Semistrict Higher Gauge Theory

We develop semistrict higher gauge theory from first principles. In particular, we describe the differential Deligne cohomology underlying semistrict principal 2-bundles with connective structures. Principal 2-bundles are obtained in terms of weak 2-functors from the Cech groupoid to weak Lie 2-groups. As is demonstrated, some of these Lie 2-groups can be differentiated to semistrict Lie 2-algebras by a method due to Severa. We further derive the full description of connective structures on semistrict principal 2-bundles including the non-linear gauge transformations. As an application, we use a twistor construction to derive superconformal constraint equations in six dimensions for a non-Abelian N=(2,0) tensor multiplet taking values in a semistrict Lie 2-algebra.

was proposed byŠevera [23]. In this construction, one considers a simplicial manifold and extracts a corresponding L ∞ -algebra as its first jet. A Lie 2-group can be encoded in terms of a simplicial manifold as the so-called Duskin nerve of its delooping. The first jet of this simplicial manifold is then constructed as a functor acting on descent data of a trivial principal 2-bundle. Finally, we would like to mention that a different proposal for semistrict higher gauge theory was given previously by Zucchini [24]. In this approach, the higher Maurer-Cartan forms are incorporated abstractly as constrained parameters into the gauge transformation. This is not the case in our approach; our detailed understanding of the differential cohomology underlying semistrict principal 2-bundles with connective structures makes the parameters of gauge transformations explicit.

Summary of results
For the reader's convenience, let us summarise our key results in an easily accessible way. In the following, we let X be a smooth manifold with covering U := {U a }. Moreover, we let G = (M, N ) be a weak Lie 2-group, which can be equivalently regarded as a smooth weak 2-groupoid with a single 0-cell e, BG = ({e}, M, N ). We denote the source and target maps by s and t. Vertical and horizontal composition in this weak 2-groupoid are denoted by • and ⊗, respectively, a stands for the associator and l and r label the left-and right-unitors.
A weak principal 2-bundle is described by a G -valuedČech 2-cocycle. Such a cocycle is given by an M -valuedČech 1-cochain {m ab } together with an N -valuedČech 0-cochain {n a } and an N -valuedČech 2-cochain {n abc } which satisfy the following cocycle conditions, cf. Definition 3.8:

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Furthermore, we define semistrict Lie 2-groups G as weak Lie 2-groups in which leftand right-unitors as well as the unit and counit are all trivial. Following [23], we then consider a functor from the category of smooth manifolds to the category of G -valued descent data on surjective submersions R 0|1 × X → X. This functor is parameterised by a 2-term L ∞ -algebra as shown in Theorem 4.24. This 2-term L ∞ -algebra is, in turn, equivalent to the semistrict Lie 2-algebra associated with the semistrict Lie 2-group G . Deriving the parametrisation of this functor is the higher equivalent of computing the Lie algebra of a Lie group.
Moreover, we demonstrate that local connective structures on principal 2-bundles with semistrict structure 2-group (as well as principal n-bundles with semistrict structure ngroup) are readily derived. To this end, we consider the tensor product of the aforementioned 2-term L ∞ -algebra with the differential graded algebra of differential forms on X. This leads to another L ∞ -algebra as well as its homotopy Maurer-Cartan equation including infinitesimal gauge transformations as shown in Propositions 5.3 and 5.9.
The finite gauge transformations are derived from an equivalence relation among the functors considered in the above differentiation of a Lie 2-group G = (M, N ) to a 2-term L ∞ -algebra v µ 1 − − → w with w := T ide M and v := ker(t) ⊆ T id ide N and higher or homotopy products µ 1,2,3 . This relation is presented in Theorem 4.26, from which Proposition 5.9 can be gleaned: a connective structure over U a ⊆ X on a semistrict principal 2-bundle is given locally on a patch U a in terms of a w-valued differential 1-form A a and a v-valued differential 2-form B a such that the fake 2-form curvature Λ pa :Ã a ⊗ p a ⇒ p a ⊗ A a − dp a , (1.5a) B a ⊗ id pa = µ(Ã a ,Ã a , p a ) + id pa ⊗ B a + µ(p a , A a , A a ) • Eventually, we combine our findings onČech cohomology with values in a semistrict Lie 2-group with those on finite gauge transformations of local connective structures and develop full semistrict Deligne cohomology of degree 2. The corresponding Deligne cocycle and coboundary relations are concisely listed in Definitions 5.16 and 5.17.
As a first application of our results, we employ semistrict Deligne cohomology of degree 2 in a twistor description of N = (2, 0) tensor multiplet equations in six dimensions. This is a generalisation of the previous results obtained in [9,10] from strict to semistrict JHEP04(2015)087 gauge 2-groups. The main result here is Theorem 6.5 in which a bijection is established between equivalence classes of certain holomorphic semistrict principal 2-bundles over a twistor space and equivalence classes of solutions to certain superconformal tensor multiplet equations in six dimensions. We hope that the latter equations may serve as an inspiration for a classical formulation of the (2, 0)-theory.

Outlook
There are a number of questions arising from this paper that we plan to address in future work. First of all, there should be an integration operation, inverse to our differentiation of a Lie 2-group to a semistrict Lie 2-algebra. An obvious question is how this integration is related to that of Getzler [21] and Henriques [22]. The answer seems to be similar to that found in [25] for the strict case. Here, straightforward Lie integration of a strict Lie 2-algebra led to a Lie 2-group which is Morita equivalent to the 2-group obtained by the method of Getzler and Henriques.
As mentioned above, we hope that the detailed description of semistrict principal 2-bundles with connective structure allows for a more detailed understanding of the framework of higher gauge theory. More general theories than those derived in this present work can be considered so that the relation to alternative approaches such as the abovementioned non-Abelian tensor hierarchies should become clearer.
The most interesting dynamical theories involving connective structures on semistrict principal 2-bundles are certainly the (2, 0)-theory and its dimensional reductions. As is common to supersymmetric theories, particular attention should be paid to the BPS subsectors of this theory. Higher analogues of instantons and monopoles, such as, for example, self-dual strings, should be studied in more detail from a mathematical perspective. Especially, the relevant topological invariants should be analysed. Some preliminary comments in this direction were already given in [26]. General considerations concerning topological invariants in higher gauge theory can be found in [27] as well as in [28] from the perspective of so-called Q-manifolds.
An important issue is to couple matter fields satisfyingly to higher gauge theories. Mathematically speaking, we would like to consider 2-vector bundles associated to our semistrict principal 2-bundles. Zucchini has suggested such a coupling in his approach to semistrict gauge theory [24]. However, the existence of so-called gauge rectifiers necessary in his approach could not be proved so far. Our twistor construction gives illuminating insights into how such couplings should be achieved. In particular, our approach yields the explicit example of the matter fields contained in the tensor multiplet, discusses the properties they satisfy, how gauge transformations act on them, and how they couple to connective structures.
The most important consistency test for a classical (2,0)-theory is to reproduce fivedimensional maximally supersymmetric Yang-Mills theory in a certain limit. This is a requirement from string theory and so far, this has neither been achieved for higher gauge theories nor for the models arising from tensor hierarchies.

JHEP04(2015)087 2 Preliminaries
In this paper, we require basics of weak 2-category theory. We shall try to be as selfcontained as possible and therefore we present the relevant definitions together with some useful examples in this section.

