Theory and Applications of Categories, Vol. 18, No. 9, 2007, pp. 240–273.

MORE MORPHISMS BETWEEN BUNDLE GERBES

KONRAD WALDORF

Abstract. Usually bundle gerbes are considered as objects of a 2-groupoid, whose 1morphisms, called stable isomorphisms, are all invertible. I introduce new 1-morphisms which include stable isomorphisms, trivializations and bundle gerbe modules. They fit into the structure of a 2-category of bundle gerbes, and lead to natural definitions of surface holonomy for closed surfaces, surfaces with boundary, and unoriented closed surfaces.

Introduction

From several perspectives it becomes clear that bundle gerbes are objects in a 2-category: from the bird’s-eye view of algebraic geometry, where gerbes appear as some kind of stack, or in topology, where they appear as one possible categorification of a line bundle, but also from a worm’s-eye view on the definitions of bundle gerbes and their morphisms, which show that there have to be morphisms between the morphisms.

In [Ste00] a 2-groupoid is defined, whose objects are bundle gerbes, and whose 1morphisms are stable isomorphisms. To explain a few details, recall that bundle gerbes are defined using surjective submersions π : Y → M, and that a stable isomorphism A : G_{1} → G_{2} between two bundle gerbes G_{1} and G_{2} with surjective submersions π_{1} : Y_{1} → M and π_{2} : Y → M consists of a certain line bundle A over the fibre product Y_{1} ×_{M} Y_{2}. 2morphisms between stable isomorphisms are morphisms β : A → A′ of those line bundles, obeying a compatibility constraint. Many examples of surjective submersions arise from open covers {U_{α}}_{α∈A} of M by taking Y to be the disjoint union of the open sets U_{α} and π to be the projection (x, α) ↦ x. From this point of view, fibre products Y_{1} ×_{M} Y_{2} correspond the common refinement of two open covers. So, the line bundle A of a stable isomorphism lives over the common refinement of the open covers of the two bundle gerbes.

Difficulties with this definition of stable isomorphisms arise when two stable isomorphisms A : G_{1} → G_{2} and A′ : G_{2} → G_{3} are going to be composed: one has to define a line bundle Ã over Y_{1} ×_{M} Y_{3} using the line bundles A over Y_{1} ×_{M} Y_{2} and A′ over Y_{2} ×_{M} Y_{3}. In [Ste00] this problem is solved using descent theory for line bundles.

In this note, I present another definition of 1-morphisms between bundle gerbes (Definition 1.4). Compared to stable isomorphisms, their definition is relaxed in two aspects:

Received by the editors 2007-03-04 and, in revised form, 2007-04-30.

Transmitted by Larry Breen. Published on 2007-05-11.

2000 Mathematics Subject Classification: 55R65, 53C29, 18B40.

Key words and phrases: 2-category, bundle gerbe, holonomy.

\begin{matrix}c \\ ◯ \end{matrix} Konrad Waldorf, 2007. Permission to copy for private use granted.

1) the line bundle is replaced by a certain vector bundle of rank possibly higher than one.

2) this vector bundle is defined over a smooth manifold Z with surjective submersion ζ : Z → Y_{1} ×_{M} Y_{2}. In terms of open covers, the vector bundle lives over a refinement of the common refinement of the open covers of the two bundle gerbes.

Stable isomorphisms appear as a particular case of this relaxed definition. I also give a generalized definition of 2-morphisms between such 1-morphisms (Definition 1.6). Two goals are achieved by this new type of morphisms between bundle gerbes. First, relaxation 1) produces many 1-morphisms which are not invertible, in contrast to the stable isomorphisms in [Ste00]. To be more precise, a 1-morphism is invertible if and only if its vector bundle has rank 1 (Proposition 1.13). The non-invertible 1-morphisms provide a new formulation of left and right bundle gerbe modules (Definition 3.6). Second, relaxation 1) erases the difficulties with the composition of 1-morphisms: the vector bundle Ã of the composition A′ ∘ A is just defined to be the tensor product of the vector bundles A and A′ of the two 1-morphisms pulled back to Z ×_{Y_{2}} Z′, where A over Z is the vector bundle of A and A′ over Z′ is the vector bundle of A′. The composition defined like that is strictly associative (Proposition 1.8). This way we end up with a strictly associative 2-category 𝔅𝔊𝔯𝔟(M) of bundle gerbes over M. The aim of this note is to show that a good understanding of this 2-category can be useful.

This note is organized as follows. Section 1 contains the definitions and properties of the 2-category 𝔅𝔊𝔯𝔟(M) of bundle gerbes over M. We also equip this 2-category with a monoidal structure, pullbacks and a duality. Section 2 relates our new definition of 1morphisms between bundle gerbes to the one of a stable isomorphism: two bundle gerbes are isomorphic objects in 𝔅𝔊𝔯𝔟(M) if and only if they are stably isomorphic (Corollary 2.2). In section 3 we present a unified view on important structure related to bundle gerbes in terms of the new morphisms of the 2-category 𝔅𝔊𝔯𝔟(M):

a) a trivialization of a bundle gerbe G is a 1-isomorphism A : G → I_{ρ} from G to a trivial bundle gerbe I_{ρ} given by a 2-form ρ on M.

b) a bundle gerbe module of a bundle gerbe G is a (not necessarily invertible) 1morphism E : G → I_{ω} from G to a trivial bundle gerbe I_{ω}.

c) a Jandl structure on a bundle gerbe G over M is a triple (k, A, φ) of an involution k of M, a 1-isomorphism A : k^{∗}G → G^{∗} and a certain 2-morphism φ : k^{∗}A ⇒ A^{∗}.

Then we demonstrate how this understanding in combination with the properties of the 2-category 𝔅𝔊𝔯𝔟(M) can be employed to give convenient definitions of surface holonomy. For this purpose we classify the morphisms between trivial bundle gerbes: there is an equivalence of categories

ℌ𝔬𝔪(I_{ρ_{1}}, I_{ρ_{2}}) ≅ 𝔅𝔲𝔫_{ρ_{2}−ρ_{1}}(M)

between the morphism category between the trivial bundle gerbes I_{ρ_{1}} and I_{ρ_{2}} and the category of vector bundles over M for which the trace of the curvature gives the 2-form ρ_{2} − ρ_{1} times its rank.

The interpretation of bundle gerbe modules and Jandl structures in terms of morphisms between bundle gerbes is one step to understand the relation between two approaches to two-dimensional conformal field theories: on the one hand the Lagrangian approach with a metric and a bundle gerbe G being the relevant structure [GR02] and on the other hand the algebraic approach in which a special symmetric Frobenius algebra object A in a modular tensor category C plays this role [FRS02]. Similarly as bundle gerbes, special symmetric Frobenius algebra objects in C form a 2-category, called Frob_{C}. In both approaches it is well-known how boundary conditions have to be imposed. In the Lagrangian approach one chooses a D-brane: a submanifold Q of the target space together with a bundle gerbe module for the bundle gerbe G restricted to Q [Gaw05]. In the algebraic approach one chooses a 1-morphism from A to the tensor unit I of C (which is trivially a special symmetric Frobenius algebra object) in the 2-category Frob_{C} [SFR06]. Now that we understand a gerbe module as a 1-morphism from G to I_{ω} we have found a common principle in both approaches. A similar success is obtained for unoriented conformal field theories. In the Lagrangian approach, the bundle gerbe G has to be endowed with a Jandl structure [SSW05], which is in particular a 1-isomorphism k^{∗}G → G^{∗} to the dual bundle gerbe G^{∗}. In the algebraic approach one has to choose a certain algebra isomorphism A → A^{op} from A to the opposed algebra [FRS04].

Acknowledgements. I would like to thank Christoph Schweigert for his advice and encouragement, and Urs Schreiber for the many helpful discussions on 2-categories.

Conventions. Let us fix the following conventions for the whole article: by vector bundle I refer to a complex vector bundle of finite rank, equipped with a hermitian structure and with a connection respecting this hermitian structure. Accordingly, a morphism of vector bundles is supposed to respect both the hermitian structures and the connections. In particular, a line bundle is a vector bundle in the above sense of rank one. The symmetric monoidal category 𝔅𝔲𝔫(M), which is formed by all vector bundles over a smooth manifold M and their morphisms in the above sense, is for simplicity tacitly replaced by an equivalent strict tensor category.

1. The 2-Category of Bundle Gerbes

Summarizing, the 2-category 𝔅𝔊𝔯𝔟(M) of bundle gerbes over a smooth manifold M consists of the following structure:

1. A class of objects – bundle gerbes over M.

2. A morphism category ℌ𝔬𝔪(G, H) for each pair G, H of bundle gerbes, whose objects are called 1-morphisms and are denoted by A : G → H, and whose morphisms are called 2-morphisms and are denoted β : A_{1} ⇒ A_{2}.

3. A composition functor

∘ : ℌ𝔬𝔪(H, K) × ℌ𝔬𝔪(G, H) ⟶ ℌ𝔬𝔪(G, K)

for each triple G, H, K of bundle gerbes.

4. An identity 1-morphism id_{G} : G → G for each bundle gerbe G together with natural 2-isomorphisms

ρ_{A} : id_{H} ∘ A ⟹ A and λ_{A} : A ∘ id_{G} ⟹ A

associated to every 1-morphism A : G → H.

This structure satisfies the axioms of a strictly associative 2-category:

(2C1) For three 1-morphisms A : G_{1} → G_{2}, A′ : G_{2} → G_{3} and A′′ : G_{3} → G_{4}, the composition functor satisfies

A′′ ∘ (A′ ∘ A) = (A′′ ∘ A′) ∘ A.

(2C2) For 1-morphisms A : G_{1} → G_{2} and A′ : G_{2} → G_{3}, the 2-isomorphisms λ_{A} and ρ_{A}

satisfy the equality

id_{A′} ∘ ρ_{A} = λ_{A′} ∘ id_{A}

as 2-morphisms from A′ ∘ id_{G_{2}} ∘ A to A′ ∘ A.

The following two subsections contain the definitions of the structure of the 2-category 𝔅𝔊𝔯𝔟(M). The two axioms are proved in Propositions 1.8 and 1.11. The reader who is not interested in these details may directly continue with section 3.

1.1. Objects and Morphisms. The definition of the objects of the 2-category 𝔅𝔊𝔯𝔟(M)

– the bundle gerbes over M – is the usual one, just like, for instance, in [Mur96, Ste00, GR02]. Given a surjective submersion π : Y → M we use the notation Y^{[k]} := Y ×_{M} ...×_{M} Y for the k-fold fibre product, which is again a smooth manifold. Here we consider fibre products to be strictly associative for simplicity. For the canonical projections between fibre products we use the notation π_{i_{1}...i_{k}} : Y^{[n]} → Y^{[k]}.

1.2. Definition. A bundle gerbe G over a smooth manifold M consists of the following structure:

1. a surjective submersion π : Y → M,

2. a line bundle L over Y^{[2]},

3. a 2-form C ∈ Ω^{2}(Y), and 4. an isomorphism

μ : π_{12}^{∗}L ⊗ π_{23}^{∗}L ⟶ π_{13}^{∗}L

of line bundles over Y^{[3]}.

This structure has to satisfy two axioms:

(G1) The curvature of L is fixed by

curv(L) = π_{2}^{∗}C − π_{1}^{∗}C.

(G2) μ is associative in the sense that the diagram

\xymatrix{π_{12}^{∗}L ⊗ π_{23}^{∗}L ⊗ π_{34}^{∗}L \ar[r]^{π_{123}^{∗}μ⊗id} \ar[d]_{id⊗π_{234}^{∗}μ} & π_{13}^{∗}L ⊗ π_{34}^{∗}L \ar[d]^{π_{134}^{∗}μ} \\ π_{12}^{∗}L ⊗ π_{24}^{∗}L \ar[r]_{π_{124}^{∗}μ} & π_{14}^{∗}L}

of isomorphisms of line bundles over Y^{[4]} is commutative.

To give an example of a bundle gerbe, we introduce trivial bundle gerbes. Just as for every 1-form A ∈ Ω^{1}(M) there is the (topologically) trivial line bundle over M having this 1-form as its connection 1-form, we find a trivial bundle gerbe for every 2-form ρ ∈ Ω^{2}(M). Its surjective submersion is the identity id : M → M, and its 2-form is ρ. Its line bundle over M ×_{M} M ≅ M is the trivial line bundle with the trivial connection, and its isomorphism is the identity isomorphism between trivial line bundles. Now, axiom (G1) is satisfied since curv(L) = 0 and π_{1} = π_{2} = id_{M}. The associativity axiom (G2) is surely satisfied by the identity isomorphism. Thus we have defined a bundle gerbe, which we denote by I_{ρ}.

