2D sigma models and differential Poisson algebras

We construct a two-dimensional topological sigma model whose target space is endowed with a Poisson algebra for differential forms. The model consists of an equal number of bosonic and fermionic fields of worldsheet form degrees zero and one. The action is built using exterior products and derivatives, without any reference to any worldsheet metric, and is of the covariant Hamiltonian form. The equations of motion define a universally Cartan integrable system. In addition to gauge symmetries, the model has one rigid nilpotent supersymmetry corresponding to the target space de Rham operator. The rigid and local symmetries of the action, respectively, are equivalent to the Poisson bracket being compatible with the de Rham operator and obeying graded Jacobi identities. We propose that perturbative quantization of the model yields a covariantized differential star product algebra of Kontsevich type. We comment on the resemblance to the topological A model.


Introduction
There are two ways to quantize a Poisson manifold depending on whether the starting point is the Poisson bracket [1] or the two-dimensional Poisson sigma model [2,3]. In the first approach, the central result is Kontsevich's formality theorem [1] that establishes the existence of a unique deformation of the Poisson bracket into a bi-differential operator, sometimes referred to as the star product, given in an expansion fixed by the conditions of general covariance and associativity. Inspired by string theory, Kontsevich also gave the star product explicitly in the case of a general Poisson structure on R n . In the second approach, this formula was derived using AKSZ path integral methods [4] from the perturbative expansion of the correlations functions of the Poisson sigma model subject to suitable boundary conditions [5] (see also [6]). More precisely, the original works of Kontsevich and later Cattaneo and Felder concern the deformation of the commutative algebra of functions, or zero-forms, on the Poisson manifold. However, the general covariance of their star product formula is not manifest, nor does it apply to differential forms of arbitrary degrees.
A natural extension of Poisson algebras to include higher form degrees, sometimes referred to as differential Poisson algebras, was defined and studied in [7]. Later, following the algebraic approach, the corresponding manifestly generally covariant form of the star product has been studied in [8,9,10] (see also [11]), though its explicit form remains to be given beyond 2 -corrections 1 . In this paper we provide a generalization of the two-dimensional Poisson sigma model in [2,3] as to include fermionic worldsheet zero-forms facilitating the mapping of target space differential forms to vertex operators of worldsheet form degree zero. We propose that its path integral quantizationà la Cattaneo and Felder covariantizes Kontsevich's star product formula.
A key feature of differential Poisson algebras is the presence of a connection oneform Γ α β that is compatible with the Poisson bi-vector Π αβ . As we shall review in Section 2, the covariantized Poisson bracket between two differential forms reads where ∇ α has connection coefficients Γ α βγ = Γ α γβ and R α β = d Γ α β + Γ α γ ∧ Γ γ β . Since the connection plays a key role in covariantizing the star product in the algebraic approach, it is natural to seek an extension of the original Poisson sigma model that couples to it as well. Another problem that one would like to address is how to map target space pforms dφ α 1 ∧· · ·∧dφ αp ω α 1 ...αp to vertex operators of form degree zero on the worldsheet. To this end, one observes that by introducing fermionic worldsheet zero-forms θ α , the vertex operators can be taken to be θ α 1 · · · θ αp ω α 1 ...αp . Thus, combining these two observations, we are lead to adding fermionic copies (θ α , χ α ) of the original bosonic worldsheet zero-and one-forms (φ α , η α ). The proposed action, which we shall study in more detail in Section 3, reads 2 The role of the additional quartic fermion coupling is to ensure a rigid supersymmetry δ f that in particular acts as δ f (φ α , θ α ) = (θ α , 0). Under additional conditions on the background, the action is also invariant under gauge transformations with one unconstrained parameter for each one-form. We shall show that these rigid and local symmetries, respectively, are equivalent to the bracket being compatible with the de Rham differential and obeying graded Jacobi identities. Thus, assuming that exist a gauge fixed action that is manifestly background covariant, we expect that the products of the aforementioned vertex operators contain the covariantized Kontsevich star product for differential forms of any degree, whose explicit construction we leave for a future work. 1 In [12], the deformation quantization procedure has been set up and studied at order in the case of more general vector bundles over Poisson manifolds. 2 In the Conclusions, we shall comment on the resemblance between the action presented here and that of the topological A model. The plan of the paper is as follows. In Section 2, we review the basic properties of differential Poisson algebras and the conditions on the Poisson bi-vector and curvature following from the Jacobi identities. In Section 3, we present the sigma model action and show that its symmetries are equivalent to the salient features of the differential Poisson algebra. In Section 4, we conclude and remark on the resemblance between our model and the topological A model, and its potential importance in higher spin theory. We give our conventions and some useful identities in Appendix A.

