Third order extensions of $3d$ Chern-Simons interacting to gravity: Hamiltonian formalism and stability

We consider inclusion of interactions between 3d Einstein gravity and the third order extensions of Chern-Simons. Once the gravity is minimally included into the third order vector field equations, the theory is shown to admit a two-parameter series of symmetric tensors with on-shell vanishing covariant divergence. The canonical energy-momentum is included into the series. For a certain range of the model parameters, the series include the tensors that meet the weak energy condition, while the canonical energy is unbounded in all the instances. Because of the on-shell vanishing covariant divergence, any of these tensors can be considered as an appropriate candidate for the right hand side of Einstein's equations. If the source differs from the canonical energy momentum, the coupling is non-Lagrangian while the interaction remains consistent with any of the tensors. We reformulate these not necessarily Lagrangian third order equations in the first order formalism which is covariant in the sense of 1+2 decomposition. After that, we find the Poisson bracket such that the first order equations are Hamiltonian in all the instances, be the original third order equations Lagrangian or not. The brackets differ from canonical ones in the matter sector, while the gravity admits the usual PB's in terms of ADM variables. The Hamiltonian constraints generate lapse, shift and gauge transformations of the vector field with respect to these Poisson brackets. The Hamiltonian constraint, being the lapse generator, is interpreted as strongly conserved energy. The matter contribution to the Hamiltonian constraint corresponds to 00-component of the tensor included as a source in the right hand side of Einstein equations. Once the 00-component of the tensor is bounded, the theory meets the usual sufficient condition of classical stability, while the original field equations are of third order.


Introduction
Various higher derivative field theories are studied once and again over many decades for several reasons. Among the most frequently mentioned advantages of the higher derivative systems are the better convergence properties at classical and quantum level comparing to the analogues without higher derivatives. For discussion of various types of higher derivative models we refer to the paper [1] and references therein. The higher derivative theories are also notorious for the instability problem. The simplest stability test -boundedness of energy -is usually failed by the models with higher order equations of been recently found that the model is multi-Hamiltonian at free level [21]. To the best of our knowledge, it is the first known explicit example of higher derivative field theory that admits multi-Hamiltonian structure. It is also found that the stable interaction can be included with spin 1/2 in this model [21]. To keep dynamics stable, the explicitly covariant interaction vertices should be non-minimal and non-Lagrangian [21]. Even though the stable interaction is non-Lagrangian, the theory still admits Hamiltonian formulation at interacting level [21], so it can be quantized.
In this paper, we study the coupling of the third order extension of Chern-Simons to Einstein's gravity. At free level, the theory admits a continuous series of conserved tensors found in [10] that includes canonical energy-momentum. The series involves bounded quantities if the free third order field equations describe reducible unitary representations, while the canonical energy-momentum is unbounded in every instance.
We begin with inclusion of minimal coupling to gravity into the third order field equations where A = A µ (x)dx µ , µ = 0, 1, 2 is the vector field, m is the constant with the dimension of mass, α 1 , α 2 are the dimensionless constant parameters, d denotes the exterior derivative, and * stands for the Hodge star operator, All the tensor indices are raised and lowered by the spacetime metric g µν . The signature of the metric is mostly negative. The 3d Levi-Civita symbol ε µνρ is the tensor density, ε 012 = 1.
Here α are the parameters involved in the field equations (1), while β, γ are the independent real parameters labeling the representatives of the series of tensors. The sign ≈ means the on-shell equality with respect to the equations (1). The explicit expressions for the series of tensors T are provided in the next section. Once the tensors (3) are on shell covariantly transverse, any representative of the series can be considered as an admissible right hand side for the Einstein equations 1 If the rhs of these equations is the canonical energy-momentum for the field A subject to the equations (1), then the equations (1), (4) form a Lagrangian system. Otherwise, it is not Lagrangian. Be the equations Lagrangian or not, the system (1), (4) is fully consistent once the tensor T on the rhs of (4) is transverse on shell (1).
