Generalized higher gauge theory

We study a generalization of higher gauge theory which makes use of generalized geometry and seems to be closely related to double field theory. The local kinematical data of this theory is captured by morphisms of graded manifolds between the canonical exact Courant Lie 2-algebroid T M ⊕ T∗M over some manifold M and a semistrict gauge Lie 2-algebra. We discuss generalized curvatures and infinitesimal gauge transformations. Finite gauge transformation as well as global kinematical data are then obtained from principal 2-bundles over 2-spaces. As dynamical principle, we consider first the canonical Chern-Simons action for such a gauge theory. We then show that a previously proposed 3-Lie algebra model for the six-dimensional (2,0) theory is very naturally interpreted as a generalized higher gauge theory.


Introduction and results
Higher gauge theory describes the parallel transport of extended objects transforming under local internal symmetries. There are well-known no-go theorems stating that in a naive setting, the internal symmetry group has to be abelian for objects with positive dimension. To avoid these theorems, one has to categorify the ingredients of usual gauge theory, see e.g. [1] for details. This leads in particular to categorified structure groups, known as n-groups, as well as categorified notions of principal bundles known as principal n-bundles.
One severe open problem in higher gauge theory is the lack of non-trivial examples of non-abelian principal n-bundles with connection. For example, one would expect categorified analogues of non-abelian monopoles and instantons to exist. Although higher analogues of the twistor descriptions of monopoles and instantons have been constructed [2][3][4], the known solutions, e.g. those of [5], do not quite fit the picture. 1 This lack of examples presents an obstacle to both mathematical as well as physical progress in the study of higher gauge theory. It is therefore important to find generalized formulations which allow for interesting examples. In this paper, we study the case in which 1 If higher gauge theory is to describe the parallel transport of some extended objects, then a condition needs to be imposed on the curvature of the principal n-bundle to ensure that the parallel transport is invariant under reparameterizations. While the solutions of [5] do not directly satisfy this curvature condition, one could argue that at least in the case of the self-dual strings considered in [5], the underlying parallel transport of strings is trivial and the fake curvature condition becomes physically irrelevant. also the base manifold is categorified to what has been called a 2-space [6]: a category internal to the category of smooth manifolds.
In recent developments in string theory, there are many pointers towards the necessity of using 2-spaces instead of ordinary space-time manifolds, in particular in relation with generalized geometry and double field theory. In both contexts, it is usually the exact Courant algebroid T M ⊕ T * M over some manifold M , which is used to give expressions a coordinate-invariant meaning, see e.g. [7][8][9]. It is therefore only natural to ask whether a definition of gauge theory involving this algebroid has some interesting features.
Recall that the local kinematical data of ordinary gauge theory over some manifold M can be described by a morphism of graded algebras between the Chevalley-Eilenberg algebra of the gauge Lie algebra and the Weil algebra of the manifold M , which is the Chevalley-Eilenberg algebra of the tangent Lie algebroid T M . For higher gauge theory, the domain of this morphism is extended to the Chevalley-Eilenberg algebra of some L ∞algebra. In this paper, we also generalize the range of this morphism to the Chevalley-Eilenberg algebra of the Courant algebroid, cf. figure 1. The latter should more properly be regarded as a symplectic Lie 2-algebroid, and thus we arrive at a notion of gauge theory over the 2-space (T * M ⇒ M ).
We discuss in detail the case where the gauge L ∞ -algebra consists of two terms, corresponding to a semistrict Lie 2-algebra. In particular, we derive the form of the gauge potential and its curvature, which are encoded in the morphism of graded algebras and its failure to be a morphism of differential graded algebras. We also give the relevant formulas for infinitesimal gauge transformations. As we show, these results can also be obtained from the homotopy Maurer-Cartan equations of an L ∞ -algebra consisting of the tensor product of the gauge L ∞ -algebra with the Weil algebra of T * M .
To glue together local kinematical data to global ones, we need a generalized principal 2-bundle structure as well as finite gauge transformations. We find both by considering principal 2-bundles over the 2-space T * M . We thus arrive at an explicit formulation of the first generalized higher Deligne cohomology class, encoding equivalence classes of these higher bundles with connection.
In a second part, we discuss two possible dynamical principles for the generalized higher connections. The first one is a Chern-Simons action, which is obtained via a straightforward generalization of the AKSZ procedure. The second one is a previously proposed set of equations for a 3-Lie algebra 2 -valued (2, 0)-tensor supermultiplet in six dimensions [10].

