Higher dimensional Lie algebroid sigma model with WZ term

We generalize the $(n+1)$-dimensional twisted $R$-Poisson topological sigma model with flux on a target Poisson manifold to a Lie algebroid. Analyzing consistency of constraints in the Hamiltonian formalism and the gauge symmetry in the Lagrangian formalism, geometric conditions of the target space to make the topological sigma model consistent are identified. The geometric condition is an universal compatibility condition of a Lie algebroid with the multi-symplectic structure. This condition is a generalization of the momentum map theory of a Lie group and is regarded as a generalization of the momentum section condition of the Lie algebroid.

The Poisson structure is not only a fundamental structure of the classical mechanics but also a generalization of a Lie algebra, which mainly appears as symmetries. It is defined by a bivector field π ∈ Γ(∧ 2 T M) satisfying [π, π] S = 0, where [−, −] S is the Schouten bracket defined on the space of multivector fields Γ(∧ • T M). A sigma model with the Poisson structure, the Poisson sigma model [27,28], describes topological aspects of the NS-flux and has many applications such as the derivation of Kontsevich formula of the deformation quantization [29]. The Poisson sigma model is generalized to the twisted Poisson sigma model by introducing the WZ term as a consistent constrained mechanical system. Consistency requires the deformation of the Poisson structure to the twisted Poisson structure [30,31,32]. The twisted Poisson structure is defined by equations, 1 2 [π, π] S = ⊗ 3 π, H , where H is a closed 3-form. For a manifold M with a Poisson or a twisted Poisson structure, the cotangent bundle T * M has a Lie algebroid structure. Thus, it is interesting to generalize a Poisson or a twisted Poisson structure to a general Lie algebroid case.
Recently, Chatzistavrakidis has proposed a higher generalization of the twisted Poisson structure and the twisted Poisson sigma model by considering a higher dimensional topological sigma model [33]. It is a topological sigma model with WZ term on a (n + 1)-dimensional worldvolume. The twisted R-Poisson structure is defined by the following condition, [π, π] S = 0, [π, J] S = ⊗ n+2 π, H , where π is the Poisson bivector field, H ∈ Ω n+2 (M) is a closed (n + 2)-form and J ∈ Γ(∧ n+1 (M)) is an (n + 1)-multivector field on M. † In this paper, we consider a new topological sigma model by generalizing the Poisson structure to a Lie algebroid in the twisted R-Poisson sigma model. The key equation is where E d is the Lie algebroid differential, J is an E-(n + 1)-form, ρ is the so called anchor map of a Lie algebroid and H is a closed (n + 2)-form. We analyze mathematical structures of Equation (6) in details in Section 3. We show that the total structure is regarded as a higher Dirac structure of a Lie (n + 1)-algebroid.
Another purpose is to generalize the so called AKSZ sigma models [34,35,36,37] adding the WZ term. The AKSZ construction of topological sigma models is a clear geometric construction method of the rather complicated BFV formalism [38,39] and the BV formalism [40,41] from a classical action based on graded symplectic geometry. The BV bracket and the BV action are directly constructed by pullbacks of the target space graded symplectic structure. For instance, refer to a review of AKSZ sigma models [15]. However, the AKSZ construction does not work if we twist the classical action adding the WZ term. In two dimensional case, the BV and BFV formalisms of the twisted Poisson sigma model have been constructed in the paper [42], and it was discussed that the correct BV action of the twisted PSM was not obtained by the genuine AKSZ procedure. In order to consider generalizations to higher dimensions, first we need to clarify background geometric structures of higher dimensional twisted topological sigma models with the WZ term.
This paper is organized as follows. In Section 2, we introduce a topological sigma model with a Lie algebroid structure and WZ term. In Section 3, we prepare geometric structures which appear in our model such as a Lie algebroid, a pre-multisymplectic structure and their compatibility condition. We also explain some related examples. In Section 4, we analyze the Hamiltonian formalism and show that the theory is consistent if and only if the geometric compatibility condition holds. In Section 5, the Hamiltonian formalism is rewritten to the target space covariant expression. All equations are described by geometric quantities of the target manifold. In Section 6, we consider the Lagrangian formalism and obtain consistent gauge transformations under the same geometric compatibility condition. In Section 7, we rewrite gauge transformations to the manifestly covariant formulation. Section 8 is devoted to discussion and outlook. In Appendix A, some formulas are summarized. We consider a smooth map from N to M, X : is an n-form taking a value on X * T * M. We consider the following sigma model action functional, Here d is the de Rham differential on Σ. For pairings of pullbacks by X, the same notation Taking local coordinates on M and E, we have four kinds of fields where i is the index of M and a is the index of the fiber of E. The action is ρ i a is local coordinate expression of the anchor map ρ, C c ab are the structure functions of the Lie bracket, J a 1 ...a n+1 and H i 1 ...i n+2 are J and H, which are completely antisymmetric tensors. We call this model the twisted Lie algebroid sigma model with flux, or the Lie algebroid sigma model with the WZ term.
The equations of motion are computed as 3 Lie algebroid and compatible E-flux on pre-multisymplectic manifold In this section, we explain the background geometry of the sigma model (7) introduced in Section 2.

