd, \end{matrix}

where a_{M} is an odd form on M. (iv) Let V_{M} = d̄_{M} − d_{M}.

T r_{M}^{s}(\sqrt{t}V_{M}e^{−tD_{M}^{2}} ) = \{ \begin{matrix}O(t), & dim(Z) even, \\ O(\sqrt{t}), & dim(Z) odd. \end{matrix}

(v) T r_{M}^{s}((1 − 2tD_{M}^{2})e^{−tD_{M}^{2}} ) = \{ \begin{matrix}ind_{w}(D_{M}) + O(t), & dim(Z) even, \\ O(\sqrt{t}), & dim(Z) odd. \end{matrix}
(vi) T r_{M}^{s}(\frac{N̄}{2} (1 − 2tD_{M}^{2})e^{−tD_{M}^{2}} ) = \{ \begin{matrix}\frac{dim(M)}{4}ind_{w}(D_{M}) + O(t), & dim(Z) even, \\ O(\sqrt{t}), & dim(Z) odd. \end{matrix}

Proof. (i) Since (19) is local, we can assume B and Z are oriented and spin manifolds. By Theorem 4.1 [BGV], the heat kernel K_{t}(x, y) of D_{M} has the following expansion,

K_{t}(x, x) ∼ (4πt)^{− \frac{dim(M)}{2}} ∑_{i=0}^{∞} t^{i}K_{i}(x),  (20)

with K_{i}(x) ∈ C^{∞}(M, C_{2i}(T^{∗}M) ⊗ End_{C(T^{∗}M)}(∧^{∗}(T^{∗}M))). Here C_{2i}(T^{∗}M) is the subset of elements in the Clifford bundle C(T^{∗}M) with degree ≤ 2i. The symbol of sum of the Ki^{′}s is

σ(K) = ∑_{i=0}^{\frac{dim(M)}{2}} σ_{2i}(K_{i}) = Â(M) exp(−R̂^{M⊗E,u}) ∈ Ω(M, End_{C(T^{∗}M)}(∧^{∗}(T^{∗}M))).

Here R̂^{M⊗E,u} = R̂^{M} ⊗ I + I ⊗ R^{E,u}, and R^{E,u} is the curvature of ∇^{E,u} = ∇^{E} + \frac{1}{2}w(∇^{E}, g^{E}), R^{E,u} = −\frac{1}{4}w(∇^{E}, g^{E})^{2}. For a ∈ C_{2i}(T^{∗}M), i < \frac{dim(M)}{2}, we know that its super trace is zero. Thus,

ind_{w}(D_{M}) = lim_{t→0}(4πt)^{− \frac{dim(M)}{2}} ∑_{i≥ \frac{dim(M)}{2}} t^{i} ∫_{M} tr_{M}^{s}(K_{i}(x))
= (4π)^{− \frac{dim(M)}{2}} ∫_{M} tr_{M}^{s}(K_{\frac{dim(M)}{2}} (x))
= (4π)^{− \frac{dim(M)}{2}} ∫_{M} Â(M)tr_{M}^{s}(e^{−R̂^{M⊗E,u}})
= (2πi)^{− \frac{dim(M)}{2}} ∫_{M} Â(M)tr_{∗,r}^{s}(e^{−R̂^{M}})tr(e^{−R^{E,u}}).

We obtain from ([BL], (3.77)) that tr^{s}(e^{−R^{E,u}}) = rk(E). This implies (19).

(ii) If Z is even dimensional, the result follows from Part (i) and (20). For odd dimensional Z, we still have Formula (20). However, the heat kernel in this case contains only the half-integer powers of t. Also tr_{M}^{s}(a) = 0 for a ≠ c^{1} ⋯ c^{dim(M)}c_{v}^{1} ⋯ c_{v}^{dim(M)}, since both tr_{∗}^{s} and tr^{s} have such a property. Thus, by local index theory techniques, we get that as t → 0,

lim_{t→0} T r_{M}^{s}(e^{−tD_{M}^{2}} ) = lim_{t→0} ∑_{i> \frac{dim(M)}{2}} t^{i− \frac{dim(M)}{2}} ∫_{M} tr_{M}^{s}(K_{i}(x)) = 0,

since tr_{M}^{s}(K_{i}(x)) is an even form and dim(M) is odd.

(iii) Let {e_{a}} and {e^{a}} be local orthonormal bases of T M and T^{∗}M, respectively. The operator N̄ can be written as

N̄ = \frac{1}{2} ∑_{i=1}^{dim(M)} c(e_{a_{i}})c_{v}(e_{a_{i}}) + \frac{dim(M)}{2},

where c(e_{a_{i}}) and c_{v}(e_{a_{i}}) are the Clifford variables associated with e_{a_{i}} [BZ]. Using the Getzler rescaling G\sqrt{t}, we get that as t → 0,

G\sqrt{t}(\frac{\sqrt{t}}{2} ∑_{i=1}^{dim(M)} c(e_{a_{i}})c_{v}(e_{a_{i}})) → \frac{1}{2} ∑_{i=1}^{dim(M)} e^{a_{i}} ∧ c_{v}(e_{a_{i}})

and

G\sqrt{t}(tD_{M}^{2}) → (∂_{a} − \frac{1}{4}R_{ab}^{M}x_{b})^{2} + R̂^{M⊗E,u}.

Therefore, for the even dimensional Z,

T r_{M}^{s}(\frac{\sqrt{t}}{2} ∑_{i=1}^{dim(M)} c(e_{a_{i}})c_{v}(e_{a_{i}})e^{−tD_{M}^{2}} ) →

\frac{(2πi)^{− \frac{dim(M)}{2}}}{2} ∫_{M} Â(M)tr_{∗,r}^{s}( ∑_{i=1}^{dim(M)} e^{a_{i}}c_{v}(e_{a_{i}})e^{−R̂^{M}} tr^{s}(e^{−R^{E,u}})).

As shown in ([BZ], XI) with the usual trace replaced by the mixed one, we obtain that the above limit is zero. Hence by Part (ii), we have

lim_{t→0} T r_{M}^{s}(N̄e^{−tD_{M}^{2}} ) = lim_{t→0} T r_{M}^{s}(\frac{dim(M)}{2}e^{−tD_{M}^{2}} ) = \frac{dim(M)}{2}ind_{w}(D_{M}).

The approximate rate is clear. Similarly, one can prove the odd dimensional case.

(iv) Let dim(Z) be even. We may apply the proof of Theorem 3.16 [BL] to the situation here. Indeed, let z be an odd Grassmann variable which anticommutes with all other Grassmann variables, z^{2} = 0. For ξ ∈ Ω(B) ⊗ ℂ[z], ξ = ξ_{0} + zξ_{1}, ξ_{i} ∈ Ω(B), we define ξ^{z} = ξ_{1}.

Then

lim_{t→0} T r_{M}^{s}(\sqrt{t}V_{M}e^{−tD_{M}^{2}} ) = lim_{t→0} T r_{M}^{s}(\frac{1}{\sqrt{t}}e^{−t(D_{M}^{2} −zV_{M})})^{z}
= (2πi)^{− \frac{dim(M)}{2}} ∫_{M} Â(M)tr_{∗,r}^{s}(e^{−R̂^{M}})tr(e^{−R^{E,u}+ \frac{z}{2} ψ})^{z}
= (2πi)^{− \frac{dim(M)}{2}} ∫_{M} Â(M)tr_{∗,r}^{s}(e^{−R̂^{M}})tr(\frac{1}{2}ψe^{−( \frac{1}{2} ψ)^{2}}) = 0,

since tr(\frac{1}{2}ψe^{−( \frac{1}{2} ψ)^{2}}) is an odd form. Similarly, by the argument of Part (ii), we get the result for the odd dimensional case.

