Transgression of the index gerbe

Ulrich Bunke^{∗}

August 5, 2002

Abstract

The holonomy of an unitary line bundle with connection over some base space B is a U(1)-valued function on the loop space LB. In a parallel manner, the holonomy of a gerbe with connection on B is a line bundle with connection over LB.

Given a family of graded Dirac operators on B and some additional geometric data one can define the determinant line bundle with Quillen metric and Bismut-Freed connection. According to Witten, Bismut-Freed the holonomy of this determinant bundle can be expressed in terms of an adiabatic limit of eta invariants of an associated family of Dirac operators over LB.

Recently, for a family of ungraded Dirac operators on B J. Lott constructed an index gerbe with connection. In the present paper we show, in analogy to the holonomy formula for the determinant bundle, that the holonomy of the index gerbe coincides with an adiabatic limit of determinant bundles of the associated family of Dirac operators over LB.

Contents

1 Families and loops 1 1.1 Geometric families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Families of Dirac operators . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Transgression in K-theory and cohomology . . . . . . . . . . . . . . . . . . 5 1.4 Transgression of line bundles with connection . . . . . . . . . . . . . . . . 5 1.5 Transgression of the determinant line bundle . . . . . . . . . . . . . . . . . 6 1.6 Gerbes with connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.7 Transgression of gerbes with connection . . . . . . . . . . . . . . . . . . . . 8 1.8 Transgression of the index gerbe . . . . . . . . . . . . . . . . . . . . . . . . 9

∗Göttingen, bunke@uni-math.gwdg.de

2 Transgression of gerbes 10 2.1 Transgression in Hitchin’s picture . . . . . . . . . . . . . . . . . . . . . . . 10 2.2 The Deligne cohomology picture . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3 Proof of Lemma 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 The determinant line bundle 20 3.1 Adiabatic limit of determinant line bundles . . . . . . . . . . . . . . . . . . 20 3.2 Adiabatic limit of the holonomy . . . . . . . . . . . . . . . . . . . . . . . . 21

4 The index gerbe 22 4.1 Construction of the index gerbe . . . . . . . . . . . . . . . . . . . . . . . . 22 4.2 An identity of curvatures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.3 Holonomy. Bounding case . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 4.4 Holonomy. Trivial K^{1}-case. . . . . . . . . . . . . . . . . . . . . . . . . . . 26 4.5 Canceling the spectral flow . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 4.6 Adiabatic limit of η-invariants . . . . . . . . . . . . . . . . . . . . . . . . . 30

1 Families and loops

1.1 Geometric families

In this subsection we combine various geometric structures on a smooth fibre bundle by introducing the notion of a geometric family.

Let π : M → B be a smooth fibre bundle with closed fibres. The base B of this bundle may be infinite dimensional, and the fibres of π may consist of several components of different dimensions. By T^{v}π we denote the vertical bundle. We assume that T^{v}π is oriented and carries a spin structure. Furthermore, let g^{T^{v}π} be a vertical Riemannian metric, and let T^{h}π be a horizontal distribution. Finally, let (V, h^{V} , ∇^{V}) be a hermitean vector bundle with metric connection over M.

Definition 1.1 A geometric family over B is a collection of objects π : M → B, spin structure of T^{v}π, metric g^{T^{v}π}, horizontal bundle T^{h}π, and (V, h^{V} , ∇^{V}) as introduced above.

Note that the choice of a spin structure induces an orientation of T^{v}π.

We now describe some natural operations with geometric families. If f : B′ → B is a smooth map and E is a geometric family over B, then we define the geometric family f^{∗}E over B′ as the collection of the following objects. As the smooth fibre bundle we take the pull-back f^{∗}π : f^{∗}M → B′, where f^{∗}M := B′ ×_{B} M. There is a natural map F : f^{∗}M → M which is a diffeomorphism fibrewise. In particular, we have an isomorphism dF_{|T^{v}f^{∗}π} : T^{v}f^{∗}π →^{∼} F^{∗}T^{v}π. Using this isomorphism we obtain the induced orientation and spin structure on T^{v}f^{∗}π. We define the vertical metric g^{T^{v}f^{∗}π} such that dF_{|T^{v}f^{∗}π} becomes an isometry. The induced horizontal distribution T^{h}f^{∗}π is given by T^{h}f^{∗}π := dF^{−1}(T^{h}π). The hermitian bundle with connection over f^{∗}M is the pull-back F^{∗}V with induced connection ∇^{F^{∗}V} and metric h^{F^{∗}V}.

If E_{0}, E_{1} are geometric families over B_{0}, B_{1}, then we can form the union E_{0} ∪E_{1} in a natural way. It is a geometric family over B_{0} ∪ B_{1}. In particular, if B = B_{0} = B_{1}, then we can compose the projection M_{0} ∪ M_{1} → B ∪ B with the natural covering map B ∪ B → B. The resulting geometric family over B is the relative union E_{0} ∪_{B} E_{1}. The fibre of E_{0} ∪_{B} E_{1} is the disjoint union of the fibres of E_{0} and E_{1}.

We can also form the product E := E_{0} × E_{1}, a geometric family over B_{0} × B_{1}. On the level of the hermitian bundles we take here V := pr_{M_{0}}^{∗}V_{0} ⊗pr_{M_{1}}^{∗}V_{1}, where pr_{M_{i}} : M_{0} ×M_{1} → M_{i} are the natural projections. Again, if B = B_{0} = B_{1}, then we can form the pull-back of E with respect to the diagonal embedding B → B × B. The resulting geometric family is the fibre product E_{0} ×_{B} E_{1}.

The main construction for the purpose of the present paper is the loop LE of a geometric family E over B. Here LE is a geometric family over LB × ℝ_{+}, where LB is the free loop space of B.

Consider the evaluation map ẽv : LB × ℝ_{+} × S^{1} → B given by ẽv(γ, ϵ, u) = γ(u). First we consider the geometric family ẽv^{∗}E over LB × ℝ_{+} × S^{1}. The bundle of LE will be the composition Lπ : ẽv^{∗}M → LB × ℝ_{+} × S^{1} →^{pr} LB × ℝ_{+}. Thus the fibre of Lπ over (γ, ϵ) is the total space of the pull-back bundle γ^{∗}M → S^{1}.

The differential dẽv^{∗}π induces an isomorphism Φ : T^{h}ẽv^{∗}π →^{∼} (ẽv^{∗}π)^{∗}(T LB⊕T ℝ_{+}⊕T S^{1}). Then we have T^{v}Lπ ≅ T^{v}ẽv^{∗}π⊕Φ^{−1}(ẽv^{∗}π)^{∗}T S^{1}, and we define T^{h}Lπ := Φ^{−1}(ẽv^{∗}π)^{∗}(T LB⊕ T ℝ_{+}). We equip T S^{1} with the bounding spin structure and the orientation obtained from the identification S^{1} := ℝ/ℤ. Together with the orientation and spin structure of T^{v}ẽv^{∗}π this induces an orientation and a spin structure on T^{v}Lπ. The description of S^{1} above also induces a metric g^{T S^{1}} of volume one. We define the vertical metric of LE by g^{T^{v}ẽv^{∗}π} ⊕ \frac{1}{ϵ^{2}} Φ^{∗}(ẽv^{∗}π)^{∗}g^{T S^{1}}. The hermitian vector bundle with connection of LE is (Ẽv^{∗}V, h^{Ẽv^{∗}V} , ∇^{Ẽv^{∗}V}), where Ẽv : ẽv^{∗}M → M is the natural map.

Let I_{ϵ} : LB → LB × ℝ_{+} be given by I_{ϵ}(γ) := (γ, ϵ). Then we set L_{ϵ}E := I_{ϵ}^{∗}LE.

Note that L(E_{0} ∪_{B} E_{1}) = LE_{0} ∪_{LB×ℝ_{+}} LE_{1}.

If π : M → B is a smooth fibre bundle and g^{T M} is a Riemannian metric on M, then it induces a Riemannian metric g^{T^{v}π} by restriction, and it determines a horizontal distribution T^{h}π as the orthogonal complement of T^{v}π. We are mainly interested in the case where there exists a metric g^{T B} on B such that dπ_{|T^{h}π} : T^{h}π → π^{∗}T B is an isometry.

On the other hand, if π : M → B is a smooth fibre bundle equipped with a vertical metric g^{T^{v}π} and a horizontal distribution T^{h}π, then any Riemannian metric g^{T B} on B gives rise to a family of Riemannian metrics g_{ϵ}^{T M} := g^{T^{v}π} ⊕ \frac{1}{ϵ^{2}} π^{∗}g^{T B}, ϵ ∈ ℝ_{+}. The definition of the fibre wise metric of the loop of a geometric family is a special case of this construction. The limit ϵ → 0 is called the adiabatic limit. In the present paper we will meet adiabatic limits of various geometric (e.g. curvature) and spectral geometric (e.g. η-invariant) quantities associated to the Riemannian manifold (M, g_{ϵ}^{T M}).

Given π : M → B, g^{T^{v}π} and T^{h}π, we construct a connection ∇^{T^{v}π} as follows. We choose any Riemannian metric g^{T B} and let ∇^{T^{v}π} be the projection of the Levi-Civita connection of g_{1}^{M} to T^{v}π. The connection ∇^{T^{v}π} is independent of the choice of g^{T B} (comp. [2], Proposition 10.2)

1.2 Families of Dirac operators

Given a geometric family E over B we have a family of twisted Dirac operators D(E) := (D_{b}(E))_{b∈B}. By Γ(E) we denote the infinite dimensional bundle over B the fibre over b ∈ B of which is the space of sections C^{∞}(M_{b}, S(M_{b}) ⊗ V_{|M_{b}}), where M_{b} := π^{−1}(b) and S(M_{b}) is the spinor bundle of M_{b}. We can consider D(E) as a section of End(Γ(E)). The bundle Γ(E) carries a hermitian scalar product and a natural hermitian connection ∇^{Γ(E)} (see [2], Proposition 9.13). Let T ∈ C^{∞}(M, Hom(Λ^{2}T^{h}π, T^{v}π)) be the curvature tensor (it is the negative of the tensor Ω defined in [2], p.321) of the horizontal distribution. We form c(T) ∈ Ω^{2}(B, End(Γ(E))) such that c(T)(X, Y) is given by Clifford multiplication by T (X^{h}, Y^{h}), where X^{h}, Y^{h} denote the horizontal lifts of X, Y.