It should not be unmentioned that the geometric nature of bundle gerbes allows explicit constructions of all (bi-invariant) bundle gerbes over all compact, connected and simple Lie groups [GR02, Mei02, GR03]. It becomes in particular essential that a surjective submersion π : Y → M is more general than an open cover of M.

An important consequence of the existence of the isomorphism μ in the structure of a bundle gerbe G is that the line bundle L restricted to the image of the diagonal embedding Δ : Y → Y^{[2]} is canonically trivializable (as a line bundle with connection):

1.3. Lemma. There is a canonical isomorphism t_{μ} : Δ^{∗}L → 1 of line bundles over Y, which satisfies

π_{1}^{∗}t_{μ} ⊗ id = Δ_{112}^{∗}μ and id ⊗ π_{2}^{∗}t_{μ} = Δ_{122}^{∗}μ

as isomorphisms of line bundles over Y^{[2]}, where Δ_{112} : Y^{[2]} → Y^{[3]} duplicates the first and Δ_{122} : Y^{[2]} → Y^{[3]} duplicates the second factor.

MORE MORPHISMS BETWEEN BUNDLE GERBES  245

Proof. The isomorphism t_{μ} is defined using the canonical pairing with the dual line bundle L^{∗} (which is strict by convention) and the isomorphism μ:

\xymatrix{Δ^{∗}L = Δ^{∗}L ⊗ Δ^{∗}L ⊗ Δ^{∗}L^{∗} \ar[r]^{Δ^{∗}μ⊗id} & Δ^{∗}L ⊗ Δ^{∗}L^{∗} = 1}  (1)

The two claimed equations follow from the associativity axiom (G2) by pullback of the diagram along Δ_{1222} and Δ_{1112} respectively. □

Now we define the category ℌ𝔬𝔪(G_{1}, G_{2}) for two bundle gerbes G_{1} and G_{2}, to whose structure we refer by the same letters as in Definition 1.2 but with indices 1 or 2 respectively. 1.4. Definition. A 1-morphism A : G_{1} → G_{2} consists of the following structure:

1. a surjective submersion ζ : Z → Y_{1} ×_{M} Y_{2},

2. a vector bundle A over Z, and

3. an isomorphism

α : L_{1} ⊗ ζ_{2}^{∗}A ⟶ ζ_{1}^{∗}A ⊗ L_{2}  (2)

of vector bundles over Z ×_{M} Z. This structure has to satisfy two axioms: (1M1) The curvature of A obeys

\frac{1}{n}tr(curv(A)) = C_{2} − C_{1},

where n is the rank of the vector bundle A. (1M2) The isomorphism α is compatible with the isomorphisms μ_{1} and μ_{2} of the gerbes

G_{1} and G_{2} in the sense that the diagram

\xymatrix{ζ_{12}^{∗}L_{1} ⊗ ζ_{23}^{∗}L_{1} ⊗ ζ_{3}^{∗}A \ar[r]^{μ_{1}⊗id} \ar[d]_{id⊗ζ_{23}^{∗}α} & ζ_{13}^{∗}L_{1} ⊗ ζ_{3}^{∗}A \ar[dd]^{ζ_{13}^{∗}α} \\ ζ_{12}^{∗}L_{1} ⊗ ζ_{2}^{∗}A ⊗ ζ_{23}^{∗}L_{2} \ar[d]_{ζ_{12}^{∗}α⊗id} \\ ζ_{1}^{∗}A ⊗ ζ_{12}^{∗}L_{2} ⊗ ζ_{23}^{∗}L_{2} \ar[r]_{id⊗μ_{2}} & ζ_{1}^{∗}A ⊗ ζ_{13}^{∗}L_{2}}

of isomorphisms of vector bundles over Z ×_{M} Z ×_{M} Z is commutative.

Here we work with the following simplifying notation: we have not introduced notation for the canonical projections Z → Y_{1} and Z → Y_{2}, accordingly we don’t write pullbacks with these maps. So in (2), where the line bundles L_{i} are pulled back along the induced map Z^{[2]} → Y_{i}^{[2]} for i = 1, 2 and also in axiom (1M1) which is an equation of 2-forms on Z.

As an example of a 1-morphism, we define the identity 1-morphism

id_{G} : G ⟶ G  (3)

of a bundle gerbe G over M. It is defined by Z := Y^{[2]}, the identity ζ := id_{Z}, the line bundle L of G over Z and the isomorphism λ defined by

\xymatrix{π_{13}^{∗}L ⊗ π_{34}^{∗}L \ar[r]^{π_{134}^{∗}μ} & π_{14}^{∗}L \ar[r]^{π_{124}^{∗}μ^{−1}} & π_{12}^{∗}L ⊗ π_{24}^{∗}L,}  (4)

where we identified Z^{[2]} = Y^{[4]}, ζ_{2} = π_{34} and ζ_{1} = π_{12}. Axiom (1M1) is the same as axiom (G1) for the bundle gerbe G and axiom (1M2) follows from axiom (G2).

The following lemma introduces an important isomorphism of vector bundles associated to every 1-morphism, which will be used frequently in the definition of the structure of 𝔅𝔊𝔯𝔟(M) and also in section 2.

1.5. Lemma. For any 1-morphism A : G_{1} → G_{2} there is a canonical isomorphism

d_{A} : ζ_{1}^{∗}A ⟶ ζ_{2}^{∗}A

of vector bundles over Z^{[2]} = Z ×_{P} Z, where P := Y_{1} ×_{M} Y_{2}, with the following properties:

a) It satisfies the cocycle condition

ζ_{13}^{∗}d_{A} = ζ_{23}^{∗}d_{A} ∘ ζ_{12}^{∗}d_{A}

as an equation of isomorphisms of vector bundles over Z^{[3]}.

b) The diagram

\xymatrix{L_{1} ⊗ ζ_{3}^{∗}A \ar[d]_{id⊗ζ_{34}^{∗}d_{A}} \ar[r]^{ζ_{13}^{∗}α} & ζ_{1}^{∗}A ⊗ L_{2} \ar[d]^{ζ_{12}^{∗}d_{A}⊗id} \\ L_{1} ⊗ ζ_{4}^{∗}A \ar[r]_{ζ_{24}^{∗}α} & ζ_{2}^{∗}A ⊗ L_{2}}

of isomorphisms of vector bundles over Z^{[2]} ×_{M} Z^{[2]} is commutative.

Proof. Notice that the isomorphism α of A restricted from Z ×_{M} Z to Z ×_{P} Z gives an isomorphism

α|_{Z×_{P} Z} : Δ^{∗}L_{1} ⊗ ζ_{2}^{∗}A ⟶ ζ_{1}^{∗}A ⊗ Δ^{∗}L_{2}.  (5)

By composition with the isomorphisms t_{μ_{1}} and t_{μ_{2}} from Lemma 1.3 we obtain the isomorphism d_{A}:

\xymatrix{ζ_{1}^{∗}A \ar[r]^{id⊗t_{μ_{2}}^{−1}} & ζ_{1}^{∗}A ⊗ Δ^{∗}L_{2} \ar[r]^{α|_{Z×_{P} Z}^{−1}} & Δ^{∗}L_{1} ⊗ ζ_{2}^{∗}A \ar[r]^{t_{μ_{1}}⊗id} & ζ_{2}^{∗}A.}  (6)

The cocycle condition a) and the commutative diagram b) follow both from axiom (1M2) for A and the properties of the isomorphisms t_{μ_{1}} and t_{μ_{2}} from Lemma 1.3. □

Now that we have defined the objects of ℌ𝔬𝔪(G_{1}, G_{2}), we come to its morphisms. For two 1-morphisms A_{1} : G_{1} → G_{2} and A_{2} : G_{1} → G_{2}, consider triples

(W, ω, β_{W})  (7)

consisting of a smooth manifold W, a surjective submersion ω : W → Z_{1} ×_{P} Z_{2}, where again P := Y_{1} ×_{M} Y_{2}, and a morphism β_{W} : A_{1} → A_{2} of vector bundles over W. Here we work again with the convention that we don’t write pullbacks along the unlabelled canonical projections W → Z_{1} and W → Z_{2}. The triples (7) have to satisfy one axiom (2M): the morphism β_{W} has to be compatible with isomorphism α_{1} and α_{2} of the 1morphisms A_{1} and A_{2} in the sense that the diagram

\xymatrix{L_{1} ⊗ ω_{2}^{∗}A_{1} \ar[r]^{α_{1}} \ar[d]_{1⊗ω_{2}^{∗}β_{W}} & ω_{1}^{∗}A_{1} ⊗ L_{2} \ar[d]^{ω_{1}^{∗}β_{W} ⊗1} \\ L_{1} ⊗ ω_{2}^{∗}A_{2} \ar[r]_{α_{2}} & ω_{1}^{∗}A_{2} ⊗ L_{2}}  (8)

of morphisms of vector bundles over W ×_{M} W is commutative. On the set of all triples (7) satisfying this axiom we define an equivalence relation according to that two triples (W, ω, β_{W}) and (W′, ω′, β_{W′}) are equivalent, if there exists a smooth manifold X with surjective submersions to W and W′ for which the diagram

\xymatrix{ & X \ar[dl] \ar[dr] \\ W \ar[dr]_{ω} &  & W′ \ar[dl]^{ω′} \\  & Z_{1} ×_{P} Z_{2}}  (9)

of surjective submersions is commutative, and the morphisms β_{W} and β_{W′} coincide when pulled back to X.

1.6. Definition. A 2-morphism β : A_{1} ⇒ A_{2} is an equivalence class of triples (W, ω, β_{W}) satisfying axiom (2M).

As an example of a 2-morphism we define the identity 2-morphism id_{A} : A ⇒ A associated to every 1-morphism A : G_{1} → G_{2}. It is defined as the equivalence class of the triple (Z^{[2]}, id_{Z^{[2]}}, d_{A}) consisting of the fibre product Z^{[2]} = Z ×_{P} Z, the identity id_{Z[2]} and the isomorphism d_{A} : ζ_{1}^{∗}A → ζ_{2}^{∗}A of vector bundles over Z^{[2]} from Lemma 1.5. Axiom (2M) for this triple is proven with Lemma 1.5 b).

Now we have defined objects and morphisms of the morphism category ℌ𝔬𝔪(G_{1}, G_{2}), and we continue with the definition the composition β′•β of two 2-morphisms β : A_{1} ⇒ A_{2} and β′ : A_{2} ⇒ A_{3}. It is called vertical composition in agreement with the diagrammatical notation

[figure: G_{1}; A_{1}; A_{2}; A_{3}; G_{2}; β; β′] .  (10)

We choose representatives (W, ω, β_{W}) and (W′, ω′, β_{W′}) and consider the fibre product W̃ := W ×_{Z_{2}} W′ with its canonical surjective submersion ω̃ : W̃ → Z_{1} ×_{P} Z_{3}, where again P := Y_{1} ×_{M} Y_{2}. By construction we can compose the pullbacks of the morphisms β_{W} and β_{W′} to W̃ and obtain a morphism

β_{W′} ∘ β_{W} : A_{1} ⟶ A_{3}  (11)

of vector bundles over W̃. From axiom (2M) for β and β′ the one for the triple (W̃ , ω̃, β_{W′} ∘ β_{W}) follows. Furthermore, the equivalence class of this triple is independent of the choices of the representatives of β and β′and thus defines the 2-morphism β′ •β. The composition • of the category ℌ𝔬𝔪(G_{1}, G_{2}) defined like this is associative.