JHEP04(2016)032
We show that these equations find a very natural interpretation within generalized higher gauge theory. In particular, the 3-Lie algebra valued vector field featuring crucially in the equations is part of a generalized higher connection.
Among the open questions we intend to study in future work are the following. First, an additional gauge algebra-valued vector field seems to be desirable in many open questions related to the six-dimensional (2,0)-theory. It would be interesting to see if such problems can be addressed within our framework. Second, the Courant algebroid appears in double field theory after imposing a section condition. One might therefore want to formulate a full double gauge theory, related to ours only after the section condition is imposed. Such a double gauge theory might have interesting applications in effectively describing string theory dualities. Third, it remains to be seen whether we can write down six-dimensional maximally superconformal gauge equations which are less restrictive than those obtained in [10], using generalized higher gauge theory. Finally, as stated above, it would be most interesting to extend the twistor descriptions of [2][3][4] to generalized higher gauge theory and to explore the possibility of genuinely non-trivial and non-abelian generalized principal 2-bundles with connection.

Kinematical description
We begin by reviewing the notion of NQ-manifolds and their relation to L ∞ -algebras. We use this language to describe ordinary gauge theory in terms of morphisms of graded manifolds and show how this extends to higher gauge theory, following [11][12][13][14][15][16]. This formulation naturally allows for a generalization to gauge theory involving the exact Courant algebroid T M ⊕ T * M . More explicitly, we can think of an N-manifold M as a tower of fibrations

NQ-manifolds
where M 0 = M is a manifold and M i for i ≥ 1 are linear spaces with coordinates of degree i, generating the structure sheaf. For more details on this, see e.g. [17]. A morphism of Nmanifolds is then a morphism of graded manifolds φ : In more detail, we have a map φ 0 : M → N between the underlying manifolds and a degree-preserving map φ * : O N → O M between the structure sheaves, which restricts to the pullback along φ 0 on the sheaf of smooth function on N , O N ⊂ O N . Note that for higher degrees, φ * is completely defined by its image on the local coordinates that generate O N . An NQ-manifold is now an N-manifold (M, O M ) together with a homological vector field Q, that is, a vector field of degree 1 squaring to zero: Q 2 = 0. The algebra of functions C ∞ (M) on M given by global sections of O M together with Q now forms a differential graded algebra. A morphism of NQ-manifolds is then a morphism φ between NQ-manifolds (M, O M , Q M ) and (N,