Differential Poisson algebras
In this section we recall the defining relations of Poisson differential algebras [7,8] and the resulting form of the Poisson bracket. The bracket consists of three compatible structures, namely a Poisson bi-vector Π αβ , a connection Γ α β and a one-form S αβ . The one-form contains the components of the bracket that are not contained in the pre-connection [8], that is, the covariant derivative along the Hamiltonian vector field defined using Π αβ . In the symplectic case, one can set S αβ = 0 by redefining Γ α β , in which case the Poisson bracket is given by Π αβ and a curvature two-form R α β constructed from the torsion. In what follows, we shall set S αβ = 0, leaving for future work the analysis of whether there exists non-trivial S tensors in the non-symplectic case.

Definition
A differential Poisson algebra is a differential algebra Ω endowed with a graded skewsymmetric and degree preserving bilinear map {·, ·}, called Poisson bracket, that is compatible with exterior differentiation and obeys the graded Leibniz rule, that is and that obeys the graded Jacobi identity where ω i ∈ Ω and deg(·) is the form degree. We shall furthermore assume that Ω is realized as the algebra Ω(N ) of differential forms on a manifold N . In what follows, we shall first use Eqs. (1)- (5) to expand the Poisson bracket in terms of Π, S and the curvature R, and then impose Eq. (6).

Poisson bi-vector and compatible connection
Introducing local coordinates φ α on N , we define which is thus an anti-symmetric tensor. The Poisson bracket between two zero-forms f and g can then be written as Next, to expand the Poisson bracket between a zero-form and a one-form in the coordinate basis, we define where Σ αβ is thus a symmetric one-form. From the Leibniz rule, it follows that Under a general coordinate transformation φ α = φ α (φ ′α ′ ) with Jacobian J α β ′ = ∂φ α /∂φ ′β ′ and inverse Jacobian J ′α ′ β , one has the non-tensorial transformation property Introducing a connection one-form one has the transformation law Using the tensorial transformation property of Π to rewrite (11) as we can thus write where U αβ is a tensorial one-form. It follows that are the components of a tensorial one-form. In summary, we can write where the first two terms are tensorial and the last term, which is non-tensorial, is sometimes referred to as the pre-connection [8].
It is convenient to choose the connection to belong to the equivalence class obeying which we shall assume henceforth. It follows that where the two-form as a consequence of (19).

Manifestly covariant Poisson bracket
Let ω and η be differential forms of any degree. From the basic Poisson brackets (7), (18) and (20) in the coordinate basis, and invoking (2) and (4), it then follows that where i denotes inner differentiation and ∇ uses the connection coefficients By construction, the above manifestly covariant form of the Poisson bracket obeys (5), that is, it is compatible with the de Rham operator.
The equivalence class of compatible connections is generated by shifts δ Γ α βγ obeying δ( ∇ α Π βγ ) = 0 and δΠ αβ = 0, that is δ Γ which is indeed symmetric in α and β. The invariance of the full Poisson bracket (22) can then be verified using δ∇ α ω = −δ Γ β α i β ω and δ R αβ = Π αγ ∇δ Γ β γ . In the symplectic case, the shift symmetry (24) can be used to set S = 0. In what follows, we shall specialize to the case S = 0, which has been studied in detail in [7,8,9,10], leaving the analysis of the general case for future work.