(v) By local index theory techniques, we have that for the even dimensional Z,

lim_{t→0} T r_{M}^{s}(−2tD_{M}^{2}e^{−tD_{M}^{2}} ) = lim_{t→0} T r_{M}^{s}(e^{−t(1+2z)D_{M}^{2}} )^{z}

= \{∫_{M}(2πi(1 + 2z))^{− \frac{dim(M)}{2}} det^{\frac{1}{2}}(\frac{(1 + 2z)\frac{R^{M}}{2}}{sinh(1 + 2z)\frac{R^{M}}{2}} )tr_{∗,r}^{s}(e^{−(1+2z)R̂^{M⊗E},u})\}^{z}
= (2πi)^{− \frac{dim(M)}{2}} 2 ∫_{M} Â(M)tr_{∗,r}^{s}(R̂^{M}e^{−R̂^{M}})tr(R^{E,u}e^{−R^{E,u}}) = 0.

The last step follows from the fact that tr(R^{E,u}e^{−R^{E,u}}) = 0 by Proposition 1.3 in [BL]. Hence by Part (ii), we obtain the result for the even dimensional Z. If Z is odd dimensional, we use the above argument together with the proof of Part (ii) to get the assertion.

(vi) This part essentially follows from Part (v) and the proof of Theorem 3.21 [BL]. Indeed, consider the following trivial fiber bundle Z′ → M′ = M × ℝ_{+}^{∗} →^{π′} ℝ_{+}^{∗} over ℝ_{+}^{∗}, where π′(x, s) = s and Z′ ≃ M. Let π′′ : M′ → M be the natural projection. Then T Z′ = (π′′)^{∗}(T M) and (π′′)^{∗}E is a flat vector bundle over M′. We define a metric g^{Z′} on T Z′ such that g^{Z′}|_{M×\{s\}} is equal to \frac{(π′′)^{∗}g^{M}}{s}. Let the horizontal distribution T^{H}M′ of M′ be given by T^{H}M′ = (π′)^{∗}T ℝ_{+}^{∗}. We have that ∇^{Z′} = (π′′)^{∗}∇^{M} + ds(\frac{∂}{∂s} − \frac{1}{2s}), R^{Z′} = (π′′)^{∗}R^{M} and R^{Z′}(\frac{∂}{∂s}, ·) = 0. Clearly,

d_{M′} = d_{M} + ds∂_{s},

and the fiberwise adjoint d_{M′}^{∗} of d_{M′} is

d_{M′}^{∗} = s^{−(N̄− \frac{dim M}{2} )}(d̄_{M} + ds∂_{s})s^{(N̄− \frac{dim(M)}{2} )}
= sd̄_{M} + ds(∂_{s} + \frac{1}{s}(N̄ − \frac{dim(M)}{2})).

Here d̄_{M} is the adjoint of d_{M}. Using the notation of Section 2, we get

2D′_{t} = t^{− \frac{N̄}{2}} d_{M′}^{∗}t^{\frac{N̄}{2}} − t^{\frac{N̄}{2}} d_{M′}t^{− \frac{N̄}{2}}

= s^{− \frac{N̄}{2}} 2D_{st}s^{\frac{N̄}{2}} + \frac{ds}{s}(N̄ − \frac{dim(M)}{2}).

By the Taylor expansion,

T r_{M}^{s}(f(2D′_{t}))(s) = T r_{M}^{s}(s^{− \frac{N̄}{2}} f(2D_{st})s^{\frac{N̄}{2}} ) + \frac{2ds}{s}T r_{M}^{s}(s^{− \frac{N̄}{2}} \frac{N̄}{2} f′(2D_{st})s^{\frac{N̄}{2}} )
− \frac{dim(M)}{2s}dsT r_{M}^{s}(s^{− \frac{N̄}{2}} f′(2D_{st})s^{\frac{N̄}{2}} ).

Using the proof of Part (iv) and the fact that R_{v}^{Z′}(\frac{∂}{∂s}, ·) = 0, we obtain that the limits of T r_{M}^{s}(f(2D′_{t}))(s) and T r_{M}^{s}(s^{− \frac{N̄}{2}} f(2D_{st})s^{\frac{N̄}{2}} )(1) as t → 0 do not contain a ds-term. Hence the ds term of the above identity together with Part (v) proves the assertion. Q.E.D.

Zhang [Zh] has recently obtained a similar index formula for the general case where T^{H}M could be any sub-bundle E_{1} of T M. In fact, for the even rank E_{1}, our method works in this general case as well, since ∧^{∗}(T^{∗}M) ≃ ∧^{∗}(E_{1}^{∗}) ˆ⊗ ∧^{∗} ((E_{1}^{⊥})^{∗}) and a super trace could similarly be defined. The point of our method is that we can directly use the usual formulas since we only change the super trace. See also [Go] for the Euler-signature type operator.

For the use of the next section, we list the following proposition whose proof is obvious.

Proposition 4 With the assumption of Theorem 1, there is a constant λ_{0} > 0 such that for t → ∞,

1) T r_{M}^{s}(e^{−tD_{M}^{2}} ) = T r_{M}^{s}(P_{Ker D_{M}}) + O(e^{−λ_{0}t}),
2) T r_{M}^{s}(N̄e^{−tD_{M}^{2}} ) = T r_{M}^{s}(N̄P_{Ker D_{M}}) + O(e^{−λ_{0}t}),
3) T r_{M}^{s}(N̄((1 − 2tD_{M}^{2})e^{−tD_{M}^{2}} ) = T r_{M}^{s}(N̄P_{Ker D_{M}}) + O(e^{−λ_{0}t}).

5 Torsion-Signature Invariants

In this section we use the mixed super trace to define a torsion type invariant for the operator D_{M}. Denote by D_{M}^{⊥} the restriction of D_{M} to the orthogonal complement of KerD_{M}. As usual, we have that T r_{M}^{s}(\frac{N̄}{2} e^{−t(D_{M}^{⊥})^{2}}) ∼ O(e^{−λ_{0}t}) as t → ∞ and for t → 0, T r_{M}^{s}(\frac{N̄}{2} e^{−t(D_{M}^{⊥})^{2}}) has an asymptotic expansion in t which is the integral powers of t for dim(M) even and the half-integer powers of t for dim(M) odd. Thus

ζ(r) = \frac{1}{Γ(r)} ∫_{0}^{∞} T r_{M}^{s}(\frac{N̄}{2} e^{−t(D_{M}^{⊥})^{2}})t^{r} \frac{dt}{t}, ℜr > \frac{dim(M)}{2},  (21)

extends to a meromorphic function of r ∈ ℂ such that ζ(r) is holomorphic near the zero. Definition 2. The torsion-signature T_{w}(M, E) of M associated with the flat vector bundle E is defined as

log T_{w}(M, E) = \frac{dζ(r)}{dr}|_{r=0}.