Weak 2-categories
We assume that the reader is familiar with elementary category theory. In the following, let C = (C 0 , C 1 ) be a category with C 0 the objects of C and C 1 the morphisms of C , respectively. In addition, the source and target maps in C are denoted by s and t, that is, s, t : C 1 → C 0 .
In higher category theory, there is always an issue concerning the level of strictness of the categorification under consideration. For example, 2-categories usually refer to strict 2-categories while weak 2-categories are often called bicategories. We shall exclusively use the terms weak 2-category, weak 2-groupoid etc. and avoid the notions of bicategory, bigroupoid etc.
We start off with the definition of a weak 2-category. The original definition stems from Benabou [29], and a good introduction to the topic can be found, for instance, in [30] and in particular in [31]. The following discussion follows mostly these references.
consists of a collection B 0 of objects a, b, . . . ∈ B 0 and, for any pair of objects a, b ∈ B 0 , an assignment )) is a category. The objects B 0 are called 0-cells, the objects C 0 (a, b) are called 1-cells or 1-morphisms, and the morphisms C 1 (a, b) are called 2-cells or 2-morphisms. Composition of 2-morphisms in C 1 (a, b) will be called vertical composition and denoted by •.
In addition, B comes equipped with a bifunctor ⊗ : C (a, b) × C (b, c) → C (a, c) for all a, b, c ∈ B 0 describing horizontal composition in B, a functor 1 id : 1 → id a ∈ C 0 (a, a) for all a ∈ B 0 , and natural isomorphisms a, l, and r defined by the following diagrams:

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Here, the 1 attached to the arrows refers to the identity functor and ∼ = denotes the natural The natural isomorphisms a, l, and r are referred to as the associator, left unitor, and right unitor, and they yield the 2-cells for x ∈ C 0 (a, b), y ∈ C 0 (b, c), and z ∈ C 0 (c, d). These isomorphisms are required to satisfy the pentagon and triangle identities, that is, the diagrams and Remark 2.2. The fact that ⊗ is a bifunctor implies the so-called interchange law, that is, the diagram a b x 1 where • denotes again vertical composition.
Remark 2.3. The naturalness of the associator a implies that diagrams of the form are commutative. There are similar commutative diagrams involving the unitors or a combination of the unitors and the associator.

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Example 2.5. The standard example of a strict 2-category is Cat, regarded as a 2-category, in which the 0-cells are given by small categories, the 1-cells are functors between those, and the 2-cells are natural transformations between the latter. Horizontal composition is then the obvious composition of functors and natural transformations.
Definition 2.6. A weak 2-category with a single 0-cell can be identified with a weak monoidal category. If, in addition, the natural isomorphisms a, l, and r are all trivial, then we shall speak of a strict monoidal category.
The process of identifying n-categories with a single object or 0-cell with (n − 1)categories is called looping. Below, we shall also encounter the inverse operation called delooping, see Example 2.18.
Example 2.7. An example of a strict monoidal category is the category of sets endowed with a monoidal product given either by the Cartesian product or the disjoint union of sets. Here, B 0 = {e} and C (e, e) is the category Set whose objects C 0 are sets and whose morphisms C 1 are functions between sets.
In weak 2-categories with a single 0-cell e, that is, in weak monoidal categories, we have the following result.
Proposition 2.8. (Kelly [32]) In a weak monoidal category B, the diagrams Morphisms between categories are called functors. Similarly, morphisms between 2categories are called 2-functors. These come in a number of variants, the most general of which are the lax 2-functors.
and a collection Φ 2 of 2-cells, where a, b, c ∈ B 0 and x ∈ C 0 (a, b) and y ∈ C 0 (b, c) such that the following diagrams are commutative: for all a, b ∈ B 0 and x ∈ C 0 (a, b).
Finally, for 2-categories, it is useful to continue the sequence of 2-categories, 2-functors, 2-transformations to 2-modifications. Definition 2.15. Let Φ, Ψ : B →B be two lax 2-functors between two weak 2-categories B andB. A 2-modification between two lax natural 2-transformations α, β : Φ → Ψ is a collection of morphisms ϕ a : α a 1 ⇒ β a 1 for each a ∈ B 0 such that is commutative. If the morphisms ϕ a are invertible, we call the 2-modification invertible.
Note that composition of 2-modifications is trivially obtained by concatenation.

Weak 2-groupoids
In this section, we would like to introduce the notion of 2-groupoids as they play key roles in the definition of principal 2-bundles. We begin by recalling the definition of a groupoid first.
Definition 2.16. A groupoid is a small category in which every morphism is invertible.
Two important examples of groupoids that we shall frequently encounter throughout this work are those of theČech groupoid and the delooping of a group.

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Example 2.17. TheČech groupoid relative to a covering U := {U a } of a topological manifold X, denoted byČ (U) in the following, is defined to be the groupoid that has the covering sets as objects and the intersection of covering sets as morphisms. Concretely, the set of objects ofČ (U) is defined to be the disjoint union˙ a U a := a {(x, a) | x ∈ U a } and the set of morphisms ofČ (U) is defined to be the disjoint union˙ Example 2.18. Let G be a group. The delooping of G, denoted by BG, is defined to be the groupoid that has only a single object, denoted by e, and the elements of the group G as its morphisms, g : e → e with g ∈ G. In BG, the composition of morphisms is then simply given by the group multiplication on G, that is, g 2 • g 1 := g 2 g 1 for any g 1,2 ∈ G.
We are interested in the categorification of the notion of a groupoid, which is defined as follows.
Definition 2.19. A weak 2-groupoid is a weak 2-category such that all morphisms are equivalences. A weak 2-groupoid with an underlying strict 2-category is a called a strict 2-groupoid.
All morphisms being equivalences implies that the 2-cells are strictly invertible and the 1-cells are invertible up to isomorphisms. Unpacking this definition further, 4 a weak 2groupoid is a weak 2-category B such that for every pair of objects a, b ∈ B 0 , the category C (a, b) is a groupoid. Moreover, for every pair a, b ∈ B 0 there is a functor· : C (a, b) → C (b, a) and for every 1-cell x ∈ C 0 (a, b) there are natural isomorphisms i x : id a ⇒ x ⊗x and e x :x ⊗ x ⇒ id b called the unit and counit. These have to satisfy coherence axioms, which state that for any 1-cell x ∈ C 0 (a, b) and a, b ∈ B 0 , the diagrams and Example 2.20. An example of a strict 2-groupoid important in our subsequent discussion is the so-calledČech 2-groupoid. The 0-and 1-cells are given by the objects and morphisms of theČech groupoid (see Example 2.17), and all 2-cells defined to be trivial.

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In Example 2.18, we have seen that any group can be viewed as a groupoid with a single object. Analogously, we give the following definition. Remark 2.22. This definition is equivalent to that given by Baez & Lauda [4]. In particular, they define weak 2-groups as weak monoidal categories in which all morphisms are invertible and all objects are weakly invertible. They also introduce so-called coherent 2groups as weak monoidal categories in which all morphisms are invertible and all objects come with an adjoint equivalence. Both notions are shown to be equivalent. Our definition 2.21 uses the looping, as the weak 2-groups we are interested in will mostly appear as deloopings of coherent 2-groups in the sense of Baez & Lauda. We shall therefore write BG = ({e}, M, N ): the single 0-cell is denoted by e in the following while the 1-and 2-cells are denoted by M and N , respectively. The (monoidal) category C (e, e) contained in BG is then the actual weak 2-group.
Definition 2.23. A strict 2-group is the looping of a strict 2-groupoid with a single 0-cell.
Put differently, a strict 2-group is a weak 2-group in which the unitors, the unit and counit, and the associator are all trivial. Furthermore, we will need the notion of a skeletal 2-group which is as follows.
Definition 2.24. A skeletal 2-group is a weak 2-group, in which the underlying category is skeletal.
Recall that a category is skeletal whenever all isomorphic objects are equal: for all morphisms f in the category, s(f ) = t(f ).
One version of Mac Lane's coherence theorem [35] states that every weak monoidal category is equivalent to a strict monoidal category. In the case of weak 2-groups, we have the following proposition from ( [4], section 8.3), which can be used to classify weak Lie 2-groups. [4]) Every weak 2-group is categorically equivalent to a 'special' weak 2-group which is skeletal and in which all unitors, units, and counits are identity natural transformations. In particular, a special weak 2-group can be given in terms of a group G, an Abelian group H, a representation α of G as automorphisms of H and an element [a] ∈ H 3 (G, H).
In addition, we have the following result. The notion of 2-groups relevant for our subsequent discussion will be the following.

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We would like to emphasise that this notion is weaker than that of a strict 2-group, because the associator remains unrestricted. For semistrict 2-groups, we have the following results. Proof. This follows trivially by combining the pentagon and triangle diagrams together with the diagrams displayed in (2.16).
Proposition 2.29. In any semistrict 2-group G = (M, N ) and for any 2-cell n ∈ N , such that n • n −1 = id t(n) and n −1 • n = id s(n) .
Proof. This follows from the proof of Proposition 20 in [4].