Jacobi identities
In order to analyze the Jacobi identities (6) (in the case that S = 0), one can use (4) to show that if they hold for functions and one-forms then they hold for forms of any degree. In the case of three functions f 1 , f 2 , f 3 , one finds from which it follows that In view of ∇ α Π γδ = 0, this condition is equivalent to that Π is a Poisson bi-vector, i.e.
In the case of two function f 1 , f 2 and a one-form ω, the Jacobi identities read from which we obtain Finally, for a single function f and two one-forms ω 1 , ω 2 , we have which implies that Finally, for three one-forms one finds that As observed in [8,10], the compatibility between the Poisson bracket and the de Rham differential implies that the independent conditions are given by the following irreducible representations 3 As for examples of non-trivial solutions, see [7,8].

Poisson sigma model
In this section, we use the Poisson bi-vector and its compatible connection to construct the couplings in a two-dimensional topological sigma model action that exhibits an extra nilpotent rigid supersymmetry δ f corresponding to the de Rham differential on N .
As we shall see, the rigid symmetry fixes the coefficient of the quartic fermion coupling while the gauge symmetries require the background fields to obey the conditions (27), (29), (31) and (32), which we recall are equivalent to that the underlying differential Poisson algebra obeys the Jacobi identities.

The action
Our action, which is formulated on a two-dimensional manifold M 2 , is given by where R γδ αβ = Π βǫ R γδ α ǫ are the components of the two-form (21) obtained from the Poisson bi-vector and its compatible connection, and the covariant exterior derivative where the connection coefficients are defined in (23). The fields are assigned form degrees deg 2 on M 2 and an additional Grassmann parity ǫ f (·) as follows: 3 From (33) the remaining conditions follow by covariant differentiation, viz.
In what follows, we shall assume M 2 to be compact and that the pull-backs of (η α , χ α ) to the boundary of M 2 vanish.
Geometrically, the action describes a sigma model with source M 2 and target space given by the N-graded bundle The sigma model map ϕ : M 2 → N has vanishing intrinsic degree and Grassmann parity, and the degree on M 2 is the form degree on M 2 . Thus, if ω n,p,ǫ denotes an n-form on N of degree p and Grassmann parity ǫ, then its pull-back ω p,ǫ := ϕ * ω n,p,ǫ is a p-form on M 2 of Grassmann parity ǫ. The de Rham differential on N has form degree one and degree one. We use the following Koszul sign convention, which is consistent with Leibniz' rule: The resulting sign convention for wedge products on M 2 reads The manifest target space covariance of the action amounts to the fact that target space diffeomorphisms which act on the worldsheet fields, induce Lie derivatives acting on the background fields, i.e. where

Nilpotent rigid fermionic symmetry
The de Rham differential d = dφ α ∂ α on N lifts to a holonomic vector field θ α ∂ α on T [0, 1]N , which in its turn can be extended to a nilpotent rigid supersymmetry δ f of the action as follows: as can be seen using ∇ α Π βγ = 0 and the Bianchi identity We note that deg 2 (δ f ) = 0, ǫ f (δ f ) = 1 and that δ 2 f η α = 0 requires the aforementioned Bianchi identity. Moreover, just as the relative coefficient between the two terms in the Poisson bracket (22) is fixed by compatibility with the d operator, the rigid supersymmetry requirement fixes the relative strength between the kinetic terms and the quartic fermion term in the action (34). In fact, the δ f -invariance of the action can be made manifest by observing that the Lagrangian is δ f -exact, viz. where as can easily be seen using ∇ α Π βγ = 0 and we note that there is no need to discard any total derivative in (45). The commutator between the rigid supersymmetry and the target space diffeomorphisms takes the form Thus the rigid supersymmetry commutes with background Killing symmetries, whose Lie derivatives by definition annihilate Π αβ and Γ α βγ and hence the action as can be seen from (41).