Lie algebroid
Since we want to consider a generalization of the R-Poisson structure, we assume that the target vector bundle is a Lie algebroid.
Local coordinate expressions of formulas in a Lie algebroid are listed in Appendix A.
A Lie algebroid is a generalization of a Lie algebra and the space of vector fields.  g acts as a differential operator, the infinitesimal action determines a map ρ : M × g → T M.
One can refer to many other examples, for instance, in [26].
Lie algebroids are described by means of Z-graded geometry [43]. A graded manifold for a vector bundle E are shifted vector bundle spanned by local coordinates x i , (i = 1, . . . , dimM) on the base manifold M of degree zero, and q a , (a = 1, . . . , rankE) on the fiber of degree one, respectively. Degree one coordinate q a has the property, q a q b = −q b q a .
E-differential forms which are sections of ∧ • E * are identified with functions on the graded which is a linear operator on a space of functions satisfying the Leibniz rule.
We define a degree plus one vector field Q on E[1]: Then, the odd vector field Q satisfies if and only if ρ, C are the anchor map and the structure function of a Lie algebroid on E.
We explain the precise correspondence of Q with E d. For e a , the basis of E * , the map (18), indices i, j are not indices of local coordinates on M , but counting of elements of Γ(E).

Compatible condition of E-differential form with pre-multisymplectic form
We introduce another geometric notion which appears in the topological sigma model (7). It is a condition on a pre-multisymplectic structure analogous to the condition of the momentum map in the symplectic manifold.
We introduce an ordinary connection ∇ on the vector bundle E. i.e., a covariant derivative for all sections µ ∈ Γ(E * ) and e ∈ Γ(E). The connection is extended to the space of differential forms and the dual connection extends to a degree 1 operator on the space of differential forms where e ∈ Γ(E) and v ∈ Γ(T M).
We introduce a new notion.
Then, a flux J is also called compatible with a pre-multisymplectic form H.
The important note is the left hand side in (24) is The condition (24) appears in many situations as we list up some examples below. This condition is regarded as one universal generalization of compatibility conditions of a Lie algebroid structure with a pre-multisymplectic form.
Some known geometric structures are regarded as special cases of Equation (24).
Example 3.6 (Twisted Poisson structure) Let (π, H) be a twisted Poisson structure on M. In this case, the cotangent bundle T * M has a Lie algebroid structure as explained in Example 3.5. Using the Lie algebroid differential E d induced from this Lie algebroid, Equation (15) is rewritten as J = π is bracket-compatible on a pre-2-plectic manifold with a pre-2-plectic form H.
Example 3.7 (twisted R-Poisson structure) Let M be a twisted R-Poisson manifold.
[33] π ∈ Γ(∧ 2 T M) is a Poisson bivector field, H is a closed (n + 2)-form, and J ∈ Γ(∧ n+1 T M) is an (n + 1)-multivector field. As explained in Example 3.4, the Poisson bivector field π induces a Lie algebroid structure on T * M. Under this Lie algebroid structure, the only nontrivial condition of R-Poisson structure (4) is written as −J is bracket-compatible for the pre-(n + 2)-plectic form H.

