Off-Shell Covariantization of Algebroid Gauge Theories

We present a generalized method to construct field strengths and gauge symmetries, which yield a Yang-Mills type action with Lie n-algebroid gauge symmetry. The procedure makes use of off-shell covariantization in a supergeometric setting. We apply this method to the system of a 1-form gauge field and scalar fields with Lie n-algebroid gauge symmetry. We work out some characteristic examples.


Introduction
Recently, many approaches for a generalization of gauge theories are being discussed. Among them, there are the so-called higher gauge theories [1], where in addition to the gauge potential higher rank forms are introduced. Such theories are expected to appear, for example, in the construction of the effective theory of multiple M5-branes where a 2-form gauge potential appears.
Another approach is the promotion of the gauge algebra to an algebroid structure. This can be thought of as a generalization of the gauged non-linear sigma model, where the structure constants of the Lie algebra become scalar field dependent [2,3,4].
A systematic way to construct higher gauge theories is to use an L ∞ -structure [5]. Any truncated L ∞ -algebra defines a gauge theory of higher form gauge fields and the corresponding gauge symmetries are generalized to Lie n-algebras [6]. As we shall see, both generalizations, i.e., to higher gauge theory and to algebroid Yang-Mills can be understood in a unified way using supergeometry. Actually, there is a common phenomenon in both approaches, i.e. the higher gauge theory using an L ∞ -structure and the approach via algebroid structure, when it comes to the formulation of the corresponding field theories. This phenomenon is the so-called fake curvature condition [7].
In a general higher gauge theory also lower form gauge fields exist. However, the field strength of the higher form gauge potential is only covariant under the condition that the field strengths associated to the lower form gauge fields vanish. This is called the fake curvature condition and results in a non-interactive theory. Therefore, it is desirable to deform the higher algebra structure to circumvent this obstruction. Such a deformation process, known as off-shell covariantization, has been analyzed in the higher gauge theory context in our previous paper [8]. There, we solved the fake curvature condition by reducing the symmetries to Lie n-subalgebras, while imposing proper conditions on the auxiliary gauge fields.
In this paper, we want to address the problem of off-shell covariantization in the context of algebroid gauge theories. We apply our method to systems consisting of a 1-form gauge field and a scalar field. We formulate the corresponding higher algebroid gauge symmetries and associated gauge invariant actions. To obtain off-shell Yang-Mills type actions, we consider deformations of gauge transformations and field strengths. Auxiliary gauge fields are projected out and field strengths are deformed by terms proportional to the lower curvatures.
In order to obtain proper gauge symmetries of gauge fields and field strengths, we use the supermanifold method on a so-called QP-manifold [9,10], which is a useful tool to generate a BRST-BV formalism of topological field theories [11]. Instead of starting from fields and an action, we start with a graded symplectic manifold and its Hamiltonian function corresponding to a BRST charge of the gauge algebra. Gauge fields, field strengths and their gauge transformations are induced from the QP-manifold structure. This idea is similar to the free differential algebra method [12,13]. In our formalism, consistency is guaranteed by the underlying QP-manifold structure [11,14,15,16].
The advantage of the supermanifold method is that the gauge transformations and field strengths can be derived in a systematic manner. The starting point of our analysis is a general theory unifying gauge theories with algebroid symmetry and those with Lie n-algebra symmetry. Examples are the Kotov-Strobl model [4] and the Ho-Matsuo model [17]. See also [18] for a gauge theory with a Lie 2-algebra symmetry.
The organization of this paper is as follows. In section 2, we briefly review QP-manifolds and explain the off-shell covariantization procedure used in this paper. In section 3, we discuss the construction of (n + 1)-dimensional higher algebroid gauge theories based on general QP-manifold structures. In section 4, we construct and analyze 4-dimensional algebroid gauge theories. We derive the relations between the structure functions necessary for off-shell covariantization. Furthermore, we discuss examples including the Stückelberg formalism, nonabelian off-shell covariantization and an example from the Kotov-Strobl models. In section 5, we examine the closure of the gauge symmetry algebra. Section 6 is devoted to discussion.