Universal Cartan integrability
Let us demonstrate that the universal Cartan integrability of the equations of motion, which is required for the validity of Cartan gauge symmetries and on-shell integration, is equivalent to that the target space background obeys the conditions (27), (29), (31) and (32), i.e. that they can be used to define a differential Poisson algebra obeying the Jacobi identity (6). To this end, we derive the generalized Bianchi identities where R i := (R φ α , R θ α , R χα , R ηα ) and M i j is a field dependent matrix, after which we require compatibility in the universal sense, that is, that the classical anomalies A i vanishes on base manifolds of arbitrary dimensions.
In particular, the second condition in (57) is equivalent to that Next, taking into account (57), the vanishing of A θ α = 0 requires the additional condition or, equivalently, which is a consequence of the previous conditions, as discussed below Eq. (31). Turning to the integrability of R χα = 0, it follows from (57) and (60) that A χα vanishes universally, noting that the χ ∧3 θ 2 -terms in ∇R χα are proportional to R ǫ[α (ρσ R βγ] λ)ǫ χ ρ ∧ χ σ ∧ χ λ θ α θ γ . Finally, using (58) one has modulo R χα , R φ α and R θ α . The second term is easily seen to be zero from condition (60). To show the vanishing of the first set of terms, we use the third condition in (57) and Employing the Ricci identity To show the vanishing of the third set of terms in (61), we rewrite the Bianchi identity On the other hand, the second condition in (57) together with Thus, contracting the above form of the Bianchi identity by Π σβ Π δλ and using (65) it follows that the third term in A ηα vanishes as well.
In summary, we have showed that the universal Cartan integrability of the equations of motion Eqs. (49)-(52) is equivalent to that the background fields Π and Γ can be used to define a differential Poisson algebra.

Gauge transformations
Relying upon the framework for generalized Poisson sigma models [13,14] (see also [15,16,17]), the universal Cartan integrability of the equations of motion implies that the action is invariant under Cartan gauge transformations. On-shell, these transformations can be obtained by first rewriting the equations of motion Eqs. (49)-(52) on the canonical form Eliminating dφ α in ∇ using R φ α = 0, we thus have The on-shell gauge transformations are then given by where ǫ i denote the gauge parameters, of which there is one for each fields with strictly positive form degree, that is, Thus, the infinitesimal gauge transformations are given by modulo R i . Off-shell, it follows from the general formalism [16,17], that the gauge transformations are given by where we have introduced the symplectic two-form of degree three on the target space N given in (37), and the pre-symplectic structure treated as a one-form of N-degree two on N . Thus, the matrix O ij can be read off from Moreover, the components P ji of the Poisson structure on N is given by Using the above four by four matrices is simple to show that P ik O kj = −δ i j . If the connection vanishes identically, then the off-shell modification of the gauge transformation (77) vanishes.