There are two obvious facts indicating the consistency of field equations (1) and (4): (i) the system has explicit gradient gauge symmetry for the field A, and the equations are diffeomorphism invariant; (ii) there are gauge identities between the 1 We adopt the following definitions for covariant derivative ∇µ, curvature tensor R µ νρτ and Ricci tensor Rµν : where Γ is the Christoffel symbol; Λ is the cosmological constant.
equations which are the same in Lagrangian and non-Lagrangian case -the divergence vanishes identically of the equations (1), while the covariant divergence of equations (4) vanishes because of (3). The orders of equations, symmetries, and identities are the same in all the cases, Lagrangian and non-Lagrangian. These data are sufficient to define the degree of freedom number for the theory being formulated in covariant form, without explicit recourse to Hamiltonian constrained analysis. The formula (8) of the paper [22] allows one to easily compute the local degree of freedom number in a covariant way. The computation gives that the number is four, 2 i.e. it is the same as for the equations (1) in Minkowski space. This means consistency, because in three dimensions, Einstein's gravity does not have local DoF by itself.
In the next section, we elaborate on the on-shell transverse tensors admitted by the equations (1). In particular, we use the ADM space decomposition to clarify the structure of the tensors. As we shall see, the series of tensors T (α, β, γ) (3) includes the representatives that meet the so-called weak energy condition (abbreviated as WEC), once the parameters α, β meet certain conditions. 3 The dynamics is stable once the tensor in the rhs of Einstein equations (4) meets the WEC. The canonical energy-momentum, being included in the series, does not meet the condition, so the stable interactions are inevitably non-Lagrangian. In the section three, we reformulate the field equations (1), (4) in the first order formalism with respect to the time derivatives. This is done making use of 1 + 2 decomposition in the ADM variables. Then, we find the Poisson bracket such that the first order equations read as a constrained Hamiltonian system in all the instances, be the original system (1), (4) Lagrangian or not. The Poisson bracket is not canonical in general, and it involves the parameters α, β, γ from the rhs of the Einstein's equations (4). All the Hamiltonian constraints are of the first class with respect to this bracket. The constraints include the Hamiltonian generators of lapse and shift transformations, and also the generator of gauge transformations for the vector field. The matter contribution to the lapse constraint can be positive for certain range of the model parameters α, β involved in the field equations (1), (4). If the rhs of Einstein's equations (4) is the canonical energy-momentum (that corresponds to β 1 = 1, β 2 = 0 ), the matter contribution to the lapse constraint will be unbounded for any α. In this case, the constrained Hamiltonian formalism is canonically equivalent to Ostrogradsky formulation of the Lagrangian theory. The formulations with the bounded Hamiltonian are inequivalent to this case, because no canonical transformation can turn any on-shell bounded quantity into an unbounded one and vice versa.

Tensors with on-shell vanishing covariant divergence and stability
In this section, we find the series of on-shell covariantly transverse second rank symmetric tensors, and study the weak energy condition for these tensors.

Let us introduce abbreviations
Obviously, F and G are gauge invariant quantities. Also notice that from the definition (5) immediately follows that the one-forms F, G are co-closed, so the covariant divergence identically vanishes of the vectors F µ , G µ , In terms of F, G, the third order equations (1) read In the paper [10], two independent on-shell conserved symmetric tensors are found for these equations in Minkowski space.
The minimal covariant extension of these tensors 4 read where α 1 , α 2 are the parameters of the third-order extension of Chern-Simons (1). Upon account for the identities (6), the covariant divergence of the tensors (8), (9) is seen to vanish on shell: Notice that any on-shell vanishing tensor is on-shell transverse. In the third-order Chern-Simons theory (1), one of these trivial tensors is relevant for constructing Hamiltonian formulation. We chose it in the form where E µ denote the lhs of equations (7), and β 1 , β 2 , γ are constant dimensionless parameters such that β 1 −β 2 α 2 −γα 1 = 0.
Once the tensor T µν (A, g; α, β, γ) is included into rhs of Einstein's equations (4), β 1 , β 2 being the factors at the on-shell non-vanishing contributions (8), (9), can be understood as coupling constants responsible for the interaction between gravity and matter. The accessory parameter γ is involved into on-shell vanishing term. So, it accounts for possible contributions to the rhs of equations (4) such that vanish on account of the field equations (1).
