Thin invariant rank-one homotopy

quantum field theories and gerbes

Ulrich Bunke^{∗} February 9, 2002

Contents

1 Introduction 1 2 Gerbes 2 3 Thin-invariant rank-one field theories 2 4 The kernel of G 4 5 G is surjective 5

1 Introduction

In a recent preprint Turner and Willerton [7] discussed the relation between thin-invariant rank-one field theories and gerbes on a smooth manifold B. If H_{1}(B,ℤ) They claimed that under the assumption that H_{1}(B,ℤ) is torsionfree there is a one-to-one correspondence between isomorphism classes thin-invariant rank-one field theories and gerbes which is induced by holonomy. Unfortunately, in general this claim is wrong, e.g. if rankH_{1}(B,ℤ) ≥ 2. The goal of the present note is to give the correct relationship. The sets of thin-invariant rank-one field theories and gerbes over B form groups which we denote by Field_{1}(B) and Gerbe(B). Holonomy induces a homomorphism

G: Field_{1}(B) → Gerbe(B) .

Theorem 1.1. Let B be a smooh finite-dimensional manifold. Then there is an exact sequence 0 → H^{2}(H_{1}(B,ℤ),U(1)) → Field_{1}(B) →^{G} Gerbe(B) → 0 .

∗Göttingen, bunke@uni-math.gwdg.de

2 Gerbes

Let Gerbe(B) denote the group of gerbes with connection on B. A geometric picture of gerbes was introduced by Hitchin [5]. 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 H_{Del}^{3}(B) 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. 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. We recall the following exact sequence from Brylinski [1], Thm. 1.5.3:

0 → A_{B}^{2}(B)/A_{B}^{2}(B,d = 0,ℤ) → Gerbe(B) →^{c} H^{3}(B,ℤ) → 0 ,  (1)

where A_{B}^{2}(B,d = 0,ℤ) denotes the space of closed forms which represent integral cohomology classes.

3 Thin-invariant rank-one field theories

The thin homotopy category T_{B} of B is defined e.g. in [7]. 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 with scalar product where the morphisms are isometries, and where the monoidal structure is implemented by the tensor product. A rank-one thin-invariant field theory (compare [7], 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. Actually, we should call such a field theory unitary since E([g]) is an isometry for any morphism (compare Turaev [6] for a similar notion in the framework of homotopy quantum field theories). Part of the structure of the monodidal functor E : T_{B}→ vect_{1} is a natural transformation Φ_{E} : ⊗ ∘ (E × E) → E ∘ ∪ of funtors from T_{B}× T_{B} → vect_{1}. Two rank-one thin-invariant field theories E,E′are isomorphic, if there exists a natural transformation ψ: E → E′ such that for any two objects γ,γ′of T_{B} we have ψ(γ ∪ γ′) ∘ Φ_{E}(γ,γ′) = Φ_{E′}(γ,γ′) ∘ (ψ(γ) ⊗ ψ(γ′)) as morphisms from E(γ) ⊗ E(γ′) to E′(γ ∪ γ′). 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]). Let Field_{1}(B) denote the set of isomorphism classes of thin-invariant rank-one field theories on B. It is a group in a natural way. Given E,F ∈ Field_{1}(B), then E ⊗ F ∈ Field_{1}(B) is given by (E ⊗ F)(γ) := E(γ) ⊗ F (γ), and

(E ⊗ F)([g]) := E([g]) ⊗ F ([g]), Φ_{E⊗F} = (Φ_{E} ⊗ Φ_{F}) ∘ T, where T : V_{1} ⊗ V_{2} ⊗ V_{3} ⊗ V_{4} →^{∼} V_{1} ⊗ V_{3} ⊗ V_{2} ⊗ V_{4} is

the flip. The identity element in 1 ∈ Field_{1}(B) is the trivial field theory given by 1(γ) := ℂ and 1([g]) := id, Φ_{1}(γ,γ′) : ℂ ⊗ ℂ →^{can}ℂ. The inverse of E is the adjoint theory E^{∗} given by E^{∗}(γ) := E(γ) and E^{∗}([g]) := E([g])^{∗}, Φ_{E^{∗}} := Φ_{E}^{∗}, where ∗ involves the hermitean structures. The natural map

G: Field_{1}(B) → Gerbe(B)

is a homomorphism of groups.