So a geometric bundle gives rise to the scaled Bismut super connection (see [2], Proposition 10.15)

A_{s}(E) := sD + ∇^{Γ(E)} + \frac{1}{4s}c(T)

if the fibres of π are even dimensional. If they are odd dimensional, then we set

A_{s}(E) := sσD + ∇^{Γ(E)} + \frac{σ}{4s}c(T) ,

where σ is an additional odd variable satisfying σ^{2} = 1.

If the base is finite dimensional, T B is oriented and has a spin structure, and g^{T B} is a Riemannian metric on the base, then T M = T^{v}π ⊕ π^{∗}T B has an induced orientation and spin structure. By D_{ϵ} we denote the V -twisted Dirac operator on M associated to the Riemannian metric g_{ϵ}^{T M} introduced in Subsection 1.1.

1.3 Transgression in K-theory and cohomology 5

1.3 Transgression in K-theory and cohomology

Given a geometric family E over a base B there is a close relation between the index index(E) ∈ K^{∗}(B) of the family D(E) and index(L_{1}E) ∈ K^{∗−1}(LB) of the family D(L_{1}E).

The choice of the spin structure of T S^{1} provides a K-orientation of the fibres of pr : LB × S^{1} → LB and therefore a map pr_{!} : K^{∗}(LB × S^{1}) → K^{∗−1}(LB). We define the transgression map T : K^{∗}(B) → K^{∗−1}(LB) by T := pr_{!} ∘ ev^{∗}. Then we have

index(L_{1}E) = T (index(E)) .  (1)

In [9] this result is attributed to Wu [13]. The orientation of S^{1} induces an integrationover-the-fibre pr_{!} : H^{∗}(LB × S^{1}, ℝ) → H^{∗−1}(B, ℝ) and therefore a transgression map in cohomology T := π_{!} ∘ ev^{∗} : H^{∗}(LB × S^{1}, ℝ) → H^{∗−1}(LB, ℝ). Transgression is compatible with the Chern character, i.e. T ∘ ch = ch ∘ T. In particular,

T ∘ ch(index(E)) = ch(index(L_{1}E)) .

The goal of the present paper is a refinement of this equality replacing the Chern classes by their Deligne cohomology valued refinements due to Cheeger-Simons [7]. To be honest, the corresponding equality for the component in degree one Deligne cohomology is known for a while, and it is equivalent to the holonomy formula for the determinant line bundle (Theorem 1.2). In the present paper we consider the degree two component.

1.4 Transgression of line bundles with connection

Let Line(B) denote the set of isomorphism classes of hermitian line bundles with connection over B. Note that the tensor product induces a group structure on Line(B). The inverse is given by the dual bundle.

We define a transgression T : Line(B) → C^{∞}(LB, U(1)) as follows. Given L := (L, h^{L}, ∇^{L}) and γ ∈ L(B), then let hol_{γ}(L) be the holonomy of L along the loop γ. We define T (L)(γ) := hol_{γ}(L).

The transgression is a group homomorphism. Note that L is completely determined by T (L). Furthermore, T (L) is invariant under the natural action of Diff(S^{1}) on LB.

1.5 Transgression of the determinant line bundle

Let E be a geometric family over B with even dimensional fibres. Then we have a determinant bundle of D(E) with hermitian metric (Quillen metric) and hermitian connection (Bismut-Freed connection) which we denote by det(E) ∈ Line(B) (see [4], or [2], Ch. 9.7) .

If D is a twisted Dirac operator on a closed oriented odd dimensional Riemannian spin manifold, then following Dai/Freed [8], we define

τ(D) := exp (2πi\frac{η(D) + dim ker D}{2}) ,

where η(D) := \frac{1}{π^{1/2}} ∫_{0}^{∞} TrDe^{−uD^{2}} \frac{du}{u^{1/2}} is the η-invariant of D first introduced by Atiyah-Patodi-Singer [1]. Applying this to the family D(L_{ϵ}E) we obtain a function τ(D(L_{ϵ}E)) ∈ C^{∞}(LB, U(1)). The adiabatic limit lim_{ϵ→0} τ(D(L_{ϵ}E)) exists locally uniformly. The main result of Bismut-Freed [4], [5] (see also [8] for a different proof) is

Theorem 1.2

T det(E) = lim_{ϵ→0} τ(D(L_{ϵ}E))

1.6 Gerbes with connection

We describe gerbes with connection on a smooth manifold B in Hitchin’s picture (see Hitchin [10] and Lott [11]).

Let U := (U_{α})_{α∈I} be an open covering of B. A gerbe G on B is given by the following data G = ((L_{αβ}), (θ_{αβγ})):

1. A hermitian line bundle L_{αβ} on each nonempty intersection U_{α} ∩ U_{β} such that L_{βα} ≅ L_{αβ}^{∗}.

2. A nowhere vanishing section θ_{αβγ} of L_{αβ} ⊗ L_{βγ} ⊗ L_{γα} over each nonempty triple intersection U_{α} ∩ U_{β} ∩ U_{γ} such that θ_{βγδ}θ_{αγδ}^{−1}θ_{αβδ}θ_{αβγ}^{−1} = 1 over each nonempty intersection U_{α} ∩ U_{β} ∩ U_{γ} ∩ U_{δ} (note that this product is a section of a bundle which is canonically trivial).

If V = (V_{μ})_{μ∈J}, s : J → I is a refinement of U, then the same gerbe G is defined by the data L′_{μν} := (L_{s(μ)s(ν)})_{|V_{μ}∩V_{ν}} and θ′_{μνσ} := (θ_{s(μ)s(ν)s(σ)})_{|V_{μ}∩V_{ν}∩V_{σ}}.

Two sets of data G = ((L_{αβ}), (θ_{αβγ})) and G′ = ((L′_{αβ}), (θ′_{αβγ})) describe isomorphic gerbes if there is a family of hermitian line bundles L_{α} over U_{α}, α ∈ I and isomorphisms L′_{αβ} ≅ L_{α}^{−1} ⊗ L_{αβ} ⊗ L_{β} such that θ′_{αβγ} corresponds to θ_{αβγ} under the induced isomorphism L′_{αβ} ⊗ L′_{βγ} ⊗ L′_{γα} ≅ L_{αβ} ⊗ L_{βγ} ⊗ L_{γα}. In general, if two gerbes G, G′ are defined with respect to coverings U, U′, then they are isomorphic if they are so when defined on a suitable common refinement.

A gerbe G gives rise to a characteristic class v(G) ∈ H^{3}(B, ℤ) which is called the Dixmir-Douady class. Two gerbes are equivalent iff their Dixmir-Douady classes coincide.

A connection of a gerbe G = ((L_{αβ}), (θ_{αβγ})) defined with respect to U consists of the following data ((∇^{L_{αβ}}), (F_{α})):

1. ∇^{L_{αβ}} is a hermitian connection on L_{αβ} such that ∇^{L_{βα}} ≅ (∇^{L_{αβ}})^{∗} and such that θ_{αβγ} is parallel with respect to the induced connection on L_{αβ} ⊗ L_{βγ} ⊗ L_{γα}.

2. F_{α} ∈ Ω^{2}(B) satisfies F_{β} − F_{α} = c_{1}(∇^{L_{αβ}}) over U_{α} ∩ U_{β}, where c_{1}(∇) := \frac{−1}{2πi}R^{∇} is the first Chern form.

If V = (V_{μ})_{μ∈J}, s : J → I is a refinement of U, then ∇^{L′_{μν}} := ∇_{|V_{μ}∩V_{ν}}^{L_{s(μ)s(ν)}} and F′_{μ} := (F_{s(μ)})_{|V_{μ}} define the same connection on G. Two gerbes with connection G = ((L_{αβ}), (θ_{αβγ}), (∇^{L_{αβ}}), (F_{α})) and G′ = ((L′_{αβ}), (θ′_{αβγ}), (∇^{L′_{αβ}}), (F′_{α})) are isomorphic, if there is family of hermitian line bundles with connection (L_{α}, ∇^{L_{α}}) over U_{α}, α ∈ I, such that ∇^{L′_{αβ}} corresponds to the induced connection ∇^{L_{α}^{−1}⊗L_{αβ}⊗L_{β}} and F′_{α} = F_{α} + c_{1}(∇^{L_{α}}). In general, if two gerbes with connection G, G′ are defined with respect to coverings U, U′, then they are isomorphic if they are so when defined on a suitable common refinement.

Let G = ((L_{αβ}), (θ_{αβγ}), (∇^{L_{αβ}}), (F_{α})) be a gerbe with connection. Then dF_{α} = dF_{β} over U_{α} ∩ U_{β}. Therefore, there is a form R^{G} ∈ Ω^{3}(B) which restricts to dF_{α} over U_{α}. This form is called the curvature of G. The cohomology class [R^{G}] ∈ H_{dR}^{3}(B) is the de Rham cohomology class corresponding to v(G) ∈ H^{3}(B, ℤ). In particular, it is independent of the connection.

Let Gerbe(B) be the set of all isomorphism classes of gerbes with connection on B. We define a product on Gerbe(B) as follows. If G, G′ ∈ Gerbe(B) are defined with respect to a covering U, then G ⊗ G′ is given by the data L_{αβ} ⊗ L′_{αβ}, θ_{αβγ} ⊗ θ′_{αβγ}, the induced connections ∇^{L_{αβ}⊗L′_{αβ}} and F_{α} + F′_{α}. The trivial element is the gerbe where L_{αβ} is the trivial bundle, θ_{αβγ} ≡ 1, ∇^{L_{αβ}} is the trivial connection, and F_{α} = 0. The inverse of G is given by L_{αβ}^{−1}, θ_{αβγ}^{−1}, the induced connection ∇^{L_{αβ}^{−1}}, and −F_{α}. We see that Gerbe(B) is an abelian group.