The equations (1) for the vector field, being considered alone, apart from Einstein's equations (4), follow from the least action principle for the functional The quantity T µν (1) (A, g; α) (8), being the first constituent of the series of on-shell transverse tensors (12), is the canonical stress-energy tensor for the action, while T µν (2) (A, g; α) (9) is a different independent tensor. If the scalar curvature of the metric and cosmological constant are added to the matter action (13), the corresponding Lagrange equations will be Einstein's ones (4) with canonical stress-energy tensor T µν (1) (A, g; α) (8) in the right hand side. Contribution of T µν (2) (A, g; α) (9) into the right hand side of equations (4) is on-shell non-trivial. This contribution is non-Lagrangian, though it is fully consistent as it has been already explained in the introduction.
Below we elaborate on the issue of stability of the system (1), (4). As we shall see, the stability can be achieved with certain representatives of the series (12) in the right hand side of Einstein's equations, while inclusion of a pure canonical stress-energy results in instability.
Various assumptions are known about energy-momentum tensor which can provide stability of dynamics of gravity coupled to the matter. These assumptions are usually referred to as energy conditions. For general discussion of energy conditions we refer to the books [23,24]. In this paper, we examine the week energy condition (WEC), which implies that the scalar is bounded from below on the mass shell (7) for arbitrary timelike vector ξ µ (x), ξ 2 > 0. This condition means that the observer will always measure a positive energy density of the matter when traveling by any timelike path. Since the Hamiltonian of the theory is the phase space equivalent of energy, the theory, whose matter satisfies the WEC (15), has a good chance to admit a Hamiltonian formulation with bounded Hamiltonian of the matter. The latter can be viewed as the stability condition in the usual sense.
Once the WEC (15) is imposed onto the tensor T (α, β, γ) (12), it restricts the admissible range of parameters α, β. Now, we are going to find these restrictions explicitly. Let us choose a special coordinate system such that the timelike vector ξ has the canonical form ξ µ = (1, 0, 0). In this coordinate system, the inequality (15) reads The lhs of this expression is a bilinear form in the variables F, G. The coefficients of the form depend on the metric. To simplify the dependence on metric, we use the ADM variables that suites well to the problems where the explicit decomposition in space and time has to be done. The ADM variables read The 3d metric g αβ and its inverse can be expressed in terms of the ADM variables, For the vector fields E (7), F , G (5), we introduce the following 1 + 2 decomposition: where E i , G i , F i are space components of the forms (7), (5), and E 0 , G 0 , F 0 are the time components of the 3d vectors where ε 12 = 1. Here, the mass shell condition θ = 0 corresponds to the Gauss law constraint in the model (1).
In the notation (20), (21), we rewrite the WEC (15) in the ADM variables We evaluate the lhs of this inequality on the mass shell (21), being equivalent to the original equations (1). On shell, give independent Cauchy data for the model (1): one of components A i can be set to zero by gradient gauge transformation, and one of components G i is fixed by the Gauss law constraint θ = 0. Choosing initial data in the form with ξ(x), ζ(x) being the arbitrary functions of spacetime coordinates, we automatically satisfy the Gauss law constraint, while the condition (22) reads Considering this expression as the quadratic form in the gradient vectors ∂ i ξ, ∂ i ζ, and using the Sylvester criterion to ensure that the form is positive semidefinite, we get two restrictions on the parameters α, β, These two conditions are also sufficient to meet WEC (15), because, in this case, (22) is a positive (semi-)definite quadratic form of the variables G 0 , F 0 , G i , F i . 5 There is a special case when the quadratic form T 00 (α, β, γ) (22) is degenerate. We skip the degenerate case in this paper. The non-degeneracy requirement restricts the parameters by the condition This restriction is assumed in all the considerations below.
Relations (25), (26) determine the range of the parameters α, β that results in the stable theory (1), (4). The consistency for these relations implies that the parameters α have to meet the condition Under this condition, the Minkowski space limit of the third order Chern-Simons equations (1)  The general conclusion is that the model (1) with any parameters α 1 , α 2 admits a series of consistent couplings with gravity described by tensors (10) included in the rhs of Einstein's equations (4). If the WEC (15) is imposed, it ensures the stability of dynamics. From the WEC, the restriction follow on the parameters α 1 , α 2 of the extended CS equations (1).
