Homotopy quantum field theory and the

index gerbe

Ulrich Bunke^{∗}

January 29, 2002

Contents

1 Introduction 1

2 The index gerbe 2

3 Thin invariant field theories 3

4 Taming 4

5 A thin-invariant rank-one field theory associated to an odd dimensional geometric family 4

1 Introduction

In the recent paper [8] Turner and Willerton study the relation between gerbes with connection and thin-invariant rank-one field theories on a space B. Their main result is that if H_{1}(B,ℤ) is torsion-free, then there is a one-to-one correspondence of gerbes with connection on B and isomorphism classes of thin-invariant rank-one field theories on B. The gerbe corresponding to a thin-invariant rank-one field theory is given by the holonomy of the field theory. In [6] Lott associates to an odd dimensional geometric family on B (see [2] and Section 2 for this notion) an index gerbe. The goal of the present paper is to analyze the construction of Turner and Willerton [8] in the case of the index gerbe. Starting from the geometric family we will construct a thin-invariant rank-one field theory on B such that its holonomy is that of the index gerbe. Our construction works without any assumption on the manifold B. If H_{1}(B,ℤ) is torsion-free, then in view of [8], Thm. 3.5, the field theory which we construct in the present paper coincides with the one obtained by applying the construction of Turner and Willerton [8] to the index gerbe.

∗Göttingen, bunke@uni-math.gwdg.de

For general B the thin-invariant field theory associated to a geometric family seems to encode slightly more structure than the index gerbe of that family. The construction of the field theory given in the present paper induces a construction of the index gerbe which is independent of the constructions in [6] and [3].

2 The index gerbe

Let Gerbe(B) denote the group of gerbes with connection on B. We refer to Hitchin [5] for a nice introduction to the geometric picture of gerbes. Lott’s index gerbe is constructed in [6] using Hitchin’s geometric picture. In the present paper we prefer to represent gerbes by Deligne cohomology classes or Cheeger-Simons differential characters. Let A_{B}^{q} denote the sheaf of real smooth q-forms on B, q ∈ ℕ_{0}. We consider the complex of sheaves on B

K (2,ℤ)_{B}: 0 → \underline{ℤ}_{B} → A_{B}^{0} → A_{B}^{1} → A_{B}^{2} → 0 ,

where the constant sheaf \underline{ℤ}_{B} sits in degree −1. The third Deligne cohomology of B is by definition the second Čech hyper cohomology group of this complex

H_{Del}^{3}(B) := Ȟ^{2}(B,K (2,ℤ)_{B}) .

There is a natural isomorphism Gerbe(B) ≅ H_{Del}^{3}(B) (see Brylinski [1] and [2]). Holonomy provides a natural isomorphism H : H_{Del}^{3}(B) →^{∼} Ĥ^{2}(B,U(1)), where Ĥ^{2}(B,U(1)) is the group of Cheeger-Simons differential characters with values in U(1) (see [4]). An element ϕ ∈ Ĥ^{2}(B,U(1)) is a homomorphism from the group of smooth cycles in B to U(1) such that there exists a closed form R^{ϕ} ∈ A_{B}^{3}(B,d = 0) with the property that

ϕ(∂C) = exp( 2πi∫_{C}R^{ϕ})

for all 3-chains C. The form R^{ϕ} is called the curvature of ϕ. A construction of H : H_{Del}^{3}(B) →^{∼} Ĥ^{2}(B,U(1)) was given in [3], Sec. 6.1. Thus we can represent gerbes in a third way, namely by Cheeger-Simons differential characters. There is a natural homomorphism R : H_{Del}^{3}(B) → A_{B}^{3}(B,d = 0), x ↦ R^{x}, such that R^{x} is the curvature of the Cheeger-Simons differential character H(x). We call R^{x} the curvature of x. An odd dimensional geometric family E_{geom} over B is given by (see [2], Sec.1.1)

• a fibre bundle π: E → B with closed odd dimensional fibres,

• a vertical orientation, a vertical spin structure, and a complex vector bundle V over E

• a vertical Riemannian metric and a horizontal distribution for π: E → B, a hermitian metric h^{V} and a metric connection ∇^{V} on V (the geometric structures).