Conclusion and remarks
We have given an action of the covariant Hamiltonian form that describes a twodimensional topological sigma model in a target space carrying the structure of a differential Poisson algebra. The kinetic term is given by the pull-back of a pre-symplectic form that is non-canonical and hence the off-shell gauge transformations contain an additional set of terms proportional to the Cartan curvatures. Besides the characteristic local symmetries of such models, whose requirements are indeed equivalent to those of the Jacobi identities of the differential Poisson algebra, our action also exhibits a rigid supersymmetry corresponding the de Rham differential on the Poisson manifold. This rigid symmetry fixes the coefficient of the quartic fermion coupling. We expect that the AKSZ quantization [5,6] of the original Poisson sigma model can be generalized to the present model in a background diffeomorphism covariant fashion, i.e. such that there exists a generalization of (41) to the gauge fixed action. Assuming furthermore that Kontsevich's formality theorem generalizes to the deformation of the graded Poisson bracket, the similarity between the structures of the Poisson bracket (22) and the action (34) suggests that the correlation functions of suitable boundary vertex operators yield the covariantized version of Kontsevich star product (at least in simple target space topology).
The action (34), which describes a classically topological theory that remains to be gauge fixed, bears a close resemblance to the complete action of the first order formulation [18,19,20] of the topological A model [21]. The latter is obtained by a topological twist of the N = (2, 2) supersymmetric sigma model, and requires the target space to be Kähler, and hence symplectic, unlike our model, whose target space is only required to be a Poisson manifold. Moreover, the type A model refers to a worldsheet metric, which enters via additional couplings to the hermitian metric and its compatible curvature of the form g αβ η α ∧ * η β and g αǫ R γδ β ǫ (g)χ α ∧ * χ β θ γ θ δ . Thus the type A model action is non-singular in the sense that it does not admit any local symmetries. Instead, the couplings are tuned such that the complete action is exact under a rigid nilpotent supersymmetry 6 , whose factorization yields a topological model. 4 Starting from a path integral weighted by exp( i S), the perturbative expansion is obtained by rescaling (η α , χ α , θ α ) → ( η α , √ χ α , √ θ α ) and expand around the two-dimensional vacuum in which φ α and θ α are constant and η α and χ α vanish. 5 In the gauge fixed theory all quantities are assigned form degrees, ghost numbers and additional Grassmann parities, and we choose the Koszul sign convention to be given by AB = (−1) |A||B|+ǫ f (A)ǫ f (B) BA where the total degree |A| = deg 2 (A) + gh(A). 6 The rigid supersymmetry generator of the A model is sometimes referred to as a BRST operator, even Our model, on the other hand, is classically topological without requiring the classical observables to be δ f -closed. Thus, in the terminology of topological field theories, our model is of the Schwarz type, while the A model is of the Witten, or cohomological, type.
It would be interesting to examine whether there are more robust relations between the type A and B models and also the interpolating A-I-B model [19], including their infinite (and possibly zero) volume limits, and our model and various deformations of it. As for the latter, one may consider adding Yukawa couplings formed out of the S-tensor defined in (17) and additional metric couplings G αβ χ α ∧ χ β (which add terms of intrinsic degree minus two to the bracket). One may also seek ways to couple of our model to two-dimensional gravity, which may be of importance for the formulation of the theory on worldsheets of higher genus, and possibly new topological open strings. To this end, besides exploring the relations to the type A and B models, it may also be fruitful to explore another route, based on the observation that prior to adding the worldsheet fermions, the Poisson sigma model exhibits vacuum bubble cancelations in simple worldsheet topologies. Adding the fermions lead to that these cancellations generalize to arbitrary topologies [23] (including boundaries). Thus, including bubbles with external matter legs, one may expect anomaly-induced topological matter-gravity couplings. We plan to address these issues in a future work.
One motivation behind the present work is Vasiliev's higher spin gravity, whose field theoretic formulation is in terms of differential star product algebras [24]. The explicit models that have been constructed so far are formulated on products of commuting manifolds, containing spacetimes, and symplectic manifolds of simple topology, quantized using the Moyal star product. The covariantized Kontsevich formalism provides a tool facilitating the formulation of higher spin gravities on manifolds of more general topology, possibly as Frobenius-Chern-Simons theories (or BF analogs thereof) following [25]. Its extension to topological open strings, with non-trivial topological expansions, may lead to complementary first-quantized descriptions of higher spin gravity. The latter perspective is supported by the recent progress in computing higher spin tree amplitudes starting from traces over oscillator algebras [26,27,28,29], in its turn motivated by the proposal made in [30] for how Vasiliev's theory arise in tensionless limits of closed strings in anti-de Sitter spacetime.
Finally, a natural part of the application to higher spin gravity as well as discretized strings, and also more general constrained systems, is the gauging of Killing symmetries of our Poisson sigma model. In principle, this procedure ought to be straightforward and leads to a natural generalization of the original gauged Poisson sigma model, which though the twisting is not a gauge fixing procedure. Attempts to identify the type A model as a gauge fixed version of a classically topological theory have been made in [22]. we expect to report on in a forthcoming publication. or in components The square of the exterior covariant derivative acting on the components of a vector field and a one-form are given by In analyzing the differential Poisson algebra, it is convenient to define a new connection ∇ with connection coefficients Γ α γβ := Γ α βγ . We make repeated us of the identity We denote the components of the curvature of ∇ by R αβ γ δ and define