If f : B′ → B is a smooth map, then it induces a group homomorphism f^{∗} : Gerbe(B) → Gerbe(B′). If G = ((L_{αβ}), (θ_{αβγ}), (∇^{L_{αβ}}), (F_{α})) ∈ Gerbe(B) is defined with respect to a covering U, then f^{∗}G := ((f^{∗}L_{αβ}), (f^{∗}θ_{αβγ}), (∇^{f^{∗}L_{αβ}}), (f^{∗}F_{α})) is defined with respect to f^{∗}U := (f^{−1}U_{α})_{α∈I}.

1.7 Transgression of gerbes with connection

In Subsection 2.1 we will consider the transgression map T : Gerbe(B) → Line(LB). We will give two equivalent definitions, one in Hitchin’s picture, and the other in the Deligne cohomology picture to be introduced below. Transgression on the level of gerbes is compatible with the corresponding notion for integral/real cohomology.

Lemma 1.3 If G ∈ Gerbe(B), then −T (v(G)) = c_{1}(T (G)).

Here c_{1}(T (G)) ∈ H^{2}(LB, ℤ) is the first Chern class of the line bundle T (G), and the map T on the right hand side is the transfer in integral cohomology T : H^{∗}(B, ℤ) → H^{∗−1}(LB, ℤ). Modulo torsion Lemma 1.3 follows from the Lemma 1.4 below. We will give a complete proof of the lemma after the description of transgression in the Deligne cohomology picture in Subsection 2.3.

Note that we can define a transfer map on differential forms T : Ω^{∗}(B) → Ω^{∗−1}(LB) which induces the transfer in real cohomology (via the de Rham isomorphism). If ω ∈ Ω^{∗}(B), then we define

T ω := ∫_{S^{1}} i_{∂t}ev^{∗}(ω)dt .

Lemma 1.4 We have 2πiT (R^{G}) = R^{T (G)}.

On the right-hand side of this equality R^{L} denotes the curvature of the hermitian line bundle with connection L. We will prove this lemma at the end of Subsection 2.2.

1.8 Transgression of the index gerbe

Let E be a geometric family over B with odd dimensional fibres. Then we have an index gerbe (which was introduced by Lott [11]) gerbe(E) ∈ Gerbe(B) (we recall the construction in Subsection 4.1).

In order to state the main result about the transgression of the index gerbe we must define the adiabatic limit of the determinant line bundle det(L_{ϵ}E) as ϵ → 0. For γ ∈ LB and ϵ ∈ (0, 1] let p(γ, ϵ) : [ϵ, 1] → LB × ℝ_{+} be the path p(γ, ϵ)(t) = (γ, t). If L is any line bundle with connection over some base and σ is a path in the base defined on [a, b] ⊂ ℝ, then by ∥_{σ}^{L} ∈ End(L_{σ(a)}, L_{σ(b)}) we denote the parallel transport along σ. The family of maps ∥_{p(γ,ϵ)}^{det(LE)}, γ ∈ LB, provides an unitary isomorphism of line bundles Φ_{ϵ} : det(L_{ϵ}E) →^{∼} det(L_{1}E). However, it is not compatible with the connections. We define a family of connections ∇^{ϵ} := Φ_{ϵ} ∘ ∇^{det(L_{ϵ}E)} ∘ Φ_{ϵ}^{−1} on det(L_{1}E).

Lemma 1.5 The limit lim_{ϵ→0} ∇^{ϵ} =: ∇^{0} exists.

We prove this Lemma in Subsection 3.1.

We define det(L_{0}E) := lim_{ϵ→0} det(L_{ϵ}E) := (det(L_{1}E), h^{det(L_{1}E)}, ∇^{0}).

We can now state the main result of the present paper.

Theorem 1.6 We have T gerbe(E) = det(L_{0}E).

Since a line bundle with connection is determined by its transgression the theorem immediately follows from

Proposition 1.7 We have T^{2}gerbe(E) = T det(L_{0}E).

As a first approximation in Subsection 4.2 we show the identity of curvatures

Lemma 1.8 R^{T gerbe(E)} = R^{det(L_{0}E)}.

The proof of Proposition 1.7 follows from the study of adiabatic limits of η-invariants and constitutes the main innovation of the present paper.

2 Transgression of gerbes

2.1 Transgression in Hitchin’s picture

In this subsection we describe the transgression map T : Gerbe(B) → Line(LB).

Let G ∈ Gerbe(B) be defined with respect to U = (U_{α})_{α∈I}. We consider on ℤ_{n} := ℤ/nℤ and S^{1} a cyclic ordering. Let t : ℤ_{n} → S^{1} be a monotone map and s : ℤ_{n} → I be any map. Then we consider the open set V (t, s) ⊂ LB consisting of all loops γ such that γ([t(i), t(i + 1)]) ∈ U_{s(i−1)} ∩ U_{s(i)}. The family V := (V (t, s))_{(n∈ℕ,t:ℤ_{n}→S^{1},s:ℤ_{n}→I)} forms an open covering of LB. We define T G by describing the restrictions T G_{|V (t,s)} and the transition maps.

We consider the map E(t, s) : V (t, s) → ∏_{i∈ℤ_{n}} U_{s(i−1)} ∩ U_{s(i)} given by E(t, s)(γ) = ∏_{i∈ℤ_{n}} γ(t(i)). Let p_{j} : ∏_{i∈ℤ_{n}} U_{s(i−1)} ∩ U_{s(i)} → U_{s(j−1)} ∩ U_{s(j)} be the canonical projection. We define the hermitian line bundle

T G_{|V (t,s)} := E(t, s)^{∗} (⨂_{i∈ℤ_{n}} p_{i}^{∗}L_{s(i−1)s(i)}) .

The transition maps are generated by two types. Assume first that (t′, s′) is a refinement of (t, s). Then there is a monotone map r : ℤ_{n} → ℤ_{m}, m ≥ n such that t = t′ ∘ r and s′(i) = s(j) for all i ∈ [r(j), r(j + 1)). In this case V (t, s) ⊂ V (t′, s′). Using the fact that L_{αα} is the trivial bundle we obtain an isomorphism

E(t′, s′)^{∗} (⨂_{i∈ℤ_{m}} p_{i}^{∗}L_{s′(i−1)s′(i)})_{|V (t,s)} ≅ E(t, s)^{∗} (⨂_{i∈ℤ_{n}} p_{i}^{∗}L_{s(i−1)s(i)}) .

The other type of transition map is related to the change of the indexing map s. We fix t : ℤ_{n} → S^{1} and consider two maps s, s′ : ℤ_{n} → I such that V (t, s) ∩ V (t, s′) ≠ ∅. Let γ ∈ V (t, s) ∩ V (t, s′). We want to construct an isomorphism of fibres

E(t, s)^{∗} (⨂_{i∈ℤ_{n}} p_{i}^{∗}L_{s(i−1)s(i)}) (γ) ≅ E(t, s′)^{∗} (⨂_{i∈ℤ_{n}} p_{i}^{∗}L_{s′(i−1)s′(i)}) (γ) .

Using the family of sections (θ_{αβγ}) we have isomorphisms

E(t, s)^{∗} (⨂_{i∈ℤ_{n}} p_{i}^{∗}L_{s(i−1)s(i)}) (γ)

= ⨂_{i∈ℤ_{n}} L_{s(i−1)s(i)}(γ(t(i)))
≅ ⨂_{i∈ℤ_{n}} L_{s(i−1)s(i)}(γ(t(i)))

⊗ ⨂_{i∈ℤ_{n}} L_{s(i−1)s′(i−1)}(γ(t(i))) ⊗ L_{s(i)s(i−1)}(γ(t(i))) ⊗ L_{s′(i−1)s(i)}(γ(t(i)))

⊗ ⨂_{i∈ℤ_{n}} L_{s′(i)s(i)}(γ(t(i))) ⊗ L_{s′(i−1)s′(i)}(γ(t(i))) ⊗ L_{s(i)s′(i−1)}(γ(t(i)))

≅ ⨂_{i∈ℤ_{n}} L_{s′(i−1)s′(i)}(γ(t(i))) ⊗ ⨂_{i∈ℤ_{n}} Hom (L_{s(i)s′(i)}(γ(t(i))), L_{s(i)s′(i)}(γ(t(i + 1)))) .

Let γ_{i} : [t(i), t(i + 1)] → U_{s(i)} ∩ U_{s′(i)} be the path obtained by restriction of γ. Then we have the element ∥_{γ_{i}}^{L_{s(i)s′(i)}} ∈ Hom (L_{s(i)s′(i)}(γ(t(i))), L_{s(i)s′(i)}(γ(t(i + 1)))). Therefore, ⨂_{i∈ℤ_{n}} ∥_{γ_{i}}^{L_{s(i)s′(i)}} induces an isomorphism

⨂_{i∈ℤ_{n}} L_{s′(i−1)s′(i)}(γ(t(i))) ⊗ ⨂_{i∈ℤ_{n}} Hom (L_{s(i)s′(i)}(γ(t(i))), L_{s(i)s′(i)}(γ(t(i + 1))))

≅ ⨂_{i∈ℤ_{n}} L_{s′(i−1)s′(i)}(γ(t(i))) .

If we vary γ ∈ V (t, s) ∩ V (t, s′), then we obtain an isomorphism

Φ((t, s′), (t, s)) : E(t, s)^{∗} (⨂_{i∈ℤ_{n}} p_{i}^{∗}L_{s(i−1)s(i)})_{|V (t,s)∩V (t,s′)}

→^{∼} E(t, s′)^{∗} (⨂_{i∈ℤ_{n}} p_{i}^{∗}L_{s′(i−1)s′(i)})_{|V (t,s)∩V (t,s′)} .