By gerbe(E_{geom}) ∈ Gerbe(B) we denote the index gerbe constructed by Lott [6]. Let index(E) ∈ K^{1}(B) be the index of the family of selfadjoint twisted Dirac operators defined by E_{geom}. We omit the subscript_{geom}since this index is independent of the choice of the geometric structures. Let ch_{1}(index(E)) ∈ H_{dR}^{1}(B) be the degree-one component of the Chern character in the de Rham cohomology of B. Under the assumption that ch_{1}(index(E)) = 0 in [3] we constructed a natural class index_{Del}^{3}(E_{geom}) ∈ H_{Del}^{3}(B) such that R^{index_{Del}^{3}(E_{geom})} = Ω^{3}(E_{geom}), where Ω(E_{geom}) = ∫_{E/B}Â(∇^{T^{v}π})ch(∇^{V}) is the local index form (see [3], Sec. 3.2 for definitions). The class index_{Del}^{3}(E_{geom}) coincides with gerbe(E_{geom}) under the natural isomorphism Gerbe(B) ≅ Ĥ^{2}(B,U(1)). The construction of the thin-invariant field theory given below will provide an independent construction of the index gerbe as an element in Ĥ^{2}(B,U(1)).

3 Thin invariant field theories

The thin homotopy category T_{B} of B is defined e.g. in [8]. Let S^{1} ⊂ ℂ be the unit circle with a fixed orientation. By S̄^{1} we denote the circle with the opposite orientation. By S_{m,n}, n,m ∈ ℕ_{0}, we denote the disjoint union of m copies of S^{1} and n copies of S̄^{1}. An object of T_{B} is a smooth map γ: S_{m,0}→ B. Let I := [0,1] denote the unit interval with the canonical orientation. We consider the oriented surface C_{m,n}:= I ×S_{m,n}which comes with a natural projection pr: C_{m,n}→ S_{m,n}. By a surface Σ we mean a compact smooth oriented surface with boundary ∂Σ together with an orientation preserving collar i_{Σ} : C_{m,n}↪ Σ such that i_{Σ}({0}×S_{m,n}) = ∂Σ. Here m,n ∈ ℕ_{0}. The image of S_{m,0} (resp. S_{0.n}) is called the ingoing (resp. outgoing ) boundary of Σ and will be denoted by ∂^{in}Σ (resp. ∂^{out}Σ). By I_{1}^{in}(Σ) and I_{1}^{out}(Σ) we denote the set of connected components of ∂^{in}Σ and ∂^{out}Σ. A B-surface is a smooth map g : Σ → B such that g ∘ i_{Σ} : C_{m,n}→ B factors over the projection pr. In particular, a B-surface determines an ingoing object γ_{in} : S_{m,0} → B and an outgoing object γ_{out}: S_{n,0} → B such that g ∘ i_{Σ} = (γ ∪ γ′) ∘ pr: C_{m,n}→ B. By a three manifold X we mean a compact three manifold with corners of codimension at most two and fixed collars. Let I_{1}(X) be the set of boundary faces ∂_{i}X, i ∈ I_{1}(X). Then ∂_{i}X, i ∈ I_{1}(X), should be a surface with collared boundary as above. The collars i_{i,X} : I × ∂_{i}X ↪ X are part of the data. Let I_{2}(X) denote the set of faces of codimension two. Let k ∈ I_{2}(X) and ∂_{i}X and ∂_{j}X, i,j ∈ I_{1}(X), be the two boundary faces which meet at ∂_{k}X. Then we require that i_{i,X}∘ (id_{I} × i_{∂_{i}X}) = i_{j,X} ∘ (id_{I} × i_{∂_{j}X}) ∘ (σ × id_{∂_{k}X}) as maps from I^{2} × ∂_{k}X ↪ X, where σ interchanges the components in I^{2}. A B-three manifold is a smooth map h : X → B such that the compositions h ∘ i_{i,X}: I × ∂_{i}X → B factor over the projections I × ∂_{i}X → ∂_{i}X and B-surfaces g_{i} : ∂_{i}X → B for all i ∈ I_{1}(X). Let γ_{in} : S_{m,0} → B and γ_{out}: S_{n,0} → B be two objects of T_{B}. A morphism γ_{in} → γ_{out} is represented by a B-surface with ingoing object γ_{in} and outgoing object γ_{out}. Two such B-surfaces g and g′define the same morphism, if they are thin homotopic, i.e. there is a homotopy h: I × Σ → B from g to g′such that rank(dh) ≤ 2 everywhere. Here I × Σ is a three manifold in the natural way, and h: I × Σ → B is required to be a B-three manifold. Composition of morphisms is induced by glueing. The category T_{B} has a product induced by disjoint union. Let vect_{1} denote the monoidal category of one dimensional complex vector spaces. A rank-one thin-invariant field theory (compare [8], Def. 3.2) is a monoidal functor E : T_{B} → vect_{1} satisfying the following condition: There is a closed form R^{E} ∈ A_{B}^{3}(B,d = 0) (the curvature of E) such that if h: C → B is a B-three manifold with boundary g: Σ → B such that Σ is closed, then