Given the parameters α 1 , α 2 , the parameters β 1 , β 2 define the admissible tensor (12) in the rhs of Einstein's equations (4). If the WEC is imposed, the parameters β have to meet the conditions (25). Under these conditions the third order theory (1) remains stable being coupled to Einstein's gravity.

The first order formulation
The existence of Hamiltonian formulation of the model turns out indifferent to stability. So, we develop the first order formalism for equations (1), (4) and seek for the Hamiltonian structure in a uniform way for any tensor of the series (12) included in the rhs of equations (4), be the model stable or unstable. Let us notice once again, that the field equations (1), (4) are non-Lagrangian in all the instances besides β 2 = 0 which results in the unstable theory, be the flat space limit stable or not. 5 As we use mostly negative signature of metric in 1+2 dimensions, the quadratic form like (k 0 ) 2 − * g ij k i k j is positive for any vector kµ.
When the Hamiltonian formulation is constructed for the diffeomorphism-invariant theories, it is convenient to represent the spacetime as a foliation whose leaves are the spacelike hypersurfaces. Locally, the spacetime is understood as a normal bundle to the spacelike hypersurface which is referred to as space. The coordinate on the fiber of normal bundle is considered as time. We suppose that we have the coordinates x µ , µ = 0, 1, 2 such that x i , i = 1, 2 are the space coordinates, while x 0 is the time. The ADM variables (17), (18), (19) are very convenient to describe the metric once the spacetime is decomposed in space and time.
Let us specify the notation related to the ADM parametrization of metric and curvature. In this section, * ∇ j denotes the covariant derivative with respect to the space metric * g ij , all the space indices are lowered and raised by * g ij and * g ij . The scalar curvature of the space metric * g ij is *

R.
Once the derivatives of metric are to be considered in terms of decomposition in space and time, the 2d tensor of extrinsic curvature of the spacelike hypersurface x 0 = const is a relevant structure. It reads Hereinafter, the dot denotes derivative by x 0 . To absorb the time derivatives of metric, we use the variable π ij , which is canonically conjugate momentum to g ij in Einstein's gravity without matter. As the matter contribution to the Einstein's equations (4), being expressed in terms of F, G (5), does not involve derivatives of metric, we expect that the inclusion of matter does not change the canonical momentum of gravity. In terms of extrinsic curvature, the momentum reads The trace of π ij is denoted by π ≡ * g sr π sr . As is seen from the definition, π ij and its trace π are correspondingly 2d tensor and scalar densities.
To depress the order of the field equations (1), we introduce the variables F i , G i , i = 1, 2 absorbing the first and second time derivatives of the vector field A: The variables F i , G i are the space components of the differential forms F, G (5) in three dimensions, i.e. these can be viewed as the reduction of the spacetime forms to the hypersurface x 0 = const.
In terms of these variables, the field equations (1), (4) read as the first order system: where F 0 , G 0 denote expressions (21) and ε 12 = ε 12 = 1. The equivalence of these first order equations to the original ones can be easily verified. The relations (32), (33), (36) allow one to express the variables G, F, π in terms of A, g in the form (28), (29), (30), (31). Upon substitution G, F, π as functions of A, * g and their derivatives into the rest of equations, one arrives at the original system (1), (4).
Notice that the first order equations (32)-(39) cannot be deduced from the original equations by any Legendre transformation, because the system (1), (4) is non-Lagrangian with the general tensor (12) inserted in the rhs of Einstein's equations (4).
In the next section, we find the Poisson bracket among the variables A i , F i , G i , * g ij , π ij such that the equations (32)-(39) represent the first class constrained Hamiltonian system, with A 0 , N, N i being the Lagrange multipliers at constraints θ, τ, τ i .