Lie 2-groups
To restrict the rather general notion of a groupoid, we can regard Lie groupoids as groupoids internal to a certain category C . In general, a category internal to C = (C 0 , C 1 ) consists of an object of objects and an object of morphisms, which are both elements in C 0 . The structure maps s, t, id, and • are given in terms of elements of C 1 and all commutative diagrams which hold in a category also hold in the internalised category. Internal functors and modifications are defined in an analogous manner. A groupoid internal to a category C is simply a category internal to C , in which all the morphisms are strictly invertible.
In this manner, we can define, for instance, topological groupoids as groupoids in Top, the category of topological spaces and continuous functions between them. Similarly, Lie groupoids are defined as follows.
Definition 2.30. A Lie groupoid is a groupoid internal to Diff, the category of smooth manifolds and smooth functions between them.
Thus, Lie groupoids are groupoids in which the sets of objects and morphisms are smooth manifolds and all the structure maps are smooth.
Remark 2.31. Recall that for any category K there exists a strict 2-category K Cat with objects being categories internal to K , morphisms being functors in K and 2-morphisms being natural transformations in K . In particular, DiffCat is the strict 2-category with categories in Diff as 0-cells, functors between these as 1-cells and natural transformations between the latter as 2-cells.
We can now define weak Lie 2-groupoids and weak Lie 2-groups by internalising weak 2-groupoids and weak 2-groups, respectively. Definition 2.32. A weak Lie 2-groupoid is a weak 2-groupoid internal to DiffCat. A weak Lie 2-group is a weak 2-group internal to DiffCat.

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Equivalently, a weak Lie 2-group is a weak Lie 2-groupoid with a single object. Specifically, such a weak Lie 2-group consists of an object C in DiffCat, a multiplication morphism ⊗ : C × C → C, an identity object 1, and an inverse map· : C → C with respect to ⊗.
Furthermore, we have for all objects x, y, and z in the category C the following natural isomorphisms: an associator a : (x⊗y)⊗z ⇒ x⊗(y⊗z), left-and right-unitors l x : 1⊗x → x and r x : x ⊗ 1 ⇒ x as well as a unit and counit i x : 1 ⇒ x ⊗x and e x :x ⊗ x → 1, such that the pentagon and triangle identities as well as the first and second zig-zag identities are satisfied, cf. [4].
For our purposes, we wish to restrict the notion of a weak Lie 2-group as given in Definition 2.32 somewhat further.
Definition 2.33. A semistrict Lie 2-group is a weak 2-group internal to DiffCat such that the unitors, the unit, and the counit are all trivial.
Note that by Proposition 2.25, semistrict Lie 2-groups are still categorically equivalent to weak Lie 2-groups.
Definition 2.34. A strict Lie 2-group is a weak 2-group in DiffCat such that the associator, the unitors, the unit, and the counit are all trivial.
We recall that there is an equivalent formulation of strict Lie 2-groups in terms of crossed modules of Lie groups.   [4] for a detailed proof. We shall use an identification between strict Lie 2-groups and crossed modules of Lie groups that slightly differs from that of [4]. Given a crossed module of Lie groups (H ∂ − → G, ), we obtain a strict Lie 2-group G = (M, N ) by identifying M := G and N := G H and setting s(g, h) := ∂(h −1 )g, t(g, h) := g, and id g = (g, 1 H ) for h, h 1,2 ∈ H and g, g 1,2 ∈ G together with g 2 ⊗ g 1 := g 2 g 1 ,

Lie 2-algebras
Apart from Lie 2-groups, we shall also be dealing with Lie 2-algebras. The most general kind of Lie 2-algebra currently in use has been defined by Roytenberg [36] as follows. for all X, Y, Z ∈ L 0 . These structure maps are subject to a number of coherence axioms, cf. [36].
In this paper, we are merely interested in so-called semistrict Lie 2-algebras.
Definition 2.38. A semistrict Lie 2-algebra is a weak Lie 2-algebra in which the alternator is trivial.
Instead of working directly with semistrict Lie 2-algebras and their rather involved coherence axioms, we can switch to a categorically equivalent formulation in terms of 2-term L ∞ -algebras, as was shown in [37]. The general definition of a strong homotopy Lie algebra is given in appendix A. Here, we just recall the following definition.
where we associate gradings −1 and 0 to elements of v and w, respectively. This complex is equipped with higher products µ 1 , µ 2 , µ 3 , which vanish except for
Example 2.41. A typical example of a semistrict Lie 2-algebra is the string Lie 2-algebra of a Lie algebra g. Here, w = g, v = R and the only non-trivial higher products are , where w 1 , w 2 , w 3 ∈ w and ·, · is the Killing form on g.
Let us briefly recall the details of the equivalence between semistrict Lie 2-algebras and 2-term L ∞ -algebras. 6 We start from a Lie 2-algebra L = (L 0 , L 1 ) and put v := ker(t) ⊆ L 1 , w := L 0 , and µ 1 := −s| v . (2.24) The higher products are defined as follows: where w 1 , w 2 , w 3 , w ∈ w and v ∈ v. This map from a semistrict Lie 2-algebra to a 2-term L ∞ -algebra can be extended to a functor Φ between the corresponding categories. Conversely, given a 2-term L ∞ -algebra v µ 1 − − → w, we obtain a semistrict Lie 2-algebra L = (L 0 , L 1 ) by putting for all v, v 1 , v 2 ∈ v and w ∈ w. In addition, we set (2.27) 6 A similar equivalence exists for weak Lie 2-algebras [36], but the resulting normalised chain complex is less convenient to work with.

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Again, this map from a 2-term L ∞ -algebra to a semistrict Lie 2-algebra can be extended to a functor Ψ between the corresponding categories. We have the following results.  [37]) There is a one-to-one correspondence between equivalence classes of semistrict Lie 2-algebras and 'special' 2-term L ∞ -algebras given in terms of a Lie algebra g, a representation of g on a vector space v, and an element J of . Here, µ 1 = 0, µ 2 is the Lie bracket in g or the action on v, and µ 3 = J.
Semistrict Lie 2-algebras can be restricted further to obtain strict Lie 2-algebras.
Definition 2.44. A strict Lie 2-algebra is a weak Lie 2-algebra with trivial alternator and trivial Jacobiator.
Our above discussion immediately implies that strict Lie 2-algebras are equivalent to 2-term L ∞ -algebras with trivial product µ 3 , which in turn, can be encoded in a differential crossed module.
Definition 2.45. The differential crossed module of a crossed module of Lie groups is obtained by applying the tangent functor to the crossed module.
In particular, given a crossed module of Lie groups (H ∂ − → G, ), the tangent functor yields a differential crossed module 7 (h ∂ − → g, ), where h := Lie(H) and g := Lie(G). The maps ∂ and satisfy where X ∈ g and Y, Y 1,2 ∈ h. The differential crossed module corresponding to a 2-term L ∞ -algebra v µ 1 − − → w with trivial µ 3 is obtained by identifying h, g, and ∂ with v, w, and µ 1 as well as This identification is readily inverted.

Principal 2-bundles with Lie 2-groups
We come now to the discussion of principal 2-bundles with weak structure 2-groups over smooth manifolds. An earlier description of general 2-bundles from a slightly different point of view can be found in Bartels [3]. In the following, let X be a smooth manifold and let U = {U a } be a covering of X.

Principal bundles as functors
Recall that aČech p-cochain with values in a group G on X relative to the covering U is a set of smooth G-valued functions on all non-empty intersections U a 0 ∩ · · · ∩ U ap . 8 We then give the following definition.
TwoČech 1-cocycles {g ab } and {g ab } are cohomologous or equivalent if and only if there is aČech 0-cochain {g a } consisting of smooth maps g a : U a → G such that The firstČech cohomology set, denoted by H 1 (U, G), is defined as the set ofČech 1-cocycles modulo this equivalence.
Cech cohomology sets can be rendered independent of the covering by taking the direct limit over all coverings U of X. We then write Elements of H 1 (X, G) are also known as (sets of) transition functions of principal bundles with structure group G (or principal G-bundles for short), and it is well-known that principal G-bundles over X can be identified with an elements in H 1 (X, G). To allow for a categorification of this picture, we switch to a functorial description of principal bundles.
Definition 3.2. A smooth principal bundle Φ with structure group G is a smooth functor Φ from theČech groupoid to the Lie groupoid BG. 9 Any two principal bundles are called equivalent if and only if there is a natural isomorphism between their defining functors.
Definition 3.2 is well-known from the description of principal bundles in terms of classifying spaces [38]. Explicitly, we have a functor and we set e a := Φ(x, a) and g ab := Φ(x, a, b). Because Φ is a functor, we immediately arrive at the cocycle conditions (3.1) as well as Φ(x, a, a) = id Φ(x,a) = 1 G ∈ G. In addition, two functors Φ and Ψ corresponding to principal bundles are equivalent if and only if 8 If not stated otherwise, we shall always assume that intersections of patches are non-empty from now on. 9 See Examples 2.17 and 2.18 for the relevant definitions.
Other conventional definitions are now also straightforwardly rephrased.