QP-manifolds and off-shell covariantization
In this section, we briefly review how to construct gauge transformations and field strengths using QP-manifolds. Then, we shortly explain the off-shell covariantization procedure of field strengths. Please refer to [19,20] for conventional details.
A QP-manifold (M, ω, Q) of degree n consists of a nonnegatively graded manifold M, a symplectic structure ω of degree n and a homological vector field Q of degree 1 on M such that L Q ω = 0. The requirement of Q to be homological is equivalent to saying Q is nilpotent, Q 2 = 0. The graded symplectic structure induces a Poisson bracket {−, −} of degree (−n).
For any QP-manifold one can find a function Θ ∈ C ∞ (M) of degree n + 1, such that (2.1) The nilpotency of Q then translates to the classical master equation A QP-manifold can also be called a symplectic NQ-manifold. The operator Q generates BRST transformations of the associated gauge theory.
Though the method can be used to construct general p-form gauge theories, in this paper we focus on a theory containing scalar fields X i (σ) and a 1-form gauge field A a = dσ µ A a µ (σ). Our set-up is as follows. We consider a QP-manifold of degree n, where the graded manifold is given by M n = T * [n]E[1] 6 , n ∈ N, and E → M is a vector bundle. M is a smooth manifold. We take the following local coordinates: x i of degree 0 on M and q a of degree 1 on the fiber of the vector bundle. When we construct the associated field theory, the degree corresponds to the ghost degree. With respect to the graded cotangent bundle T * [n], we take coordinates (ξ i , p a ) of degree (n, n − 1) conjugate to (x i , q a ). To summarize, the local coordinates on M n are (x i , q a , ξ i , p a ) of degree (0, 1, n, n − 1).
The symplectic form on M n is defined by This induces the following graded Poisson bracket, where f, g ∈ C ∞ (M n ). 7 By making use of the BV-ASKZ formalism [11,19], a topological field theory in n + 1 dimensions can be constructed from a QP-manifold of degree n. Let Σ be the worldvolume.
The starting point of the construction is the promotion of the worldvolume to a graded space 6 [1] and [n] denote shifting of degree by 1 and n, respectively. space T [1]Σ. We denote the local coordinates of Σ by σ µ , which are of degree 0, and those of the fiber by θ µ , which are of degree 1.
Let M n be our QP-manifold. Then we can define the map a : T [1]Σ → M n , such that is a superfield. Here, z is a coordinate of degree k on M n . The map a is degree-preserving so that |Z| = k. Since the resulting object is a superfield in the BV sense, it contains associated Here, d = θ µ ∂ µ denotes the superderivative and | |z|+1 denotes projection to the degree |z| + 1 part, while setting all antifield components to zero. We get the super Bianchi identity for free, The associated gauge transformation is encoded in the super field strength as the degree |z| Again, while projecting, we set all antifields to zero. 8 This formula gives a BRST transformation.
To extract the physical field directly, we define the mapã : where z is a coordinate of degree k on M n . Note that we have identified θ µ with dσ µ . Using this map, we can rewrite the physical field strength by For the QP-manifold M n under consideration, we get a scalar field associated to the degree 0 coordinate x i and a 1-form gauge field associated to the degree 1 coordinate q a , and In addition to that, we find (n − 1)-and n-form auxiliary gauge fields C a and Ξ i associated with the conjugate coordinates on our graded symplectic manifold, In this very scenario, we have gauge transformations with three independent gauge parameters corresponding to the fields A a , Ξ i and C a : However, in general, the field strength F z of a gauge fieldZ =ã * (z) is transformed adjointly, δF z ∼ F z , only on-shell since the above procedure is derived from the theory of AKSZ sigma models [11]. An action of the AKSZ sigma models is a topological field theory of BF type and the equation of motion is F z = 0. If F z transforms adjointly without use of the equations of motion, we call F z off-shell covariant. If F z is off-shell covariant, the construction of a gauge invariant Yang-Mills type action S ∼ F z ∧ * F z is possible.
The procedure to obtain off-shell covariant field strengths is as follows. First, we drop the auxiliary gauge fields (Ξ i , C a ) and extra gauge degrees of freedom µ ′ i and ǫ ′ a by the projection Ξ i = C a = 0. Then, we deform the field strengths F z and gauge symmetries δ by adding deformation terms proportional to lower field strengths. Note that the algebra of the structure constants (or functions) is not deformed. Choosing proper coefficients for the deformation leads to off-shell covariantized field strengths without changing the original gauge symmetry algebra.