Poisson brackets and Hamiltonian equations
In previous section, we have described the first order formulation for the system of the third order extension of Chern-Simons (1) coupled to Einstein's gravity (4) through transverse not necessarily canonical energy-momentum tensor (12). The first order formulation includes the evolutionary equations resolved with respect to the time derivatives of T a (y) = 0 .
These equations constitute the constrained Hamiltonian system if the Poisson bracket {y I , y J } exists of the variables y I such that where (mod T a (y)) means that the equality holds true up to the terms vanishing on constraints T . Once the Poisson brackets exist obeying (42), the Hamiltonian is defined as the linear combination of constraints, where the constraints τ, τ i , θ are defined by relations (38), (39), (35). This ansatz includes five independent parameters α 1 , α 2 , β 1 , β 2 , γ involved in the constraints. The parameters α specify the equations for the vector field (1). The parameters β, γ define the matter contribution to the rhs of Einstein's equations (4). These parameters are involved into the lapse and shift constraints τ, τ i (38), (39). The coefficient k 0 is a constant factor at the Lagrange multiplier A 0 . We introduce k 0 to conveniently control an overall multiplier at the gradient term ∂ i A 0 in equations (32). As the constraints are defined modulo overall non-vanishing factors, the redefinition is inessential.
We seek for the Poisson bracket assuming that the evolutionary equations (32) Given the Hamiltonian (45), the solution to this system reads where the vectors x, y label the space points, and δ( x − y) is the 2d δ-function. As is seen from the equations (51) Then, these functional have the following Poisson brackets: , The The admixture of gauge transformation seems to be admissible because the vector potential is not a gauge invariant quantity.
It is not even 1-form, being rather 1-gerbe. Because of that, it seems natural that the lapse and shift transformations for A may involve an admixture of the gauge transformation adding the exact 1-form to the gerbe. The gauge invariant physical observables F and G transform under the lapse and shift in the usual way, as one-forms. The latter fact identifies the constraints τ , τ i with the lapse and shift Hamiltonian generators.
In constrained Hamiltonian formalism, the strongly conserved energy of the model is usually identified with the constraint that generates lapse transformations. In the model (1), (4), the matter contribution to the lapse constraint τ (38) is given by the 00-component T 00 (α, β, γ) (22) of the on-shell covariantly transverse tensor T µν (α, β, γ) (12). Under the relations (25), the matter contribution to the lapse constraint is bounded. This fact provides another evidence of stability of the model.

Concluding remarks
In this paper we focus at three aspects of the third order extension of Chern-Simons. At first, we introduce gravity into the field equations (1) in a minimal way. Then we notice that the theory admits a two-parameter series of on-shell covariantly transverse tensors (12). This leaves some freedom in consistent inclusion of the matter into the rhs of Einstein's equations (4) because any of the transverse tensors fits well to this role. Second, we see that some of the admissible couplings with gravity meet the weak energy condition, so they are stable, while the interaction through the canonical energy-momentum breaks stability. The third point is that the inclusion of interaction with gravity through the non-canonical energy-momentum, being a non-Lagrangian coupling, still admits constrained Hamiltonian formulation of the corresponding first order equations. So, the theory admits quantization while the interaction is not necessarily Lagrangian.
Let us mention in the end, that even though Einstein's gravity does not have its own local degrees of freedom in 3d, inclusion of interaction with matter subject to higher derivative equations is a nontrivial issue, especially from the viewpoint of maintaining stability. Even if the matter dynamics is stable in the flat space due to conservation of certain tensor which differs from the canonical energy-momentum, it is unclear a priori why and how the stability can persist in the non-flat Einstein's space that corresponds to the non-trivial energy-momentum in the rhs. This work suggests the pattern of construction that can answer to this question. The construction, in fact, is not too sensitive to the dynamical content of gravity. One can expect that the similar pattern should work in higher dimensions, where Einstein's gravity has its own local degrees of freedom.
In d > 3, however, the explicit construction of consistent and stable coupling of gravity with higher derivative matter can become more complicated in certain respect. The construction of transverse tensors meeting the WEC for higher derivative field equations seems basically following the same pattern in d > 3 as in d = 3. It is the non-canonical construction of constrained Hamiltonian formalism which may seem a more complex issue in d > 3.