T_{w}(M, E) is very special for M = B × Z and E = π_{1}^{∗}(E_{B}) ⊗ π_{2}^{∗}(E_{Z}) , where π_{2} : B × Z → Z is the natural projection and E_{B} and E_{Z} are two flat vector bundles over B and Z, respectively. In this case, D_{M}^{2} = D_{B}^{2} ⊗ 1 + 1 ⊗ D_{Z}^{2} and

T r_{M}^{s}(\frac{N̄}{2} e^{−t(D_{M}^{2} )^{⊥}}) = T r_{∗}^{s}(P_{KerD_{B}})T r^{s}(\frac{N}{2} e^{−t(D_{Z}^{2})^{⊥}}) + T r^{s}(P_{KerD_{Z}})T r_{∗}^{s}(\frac{N′}{2} e^{−t(D_{B}^{2})^{⊥}}),

where N′ is the number operator of B. By the McKean-Singer index formula,

T r_{∗}^{s}(\frac{N′}{2} e^{−t(D_{B}^{⊥})^{2}}) = const.T r(∗_{B} \frac{N′}{2} e^{−t(D_{B}^{⊥})^{2}}) = const.\frac{dim(B)}{4}T r(∗_{B}e^{−t(D_{B}^{⊥})^{2}}) = 0,

where we used ∗_{B} \frac{N′}{2} + \frac{N′}{2} ∗_{B} = \frac{dim(B)}{2}∗_{B}. We get

log T_{w}(M, E) = sig(B, E_{B}) log T (Z, E_{Z}).

Hence T_{w}(M, E) is not trivial in general. Because of this, we call T_{w}(M, E) the torsionsignature invariant of M associated with E. In particular, we consider it for the fiber bundle M = B̃×_{Γ}Z and the flat vector bundle E associated with two representations ρ_{i} of Γ = π_{1}(B) as we discussed in Section 2. In this case, we denote T_{w}(M, E) by T_{w}(B, Z, ρ_{1}, ρ_{2}).

Proposition 5 (i) For B = {pt}, T_{w}({pt}, Z, id, id) is equal to the analytic torsion T (Z, E) of Z associated with the trivial vector bundle E. (ii) For Z = {pt}, log T_{w}(B, {pt}, id, id) = 0. (iii) log T_{w}(B, Z, id, id) = sig(B, E_{B}) log T (Z, E_{Z}), where id means the trivial representation of Γ.

Proof. (i) For B = {pt}, T r_{M}^{s} is the usual super trace on L^{2}(Z, ∧^{∗}(T^{∗}Z) ⊗ E). The result is obviously true.

(ii) Since T r_{M}^{s} = T r_{∗}^{s} on L^{2}(B, ∧^{∗}(T^{∗}B) ⊗ E) and T r_{∗}^{s}(\frac{N′}{2} e^{−t(D_{B}^{2})^{⊥}}) = 0 by the above argument, the assertion clearly holds.

(iii) This is the trivial case that we just discussed above. Q.E.D.

Part (iii) shows that T_{w}(M, E) might be independent of the metric in general only when the cohomology H^{∗}(Z, E|_{Z}) = 0. An unsolved problem is to find a formula for the dependence of T_{w}(M, E) on the metric.

The following version of T_{w}(M, E) will be used in the next section. Let u(x) = (1−2x)e^{−x}.

Proposition 6

log T_{w}(M, E) = ∫_{0}^{∞}[T r_{M}^{s}(\frac{N̄}{2} u(tD_{M}^{2})) − T r_{M}^{s}(\frac{N̄}{2} P_{KerD_{M}})
− (\frac{dim(M)}{4}ind_{w}(D_{M}) − T r_{M}^{s}(\frac{N̄}{2} P_{KerD_{M}}))u(t)]\frac{dt}{t}.  (22)

Proof. By Theorem 1 and Proposition 4, the right hand side of (22) is well defined. For an even dimensional Z, we have

RHS of (22) = ∫_{0}^{∞}[T r_{M}^{s}(\frac{N̄}{2} u(tD_{M}^{2})^{⊥}) − (\frac{dim(M)}{4}ind_{w}(D_{M}) − T r_{M}^{s}(\frac{N̄}{2} P_{KerD_{M}}))u(t)]\frac{dt}{t}
= \frac{d}{dr}\{\frac{1}{Γ(r)} ∫_{0}^{∞}[T r_{M}^{s}(\frac{N̄}{2} e^{−t(D_{M}^{2} )^{⊥}}) − (\frac{dim(M)}{4}ind_{w}(D_{M})
− T r_{M}^{s}(\frac{N̄}{2} P_{KerD_{M}}))u(t)]t^{r} \frac{dt}{t}\}_{r=0} + ∫_{0}^{∞} 2\frac{d}{dt}T r_{M}^{s}(\frac{N̄}{2} e^{−tD_{M}^{2}} )dt
= log T_{w}(M, E) + T r_{M}^{s}(N̄P_{KerD_{M}}) − \frac{dim(M)}{2}ind_{w}(D_{M})
− \frac{d}{dr}\{\frac{1}{Γ(r)} ∫_{0}^{∞}(\frac{dim(M)}{4}ind_{w}(D_{M}) − T r_{M}^{s}(\frac{N̄}{2} P_{KerD_{M}}))u(t)]t^{r} \frac{dt}{t}\}_{r=0}
= log T_{w}(M, E).

Here we used Theorem 1 (iii). The above proof is also valid for the odd dimensional case except that we have to consider

\frac{d}{dr}\{\frac{1}{Γ(r)} ∫_{0}^{∞} T r_{M}^{s}(\frac{N̄}{2} (−2tD_{M}^{2})e^{−tD_{M}^{2}} )t^{r} \frac{dt}{t}\}_{r=0}

= ∫_{ε}^{∞} \frac{d}{dt}T r_{M}^{s}(N̄e^{−tD_{M}^{2}} )dt + \frac{d}{dr}\{\frac{1}{Γ(r)} ∫_{0}^{ε} t^{r} \frac{d}{dt}T r_{M}^{s}(N̄e^{−tD_{M}^{2}} )dt\}_{r=0}
= T r_{M}^{s}(N̄P_{KerD_{M}}).

Q.E.D.

We will link the T_{w}(M, E) to the analytic torsion form and the invariant T s(B, M, E) by an adiabatic limit formula in the next section. We refer to [GP] for the corresponding invariant in the operator situation.

6 Adiabatic Limits of Torsion-Signature Invariants

Our purpose of this section is to find an adiabatic limit formula of the invariant T_{w}(M, E) which is related to the analytic torsion form and the invariant T s(B, M, E) defined in Section 2. Unless specifically stated, we assume throughout this section that B is an even dimensional closed Riemannian manifold, M is a compact fiber bundle over B with a closed Riemannian manifold Z as typical fiber, and that E is a flat vector bundle over M.

Recall the exterior differential d_{M} of M with coefficients in E is equal to

d_{M} = d_{Z} + ∇^{E} + i_{T} .

Let d̄_{M}, d̄_{Z}, ¯∇^{E} and ī_{T} be the adjoints of these operators with respect to the Hermitian structure on the total space M. We know Ω(M, E) ≃ Ω(B) ˆ⊗C^{∞}(B, ∧^{∗}(T^{∗}Z, E|_{Z})). As before, denote by N̄ and N′ the number operators on Ω(M, E) given by N̄ξ = jξ and N′ξ = kξ for ξ ∈ Ω^{k}(B) ˆ⊗C^{∞}(B, ∧^{l}(T^{∗}Z, E|_{Z})), k + l = j. Now the adiabatic limit amounts to blowing up the metric of the base space B by considering the metric g_{ε}^{M} = \frac{π^{∗}(g^{B})}{ε} ⊕ g^{Z} on M and then taking the limit as ε → 0. With respect to this new metric, the adjoint of the exterior differential d_{M,ε} of M is

d̄_{M,ε} = ε^{−N′} d̄_{M}ε^{N′} = d̄_{Z} + ε^{−1} ¯∇^{E}ε + ε^{−2}ī_{T} ε^{2}.