Example 3.8 (Momentum section)
The terminology 'bracket-compatible' comes from the momentum section theory with a Lie algebroid action on a symplectic manifold, which is a generalization of the moment map theory on a symplectic manifold with a Lie group (Lie algebra) action. [44] See also [46,14,47].
Suppose that a base manifold M is a pre-symplectic manifold, i.e., M has a a closed 2-form ω = H ∈ Ω 2 (M), which is not necessarily nondegenerate. Moreover, suppose a Lie algebroid For an action Lie algebroid E = M × g, a momentum section reduces a momentum map.
Since we can take the zero connection ∇ = d for the trivial bundle, the condition (M1) is dµ = −ι ρ ω. The condition (M2) reduces to the equivariant condition, for e 1 , e 2 ∈ g using (27).
If we take n = 0 in Definition 3.4 and J = µ, Equation (24) coincides with the condition (M2). Therefore Equation (24) is a generalization of the bracket-compatible condition of the momentum section to a pre-multisymplectic manifold.
We make several comments about relations with our theory to the above definitions of momentum sections. The condition corresponding to (M1), Equation (27), does not appear in our model. It is because our model is purely a topological sigma model. Refer to [14] about relations of the conditions (M1) and (M2) with the Hamiltonian mechanics. Since the Hamiltonian is zero, H = 0, we obtain only the consistency conditions of constraints, which is identified to the condition (M2). The condition (M1) is related to consistency with the Hamiltonian and constraints as discussed in [14]. If we consider non topological gauged § The connection is denoted by D in the papers [44] and [14].
nonlinear sigma models, the condition (M1) is needed as the consistency condition of gauge invariance.
The following one more condition (M0) is imposed in the paper [44].
(M0) E is presymplectically anchored with respect to ∇ if The condition (M0) is regarded as a flatness condition of the connection ∇ on µ. We do not require the condition (M0) for J in our paper.

Lie (n + 1)-algebroid and higher Dirac structure
The compatibility condition (24) One can refer to some references of mathematics of a QP-manifold [37,15].
. Let me explain the precise correspondence of two spaces. We consider a QP-manifold Let ∂ i , e a and e a be the basis of T M, E and E * , respectively. The map is given by j : where δ is the graded de Rham differential.
is called the (higher) Dorfman bracket. In the QP-manifold description, they are defined by for e, e 1 , e 2 ∈ Γ(E ⊕ ∧ m−1 E * ) and f ∈ C ∞ (M). Here e = j * e is the super function corresponding to e ∈ Γ(E). j * and j * are the pushforward and the pullback with respect to the map j defined in Equation (32). All the identities of three operations are induced from one equation (31). We identify the graded manifold description and the normal vector bundle description and drop the operation j.

Now let the vector bundle E be a Lie algebroid and M be an m-plectic manifold. Then
E has the anchor map and the Lie bracket ρ, [−, −] and M has a closed (m + 1)-form H. If we define Θ satisfies {Θ, Θ} = 0, where Note that Equation (37) does not include J. Equation E dJ = − ⊗ m+1 ρ, H , is described as a higher Dirac structure, which is explained next.
Three operations of this Lie m-algebroid are as follows.
ρ(e)f = ρ(u)f, where the bracket (−, −) in the right hand side of (39) is the pairing of E and E * . ρ in the right hand side of (40) is the anchor map of the Lie algebroid E. The interior product of the right hand side of (41) is the contraction with respect to E and E * , The Lie derivative is The higher Dirac structure is the subbundle L of the Lie m-algebroid satisfying the con- Proof In fact, the inner product of two elements of Γ(L), u + (J, u) and v + (J, v) for from completely antisymmetricity of J. Moreover the Dorfman bracket is computed by the derived bracket of the graded functions,