JHEP04(2016)032
Physicists may be familiar with NQ-manifolds from BRST quantization, where the coordinate degree and Q correspond to the ghost number and the BRST charge, respectively.
A basic example of an NQ-manifold is given by T [1]M , where we always use [n] to denote a shift of the degree of some linear space (often the fibers of a vector bundle) by n. On T [1]M , we have coordinates (x µ , ξ µ ) on the base and the fibers of degree 0 and 1 respectively, i.e. we have an N-manifold concentrated in the lowest two degrees. Note that the algebra of functions on T which can be "twisted," e.g., tõ [17]. We shall work mostly with the case T = 0. Altogether, we arrive at an NQ-manifold concentrated in degrees 0 to 2. This NQ-manifold is the one underlying the exact Courant algebroid T M ⊕ T * M , and we will come back to this point later. Also, this example is part of a larger class of NQ-manifolds given by V n := T * [n] T [1]M containing the Vinogradov algebroids T M ⊕ ∧ n−1 T * M . For more details, see e.g. [18].
Another important example of NQ-manifolds is that of a grade-shifted Lie algebra g [1] with basis τ α of degree 0 and coordinates w α of degree 1. The algebra of functions is given by ∧ • g * ∼ = Sym(g[1] * ) and Q is necessarily of the form where f α βγ are the structure constants of the Lie algebra g. The condition Q 2 = 0 directly translates to the Jacobi identity. This alternative description of a (finite-dimensional) Lie algebra is the well-known Chevalley-Eilenberg algebra CE(g [1]) of g and we can thus think of a Lie algebra as an NQ-manifold concentrated in degree 1. Analogously, we will refer to the differential graded algebra consisting of the algebra of functions on an NQ-manifold M together with the differential given by the homological vector field as the Chevalley-Eilenberg algebra CE(M) of M.
We can readily extend the last example, replacing the shifted Lie algebra by some shifted graded vector space, which we also denote by g [1]. On the latter, we introduce a JHEP04(2016)032 basis τ A of degree 0 and coordinates Z A of degree |A| ∈ N in g [1]. The vector field Q is then of the form The minus signs and normalizations are chosen for convenience.
We now also introduce a basisτ A on the unshifted g, where we absorb all grading in the basis instead of the coordinates. Thus,τ A has degree |A| − 1. The structure constants m A B 1 ...B k can then be used to define the following graded antisymmetric, k-ary brackets µ k on g of degree k − 2: For an NQ-manifold concentrated in degrees 1 to n, the condition Q 2 = 0 amounts to the homotopy Jacobi relations of an n-term L ∞ -algebra with higher products µ k , cf. [19,20]. Such n-term L ∞ -algebras are expected to be categorically equivalent to semistrict Lie n-algebras.

JHEP04(2016)032
The natural notion of inner product on an L ∞ -algebra arises from an additional symplectic structure on the underlying NQ-manifold. A symplectic NQ-manifold of degree n is an NQ-manifold M = (M, O M , Q, ω) endowed with a closed, non-degenerate 2-form ω of degree 3 n satisfying L Q ω = 0. If the degree of ω is odd, such symplectic NQ-manifolds are also known as QP-manifolds [21] or P-manifolds [22]. In the general case, symplectic NQ-manifolds of degree n are also called Σ n -manifolds [23].
We are mostly interested in the NQ-manifold V 2 becomes a symplectic NQ-manifold of degree 2: we have where Q V 2 is the homological vector field (2.2). The symplectic structure (2.9) is also compatible with the twisted homological vector field (2.3). As shown in [17], the data specifying a symplectic NQ-structure on T * where ω AB is the inverse matrix to ω AB . With our choice of symplectic structure (2.9), we have For simplicity, we will refer to both the symplectic NQ-manifolds V 2 = T * [2]T [1]M and the vector bundle T M ⊕ T * M with Courant algebroid structure as Courant algebroid. Note that the exact Courant algebroid T M ⊕ T * M features prominently in generalized geometry and double field theory. We therefore expect our following constructions to be relevant in this context.

Gauge connections as morphisms of N-manifolds
In ordinary gauge theory, we consider connections on principal G-bundles over some manifold M and encode them locally as 1-forms A taking values in a Lie algebra g = Lie(G).

JHEP04(2016)032
The curvature of A is F := dA + 1 2 [A, A] and gauge transformations are parameterized by G-valued functions g and act on A via A →Ã = g −1 Ag + g −1 dg. At infinitesimal level, these are given by A → A + δA, where δA = dλ + [A, λ] with λ a g-valued function.
Let us now reformulate the local description of gauge theory using morphisms of Nmanifolds. As discussed in section 2.1, differential forms can be encoded as functions on the NQ-manifold T [1]M . In terms of coordinates (x µ , ξ µ ) of degree 0 and 1, respectively, the de Rham differential corresponds to the homological vector field Q T [1]M = ξ µ ∂ ∂x µ . We also regard g as an NQ-manifold g [1] with coordinates w α of degree 1 and Q g = − 1 2 f α βγ w β w γ ∂ ∂w α , cf. again section 2.1.
A local connection 1-form A is then encoded in a morphism of N-manifolds a between these two NQ-manifolds .
Recall that it suffices to define the action of a on the local coordinates of g [1], so we define where (τ α ) is a basis on g. The curvature F of A then describes the failure of a to be a morphism of NQ-manifolds: (2.14) Indeed, we have where µ 2 denotes the Lie bracket on g.
Gauge transformations between a andã are encoded in flat homotopies between these, that is, morphisms [1] which are flat along the additional direction [24]. More precisely, given coordinates r along [0, 1] and ρ on T [1][0, 1], we havê Note thatâ defines a gauge potential and a curvaturê . For the homotopŷ a to be flat, we require F ⊥ = 0, which implies that