Consider

C_{ε} = \frac{1}{2}(ε^{−N′} d̄_{M}ε^{N′} + d_{M})

and

D_{ε} = \frac{1}{2}(ε^{−N′} d̄_{M}ε^{N′} − d_{M}).

We have that with a local orthonormal frame {e_{i}, e_{α}} of T M as in Section 2 where we used f_{α} for e_{α},

C_{ε} = \frac{1}{2}[(m^{j}∇_{e_{j},ε}^{M⊗E} + m^{α}\sqrt{ε}∇_{e_{α},ε}^{M⊗E})^{∗} + (m^{j}∇_{e_{j},ε}^{M⊗E} + m^{α}\sqrt{ε}∇_{e_{α},ε}^{M⊗E})]
= \frac{1}{2}(c^{j}∇_{e_{j},ε}^{M⊗E,u} + \sqrt{ε}c^{α}∇_{e_{α},ε}^{M⊗E,u}) − \frac{1}{4}c_{v}^{j}ψ_{j} − \frac{\sqrt{ε}}{4}c_{v}^{α}ψ_{α},  (23)

where ∇_{,ε}^{M⊗E} = ∇_{,ε}^{M} ⊗ 1 + 1 ⊗ ∇_{,ε}^{E}, ∇_{,ε}^{M⊗E,u} = ∇_{,ε}^{M⊗E} + \frac{1}{2}ψ, and ∇_{,ε}^{M} is the Levi-Civita connection of M with respect to the metric g_{ε}^{M}. Using the usual Lichnerowicz formula ([BGV], P.126), we get

C_{ε}^{2} = \frac{1}{4}\{(c^{j}∇_{e_{j},ε}^{M⊗E,u} + \sqrt{ε}c^{α}∇_{e_{α},ε}^{M⊗E,u})^{2} − \frac{1}{2}[c^{j}∇_{e_{j},ε}^{M⊗E,u} + \sqrt{ε}c^{α}∇_{e_{α},ε}^{M⊗E,u}, c_{v}^{j}ψ_{j} + \sqrt{ε}c_{v}^{α}ψ_{α}]
+ \frac{1}{4}(c_{v}^{j}ψ_{j} + \sqrt{ε}c_{v}^{α}ψ_{α})^{2}\}
= \frac{1}{4}\{−Δ_{ε}^{M⊗E,u} + \frac{K_{ε}}{4} + \frac{1}{2}c^{i}c^{j}R_{ε}(e_{i}, e_{j}) + \sqrt{ε}c^{i}c^{α}R_{ε}(e_{i}, e_{α})
+ \frac{1}{2}εc^{α}c^{β}R_{ε}(e_{α}, e_{β}) − \frac{1}{2}[c^{j}(∇_{e_{j},ε}^{M⊗E,u}c_{v}^{l})ψ_{l} + \sqrt{ε}c^{j}(∇_{e_{j},ε}^{M⊗E,u}c_{v}^{α})ψ_{α}
+ c^{j}(c_{v}^{l}(∇_{e_{j},ε}^{M⊗E,u}ψ_{l}) + \sqrt{ε}c_{v}^{α}(∇_{e_{j},ε}^{M⊗E,u}ψ_{α}))
+ \sqrt{ε}c^{α}((∇_{e_{α},ε}^{M⊗E,u}c_{v}^{l})ψ_{l} + \sqrt{ε}(∇_{e_{α},ε}^{M⊗E,u}c_{v}^{β})ψ_{β})
+ \sqrt{ε}c^{α}(c_{v}^{l}(∇_{e_{α},ε}^{M⊗E,u}ψ_{l}) + \sqrt{ε}c_{v}^{β}(∇_{e_{α},ε}^{M⊗E,u}ψ_{β}))]
+ \frac{1}{4}(c_{v}^{j}c_{v}^{k}ψ_{j}ψ_{k} + \sqrt{ε}c_{v}^{j}c_{v}^{α}ψ_{j}ψ_{α} + \sqrt{ε}c_{v}^{α}c_{v}^{k}ψ_{k}ψ_{α} + εc_{v}^{α}c_{v}^{β}ψ_{α}ψ_{β})\},  (24)

where Δ_{ε}^{M⊗E,u} = ∑_{a}((∇_{e_{a},ε}^{M⊗E,u})^{2} − ∇_{∇_{ea,ε}^{M}e_{a},ε}^{M⊗E,u}), K_{ε} is the scalar curvature of (M, g_{ε}^{M}), and R_{ε} = R_{ε}^{M} ⊗ 1 + 1 ⊗ R^{E,u}.To further compute the terms in (24), let ∇^{⊕} = π^{∗}(∇^{B}) ⊕ ∇^{Z} and S = ∇^{M} − ∇^{⊕}. The torsion S has the following useful properties.

(i)2 < S(X_{1})X_{2}, X_{3} > + < T (X_{1}, X_{2}), X_{3} > + < T (X_{3}, X_{1}), X_{2} > − < T (X_{2}, X_{3}), X_{1} >= 0, where T (X_{1}, X_{2}) = ∇_{X_{1}}^{⊕}X_{2} − ∇_{X_{2}}^{⊕}X_{1} − [X_{1}, X_{2}] is the torsion of ∇^{⊕}. (ii) S is a one-form with values in antisymmetric elements of End(T M). If X ∈ T M , then S(X) maps T Z to T^{H}M. If X ∈ T^{H}M, then S(X) maps T^{H}M to T Z. (iii) S(Ũ)Ũ = 0, for Ũ ∈ T^{H}M. (iv) For Ũ_{i} ∈ T^{H}M and X_{i} ∈ T Z,

< S(X_{1})X_{2}, Ũ_{1} > = − < S(X)Ũ_{1}, X_{2} >,
< S(Ũ_{1})X_{1}, Ũ_{2} > = − < S(Ũ_{1})Ũ_{2}, X_{1} >=< S(X_{1})Ũ_{1}, Ũ_{2} > .

All other components < S(·), · > vanish. These numbers are related to w_{abc} in Section 2.

(v) < S(·), · > is independent of the metric on B. We refer to [Bi][BC] for more details about these facts. Note that T takes values in T Z and T (X_{1}, X_{2}) = 0 for X_{i} ∈ T Z and T (Ũ, Ũ) = 0 for Ũ ∈ T^{H}M. In view of these facts, we have the following.

∇_{e_{i},ε}^{M⊗E,u} = ∇_{e_{i}}^{Z⊗E,u} + \frac{1}{2} < S(e_{i})e_{j}, e_{α} > c^{j}\sqrt{ε}c^{α} + \frac{1}{4} < S(e_{i})e_{α}, e_{β} > εc^{α}c^{β},

∇_{e_{α},ε}^{M⊗E,u} = \sqrt{ε}(∇_{e_{α}}^{⊕⊗E,u} + \frac{1}{2} < S(e_{α})e_{i}, e_{β} > c^{i}\sqrt{ε}c^{β}),
∇_{∇_{e_{i},ε}^{M}e_{i},ε}^{M⊗E,u} = (∇^{Z⊗E,u} + S^{ε})_{∇_{e_{i}}^{Z}e_{i}} + (∇^{Z⊗E,u} + S^{ε})_{εS(e_{i})e_{i}},

and

∇_{∇_{eα,ε}^{M}e_{α},ε}^{M⊗E,u} = \sqrt{ε}(∇^{M⊗E,u} + S^{ε})_{\sqrt{ε}∇_{eα}^{M}e_{α}}.