It remains to check that the 2-isomorphism id_{A} : A ⇒ A defined above is the identity under the composition •. Let β : A ⇒ A′ be a 2-morphism and (W, ω, β_{W}) a representative. The composite β • id_{A} can be represented by the triple (W′, ω′, β ∘ d_{A}) with W′ = Z ×_{P} W, where ω′ : W′ → Z ×_{P} Z′ is the identity on the first factor and the projection W → Z′ on the second one. We have to show, that this triple is equivalent to the original representative (W, ω, β_{W}) of β. Consider the fibre product

X := W ×_{(Z×_{P} Z′)} W′ ≅ W ×_{Z′} W,  (12)

so that condition (9) is satisfied. The restriction of the commutative diagram (8) of morphisms of vector bundles over W ×_{M} W from axiom (2M) for β to X gives rise to the commutative diagram

\xymatrix{ζ_{2}^{∗}A \ar[r]^{d_{A}^{−1}} \ar[d]_{ω_{2}^{∗}β_{W}} & ζ_{1}^{∗}A \ar[d]^{ω_{1}^{∗}β_{W}} \\ A′ \ar[r]_{Δ^{∗}d_{A′}^{−1}} & A′}  (13)

of morphisms of vector bundles over X, where d_{A} and d_{A′} are the isomorphisms determined by the 1-morphisms A and A′ according to Lemma 1.5. Their cocycle condition from Lemma 1.5 a) implies Δ^{∗}d_{A′} = id, so that diagram (13) is reduced to the equality ω_{2}^{∗}β_{W} ∘ d_{A} = ω_{1}^{∗}β_{W} of isomorphisms of vector bundles over X. This shows that the triples (W, ω, β_{W}) and (W′, ω′, β_{W} ∘ d_{A}) are equivalent and we have β • id_{A} = β. The equality id_{A′} • β = β follows analogously.

Now the definition of the morphism category ℌ𝔬𝔪(G_{1}, G_{2}) is complete. A morphism in this category, i.e. a 2-morphism β : A ⇒ A′, is invertible if and only if the morphism β_{W} : A → A′ of any representative (W, ω, β_{W}) of β is invertible. Since – following our convention – morphism of vector bundles respect the hermitian structures, β_{W} is invertible if and only if the ranks of the vector bundles of the 1-morphisms A and A′ coincide. In the following, we call two 1-morphisms A : G_{1} → G_{2} and A′ : G_{1} → G_{2} isomorphic, if there exists a 2-isomorphism β : A ⇒ A′ between them.

1.7. The Composition Functor. Let G_{1}, G_{2} and G_{3} be three bundles gerbes over M. We define the composition functor

∘ : ℌ𝔬𝔪(G_{2}, G_{3}) × ℌ𝔬𝔪(G_{1}, G_{2}) ⟶ ℌ𝔬𝔪(G_{1}, G_{3})  (14)

on objects in the following way. Let A : G_{1} → G_{2} and A′ : G_{2} → G_{3} be two 1-morphisms. The composed 1-morphism

A′ ∘ A : G_{1} ⟶ G_{3}  (15)

consists of the fibre product Z̃ := Z ×_{Y_{2}} Z′ with its canonical surjective submersion ζ̃: Z̃ → Y_{1} ×_{M} Y_{3}, the vector bundle Ã := A ⊗ A′ over Z̃, and the isomorphism

α̃ := (id_{ζ_{1}^{∗}A} ⊗ α′) ∘ (α ⊗ id_{ζ_{2}^{′∗}A′})  (16)

of vector bundles over Z̃ ×_{M} Z̃.

Indeed, this defines a 1-morphism from G_{1} to G_{3}. Recall that if ∇_{A} and ∇_{A′} denote the connections on the vector bundles A and A′, the tensor product connection ∇ on A ⊗ A′ is defined by

∇(σ ⊗ σ′) = ∇_{A}(σ) ⊗ σ′ + σ ⊗ ∇_{A′}(σ′)  (17)

for sections σ ∈ Γ(A) and σ′ ∈ Γ(A′). If we take n to be the rank of A and n′ the rank of A′ the curvature of the tensor product vector bundle is

curv(A ⊗ A′) = curv(A) ⊗ id_{n′} + id_{n} ⊗ curv(A′).  (18)

Hence its trace

\frac{1}{nn′} tr(curv(Ã)) = \frac{1}{n}tr(curv(A)) + \frac{1}{n′} tr(curv(A′))
= C_{2} − C_{1} + C_{3} − C_{2}
= C_{3} − C_{1}  (19)

satisfies axiom (1M1). Notice that equation (19) involves unlabeled projections from Z̃ to Y_{1}, Y_{2} and Y_{3}, where the one to Y_{2} is unique because Z̃ is the fibre product over Y_{2}. Furthermore, α̃ is an isomorphism

[figure: L_{1} ⊗ ζ̃_{2}^{∗}Ã; L_{1} ⊗ ζ_{2}^{∗}A ⊗ ζ_{2}^{′∗}A′; α⊗id; ζ_{1}^{∗}A ⊗ L_{2} ⊗ ζ_{2}^{′∗}A′; id⊗α′; ζ_{1}^{∗}A ⊗ ζ_{1}^{′∗}A′ ⊗ L_{3}; ζ̃_{1}^{∗}Ã ⊗ L_{3}.]  (20)

Axiom (1M2) follows from axioms (1M2) for A and A′.

1.8. Proposition. The composition of 1-morphisms is strictly associative: for three 1morphisms A : G_{1} → G_{2}, A′ : G_{2} → G_{3} and A′′ : G_{3} → G_{4} we have

(A′′ ∘ A′) ∘ A = A′′ ∘ (A′ ∘ A).

Proof. By definition, both 1-morphism (A′′ ∘ A′) ∘ A and A′′ ∘ (A′ ∘ A) consist of the smooth manifold X = Z ×_{Y_{2}} Z′×_{Y_{3}} Z′′ with the same surjective submersion X → Y_{1}×_{M} Y_{4}. On X, they have the same vector bundle A ⊗ A′ ⊗ A′′, and finally the same isomorphism

(id ⊗ id ⊗ α′′) ∘ (id ⊗ α′ ⊗ id) ∘ (α ⊗ id ⊗ id)  (21)

of vector bundles over X ×_{M} X. □

Now we have to define the functor ∘ on 2-morphisms. Let A_{1}, A′_{1} : G_{1} → G_{2} and A_{2}, A′_{2} : G_{2} → G_{3} be 1-morphisms between bundle gerbes. The functor ∘ on morphisms is called horizontal composition due to the diagrammatical notation

[figure: G_{1}; A_{1}; A′_{1}; β_{1}; G_{2}; A_{2}; A′_{2}; β_{2}; G_{3}] = [figure: G_{1}; A_{2}∘A_{1}; A′_{2}∘A′_{1}; β_{2}∘β_{1}; G_{3}] .  (22)

Recall that the compositions A_{2}∘A_{1} and A′_{2}∘A′_{1} consist of smooth manifolds Z̃ = Z_{1}×_{Y_{2}}Z_{2} and Z̃′ = Z′_{1} ×_{Y_{2}} Z′_{2} with surjective submersions to P := Y_{1} ×_{M} Y_{3}, of vector bundles Ã := A_{1} ⊗ A_{2} over Z̃ and Ã′ := A′_{1} ⊗ A′_{2} over Z̃′, and of isomorphisms α̃ and α̃′ over Z̃ ×_{M} Z̃ and Z̃′ ×_{M} Z̃′.

To define the composed 2-morphism β_{2} ∘ β_{1}, we first need a surjective submersion

ω : W ⟶ Z̃ ×_{P} Z̃′.  (23)

We choose representatives (W_{1}, ω_{1}, β_{W_{1}}) and (W_{2}, ω_{2}, β_{W_{2}}) of the 2-morphisms β_{1} and β_{2} and define

W := Z̃ ×_{P} (W_{1} ×_{Y_{2}} W_{2}) ×_{P} Z̃′  (24)

with the surjective submersion ω := z̃ × z̃′ projecting on the first and the last factor. Then, we need a morphism β_{W} : z̃^{∗}Ã → z̃^{′∗}Ã′ of vector bundles over W. Notice that we have maps

u : W_{1} ×_{Y_{2}} W_{2} ⟶ Z̃ and u′ : W_{1} ×_{Y_{2}} W_{2} ⟶ Z̃′  (25)

such that we obtain surjective submersions

z̃ × u : W ⟶ Z̃^{[2]} and u′ × z̃′ : W ⟶ Z̃^{′[2]}.  (26)

Recall from Lemma 1.5 that the 1-morphisms A_{2} ∘ A_{1} and A′_{2} ∘ A′_{1} define isomorphisms d_{A_{2}∘A_{1}} and d_{A′_{2}∘A′_{1}} of vector bundles over Z̃^{[2]} and Z̃^{′[2]}, whose pullbacks to W along the above maps are isomorphisms

d_{A_{2}∘A_{1}} : z̃^{∗}Ã ⟶ u^{∗}Ã and d_{A′_{2}∘A′_{1}} : u^{′∗}Ã′ ⟶ z̃^{′∗}Ã′  (27)

of vector bundles over W. Finally, the morphisms β_{W_{1}} and β_{W_{2}} give a morphism

β̃ := β_{W_{1}} ⊗ β_{W_{2}} : u^{∗}Ã ⟶ u^{′∗}Ã′  (28)

of vector bundles over W so that the composition

β_{W} := d_{A′_{2}∘A′_{1}} ∘ β̃ ∘ d_{A_{2}∘A_{1}}  (29)

is a well-defined morphism of vector bundles over W. Axiom (2M) for the triple (W, ω, β_{W}) follows from Lemma 1.5 b) for A_{2} ∘ A_{1} and A′_{2} ∘ A′_{1} and from the axioms (2M) for the representatives of β_{1} and β_{2}. Furthermore, the equivalence class of (W, ω, β_{W}) is independent of the choices of the representatives of β_{1} and β_{2}.

1.9. Lemma. The assignment ∘, defined above on objects and morphisms, is a functor

∘ : ℌ𝔬𝔪(G_{2}, G_{3}) × ℌ𝔬𝔪(G_{1}, G_{2}) ⟶ ℌ𝔬𝔪(G_{1}, G_{3}).

Proof. i) The assignment ∘ respects identities, i.e. for 1-morphisms A_{1} : G_{1} → G_{2} and A_{2} : G_{2} → G_{3},

id_{A_{2}} ∘ id_{A_{1}} = id_{A_{2}∘A_{1}}.  (30)

To show this we choose the defining representatives (W_{1}, id, d_{α_{1}}) of id_{A_{1}} and (W_{2}, id, d_{α_{2}}) of id_{A_{2}}, where W_{1} = Z_{1}×_{(Y_{1}×_{M} Y_{2})}Z_{1} and W_{2} = Z_{2}×_{(Y_{2}×_{M} Y_{3})}Z_{2}. Consider the diffeomorphism

f : W_{1} ×_{Y_{2}} W_{2} → Z̃ ×_{Y_{1}×_{M} Y_{2}×_{M} Y_{3}} Z̃ : (z_{1}, z′_{1}, z_{2}, z′_{2}) ↦ (z_{1}, z_{2}, z′_{1}, z′_{2}),  (31)

where Z̃ = Z_{1} ×_{Y_{2}} Z_{2}. From the definitions of the isomorphisms d_{A_{1}}, d_{A_{2}} and d_{A_{2}∘A_{2}} we conclude the equation

d_{A_{1}} ⊗ d_{A_{2}} = f^{∗}d_{A_{2}∘A_{1}}  (32)

of isomorphisms of vector bundles over W_{1} ×_{Y_{2}} W_{2}. The horizontal composition id_{A_{2}} ∘id_{A_{1}} is canonically represented by the triple (W, ω, β_{W}) where W is defined in (24) and β_{W} is defined in (29). Now, the diffeomorphism f extends to an embedding f : W → Z̃^{[4]} into the four-fold fibre product of Z̃ over P = Y_{1} ×_{M} Y_{3}, such that ω : W → Z̃^{[2]} factorizes over f,

ω = ζ̃_{14} ∘ f.  (33)

From (29) and (32) we obtain

β_{W} = d_{A_{2}∘A_{1}} ∘ (d_{A_{1}} ⊗ d_{A_{2}}) ∘ d_{A_{2}∘A_{1}}
= f^{∗}(ζ̃_{34}^{∗}d_{A_{2}∘A_{1}} ∘ ζ̃_{23}^{∗}d_{A_{2}∘A_{1}} ∘ ζ̃_{12}^{∗}d_{A_{2}∘A_{1}}).  (34)

The cocycle condition for d_{A_{2}∘A_{1}} from Lemma 1.5 a) and (33) give

β_{W} = f^{∗}ζ̃_{14}^{∗}d_{A_{2}∘A_{1}} = ω^{∗}d_{A_{2}∘A_{1}}.  (35)

We had to show that the triple (W, ω, β_{W}) which represents id_{A_{2}} ∘ id_{A_{2}} is equivalent to the triple (Z̃^{[2]}, id, d_{A_{2}∘A_{1}}) which defines the identity 2-morphism id_{A_{2}∘A_{1}}. For the choice X := W with surjective submersions id : X → W and ω : X → Z̃^{[2]}, equation (35) shows exactly this equivalence.