E([g]) = exp( 2πi∫_{C}h^{∗}R^{E}) .

Here [g] denotes the class of g in the endomorphisms of the empty object ∅ of T_{B}, and E([g]) ∈ ℂ under the natural identification of the linear endomorphisms of E(∅) ≅ ℂ with ℂ.

To any thin-invariant rank-one field theory E with curvature R^{E} on B we can associate a gerbe on B with the same curvature. If g : Σ → B is a closed B-surface, then let [g,Σ] denote the corresponding cycle on B. The Cheeger-Simons differential character ϕ ∈ Ĥ^{2}(B,U(1)) corresponding to the gerbe satisfies ϕ([g,Σ]) := E([g]).

4 Taming

Let E_{geom} be an odd dimensional geometric family over B. By D(E_{geom}) := (D_{b})_{b∈B} we denote the family of twisted Dirac operators given by E_{geom}. Let g : M → B be any smooth map from some manifold M to B. A taming of the family g^{∗}E_{geom} is given by a smooth family of selfadjoint smoothing operators (Q_{m})_{m∈M} such that D_{m}+ Q_{m} is invertible for all m ∈ M. It was shown in [7], that g^{∗}E_{geom} admits a taming exactly if index(g^{∗}E) = 0 holds in K^{1}(M). Let N ⊂ M and γ: N → B be the restriction of g to N. Assume that we already have a taming (γ^{∗}E_{geom})_{t}. Then the obstruction against extending the taming to M is the index element index(g^{∗}E,(γ^{∗}E_{geom})_{t}) ∈ K^{1}(M,N). In order to construct this element we let C_{N}(M) be the algebra of continuous functions on M which vanish on N. Let U be some neighborhood of N to which we can extend the family (Q_{n})_{n∈N} giving the taming (γ^{∗}E_{geom})_{t}. Let χ ∈ C^{∞} (M) be such that χ_{|N} = 1 and χ_{|M\U} = 0. Then the family of operators (D_{m}+ χ(m)Q_{m})_{m∈M} defines the element of KK_{1}(ℂ,C_{N}(M)) ≅ K^{1}(M,N).

5 A thin-invariant rank-one field theory associated to an odd dimensional

geometric family

Theorem 5.1. If E_{geom}is an odd dimensional geometric family on a connected manifold B such that ch_{1}(E_{geom}) = 0, then there is a natural rank-one thin invariant field theory E with curvature R^{E} = Ω^{3}(E_{geom}) such that the associated gerbe is the index gerbe gerbe(E_{geom}).