Here S^{ε} = ∇_{ε}^{M} − ∇_{ε}^{⊕}. We use the convention that (∇_{e_{a}} + A(e_{a}))^{2} = ∑_{a}(∇_{e_{a}} + A(e_{a}))_{r}^{2} − (∇ + A)_{∑_{a} ∇_{ea}e_{a}}, where (.)_{r} means an honest square. Then

Δ_{ε}^{M⊗E,u} = (∇_{e_{i}}^{Z⊗E,u} + \frac{1}{2} < S(e_{i})e_{j}, e_{α} > c^{j}\sqrt{ε}c^{α} + \frac{1}{4} < S(e_{i})e_{α}, e_{β} > εc^{α}c^{β})^{2}
− ∇_{∑_{i} εS(e_{i})e_{i}}^{Z⊗E,u} − S^{ε}(∑_{i} εS(e_{i})e_{i})
+ (\sqrt{ε}(∇_{e_{α}}^{⊕⊗E,u} + \frac{1}{2} < S(e_{α})e_{i}, e_{β} > c^{i}\sqrt{ε}c^{β}))^{2}.  (25)

By (24) and (25), we obtain the following important Lichnerowicz type formula. Theorem 2

C_{ε}^{2} = \frac{1}{4}\{−(∇_{e_{i}}^{Z⊗E,u} + \frac{1}{2} < S(e_{i})e_{j}, e_{α} > c^{j}\sqrt{ε}c^{α} + \frac{1}{4} < S(e_{i})e_{α}, e_{β} > εc^{α}c^{β})^{2}
+ ∇_{∑_{i} εS(e_{i})e_{i}}^{Z⊗E,u} + \frac{ε}{2} < S(S(e_{i})e_{i})e_{j}, e_{α} > c^{j}\sqrt{ε}c^{α}
− (\sqrt{ε}(∇_{e_{α}}^{⊕⊗E,u} + \frac{1}{2} < S(e_{α})e_{i}, e_{β} > c^{i}\sqrt{ε}c^{β}))^{2}
+ \frac{K_{ε}}{4} + \frac{1}{2}c^{i}c^{j}R_{ε}(e_{i}, e_{j}) + \sqrt{ε}c^{i}c^{α}R_{ε}(e_{i}, e_{α}) + \frac{1}{2}εc^{α}c^{β}R_{ε}(e_{α}, e_{β})
− \frac{1}{2}[c^{j}c_{v}(∇_{e_{j},ε}^{M}e_{l})ψ_{l} + \sqrt{ε}c^{j}c_{v}(∇_{e_{j},ε}^{M}e_{α})ψ_{α} + c^{j}c_{v}^{l}(∇_{e_{j},ε}^{M⊗E,u}ψ_{l})
+ \sqrt{ε}c^{j}c_{v}^{α}(∇_{e_{j},ε}^{M⊗E,u}ψ_{α}) + \sqrt{ε}c^{α}c_{v}(∇_{e_{α},ε}^{M}e_{l})ψ_{l} + εc^{α}c_{v}(∇_{e_{α},ε}^{M}e_{β})ψ_{β}
+ \sqrt{ε}c^{α}c_{v}^{l}(∇_{e_{α},ε}^{M⊗E,u}ψ_{l}) + εc^{α}c_{v}^{β}(∇_{e_{α},ε}^{M⊗E,u}ψ_{β})]
+ \frac{1}{4}(c_{v}^{j}c_{v}^{k}ψ_{j}ψ_{k} + εc_{v}^{α}c_{v}^{β}ψ_{α}ψ_{β})\}.  (26)

Q.E.D.

More generally, let z be an auxiliary odd Grassmann variable which anticommutes with all other odd Grassmann variables and z^{2} = 0. We want to compute 4tC_{ε}^{2} − 2zt^{\frac{1}{2}} D_{ε} . We have

D_{ε} = \frac{1}{2}[(m^{j}∇_{e_{j},ε}^{M⊗E} + m^{α}\sqrt{ε}∇_{e_{α},ε}^{M⊗E})^{∗} − (m^{j}∇_{e_{j},ε}^{M⊗E} + m^{α}\sqrt{ε}∇_{e_{α},ε}^{M⊗E})]
= \frac{1}{2}[−c_{v}^{j}∇_{e_{j},ε}^{M⊗E,u} − \sqrt{ε}c_{v}^{α}∇_{e_{α},ε}^{M⊗E,u} + \frac{1}{2}c^{j}ψ_{j} + \frac{\sqrt{ε}}{2}c^{α}ψ_{α}].  (27)

Let

D_{i,ε} = ∇_{e_{i}}^{Z⊗E,u} + \frac{1}{2} < S(e_{i})e_{j}, e_{α} > c^{j}\sqrt{ε}c^{α} + \frac{1}{4} < S(e_{i})e_{α}, e_{β} > εc^{α}c^{β},

and

D_{α,ε} = \sqrt{ε}(∇_{e_{α}}^{⊕⊗E,u} + \frac{1}{2} < S(e_{α})e_{i}, e_{β} > c^{i}\sqrt{ε}c^{β}).

D_{j,ε} \frac{z}{2\sqrt{t}}c_{v}^{j} + \frac{z}{2\sqrt{t}}c_{v}^{j}D_{j,ε} − \frac{z}{2\sqrt{t}}c_{v}(∇_{e_{j}}^{Z}e_{j})

= \frac{z}{2\sqrt{t}}[c_{v}(∇_{e_{j}}^{Z}e_{j}) + 2c_{v}^{j}∇_{e_{j}}^{Z⊗E,u} + \frac{1}{2} < S(e_{j})e_{l}, e_{α} > \sqrt{ε}(c_{v}^{j}c^{l} − c^{l}c_{v}^{j})c^{α}
+ \frac{1}{4} < S(e_{j})e_{α}, e_{β} > ε2c^{α}c^{β}c_{v}^{j} − c_{v}(∇_{e_{j}}^{Z}e_{j})].  (28)

D_{α,ε} \frac{z}{2\sqrt{t}}c_{v}^{α} + \frac{z}{2\sqrt{t}}c_{v}^{α}D_{α,ε} − \frac{z}{2\sqrt{t}}c_{v}(\sqrt{ε}∇_{e_{α}}^{B}e_{α})

= \frac{z\sqrt{ε}}{2\sqrt{t}}[c_{v}(∇_{e_{α}}^{B}e_{α}) + 2c_{v}^{α}∇_{e_{α}}^{⊕⊗E,u} + \frac{1}{2} < S(e_{α})e_{i}, e_{β} > \sqrt{ε}2c^{i}c^{β}c_{v}^{α}
− c_{v}(\sqrt{ε}∇_{e_{α}}^{B}e_{α})].  (29)

Hence by (27) − (29), we obtain

t[D_{j,ε} \frac{z}{2\sqrt{t}}c_{v}^{j} + \frac{z}{2\sqrt{t}}c_{v}^{j}D_{j,ε} − \frac{z}{2\sqrt{t}}c_{v}(∇_{e_{j}}^{Z}e_{j}) + D_{α,ε} \frac{z}{2\sqrt{t}}c_{v}^{α} + \frac{z}{2\sqrt{t}}c_{v}^{α}D_{α,ε}

− \frac{z}{2\sqrt{t}}c_{v}(\sqrt{ε}∇_{e_{α}}^{B}e_{α})] = −2z\sqrt{t}D_{ε} + z\sqrt{t}(\frac{1}{2}c^{j}ψ_{j} + \frac{\sqrt{ε}}{2}c^{α}ψ_{α}).  (30)

Therefore, we have proved by (26) and (30) the following Lichnerowicz type formula, which is useful in many situations. Theorem 3