ii) The assignment ∘ respects the composition •, i.e. for 2-morphisms β_{i} : A_{i} ⇒ A′_{i} and β′_{i} : A′_{i} ⇒ A′′_{i} between 1-morphisms A_{i}, A′_{i} and A′′_{i} from G_{i} to G_{i+1}, everything for i = 1, 2, we have an equality

(β′_{2} • β_{2}) ∘ (β′_{1} • β_{1}) = (β′_{2} ∘ β′_{1}) • (β_{2} ∘ β_{1})  (36)

of 2-morphisms from A_{2} ∘ A_{1} to A′′_{2} ∘ A′′_{1}. This equality is also known as the compatibility of vertical and horizontal compositions. To prove it, let us introduce the notation Z̃ := Z_{1} ×_{Y_{2}} Z_{2}, and analogously Z̃′ and Z̃′′, furthermore we write P := Y_{1} ×_{M} Y_{3}. Notice that the 2-morphism on the left hand side of (36) is represented by a triple (V, ν, β_{V}) with

V = Z̃ ×_{P} (W̃_{1} ×_{Y_{2}} W̃_{2}) ×_{P} Z̃′′,  (37)

where the fibre products W̃_{i} := W_{i} ×_{Z′_{i}} W′_{i} arise from the vertical compositions β′_{i} • β_{i}. The surjective submersion ν : V → Z̃ ×_{P} Z̃′′ is the projection on the first and the last factor, and

β_{V} = d_{A′′_{2}∘A′′_{1}} ∘ ((β′_{1} ∘ β_{1}) ⊗ (β′_{2} ∘ β_{2})) ∘ d_{A_{2}∘A_{1}}  (38)

is a morphism of vector bundles over V. The 2-morphism on the right hand side of (36) is represented by the triple (V′, ν′, β_{V′}) with

V′ = (Z̃ ×_{P} (W_{1} ×_{Y_{2}} W_{2}) ×_{P} Z̃′) ×Z̃′ (Z̃′ ×_{P} (W′_{1} ×_{Y_{2}} W′_{2}) ×_{P} Z̃′′)
≅ Z̃ ×_{P} (W_{1} ×_{Y_{2}} W_{2}) ×_{P} Z̃′ ×_{P} (W′_{1} ×_{Y_{2}} W′_{2}) ×_{P} Z̃′′,  (39)

where ν′ is again the projection on the outer factors, and

β_{V′} = d_{A′′_{2}∘A′′_{1}} ∘ (β′_{1} ⊗ β′_{2}) ∘ d_{A′_{2}∘A′_{1}} ∘ (β_{1} ⊗ β_{2}) ∘ d_{A_{2}∘A_{1}},  (40)

where we have used the cocycle condition for d_{A′_{2}∘A′_{1}} from Lemma 1.5 b).

We have to show that the triples (V, ν, β_{V}) and (V′, ν′, β_{V′}) are equivalent. Consider the fibre product

X := V ×_{Z̃×_{P} Z̃′′} V′  (41)

with surjective submersions v : X → V and v′ : X → V′. To show the equivalence of the two triples, we have to prove the equality

v^{∗}β_{V} = v^{′∗}β_{V′}.  (42)

It is equivalent to the commutativity of the outer shape of the following diagram of isomorphisms of vector bundles over X:

[figure: A_{1} ⊗ A_{2}; d_{A_{2}∘A_{1}}; d_{A_{2}∘A_{1}}; v^{∗}(A_{1} ⊗ A_{2}); d_{A_{2}∘A_{1}}; β_{1}⊗β_{2}; v^{′∗}(A_{1} ⊗ A_{2}); β_{1}⊗β_{2}; v^{′∗}(A′_{1} ⊗ A′_{2}); d_{A′_{2}∘A′_{1}}; v^{∗}(A′_{1} ⊗ A′_{2}); β′_{1}⊗β′_{2}; d_{A′_{2}∘A′_{1}}; d_{A′_{2}∘A′_{1}}; v^{′∗}(A′_{1} ⊗ A′_{2}); β′_{1}⊗β′_{2}; v^{∗}(A′′_{1} ⊗ A′′_{2}); d_{A{′′ \atop 2}∘A{′′ \atop 1}}; d_{A{′′ \atop 2}∘A{′′ \atop 1}}; v^{′∗}(A′′_{1} ⊗ A′′_{2}); d_{A{′′ \atop 2}∘A{′′ \atop 1}}; A′′_{1} ⊗ A′′_{2}]  (43)

The commutativity of the outer shape of this diagram follows from the commutativity of its five subdiagrams: the triangular ones are commutative due to the cocycle condition from Lemma 1.5 a), and the commutativity of the foursquare ones follows from axiom (2M) of the 2-morphisms. □

To finish the definition of the 2-category 𝔅𝔊𝔯𝔟(M) we have to define the natural 2isomorphisms λ_{A} : A∘id_{G} ⇒ A and ρ_{A} : id_{G′} ∘A ⇒ A for a given 1-morphism A : G → G′, and we have to show that they satisfy axiom (2C2). We define the 1-morphism A ∘ id_{G} as follows: it has the canonical surjective submersion from Z̃ = Y^{[2]} ×_{Y} Z ≅ Y ×_{M} Z to P := Y ×_{M} Y′ and the vector bundle L ⊗ A over Z̃. Consider

W := Z̃ ×_{P} Z ≅ Z ×_{Y′} Z  (44)

and the identity ω := id_{W}. Under this identification, let us consider the restriction of the isomorphism α of the 1-morphism A from Z ×_{M} Z to W = Z ×_{Y′} Z. If s : W → W denotes the exchange of the two factors, we obtain an isomorphism

s^{∗}α|_{W} : L ⊗ ζ_{1}^{∗}A ⟶ ζ_{2}^{∗}A ⊗ Δ^{∗}L′  (45)

of vector bundles over W. By composition with the canonical trivialization of the line bundle Δ^{∗}L′ from Lemma 1.3 it gives an isomorphism

λ_{W} := (id ⊗ t_{μ′}) ∘ s^{∗}α|_{W} : L ⊗ ζ_{1}^{∗}A ⟶ ζ_{2}^{∗}A  (46)

of vector bundles over W. The axiom (2M) for the triple (W, ω, λ_{W}) follows from axiom (1M2) for the 1-morphism A and from the properties of t_{μ′} from Lemma 1.3. So, λ_{A} is defined to be the equivalence class of this triple. The definition of ρ_{A} goes analogously: we take W = Z ×_{Y} Z and obtain by restriction the isomorphism

α|_{W} : Δ^{∗}L ⊗ ζ_{2}^{∗}A ⟶ ζ_{1}^{∗}A ⊗ L′.  (47)

Then, the 2-isomorphism ρ_{A} is defined by the triple (W, ω, ρ_{W}) with the isomorphism

ρ_{W} := (t_{μ} ⊗ id) ∘ α|_{W}^{−1} : ζ_{1}^{∗}A ⊗ L′ ⟶ ζ_{2}^{∗}A  (48)

of vector bundles over W.

1.10. Lemma. The 2 -isomorphisms λ_{A} and ρ_{A} are natural in A, i.e. for any 2-morphism β : A ⇒ A′ the naturality squares

\xymatrix{id_{G′} ∘ A \ar@{=>}[dd]_{id_{_{id}_{G′}} ∘β} \ar@{=>}[r]^{ρ_{A}} & A \ar@{=>}[dd]^{β} &  & A ∘ id_{G} \ar@{=>}[dd]_{β∘id_{id_{G}}} \ar@{=>}[r]^{λ_{A}} & A \ar@{=>}[dd]^{β} \\  &  & and \\ id_{G′} ∘ A′ \ar@{=>}[r]_{ρ_{A′}} & A′ &  & A′ ∘ id_{G} \ar@{=>}[r]_{λ_{A′}} & A′}

are commutative.

Proof. To calculate for instance the horizontal composition id_{id_{G′}} ∘β in the diagram on the left hand side first note that id_{id_{G′}} is canonically represented by the triple (Y^{′[2]}, id, id_{L}). The isomorphism

d_{id_{G′}∘A} : ζ̃_{1}^{∗}(A ⊗ L′) → ζ̃_{2}^{∗}(A ⊗ L′),  (49)

which appears in the definition of the horizontal composition, is an isomorphism of vector bundles over Z̃×_{Y ×_{M} Y′} Z̃, where ζ̃: Z̃ := Z ×_{M} Y′ → Y ×_{M} Y′ is the surjective submersion of the composite id_{G′} ∘ A. Here it simplifies to

d_{id_{G′}∘A} = (t_{μ} ⊗ id ⊗ id) ∘ (α^{−1} ⊗ id) ∘ (1 ⊗ ζ̃_{1}^{∗}μ^{′−1}).  (50)

With these simplifications and with axiom (1M2) for A and A′, the naturality squares reduce to the compatibility axiom (2M) of β with the isomorphisms α and α′ of A and A′ respectively. □

It remains to show that the isomorphisms λ_{A} and ρ_{A} satisfy axiom (2C2) of a 2category.

1.11. Proposition. For 1-morphisms A : G_{1} → G_{2} and A′ : G_{2} → G_{3}, the 2-isomorphisms λ_{A} and ρ_{A} satisfy

id_{A′} ∘ ρ_{A} = λ_{A′} ∘ id_{A}.

Proof. The equation to prove is an equation of 2-morphisms from A′ ∘ id_{G_{2}} ∘ A to A′ ∘ A. The first 1-morphism consists of the surjective submersion Z̃ := Z ×_{M} Z′ → P_{13}, where we define P_{ij} := Y_{i} ×_{M} Y_{j}, further of the vector bundle A ⊗ L_{2} ⊗ A′ over Z̃. The second 1-morphism A′ ∘ A consists of the surjective submersion Z̃′ := Z ×_{Y_{2}} Z′ → P_{13} and the vector bundle A ⊗ A′ over Z̃′. Let us choose the defining representatives for the involved 2-morphisms: we choose (Z^{′[2]}, id, d_{A′}) for id_{A′}, with W := Z ×_{Y_{1}} Z we choose (W, id, ρ_{W}) for ρ_{A}, with W′ := Z′ ×_{Y_{3}} Z′ we choose (W′, id, λ_{W′}) for λ_{A′}, and we choose (Z^{[2]}, id, d_{A′}) for id_{A}.

Now, the horizontal composition id_{A′} ∘ ρ_{A} is defined by the triple (V, ν, β_{V}) with

V = Z̃ ×_{P_{13}} (W ×_{Y_{2}} Z^{′[2]}) ×_{P_{13}} Z̃′,  (51)

the projection ν : V → Z̃ ×_{P_{13}} Z̃′ on the first and the last factor, and the isomorphism

β_{V} = d_{A′∘A} ∘ (ρ_{W} ⊗ d_{A′}) ∘ d_{A′∘id∘A}  (52)

of vector bundles over V. The horizontal composition λ_{A′} ∘ id_{A} is defined by the triple (V′, ν′, β_{V′}) with

V′ = Z̃ ×_{P_{13}} (Z^{[2]} ×_{Y_{2}} W′) ×_{P_{13}} Z̃′,  (53)

again the projection ν′ on the first and the last factor, and the isomorphism

β_{V′} = d_{A′∘A} ∘ (d_{A} ⊗ λ_{W′}) ∘ d_{A′∘id∘A}  (54)

of vector bundles over V.