We now describe the connection on the hermitian line bundle constructed above. Note that the restrictions T G_{|V (t,s)} come with induced hermitian connections. However, these connections are not compatible with the transition maps Φ((t, s′), (t, s)). The point is that γ ↦ ⨂_{i∈ℤ_{n}} ∥_{γ_{i}}^{L_{s(i)s′(i)}} is not parallel. We will fix this problem by introducing a correction using the forms F_{α}. For our purpose it turns out to be useful to describe the parallel transport of the connection ∇^{T G}. Let Γ : [a, b] → V (t, s) be a path. For i ∈ ℤ_{n} we define the path Γ_{i} : [a, b] → U_{s(i−1)} ∩ U_{s(i)} by Γ_{i}(x) = Γ(x)(t(i)). Then we define

∥_{Γ}^{T G} := ⨂_{i∈ℤ_{n}} ∥_{Γ_{i}}^{L_{s(i−1)s(i)}} ∏_{i∈ℤ_{n}} exp (2πi ∫_{[a,b]×[t(i),t(i+1)]} Γ^{∗}F_{s(i)}) .

We leave it as an exercise to the interested reader to check that the transgression is well defined. In fact, in Subsection 2.2 we will show that this construction coincides with the explicit description of the transgression defined in the Deligne cohomology picture. This also implies well-definedness (see Corollary 2.3).

If f : B′ → B is a smooth map, then in induces a map Lf : LB′ → LB. One can check that transgression is compatible with pull-back, i.e.

T ∘ f^{∗} = Lf^{∗} ∘ T ,  (2)

where Lf^{∗} : Line(LB) → Line(LB′) is the pull-back of hermitian line bundles with connection.

2.2 The Deligne cohomology picture

If (R^{∗}, d) is a complex of sheaves on a space B, then under certain conditions we can compute its hypercohomology using Čech cohomology. If U is a covering of B, then we form the double complex C^{p,q} := C_{U}^{p}(R^{q}), d_{0} := δ, d_{1} := d, where C_{U}^{p}(R^{q}) is the space of Čech p-cochains of R^{q} and δ is the differential of the Čech complex. Its action on a Čech p-cochain X is given by

δX_{i_{0}...i_{p+1}} := ∑_{j=0}^{p+1}(−1)^{j}X_{i_{0}...ǐ_{j}...i_{p+1}}

in the usual notation. We write this down in order to fix our sign conventions. The hypercohomology of the complex (R^{∗}, d) is approximated (one has to go to the limit over all coverings) by the total complex complex (C_{tot}^{∗}, d_{tot}) associated to the double complex above. In order to fix signs, d_{tot}c^{p,q} := δc^{p,q} + (−1)^{p}dc^{p,q} for c^{p,q} ∈ C^{p,q}.

Let K_{B}^{p} be the complex of sheaves on a manifold B

0 → \underline{U(1)}_{B} →^{\frac{1}{2πi} d log} Ω^{1} →^{d} … →^{d} Ω^{p} → 0 ,

where \underline{U(1)}_{B} denotes the sheaf of functions with values in U(1), and Ω^{q} is the sheaf of real q-forms. By H^{k}(B, K_{B}^{p}) we denote its k’th hypercohomology. There is an integration over the fibre

∫_{S^{1}} : H^{k}(B × S^{1}, K_{B×S^{1}}^{p}) → H^{k−1}(B, K_{B}^{p−1})

which was introduced by Gawedzki and is described in the Čech picture in Brylinski‘s book [6], Proposition 6.5.2. We can now define the transgression

T : H^{k}(B, K_{B}^{p}) → H^{k−1}(LB, K_{LB}^{p−1})

as the composition T := ∫_{S^{1}} ∘ev^{∗}.

There are natural isomorphisms u_{B} : C^{∞}(B, U(1)) →^{∼} H^{0}(K_{B}^{0}), l_{B} : Line(B) →^{∼} H^{1}(B, K_{B}^{1}) and g_{B} : Gerbe(B) →^{∼} H^{2}(B, K_{B}^{2}). We will show that the transgression in Deligne cohomology is compatible with the transgression defined on the level of geometric objects.

First we describe the maps u_{B}, l_{B}, g_{B} explicitly. A function f ∈ C^{∞}(B, U(1)) represents a cohomology class u_{B}(f) ∈ H^{0}(B, K_{B}^{0}) = H^{0}(B, \underline{U(1)}_{B}) in a natural way.

Let L = (L, h^{L}, ∇^{L}) ∈ Line(B). We choose a covering U := (U_{α})_{α∈I} such that L_{|U_{α}} is trivial. We fix sections s_{α} ∈ C^{∞}(U_{α}, L) of unit length for all α ∈ I. Then we define A_{α} := ∇^{L} log s_{α}, A_{α} ∈ iΩ^{1}(U_{α}). Let U_{αβ} ∈ C^{∞}(U_{α} ∩ U_{β}, U(1)) be the transition function U_{αβ} := \frac{s_{α}}{s_{β}}. Then ((U_{αβ}), (\frac{−1}{2πi}A_{α})) is a Čech cocycle representing the hypercohomology class l_{B}(L). Note that over U_{α} ∩ U_{β} we have A_{α} − A_{β} = d log U_{αβ}.

Let G = ((L_{αβ}), (θ_{αβγ}), (∇^{L_{αβ}}), (F_{α})) be a gerbe with connection on B defined with respect to U. We assume that L_{αβ} is trivial, and we fix unit length sections s_{αβ}. Then we define A_{αβ} := ∇^{L_{αβ}} log s_{αβ}, A_{αβ} ∈ iΩ^{1}(U_{α} ∩ U_{β}). Furthermore, we can identify the section θ_{αβγ} with a function \frac{θ_{αβγ}}{s_{αβ}s_{βγ}s_{γα}} ∈ C^{∞}(U_{α} ∩ U_{β} ∩ U_{γ}, U(1)). Then ((θ_{αβγ}), (\frac{1}{2πi}A_{αβ}), (−F_{α})) is a Čech cocycle representing the hypercohomology class g_{B}(G).

Lemma 2.1 T ∘ l_{B} = u_{LB} ∘ T.

Proof. Let L = (L, h^{L}, ∇^{L}) ∈ Line(B). Then u_{LB} ∘ T (L) is just the U(1)-valued function which maps γ ∈ LB to the holonomy hol_{γ}(L) of L along γ. Let l_{B}(L) be represented by the Čech cocycle ((U_{αβ}), (\frac{−1}{2πi}A_{α})) as above. We consider γ ∈ LB. We choose n ∈ ℕ, a monotone map t : ℤ_{n} → S^{1} and s : ℤ_{n} → I such that γ([t(i), t(i + 1)]) ⊂ U_{s(i)}. We choose small neighborhoods W_{i} of [t(i), t(i + 1)] such that γ(W̄_{i}) ⊂ U_{s(i)}. We define the neighborhood Ṽ of γ to be the space of all γ′ ∈ LB satisfying γ′(W̄_{i}) ⊂ U_{s(i)}. Then ev(Ṽ × W_{i}) ⊂ U_{s(i)} for all i ∈ ℤ_{n}. The restriction of ev^{∗}l_{B}(L) is represented by ((u_{ij}), (\frac{−1}{2πi}a_{i})) where u_{ij} is defined if j = i ± 1 by u_{(i+1)i}(γ, x) = U_{s(i+1)s(i)}(γ(x)), x ∈ W_{i} ∩ W_{i+1}, and a_{i} := ev^{∗}A_{s(i)} ∈ Ω^{1}(Ṽ × W_{i}). The description of ∫_{S^{1}} given in [6], Equation (6.21), yields

T ∘ l_{B}(L)(γ) = ∏_{i∈ℤ_{n}} exp (− ∫_{[t(i),t(i+1)]} i_{∂t}a_{i}dt) ∏_{j∈ℤ_{n}} u_{j,j+1}(γ, t(i + 1))^{−1} .

The right hand side can be rewritten as

∏_{i∈ℤ_{n}} exp (− ∫_{[t(i),t(i+1)]} i_{∂γ_{t}}A_{s(i)}dt) ∏_{j∈ℤ_{n}} U_{s(j+1)s(j)}(γ(t(i + 1))) = hol_{γ}(L) .

□

Lemma 2.2 T ∘ g_{B} = l_{LB} ∘ T

Proof. It suffices to show that T^{2} ∘ g_{B} = T ∘ l_{LB} ∘ T holds true in H^{0}(L^{2}B, K_{L^{2}B}^{0}) ≅ C^{∞}(L^{2}B, U(1)). Let G = ((L_{αβ}), (θ_{αβγ}), (∇^{L_{αβ}}), (F_{α})) be a gerbe with connection on B defined with respect to U. In Subsection 2.1 we described the parallel transport of T G. Let Γ ∈ L^{2}B. Then we choose a monotone map u : ℤ_{m} → S^{1} such that there is a fixed monotone function t : ℤ_{n} → S^{1} and a family s_{i} : ℤ_{n} → I such that Γ([u(i), u(i + 1)]) ⊂ V (t, s_{i}) for all i ∈ ℤ_{n}. We define γ_{ij} : [u(i), u(i + 1)] → B by γ_{ij}(x) := Γ(x, t(j)). Then we have

hol_{Γ}(T G) = ∘_{i∈ℤ_{n}}∥_{Γ_{|[u(i),u(i+1)]}}^{T G}
= ∘_{i∈ℤ_{n}} (Φ((t, s_{i+1}), (t, s_{i})) ∘ ⨂_{j∈ℤ_{m}} ∥_{γ_{ij}}^{L_{s_{i}(j−1)s_{i}(j)}})
∏_{i∈ℤ_{n}} ∏_{j∈ℤ_{m}} exp (2πi ∫_{[u(i),u(i+1)]×[t(j),t(j+1)]} Γ^{∗}F_{s_{i}(j)})

Writing out the first factor and the transition maps Φ with respect to the given trivializations of the L_{αβ} we get

hol_{Γ}(T G) = ∏_{i∈ℤ_{n}} ∏_{j∈ℤ_{m}} θ_{s_{i}(j−1)s_{i+1}(j−1)s_{i}(j)}(Γ(u(i + 1), t(j)))θ_{s_{i}(j)s_{i+1}(j−1)s_{i+1}(j)}(Γ(u(i + 1), t(j)))
∏_{i∈ℤ_{n}} ∏_{j∈ℤ_{m}} exp (− ∫_{[t(j),t(j+1)]} Γ(u(i + 1))^{∗}A_{s_{i}(j)s_{i+1}(j)})