4tC_{ε}^{2} − 2z\sqrt{t}D_{ε} = t\{−(∇_{e_{i}}^{Z⊗E,u} + \frac{1}{2} < S(e_{i})e_{j}, e_{α} > c^{j}\sqrt{ε}c^{α} + \frac{1}{4} < S(e_{i})e_{α}, e_{β} > εc^{α}c^{β}
− \frac{z}{2\sqrt{t}}c_{v}^{j})^{2} + ∇_{∑_{i} εS(e_{i})e_{i}}^{Z⊗E,u} + \frac{ε}{2} < S(S(e_{i})e_{i})e_{j}, e_{α} > c^{j}\sqrt{ε}c^{α}
− (\sqrt{ε}(∇_{e_{α}}^{⊕⊗E,u} + \frac{\sqrt{ε}}{2} < S(e_{α})e_{i}, e_{β} > c^{i}\sqrt{ε}c^{β} − \frac{z}{2\sqrt{t}}c_{v}^{α}))^{2}
+ \frac{K_{ε}}{4} + \frac{1}{2}c^{i}c^{j}R_{ε}(e_{i}, e_{j}) + \sqrt{ε}c^{i}c^{α}R_{ε}(e_{i}, e_{α}) + \frac{1}{2}εc^{α}c^{β}R_{ε}(e_{α}, e_{β})
− \frac{1}{2}[c^{j}c_{v}(∇_{e_{j},ε}^{M}e_{l})ψ_{l} + \sqrt{ε}c^{j}c_{v}(∇_{e_{j},ε}^{M}e_{α})ψ_{α} + c^{j}c_{v}^{l}(∇_{e_{j},ε}^{M⊗E,u}ψ_{l})
+ \sqrt{ε}c^{j}c_{v}^{α}(∇_{e_{j},ε}^{M⊗E,u}ψ_{α}) + \sqrt{ε}c^{α}c_{v}(∇_{e_{α},ε}^{M}e_{l})ψ_{l} + εc^{α}c_{v}(∇_{e_{α},ε}^{M}e_{β})ψ_{β}
+ \sqrt{ε}c^{α}c_{v}^{l}(∇_{e_{α},ε}^{M⊗E,u}ψ_{l}) + εc^{α}c_{v}^{β}(∇_{e_{α},ε}^{M⊗E,u}ψ_{β})] + \frac{1}{4}(c_{v}^{j}c_{v}^{k}ψ_{j}ψ_{k} + εc_{v}^{α}c_{v}^{β}ψ_{α}ψ_{β})\}
+ \frac{z}{2\sqrt{t}}(c^{j}ψ_{j} + \sqrt{ε}c^{α}ψ_{α}).  (31)

Q.E.D.

We now apply the Getzler rescaling G_{(εt)^{\frac{1}{2}}} to 4tC_{ε}^{2}. Namely, we make the following changes: ∇_{e_{α}} → (εt)^{− \frac{1}{2}} ∇_{e_{α}}, c^{α} → ((εt)^{− \frac{1}{2}} e_{α}∧) − (εt)^{\frac{1}{2}} ι_{e_{α}} and c_{v}^{α} → c_{v}^{α}. Then by (26),

lim_{ε→0} G_{(εt)^{\frac{1}{2}}} (4tC_{ε}^{2}) = t\{−(∇_{e_{i}}^{Z⊗E,u} + \frac{1}{2} < S(e_{i})e_{j}, e_{α} > c^{j} \frac{e_{α}}{\sqrt{t}} + \frac{1}{4t} < S(e_{i})e_{α}, e_{β} > e_{α}e_{β})^{2}

− \frac{1}{t}(∂_{α} − \frac{1}{4}R_{αβ}^{B}x_{β})^{2} + \frac{K}{4} + \frac{1}{2}c^{i}c^{j}R(e_{i}, e_{j}) + \frac{c^{i}m^{α}}{\sqrt{t}} R(e_{i}, e_{α}) + \frac{1}{2t}m^{α}m^{β}R_{0}(e_{α}, e_{β})
− \frac{1}{2}[c^{j}c_{v}^{l}(∇_{e_{j},0}^{M⊗E,u}ψ_{l}) + \frac{m^{α}}{\sqrt{t}} c_{v}(∇_{e_{α},0}^{M}e_{l})ψ_{l} + \frac{m^{α}}{\sqrt{t}} c_{v}^{l}(∇_{e_{α},0}^{M⊗E,u}ψ_{l})] + \frac{1}{4}c_{v}^{j}c_{v}^{k}ψ_{j}ψ_{k}\}.  (32)

Here we used the facts that with appropriate local orthonormal bases of T M and T^{∗}M, ∇_{e_{j},ε}^{M}e_{l} = ε(∇_{e_{j}}^{M}e_{l})^{H} (horizontal component) and ∇_{e_{j},ε}^{M}e_{α} = (∇_{e_{j}}^{M}e_{α})^{v} + ε(∇_{e_{j}}^{M}e_{α})^{H}. (·)^{v} is the vertical component. Note that ∇_{,0}^{M⊗E,u} = lim_{ε→0} ∇_{,ε}^{M⊗E,u} = ∇^{M⊗E,u} − P^{H}S = ∇^{Z⊗E,u} + ∇^{B}, where P^{H} : T M = T^{H}M ⊕ T Z → T^{H}M is the projection. R_{0}(e_{α}, e_{β}) = R(e_{α}, e_{β}) + R̂^{B}(e_{α}, e_{β}). Let C_{t} = \frac{1}{2}(t^{−N/2}d_{M}^{∗}t^{N/2} + t^{N/2}d_{M}t^{−N/2}). Here d_{M}^{∗} is the fiberwise adjoint of d_{M}. Theorem 3.11 [BL] together with (32) proves

lim_{ε→0} G_{(εt)^{\frac{1}{2}}} (4tC_{ε}^{2}) = C_{4t}^{2} − \frac{\sqrt{t}m^{α}}{2} c_{v}^{l}(∇_{e_{α}}^{B}ψ_{l})
− (∂_{α} − \frac{1}{4}R_{αβ}^{B}x_{β})^{2} + \frac{1}{2}m^{α}m^{β}R̂^{B}(e_{α}, e_{β})
= C_{4t}^{2} − \sqrt{t}C′ − (∂_{α} − \frac{1}{4}R_{αβ}^{B}x_{β})^{2} + \frac{1}{2}m^{α}m^{β}R̂^{B}(e_{α}, e_{β})
≡^{def} B_{4t},  (33)

where

C′ = \frac{m^{α}}{2} c_{v}^{l}(∇_{e_{α}}^{B}ψ_{l}).  (34)

C′ comes from the term m^{α}c_{v}^{l}(∇_{e_{α},0}^{M⊗E,u}ψ_{l}) = m^{α}c_{v}^{l}(∇_{e_{α}}^{Z⊗E,u}ψ_{l} + ∇_{e_{α}}^{B}ψ_{l}) in (32). Note that if the metric g^{E} on E is either locally independent of B or covariantly-constant with respect to ∇^{E}, then C′ = 0. In particular, if ∇^{E} is fiberwise unitary, then C′ = 0. This is the case where the adiabatic limit of η-invariants is discussed [BC].

We now come to one of our main results. Let M_{ε} be the manifold M with the metric g_{ε}^{M}. Theorem 4 Let dim(B) be even, C′ = 0 and H^{∗}(Z, E|_{Z}) = 0. If Z is even dimensional, assume further ind_{w}(D_{M}) = 0. Then

lim_{ε→0} log T_{w}(M_{ε}, E) = 2^{\frac{dim(B)}{2}} ∫_{B} φL(B)tr(−\frac{R^{B}}{2} coth \frac{R^{B}}{2} )T_{1}(T^{H}M, g^{Z}, g^{E})

+ 2^{\frac{dim(B)}{2}} ∫_{B} φL(B)T (T^{H}M, g^{Z}, g^{E}) + ∫_{B} φÂ(B)tr_{∗}(−2R̂^{B}e^{−R̂^{B}})T_{1}(T^{H}M, g^{Z}, g^{E}),(35)

where L(B) = det^{\frac{1}{2}} (\frac{R^{B}/2}{tanh R^{B}/2}) is the stable Hirzebruch L-class of B.