Proof. We fix a base point b_{0} and a taming E_{b_{0},t} of the fibre of E_{geom} over b_{0}. Let γ: S^{1} → B be a smooth map. Since S^{1} is one dimensional we have an inclusion ch_{1} : K^{1}(S^{1}) ↪ H_{dR}^{1}(S^{1}). Since ch_{1}(index(γ^{∗}E)) = γ^{∗}ch_{1}(index(E)) = 0 the family γ^{∗}E_{geom} admits a taming (γ^{∗}E_{geom})_{t}. The set of homotopy classes of tamings of γ^{∗}E_{geom}is a torsor over K^{1}(S^{1}×I,∂(S^{1}×I)) ≅ ℤ. We employ the choice E_{b_{0},t} in order to distinguish one of these classes. Let M := S^{1}∪_{1}I, where we identify 1 ∈ S^{1} and 1 ∈ I. We extend γ to γ̃: M → B such that γ̃maps 0 ∈ I to b_{0}. Note that K^{1}(S^{1}∪_{1}I,S^{1}∪{0}) ≅ ℤ. The distinguished homotopy class of tamings on γ^{∗}E_{geom} is now characterized by the property that it can be connected with E_{b_{0},t} along the path γ̃_{|I}. Because of our assumption ch_{1}(index(E)) = 0 this class is independent of the choice of the path γ̃_{|I}. We now construct the space E(γ). Let γ: S^{1} → B be a smooth map. We consider the set Ẽ(γ) of pairs (λ,(γ^{∗}E_{geom})_{t}), where λ ∈ ℂ and (γ^{∗}E_{geom})_{t} is a taming of γ^{∗}E_{geom} in the distinguished homotopy class. Let (γ^{∗}E_{geom})_{t} and (γ^{∗}E_{geom})_{t′}be two tamings in the distinguished class. We consider γ̃: I × S^{1} →^{pr} S^{1} →^{γ} B. Then we have a taming of γ̃^{∗}E_{geom}over {0}×S^{1} given by (γ^{∗}E_{geom})_{t} and over {1}×S^{1} given by (γ^{∗}E_{geom})_{t′}. Since these belong to the same homotopy class we can extend this taming to a taming (γ̃^{∗}E_{geom})_{t}. Any two of these extensions are homotopic since K^{1}(I × I × S^{1},∂(I × I × S^{1})) = 0.

By η^{2}((γ̃^{∗}E_{geom})_{t}) ∈ A_{I×S^{1}}^{2} (I × S^{1}) we denote the 2-form component of the η form as defined in [3], Sec. 3.3. We claim that ∫_{I×S^{1}} η^{2}((γ̃^{∗}E_{geom})_{t}) is independent of the choice of the taming. We consider a homotopy between two choices (γ̃^{∗}E_{geom})_{t}, (γ̃^{∗}E_{geom})_{t′}, i.e. γ̂: I × I × S^{1} →^{pr} S^{1} →^{γ} B and a taming (γ̂^{∗}E_{geom})_{t}. Then we have by [3], Prop. 3.2, that dη^{2}((γ̂^{∗}E_{geom})_{t}) = Ω^{3}(γ̂^{∗}E_{geom}). Moreover Ω^{3}(γ̂^{∗}E_{geom}) = 0 since γ̂factors over a one dimensional manifold. Therefore by Stoke’s Lemma

0 = ∫_{{1}×I×S^{1}} η^{2}((γ̂^{∗}E_{geom})_{t}) − ∫_{{0}×I×S^{1}} η^{2}((γ̂^{∗}E_{geom})_{t})  (1)
−∫_{I×{1}×S^{1}} η^{2}((γ̂^{∗}E_{geom})_{t}) + ∫_{I×{0}×S^{1}} η^{2}((γ̂^{∗}E_{geom})_{t}) .  (2)

Here (1) is equal to

∫_{I×S^{1}} η^{2}((γ̃^{∗}E_{geom})_{t′}) − ∫_{I×S^{1}} η^{2}((γ̃^{∗}E_{geom})_{t}) .

Moreover, ((γ̂^{∗}E_{geom})_{t})_{|I×{0}×S^{1}} = pr^{∗}(γ^{∗}E_{geom})_{t}, where pr: I × {0} × S^{1} → S^{1}. Thus for dimensional reasons η^{2}((γ̂^{∗}E_{geom})_{t})_{|I×{0}×S^{1}} = pr^{∗}η^{2}((γ^{∗}E_{geom})_{t}) = 0. In a similar manner we have η^{2}((γ̂^{∗}E_{geom})_{t})_{|I×{1}×S^{1}} = 0. Thus the terms in (2) vanish. This finishes the proof of the claim. We now define an equivalence relation on Ẽ(γ) such that (λ,(γ_{geom}^{E})_{t}) ∼ (λ′,(γ_{geom}^{E})_{t′}) if