Definition 3.4. A principal bundle is called trivial if and only if its defining functor is equivalent to the functor
Concretely, a principal bundle is trivial whenever there is a natural isomorphism α = {g a } such that Finally, let φ : X → Y be a smooth map between two smooth manifolds X and Y . Let U Y be a covering of Y . Then we can construct a covering U X of X from the pre-images of the patches in U Y under φ. This yields a morphism of groupoidsČ (U X ) →Č (U Y ).
Definition 3.6. The restriction of a principal bundle Φ over a manifold X to a submanifold Y of X is the pullback of Φ along the embedding map Y →X.

Principal 2-bundles as 2-functors
The reformulation of principal bundles with structure group G in terms of functors between theČech groupoid and the Lie groupoid BG is a good starting point for categorifying the notion of principal bundles. We can simply regard theČech groupoid as an n-groupoid and take an n-functor to a Lie n-groupoid with a single 0-cell. In the following, we shall develop the case n = 2 in detail. Note that our discussion will first centre around weak principal 2-bundles which we shall define in terms of weak 2-functors. In the following, we

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shall consider the delooping BG = ({e}, M, N ) of a weak Lie 2-group G = (M, N ), which is a weak Lie 2-groupoid with a single object e. As in section 2, we shall denote horizontal and vertical composition in BG by ⊗ and •, respectively. Principal 2-bundles will be described byČech cocycles with values in G . We therefore start by giving the following definition.
In the following, we are interested in the case p = 2, for which we have a triple To derive the explicit cocycle and coboundary conditions appropriate for weak Lie 2-groups, we again employ the functorial approach.
Let us be more specific. We have a weak 2-functor 10 consisting of a function Φ 0 (x, a), functors Φ 1 (x, a, b) and 2-cells Φ 2 . Note that the 0-cells of BG = ({e}, M, N ) and the 2-cells ofČ (U) are trivial and we shall denote them by e. We can therefore specify Φ in terms of constant functions e a := Φ 0 (x, a) : U a → e, functions m ab : with id ea ∈ M . 10 Cf. Definition 2.10.

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The following diagrams, which arise from (2.9) with a, r, and l being trivial inČ (U), are commutative: and In formulae, this reads as and Pushing the analogy with the case of principle bundles further, we derive equivalence relations between weak principal 2-bundles from natural 2-transformations. Explicitly, for weak principal 2-bundles Φ andΦ, such a natural 2-transformation α :Φ → Φ is given by the following data: we have 1-cells {m a } and 2-cells {n ab },

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The coherence conditions for natural 2-transformations require also the diagrams to be commutative. In formulae, this amounts to and Definition 3.12. A weak principal 2-bundle that is equivalent to the weak principal 2bundle specified by the functor is called trivial.
We shall give explicit formulae for the transition functions of trivial bundles in the case of semistrict principal 2-bundles later on. Note that for strict 2-bundles, the 2-cells {n a } can always be chosen to be trivial, as was done, for instance, in [9,10]. The same is true here, as we verify now. defines another weak principal 2-bundleΦ. In addition, these equations imply Proof. One readily checks that the cocycle conditions (3.12) and (3.14) are satisfied for any possible doubling of indices.
Definition 3.14. For every weak principal 2-bundle Φ, the weak principal 2-bundleΦ obtained from the construction of Lemma 3.13 is called the normalisation of Φ.
Proposition 3.15. Every weak principal 2-bundle is equivalent to its normalisation.
Proof. A natural 2-transformation that yields the equivalence is given by Note thatm ab = m ab for a = b. As one may check, the coboundary conditions (3.15) and (3.18) are indeed satisfied. Note that the choice of n ab in this transformation for a = b is not unique.
Proof. By Proposition 3.15, a weak principal 2-bundle Φ is equivalent to its normalisation, for which we have on any U a ∈ U. Thus, the weak principal 2-bundle is locally equivalent to a trivial one.
Recall that trivial principal bundles with structure group G are described by transition functions {g ab } of the form g ab = g a g −1 b , where {g a } is a G-valuedČech 0-cochain. Note that the g a can be multiplied by a (global) G-valued function from the right, leaving g ab = g a g −1 b invariant. This is an equivalence relation, which is described by modifications in functorial language.
The corresponding equivalence relations are more comprehensive in the case of principal 2-bundles, as we shall see in the following. Consider two equivalent weak principal 2-bundles Φ andΦ with natural 2-transformations α :Φ → Φ andα :Φ → Φ between them. A weak 2-modification ϕ : α ⇒α is given by a smooth map ϕ : α →α that assigns to every object (x, a) ∈Č (U) a 2-morphism ϕ (x,a) : α (x,a) ⇒α (x,a) . We set o a := ϕ (x,a) so that o a : m a ⇒m a . Moreover, the following diagram is required to be commutative: To define pullbacks and restrictions of weak principal 2-bundles, we proceed just as in the case of the functorial description of principal bundles; see Definitions 3.5 and 3.6. Recall that given a smooth map φ : X → Y and a covering U Y of Y , the pre-images of the patches in U Y form a covering of U X . The resulting groupoid morphismsČ (U X ) →Č (U Y ) can be extended to a strict 2-functor φ U . Therefore, we give the following definitions.
Definition 3.19. The restriction of a weak principal 2-bundle Φ over a manifold X to a submanifold Y inside X is the pullback of Φ along the embedding map Y →X.

Semistrict and strict principal 2-bundles
We shall be specifically interested in weak principal 2-bundles with semistrict structure 2-groups. This implies a number of simplifications, which we shall discuss in the following. (3.26a) The cocycle conditions for this type of principal 2-bundle then read as while the coboundary conditions and modifications are given by and o a : m a ⇒m a , respectively.

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Remark 3.21. A trivial semistrict principal 2-bundle is described by transition functions ({m ab }, {n abc }) given in terms of coboundary data ({m a }, {n ab }) according to m a :ẽ a → e a and n ab : where n aa = id ma .
To recover principal 2-bundles based on crossed modules as discussed in most of the current literature, we define the following. A well-known result is then the following.
Proof: Let us again sketch the identification. For a strict principal 2-bundle, the cocycle and coboundary conditions, as well as the coherence equation for modifications, reduce to and and o a : m a ⇒m a ,

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Next, recall the identification of strict Lie 2-groups with crossed modules of Lie groups of Proposition 2.36. To go from a crossed module of Lie groups H ∂ − → G to a strict Lie 2-group G , we identify G with (G, G H) in terms of the Lie groups G and H contained in the equivalent crossed module, we can identify m ab = g ab and n abc = (g abc , h abc ). From we immediately obtain the first equation in (3.28a). Likewise, using id m ab = (g ab , 1 H ) and (2.19), it is a straightforward exercise to show that (3.29a) simplifies to the second equation in (3.28a). We skip the inverse transition from Lie 2-groups to crossed modules here; details on this point can be found in the proof of Proposition 4.30.
Remark 3.24. In the strict setting, we may define where m ⊗ m = id e . It is easy to see that ab : m ab ⇒ m a ⊗m ab ⊗ m b , and, in particular, if the bundle is trivial, then ab : m ab ⇒ m a ⊗ m b . In this case, one may show that n abc can be rewritten in terms of ab as It is amusing to note the resemblance with a trivial Abelian gerbe: the only difference is that ordinary products are replaced by • and ⊗.