To prove the proposition, we show that the triples (V, ν, β_{V}) and (V′, ν′, β_{V′}) are equivalent. Consider the fibre product

X := V ×_{(Z̃×_{P_{13}} Z̃′)} V′  (55)

with surjective submersions v : X → V and v′ : X → V′. The equivalence of the two triples follows from the equation

v^{∗}β_{V} = v^{′∗}β′  (56)

of isomorphisms of vector bundles over X. It is equivalent to the commutativity of the outer shape of the following diagram of isomorphisms of vector bundles over X:

\xymatrix{ & A ⊗ L_{2} ⊗ A′ \ar[dl]_{d_{A′∘id∘A}} \ar[dr]^{d_{A′∘id∘A}} \\ v^{∗}(A ⊗ L_{2} ⊗ A′) \ar[d]_{ρ_{W} ⊗d_{A′}} \ar[rr]|{d_{A′∘id∘A}} &  & v^{′∗}(A ⊗ L_{2} ⊗ A′) \ar[d]^{d_{A}⊗λ_{W′}} \\ v^{∗}(A ⊗ A′) \ar[dr]_{d_{A′∘A}} \ar[rr]|{d_{A′∘A}} &  & v^{′∗}(A ⊗ A′) \ar[dl]^{d_{A′∘A}} \\  & A ⊗ A′}  (57)

The diagram is patched together from three subdiagrams, and the commutativity of the outer shape follows because the three subdiagrams are commutative: the triangle diagrams are commutative due to the cocycle condition from Lemma 1.5 b) for the 1-morphisms A′ ∘ id_{G_{2}} ∘ A and A′ ∘ A respectively. The commutativity of the rectangular diagram in the middle follows from Lemma 1.3 and from axioms (1M2) for A and A′. □

1.12. Invertible 1-Morphisms. In this subsection we address the question, which of the 1-morphisms of the 2-category 𝔅𝔊𝔯𝔟(M) are invertible. Let G_{1} and G_{2} be two bundle gerbes over M. In a (strictly associative) 2-category, a 1-morphism A : G_{1} → G_{2} is called invertible or 1-isomorphism, if there is a 1-morphism A^{−1} : G_{2} → G_{1} in the opposite direction, together with 2-isomorphisms i_{l} : A^{−1} ∘ A ⇒ id_{G_{1}} and i_{r} : id_{G_{2}} ⇒ A ∘ A^{−1} such that the diagram

[figure: A ∘ A^{−1} ∘ A; i_{r}∘id_{A}; id_{A}∘i_{l}; A ∘ id_{G_{1}}; λ_{A}; id_{G_{2}} ∘ A; ρ_{A}; A]  (58)

of 2-isomorphisms is commutative. The inverse 1-isomorphism A^{−1} is unique up to isomorphism.

Notice that if β : A ⇒ A′ is a 2-morphism between invertible 1-morphisms we can form a 2-morphism β^{#} : A^{′−1} ⇒ A^{−1} using the 2-isomorphisms i_{r} for A^{−1} and i_{l} for A^{′−1}. Then, diagram (58) induces the equation id_{A}^{#} = id_{A^{−1}}.

1.13. Proposition. A 1-morphism A : G_{1} → G_{2} in 𝔅𝔊𝔯𝔟(M) is invertible if and only if the vector bundle of A is of rank 1.

Proof. Suppose that A is invertible, and let n be the rank of its vector bundle. Let A^{−1} be an inverse 1-morphism with a vector bundle of rank m. By definition, the composed 1-morphisms A ∘ A^{−1} and A^{−1} ∘ A have vector bundles of rank nm, which has – to admit the existence of the 2-isomorphisms i_{l} and i_{r} – to coincide with the rank of the vector bundle of the identity 1-morphisms id_{G_{1}} and id_{G_{2}} respectively, which is 1. So n = m = 1. The other inclusion is shown below by an explicit construction of an inverse 1-morphism A^{−1} to a 1-morphism A with vector bundle of rank 1. □

Let a 1-morphism A consist of a surjective submersion ζ : Z → Y_{1} ×_{M} Y_{2}, of a line bundle A over Z and of an isomorphism α of line bundles over Z ×_{M} Z. We explicitly construct an inverse 1-morphism A^{−1}: it has the surjective submersion Z → Y_{1} ×_{M} Y_{2} → Y_{2} ×_{M} Y_{1}, where the first map is ζ and the second one exchanges the factors, the dual line bundle A^{∗} over Z and the isomorphism

[figure: L_{2} ⊗ ζ_{2}^{∗}A^{∗}; ζ_{1}^{∗}A^{∗} ⊗ ζ_{1}^{∗}A ⊗ L_{2} ⊗ ζ_{2}^{∗}A^{∗}; id⊗α^{−1}⊗id; ζ_{1}^{∗}A^{∗} ⊗ L_{1} ⊗ ζ_{2}^{∗}A ⊗ ζ_{2}^{∗}A^{∗}; ζ_{1}^{∗}A^{∗} ⊗ L_{1}.]  (59)

Axiom (1M1) for the 1-morphism A^{−1} is satisfied because A^{∗} has the negative curvature, and axiom (1M2) follows from the one for A.

To construct the 2-isomorphism i_{l} : A^{−1}∘A ⇒ id_{G_{1}} notice that the 1-morphism A^{−1}∘A consists of the line bundle ζ_{1}^{∗}A ⊗ ζ_{2}^{∗}A^{∗} over Z̃ = Z ×_{Y_{2}} Z. We identify Z̃ ≅ Z̃ ×_{P} Y_{1}^{[2]}, where P = Y_{1}^{[2]}, which allows us to choose a triple (Z̃, idZ̃, βZ̃) defining i_{l}. In this triple, the isomorphism βZ̃ is defined to be the composition

\xymatrix{ζ_{1}^{∗}A ⊗ ζ_{2}^{∗}A^{∗} \ar[r]^{id⊗t_{μ_{2}}^{−1}⊗id} & ζ_{1}^{∗}A ⊗ Δ^{∗}L_{2} ⊗ ζ_{2}^{∗}A^{∗} \ar[r]^{α^{−1}⊗id} & L_{1} ⊗ ζ_{2}^{∗}A ⊗ ζ_{2}^{∗}A^{∗} = L_{1}.} (60)

Axiom (2M) for the isomorphism βZ̃ follows from axiom (1M2) of A, so that the triple (Z̃, idZ̃, βZ̃) defines a 2-isomorphism i_{l} : A^{−1} ∘ A ⇒ id_{G_{1}}. The 2-isomorphism i_{r} : id_{G_{2}} ⇒ A ∘ A^{−1} is constructed analogously: here we take the isomorphism

\xymatrix{L_{2} = ζ_{1}^{∗}A^{∗} ⊗ ζ_{1}^{∗}A ⊗ L_{2} \ar[r]^{id⊗α^{−1}} & ζ_{1}^{∗}A^{∗} ⊗ Δ^{∗}L_{1} ⊗ ζ_{2}^{∗}A \ar[r]^{id⊗t_{μ_{1}}⊗id} & ζ_{1}^{∗}A^{∗} ⊗ ζ_{2}^{∗}A.} (61)

of line bundles over W. Notice that by using the pairing A^{∗} ⊗ A = 1 we have used that A is a line bundle as assumed. Finally, the commutativity of diagram (58) follows from axiom (1M2) of A.

Proposition 1.13 shows that we have many 1-morphisms in 𝔅𝔊𝔯𝔟(M) which are not invertible, in contrast to the 2-groupoid of bundle gerbes defined in [Ste00]. Notice that we have already benefited from the simple definition of the composition A^{−1} ∘ A, which makes it also easy to see that it is compatible with the construction of inverse 1-morphisms A^{−1}:

(A_{2} ∘ A_{1})^{−1} = A_{1}^{−1} ∘ A_{2}^{−1}.  (62)

1.14. Additional Structures. The 2-category of bundle gerbes has natural definitions of pullbacks, tensor products and dualities; all of them have been introduced for objects in [Mur96, MS00].

Pullbacks and tensor products of 1-morphisms and 2-morphisms can also be defined in a natural way, and we do not carry out the details here. Summarizing, the monoidal structure on 𝔅𝔊𝔯𝔟(M) is a strict 2-functor

⊗ : 𝔅𝔊𝔯𝔟(M) × 𝔅𝔊𝔯𝔟(M) ⟶ 𝔅𝔊𝔯𝔟(M),  (63)

for which the trivial bundle gerbe I_{0} is a strict tensor unit, i.e.

I_{0} ⊗ G = G = G ⊗ I_{0}.  (64)

The idea of the definition of ⊗ is to take fibre products of the involved surjective submersions, to pull back all the structure to this fibre product and then to use the monoidal structure of the category of vector bundles over that space. This was assumed to be strict, and so is ⊗. Pullbacks for the 2-category 𝔅𝔊𝔯𝔟(M) are implemented by strict monoidal 2-functors

f^{∗} : 𝔅𝔊𝔯𝔟(M) ⟶ 𝔅𝔊𝔯𝔟(X)  (65)

associated to every smooth map f : X → M in the way that g^{∗} ∘f^{∗} = (f ∘g)^{∗} for a second smooth map g : Y → X. The idea of its definition is, to pull back surjective submersions, for instance

\xymatrix{f^{−1}Y \ar[r]^{f̃} \ar[d] & Y \ar[d]^{π} \\ X \ar[r]_{f} & M}  (66)

and then pull back the structure over Y along the covering map f̃. The 2-functors ⊗ and f^{∗} are all compatible with the assignment of inverses A^{−1} to 1-morphisms A from subsection 1.12:

f^{∗}(A^{−1}) = (f^{∗}A)^{−1} and (A_{1} ⊗ A_{2})^{−1} = A_{1}^{−1} ⊗ A_{2}^{−1}.  (67)

Also the trivial bundle gerbes I_{ρ} behave naturally under pullbacks and tensor products:

f^{∗}I_{ρ} = I_{f^{∗}ρ} and I_{ρ_{1}} ⊗ I_{ρ_{2}} = I_{ρ_{1}+ρ_{2}}.  (68)

To define a duality we are a bit more precise, because this has yet not been done systematically in the literature. Even though we will strictly concentrate on what we need in section 3.8. For those purposes, it is enough to understand the duality as a strict 2-functor

()^{∗} : 𝔅𝔊𝔯𝔟(M)^{op} → 𝔅𝔊𝔯𝔟(M)  (69)

where the opposed 2-category 𝔅𝔊𝔯𝔟(M)^{op} has all 1-morphisms reversed, while the 2morphisms are as before. This 2-functor will satisfy the identity

()^{∗∗} = id_{𝔅𝔊𝔯𝔟(M)}.  (70)

We now give the complete definition of the functor ()^{∗} on objects, 1-morphisms and 2-morphisms. For a given bundle gerbe G, the dual bundle gerbe G^{∗} consists of the same surjective submersion π : Y → M, the 2-form −C ∈ Ω^{2}(Y), the line bundle L^{∗} over Y^{[2]} and the isomorphism

μ^{∗−1} : π_{12}^{∗}L^{∗} ⊗ π_{23}^{∗}L^{∗} → π_{13}^{∗}L^{∗}  (71)

of line bundles over Y^{[3]}. This structure clearly satisfies the axioms of a bundle gerbe. We obtain immediately

G^{∗∗} = G and (G ⊗ H)^{∗} = H^{∗} ⊗ G^{∗},  (72)

and for the trivial bundle gerbe I_{ρ} the equation

I_{ρ}^{∗} = I_{−ρ}.  (73)

For a 1-morphisms A : G_{1} → G_{2} consisting of a vector bundle A over Z with surjective submersion ζ : Z → P with P := Y_{1} ×_{M} Y_{2} and of an isomorphism α of vector bundles over Z ×_{M} Z, we define the dual 1-morphism

A^{∗} : G_{2}^{∗} ⟶ G_{1}^{∗}  (74)

as follows: its surjective submersion is the pullback of ζ along the exchange map s : P′ → P, with P′ := Y_{2} × Y_{1}; that is a surjective submersion ζ′ : Z′ → P′ and a covering map s_{Z} in the commutative diagram

\xymatrix{Z′ \ar[d]_{ζ′} \ar[r]^{s_{Z}} & Z \ar[d]^{ζ} \\ P′ \ar[r]_{s} & P.}  (75)

The vector bundle of A^{∗} is A′ := s_{Z}^{∗}A over Z′ and its isomorphism is

[figure: L_{2}^{∗} ⊗ ζ_{2}^{′∗}A′; L_{2}^{∗} ⊗ L_{1} ⊗ ζ_{2}^{′∗}s_{Z}^{∗}A ⊗ L_{1}^{∗}; id⊗s̃^{∗}α⊗id; L_{2}^{∗} ⊗ ζ_{1}^{′∗}s_{Z}^{∗}A ⊗ L_{2} ⊗ L_{1}^{∗}; ζ_{1}^{′∗}A′ ⊗ L_{1}^{∗}.]  (76)

Axiom (1M1) is satisfied since the dual bundle gerbes have 2-forms with opposite signs,

curv(A′) = s_{Z}^{∗}curv(A) = s_{Z}^{∗}(C_{2} − C_{1}) = C_{2} − C_{1} = (−C_{1}) − (−C_{2}).  (77)

Axiom (1M2) relates the isomorphism (76) to the isomorphisms μ_{1}^{∗−1} and μ_{2}^{∗−1} of the dual bundle gerbes. It can be deduced from axiom (1M2) of A using the following general fact, applied to μ_{1}^{∗} and μ_{2}^{∗}: the dual f^{∗} of an isomorphism f : L_{1} → L_{2} of line bundles coincides with the isomorphism

\xymatrix{L_{2}^{∗} = L_{2}^{∗} ⊗ L_{1} ⊗ L_{1}^{∗} \ar[r]^{id⊗f⊗id} & L_{2}^{∗} ⊗ L_{2} ⊗ L_{1}^{∗} = L_{1}^{∗},}  (78)

defined using the duality on line bundles.