λ′= λexp( −2πi∫_{I×S^{1}} η^{2}((γ̃^{∗}E_{geom})_{t})) ,

where (γ̃^{∗}E_{geom})_{t} is any homotopy between (γ^{∗}E_{geom})_{t} and (γ^{∗}E_{geom})_{t′}. We set E(γ) := Ẽ(γ)/ ∼. Then E(γ) is a one dimensional complex vector space. If γ = ∪_{i=1,...,m}γ_{i} : S_{m,0}→ B, then we set E(γ) := ⊗_{i=1}^{m} E(γ_{i}). We now define E([g]) for a morphism [g] in T_{B} represented by a B-surface g: Σ → B. First assume that Σ is closed. We again have an inclusion ch_{1} : K^{1}(Σ) ↪ H_{dR}^{1}(Σ). Thus index(g^{∗}E) = 0 because of our assumption ch_{1}(index(E)) = 0, and there exists a taming (g^{∗}E_{geom})_{t}. The set of homotopy classes of tamings is parameterized by K^{1}(I × Σ,∂I × Σ) ≅ K^{0}(Σ) ≅ ℤ ⊕ ℤ. We restrict the choices of tamings (g^{∗}E_{geom})_{t} by the following condition. Let s ∈ Σ be any point. We consider M := Σ ∪_{s∼1} I. We choose some extension g̃: M → B of g by choosing a path from b_{0} to g(s). We will only consider tamings (g^{∗}E_{geom})_{t} which can be connected with g̃_{|{0}}^{∗}E_{b_{0},t} along M. This condition is independent of the choice of s and the path. The remaining choices correspond to the reduced K-theory K̃^{0}(Σ) ≅ ℤ. We define

E(g) := exp( 2πi∫_{Σ} η^{2}((g^{∗}E_{geom})_{t})) ∈ ℂ ≅ End(E(∅)) .

If Σ has a boundary, then for each boundary component i ∈ I_{1}(Σ) we obtain an object γ_{i} : S^{1} → B. We choose tamings (γ_{i}^{∗}E_{geom})_{t}, i ∈ I_{1}(Σ), in the distinguished components. The obstruction against extending these tamings to g^{∗}E_{geom} belongs to K^{1}(Σ,∂Σ). Since this group is trivial we can extend the given taming over the boundaries to a taming (g^{∗}E_{geom})_{t}. The set of homotopy classes of these extensions is parameterized by K^{1}(I × Σ,∂(I × Σ)) ≅ K^{0}(Σ,∂Σ) ≅ ℤ. Let z ∈ E(γ_{in}) be represented by

⊗_{i∈I_{1}^{in}(Σ)} (λ_{i},(γ_{i}^{∗}E_{geom})_{t}) .

Then we define

E(g)z:= exp( 2πi∫_{Σ} η^{2}((g^{∗}E_{geom})_{t})) z′∈ E(γ_{out}) ,

where z′∈ E(γ_{out}) is represented by

⊗_{i∈I_{1}^{out}(Σ)} (λ_{i},(γ_{i}^{∗}E_{geom})_{t}) .

We must show that E(g) only depends on the class [g] ∈ T_{B}. Let g and g′ be thin homotopic by a homotopy h : I × Σ → B. Furthermore, let (h^{∗}E_{geom})_{t} be a taming which restricts to the tamings (g^{∗}E_{geom})_{t} on {0} × Σ and (g^{′∗}E_{geom})_{t′} on {1} × Σ. Then we have by [3], Prop. 3.2., that dη^{2}((h^{∗}E_{geom})_{t}) = Ω^{3}(h^{∗}E_{geom}). Since the homotopy h is thin we have Ω^{3}(h^{∗}E_{geom}) = h^{∗}Ω^{3}(E_{geom}) = 0. It follows from Stoke’s Lemma that

∫_{Σ} η^{2}((g^{∗}E_{geom})_{t}) − ∫_{I×∂_{in}Σ} η^{2}((h^{∗}E_{geom})_{t}) + ∫_{I×∂_{out}Σ} η^{2}((h^{∗}E_{geom})_{t}) = ∫_{Σ} η^{2}((g^{′∗}E_{geom})_{t′}) .