Differentiating semistrict Lie 2-groups
In order to define connective structures on semistrict principal 2-bundles, we first need to develop a way of differentiating a semistrict Lie 2-group to a semistrict Lie 2-algebra. The approach we shall develop is based on an idea ofŠevera's [23] (see also Jurčo [39]). As before, we let X be a smooth manifold. The sheaf of smooth differential p-forms on X is denoted by Ω p X , and we set Ω • X := p≥0 Ω p X . In general, given a module v = k∈Z v k with a Z-grading, one may always introduce a Z 2 -grading referred to as the Graßmann parity in terms of the parity of degrees: v = k∈Z v 2k ⊕ k∈Z v 2k−1 . Elements of k∈Z v 2k are said to be Graßmann-even while elements of k∈Z v 2k−1 are said to be Graßmannodd, respectively. We shall also make use of the Graßmann-parity changing functor Π.
Specifically, a given descent datum describes the descent of a trivial principal G-bundle over Y to a non-trivial principal G-bundle over X. The following example makes this more transparent.
Example 4.2. Let X be a smooth manifold with covering U = {U a } a∈I indexed by the index set I. Consider the trivial projection σ : I × X → X. A G-valued descent datum is then given by a map g : I×I×X → X such that g(a, a, x) = 1 G and g(a, b, x)g(b, c, x) = g(a, c, x) for all a, b ∈ I and x ∈ X. Setting g ab (x) := g(a, b, x), we have obtained a G-valuedČech 1cocycle {g ab } on X relative to the covering U. This, in turn, describes a principal G-bundle over X.
Below, we shall be interested in the trivial projection σ : R 0|1 × X → X, so a G-valued decent datum is in this case given by a map g : We can regard the maps from the surjective submersion R 0|1 × X → X to a descent datum as a contravariant functor from the category of smooth manifolds to the category of sets. As we shall see below, this functor is representable by g[−1], where g is the Lie algebra of G. In particular, calculating the moduli of this functor yields the Lie algebra g as a vector space. To describe its Lie bracket, one needs to compute the action of its Chevalley-Eilenberg differential 12 d CE . This differential is governed by a generator of the natural action of C ∞ (R 0|1 , R 0,1 ) on the descent data, as was first discussed by Kontsevich [40] (see also [23]). Let us now review this in some more detail.

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Furthermore, the de Rham differential d on H 0 (X, Ω • X ) follows from the action of X). Concretely, transformations of the form θ →θ = bθ + β for b ∈ R, β ∈ R 0|1 induce an action on elements of C ∞ (R 0|1 , X) which in local coordinates (x 1 , . . . , x n ) of X is given by for i = 1, . . . , n. Translated into differential forms, this means that x i → x i + βdx i and dx i → bdx i . We thus arrive at the following result.
We obtain from (4.5). Comparing coefficients in the Graßmann-odd coordinates, we can read off the action of an induced operator, again denoted by d K on the components Proposition 4.6. The operator d K is a differential. That is, it has the following properties: obeys a graded Leibniz rule, where |f | denotes the Graßmann parity of f .
Proof. These properties are an immediate consequence of the definition of d K .

Lie algebra of a Lie group
Having collected all relevant ideas, let us put them to use and start by computing the Lie algebra of a Lie group as a guiding example for the case of Lie 2-groups. This has been done in [23,39], and our discussion below is an expanded version of the one found in these references. Consider a Lie group G with Lie algebra g = T 1 G G. To prepare our discussion for semistrict Lie 2-groups, we shall not assume that G is a matrix group, rather we only make use of the fact that there is a local diffeomorphism ϕ between a neighbourhood U g of 0 ∈ g and a neighbourhood U G of 1 G ∈ G with ϕ(a) = g for a ∈ U g and g ∈ U G , ϕ(0) = 1 G , and ϕ * | 0 is the identity. In addition, we wish to restrict ourselves to infinitesimal neighbourhoods by considering elements of g[−1] multiplied by a Graßmannodd coordinate.
Proposition 4.7. Let ϕ : U g → U G be the above-described local diffeomorphism. For a, a 1,2 ∈ g[−1], we have the following relations: where the operation · : is defined by the second equation. This operation is bilinear and a 1 · a 2 + a 2 · a 1 = [a 1 , a 2 ] is the Lie bracket shifted by one degree.
Remark 4.8. For matrix Lie groups, we may suggestively write In addition, one may also define products between elements g and a of G and g[−1], respectively. For matrix Lie groups, we simply write ga. For general Lie groups, one replaces such expressions by the pullback L * g a of a, where L g denotes left multiplication on G.

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We are now ready to discuss the computation of the Lie algebra of a Lie group by Severa's construction [23]. Consider a G-valued descent datum on the trivial projection R 0|1 × X → X. That is, we a have smooth map g : R 0|1 × R 0|1 × X → G satisfying the cocycle condition (4.2). Since we are interested in the functor from the category of smooth manifolds to the category of descent data in the following, we shall suppress the explicit dependence on x ∈ X and simply write {g 01 := g(θ 0 , θ 1 )} with g 01 g 12 = g 02 and g 10 = g −1 01 . Then, we have the following result. Lemma 4.9. Letting g(θ) := g(θ, 0), we have Proof. This is an immediate consequence of (4.2).
Together with the Propositions 4.4 and 4.7, we get the following result. 14) The induced differential is given by As stated previously, we wish to identify the induced action of the differential d K with the Chevalley-Eilenberg differential d CE on g. Recall that the Chevalley-Eilenberg differential of a Lie algebra g acts as Altogether, we have proved the following theorem.
Theorem 4.11. The functor from the category of smooth manifolds X to the category of G-valued descent data on surjective submersions R 0|1 × X → X is parameterised by elements of g[−1] with g = Lie(G). The action of the differential d K on descent data yields the action of the Chevalley-Eilenberg differential corresponding to g.
Proposition 4.12. Consider two equivalent G-valued descent data that are parametrised by a ∈ g[−1] andã ∈ g[−1], respectively. Then there is aČech coboundary transformations between these, which is parametrised by p : R 0|1 → G with p(θ) = p + πθ for some p ∈ G and π ∈ T p [−1]G, such that Remark 4.13. Note that by replacing d K by the de Rham differential in all of the above, we recover the definition of the curvature of a connection 1-form on a principal bundle with structure group G as well as its gauge transformation. We will make use of this observation later on.

Semistrict Lie 2-algebra of a semistrict Lie 2-group
Now we generalise the previous discussion to the case of semistrict Lie 2-groups G = (M, N ), which we shall regard as a weak Lie 2-groupoid BG ({e}, M, N ) in the following. In this case, the local diffeomorphism ϕ = (ϕ M , ϕ N ) goes between neighbourhoods U m of m := T ide M and U n of n := T id ide N as well as neighbourhoods U M of id e and U N of id ide . As before, ϕ(0) = (id e , id ide ) and ϕ * | 0 is the identity. Following our previous discussion, we shall again be interested in infinitesimal neighbourhoods and we shall always write suggestively id e + aθ and id ide + bθ for ϕ M (aθ) and ϕ N (bθ), where a ∈ m[−1] and b ∈ n[−1]. (id e + a 1 θ 1 ) ⊗ (id e + a 2 θ 2 ) = id e + a 1 θ 1 + a 2 θ 2 − a 1 ⊗ a 2 θ 1 θ 2 , where a 1,2 ∈ m[−1] and b 1,2 ∈ n[−1], respectively.
Proof. The proof is essentially the same as the one given for Proposition 4.7.

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We now turn to the maps induced by the structure maps s, t, and id on n[−1] and m[−1]. Note that for elements a ∈ m[−1] and b ∈ n[−1], we have where the differentials are to be taken at id ide and id e , respectively. More generally, the following result holds.

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Remark 4.19. In the following, we shall simply write s, t, and id for s * , t * , and id * , slightly abusing notation. We shall also write id a instead of id * (a). The distinction between these linear maps and the finite maps on M and N should always be clear from the context.
This completes the preliminary discussion, and we can turn to the differentiation of a semistrict Lie 2-group G = (M, N ) to a 2-term L ∞ -algebra. Following our discussion for Lie groups, we consider the functor from the category of smooth manifolds X to the category of G -valued descent data on surjective submersions R 0|1 × X → X that are represented by    according to the following expansions in the Graßmann-odd coordinates: where µ(α, α, α) : α ⊗ (α ⊗ α) − (α ⊗ α) ⊗ α ⇒ 0.
To derive the expansion (4.30d), we use n(θ 0 , θ 1 , 0) = n(θ 0 , θ 1 ) together with the normalisation n 001 = id m 01 and n 011 = id m 01 . Hence, n 012 must be of the form In order to evaluate (4.27) for coboundaries given in (4.30), we note that (4.32) can be rewritten as and likewise for n 01 = id ide + (id α − βθ 1 )θ 0 and all the other terms appearing in (4.27). Thus, our definitions of the induced concatenation and products ⊗ to linear order are sufficient to evaluate (4.27). For example, we compute Comparing the coefficient of θ 0 θ 1 θ 2 of both sides of equation (4.27), we obtain In deriving the latter, we have used β • (id s(β) − β) = 0, which follows immediately from Proposition 2.29.