Hamiltonian functions
A Hamiltonian function Θ on a general QP-manifold M n of degree n is of degree n + 1. In this section, we examine the most general Hamiltonian function on M n by expanding it in conjugate coordinates (ξ i , p a ), where Θ (k) is a k-th order function in (ξ i , p a ).
The following cases occur.
A) n ≥ 4: Since the degrees of (ξ i , p a ) are (n, n − 1), the degree of Θ (k) for k ≥ 2 is larger than 2n − 2. Therefore, if n ≥ 4, then Θ (k) = 0 for k ≥ 2 by degree counting, i.e. the general form of the Hamiltonian function is B) n = 3: In this case, Θ (k) = 0 for k ≥ 3 by degree counting. Therefore, the expansion stops at second order, this QP-manifold defines a Lie 3-algebroid structure on E, see [21]. Only for n ≤ 3 the Hamiltonian Θ provides freedom for deformations. We discuss case n = 3 in detail in section 4.
For n = 1, Θ defines a Poisson structure on E.

Gauge fields and field strengths induced from Hamiltonian functions
First, the Hamiltonian function Θ (1) reproduces a Lie algebroid for general n. It contains the following terms, are structure functions depending on x. Lie algebroid operations are given by the following derived brackets, where e, e 1 , e 2 ∈ Γ(E) are sections of a Lie algebroid which is locally expressed by e = e a (x)p a and f ∈ C ∞ (M).
For details and notation, see appendix A.
The classical master equation, {Θ (1) , Θ (1) } = 0, implies the following conditions on the structure constants, The pullback a * maps the four coordinates to superfields as follows,
where F (C) and F (Ξ) are the super field strengths of C and Ξ, respectively. When we substitute the component expansions to (3.30)-(3.33), then the corresponding degree |z| + 1 parts are the field strengths: The degree |z| parts of the component expansions of the super field strengths yield the gauge transformations, The gauge transformations of the gauge field strengths are In general, F a A is on-shell covariant. are (x i , q a , ξ i , p a ) of degree (0, 1, 3, 2), respectively. Since Θ is of degree 4, the Hamiltonian function is at most a second order function in (ξ i , p a ), by degree counting, and can be expanded as Θ = Θ (0) + Θ (1) + Θ (2) . Therefore, the concrete expressions are with additional structure functions h abcd (x), f c ab (x), ρ i a (x) and k ab (x). From the classical master equation, {Θ, Θ} = 0, we obtain the following identities, which define a Lie 3-algebroid [21].
Based on the general theory that we explained in the beginning, we consider the restriction of the 4-dimensional theory. The pullback a * maps the four coordinates to superfields as follows, where (x, A, Ξ, C) are of degree (0, 1, 3, 2). The super field strengths are given by where F (C) and F (Ξ) are the super field strengths of C and Ξ, respectively. When we substitute the component expansions to (4.54)-(4.57), then the corresponding degree |z| + 1 parts are the field strengths: The gauge transformations of the field strengths are One recognizes from (4.67), that F a A does not transform off-shell covariantly unless k ab (X) and f a bc (X) are constants. We seek nontrivial deformations of gauge transformations and field strengths, that lead to off-shell covariant gauge structures. This is done by adding terms to the field strengths and gauge transformations using the fundamental fields and lower form field strengths. Before introducing deformation terms, the auxiliary gauge fields are projected out by imposing Ξ i = C a = 0.
By form degree counting, we assume the following structure of deformations of the field strengths in terms of X i and A a , where K a ci (X) and L a ij (X) are functions. The gauge transformations of (X i , A a ) should be of the following form,δ where N a ci (X) is a function. Let us compute the gauge transformations of (4.68) and (4.69) using (4.70) and (4.71).
Employing the Bianchi identities derived from (2.8), we can computeδF a A . The requirement that the coefficients of  The gauge invariant action, is the so-called Stückelberg formalism of the massive vector field A a µ . We conclude that our formalism provides a nonlinear generalization of the Stückelberg formalism.