Higher gauge connections
We can readily extend the picture of the previous section to the case of higher gauge theory.
Here, we simply replace the gauge Lie algebra by a general L ∞ -algebra g. 4 The morphism of N-manifolds a : now also contains forms of higher degree. Similarly, the curvature, which is again given by the failure of a to be a morphism of NQ-manifolds, leads to higher curvature forms.
As an instructive example let us look at the 2-term L ∞ -algebra W ← V [1] introduced before in section 2.1. The image of the pullback morphism a * on the coordinates w α and v a of degree 1 and 2 on the shifted vector space W where in addition to the W -valued 1-form potential A we now also have a V [1]-valued 2-form potential B. We combine both into the 2-connection With Q g from (2.7), we compute the curvature components (2.24) which we combine into the 2-curvature Again, the infinitesimal gauge transformations between a andã are encoded in homotopiesâ : that are flat in the extra homotopy direction. We use coordinates (x, ξ, r, ρ) on T [1](M × [0, 1]) and we haveâ| r=0 = a as well asâ| r=1 =ã. Thenâ defines a gauge potentials as before, that is, (2.26) 4 In principle, we could also allow for L∞-algebroids, which would lead us to higher gauged sigma models.

JHEP04(2016)032
Using the extended vector fieldQ T [1]M = ξ µ ∂ ∂x µ + ρ ∂ ∂r , we calculate the curvature defined byâ along the additional direction to bê As before, the infinitesimal gauge transformations are encoded in the flat homotopies for which the above curvature in the directions including ρ vanishes. This leads to the transformations which are parameterized by two infinitesimal gauge parameters: the W -valued function A r (x, 0) and a V [1]-valued 1-formB νr (x, 0). We thus obtain the infinitesimal gauge transformations of semistrict higher gauge theory as found e.g. in [4]. 5 Putting µ 3 = 0, we obtain the infinitesimal gauge transformations of strict higher gauge theory, which can be integrated as done in [26]. SettingB, µ 1 and µ 3 to zero reduces the transformation back to the case of ordinary gauge theory.

Local description of generalized higher gauge theory
We now come to our extension of higher gauge theory to generalized higher gauge theory.
To this end, we replace the domain of the morphism of N-manifolds, which has been T [1]M so far, by the Courant algebroid V 2 = T * [2]T [1]M with coordinates x µ , ξ M , p µ of degree 0, 1 and 2, respectively, and homological vector field Q V 2 = ξ µ ∂ ∂x µ + p µ ∂ ∂ξµ , see section 2.1. Generalized higher gauge theory is thus given by a morphism of N-manifolds a : V 2 → g [1], where g is an arbitrary L ∞ -algebra. We again focus on the example where g is a 2-term L ∞ -algebra W ← V [1] and we introduce a basis (τ α , t a ) and coordinates (w α , v a ) of degree 1 and 2, respectively, on g [1]. The homological vector field Q g is given in (2.7). The images of the coordinates of g [1] under the morphism a are where A = A M ξ M = A µ ξ µ + A µ ξ µ can now be regarded as the sum of a 1-form and a vector field, which are both W -valued. Similarly, B consists of a 2-form, a bivector, a tensor of 5 An alternative approach to finite gauge transformations of semistrict higher gauge theory is found in [25].