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Proof. This is a direct consequence of the application of the differential d K to {n 012 } as given in Proposition 4.22. Alternatively, the first of these equations can also be obtained from the application of d K to {m 01 }.
From equations (4.37), we can now extract the Chevalley-Eilenberg algebra of a 2term L ∞ -algebra. In particular, let (τ i ) and (σ m ) be bases of w := m = T ide M and v := ker(t) ⊆ n = T id ide N , respectively, and let (τ i ) and (σ m ) be the corresponding dual bases of w ∨ and v ∨ . The equations (4.37) should be regarded as the evaluation of (4.39) The additional signs are included to match our overall conventions, cf. Remark 2.40. The higher homotopy Jacobi identities follow from the fact that d 2 CE = d 2 K = 0 [41]. We sum up our findings in the following theorem.
where v → w is the 2-term L ∞ -algebra for which w := T ide M and v := ker(t) ⊆ T id ide N . The action of the differential d K on the descent data yields the Chevalley-Eilenberg differential of the 2-term L ∞ -algebra v → w.
Analogously to Lie groups, we would like to consider an equivalent descent datum and compare the change of the resulting Chevalley-Eilenberg algebra. This will eventually give us equivalent an parameterisation (α,β) for some p ∈ N and λ p ∈ T p [−1]N .
Note that contrary to the previously considered coboundaries, p 0 and q 01 are points in M near p and in N near id p , respectively. Our formulae for linearising the structure maps at p and id p , however, remain essentially the same, cf. Remark 4.18.
Proof. Due to the naturalness of the associator, it is straightforward to see that λ p can be expressed in terms of λ in the above way.
which yields the following.
Lemma 4.29. For strict Lie 2-groups, the functor between the category of smooth manifolds X and the category of G -valued descent data on R 0|1 × X → X reads as To compare with the literature, we need to translate these results into expressions using crossed modules of Lie groups. and (4.66b) The action of the differential d K translates to In the strict case, α and β take values in a 2-term L ∞ -algebra with trivial associator, which forms a differential crossed module. From the actions of d K given in (4.65) as well as equations (4.38) and (4.39), we read off that the tensor products α ⊗ α and id α ⊗ β − β ⊗ id α turn into the commutator and the action of G onto H.
These are the expressions that were already obtained in Jurčo [39]. Furthermore, combining the results of Theorem 4.26 and Definition 4.27 with the interchange law (2.5), we arrive after a few algebraic manipulations at (4.69) Translated into crossed modules of Lie groups, this takes the following form.

Comment on differentiation and categorical equivalence
Recall from Proposition 2.25 that every weak 2-group is categorically equivalent to a special weak 2-group given in terms of a group G, an Abelian group H, a representation of G on H, and an element [a] ∈ H 3 (G, H). The corresponding Proposition 2.43 for Lie 2-algebras states that semistrict Lie 2-algebras are categorically equivalent to special Lie 2-algebras given in terms of a Lie algebra g, a representation of g on a vector space v, and an element It is now tempting to assume that the natural integration process factors through categorical equivalence and therefore special Lie 2-algebras can be integrated to special Lie JHEP04(2015)087 2-groups. However, Baez & Lauda proved a no-go theorem ( [4], section 8.5), which shows that certain special Lie 2-algebras can be integrated to 2-groups, which, however, can be turned into topological 2-groups only for the strict case a = 0. In particular, consider the case of a special Lie 2-algebra with v = u(1). We have H 3 (g, u(1)) ∼ = R. The latter contains a lattice ∼ = Z, which can be embedded into H 3 (G, U(1)), yielding the integration to a 2-group. In the topological case, however, we have to use continuous cohomology, for which H 3 cont. (G, U(1)) = 0. The differentiation of Lie 2-groups we performed in this section is the inverse operation to this integration. As integration does not commute with categorical equivalence, neither will differentiation.

Semistrict higher gauge theory
We now put the results of the previous section together and develop a description of semistrict principal 2-bundles with connective structure. We first discuss the local case, 14 which can be readily derived from the Maurer-Cartan equation of an L ∞ -algebra. We then give the global description in terms of non-Abelian Deligne cohomology sets.
As before, let X be a smooth manifold with covering U = {U a } and let U ⊆ X be an open subset of X. Furthermore, let Ω p X be the sheaf of smooth differential p-forms on X and set Ω • X = p≥0 Ω p X .

Local semistrict higher gauge theory
Local semistrict higher gauge theory corresponds to the Maurer-Cartan equation (A.7) for a degree-1 element of the L ∞ -algebra arising from the tensor product of Ω • X and a gauge L ∞ -algebra L. The corresponding infinitesimal gauge transformations are the gauge transformations of the Maurer-Cartan equation (A.8). To make this explicit, we wish to recall the following proposition.
Proposition 5.1. A tensor product of a differential graded algebra a and an L ∞ -algebra L comes with a natural L ∞ -structure. The grading of an element of a ⊗ L is the sum of its individual gradings. Moreover, for a tuple of elements (a 1 ⊗ 1 , . . . , a i ⊗ i ) of a ⊗ L, the higher productsμ i read as Here, the µ i are the higher products in L, deg denotes the degrees in a, and χ = ±1 is the so-called Koszul sign arising from moving graded elements of a past graded elements of L.
Proof. The higher homotopy Jacobi identities, displayed in the appendix in (A.2), for the higher productsμ i are readily checked.

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Example 5.2. As an example, let us work out the details for the case where a is the de Rham complex on X and L is a 2-term L ∞ -algebra. Let U ⊆ X be an open subset. The tensor product of H 0 (U, Ω • X ) and the 2-term L ∞ -algebra v µ 1 − − → w consists of the following graded subspaces , the homotopy Maurer-Cartan equation (A.7) reads as
Let us now generalise from gauge potential 1-and 2-forms A and B satisfying the Maurer-Cartan equation to general kinematic data for local semistrict higher gauge theory. It makes sense to relax the equation H = 0: a trivial calculation shows that in this case, H transforms under under gauge transformations (5.6) covariantly according to δH = µ 2 (H, ω). There are a number of reasons, however, why we cannot relax F = 0. Firstly, consistency of the underlying parallel transport requires F to vanish, just as it did in the strict case. Secondly, the above covariant transformation law is broken for non-vanishing F , which makes it impossible to impose a self-duality condition on H. Such a condition, however, is expected to arise in the N = (2, 0) superconformal field theory in six dimensions. We therefore arrive at the following definition.
Remark 5.5. For trivial µ 3 , the equations (5.5) reduce to the field equations for a flat connective structure of a principal 2-bundle with strict structure 2-group and equations (5.6) describe infinitesimal gauge transformations.
Note also that there are equivalence relations between gauge transformations which have the same effect on A and B. These are given by δω = µ 1 (σ) and δΛ = dσ + µ 2 (A, σ) , where σ ∈ H 0 (U, Ω 0 X ⊗ v).
Remark 5.6. Finally, we would like to stress that the kinematic data, the local flatness conditions and the infinitesimal gauge transformations for local semistrict higher gauge theory based on an n-term L ∞ -algebras L are similarly derived by considering the tensor product of Ω • X with L.

Finite gauge transformations
Having derived curvature and infinitesimal gauge transformation for semistrict higher gauge theory, let us now turn to the finite gauge transformations. Here, we rely on the results of section 4, and the lift to Lie n-algebra valued potential and curvature forms is readily obtained. In Proposition 4.12, we showed that the equation d K a + 1 2 [a, a] = 0 was invariant under a →ã = pap −1 + pd K p −1 . Since d K and the de Rham differential d have the same algebraic properties, we derived the well-known statement Note also the following consequence. where π ∈ H 0 (U, Ω 0 X ⊗ g). They match the gauge transformations in Proposition 5.3 for the 2-term L ∞ -algebra {0} → g.