Proof. Since H^{∗}(Z, E|_{Z}) = 0, the operator d_{Z}^{∗} + d_{Z} is invertible. We can apply Proposition 4.41 [BC] to conclude that D_{M_{ε}} is invertible for ε very close to zero. Thus, P_{KerD_{M}} = 0. Hence T r_{M}^{s}(\frac{N̄}{2} P_{KerD_{M}}) = 0. By Proposition 6, this implies

lim_{ε→0} log T_{w}(M_{ε}, E) = lim_{ε→0} ∫_{0}^{∞} T r_{M}^{s}(\frac{N̄}{2} f′(i\sqrt{t}D_{M_{ε}}))\frac{dt}{t}.  (36)

Recall that N′ is the number operator of B. N′ = \frac{1}{2} ∑_{i=0}^{dim(B)} c(e_{a_{i}})c_{v}(e_{a_{i}}) + \frac{dim(B)}{2}. Then

N̄ = (N + \frac{dim(B)}{2}) + \frac{1}{2} ∑_{i=0}^{dim(B)} c(e_{a_{i}})c_{v}(e_{a_{i}}).

We thus break up (36) into two terms which correspond to the two terms of N̄. Let us first consider the term associated to (N + \frac{dim(B)}{2}). Let g(x) = (1 + 2x)e^{x}. By the fact that V_{M,ε}^{2} = −C_{ε}^{2} and (33), we have

lim_{ε→0} T r_{M}^{s}(\frac{2N + dim(B)}{4}g(4tV_{M,ε}^{2})) = lim_{ε→0} T r_{M}^{s}(\frac{2N + dim(B)}{4}g(G_{(εt)^{\frac{1}{2}}} (−4tC_{ε}^{2})))
= T r_{M}^{s}(\frac{2N + dim(B)}{4}g(−B_{4t}^{2}))  (37)

uniformly for t ∈ [δ, T] and arbitrary δ, T > 0. (37) also holds uniformly for t ∈ [0, δ]. Indeed by (33), lim_{ε→0} G_{ε^{\frac{1}{2}}} (4C_{ε}^{2}) = B_{4}. The asymptotic expansion of T r_{M}^{s}(\frac{2N+dim(B)}{4}g(−tG\sqrt{ε}(4C_{ε}^{2}))) at the origin depends on the local symbol of the operator G\sqrt{ε}(4C_{ε}^{2}), and its coefficients approximate to those in the expansion of T r_{M}^{s}(\frac{2N+dim(B)}{4}g(−tB_{4})) with remainder bounded by a number independent of ε. Hence we obtain

∫_{0}^{T} T r_{M}^{s}(\frac{2N + dim(B)}{4}g(4tV_{M,ε}^{2}))\frac{dt}{t} → ∫_{0}^{T} T r_{M}^{s}(\frac{2N + dim(B)}{4}g(−B_{4t}))\frac{dt}{t}, ε → 0.

To estimate the large time contribution of T r_{M}^{s}(\frac{2N+dim(B)}{4}g(4tV_{M,ε}^{2})), we need the assumption that H^{∗}(Z, E|_{Z}) = 0. Thus there exists a λ_{0} > 0 such that the eigenvalues of −V_{M,ε}^{2} are greater than λ_{0} (cf. [BC], Prop. 4.41).

T r_{M}^{s}(\frac{2N + dim(B)}{2}e^{4tV_{M,ε}^{2}}) ≤ const.e^{−(4t−1)λ_{0}}T r_{M}^{s}(e^{V_{M,ε}^{2}})
≤ const.e^{−(4t−1)λ_{0}}ε^{− \frac{dim(B)}{2}} , t → ∞.  (38)

Here we used the fact that the metrics g_{ε}^{M} have uniformly bounded geometry with volume growth as ε^{− \frac{dim(B)}{2}} . Therefore,

| ∫_{T}^{∞} T r_{M}^{s}(\frac{2N + dim(B)}{4}e^{4tV_{M,ε}^{2}})\frac{dt}{t}| ≤ const.ε^{− \frac{dim(B)}{2}} \frac{1}{4T λ_{0}} e^{−(4T −1)λ_{0}}.

By choosing T such that (4T − 1)λ_{0} = \frac{dim(B)}{2}| ln ε|, we get

∫_{T}^{∞} T r_{M}^{s}(\frac{2N + dim(B)}{2}e^{4tV_{M,ε}^{2}})\frac{dt}{t} → 0, ε → 0.  (39)

Similarly, one can prove that as ε → 0,

∫_{T}^{∞} T r_{M}^{s}(\frac{2N + dim(B)}{2}2(4tV_{M,ε}^{2})e^{4tV_{M,ε}^{2}})\frac{dt}{t} → 0.

Note that ∫_{T}^{∞} T r_{M}^{s}(\frac{2N+dim(B)}{4}g(−B_{4t}))\frac{dt}{t} → 0 as T → ∞ (see the proof of Lemma 3.2 [GR]). Hence, we obtain

lim_{ε→0} ∫_{0}^{∞} T r_{M}^{s}(\frac{2N + dim(B)}{4}g(4tV_{M,ε}^{2}))\frac{dt}{t} = ∫_{0}^{∞} T r_{M}^{s}(\frac{2N + dim(B)}{4}g(−B_{4t}^{2}))\frac{dt}{t}. (40)

We now use local index theory techniques (cf. [BGV], Chapter 4) and get

lim_{ε→0} G_{ε^{\frac{1}{2}}} tr_{M}^{s}(e^{4tV_{M,ε}^{2}}) = (2πi)^{− \frac{dim(B)}{2}} Â(B)tr_{∗}(e^{−R̂^{B}})tr^{s}(e^{−C_{4t}^{2}+\sqrt{t}C′}).

and

lim_{ε→0} G_{ε^{\frac{1}{2}}} tr_{M}^{s}(2(4tV_{M,ε}^{2})e^{4tV_{M,ε}^{2}}) = [det^{\frac{1}{2}}(\frac{(1 + 2z)\frac{R^{B}}{2}}{sinh(1 + 2z)\frac{R^{B}}{2}} )(2πi(1 + 2z))^{− \frac{dim(B)}{2}}
· tr_{∗}(e^{−(1+2z)R̂^{B}})tr^{s}(e^{(1+2z)(−C_{4t}^{2}+\sqrt{t}C′)})]^{z}.