∏_{i∈ℤ_{n}} ∏_{j∈ℤ_{m}} exp (− ∫_{[u(i),u(i+1)]} Γ(., t(j))^{∗}A_{s_{i}(j−1)s_{i}(j)})
∏_{i∈ℤ_{n}} ∏_{j∈ℤ_{m}} exp (2πi ∫_{[u(i),u(i+1)]×[t(j),t(j−1)]} Γ^{∗}F_{s_{i}(j)})

We now describe T ∘ g_{B}(G). We choose neighborhoods W_{j} of [t(j), t(j + 1)] and R_{i} of [u(i), u(i + 1)] such that Γ(R̄_{i} × W̄_{j}) ⊂ U_{s_{i}(j−1)} ∩ U_{s_{i}(j)}. Let Ṽ_{i} ⊂ LB be the set of all γ such that γ(W̄_{j}) ⊂ U_{s_{i}(j−1)} ∩ U_{s_{i}(j)} for all j ∈ ℤ_{m}. We now describe the restriction of T ∘ g_{B}(G) to ∪_{i∈ℤ_{n}} Ṽ_{i}. The restriction of ev^{∗}g_{B}(G) to ∪_{i∈ℤ_{n},j∈ℤ_{m}} Ṽ_{i} × W_{j} is given by

((ev^{∗}θ_{s_{i_{1}}(j_{1})s_{i_{2}}(j_{2})s_{i_{3}}(j_{3})}), (\frac{1}{2πi}ev^{∗}A_{s_{i_{1}}(j_{1})s_{i_{2}}(j_{2})}), (−ev^{∗}F_{s_{i}(j)})) .

Applying ∫_{S^{1}} we obtain a cocycle ((X_{i_{1}i_{2}}), (Y_{i})) representing the restriction of T ∘ g_{B}(G), where

X_{i_{1}i_{2}}(γ) = ∏_{j∈ℤ_{m}} θ_{s_{i_{1}}(j−1)s_{i_{1}}(j)s_{i_{2}}(j)}(γ(t(j))θ_{s_{i_{1}}(j−1)s_{i_{2}}(j−1)s_{i_{2}}(j)}(γ(t(j))^{−1}
∏_{j∈ℤ_{m}} exp (∫_{[t(j),t(j+1)]} γ^{∗}A_{s_{i_{1}}(j)s_{i_{2}}(j)}))
Y_{i} = − ∑_{j∈ℤ_{m}} ∫_{[t(j),t(j+1)]} i_{∂_{t}}ev^{∗}F_{s_{i}(j)}dt

− ∑_{j∈ℤ_{m}} ev_{|.×{t(j+1)}}^{∗}A_{s_{i}(j)s_{i}(j+1)} .

It now suffices to compute ev_{{Γ}×S^{1}}^{∗}((X_{i_{1}i_{2}}), (Y_{i})) with respect to the covering ({Γ}×Ṽ_{i})_{i∈ℤ_{n}} and apply ∫_{S^{1}} again. The result is then

T^{2} ∘ g_{B}(G)

= ∏_{i∈ℤ_{n}} ∏_{j∈ℤ_{m}} θ_{s_{i}(j−1)s_{i}(j)s_{i+1}(j)}(Γ(u(i + 1), t(j))^{−1}θ_{s_{i}(j−1)s_{i+1}(j−1)s_{i+1}(j)}(Γ(u(i + 1), t(j))
∏_{i∈ℤ_{n}} ∏_{j∈ℤ_{m}} exp (− ∫_{[t(j),t(j+1)]} Γ(u(i + 1))^{∗}A_{s_{i}(j)s_{i+1}(j)})
∏_{i∈ℤ_{n}} ∏_{j∈ℤ_{m}} exp (− ∫_{[u(i),u(i+1)]} Γ(., t(j + 1))^{∗}A_{s_{i}(j−1)s_{i}(j)})

∏_{i∈ℤ_{n}} ∏_{j∈ℤ_{m}} exp (2πi ∫_{[u(i),u(i+1)]×[t(j),t(j+1)]} Γ^{∗}F_{s_{i}(j)}) .

Using that δ(θ_{αβγ}) = 1 it is now easy to see that T^{2} ∘ g_{B}(G)(Γ) = hol_{Γ}(T G). □

An immediate consequence of this proof is :

Corollary 2.3 The transgression T : Gerbe(B) → Line(L) is well-defined by the construction given in Subsection 2.1.

It now follows from [6], Proposition 6.5.1, that the transgression of gerbes with connection constructed in 2.1 is equivalent to the negative of the construction given in [6], 6.2.1, which is due to Deligne and Brylinski.

The equation [6], (6.8) for the curvature of T G shows

R^{T G} = 2πiT R^{G} .

This proves the assertion of Lemma 1.4. □

2.3 Proof of Lemma 1.3

We start with a description of the characteristic classes

c_{1} : H^{1}(X, K_{X}^{1}) → H^{2}(X, \underline{ℤ}_{X})

and

v : H^{2}(X, K_{X}^{2}) → H^{2}(X, \underline{ℤ}_{X})

in the Čech picture. Let U = {U_{α}}_{α∈L} be a good open covering of X. If L ∈ H^{1}(X, K_{X}^{1}) is represented by the Čech cocycle ((u_{αβ}), (f_{α}^{1})), then c_{1}(L) is represented by \frac{1}{2πi}δ(log u_{αβ}). Analogously, if G ∈ H^{2}(X, K_{X}^{2}) is represented by ((u_{αβγ}), (f_{αβ}^{1}), (f_{α}^{2})), then v(G) is represented by −\frac{1}{2πi}δ(log u_{αβγ}). It follows from [6] Theorem 1.5.3, (1-33), that the classes c_{1}(L) and v(G) are exactly the obstructions against representing L and G by a global one-form or two-form, respectively.

We now let X = B × S^{1}. If v(G) = 0, then we represent G by a form f^{2} ∈ Ω^{2}(B × S^{1}). It follows from [6], Prop. 6.5.2, that ∫_{S^{1}} G is represented by the global one-form ∫_{S^{1}} f^{2} ∈ Ω^{1}(B). In particular, c_{1}(∫_{S^{1}} G) = 0.

Note that H^{3}(B × S^{1}, \underline{ℤ}_{B×S^{1}}) ≅ H^{2}(B, \underline{ℤ}_{B}) ⊗ H^{1}(S^{1}, \underline{ℤ}_{S^{1}}) ⊕ H^{3}(B, \underline{ℤ}_{B}) ⊗ H^{0}(S^{1}, \underline{ℤ}_{S^{1}}), where the isomorphism is induced by the cup product. Let ω_{S^{1}} be the generator of H^{1}(S^{1}, \underline{ℤ}_{S^{1}}). If z ∈ H^{3}(B × S^{1}, \underline{ℤ}_{B×S^{1}}), then the integral ∫_{S^{1}} z ∈ H^{2}(B, \underline{ℤ}_{B}) is given as follows. First we project z to the first summand and write the result as a ⊗ ω_{S^{1}}. Then ∫_{S^{1}} z := a.

In order to prove Lemma 1.3 it now suffices to verify the following two equations. Let a ∈ H^{2}(B, \underline{ℤ}_{B}) be given. Assume, that G is such that v(G) = −a ⊗ ω_{1}. Then

c_{1}(∫_{S^{1}} G) = a .  (3)

Furthermore, if ã ∈ H^{3}(B, \underline{ℤ}_{B}) and G̃ is such that v(G̃) = ã ⊗ 1_{S^{1}}, then

c_{1}(∫_{S^{1}} G̃) = 0 .  (4)

In order to construct a class G such that v(G) = a⊗ω_{S^{1}} we use the multiplicative structure of Deligne cohomology. In order to describe the product we use the complexes of sheaves

B_{X}^{k} : 0 → \underline{ℤ}_{X} → Ω^{0} → … → Ω^{k−1} → 0

which are quasi-isomorphic to the complexes K_{X}^{k}. Note that \underline{ℤ}_{X} sits in degree −1. The quasi-isomorphism B_{X}^{k} → K_{X}^{k} is given by f^{0} ↦ exp(2πif^{0}) in degree zero, and by the identity in higher degrees.

Let U := {U_{α}}_{α∈L} be a good open covering of B, and V := {V_{i}}_{i∈K} be a good open covering of S^{1}. Let a be represented by the Čech cocycle (z_{αβγ}), and let ω_{1} be represented by (v_{ik}). We choose chains (f_{αβ}^{0}) and (f_{α}^{1}) such that ((z_{αβγ}), (f_{αβ}^{0}), (f_{α}^{1})) represents L ∈ H^{1}(B, B_{B}^{1}). Then c_{1}(L) = a. Furthermore, we choose a chain (g_{i}^{0}) such that ((v_{ik}), (g_{i}^{0})) represents U ∈ H^{0}(S^{1}, B_{S^{1}}^{0}). In general, there is a characteristic class w : H^{0}(X, B_{X}^{0}) → H^{1}(X, \underline{ℤ}_{X}). In our case w(U) = ω_{S^{1}}.