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Analogously, we treat the kinematic data of local semistrict higher gauge theory. In Theorem 4.26, we showed that the equations are invariant under (4.50a) and (4.50b). Again, since d K and d have the same algebraic properties, we have derived the following statement.
Proposition 5.9. If the curvatures F and H of local gauge potential 1-and 2-forms A and B as defined in Proposition 5.3 vanish, then they are invariant under the transformation where p ∈ H 0 (U, M ) and 15 Λ p ∈ H 0 (U, Ω 1 X ⊗ T p N ). We shall refer to such transformations as gauge transformations.
As a consistency check, we can linearise these gauge transformations, obtaining the transformations (5.6): Proposition 5.10. At the infinitesimal level, the gauge transformations (5.11) become where w ∈ H 0 (U, Ω 0 X ⊗ w) and v ∈ H 0 (U, Ω 1 X ⊗ v). Hence, they agree with the gauge transformations in Proposition 5.3 for the 2-term L ∞ -algebra v Proof. We linearise p = id e + δp and Λ = id A + δΛ such that equation (5.11a) reads as we immediately obtain the first equation in (5.12). The derivation of the second equation in (5.12) from linearising (5.11b) is somewhat more involved. We start from The remaining calculation is rather lengthy but straightforward, if one makes use of the (linearised) interchange law, Proposition 4.17 and the identity s(B) = −dA + A ⊗ A. 15 Here, TpN denotes the sheaf over U ⊆ X with stalks T p(x) N over x ∈ U . JHEP04(2015)087

Connective structure
Consider a semistrict principal 2-bundle Φ with a semistrict structure 2-group G = (M, N ) over a smooth manifold X with covering U = {U a }. We use again the notation w := T ide M and v := ker(t) ⊆ T id ide N . The bundle Φ is characterised by G -valued transition functions ({m ab }, {n abc }). Next, we would like to equip Φ with a connective structure. From the discussion of strict principal 2-bundles, it is clear that a connective structure will consist locally of a w- valued 1-form A a , a v-valued 2-form B a , and, on intersections (A a , B a )  and (A b , B b ) are related by a gauge transformation on U a ∩ U b , which is parameterised by (m ab , Λ ab ). The explicit formula is then clear from Proposition 5.9 and reads as follows: provided the fake curvature F a := dA a + A a ⊗ A a + s(B a ) vanishes on all coordinate patches U a . Note that the condition that two transformations of the form (5.16) combine to a third one on non-empty triple intersection of coordinate patches yields the cocycle condition for {Λ ab }. To derive this condition, let us consider Chasing the commutative diagram relating the two possible ways of going from (A b ⊗m ba )⊗ m ac to m bc ⊗ A c − dm bc , we obtain the following proposition.
Proposition 5.11. The 1-forms {Λ ab } are consistent over triple overlaps U a ∩ U b ∩ U c , if the following holds: (5.20) In the above equation, we have again used our intuitive notation: for instance, n bac ⊗id Ac − dn bac has to be understood as We now have all the ingredients for defining the notion of a connective structure.

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Definition 5.12. A connective structure on a semistrict principal 2-bundle Φ with semistrict structure 2-group G = (M, N ) with associated 2-term L ∞ -algebra v or, equivalently, To derive the action on {Λ ab }, we compare the two expressions, Again, chasing the corresponding commutative diagram relating the two possible ways of going from (Ã a ⊗m a )⊗m ab to (m ab ⊗Ã b )⊗m b −dm ab ⊗m b yields the following proposition.
As before, we have used our intuitive notation here.

Semistrict non-Abelian Deligne cohomology
Deligne cohomology describes gauge configurations on a principal bundle with connection modulo gauge transformations, which act simultaneously on the connection and the transition functions of the bundle. Deligne cohomology for categorified bundles was described previously in some special cases. In particular, the case of Abelian gerbes was discussed in [43], the case of principal 2-bundles with strict structure 2-group was given in [44], and the case of principal 3-bundles was presented in [10] (see also [45]). Here, we wish to describe the low-lying sets of the Deligne cohomology with values in a semistrict Lie 2-group. In the special case of the 2-group BU(1), this reduces to ordinary, Abelian Deligne cohomology.
As before, we consider a smooth manifold X with covering U = {U a }. We shall write C p,q (U, S) for the Ω q X ⊗ S-valuedČech p-cochains relative to the covering U, where S is a some sheaf on X. Now, let G = (M, N ) be a semistrict Lie 2-group. We again make use of the notation w := T ide M and v := ker(t) ⊆ T id ide N and denote the corresponding 2-term The sum of theČech and de Rham degrees of ({n a 0 ···ap }, . . . , {n a 0 }) is p while for ({m a 0 ···a p−1 }, . . . , {m a 0 }) it is p − 1. Compared to the analogous discussions of Deligne cochains for strict 2-groups in Schreiber & Waldorf [44], we have droppedČech cochains that are always cohomologous to trivial ones, cf. [10] and Proposition 3.15.
Using our results from the previous sections as well as appendix B, we can describe Deligne cohomology with semistrict 2-groups up to degree 2. In particular, we have provided ample motivation for giving the following definition.
Definition 5.16. A degree-p Deligne cocycle is a degree-p Deligne cochain satisfying a cocycle relation. Here, we restrict ourselves to the case p ≤ 2, and define the following: (i) A degree-0 Deligne cocycle is an element {n a } ∈ C 0,0 (U, N ) such that on non-empty intersections U a ∩ U b n a = n b .
or, equivalently, and Note that there are further equivalences between Deligne coboundaries arising from modification transformations. These are not relevant for our discussion of Deligne cohomology and we therefore do not wish to go into any further detail.
Let us end this section by briefly commenting on the interpretation of elements of Deligne cohomology sets. The first case of degree-0 Deligne cocycles is readily understood. A degree-0 Deligne cocycles describes an N -valued function on X, which could be regarded as a principal 0-bundle.
The case of Deligne 1-cocycles is slightly more involved. If N is a group, then a degree-1 Deligne cocycle defines a principal (1-)bundle with connection one-form B and a preferred section m. This data was called a crossed module bundle, from which crossed module bundle gerbes were constructed in [2], see also [46,47]. Recall that an Abelian bundle (p + 1)-gerbe over a manifold X can be constructed from the notion of an Abelian bundle JHEP04(2015)087 p-gerbe, by considering a surjective submersion Y → X together with Abelian bundle pgerbes over Y × X Y . The analogous construction for crossed module bundle gerbes starts from a crossed module bundle. If N is not a group, then a Deligne 1-cocycle describes a 2group principal bundle, which is a special form of a groupoid principal bundle. Considering 2-group principal bundles over Y × X Y yields then to 2-group bundle gerbes or the principal 2-bundles described by Deligne 2-cocycles.
A degree-2 Deligne cocycle describes a semistrict principal 2-bundle with connective structure. Again, gauge equivalence is captured by the cohomology. To study such Deligne 2-cocycles further, it is useful to introduce the curvature 3-form, apart from the 2-form fake curvature (5.22) that vanishes; see also Proposition 5.3. (5.36)

Application: Penrose-Ward transform
As an application of the theory of principal 2-bundles which we have developed in the previous sections, we now show how to generalise the results of [9]. Specifically, [9] established a Penrose-Ward transform that yields a bijection between holomorphic principal 2-bundles with strict structure 2-group over a twistor space and non-Abelian self-dual tensor fields on six-dimensional flat space-time. We can now replace the strict principal 2-bundles by semistrict ones in this construction. In the following, we denote by O X the sheaf of holomorphic functions and by Ω p X the sheaf of holomorphic differential p-forms on a complex (super)manifold X.
The starting point is the chiral superspace M 6|8n := C 6|8n with n = 0, 1, 2. This space can be equipped with the coordinates (x AB , η A I ), where x AB = −x BA with A, B, . . . = 1, . . . , 4 are the usual Graßmann-even coordinates in spinor notation, η A I are the Graßmannodd coordinates and I, J, . . . = 1, . . . , 2n are the R-symmetry indices. We may raise and lower the spinor indices using the Levi-Civita symbol, that is,

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Remark 6.1. In our subsequent discussion, we shall always choose the standard Stein coverÛ = {Û a } on the twistor space P 6|2n → P 3 (generated by the standard Stein cover on P 3 ) and the induced cover U := {U a := π −1 1 (U a )} on the correspondence space F 9|8n , respectively.