Dual 1-morphisms defined like this have the properties

A^{∗∗} = A , (A′ ∘ A)^{∗} = A^{∗} ∘ A^{′∗} and (A_{1} ⊗ A_{2})^{∗} = A_{2}^{∗} ⊗ A_{1}^{∗}.  (79)

Finally, for a 2-morphism β : A_{1} ⇒ A_{2} we define the dual 2-morphism

β^{∗} : A_{1}^{∗} ⟹ A_{2}^{∗}  (80)

in the following way. If β is represented by a triple (W, ω, β_{W}) with an isomorphism β_{W} : A_{1} → A_{2} of vector bundles over W, we consider the pullback of ω : W → Z_{1} ×_{P} Z_{2} along s_{Z_{1}} × s_{Z_{2}} : Z′_{1} ×_{P′} Z′_{2} → Z_{1} ×_{P} Z_{2}, where Z_{1}, Z′_{2} and P′ are as in (75), and s_{Z_{1}} and s_{Z_{2}} are the respective covering maps. This gives a commutative diagram

\xymatrix{W′ \ar[d]_{ω′} \ar[r]^{s_{W}} & W \ar[d]^{ω} \\ Z′_{1} ×_{P′} Z′_{2} \ar[r]_{s_{Z_{1}}×s_{Z_{2}}} & Z_{1} ×_{P} Z_{2}.}  (81)

Now consider the triple (W′, ω′, s_{W}^{∗} β_{W}) with the isomorphism

s_{W}^{∗} β_{W} : s_{Z_{1}}^{∗}A_{1} ⟶ s_{Z_{2}}^{∗}A_{2}  (82)

of vector bundles over W′. It satisfies axiom (2M), and its equivalence class does not depend on the choice of the representative of β. So we define the dual 2-morphism β^{∗} to be this class. Dual 2-morphisms are compatible with vertical and horizontal compositions

(β_{2} ∘ β_{1})^{∗} = β_{1}^{∗} ∘ β_{2}^{∗} and (β • β′)^{∗} = β^{∗} • β^{′∗}  (83)

and satisfy furthermore

β^{∗∗} = β and (β_{1} ⊗ β_{2})^{∗} = β_{2}^{∗} ⊗ β_{1}^{∗}.  (84)

We can use adjoint 2-morphisms in the following situation: if A : G → H is an invertible 1-morphism with inverse A^{−1} and associated 2-isomorphisms i_{l} : A^{−1} ∘A ⇒ id_{G} and i_{r} : id_{H} ⇒ A ∘ A^{−1}, their duals i_{l}^{∗} and i_{r}^{∗} show that (A^{−1})^{∗} is an inverse to A^{∗}. Since inverses are unique up to isomorphism,

(A^{∗})^{−1} ≅ (A^{−1})^{∗}.  (85)

Summarizing, equations (72), (79), (83) and (84) show that ()^{∗} is a monoidal strict 2-functor, which is strictly involutive. Let us finally mention that it is also compatible with pullbacks:

f^{∗}(G^{∗}) = (f^{∗}G)^{∗} , f^{∗}A^{∗} = (f^{∗}A)^{∗} and f^{∗}β^{∗} = (f^{∗}β)^{∗}.  (86)

2. Descent Theory for Morphisms

In this section we compare 1-morphisms between bundle gerbes in the sense of Definition 1.4 with 1-morphisms whose surjective submersion ζ : Z → Y_{1}×_{M} Y_{2} is the identity, like in [Ste00]. For this purpose, we introduce the subcategory ℌ𝔬𝔪_{F P} (G_{1}, G_{2}) of the morphism category ℌ𝔬𝔪(G_{1}, G_{2}), where all smooth manifolds Z and W appearing in the definitions of 1- and 2-morphisms are equal to the fibre product P := Y_{1} ×_{M} Y_{2}. Explicitly, an object in ℌ𝔬𝔪_{F P} (G_{1}, G_{2}) is a 1-morphism A : G_{1} → G_{2} whose surjective submersion is the identity id_{P} and a morphism in ℌ𝔬𝔪_{F P} (G_{1}, G_{2}) is a 2-morphism β : A_{1} ⇒ A_{2} which can be represented by the triple (P, ω, β) where ω : P → P ×_{P} P ≅ P is the identity.

2.1. Theorem. The inclusion functor

D : ℌ𝔬𝔪_{F P} (G_{1}, G_{2}) ⟶ ℌ𝔬𝔪(G_{1}, G_{2})

is an equivalence of categories.

In the proof we will make use of the fact that vector bundles form a stack, i.e. fibred category satisfying a gluing condition. To make this gluing condition concrete, we define for a surjective submersion ζ : Z → P a category 𝔇𝔢𝔰(ζ) as follows. Its objects are pairs (A, d), where A is a vector bundle over Z and

d : ζ_{1}^{∗}A ⟶ ζ_{2}^{∗}A  (87)

is an isomorphism of vector bundles over Z^{[2]} such that

ζ_{13}^{∗}d = ζ_{23}^{∗}d ∘ ζ_{12}^{∗}d.  (88)

A morphism α : (A, d) → (A′, d′) in 𝔇𝔢𝔰(ζ) is an isomorphism α : A → A′ of vector bundles over Z such that the diagram

\xymatrix{ζ_{1}^{∗}A \ar[d]_{d} \ar[r]^{ζ_{1}^{∗}α} & ζ_{1}^{∗}A′ \ar[d]^{d′} \\ ζ_{2}^{∗}A \ar[r]_{ζ_{2}^{∗}α} & ζ_{2}^{∗}A′}  (89)

of isomorphisms of vector bundles over Z^{[2]} is commutative. The composition of morphisms is just the composition of isomorphisms of vector bundles. Now, the gluing condition for the stack of vector bundles is that the pullback along ζ is an equivalence

ζ^{∗} : 𝔅𝔲𝔫(P) ⟶ 𝔇𝔢𝔰(ζ)  (90)

between the category 𝔅𝔲𝔫(P) of vector bundles over P and the category 𝔇𝔢𝔰(ζ).

Proof of Theorem 2.1. We show that the faithful functor D is an equivalence of categories by proving (a) that it is essentially surjective and (b) that the subcategory ℌ𝔬𝔪_{F P} (G_{1}, G_{2}) is full.

For (a) we have to show that for every 1-morphism A : G_{1} → G_{2} with arbitrary surjective submersion ζ : Z → P there is an isomorphic 1-morphism S_{A} : G_{1} → G_{2} with surjective submersion id_{P}. Notice that the isomorphism d_{A} : ζ_{1}^{∗}A → ζ_{2}^{∗}A of vector bundles over Z^{[2]} from Lemma 1.5 satisfies the cocycle condition (88), so that (A, d_{A}) is an object in 𝔇𝔢𝔰(ζ). Now consider the surjective submersion ζ^{2} : Z ×_{M} Z → P^{[2]}. By Lemma 1.5 b) and under the identification of Z^{[2]} ×_{M} Z^{[2]} with (Z ×_{M} Z) ×_{P^{[2]}} (Z ×_{M} Z) the diagram

\xymatrix{L_{1} ⊗ ζ_{2}^{∗}A \ar[d]_{1⊗ζ_{24}^{∗}d_{A}} \ar[r]^{ζ_{12}^{∗}α} & ζ_{1}^{∗}A ⊗ L_{2} \ar[d]^{ζ_{13}^{∗}d_{A}⊗1} \\ L_{1} ⊗ ζ_{4}^{∗}A \ar[r]_{ζ_{34}^{∗}α} & ζ_{3}^{∗}A ⊗ L_{2}}  (91)

of isomorphisms of vector bundles over (Z ×_{M} Z) ×_{P^{[2]}} (Z ×_{M} Z) is commutative, and shows that α is a morphism in 𝔇𝔢𝔰(ζ^{2}). Now we use that ζ^{∗} is an equivalence of categories: we choose a vector bundle S over P together with an isomorphism β : ζ^{∗}S → A of vector bundles over Z, and an isomorphism

σ : L_{1} ⊗ ζ_{2}^{∗}S ⟶ ζ_{1}^{∗}S ⊗ L_{2}  (92)

of vector bundles over P ×_{M} P such that the diagram

\xymatrix{L_{1} ⊗ ζ_{2}^{∗}ζ^{∗}S \ar[d]_{id⊗ζ_{2}^{∗}β} \ar[r]^{ζ^{∗}σ} & ζ_{1}^{∗}ζ^{∗}S ⊗ L_{2} \ar[d]^{ζ_{1}^{∗}β⊗id} \\ L_{1} ⊗ ζ_{2}^{∗}A \ar[r]_{α} & ζ_{1}^{∗}A ⊗ L_{2}}  (93)

of isomorphisms of vector bundles over Z×_{M} Z is commutative. Since ζ is an equivalence of categories, the axioms of A imply the ones of the 1-morphism S_{A} defined by the surjective submersion id_{P}, the vector bundle S over P and the isomorphism σ over P^{[2]}. Finally, the triple (Z ×_{P} P, id_{Z}, β) with Z ≅ Z ×_{P} P defines a 2-morphism S_{A} ⇒ A, whose axiom (2M) is (93).

(b) We have to show that any morphism β : A ⇒ A′ in ℌ𝔬𝔪(G_{1}, G_{2}) between objects A and A′ in ℌ𝔬𝔪_{F P} (G_{1}, G_{2}) is already a morphism in ℌ𝔬𝔪_{F P} (G_{1}, G_{2}). Let (W, ω, β_{W}) be any representative of β with a surjective submersion ω : W → P and an isomorphism β_{W} : ω^{∗}A → ω^{∗}A′ of vector bundles over W. The restriction of axiom (2M) for the triple (W, ω, β_{W}) from W ×_{M} W to W ×_{P} W shows ω_{1}^{∗}β_{W} = ω_{2}^{∗}β_{W}. This shows that β_{W} is a morphism in the descent category 𝔇𝔢𝔰(ω). Let β_{P} : A → A′ be an isomorphism of vector bundles over P such that

ω^{∗}β_{P} = β_{W}  (94)

Because ω is an equivalence of categories, the triple (P, id_{P} , β_{P}) defines a 2-morphism from A to A′ being a morphism in ℌ𝔬𝔪_{F P} (G_{1}, G_{2}). Equation (94) shows that the triples (P, id_{P} , β_{P}) and (W, ω, β_{W}) are equivalent. □

In the remainder of this section we present two corollaries of Theorem 2.1. First, and most importantly, we make contact to the notion of a stable isomorphism between bundle gerbes. By definition [MS00], a stable isomorphism is a 1-morphism, whose surjective submersion is the identity id_{P} on the fibre product of the surjective submersions of the two bundle gerbes, and whose vector bundle over P is a line bundle. From Proposition 1.13 and Theorem 2.1 we obtain

2.2. Corollary. There exists a stable isomorphism A : G_{1} → G_{2} if and only if the bundle gerbes are isomorphic objects in 𝔅𝔊𝔯𝔟(M).

It is shown in [MS00] that the set of stable isomorphism classes of bundle gerbes over M is a group (in virtue of the monoidal structure) which is isomorphic to the Deligne cohomology group H^{2}(M, D(2)). This is a very important fact which connects the theory of bundle gerbes to other theories of gerbes, for instance, to Dixmier-Douady sheaves of groupoids [Bry93]. Corollary 2.2 states that although our definition of morphisms differs from the one of [MS00], the bijection between isomorphism classes of bundle gerbes and the Deligne cohomology group persists.

Second, Theorem 2.1 admits to use existing classification results for 1-isomorphisms. Consider the full subgroupoid 𝔄𝔲𝔱(G) of ℌ𝔬𝔪(G, G) associated to a bundle gerbe G, which consists of all 1-isomorphisms A : G → G, and all (necessarily invertible) 2-morphisms between those. From Theorem 2.1 and Lemma 2 of [SSW05] we obtain

2.3. Corollary. The skeleton of the groupoid 𝔄𝔲𝔱(G), i.e. the set of isomorphism classes of 1-isomorphisms A : G → G is a torsor over the group Pic_{0}(M) of isomorphism classes of flat line bundles over M.