We now let G := pr_{B}^{∗}L ∪ pr_{S^{1}}^{∗}U, where

∪ : H^{1}(B × S^{1}, B_{B×S^{1}}^{1}) ⊗ H^{0}(B × S^{1}, B_{B×S^{1}}^{0}) → H^{2}(B × S^{1}, B_{B×S^{1}}^{2})

is the product defined in [6], (1-36). We describe an explicit Čech cocycle

((c_{(α,i)(β,j)(γ,k)(δ,l)}^{−1}), (c_{(α,i)(β,j)(γ,k)}^{0}), (c_{(α,i)(β,j)}^{1}), (c_{(α,i)}^{2})) .

representing G with respect to the covering {W_{(α,i)}}_{(α,i)∈L×K} with W_{(α,i)} := U_{α} × V_{i} ⊂ B × S^{1}. Here we use the formula for the product in Čech cohomology [6], (1-18), and (1-36).

c_{(α,i)(β,j)(γ,k)(δ,l)}^{−1} := z_{αβγ}v_{kl}
c_{(α,i)(β,j)(γ,k)}^{0} := z_{αβγ}g_{k}^{0}
c_{(α,i)(β,j)}^{1} := f_{αβ}^{0}dg_{j}^{0}
c_{(α,i)}^{2} := f_{α}^{1}dg_{i}^{0}

We see again that (c_{(α,i)(β,j)(γ,k)(δ,l)}^{−1}) represents −v(G) = pr_{B}^{∗}a ∪ pr_{S^{1}}^{∗}ω_{1}. We now apply the map ∫_{S^{1}} given in [6], Prop. 6.5.2, to the cocycle

((exp(2πic_{(α,i)(β,j)(γ,k)}^{0})), (c_{(α,i)(β,j)}^{1}), (c_{(α,i)}^{2}))

representing G in H^{2}(B × S^{1}, K_{B×S^{1}}^{2}). We are only interested in its part (r_{αβ}) in Čech degree one. We can assume that K ≅ ℤ/nℤ for an appropriate n ∈ ℕ, and that there are points t_{i} ∈ S^{1}, i ∈ K, in cyclic order such that I_{i} := [t_{i}, t_{i+1}] ⊂ V_{i}. Then (r_{αβ}) is given by

r_{αβ} = ∏_{j∈K} exp(2πi ∫_{I_{j}} c_{(α,j)(β,j)}^{1})

∏_{j∈K} exp(2πic_{(α,j)(α,j+1)(β,j+1)}^{0}(t_{j+1})) exp(−2πic_{(α,j)(β,j)(β,j+1)}^{0}(t_{j+1}))

= ∏_{j∈K} exp(2πi ∫_{I_{j}} f_{αβ}^{0}dg_{j}^{0})

∏_{j∈K} exp(2πiz_{ααβ}g_{j+1}(t_{j+1})) exp(−2πiz_{αββ}g_{j+1}(t_{j+1}))

= exp(2πif_{αβ}^{0} ∑_{j∈K} ∫_{I_{j}} dg_{j}^{0})
= exp(2πif_{αβ}^{0})

since without loss of generality z_{ααβ} = 0 = z_{αββ} and ∑_{j∈K} ∫_{I_{j}} dg_{j}^{0} = 1. Now we have

δ(\frac{1}{2πi} log exp(2πif_{αβ}^{0})) = δf_{αβ}^{0}
= (z_{αβγ}) .

This shows that c_{1}(∫_{S^{1}} G) = a. We have verified (3).

The other case (4) can be shown in a similar manner. The details are left to the reader. □

3 The determinant line bundle

3.1 Adiabatic limit of determinant line bundles

The goal of this subsection is the proof of Lemma 1.5. We first recall the Bismut-Freed formula for the curvature of the determinant line bundle det(E) associated to a geometric family E over some base B with even dimensional fibres.

R^{det(E)} = [∫_{M/B} Â(∇^{T^{v}π})ch(∇^{V})]_{(2)} .  (5)

Here [.]_{(2)} stands for the two-form component. For a proof of this formula we refer to [2], 10.35. Note that in the present paper the Chern character form ch and the Â-genus already include the 2πi-factors (we use the topologist’s convention) whereas in [2] they are not included.

Let now E be a geometric family with odd dimensional fibres and consider the loop LE over LB × ℝ_{+}. The form Â(∇^{T^{v}Lπ}) depends polynomially on ϵ ∈ ℝ_{+}. In particular, its limit as ϵ → 0 is Â(∇^{T^{v}ẽv^{∗}π}) using the notation introduced in Subsection 1.1.

Recall the definition of the connection ∇^{ϵ} on det(L_{1}E) given in Subsection 1.8. We can express the parallel transport ∥^{∇^{ϵ}} of this connection in terms of the parallel transport ∥^{det(L_{1}E)} as follows. Let Γ : [a, b] → LB be any path. Then

∥_{Γ}^{∇^{ϵ}} = ∥_{Γ}^{det(L_{1}E)} exp (− ∫_{[ϵ,1]×[a,b]} Ψ^{∗}R^{det LE}) ,

where Ψ : [ϵ, 1] × [a, b] → LB × ℝ_{+} is given by Ψ(δ, x) := (Γ(x), δ). We now employ (5) and perform the limit ϵ → 0. We obtain

∥_{Γ}^{∇^{0}} = ∥_{Γ}^{det(L_{1}E)} exp (− ∫_{(0,1]×[a,b]} Ψ^{∗} [∫_{ev^{∗}M/B} Â(∇^{T^{v}Lπ})ch(∇^{Ev^{∗}V})]_{(2)}) ,

where the integral converges in view of the remark above. □

3.2 Adiabatic limit of the holonomy

In Subsection 3.1 we have seen that the limit lim_{ϵ→0} ∇^{ϵ} =: ∇^{0} exists. Recall the definition det(L_{0}E) = (det(L_{1}E), h^{det(L_{1}E)}, ∇^{0}). By Theorem 1.2 we know that T det(L_{ϵ}E) = lim_{δ→0} τ(D(L_{δ}L_{ϵ}E)) and therefore

T det(L_{0}E) = lim_{ϵ→0} lim_{δ→0} τ(D(L_{δ}L_{ϵ}E)) .

Lemma 3.1 We have T det(L_{0}E) = lim_{ϵ→0} τ(D(L_{ϵ}L_{ϵ}E)).

Proof. We fix Γ ∈ L^{2}(B). Then we define the map f : ℝ_{+} × ℝ_{+} → L(LB × ℝ_{+}) × ℝ_{+} by f(ϵ, δ)(t) = ((Γ(t), δ), ϵ). We consider the geometric family f^{∗}L^{2}E over ℝ_{+} × ℝ_{+}. Then we have τ(D(L_{ϵ}L_{δ}E))(Γ) = τ(D(f^{∗}L^{2}E))(ϵ, δ). The differential of the τ-function is given by [8], Thm. 1.9,

dτ(D(f^{∗}L^{2}E)) = 2πi [∫_{fibre} Â(∇^{T^{v}f^{∗}L^{2}π})ch(∇^{W})]_{(1)} ,

where ∫_{fibre} means integration over the fibre of the family f^{∗}L^{2}E, and W is the induced hermitian bundle with connection of the family f^{∗}L^{2}E.

Again, Â(∇^{T^{v}f^{∗}L^{2}π}) depends polynomially on ϵ, δ and is in particular uniformly bounded for small (ϵ, δ). This justifies the change of the path of integration below. Let γ, γ′ be the path‘s in ℝ_{+} × ℝ_{+} given by γ(t) := (1 − 2t, 1), t ∈ [0, \frac{1}{2}], and γ(t) := (0, 1 − 2(t − \frac{1}{2})) for t ∈ [\frac{1}{2}, 1], γ′(t) := (1 − t, 1 − t), t ∈ [0, 1]. Then we can compute

lim_{ϵ→0} lim_{δ→0} τ(D(L_{δ}L_{ϵ}E))(Γ) = τ(D(L_{1}L_{1}E))(Γ) + ∫_{γ} dτ(D(f^{∗}L^{2}E))
= τ(D(L_{1}L_{1}E))(Γ) + ∫_{γ′} dτ(D(f^{∗}L^{2}E))
= lim_{ϵ→0} τ(D(L_{ϵ}L_{ϵ}E))(Γ) .
□

4 The index gerbe

4.1 Construction of the index gerbe

Let E be a geometric family over some base B with odd dimensional fibres. Furthermore, let Q ∈ End(Γ(E)) be a family of finite dimensional selfadjoint operators such that Q_{b} = E_{D_{b}(E)}(−R(b), R(b))Q_{b}E_{D_{b}(E)}(−R(b), R(b)), b ∈ B, for some continuous function R ∈ C(B), where E_{D_{b}(E)} denotes the spectral projection.

If D(E) + Q is invertible for every b ∈ B, then following [9] we call Q a Melrose-Piazza operator. It was shown by Melrose-Piazza in [12] that a Melrose-Piazza operator exists iff index(E) = 0.

The index gerbe gerbe(E) ∈ Gerbe(B) was introduced by Lott in [11]. Here we recall his definition and discuss a slight generalization gerbe(E, Q), which is the index gerbe of the perturbed family D(E, Q) := D(E) + Q.

Let U = (U_{α})_{α∈I} be a covering of B. If we choose these sets sufficiently small then there exists functions h_{α} ∈ C_{c}^{∞}(ℝ), α ∈ I, such that D_{b,α} := D_{b}(E, Q) + h_{α}(D_{b}(E, Q)) is invertible for all b ∈ U_{α}. In fact, in order to construct h_{α} it suffices to find a family of open intervals I_{α}, J_{α} ⊂ ℝ_{+}, α ∈ I, such that (I_{α} ∪ −J_{α}) ∩ spec(D_{b}(E, Q)) = ∅ for all b ∈ U_{α}. Then one takes for h_{α} a function which is zero outside [−J_{α}, I_{α}], and is equal to one in the region between −J_{α} and I_{α}

With this definition we can now repeat Lott’s definition and arguments line by line. The only difference is that [11], (2.4), holds if one replaces ”lim” by ”LIM”. But the conclusion [11], Proposition 1., is still valid.

Let us recall the construction of gerbe(E, Q). Let E_{αβ}^{±} → U_{α}∩U_{β} denote the ±2-eigenspace bundle of \frac{D_{β}}{|D_{β}|} − \frac{D_{α}}{|D_{α}|}. This bundle comes with a natural hermitian metric and a connection induced by ∇^{Γ(E)}. We define the hermitian line bundle L_{αβ} := det(E_{αβ}^{+})⊗det(E_{αβ}^{−})^{−1}, and we let ∇^{L_{αβ}} be the induced connection. Over U_{α} ∩ U_{β} ∩ U_{γ} the product L_{αβ} ⊗ L_{βγ} ⊗ L_{γα} is canonically trivial. This gives the sections θ_{αβγ}. Finally, we define the two-forms

F_{α} := \left[{ \atop η̂_{α}}\right]_{(2)}, where

η̂_{α} = R LIM_{t→0} \frac{1}{π^{1/2}} ∫_{t}^{∞} Tr_{σ} (\frac{dA_{s}}{ds} e^{−A_{s}^{2}}) ds .

with the super connection

A_{s} = sσD_{α} + ∇^{Γ(E)} + \frac{σ}{4s}c(T)

(see Subsection 1.2 for notation), and R ∈ End(Ω(B)) multiplies p-forms by (2πi)^{−p/2}.