Penrose-Ward transform
To formulate the Penrose-Ward transform, we first need to introduce a few basic notions. In particular, we will need the sheaf of holomorphic relative differential p-forms, denoted by Ω p π 1 , on F 9|8n along the fibration π 1 : F 9|8n → P 6|2n . It is defined by the short exact sequence In addition, if pr π 1 : Ω p F 9|8n → Ω p π 1 denotes the quotient mapping, we can define the relative exterior derivative d π 1 by setting d π 1 := pr π 1 • d : Ω p π 1 → Ω p+1 π 1 , (6.11) where d denotes the usual holomorphic exterior derivative on the correspondence space. In the local coordinates (x AB , η A I λ A ) on F 9|8n , d π 1 is presented in terms of the vector fields of the twistor distribution (6.5); see also (6.21) below. The relative exterior derivative characterises the so-called relative holomorphic de Rham complex, which is the complex that is given in terms of an injective resolution of the topological inverse π −1 1 O P 6|2n of the sheaf O P 6|2n on the correspondence space F 9|8n : Note that π −1 1 O P 6|2n consists of those holomorphic functions that are locally constant along the fibres of π 1 : F 9|8n → P 6|2n .
Next, let Φ be a holomorphic semistrict principal 2-bundles on the correspondence space F 9|8n , with G = (M, N ) as its semistrict structure 2-group. As before, we denote the 2-term L ∞ -algebra associated with G by v µ 1 −→ w, where w := T ide M and v := ker(t) ⊆ T id ide N . The bundle Φ is described by holomorphic G -valued transition functions ({m ab }, {n abc }) relative to the cover U .
As we shall see momentarily, the Penrose-Ward transform will be based on so-called relative degree-2 Deligne cohomology. For this reason, we wish to equip Φ with a relative connective structure and study its behaviour under equivalence transformations. Concretely, Φ is then described by a degree-2 Deligne cocycle 17 Here, the subscript 'π 1 ' indicates that these are relative differential forms. For instance, the Λ ab and A a take values in Ω 1 π 1 ⊗ v and Ω 1 π 1 ⊗ w, respectively, while 17 To simplify notation, we shall suppress the superscript 0 in the Λ-part of the cocycle here and in the following.

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the B a take values in Ω 2 π 1 ⊗ v. In addition, we call the relative connective structure flat whenever, apart from the vanishing of 2-form fake curvature, inherent to 2-degree Deligne cocycles, also the 3-form curvature vanishes The final ingredient we shall need is a holomorphic semistrict principal 2-bundleΦ on P 6|2n with G = (M, N ) as its semistrict structure 2-group. The bundleΦ is described by holomorphic G -valued transition functions ({m ab }, {n abc }) relative to the coverÛ. Following Manin [55],Φ will be called M 6|8n -trivial whenever it is holomorphically trivial on x = π 1 (π −1 2 (x)) → P 6|2n for all x ∈ M 6|8n ; see also Definition 3.19. Then we have the following result. Proposition 6.2. Consider π 1 : F 9|8n → P 6|2n of the double fibration (6.6). There is a bijection between (i) equivalence classes of topologically trivial M 6|8n -trivial holomorphic semistrict principal 2-bundles on P 6|2n and (ii) equivalence classes of holomorphically trivial semistrict principal 2-bundles on F 9|8n equipped with a relative connective structure which is globally flat.

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where the A a define a globally defined w-valued relative 1-form A π 1 ∈ H 0 (F 9|8n , Ω 1 π 1 ⊗ w), that is, A a = A π 1 | U a and A a = A b on U a ∩ U b . Thus, using (5.35) with Λ a , we see that the degree-2 Deligne cocycle (6.17) is cohomologous to where the B a yield a globally defined v-valued relative 2-form B π 1 ∈ H 0 (F 9|8n , Ω 2 Altogether, we have obtained a holomorphically trivial semistrict principal 2-bundle Φ on the correspondence space, equipped with a globally defined relative connective structure represented by (A π 1 , B π 2 ). As this relative connective structure is pure gauge, its curvatures necessarily vanish, and, therefore, the relative connective structure is globally flat.
(ii) → (i) Conversely, starting from a holomorphically trivial semistrict principal 2bundle Φ on the correspondence space represented by a relative degree-2 Deligne cocycle of the form (6.20) with a relative connective structure that is flat, we can use a generalised Poincaré lemma [56] to find a relative degree-2 Deligne cochain to transform (6.20) into a cocycle of the form (6.17). This cocycle descends down to twistor space to a relative degree-2 Deligne cocycle of the form (6.16).
Note that there are equivalence transformations acting on the ingredients of this construction. For instance, constructing the degree-2 Deligne cochains explicitly that mediate between the different degree-2 Deligne cocycles amounts to solving Riemann-Hilbert problems whose solutions are not unique. We shall come back to this in Remark 6.6.
Remark 6.4. Note that differential 1-, 2-and 3-forms α, β, and γ on chiral superspace M 6|8n have components where γ A B I C is traceless over the AB indices. By virtue of Lemma 6.3, we realise that all of these components for the 1-and 2-forms and some of these components for the 3-form appear in the expansion of relative 1-, 2-and 3-forms α π 1 , β π 1 , and γ π 1 on the correspondence F 9|8n . Note further that the components (γ AB , γ AB ) represent the self-dual and anti-self dual parts of a Graßmann-even differential 3-form γ on M 6|0 .
These considerations then enable us to prove the following Penrose-Ward transform.  Before proving the theorem, let us make a few comments. The fields ψ I A are Graßmannodd spinor fields while the fields φ IJ are Graßmann-even scalar fields. The condition H AB = 0 implies that the Graßmann-even part of the 3-form H is self-dual, cf. Remark 6.4. Altogether, (H AB , ψ I A , φ IJ ) constitutes an N = (n, 0) tensor multiplet for n = 0, 1, 2. Note that only for n = 2, the condition φ IJ Ω IJ = 0 arises, so that we always find the correct number of scalar fields. See also Saemann & Wolf [9][10][11] for more details on this point.
(6.26) Upon using (6.21) and the expansions given in Lemma 6.3, we arrive at the constraint equations (6.24) and (6.25) after a few algebraic manipulations.

(6.27)
Such Deligne 1-cochains are therefore described by p(x, η), Λ AB (x, η), and Λ I A (x, η) which themselves form a Deligne 1-cochain encoding an equivalence relation between Deligne 2cocycles on the chiral superspace M 6|8n . The gauge transformations are then simply of the form given in Proposition 5.9.

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Definition A.2. A Z-graded coalgebra is a Z-graded vector space L = ⊕ p∈Z L p endowed with a coproduct ∆ : A → A ⊗ A of degree 0 such that (1 ⊗ ∆) • ∆ = (∆ ⊗ 1) • ∆. A coderivation of degree k on a coalgebra C is a linear map D : C → C of degree k such that ∆ • D = (1 ⊗ D + D ⊗ 1) • ∆. A differential graded coalgebra is a graded coalgebra endowed with a coderivation D of degree 1 such that D • D = 0.
On the other hand, given a commutative differential graded coalgebra, we can derive a corresponding L ∞ -algebra. Altogether, we arrive at the following proposition.
Proposition A.3. An L ∞ -algebra is equivalent to a commutative differential graded coalgebra.
Instead of working with coalgebras, it is usually more convenient to work directly with differential graded algebras. Assuming that the vector subspaces L p ⊆ L are finite dimensional, we can consider the dual complex It is straightforward to verify the CE(L) is indeed a differential graded algebra.
The Chevalley-Eilenberg algebra of a Lie algebra g is a differential graded algebra that encodes the Lie bracket via the equation where theτ i form a basis of the dual g ∨ of g and f k ij are the structure constants of g with respect to the dual basis (τ i ) withτ i (τ j ) = δ i j . Evaluated at an element a ∈ the Maurer-Cartan equation of the differential graded algebra. This equation can be generalised to the case of L ∞ -algebras.

JHEP04(2015)087 B Groupoid bundles
In this appendix, we present the parameterisation of a functor from the category of supermanifolds to the category of groupoid bundles with preferred section, completing the discussion of Deligne 1-cocycles with values in a semistrict Lie 2-group. Such cocycles arise from functors between theČech groupoid and the Lie 2-group (regarded as a monoidal category). Our discussion follows closely the lines of that in section 4, and we shall therefore be concise.
We where n 0 := n(θ 0 ) = n(0, θ 0 ). Such a coboundary, and therefore the whole functor under consideration, is parameterised by a β ∈ v = ker(t) ⊆ T id ide N according to and we conclude that m 0 = id e + s(β)θ 0 . Equivalence relations on such descent data are described by degree-1 Deligne coboundaries according to To compare this coboundary with n 0 , we have to bring it to the formñ 0 =ñ(0, θ 0 ) by a modification transformation. Note that the coboundary relatioñ is invariant under the modification transformation
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