Hamiltonian formalism
In this section, the Hamiltonian formalism and constraints are analyzed to make the action functional (7) consistent. We show that the classical action (7) is consistent if the target space geometric data satisfy Equation (24), i.e., the target space is a pre-multisymplectic manifold with a Lie algebroid action and a bracket-compatible E-flux.
Take the worldvolume, Σ = R × T n or Σ = S 1 × T n . Canonical conjugate momenta of X i and A i ¶ are where (s) means coefficient functions of the space components of the differential forms on Σ.
Substituting Equations (45) and (46) to the basic Poisson bracket of canonical quantities, , we obtain Poisson brackets of fundamental fields, The symplectic form corresponding to these Poisson brackets (47)- (49) is The canonical conjugates of time components A (0)a , Y where (0) denotes the time component of the field. The Hamiltonian is proportional to constraints, (52) ¶ Z i and Y a appear as canonical conjugates of X i and A i .
Here G's are constraints without time derivatives, which are spatial parts of equations of motion. The secondary constraints are calculated by computing Poisson brackets with primary constraints (51) and the Hamiltonian H. The For the consistency condition of the mechanics, we require that G i X , G a A and G Y a are first class constraints, i.e., Eqs. (53)-(55) generate a closed algebra under Poisson brackets.
We suppose that a Lie algebroid structure on the target space vector bundle E. ρ i a and C c ab are local coordinate expressions of the anchor map and structure functions satisfying Equations (94) and (95). Moreover suppose that H in the WZ term is a closed (n + 2)-form.
Under the above assumptions, Poisson brackets of constraints G i X , G a A and G Y a are computed using the fundamental Poisson brackets (47)- (49). They are the first class if and only if J satisfies the bracket-compatible condition (24). In fact, under Equation (24), we obtain the following Poisson brackets of three constraints, which shows that all the constraints are the first class. Here σ µ , σ ′ µ are local coordinates on T n and all the fields are spatial components. Equation (24)

Target space covariantization
Constraints and Poisson brackets are rewritten by geometric quantities of the target Lie algebroid by introducing a connection ∇ on E.
Let ω = ω b ai dx i ⊗ e a ⊗ e b be the connection 1-form for the connection ∇. Let s, s ′ ∈ Γ(E). Additional to the following ordinary curvature, in a Lie algebroid, the following E-torsion T , the E-curvature and the basic curvature S are defined, [49,46] T (s, Local coordinate expressions appear in Appendix A. We can rewrite constraints as follows. Since G i X is already covariant under the target space diffeomorphism, the local coordinate expression is the same as Equation (53). G a A and G Y a are written as where and the covariantized constraints are given by If we impose the bracket-compatible condition (24), we obtain the following Poisson brackets, which shows all the constraints are the first class. Here ∇ i ρ j a = ∂ i ρ j a + ω b ai ρ j b . The coefficients of Poisson brackets are written by ρ, H, J, ∇, T and S. Therefore we obtain the following result from Equations (73)-(78).
Theorem 5.1 Suppose that the target space has a Lie algebroid structure and dH = 0. Then, constraints G i X , G a A and G Y a are the first class if and only if J satisfies the bracket-compatible condition (24).