JHEP04(2016)032
degree (1,1) and a vector field, all taking values in V [1]. We combine all these into the generalized 2-connection The generalized 2-curvature F is again obtained from the failure of a to be a morphism of NQ-manifolds, and splits into components according to (2.31) The components of F are computed to be (2.32) Flat homotopies between morphism of N-manifolds a andã give the generalized higher gauge transformations. These are encoded in morphismsâ from T * [2]T [1]M × T [1]I to g [1], where we introduce additional coordinates (r, ρ) of degrees (0, 1) in the new direction and the vector field Q V 2 is amended tô The morphismâ then defines gauge potentialŝ The requirement that these terms vanish yields the infinitesimal gauge transformations which are parameterized by a W -valued functionÂ r , as well as a 1-formB µr and a vector fieldB µ r , both taking values in V [1]. Note that generalized higher gauge theory contains higher gauge theory. In particular, if we put the fields A µ , B µ ν , B µν and B µ to zero, we obtain the usual 2-connection.
Analogously, we can restrict the gauge transformations.

Equivalent description from Maurer-Cartan equations
The above gauge potentials, field strengths and gauge transformations can also be derived in a different manner as we explain now, following a similar discussion as that in [4]. First, note that functions C ∞ (V 2 ) on the Courant algebra V 2 form a differential graded algebra with differential Q V 2 . It is well known that the tensor product of such a differential graded algebra with an L ∞ -algebra carries a natural L ∞ -algebra structure. For us, the relevant L ∞ -algebra is g = (W ← V [1]) and the tensor productL = C ∞ (V 2 ) ⊗ g carries the higher JHEP04(2016)032 products 37) where µ i are the higher products in g, f i ∈ C ∞ (V 2 ) and i ∈ g, deg denotes the degree and χ = ±1 is the Koszul sign arising from moving functions on V 2 past elements of g. Note that the total degree of an element f ⊗ in C ∞ (V 2 ) ⊗ g is deg(f ) − deg( ) and we truncate C ∞ (V 2 ) ⊗ g to non-negative degrees.
Recall that an element φ of an L ∞ -algebraL is called a Maurer-Cartan element, if it satisfies the homotopy Maurer-Cartan equation This equations is invariant under infinitesimal gauge symmetries parameterized by an element λ ∈L of degree 0 according to cf. [4,19,27]. Equation (2.38) states that the higher curvature vanishes and therefore, it can be used to identify the correct notion of curvature. Equation (2.39) then gives the appropriate infinitesimal gauge transformations. Following [4], we now consider an element φ of degree 1 in L = C ∞ (V 2 ) ⊗ g with g = W ← V [1], which we can identify with the generalized 2-connection of equation (2.30). That is, we write φ = A α µ ξ µ ⊗τ α + A µα ξ µ ⊗τ α + 1 2 B a µν ξ µ ξ ν ⊗t a + B µ νa ξ µ ξ ν ⊗t a + 1 2 B µνa ξ µ ξ ν ⊗t a + B µa p µ ⊗t a . (2.40) The homotopy Maurer-Cartan equations (2.38) defining the various curvatures are then −F = 0, where F is the generalized 2-curvature (2.32) as found before. Infinitesimal gauge transformations are parameterized by whereÂ r takes values in W andB µr andB µ r are V [1]-valued. Their general action (2.39) amounts to (2.36) for a generalized 2-connection. Altogether, we recovered the gauge potential, the curvatures and the infinitesimal gauge transformations of generalized higher gauge theory.