The index gerbe gerbe(E, Q) is now given by ((L_{αβ}), (θ_{αβγ}), (∇^{L_{αβ}}), (F_{α})). Formally, gerbe(E) := gerbe(E, 0).

The curvature of gerbe(E, Q) is independent of Q and given by [11], Thm. 1,

R^{gerbe(E,Q)} = [∫_{M/B} Â(∇^{T^{v}π})ch(∇^{V})]_{(3)} ∈ Ω^{3}(B) .  (6)

Note that gerbe(E, Q) may depend on Q. But its transgression does not.

Lemma 4.1 We have T gerbe(E, Q) = T gerbe(E).

Proof. Since the transgression of line bundles with connections is injective it suffices to show that T^{2}gerbe(E, Q) = T^{2}gerbe(E).

Let Γ ∈ L^{2}B. Then Γ : S^{1} × S^{1} → B. Furthermore, let 1 ∈ L^{2}(S^{1} × S^{1}) be given by the identity map. Then by (2) we can write T^{2}gerbe(E, Q)(Γ) = T^{2}gerbe(Γ^{∗}E, Γ^{∗}Q)(1).

Let pr : S^{1} × S^{1} × ℝ → S^{1} × S^{1} be the projection. We define Q̃ ∈ End(Γ(pr^{∗}Γ^{∗}E)) such that Q̃_{|S^{1}×S^{1}×{ϵ}} = ϵΓ^{∗}Q. By (6) we have on the one hand R^{gerbe(pr^{∗}Γ^{∗}E,Q̃)} = pr^{∗}R^{gerbe(Γ^{∗}E)}. On the other hand, R^{gerbe(Γ^{∗}E)} = 0 since dim(S^{1} × S^{1}) = 2 < 3.

Furthermore, H^{3}(S^{1} × S^{1} × ℝ, ℤ) = 0. Therefore g_{S^{1}×S^{1}×ℝ}(gerbe(pr^{∗}Γ^{∗}E, Q̃)) can be represented by a cocycle of the form ((1), (0), (−F_{α})), where F_{α} is the restriction of a global two form which is closed since the curvature of gerbe(pr^{∗}Γ^{∗}E, Q̃) is trivial. It is now obvious that

T^{2}gerbe(Γ^{∗}E, Γ^{∗}Q)(1) = exp (2πi ∫_{S^{1}×S^{1}×{1}} F )
= exp (2πi ∫_{S^{1}×S^{1}×{0}} F )
= T^{2}gerbe(Γ^{∗}E)(1)

Therefore, T^{2}gerbe(E, Q)(Γ) = T^{2}gerbe(E)(Γ). □

It follows immediately from the definitions that if E = E_{0} ∪_{B} E_{1}, then gerbe(E) = gerbe(E_{0}) ⊗ gerbe(E_{1}).

4.2 An identity of curvatures

In this Subsection we prove Lemma 1.8. By equation (6) and Lemma 1.4 we have

R^{T gerbe(E)} = 2πiT R^{gerbe(E)}
= 2πi ∫_{LB×S^{1}/LB} ev^{∗} [∫_{M/B} Â(∇^{T^{v}π})ch(∇^{V})]_{(3)}
= 2πi [∫_{ev^{∗}M/LB} Â(∇^{T^{v}ev^{∗}π})ch(∇^{Ev^{∗}V})]_{(2)} .

The curvature of det(L_{ϵ}E) is given by

R^{det(L_{ϵ}E)} := I_{ϵ}^{∗} [∫_{ẽv^{∗}M/LB×ℝ_{+}} Â(∇^{T^{v}L̃π})ch(∇^{Ẽv^{∗}V})]_{(2)} .

Since lim_{ϵ→0} Â(∇^{T^{v}Lπ}) = Â(∇^{T^{v}ev^{∗}π}) we conclude that R^{det(L_{0}E)} = lim_{ϵ→0} R^{det(L_{ϵ}E)} = 2πiT R^{gerbe(E)}. □

4.3 Holonomy. Bounding case

Consider again a loop of loops Γ ∈ L^{2}B, i.e. a map Γ : S^{1} × S^{1} → B. We say that Γ bounds, iff [Γ : S^{1} × S^{1} → B] is trivial in the bordism group Ω_{2}^{spin}(B), i.e. there is a 3-dimensional oriented spin manifold W such that ∂W ≅ S^{1} × S^{1} as oriented spin manifolds, and such that Γ extends to Γ̃ : W → B.

We now show the following partial case of Proposition 1.7.

Proposition 4.2 If Γ bounds, then T^{2}(gerbe(E))(Γ) = T (det(L_{0}E))(Γ).

Proof. We employ the extension Γ̃ : W → B in order to compute both sides in terms of curvatures.

Let 1 ∈ L^{2}(S^{1}×S^{1}) be given by the identity. Then we have T^{2}(gerbe(E))(Γ) = T^{2}(gerbe(Γ^{∗}E))(1). Now gerbe(Γ^{∗}E) is the restriction of gerbe(Γ̃^{∗}E) to the boundary of W. Since H^{3}(W, ℤ) = 0 we can represent gerbe(Γ̃^{∗}E) by a global two-form F ∈ Ω^{2}(W) such that dF = R^{Γ̃^{∗}E}. We have

T^{2}(gerbe(Γ^{∗}E))(1) = exp (2πi ∫_{∂W} F )
= exp (2πi ∫_{W} dF )
= exp (2πi ∫_{W} R^{Γ̃^{∗}E}) .

By (6) we have

T^{2}(gerbe(E))(Γ) = exp (2πi ∫_{W} Γ̃^{∗} [∫_{M/B} Â(∇^{T^{v}π})ch(∇^{V})]_{(3)}) .

We now compute

T (det(L_{0}E))(Γ) = lim_{ϵ→0} T (det(L_{ϵ}E))(Γ) = lim_{ϵ→0} lim_{δ→0} τ(D(L_{δ}L_{ϵ}E))(Γ) ,

and hence by Lemma 3.1, T (det(L_{0}E))(Γ) = lim_{ϵ→0} τ(D(L_{ϵ}L_{ϵ}E))(Γ). In other words, T (det(L_{0}E))(Γ) is equal to the adiabatic limit lim_{ϵ→0} τ(D_{ϵ}), where D_{ϵ} is the total Dirac operator on Γ^{∗}M (see Subsection 1.2). We choose Γ̃ such that it is independent of the normal variable in a tubular neighborhood of ∂W. Furthermore, we choose a Riemannian metric g^{W} as a product in that neighborhood. Then the total Dirac operator D̃_{ϵ} on Γ̃^{∗}M has a product structure at ∂Γ̃^{∗}M.

The index theorem of Atiyah-Patodi-Singer [1] gives

τ(D_{ϵ}) = exp (2πi ∫_{Γ̃^{∗}M} Â(∇_{ϵ}^{T Γ̃^{∗}M})ch(∇^{Γ̃^{∗}V})) ,

where ∇_{ϵ}^{T Γ̃^{∗}M} is the Levi Civita connection to the metric g_{ϵ}^{T Γ̃^{∗}M}. Now lim_{ϵ→0} Â(∇_{ϵ}^{T Γ̃^{∗}M}) = Â(∇^{T^{v}Γ̃^{∗}π}). We conclude that

T (det(L_{0}E))(Γ) = exp (2πi ∫_{Γ̃^{∗}M} Â(∇^{T^{v}Γ̃^{∗}π})ch(∇^{Γ̃^{∗}V}))
= exp (2πi ∫_{W} Γ̃^{∗} [∫_{M/B} Â(∇^{T^{v}π})ch(∇^{V})]_{(3)}) .
□

4.4 Holonomy. Trivial K^{1}-case.

In this subsection we consider another special case of Proposition 1.7. Again, let Γ ∈ L^{2}B.

4.4 Holonomy. Trivial K^{1}-case. 27

Proposition 4.3 If Γ^{∗}index(E) = 0, then T^{2}(gerbe(E))(Γ) = T (det(L_{0}E))(Γ).

Proof. In the present subsection we will prove this proposition up to a certain assertion about adiabatic limits of η-invariants Proposition 4.5.

Again, we write T^{2}(gerbe(E))(Γ) = T^{2}(gerbe(Γ^{∗}E))(1). Since index(Γ^{∗}E) = 0 we can find a Melrose-Piazza operator Q ∈ End(Γ(Γ^{∗}E)) (see Subsection 4.1). By Lemma 4.1 we have

T^{2}(gerbe(Γ^{∗}E))(1) = T^{2}(gerbe(Γ^{∗}E, Q))(1) .  (7)

Since D(Γ^{∗}E, Q) is invertible we can represent gerbe(Γ^{∗}E, Q) with respect to the covering U = (S^{1} × S^{1}), i.e. by the global two form F := \left[{ \atop η̂}\right]_{(2)}, the two form component of the eta form of D(Γ^{∗}E, Q). In particular, we have

T^{2}(gerbe(Γ^{∗}E, Q))(1) = exp (2πi ∫_{S^{1}×S^{1}} F ) .  (8)

Let D_{ϵ} be the total Dirac operator on Γ^{∗}E with respect to the metric g_{ϵ}^{T Γ^{∗}M}. Then as in the proof of Proposition 4.2 we have

T (det(L_{0}E))(Γ) = lim_{ϵ→0} τ(D_{ϵ}) .  (9)

We choose some function ρ ∈ C^{∞}(ℝ_{+}) such that ρ(s) = 0 for s ≤ 1 and ρ(s) = const for t ≥ 2. Let D_{ϵ}(s) = sD_{ϵ} + sρ(s)Q. We define

η(ϵ, ρ) := \frac{2}{π^{1/2}} ∫_{0}^{∞} Tr\frac{d}{ds}D_{ϵ}(s)e^{−D_{ϵ}(s)^{2}}ds .