Gauge transformation
In this section, we discuss the Lagrangian formalism.
where we should carefully fix freedom of the term τ a (Φ)G a , which is the freedom of onshell vanishing trivial gauge transformations. τ a (Φ) is an arbitrary function of fields. These ambiguities and problems were discussed in the paper [50] for the twisted Poisson sigma model.
In the twisted Poisson sigma model, τ a (Φ) is a nonzero function. The situation for our twisted Lie algebroid topological sigma model is similar to the twisted Poisson sigma model. We need a nontrivial term τ a (Φ) and it is fixed by imposing the Lorentz, or diffeomorphism covariance of gauge transformations on Σ.
Using this formula, we can compute gauge transformations of each field from constraints in Section 4. We need three gauge parameters corresponding to constraints G Y a , G a A and G i X , c a ∈ Γ(Σ, X * (E)), t a ∈ Γ(∧ n−2 T * Σ, X * (E * )), w i ∈ Γ(∧ n−1 T * Σ, X * (T * M)). c a is a function, t a is an (n − 2)-form and w i is an (n − 1)-form.
Gauge transformations of fundamental fields are given by In fact, the action functional (8)  ∇ on E as in Section 5. In gauge transformations of the basis of E and E * , terms using the connection 1-form ω b ai appear as follows, The gauge transformation of X i , Equation (80), is already covariant δ ∇ X i = δX i . The In fact, using transformations of basis (84), the gauge transformation of the coordinate independent form A = A a ⊗ e a is calculated as follows: where ∇c a = dc a + ω a bi dX i c b . Equation (87) is covariant under the diffeomorphism on M and coordinate transformations on the fiber of E. For instance, ω b ai is transformed as ω For Y , a similar calculation gives the following covariant gauge transformation, We can check the coordinate independent covariant gauge transformation, where Similarly, we obtain the covariant gauge transformation of Z as The coordinate independent form is We obtain invariant coordinate independent gauge transformations (87), (89) and (93).

Conclusion and discussion
We have constructed an (n + 1) dimensional topological sigma model with a Lie algebroid structure, an E-flux and the WZ term, generalizing the twisted Poisson sigma model and the twisted R-Poisson sigma model. The Poisson manifold target space is generalized to a Lie algebroid target space. Moreover, from the consistency condition of constraints, we fixed a consistency condition of the E-flux, the WZ term and other coefficient functions. They are universal geometric conditions of compatibility of E-differential forms with a pre-multisymplectic structure under a Lie algebroid action. We pointed out that they were regarded as a Lie algebroid generalization of parts of the momentum map theory on the multi-symplectic manifold.
We will be able to understand and apply this result to geometric description of higher fluxes and dualities in higher dimensions.
In general, a higher dimensional topological sigma model of AKSZ type has a higher L ∞algebroid structure. If we deform the theory adding the WZ term to the action, the AKSZ construction does not work. We need to modify the AKSZ construction of the BV formalism for topological sigma models with the WZ term. Though the BFV and BV formalisms of the two dimensional twisted Poisson sigma model were geometrically constructed [42], they are still open in higher dimensional topological sigma models with WZ term. In order to construct the BFV and BV in higher dimensions, geometric analysis of compatibility conditions of the Lie n-, or L ∞ -algebroid structure with the pre-multisymplectic structure may be a key point.
The result in this paper gives a new insight and is one step. The construction of the BV and BFV formalism of the twisted Lie algebroid sigma model and the twisted Lie-n, or L ∞algebroid sigma model are an important future problem for analysis of higher dimensional duality physics.

Acknowledgments
The author would like to thank Athanasios Chatzistavrakidis and Yuji Hirota and for useful comments and discussion. This work was supported by the research promotion program for acquiring grants in-aid for scientific research(KAKENHI) in Ritsumeikan university.
A Geometry of Lie algebroid C e ad C d bc + ρ i a ∂ i C e bc + Cycl(abc) = 0.
(98) ω = ω b ai dx i ⊗ e a ⊗ e b be a connection 1-form. Then, local coordinate expressions of covariant derivatives and the E-covariant derivative are The following local coordinate expressions are given as where the covariant derivative ∇ i T c ab is the E-curvature is given from the basic curvature as

B Computation of Equation (62)
The derivation of Poisson bracket, {G Y a (σ), G Y b (σ ′ )} P B is slightly complicated. Using Lie algebroid identities of ρ i a and C a bc , identities of J and H, we obtain The final two terms are rewritten using the constraint G k X as (−1) n−1 (n + 1)!