4 The kernel of G

Let Γ be any group. We can consider Γ as a monoidal category where the only morphisms are the identities, and the product is given by the group structure.

Lemma 4.1. There is a natural bijection K between the set of isomorphism classes of monoidal functors E : Γ → vect_{1} and H^{2}(Γ,U(1)).

Proof. Let E be a monoidal functor E : Γ → vect_{1}. We fix isometries ϕ_{x}: E(x) →^{∼} ℂ for all x ∈ Γ. If x,y ∈ Γ, then we define the number c(x,y) ∈ U(1) such that the following diagram becomes commutative:

\begin{matrix}ℂ ⊗ ℂ & →^{can} & ℂ & →^{c(x,y)} & ℂ \\ ↑ ϕ_{x}⊗ ϕ_{y} & & & & ↑ ϕ_{xy} \\ E(x) ⊗ E(y) & & −→^{Φ_{E}(x,y)} & & E(xy) \end{matrix} .

Associativity of the monoidal structure yields the cocycle relation

c(y,z)c(xy,z)^{−1} c(x,yz)c(x,y)^{−1} = 1 .

If ϕ′_{x}, x ∈ Γ, is another family of identifications, then we can write ϕ′_{x} = d(x)ϕ_{x} for some d(x) ∈ U(1). The corresponding cocyle is given by c′(x,y) = d(y)^{−1} d(xy)d(x)^{−1} c(x,y). Therefore we have a well-defined group cohomology class K(E) := [c] ∈ H^{2}(Γ,U(1)) represented by c for some choice of the identifications ϕ_{x}. On the one hand, if K(E) = 0, then we can adjust the identifications ϕ_{x} such that c ≡ 0. Therefore, E is isomorphic to the trivial functor. On the other hand, let [c] ∈ H^{2}(Γ,U(1)) be any class represented by a cocycle c. Then we define a monoidal functor E as follows. We let E(x) := ℂ, and we define the monoidal structure Φ_{E} such that

\begin{matrix}ℂ ⊗ ℂ & →^{can} & ℂ & →^{c(x,y)} & ℂ \\ ∥ & & & & ∥ \\ E(x) ⊗ E(y) & & −→^{Φ_{E}(x,y)} & & E(xy) \end{matrix} .

commutes. Then K(E) = [c]. □

Proposition 4.2. The kernel of G: Field_{1}(B) → Gerbe(B) is naturally isomorphic to H^{2}(H_{1}(B,ℤ),U(1)).

Proof. We construct a correspondence between ker(G) and the monoidal functors H_{1}(B,ℤ) → vect_{1}. Then we apply Lemma 4.1. Consider E ∈ ker(G). Let γ_{in},γ_{out} be two objects in T_{B}. If g: Σ → B and g′: Σ′→ B are B-surfaces representing morphisms [g],[g′] from γ_{in} to γ_{out}, then we claim that E([g]) = E([g′]). Let ḡ′: Σ̄′→ B represent the inverse of [g′], where Σ̄′ is Σ′with the opposite orientation, and ḡ′= g′as maps. Then E([g]) ∘ E([ḡ′]) = E([g] ∘ [ḡ′]) = G(E)(< g∘ḡ′>) = 1, where < g∘ḡ′> denotes the closed surface obtained from g∘ḡ′: Σ∪_{∂_{in}Σ=∂_{out}Σ̄′} Σ̄′→ B by identifying ingoing and outgoing boundaries.