Nonabelian gauged nonlinear sigma models
We list a simple but nontrivial example, taking again M 3 as a starting point. Let the structure constants be f a bc = constant, ρ i a = h abcd = 0, k ab (x) = arbitrary.
Then, the Hamiltonian function is The resulting field strengths are The gauge transformations of the gauge fields are (4.97) Using these equations, we compute the gauge transformations of the field strengths as The gauge transformation of F a A is not off-shell covariant. Let us apply our formalism to this system. A solution of (4.73)-(4.75) in this example is where w(x) is an arbitrary function. The covariantized field strengths and gauge transforma- Assume that M is 1-dimensional. Then, we drop the index i and take which yields

Kotov-Strobl model
As third example we formulate the model proposed in [4].
Here, we consider a QP-manifold of degree two, M 2 = T * [2]E [1], in order to demonstrate the covariantization procedure for the Kotov-Strobl model. Note that the resulting gauge theory is not restricted to any dimension. The local coordinates of M 2 are denoted by (x i , ξ i , q a ) of degree (0, 2, 1). The fiber coordinates of E [1] and E * [1] are identified by introducing a fiber metric λ ab . 9 The graded symplectic form is defined by The most general form of the Hamiltonian is given by where λ is a constant. That corresponds to choosing h abc = 0, λ 11 = 1, The associated superfields are defined as x ≡ a * (x), y ≡ a * (y), Using the formulas (2.11) and (2.9) , we obtain the following field strengths, Here, ǫ is the 0-form gauge parameter corresponding to A, and µ ′ 1 and µ ′ 2 are the 1-form gauge parameters corresponding to Ξ and H, respectively. We are only interested in the gauge transformations and field strengths of the fields (X, Y, A). The gauge transformations of the field strengths are computed as The gauge transformations of F Y and F A are not off-shell covariant.
We apply the off-shell covariantization procedure to this theory. The possible deformations of the field strengths and gauge transformations arê One solution is M = − λ 2 Y and N = 0. In this case,δF Y is covariantized aŝ In the next step, we require off-shell covariance ofδF A . This determines J = − λ 2 Y , K = 0 and L = − λ 2 Y e λ 2 XY . The resulting field strengths and gauge transformations arê The gauge transformation ofF A is computed aŝ which is off-shell covariant.
Invariant Action Since the scalar field strength F i X = dX i − ρ i a (X)A a transforms off-shell covariantly,δ the action In this example, the action is given by and is invariant under gauge transformations. The gauge transformation of the third term is given byδ Correspondence to the Kotov-Strobl model By the off-shell covariantization procedure, we obtain the field strengths F X = dX, (4.144)  A ≡ e − λ 2 XY A, (4.154) we obtain the following field strengths from eqs. (4.144)-(4.146), These are the field strengths discussed in [4]. We can rewrite the gauge transformations of the gauge fields usingǫ byδ Then, the gauge transformations of the field strengths are given bŷ δF X = 0, (4.162) which are the same expressions as in [4].

Gauge algebras
Finally, we discuss the closure of the gauge symmetry algebra. For this, we write the gauge transformations asδ where the gauge parameterǫ a is an ordinary function. We find, that two gauge transformationsδ 1 andδ 2 close toδ 3 by [δ 1 ,δ 2 ] =δ 3 withǫ a 3 = f a bcǫ b 1ǫ c 2 , whereδ i denotes the gauge transformation with respective gauge parametersǫ i ,

Discussion
In this paper, we generalized the method to obtain off-shell covariant gauge transformations and field strengths of higher gauge theories in [8] and applied it to a system of algebroid gauge theory with 1-form gauge field and scalars. We demonstrated off-shell covariantization of a gauge theory based on a Lie 2-algebroid and a Lie 3-algebroid. Recall that the resulting gauge theory is not restricted to any dimension. For covariantization, we deform field strengths and gauge transformations. The starting point of this procedure is an on-shell (i.e. F Z = 0) covariant theory. Since the gauge transformations and field strengths are deformed proportional to the lower field strengths, they are consistent if the theory is kept on-shell.
There are several directions to develop the approach presented in this paper. The extension of the method to gauge theories with Lie n-algebroid gauge symmetry induced from a QPmanifold of degree n is straightforward. Similar conditions corresponding to (4.73)-(4.75) can be computed for arbitrary n.
Here, we have formulated the Kotov-Strobl model using a QP-manifold of degree two.
However, we can also construct the Kotov-Strobl model from a QP-manifold of degree three.
For this, a further generalization of the procedure is necessary. Another possible application of our method is to investigate multiple M5-brane systems [24,25]. We can add scalar fields to the analysis conducted in [8]. The procedure in this paper can also be applied to investigate the properties of supergravity in connection with tensor hierarchy. Furthermore, gauge theoretical formulations of gravity such as the vielbein formalism or the gauge theory of the Poincaré group can also be treated in this formalism. It would also be interesting to compare the present formalism with the approach taken in [26]. We expect that our approach will shed new light on the analysis of such systems.