Global description
Finite (small) gauge transformations can be obtained from the infinitesimal ones described above by using the integration method of [24], which follows an idea of [28]. The local JHEP04(2016)032 kinematical data can then be glued together on overlaps of patches of a cover by these finite gauge transformations. One disadvantage of this approach is the following. In certain simple cases as e.g. that of crossed modules of Lie algebras, there is a straightforward integration available, as e.g. that to a crossed module of Lie groups. The integration of [28], however, usually yields a different result, which is only categorically equivalent to that of the straightforward integration.
Here, we follow a slightly different route, starting from a description of the generalized principal 2-bundle without connection in terms ofČech cochains. Based on this description, the infinitesimal gauge transformations (2.36) are then readily integrated. We shall restrict ourselves to the case of strict Lie 2-algebras, which will simplify our discussion drastically.
First, recall that a crossed module of Lie groups H ∂ − − → G is a pair of Lie groups (H, G) together with a Lie group homomorphism ∂ : H → G as well as an action of G on H by automorphisms satisfying for all g ∈ G and h, h 1 , h 2 ∈ H. The first equation is simply equivariance of ∂, while the second relation is known as the Peiffer identity. Moreover, such a crossed module of Lie groups is categorically equivalent to a strict Lie 2-group 6 as follows, cf. [29]. The underlying category is given by the groupoid G H ⇒ G with structure maps The monoidal product is given by Recall that a 2-space X is a category internal to Man ∞ and therefore consists of a manifold X 0 of objects, a manifold X 1 of morphisms as well as smooth maps s, t : X 1 ⇒ X 0 and id : X 0 → X 1 as well as a composition map • : X 1 × X 0 X 1 such that the usual axioms for the structure map in a category are satisfied, cf. [6]. In the case of T * M ⇒ M , we have the structure maps 7 Given a cover U = a U a of M , we have an obvious induced 2-cover by the 2-space T * U ⇒ U . This 2-cover gives rise to aČech double groupoiď That is a monoidal category internal to the category of mooth manifolds in which the product is associative and unital and in which objects and morphisms are (strictly) invertible. 7 Such a category in which source and target maps agree, is called skeletal. This property will simplify some aspects of the subsequent discussion.

JHEP04(2016)032
Given a crossed module of Lie groups, H ∂ − − → G, there is also a natural double groupoid corresponding to the delooping of the strict Lie 2-group G = (G H ⇒ G) as a double groupoid BG := (G H ⇒ G) ⇒ ( * ⇒ * ) , (2.46) where * denotes the one-element or singleton set. A "generalized" principal 2-bundle on the 2-space T * M ⇒ M is naturally defined as a lax (double) functor fromČ to BG, see e.g. [4] for a very detailed related discussion. Explicitly, such a lax double functor consists of an ordinary functor together with a double natural isomorphism The ordinary functor (Φ ab ) is encoded in maps (g ab , h ab ) : T * U [2] → G H, where necessarily (g ab ) is independent of the fibers in T * U [2] and ∂(h ab ) = 1 G . The double natural isomorphism (Ξ abc ) gives rise to maps (g abc , h abc ) : where the map g abc is fully fixed by the source of the double natural isomorphism. The fact that (Φ ab ) and (Ξ abc ) form a lax double functor yields the equations (g ac (x), h ac (x, p 1 + p 2 )) • (g abc (x), h abc (x)) = (g abc (x), h abc (x)) • (g ab (x), h ab (x, p 1 )) ⊗ (g bc (x), h bc (x, p 2 )) , (2.50) where x ∈ U [2] and p 1,2 ∈ T * U [2] . Using relations (2.43), this is readily translated into (2.51) We will refer to a set of maps (g ab , h ab , h abc ) satisfying the above equations as generalized 1-cocycles. As a consistency check of our derivation, we can imagine replacing the 2-space T * M ⇒ M by a discrete 2-space M ⇒ M . In this case, the cocycle relations (2.51) restrict to the usual ones of a principal 2-bundle with strict structure 2-group as found e.g. in [6]. Also, note that the cocycle relations (2.51) were also derived in [6] using a different approach.
Analogously, we can now derive coboundary relations as double natural isomorphisms between the lax double functors. These give rise to isomorphism classes of generalized principal 2-bundles. Such a cocycle consists of maps (γ a , χ a ) : T * U → G H, where (γ a ) is independent of the fibers in T * U and ∂(χ a ) = 1 G as well as maps (γ ab , χ ab ) : U [2] → G H, where (γ ab ) is again fully fixed. Instead of listing the general coboundary relations, let us JHEP04(2016)032 just state that a trivial generalized principal 2-bundle has generalized 1-cocycles (2.52) To endow the principal 2-bundle with connection, we now put local kinematic data of generalized higher gauge theory as discussed above on each patch. On overlaps of patches U a ∩ U b , the components are then glued together via gauge transformations. The latter arise from integrating the infinitesimal gauge transformations (2.36) as done in [26] and lead to (2.53b) Here, the g ab are part of the generalized 1-cocycle and the (Λ ab,µ ξ µ ) and (Λ µ ab ξ µ ) are additional Lie(H)-valued 1-forms and vector fields on U [2] satisfying the cocycle condition Λ ac,µ ξ µ = Λ bc,µ + g −1 bc Λ ab,µ − g −1 Finite gauge transformations are also readily read off from (2.53).
Recall that principal bundles with connection and their isomorphisms are captured by the first (non-abelian) Deligne cohomology class. Our formulas for generalized principal 2-bundles with 2-connections and their gauge transformations thus gives a complete description of the first generalized (non-abelian) Deligne cohomology class.