Since γ_{in} and γ_{out} are cobordant exactly if they belong to the same homology class we can therefore associate in a natural way a one-dimensional vector space E(x) to each homology class x ∈ H_{1}(B,ℤ). Moreover, there is a natural isometry Φ_{E}(x,x′) : E(x)⊗E(x′) → E(x +x′) obtained as follows: If γ,γ′represent x,x′, then E(x) ≅ E(γ), E(x′) ≅ E(γ′) and Φ_{E}(x,x′) corresponds to Φ_{E}(γ,γ′). We thus have defined the required monoidal functor E : H_{1}(B,ℤ) → vect_{1}. On the one hand, if this functor is isomorphic to the trivial one, then the field theory E was isomorphic to the trivial one, too. On the other hand, any monoidal functor E : H_{1}(B,ℤ) → vect_{1} can be extended to a field theory in ker(G). □

Let B = T^{2} be the two dimensional torus. Then H_{1}(T^{2},ℤ) = ℤ^{2}. We have H^{2}(ℤ^{2},U(1)) ≅ U(1). Therefore, even though H_{1}(T^{2},ℤ) is torsion-free there is a family of pairwise non-equivalent thin-invariant rank-one field theories parametrized by U(1) which have trivial holonomy. The argument given in the proof of [7], Thm. 3.5, (p7-b10), breaks down since the group ℤ^{2} is not freely generated by two generators as a group.

5 G is surjective

Proposition 5.1. If B is finite-dimensional, then the homomorphism G is surjective.

Proof. Let G ∈ Gerbe(B) be a gerbe such that c(G) = 0. By (1) G is represented by a form ω ∈ A_{B}^{2}(B). We now define a field theory E ∈ Field_{1}(B) such that G(E) = G. We set E(γ) := ℂ for all objects γ of T_{B}. We fix the canonical monoidal structure ℂ ⊗ ℂ ≅ ℂ. If g: Σ → B represents a morphism [g] ∈ Hom_{T_{B}}(γ,γ′), then we define E([g]) : ℂ → ℂ to be multiplication by

exp( 2πi∫_{Σ} g^{∗}ω) .

Indeed, this number only depends on the thin homotopy class of g. This field theory satisfies by construction G(E) = G. Now assume that B is connected and π_{1}(B) = 0. Let G ∈ Gerbe(B). Then we can apply the construction given in the proof of [7], Thm. 4.1, in order to construct a field theory E such that G(E) = G. For an object γ of T_{B} we let Ẽ(γ) be the free vector space over ℂ generated by all B-surfaces g : Σ → B which represent a morphism in T_{B} from γ to the empty object. We define an equivalence relation on Ẽ(γ) such that g′∼ H(G)(< g′∘ḡ>)g. Then we define E(γ) := Ẽ(γ)/ ∼. The monoidal structure E(γ)⊗E(γ′) → E(γ∪γ′) is induced by g ⊗ g′↦ g ∪ g′. If h: Σ → B represents a morphism [h] ∈ Hom_{T_{B}}(γ,γ′), then the map E([h]) : E(γ) → E(γ′) is induced by g ↦ h ∘ g. In this way we obtain a field theory E ∈ Field_{1}(B) such that G(E) = G. Finally, let B by any manifold and G ∈ Gerbe(B). Let M be a smooth manifold which is dim(B) + 1-homotopy equivalent to K(3,ℤ). Then H^{3}(M,ℤ) = ℤ. Let 1 ∈ H^{3}(M,ℤ) be a fixed generator. Let G_{1} ∈ Gerbe(M) be a gerbe such that c(G_{1}) = 1. Let E_{1} ∈ Field_{1}(M) be the field theory such that G(E_{1}) = G_{1}. It exists since π_{1}(M) = 0, and it is unique by Proposition 4.2 since H_{1}(M,ℤ) = 0. Since [B,M] →^{∼} H^{3}(B,ℤ), [f] ↦ f^{∗}1, there exists f : B → M such that c(G) = f^{∗}1. Then c(G ⊗ f^{∗}G_{1}^{−1} ) = 0. Thus there exists a field theory E_{0} ∈ Field_{1}(B) such that G(E_{0}) = G ⊗ f^{∗}G_{1}^{−1} . Now E := E_{0}⊗ f^{∗}E_{1} ∈ Field_{1}(B) satisfies G(E) = G. □

Theorem 1.1 follows from Propositions 4.2 and 5.1. □

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