On the other hand, by Theorem 3.15 [BL] and the assumption H^{∗}(Z, E|_{Z}) = 0, we have that φT r^{s}(e^{−C_{4t}^{2}}) is an exact form. This together with the proof of Theorem 1(v) implies that φT r^{s}((1 − 2C_{4t}^{2})e^{−C_{4t}^{2}}) is also exact. Hence, by (40) and C′ = 0, we get

lim_{ε→0} ∫_{0}^{∞} T r_{M}^{s}(\frac{2N + dim(B)}{4}f′(i\sqrt{t}D_{M_{ε}}))\frac{dt}{t} = ∫_{B} φÂ(B)tr_{∗}(e^{−R̂^{B}})T_{1}(T^{H}M, g^{Z}, g^{E})

+ \{∫_{B}[(1 + 2z)^{− \frac{dim(B)}{2}} det^{\frac{1}{2}}(\frac{(1 + 2z)\frac{R^{B}}{2}}{sinh(1 + 2z)\frac{R^{B}}{2}} )tr_{∗}(e^{−(1+2z)R̂^{B}})
· \frac{d}{dr}(\frac{1}{Γ(r)} ∫_{0}^{∞}(\frac{1}{2})T r^{s}(Ne^{(1+2z)(−C_{4t}^{2})})t^{r} \frac{dt}{t})_{r=0}]^{z}\}(2πi)^{− \frac{dim(B)}{2}} .  (41)

Now, since z^{2} = 0, (1+2z)^{− \frac{dim(B)}{2}} = 1− \frac{dim(B)}{2}2z. Let h(x) = \frac{1}{2} log \frac{\frac{x}{2}}{sinh \frac{x}{2}} . h′(x) = \frac{1}{2x}[1− \frac{\frac{x}{2}}{tanh \frac{x}{2}} ].

det^{\frac{1}{2}}(\frac{(1 + 2z)\frac{R^{B}}{2}}{sinh(1 + 2z)\frac{R^{B}}{2}} ) = exp tr(h((1 + 2z)R^{B}))
= exp tr(h(R^{B}) + z(1 − \frac{\frac{R^{B}}{2}}{tanh \frac{R^{B}}{2}} ))
= exp trh(R^{B})(1 + ztr(1 − \frac{R^{B}}{2} coth \frac{R^{B}}{2} )).

Since Â(B)tr_{∗}(e^{−R̂^{B}}) = 2^{\frac{dim(B)}{2}} L(B), a computation proves that the second term in the right hand side of (41) is equal to

\{∫_{B} φL(B)(− dim(B) − 1 + tr(1 − \frac{R^{B}}{2} coth \frac{R^{B}}{2} ))T_{1}(T^{H}M, g^{Z}, g^{E})

+ ∫_{B} φL(B) ∫_{0}^{∞}(\frac{1}{2})φT r^{s}(N(1 + 2(−C_{4t}^{2}))e^{−C_{4t}^{2}})\frac{dt}{t}\}2^{\frac{dim(B)}{2}}
− 2 ∫_{B} φÂ(B)tr_{∗}(R̂^{B}e^{−R̂^{B}})T_{1}(T^{H}M, g^{Z}, g^{E}).

From this we get that the left hand side of (41) is equal to the right hand side of (35). Therefore, by (36) it remains to check

lim_{ε→0} ∫_{0}^{∞} T r_{M}^{s}(\frac{1}{2} ∑_{i=0}^{dim(B)} c(e_{a_{i}})c_{v}(e_{a_{i}})g(4tV_{M,ε}^{2}))\frac{dt}{t} = 0.

But this clearly follows from the proofs of Theorem 1 (iii) and of (41). Q.E.D. By the above proof, we get the following corollary.

Corollary 1 With the condition of Theorem 4,

lim_{ε→0} log T_{w}(M_{ε}, E) = 2^{\frac{dim(B)}{2}} ∫_{B} φL(B)T_{1}(T^{H}M, g^{Z}, g^{E}).

Proof. By assumption, we have that for τ > 0 very small and ε → 0,

log T_{w}(M_{ε}, E) = \frac{d}{dr}(\frac{1}{Γ(r)} ∫_{0}^{τ} t^{r}T r_{M}^{s}(\frac{N̄}{2} e^{−tD_{Mε}^{2}})\frac{dt}{t})_{r=0}
+ ∫_{τ}^{∞} T r_{M}^{s}(\frac{N̄}{2} e^{−tD_{Mε}^{2}})\frac{dt}{t}.

The result follows from the proof of Theorem 4. Q.E.D.

Remark 1 (i) Formula (35) is much more complicated than the formula in Corollary 1. This is due to the fact that compared to T_{1}(T^{H}M, g^{Z}, g^{E}), the analytic torsion form has an extra term. Up to a constant, the second term of (35) is the invariant T s(B, M, E) in Section 2.

(ii) The argument above also works for the adiabatic limit of the analytic torsion invariant of the total space. But for this case one can only see the contribution of the degree zero component of the analytic torsion form, since the Euler form e(T B, ∇^{B}) is in the top dimension dim(B). The formula under the assumption of Theorem 4 is

lim_{ε→0} log T (M_{ε}, E) = (2πi)^{\frac{dimB}{2}} ∫_{B} φe(T B, ∇^{B})T (T^{H}M, g^{Z}, g^{E})

+ (2πi)^{\frac{dimB}{2}} ∫_{B} φe(T B, ∇^{B})tr(−\frac{R^{B}}{2} coth \frac{R^{B}}{2} )T_{1}(T^{H}M, g^{Z}, g^{E})
+ ∫_{B} φÂ(B)tr_{r}^{s}(−2R̂^{B}e^{−R̂^{B}})T_{1}(T^{H}M, g^{Z}, g^{E}).  (42)

Indeed, to apply the proof of Theorem 4 to the situation here, we need only to change the trace T r_{M}^{s} to T r^{s} and ind_{w}(D_{M}) to ind(D_{M}) and note that Â(B)tr_{r}^{s}(e^{−R̂^{B}}) = (2πi)^{\frac{dim(B)}{2}} P f(−\frac{R^{B}}{2π}). Similar formula to that in Corollary 1 also holds.

(iii) For C′ ≠ 0, one needs other forms to get an adiabatic limit formula. The case where H^{∗}(Z, E|_{Z}) ≠ 0 is much more involved, and is considered by Dai-Melrose [DM] and Forman [Fo] for the analytic torsion invariant. Theorem 4 can be extended to the L^{2}-case.

References

[BGV] N. Berline, E. Getzler, M. Vergne, Heat Kernels and Dirac operators, Springer, NY., 1992.

[Bi] J-M. Bismut, The Atiyah-Singer index theorem for families of Dirac operators: two heat equation proofs, Invent. Math., 93 (1986), 91-151.

[BL] J-M. Bismut and J. Lott, Flat vector bundle, direct images and higher real analytic torsion, J. Amer. Math. Soc., 8 (1995), 291-363.

[BC] J-M. Bismut and J. Cheeger , η-invariants and their adiabatic limits, J. Amer. Math. Soc., 2 (1989), 33-70.

[BZ] J-M. Bismut and W. Zhang, An extension of the Cheeger-Müller theorem, Astérisque, 205 (1992), Paris.

[DM] X. Dai and R. Melrose, The adiabatic limit of analytic torsion, preprint, 1995.

[Fo] R. Forman, Spectral sequences and adiabatic limits, Comm. Math. Phys., 168 (1995), 57-116.

[Go] D. Gong, Index theory for parabolic manifolds, preprint, 1996.

[GP] D. Gong and J. Pincus, Torsion invariants for finite von Neumann algebras, Contemp. Math., 185 (1995), 151-186.

[GR] D. Gong and M. Rothenberg, Analytic torsion forms for noncompact manifolds, preprint, 1997.

[Ka] P. Kahn, Characteristic numbers and oriented homotopy type, Topology, 3 (1965), 81-95.

[RS] D. Ray and I. Singer, R-torsion and the Laplacian on Riemannian manifolds, Adv. Math., 7 (1971), 145-210.

[Zh] W. Zhang, A local index theorem for sub-signature operators, preprint, 1995; and Chinese Sci. Bull., 41 (1996), 294-295.

Max-Planck-Institut für Mathematik, Gottfried Claren Str. 26, 53225 Bonn, Germany