This implies that E(g) = E(g′). Assume that we are given two tamings (g^{∗}E_{geom})_{t} and (g^{∗}E_{geom})_{t′}which coincide over the boundary of Σ, and which are not homotopic. Let Σ̃ = Σ ∪_{∂Σ} (Σ̄) be the surface obtained by doubling Σ along the boundary, and let g̃: Σ̃ → B be the map induced by g. We consider the family of cylinders I ×g̃^{∗}E → Σ̃ which is boundary tamed by (g^{∗}E_{geom})_{t} on {0} ×g̃^{∗}E, and by (g^{∗}E_{geom})_{t′}on {1} ×g̃^{∗}E over the copy Σ, and similarly by (g^{∗}E_{geom})_{t} and again (g^{∗}E_{geom})_{t} over the other copy Σ̄. This works since the two tamings (g^{∗}E_{geom})_{t} and (g^{∗}E_{geom})_{t′}coincide near ∂Σ. The index index((I ×g̃^{∗}E_{geom})_{bt}) ∈ K^{0}(Σ̃) is the obstruction against extending the boundary taming to a taming (we refer to [3], Sec. 2.4 and 2.5 for definitions). It follows from our restrictions on the choice of tamings (g^{∗}E_{geom})_{t} and (g^{∗}E_{geom})_{t′}that dimindex((I ×g̃^{∗}E_{geom})_{bt}) = 0. Let F_{geom} be any geometric family over Σ̃ with closed even dimensional fibres such that

index(F) = −index((I ×g̃^{∗}E_{geom})_{bt}) .

In fact we can realize F_{geom}with zero dimensional fibres (see [3], Sec. 3.1). Then index((I ×g̃^{∗}E_{geom}+F_{geom})_{bt}) = 0, and the boundary taming (I ×g̃^{∗}E_{geom}+ F_{geom})_{bt} can be extended to a taming. Using [3], Prop. 3.2, (at ∗), we obtain

∫_{Σ} η^{2}((g^{∗}E_{geom})_{t′}) − ∫_{Σ} η^{2}((g^{∗}E_{geom})_{t}) = ∫Σ̃η^{2}(∂(I ×g̃^{∗}E_{geom}+ F_{geom})_{t})
=^{∗} −∫Σ̃dη^{1}((I ×g̃^{∗}E_{geom}+ F_{geom})_{t}) + ∫Σ̃Ω^{2}(F_{geom})
= < ch_{2}(index(F)),[Σ̃] >
∈ ℤ .

We conclude that

exp( 2πi∫_{Σ} η^{2}((g^{∗}E_{geom})_{t})) = exp( 2πi∫_{Σ} η^{2}((g^{′,∗}E_{geom})′_{t})) .

We now have seen that E(g) only depends on the class [g] ∈ T_{B}. We have constructed for each choice of a taming E_{b_{0},t} a thin-invariant rank-one field theory. We must now show that this theory is independent of this choice. Let E_{b_{0},t′}be another choice, and let E′ be the corresponding field theory. We must define an equivalence Q : E → E′. Let γ : S^{1} → B be an object of T_{B}. Let (λ,(γ^{∗}E_{geom})_{t}) represent x ∈ E(γ), and let (γ^{∗}E_{geom})_{t′} be a taming in the distinguished class for the choice E_{b_{0},t′}. Then we consider the geometric family I × γ^{∗}E_{geom} over S^{1} with a boundary taming by (γ^{∗}E_{geom})_{t} and (γ^{∗}E_{geom})_{t′}at {0} × E_{geom} and {1} × E_{geom}. We consider index((I × γ^{∗}E_{geom})_{bt}) ∈ K^{0}(S^{1}) ≅ ℤ. Let F be a geometric family over S^{1} with underlying bundle S^{1} → S^{1} and vector bundle ℂ^{n}× S^{1} → S^{1}, where we adjust n and the fibre wise orientation such that index(F) = −index((I × γ^{∗}E_{geom})_{bt}). Since index((I ×γ^{∗}E_{geom}∪F_{geom})_{bt}) = 0 this boundary taming can be extended to a taming (I ×γ^{∗}E_{geom}∪F_{geom})_{t}. We define

Q(γ)(x) = ( λexp( −2πi∫_{S^{1}}η^{1}((I × γ^{∗}E_{geom}∪ F_{geom})_{t})) ,(γ^{∗}E_{geom})_{t′}) .