In 2-dimensional conformal field theory, where a bundle gerbe G is considered to be a part of the background field, the groupoid 𝔄𝔲𝔱(G) may be called the groupoid of gauge transformations of G. The above corollary classifies such gauge transformation up to equivalence.

3. Some important Examples of Morphisms

To discuss holonomies of bundle gerbes, it is essential to establish an equivalence between the morphism categories between trivial bundle gerbes over M and vector bundles of certain curvature over M. Given two 2-forms ρ_{1} and ρ_{2} on M, consider the category ℌ𝔬𝔪_{F P} (I_{ρ_{1}}, I_{ρ_{2}}). An object A : I_{ρ_{1}} → I_{ρ_{2}} consists of the smooth manifold Z = M with the surjective submersion ζ = id_{M}, a vector bundle A over M and an isomorphism α : A → A. Axiom (1M2) states

\frac{1}{n}tr(curv(A)) = ρ_{2} − ρ_{1}  (95)

with n the rank of A, and axiom (1M2) reduces to α^{2} = α, which in turn means α = id_{A}. Together with Theorem 2.1, this defines a canonical equivalence of categories

Bun : ℌ𝔬𝔪(I_{ρ_{1}}, I_{ρ_{2}}) ⟶ 𝔅𝔲𝔫_{ρ_{2}−ρ_{1}}(M),  (96)

where 𝔅𝔲𝔫_{ρ}(M) is the category of vector bundles A over M whose curvature satisfies (95). Its following properties can directly be deduced from the definitions.

3.1. Proposition. The functor Bun respects the structure of the 2-category of bundle gerbes, namely:

a) the composition of 1-morphisms,

Bun(A_{2} ∘ A_{1}) = Bun(A_{1}) ⊗ Bun(A_{2}) and Bun(id_{I_{ρ}}) = 1.

b) the assignment of inverses to invertible 1-morphisms,

Bun(A^{−1}) = Bun(A)^{∗}.

c) the monoidal structure,

Bun(A_{1} ⊗ A_{2}) = Bun(A_{1}) ⊗ Bun(A_{2}).

d) pullbacks,

Bun(f^{∗}A) = f^{∗}Bun(A) and Bun(f^{∗}β) = f^{∗}Bun(β).

e) the duality

Bun(A^{∗}) = Bun(A) and Bun(β^{∗}) = Bun(β).

In the following subsections we see how the 2-categorial structure of bundle gerbes and the functor Bun can be used to give natural definitions of surface holonomy in several situations.

3.2. Trivializations. We give the following natural definition of a trivialization.

3.3. Definition. A trivialization of a bundle gerbe G is a 1-isomorphism

T : G ⟶ I_{ρ}.

Let us write out the details of such a 1-isomorphism. By Theorem 2.1 we may assume that the surjective submersion of T is the identity id_{P} on P := Y ×_{M} M ≅ Y with projection π to M. Then, T consists further of a line bundle T over Y, and of an isomorphism τ : L ⊗ π_{2}^{∗}T → π_{1}^{∗}T of line bundles over Y^{[2]}. Axiom (1M2) gives π_{13}^{∗}τ ∘ (μ ⊗ id) = π_{12}^{∗}τ ∘ π_{23}^{∗}τ as an equation of isomorphisms of line bundles over Y^{[3]}. This is exactly the definition of a trivialization one finds in the literature [CJM02]. Additionally, axiom (1M2) gives curv(T) = π^{∗}ρ − C. If one does not specify ρ as a part of the definition of a trivialization, it is uniquely determined by this equation.

Trivializations are essential for the definition of holonomy around closed oriented surfaces.

3.4. Definition. Let ϕ : Σ → M be a smooth map from a closed oriented surface Σ to a smooth manifold M, and let G a bundle gerbe over M. Let

T : ϕ^{∗}G ⟶ I_{ρ}

be any trivialization. The holonomy of G around ϕ is defined as

hol_{G}(ϕ) := exp (i ∫_{Σ} ρ) ∈ U(1).

In this situation, the functor Bun is a powerful tool to prove that this definition does not depend on the choice of the trivialization: if T′ : ϕ^{∗}G → I_{ρ′} is another trivialization, the composition T ∘ T^{′−1} : I_{ρ′} → I_{ρ} corresponds to a line bundle Bun(T ∘ T^{′−1}) over M with curvature ρ − ρ′. In particular, the difference between any two 2-forms ρ is a closed 2-form with integer periods and vanishes under the exponentiation in the definition of hol_{G}(ϕ).

3.5. Bundle Gerbe Modules. For oriented surfaces with boundary one has to choose additional structure on the boundary to obtain a well-defined holonomy. This additional structure is provided by a vector bundle twisted by the bundle gerbe G [Gaw05], also known as a G-module. In our formulation, its definition takes the following form:

3.6. Definition. Let G be a bundle gerbe over M. A left G-module is a 1-morphism E : G → I_{ω}, and a right G-module is a 1-morphism F : I_{ω} → G.

Let us compare this definition with the original definition of (left) bundle gerbe modules in [BCM^{+}02]. Assume – again by Theorem 2.1 – that a left G-module E : G → I_{ω} has the surjective submersion id_{P} with P ≅ Y. Then, it consists of a vector bundle E over Y and of an isomorphism ϵ : L ⊗ π_{2}^{∗}E → π_{1}^{∗}E of vector bundles over Y^{[2]} which satisfies

π_{13}^{∗}ϵ ∘ (μ ⊗ id) = π_{23}^{∗}ϵ ∘ π_{12}^{∗}ϵ  (97)

by axiom (1M2). The curvature of E is restricted by axiom (1M2) to

\frac{1}{n}tr(curv(E)) = π^{∗}ω − C  (98)

with n the rank of E.

The definition of bundle gerbe modules as 1-morphisms makes clear that left and right G-modules form categories 𝔏𝔐𝔬𝔡(G) and ℜ𝔐𝔬𝔡(G). This is useful for example to see that a 1-isomorphism A : G → G′ defines equivalences of categories

𝔏𝔐𝔬𝔡(G) ≅ 𝔏𝔐𝔬𝔡(G′) and ℜ𝔐𝔬𝔡(G) ≅ ℜ𝔐𝔬𝔡(G′)  (99)

and that there are equivalences between left modules of G and right modules of G^{∗} (and vice versa), by taking duals of the respective 1-morphisms. Moreover, for a trivial bundle gerbe I_{ρ} the categories 𝔏𝔐𝔬𝔡(I_{ρ}) and ℜ𝔐𝔬𝔡(I_{ρ}) become canonically equivalent to the category 𝔅𝔲𝔫(M) of vector bundles over M via the functor Bun. We can combine this result with the equivalences (99) applied to a trivialization T : G → I_{ρ} of a bundle gerbe G over M. In detail, a left G-module E : G → I_{ω} first becomes a left I_{ρ}-module

E ∘ T^{−1} : I_{ρ} ⟶ I_{ω}  (100)

which in turn defines the vector bundle E := Bun(E ∘ T^{−1}) over M. The same applies to right G-modules F : I_{ω} → G which defines a vector bundle Ē := Bun(T ∘ F) over M.

A D-brane for the bundle gerbe G is a submanifold Q of M together with a left G|_{Q}-module. Here G|_{Q} means the pullback of G along the inclusion Q ↪ M.

3.7. Definition. Let G be a bundle gerbe over M with D-brane (Q, E) and let ϕ : Σ → M be a smooth map from a compact oriented surface Σ with boundary to M, such that ϕ(∂Σ) ⊂ Q. Let

T : ϕ^{∗}G ⟶ I_{ρ}

be any trivialization of the pullback bundle gerbe ϕ^{∗}G and let

E := Bun(ϕ^{∗}E ∘ T^{−1})  (101)

be the associated vector bundle over ∂Σ. The holonomy of G around ϕ is defined as

hol_{G,E}(ϕ) := exp (i ∫_{Σ} ρ) · tr (hol_{E}(∂Σ)) ∈ ℂ.

The definition does not depend on the choice of the trivialization: for another trivialization T′ : ϕ^{∗}G → I_{ρ′} and the respective vector bundle E′ := Bun(E ∘ T^{′−1}) we find by Proposition 3.1 a)

E′ = Bun(E ∘ T^{′−1}) ≅ Bun(E ∘ T^{−1} ∘ T ∘ T^{′−1}) = E ⊗ Bun(T ∘ T^{′−1}).  (102)

Because isomorphic vector bundles have the same holonomies, and the line bundle Bun(T ∘ T^{′−1}) has curvature ρ − ρ′ we obtain

tr (hol_{E′}(∂Σ)) = tr (hol_{E}(∂Σ)) · exp (i ∫_{Σ} ρ − ρ′).  (103)

This shows the independence of the choice of the trivialization.

3.8. Jandl Structures. In this last section, we use the duality on the 2-category 𝔅𝔊𝔯𝔟(M) introduced in section 1.14 to define the holonomy of a bundle gerbe around unoriented, and even unorientable surfaces (without boundary). For this purpose, we explain the concept of a Jandl structure on a bundle gerbe G, which has been introduced in [SSW05], in terms of 1- and 2-isomorphisms of the 2-category 𝔅𝔊𝔯𝔟(M).

3.9. Definition. A Jandl structure J on a bundle gerbe G over M is a collection (k, A, φ) of an involution k : M → M, i.e. a diffeomorphism with k ∘ k = id_{M}, a 1-isomorphism

A : k^{∗}G ⟶ G^{∗}

and a 2-isomorphism

φ : k^{∗}A ⟹ A^{∗}

which satisfies the condition

k^{∗}φ = φ^{∗−1}.

Notice that the existence of the 2-isomorphism φ is only possible because G^{∗∗} = G from (72), and that the equation k^{∗}φ = φ^{∗−1} only makes sense because A^{∗∗} = A from (79). Let us now discuss the relation between Definition 3.9 and the original definition of a Jandl structure from [SSW05]. For this purpose we elaborate the details. We denote the pullback of the surjective submersion π : Y → M along k by π_{k} : Y_{k} → M; for simplicity we take Y_{k} := Y and π_{k} := k ∘ π. Now, we assume by Theorem 2.1 that the 1-isomorphism A consists of a line bundle A over Y_{k} ×_{M} Y. As smooth manifolds, we can identify Y_{k} ×_{M} Y with P := Y^{[2]}; to have an identification as smooth manifolds with surjective submersions to M, we define the projection p : P → M by p := π ∘ π_{2}. Under this identification, the exchange map s : Y ×_{M} Y_{k} → Y_{k} ×_{M} Y becomes an involution of P which lifts k,

\xymatrix{P \ar[d]_{p} \ar[r]^{s} & P \ar[d]^{p} \\ M \ar[r]_{k} & M.}  (104)

The dual 1-isomorphism A^{∗} has by definition the line bundle s^{∗}A over P. Now, similarly as for the pullback of π : Y → M we denote the pullback of p : P → M by p_{k} : P_{k} → M and choose P_{k} := P and p_{k} := k ∘ p. This way, the pullback 1-isomorphism k^{∗}A has the line bundle A over P. Again by Theorem 2.1, we assume that the 2-isomorphism φ can be represented by a triple (P, id_{P} , φ_{P}) with an isomorphism φ_{P} : A → s^{∗}A of line bundles over P satisfying the compatibility axiom (2M) with the isomorphism α of A:

\xymatrix{L ⊗ ζ_{2}^{∗}A \ar[d]_{id⊗ζ_{2}^{∗}φ_{P}} \ar[r]^{α} & ζ_{1}^{∗}A ⊗ L \ar[d]^{ζ_{1}^{∗}φ_{P} ⊗id} \\ L ⊗ ζ_{2}^{∗}s^{∗}A \ar[r]_{s^{∗}α} & ζ_{1}^{∗}s^{∗}A ⊗ L}  (105)

The dual 2-isomorphism φ^{∗} is given by (P, id_{P} , s^{∗}φ_{P}), and the equation φ = k^{∗}φ^{∗−1} becomes φ_{P} = s^{∗}φ_{P}^{−1}. So, φ_{P} is an s-equivariant structure on A. This is exactly the original definition [SSW05]: a stable isomorphism A : k^{∗}G → G^{∗}, whose line bundle A is equipped with an s-equivariant structure which is compatible with the isomorphism α of A in the sense of the commutativity of diagram (105).