Dynamics
Having fixed the kinematical background, we are now in a position to consider dynamical principles.

Weil algebra and higher Chern-Simons theory
A very natural action functional arising directly within the framework of NQ-manifolds is that of (higher) Chern-Simons theory, constructed via the AKSZ mechanism. The methods

JHEP04(2016)032
from the original paper [21] can be easily generalized and applied to the higher, L ∞ -algebra, scenario, see e.g. [12][13][14][15][16]. In the following we give a quick review of the necessary tools for ordinary higher gauge theory, referring to the references for any further details.
Recall from section 2.2 that the map a : T [1]M → g [1] is not a morphism of NQmanifolds. It can, however, be lifted to a map f : [1], such that where (Z A ) are coordinates on g [1] and d g is just the exterior differential on g [1] of weight 1. Note that we have the homological vector field on T [1]g [1]. It acts on coordinates of T [1]g [1] as follows: With respect to Q W , the map f is now indeed a morphism of NQ-manifolds. The algebra of functions C ∞ (T [1]g [1]) ∼ = Sym(g[1] * ⊕ g[2] * ) together with Q W is known as the Weil algebra W (g [1]) of g [1] and for g an ordinary Lie algebra, this reproduces the conventional definition of the Weil algebra.
It is reasonable to assume that we are interested in objects that are invariant under the action of the gauge L ∞ -algebra g: these are called invariant polynomials, inv(g [1]), and they are described as follows. The Weil algebra fits into the sequence inv(g [1]) → W(g [1]) where π W is the obvious projection by pulling back along the embedding g [1] →T [1]g [1] as zero sections of the vector bundle. The invariant polynomials inv(g [1]) are then elements in W (g [1]) that sit completely in Sym(g * [2]) and are closed under Q W . In other words, for p ∈ inv(g [1]), we have that π W (p) = 0 and, using d 2 g = 0, Q W p = Q g p = 0. It is clear that contraction with a generic element X ∈ g [1] vanishes, so that also L X p = 0. It is therefore these types of objects that are relevant for constructing topological invariants or even physical models.
In the case of the AKSZ mechanism, we are interested in the invariant polynomial corresponding to the symplectic structure on g [1], This symplectic structure has a local symplectic potential where ε = A |Z A |Z A ι ∂ A is the Euler vector field, ∂ A := ∂ ∂Z A and |Z A | indicates the degree of the coordinate Z A . We saw above that invariant polynomials on T [1]g [1] have to be of the form p = k p α 1 ···α k d g ξ α 1 ∧ · · · ∧ d g ξ α k , so that the invariant part of the lift of ω to the bundle isω = 1 2 d g ξ α ∧ω αβ d g ξ β . One can therefore ask whether an object χ exists, such that Q W χ =ω, and the projection to the Chevalley-Eilenberg cohomology on JHEP04(2016)032 g [1] gives a cocycle κ on CE(g [1]), i.e. χ| CE (g [1]) = κ with Q g κ = 0. Such an object χ is called a transgression element. It allows to map the cohomology of T [1]g [1] onto that of [1] get pulled back to exact objects in T [1]M , in particular df * (χ) = f * (ω).
A particularly interesting cocycle is the Hamiltonian S of Q g , which satisfies for any ψ ∈ C ∞ (g [1]), where {−, −} is the Poisson bracket induced by the symplectic structure ω and the relation Q 2 = 0 amounts to {S, S} = 0. The transgression element for this cocycle will be called a Chern-Simons element, and can be found in the following way.
Starting with the lift of α to the tangent bundle,α = B |Z B |Z B ω BC Q W Z C , we see by how much Q Wα fails to be in inv(g [1]): (3.7) Combining this with where in the first equality we used that {S, S} = Q g S = 0, we obtain the Chern-Simons element as χ = 1 n + 1 (α − S) . (3.9) The Lagrangian for higher Chern-Simons theory 8 is now simply the pullback of the Chern-Simons element of the gauge L ∞ -algebra along f : where we identified polynomials in the Z A of degree n with n-forms. The field content consists of an n-connection encoded in the morphism of NQ-manifolds a which was lifted to f . The equations of motion of (3.10) are simply the homotopy Maurer-Cartan equations (2.38) yielding a flat higher connection. For details on such models see again [32] and references therein.