We claim that Q is well-defined. Let (γ^{∗}E_{geom})t̃and (γ^{∗}E_{geom})t̃′other choices of the tamings in the distinguished classes. Let (γ̃^{∗}E_{geom})_{t} and (γ̃^{∗}E_{geom})_{t′}be corresponding homotopies of tamings from (γ^{∗}E_{geom})_{t} to (γ^{∗}E_{geom})t̃

and from (γ^{∗}E_{geom})_{t′}to (γ^{∗}E_{geom})t̃′, where γ̃: I × S^{1} →^{pr} S^{1} →^{γ} B.

Then we obtain a boundary tamed family (I × γ̃^{∗}E_{geom})_{bt} over I × S^{1}. Let F̃_{geom}:= pr^{∗}F_{geom}. Then we can extend the boundary taming of I × γ̃^{∗}E_{geom}∪ F̃_{geom} to a taming (I × γ̃^{∗}E_{geom}∪ F̃_{geom})_{t}. By [3], Prop. 3.2, we have

dη^{1}((I × γ̃^{∗}E_{geom}∪ F̃_{geom})_{t}) = −η^{2}((γ̃^{∗}E_{geom})_{t′}) + η^{2}((γ̃^{∗}E_{geom})_{t}) .

By Stoke’s Lemma

−∫_{S^{1}}η^{1}((I ×γ^{∗}E_{geom}∪F_{geom})t̃)+∫_{S^{1}}η^{1}((I ×γ^{∗}E_{geom}∪F_{geom})_{t}) = −∫_{I×S^{1}} η^{2}((γ̃^{∗}E_{geom})_{t′})+∫_{I×S^{1}} η^{2}((γ̃^{∗}E_{geom})_{t}) .

An inspection of the definition of E(γ) shows that this relation implies that Q(γ) is well-defined independent of the choice of representatives of E(γ), E′(γ), and of the additional tamings. We use the monoidal structures of T_{B} and vect_{1} in order to extend Q to general objects of T_{B}. It remains to show that Q is natural. Let g : Σ → B be an X-surface representing a morphism [g] from γ_{in} to γ_{out}. Let (g^{∗}E_{geom})_{t} and (g^{∗}E_{geom})_{t′}be tamings in the distinguished classes with respect to E_{b_{0},t} and E_{b_{0},t′}. Then we consider the geometric family I × g^{∗}E_{geom} over Σ which is boundary tamed by (g^{∗}E_{geom})_{t} and (g^{∗}E_{geom})_{t′}at {0} × g^{∗}E_{geom} and {1} × g^{∗}E_{geom}. We now want to kill index((I × g^{∗}E_{geom})_{bt}) ∈ K^{0}(Σ). Let L_{0} := ℂ^{|dim(index((I×g^{∗}E_{geom})_{bt}))|} × Σ → Σ be a trivial vector bundle. If Σ is closed, then furthermore let L_{1} → Σ be a line bundle with c_{1}(L_{1}) = −c_{1}(index((I × g^{∗}E_{geom})_{bt})). We equip L_{1} with a hermitian metric and a metric connection. We let F_{0,geom} be the geometric family with underlying fibre bundle Σ → Σ and vector bundle L_{0}, where we flip the orientation of the fibres if dim(index((I × g^{∗}E_{geom})_{bt})) > 0. If Σ is closed, then we let F_{1,geom} be the union of geometric families F_{1a,geom} and F_{1b,geom}. Here F_{1a,geom} has the underlying fibre bundle Σ → Σ and the vector bundle is L_{1}, and F_{1b,geom} has underlying fibre bundle Σ → Σ with flipped orientation and the bundle is ℂ × Σ → Σ. Finally we set F_{geom}:= F_{0,geom} if Σ has a non-trivial boundary, and F_{geom}:= F_{0,geom}∪ F_{1,geom} if Σ is closed. Then index((I × g^{∗}E_{geom}∪ F_{geom})_{bt}) = 0 and we can extend the boundary taming to a taming (I × g^{∗}E_{geom}∪F_{geom})_{t}. Note that Ω^{2}(I ×g^{∗}E_{geom}∪F_{geom}) = Ω^{2}(F_{geom}) and ∫_{Σ} Ω^{2}(F_{geom}) =< c_{1}(index(F)),[Σ] >∈ ℤ, if Σ is closed, and ∫_{Σ} Ω^{2}(F_{geom}) = 0, if Σ is not closed. By [3], Prop. 3.2, we have

dη^{1}((I × g^{∗}E_{geom}∪ F_{geom})_{t}) = −η^{2}((g^{∗}E_{geom})_{t′}) + η^{2}((g^{∗}E_{geom})_{t}) + Ω^{2}(F_{geom}) .