Since ρ(s) ≡ 0 for small s this integral converges at s = 0. Moreover, since ρ(s) is constant for large s the integrand vanishes exponentially if s → ∞. We further define

τ(ϵ, ρ) := exp (2πi\frac{η(ϵ, ρ) + dim ker D_{ϵ}(∞)}{2}) .

It is now easy to see that τ(ϵ, ρ) depends smoothly on the parameters. In fact a change of the kernel of D_{ϵ}(∞) gives rise to an integer jump of η(ϵ, ρ), and the combination η(ϵ, ρ) + dim ker D_{ϵ}(∞) jumps by an even integer.

Lemma 4.4 τ(ϵ, ρ) is independent of ρ.

Proof. The usual derivation for the variation of the η-invariant gives

\frac{δ}{δρ} log τ(ϵ, ρ) = 2πi LIM_{t→0} \frac{1}{π^{1/2}} Tr (\frac{δ}{δρ}D_{ϵ}(t)) e^{−D_{ϵ}(t)^{2}} = 0

since \frac{δ}{δρ}D_{ϵ}(t) ≡ 0 for small t. □

From now on we assume that ρ(s) ≡ 1 for large s. We form the family of super connections

A_{s}(ρ) := sσD(E) + sρ(s)σQ + ∇^{Γ(E)} + \frac{σ}{4s}c(T)

and define the eta form

η̂(ρ) := R \frac{1}{π^{1/2}} ∫_{0}^{∞} Tr_{σ} (\frac{dA_{s}(ρ)}{ds}e^{−A_{s}^{2}(ρ)}) ds .  (10)

For small s the super connection A_{s}(ρ) = A_{s}(E) is the Bismut super connection and therefore the integral converges at s = 0. For large s the operator D(E)+ρ(s)Q = D(E)+Q is invertible, and therfore the integrand vanishes exponentially for s → ∞.

The following proposition is an analog of the result of Bismut-Cheeger [3].

Proposition 4.5 We have

lim_{ϵ→0} τ(ϵ, ρ) = exp (2πi ∫_{S^{1}×S^{1}} η̂(ρ)) .

We will prove this proposition in Subsection 4.6.

If we replace the definition (10) of η̂(ρ) by

η̂(ρ) := R LIM_{t→0} \frac{1}{π^{1/2}} ∫_{t}^{∞} Tr_{σ} (\frac{dA_{s}(ρ)}{ds}e^{−A_{s}^{2}(ρ)}) ds ,

then it extends to all functions ρ ∈ C^{∞}[0, ∞) satisfying ρ(s) ≡ 1 for large s and ρ(s) ≡ const for small s (this is completely parallel to the definition of η̂_{α} in [11], (2.19)).

Lemma 4.6 The difference η̂(ρ) − η̂ is exact.

Proof. Let ρ_{u}, u ∈ [0, 1] be a smooth family of functions as above interpolating between ρ = ρ_{0} and the constant function ρ_{1} ≡ 1. We now repeat the argument of the proof of [11], Proposition 3. Fortunately, Lott has written the argument in a way which does not use the precise s-dependence of A_{s}(ρ). In order to get [11], equation (2.11), one only needs that ρ(s) is constant for small s. □

Combining Lemma 4.6 with Proposition 4.5, and then further with Lemma 4.4 and Equation (9) we get

exp (2πi ∫_{S^{1}×S^{1}} F ) = exp (2πi ∫_{S^{1}×S^{1}} η̂(ρ))
= lim_{ϵ→0} τ(ϵ, ρ)
= lim_{ϵ→0} τ(D_{ϵ})
= T (det(L_{0}E))(Γ) .

In view of Equation (8) this shows Proposition 4.3. □

4.5 Canceling the spectral flow

In this Subsection we combine Propositions 4.2 and 4.3 in order to deduce Proposition 1.7.

Let E_{1} be a geometric family over S^{1} with odd dimensional fibres such that index(E_{1}) is the generator of K^{1}(S^{1}) ≅ ℤ. In fact, for E_{1} one can take the family given by pr : S^{1} × S^{1} → S^{1} with standard metrics and horizontal subspaces such that L is a line bundle with connection over S^{1} × S^{1} with first Chern class satisfying c_{1}(L)[S^{1} × S^{1}] = 1.

If we choose generators γ_{1}, γ_{2} of π_{1}(S^{1} × S^{1}) ≅ ℤ^{2}, then we obtain an identification K^{1}(S^{1} × S^{1}) ≅ ℤ^{2}. If E is a geometric family over S^{1} × S^{1} with odd dimensional fibres, then index(E) = (sf(E)(γ_{1}), sf(E)(γ_{2})), where sf(E)(γ) denotes the spectral flow along γ.

Let now E be any geometric family with odd dimensional fibres over some base B. The spectral flow provides a homomorphism π_{1}(B) → ℤ. This homomorphism corresponds to an element sf(E) ∈ H^{1}(B, ℤ). Let S : B → S^{1} be the classifying map of −sf(E). Then the spectral flow of S^{∗}E_{1}∪E vanishes. In particular, if Γ ∈ L^{2}B, then index(Γ^{∗}(S^{∗}E_{1}∪_{B}E)) = 0.

Therefore we can apply Proposition 4.3 in order to conclude

T^{2}(gerbe(S^{∗}E_{1} ∪_{B} E))(Γ) = T (det(L_{0}(S^{∗}E_{1} ∪_{B} E)))(Γ) .

Using multiplicativity of gerbes and determinant bundles under relative disjoint sum we obtain

\frac{T^{2}(gerbe(E))(Γ)}{T (det(L_{0}(E)))(Γ)} = \frac{T^{2}(gerbe(S^{∗}E_{1}))(Γ)}{T (det(L_{0}(S^{∗}E_{1})))(Γ)} .  (11)

Now the map S ∘ Γ : S^{1} × S^{1} → S^{1} bounds. In fact, let u ∈ π_{1}(S^{1} × S^{1}) be a primitive generator of ker(S ∘ Γ)_{∗} : π_{1}(S^{1} × S^{1}) → π_{1}(S^{1}). Then can find a diffeomorphism F : S^{1} × S^{1} → S^{1} × S^{1} such that F (S^{1} × {∗}) represents u. The map S ∘ Γ ∘ F now extends to D^{2} × S^{1}. It follows that we can apply Proposition 4.2 in order to conclude that the right hand side of (11) is equal to one. This proves Proposition 1.7 in general (up to the verification of Proposition 4.5). □

4.6 Adiabatic limit of η-invariants

In this subsection we prove Proposition 4.5. In fact we can repeat the proof of Bismut-Cheeger [3], Thm. 4.35. We will just explain the changes in various places. The main point is that we must replace u^{1/2}D_{ϵ} by u^{1/2}(D_{ϵ}(E) + ρ(u^{1/2})Q) =: D_{ϵ}(u^{1/2}). The righthand side of formula [3], (4.39), gets correspondingly replaced by

\frac{1}{2} [dim ker D_{ϵ}(∞) + \frac{2}{π^{1/2}} ∫_{0}^{∞} Tr (\frac{∂}{∂u}D_{ϵ}(u^{1/2})) e^{−D_{ϵ}^{2}(u^{1/2})}du] .

The proof of [3], Proposition 4.41, can be applied to D_{ϵ}(u^{1/2}) for u ≥ 4 and yields a lower bound spec(D_{ϵ}^{2}(u^{1/2})) ≥ λ_{0} > 0 which is uniform for u ≥ 4 and ϵ ∈ (0, ϵ_{0}], ϵ_{0} sufficiently small.

In step ii of the proof of [3], Thm. 4.35 we must replace, of course, uD_{ϵ}^{2} − zu^{1/2}D_{ϵ} by D_{ϵ}(u^{1/2})^{2} − z2u\frac{∂}{∂u}D_{ϵ}(u^{1/2}). From [3], (4.45), we get (note that D^{Z} in [3] is D(E) here)

D_{ϵ}(u^{1/2}) − u^{1/2}D_{ϵ} = u^{1/2}ρ(u^{1/2})Q
D_{ϵ}(u^{1/2})^{2} − uD_{ϵ}^{2} = ϵ^{1/2}u^{1/2}ρ(u^{1/2})[E_{ϵ}, Q] + uρ(u^{1/2}){D(E), Q} + uρ(u^{1/2})^{2}Q^{2}
2u\frac{∂}{∂u}D_{ϵ}(u^{1/2}) − u^{1/2}D_{ϵ} = uρ′(u^{1/2})Q

Therefore, in the right hand side of [3], (4.51), we have the following additional terms

ϵ^{1/2}u^{1/2}ρ(u^{1/2})[E_{ϵ}, Q] + uρ(u^{1/2}){D(E), Q} + uρ(u^{1/2})^{2}Q^{2}
−zuρ′(u^{1/2})Q .

We now proceed as in [3] and apply Getzler’s rescaling along the base. This is possible since the perturbation Q is local with respect to the base. Formula [3], (4.70), gets replaced by

H + A_{u^{1/2}}^{2}(ρ) − z2u\frac{∂}{∂u}A_{u^{1/2}}(ρ)

(after setting σ = 1). The Hermite operator H simplifies to − ∑_{α}(∂_{α})^{2} since the torus S^{1} × S^{1} is flat. Equation [3] (4.74) simplifies to

\frac{1}{π^{1/2}} Tr (\frac{∂}{∂u}D_{ϵ}(u)) e^{−D_{ϵ}^{2}(u)} ⟶^{ϵ→0} \frac{1}{2πi} \frac{1}{π^{1/2}} ∫_{S^{1}×S^{1}} Tr_{σ} \frac{∂}{∂u}A_{u^{1/2}}(ρ)e^{−A_{u^{1/2}}^{2}(ρ)} .  (12)

Since ρ(s) vanishes for small s, the discussion in [3] of uniformity of the convergence (12) on intervals (0, T], T > 0 applies to the present case. Again using the fact, that Q is local with respect to the base we repeat the derivation of the remainder term O(ϵ^{1/2}(1 + T^{N})) in (12). Thus we have verified the analog of [3] (4.40). The final step iii. of the proof of [3], Thm. 4.35, can be taken without change. □

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