Generalized higher Chern-Simons theory
Let us now apply the AKSZ construction to obtain the Chern-Simons form of generalized higher gauge theory. As discussed in the previous section, we will have to pull back the Chern-Simons element on a Lie 2-algebra g = W

JHEP04(2016)032
which is a lift of the map a : T * [2]T [1]R 4 −→ g [1]. Note that in this setup, the gauge connection has significantly more components than in ordinary higher gauge theory. Recall from section 2.4 that a generalized 2-connection is of the form where A takes values in W , B is V [1]-valued and we used again coordinates {x µ , ξ µ , ξ µ , p µ } of degree (0, 1, 1, 2), respectively, on V 2 . We will work with the general twisted homological vector fieldQ where T µνκ are the components of a closed 3-form on M .
The sign conventions for the Q-structure are chosen as follows: so that we get the corresponding Hamiltonian Its pullback along f then yields the generalized higher Chern-Simons action: where vol is the volume form on R 4 and S pT is a further contribution coming mostly from the twist term T , Note that the action functional is a function on T * [2]T [1]R 4 of degree 4. This is due to the following fact: in ordinary (higher) gauge theory where we use T [1]M , we can identify functions of degree 4 with the volume form on R 4 . In generalized higher gauge theory, however, this identification is no longer possible, and one should integrate each component of the Lagrangian f * ( 1 3 (α − S)) separately. The stationary points of the action functional S ghCS are now given by totally flat generalized 2-connections. That is, the equations of motion simply read as F = 0 with F given in (2.32).

3-Lie algebra based (2,0)-model
Another interesting application of generalized higher gauge theory is an interpretation of the effective M5-brane dynamics proposed in [10]. In these equations, the field content consists of a six-dimensional (2,0)-supermultiplet and an additional vector field, both taking values in a 3-Lie algebra a, as well as a gauge potential taking values in the inner derivations of a.

JHEP04(2016)032
Furthermore, the component 1 3! H µνκ ξ µ ξ ν ξ κ encodes a self-dual 3-form on R 1,5 . Identifying the vector field C µ with the component B µ of the generalized 2-connection, we also impose the constraint D(B µ , B ν ) = 0. This implies the above described factorization B µ = b µ v for some vector field b µ on R 1,5 and v the constant element in R 4 [1] defining µ 1 in the Lie 2algebra A v 4 discussed in appendix A. Since the component F µ p µ of F vanishes and µ 1 (B µ ) = 0, we conclude that the component A µ ξ µ of A vanishes. Considering the component H µν , we learn that also the component 1 2 B µν ξ µ ξ ν of A vanishes. Vanishing of H µν κ implies that B µ ν is covariantly constant, which implies that it can be gauged away. Finally, H µ ν = 0 and F µν = 0 lead to equations (3.23e) and (3.23f), respectively. The only remaining non-trivial component of F is then indeed H µνκ , which describes the self-dual 3-form. Altogether, we thus recover the gauge part of equations (3.23).

JHEP04(2016)032
Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.