Defining a Jandl structure in terms of 1- and 2-morphisms has – just like for gerbe modules – several advantages. For example, it is easy to see that Jandl structures are compatible with pullbacks along equivariant maps, tensor products and duals of bundle gerbes. Furthermore, we have an obvious definition of morphisms between Jandl structures, which induces exactly the notion of equivalent Jandl structures we introduced in [SSW05].

3.10. Definition. A morphism β : J → J′ between Jandl structures J = (k, A, φ) and J′ = (k, A′, φ′) on the same bundle gerbe G over M with the same involution k is a 2-morphism

β : A ⟹ A′

which commutes with φ and φ′ in the sense that the diagram

\xymatrix{A \ar@{=>}[r]^{φ} \ar@{=>}[d]_{β} & k^{∗}A^{∗} \ar@{=>}[d]^{k^{∗}β^{∗}} \\ A′ \ar@{=>}[r]_{φ′} & k^{∗}A^{′∗}}

of 2-morphisms is commutative.

Since A is invertible, every morphism of Jandl structures is invertible. We may thus speak of a groupoid 𝔍𝔡𝔩(G, k) of Jandl structures on the bundle gerbe G with involution k. The skeleton of this groupoid has been classified [SSW05]: it forms a torsor over the group of flat k-equivariant line bundles over M. The following proposition relates these groupoids of Jandl structures on isomorphic bundle gerbes on the same space with the same involution. This relation is a new result, coming and benefiting very much from the 2-categorial structure of bundle gerbes we have developed.

3.11. Proposition. Any 1-isomorphism B : G → G′ induces an equivalence of groupoids

J_{B} : 𝔍𝔡𝔩(G′, k) ⟶ 𝔍𝔡𝔩(G, k)

with the following properties:

a) any 2-morphism β : B ⇒ B′ induces a natural equivalence J_{B} ≅ J_{B′}.

b) there is a natural equivalence J_{id_{G}} ≅ id_{𝔍𝔡𝔩(G,k)}.

c) it respects the composition of 1-morphisms in the sense that

J_{B′∘B} = J_{B} ∘ J_{B′}.

Proof. The functor J_{B} sends a Jandl structure (k, A, φ) on G′ to the triple (k, A′, φ′) with the same involution k, the 1-isomorphism

A′ := B^{∗} ∘ A ∘ k^{∗}B : k^{∗}G ⟶ G^{∗}  (106)

and the 2-isomorphism

[figure: k^{∗}A′; k^{∗}B^{∗} ∘ k^{∗}A ∘ B; id_{k^{∗}B^{∗}}∘φ∘id_{B}; k^{∗}B^{∗} ∘ A^{∗} ∘ B; k^{∗}A^{′∗}]  (107)

where we use equation (79). The following calculation shows that (k, A′, φ′) is a Jandl structure:

k^{∗}φ^{′∗} =^{def} k^{∗}(id_{k^{∗}B^{∗}} ∘ φ ∘ id_{B})^{∗}
=^{(79)} id_{k^{∗}B^{∗}} ∘ k^{∗}φ^{∗} ∘ id_{B}
= id_{B} ∘ φ^{−1} ∘ id_{B^{∗}}
=^{def} φ^{′−1}.  (108)

A morphism β of Jandl structures on G′ is sent to the morphism

J_{B}(β) := id_{B^{∗}} ∘ β ∘ id_{k^{∗}B}  (109)

of the respective Jandl structures on G′. The two axioms of the composition functor ∘ from Lemma 1.9 show that the composition of morphisms of Jandl structures is respected, so that J_{B} is a functor. It is an equivalence because J_{B^{−1}} is an inverse functor, where the natural equivalences J_{B^{−1}} ∘ J_{B} ≅ id and J_{B} ∘ J_{B^{−1}} ≅ id use the 2-isomorphisms i_{r} and i_{l} from section 1.12 associated to the inverse 1-morphism B^{−1}.

To prove a), let β : B ⇒ B′ be a 2-morphism. We define the natural equivalence J_{B} ≅ J_{B′}, which is a collection of morphisms β_{J} : J_{B}(J) → J_{B′}(J) of Jandl structures on G for any Jandl structure J on G′ by

β_{J} := β^{∗} ∘ id_{A} ∘ k^{∗}β.  (110)

This defines indeed a morphism of Jandl structures and makes the naturality square

\xymatrix{J_{B}(J) \ar[r]^{β_{J}} \ar[d]_{J_{B}(β)} & J_{B′}(J) \ar[d]^{J_{B′}(β)} \\ J_{B}(J′) \ar[r]_{β_{J′}} & J_{B′}(J′)}  (111)

commutative. The natural equivalence for b) uses the 2-isomorphisms λ_{A} and ρ_{A} of the 2-category 𝔅𝔊𝔯𝔟(M) and the fact that id_{G}^{∗} = id_{G^{∗}}. Finally, c) follows from the definition of J_{B} and the fact that the duality functor ()^{∗} respects the composition of 1-morphisms, see (79). □

It is worthwhile to consider a Jandl structure J = (k, A, φ) over a trivial bundle gerbe I_{ρ}. By definition, this is a 1-isomorphism

A : I_{k^{∗}ρ} ⟶ I_{−ρ}  (112)

and a 2-isomorphism φ : k^{∗}A ⇒ A^{∗} satisfying φ = k^{∗}φ^{∗−1}. Now we apply the functor Bun and obtain a line bundle R̂ := Bun(A) over M of curvature −(ρ + k^{∗}ρ) and an isomorphism φ̂ := Bun(φ) : k^{∗}R̂ → R̂ of line bundles over M which satisfies φ̂ = k^{∗}φ̂^{−1}, summarizing: a k-equivariant line bundle. So, the functor Bun induces an equivalence of groupoids

Bun_{ρ}^{k} : 𝔍𝔡𝔩(I_{ρ}, k) ⟶ 𝔏𝔅𝔲𝔫_{−(ρ+k^{∗}ρ)}^{k}(M)  (113)

between the groupoid of Jandl structures on I_{ρ} with involution k and the groupoid of k-equivariant line bundles over M with curvature −(ρ + k^{∗}ρ). In particular, if G is a bundle gerbe over M and T : G → I_{ρ} a trivialization, we obtain a functor

\xymatrix{𝔍𝔡𝔩(G, k) \ar[r]^{J_{T ^{−1}}} & 𝔍𝔡𝔩(I_{ρ}, k) \ar[r]^{^{Bun^{k}^{ρ}}} & 𝔏𝔅𝔲𝔫_{−(ρ+k^{∗}ρ)}^{k}(M)}  (114)

converting a Jandl structure on the bundle gerbe G into a k-equivariant line bundle over M. It becomes obvious that the existence of a Jandl structure with involution k on the trivial bundle gerbe I_{ρ} constraints the 2-form ρ: as the curvature of a line bundle, the 2-form −(ρ + k^{∗}ρ) has to be closed and to have integer periods.

Let us now explain how Jandl structures enter in the definition of holonomy around unoriented surfaces, and how we can take further advantage of the 2-categorial formalism. We have learned before that to incorporate surfaces with boundary we had to do two steps: we first specified additional structure, a D-brane of the bundle gerbe G, and then specified which maps ϕ : Σ → M are compatible with this additional structure: those who send the boundary of Σ into the support of the D-brane. To discuss unoriented surfaces (without boundary), we also do these two steps: the additional structure we choose here is a Jandl structure J = (k, A, φ) on the bundle gerbe G. To describe the space of maps we want to consider, we have to introduce the following geometric structures [SSW05]:

• For any (unoriented) closed surface Σ there is an oriented two-fold covering pr : Σ̂ → Σ. It is unique up to orientation-preserving diffeomorphisms and it is connected if and only if Σ is not orientable. It has a canonical, orientation-reversing involution σ, which permutes the sheets and preserves the fibres. We call this two-fold covering the orientation covering of Σ.

• A fundamental domain of Σ in Σ̂ is a submanifold F of Σ̂ with ( possibly only piecewise smooth) boundary, such that

F ∪ σ(F) = Σ̂ and F ∩ σ(F) = ∂F.  (115)

A key observation is that the involution σ restricts to an orientation-preserving involution on ∂F ⊂ Σ̂. Accordingly, the quotient \overline{∂F} is an oriented closed 1-dimensional submanifold of Σ.

Now, given a closed surface Σ, we consider maps ϕ̂: Σ̂ → M from the orientation covering Σ̂ to M, which are equivariant with respect to the two involutions on Σ̂ and M, i.e. the diagram

\xymatrix{Σ̂ \ar[r]^{ϕ̂} \ar[d]_{σ} & M \ar[d]^{k} \\ Σ̂ \ar[r]_{ϕ̂} & M}

has to be commutative.

3.12. Definition. Let J be a Jandl structure on a bundle gerbe G over M, and let ϕ̂: Σ̂ → M be an equivariant smooth map. For a trivialization

T : ϕ̂^{∗}G ⟶ I_{ρ}

let R̂ be the σ-equivariant line bundle over Σ̂ determined by the functor

Bun_{σ}^{ρ} ∘ J_{T^{−1}} : 𝔍𝔡𝔩(ϕ̂^{∗}G, σ) ⟶ 𝔏𝔅𝔲𝔫_{−(ρ+σ^{∗}ρ)}^{σ}(Σ̂)  (116)

from (114). In turn, R̂ defines a line bundle R over Σ. Choose any fundamental domain F of Σ. Then, the holonomy of G with Jandl structure J around ϕ̂ is defined as

hol_{G,J} (ϕ̂) := exp (i ∫_{F} ρ) · hol_{R}(\overline{∂F}).

Definition 3.12 is a generalization of Definition 3.4 of holonomy around an oriented surface: for an orientable surface Σ and any choice of an orientation, they coincide [SSW05]. To show that Definition 3.12 does not depend on the choice of the trivialization T , we combine all the collected tools. Let T′ : ϕ̂^{∗}G → I_{ρ′} be any other trivialization. We consider the 1-isomorphism

B := T ∘ T^{′−1} : I_{ρ′} ⟶ I_{ρ}  (117)

and the corresponding line bundle T := Bun(B). To compare the two σ-equivariant line bundles R̂ and R̂′ corresponding to the two trivializations, we first compare the Jandl structures J_{T^{−1}}(J) on I_{ρ} and J_{T^{′−1}}(J) on I_{ρ′}. By Proposition 3.11 a), b) and c), there exists an isomorphism

J_{T^{′−1}}(J) ≅ J_{B}(J_{T^{−1}}(J))  (118)

of Jandl structures on I_{ρ}. By definition of the functor J_{B}, this isomorphism is a 2isomorphism

A′ ≅ B^{∗} ∘ A ∘ σ^{∗}B,  (119)

where A is the 1-morphism of J_{T^{−1}}(J) and A′ is the 1-morphism of J_{T^{′−1}}(J). Now we apply the functor Bun and obtain an isomorphism

R̂′ ≅ T ⊗ R̂ ⊗ σ^{∗}T  (120)

of σ-equivariant line bundles over Σ̂, where Q̂ := σ^{∗}T ⊗ T has the canonical σ-equivariant structure by exchanging the tensor factors. Thus, we have isomorphic line bundles

R′ ≅ R ⊗ Q  (121)

over Σ. Notice that the holonomy of the line bundle Q is

hol_{Q}(\overline{∂F}) = hol_{T} (∂F) = exp (i ∫_{F} ρ − ρ′)  (122)

This shows

exp (i ∫_{F} ρ′) · hol_{R′}(\overline{∂F}) = exp (i ∫_{F} ρ′) · hol_{Q}(\overline{∂F}) · hol_{R}(\overline{∂F})
= exp (i ∫_{F} ρ) · hol_{R}(\overline{∂F})  (123)

so that Definition 3.12 does not depend on the choice of the trivialization. In [SSW05] we have deduced from the equation curv(R̂) = −(ρ + σ^{∗}ρ) that it is also independent of the choice of the fundamental domain.

Unoriented surface holonomy, defined in terms of Jandl structures on bundle gerbes, provides a candidate for the Wess-Zumino term in two-dimensional conformal field theory for unoriented worldsheets, as they appear in type I string theories. Following the examples of M = SU(2) and M = SO(3) we give in [SSW05], we reproduce results known from other approaches. This indicates, that a bundle gerbe with Jandl structure, together with a metric, is the background field for unoriented WZW models. In this setup, Proposition 3.11 assures, that – just like for oriented WZW models – only the isomorphism class of the bundle gerbe is relevant.

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Department Mathematik Schwerpunkt Algebra und Zahlentheorie Universität Hamburg Bundesstraße 55 D–20146 Hamburg

Email: konrad.waldorf@math.uni-hamburg.de

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