By Stoke’s Lemma

∫_{∂Σ} η^{1}((I × g^{∗}E_{geom}∪ F_{geom})_{t}) = −∫_{Σ} η^{2}((g^{∗}E_{geom})_{t′}) + ∫_{Σ} η^{2}((g^{∗}E_{geom})_{t}) + ℤ .

This relation implies

Q(γ)E(g) = E′(g)Q(γ) .

Next we show that the curvature of E is Ω^{3}(E_{geom}). Let h: C → B be a B-three manifold with boundary g: Σ → B such that Σ is closed. If ∂C = ∅, then

∫_{C}Ω^{3}(E_{geom}) =< ch_{3}(index(E)),[C] >∈ ℤ

(see Lott [6], Prop. 8) so that

exp( 2πi∫_{C}Ω^{3}(E_{geom})) = 1 = E([g]) .

If ∂C ≠ 0, then C is homotopy equivalent to a two dimensional space. In this case vanishing of ch_{1}(index(h^{∗}E)) = h^{∗}ch_{1}(index(E)) implies that there exists a taming (h^{∗}E_{geom})_{t}. We have by [3], Prop. 3.2,

E([g]) = exp( 2πi∫_{∂C}η^{2}((h^{∗}E_{geom})_{t}))
= exp( 2πi∫_{C}dη^{2}((h^{∗}E_{geom})_{t}))
= exp( 2πi∫_{C}Ω^{3}(E_{geom})) .

Finally we show that the gerbe associated to the rank-one thin-invariant field theory E is the index gerbe gerbe(E_{geom}). Let g: Σ → B be a closed B-surface defining the cycle [g,Σ]. Let U be a tubular neighborhood of g(Σ). We choose a taming E_{|U,t}which can be connected with E_{b_{0},t}along some path from U to b_{0}. Then we obtain a taming (g^{∗}E_{geom})_{t}. Then we have

H(gerbe(E_{geom}))([g,Σ]) = exp( 2πi∫_{Σ} g^{∗}η^{2}(E_{|U,t}))
= exp( 2πi∫_{Σ} η^{2}((g^{∗}E_{geom})_{t}))
= E([g]) .
□

\underline{Remark:} Let E_{geom} be an odd-dimensional geometric family over B. Let f : B → S^{1} by any classifying map of −c_{1}(index(E)) ∈ H^{1}(B,ℤ). Furthermore, let F_{geom} be some odd dimensional geometric family over S^{1} such that c_{1}(index(F)) = 1 ∈ H^{1}(S^{1},ℤ) ≅ ℤ. Then we have ch_{1}(index(E + f^{∗}F)) = 0. One can show that the rank-one thin-invariant field theory associated to E_{geom}+ f^{∗}F_{geom} is independent of the choice of f and F_{geom}.

References

[1] J. L. Brylinski. Loop Spaces, Characteristic Classes, and Geometric Quantization. Birkäuser, Progress in Math. 107, 1993.

[2] U. Bunke. Transgression of the index gerbe. arXiv:math.DG/0109052, 2001.

[3] U. Bunke. Index theory, eta forms, and Deligne cohomology. arXiv:math.DG/0201112, 2002.

[4] J. Cheeger and J. Simons. Differential characters and geometric invariants. In LNM1167, pages 50–80. Springer Verlag, 1985.

[5] N. Hitchin. Lectures on special lagrangian submanifolds. arXiv:math.DG/9907034, 1999.

[6] J. Lott. Higher-degree analogs of the determinant line bundle. arXiv:math.DG/0106177, 2001.

[7] R. B. Melrose and P. Piazza. Families of Dirac operators, boundaries, and the b-calculus. J. Differential. Geom., 46 (1997), 99–180.

[8] P. Turner and S. Willerton. Gerbes and homotopy quantum field theory. arXiv:math.AT/0201116, 2002.