General structure of Thomas$-$Whitehead gravity

Thomas-Whitehead (TW) gravity is a projectively invariant model of gravity over a d-dimensional manifold that is intimately related to string theory through reparameterization invariance. Unparameterized geodesics are the ubiquitous structure that ties together string theory and higher dimensional gravitation. This is realized through the projective geometry of Tracy Thomas. The projective connection, due to Thomas and later Whitehead, admits a component that in one dimension is in one-to-one correspondence with the coadjoint elements of the Virasoro algebra. This component is called the diffeomorphism field $\mathcal{D}_{ab }$ in the literature. It also has been shown that in four dimensions, the TW\ action collapses to the Einstein-Hilbert action with cosmological constant when $\mathcal{D}_{ab}$ is proportional to the Einstein metric. These previous results have been restricted to either particular metrics, such as the Polyakov 2D\ metric, or were restricted to coordinates that were volume preserving. In this paper, we review TW gravity and derive the gauge invariant TW action that is explicitly projectively invariant and general coordinate invariant. We derive the covariant field equations for the TW action and show how fermionic fields couple to the gauge invariant theory. The independent fields are the metric tensor $g_{ab}$, the fundamental projective invariant $\Pi^{a}_{\,\,\,bc}$, and the diffeomorphism field $\mathcal D_{ab}$.


I. INTRODUCTION
The Virasoro algebra [1] is usually considered at the heart of string theory through its relationship with conformal symmetry, where two copies of the Virasoro algebra define the conformal algebra. However, the relationship between string theory and the Virasoro algebra also has an even more primitive origin through its identity as a one dimensional vector space [2] and projective structure [3][4][5]. Since the coadjoint orbits admit a natural symplectic structure, their geometric actions provide an avenue to the two dimensional field theories that can be associated with quantum gravity [6][7][8]. Furthermore, when married with an affine Lie Algebra (Kac-Moody algebra), one finds that the coadjoint elements appear as background sources for the two-dimensional gravitation (Virasoro sector) and gauge (Kac-Moody sector) theories. The background fields in the Kac-Moody sector correspond to the vector potentials which serve as the gauge connections, A a , for Yang-Mills theories. It was suggested in [9] that the coadjoint elements of the Virasoro sector also could be put on an equivalent footing with the Kac-Moody sector if the coadjoint elements of the Virasoro algebra could also have an associated "gauge" field in higher dimensions. The posited field was dubbed the diffeomorphism field, D ab . This realization was recently established in [10], when Kirillov's observation [4,5] that the coadjoint elements of the Virasoro algebra are in one-to-one correspondence with Sturm-Liouville [11,12] operators, was reexamined. The authors were able to use the one dimensional projective structure to provide a bridge between the Virasoro algebra and projective geometry in higher dimensions. Thus the analogous "gauge" symmetry due to reparameterization invariance in the Virasoro sector is projective invariance and the diffeomorphism field corresponds to projective connections. With this, the diffeomorphism field that appears in two dimensions through the geometric action as a background field has a different interpretation than that of expectation values of external energy-momentum tensors, as in conformal field theories. Furthermore the diffeomorphism field can acquire dynamics as a fundamental field through the projective curvature squared terms. Some of the entangled relationship between conformal geometry and projective geometry has been studied in [13][14][15][16][17]. For a good review see [18].
So far, discussions of dynamical projective connections [10,19] have been restricted to particular metrics that are focused on the 2D Polyakov metric [20,21] or Einstein geometries in four dimensions where compatibility has been enforced. In this paper we generalize those considerations for any space-time dimensions and exhibit a Lagrangian that is explicitly projectively invariant and general coordinate invariant, i.e. gauge invariant. We will briefly review the salient features of the study of geodesics through the Thomas-Whitehead connection, the Thomas Cone and tensor and fermion representations on the Thomas Cone. Then, by using the Palatini [22] formalism, we explicitly construct the gauge invariant Thomas -Whitehead gravitational action (TW) [10], the gauge invariant Dirac action and covariant field equations, its coupling to arbitrary Yang-Mills theories, and the energy-momentum tensor. This work can be extended to include higher-order interactions, using the projective version of Lovelock Gravity [23] to classically maintain an initial value formulation. We will conclude with remarks on geodesic deviations as it is there that contributions through gravitational radiation may become manifest.

II. FROM GEODESICS TO PROJECTIVE CURVATURE
In its most pragmatic form, string theory can be thought of as regulating the Feynman diagrams in gravitational theories by adding a small space-like curve to the point particle.
This activity already endows the string with a projective structure. The curves are parameterized by vector fields, say ζ a = dx a dσ , which allows one to take the intrinsic or absolute derivative of any vector field along these curves. In one-dimension the Virasoro algebra is the algebra of centrally extended vector fields on a line or circle and a projective structure emerges [3,5,24].

A. Geodetics
In any dimension, the intrinsic (or absolute) derivative of a vector field v a along a curve C parameterized by σ is given by, where Γ a bc are connection coefficients associated with a connection ∇ a and ζ a is the tangent vector dx a dσ along the curve C. An affine geodesic generalizes the notion of a straight line and ζ a is said to be geodesic if the change of ζ a along the curve C parameterized by σ is be proportional to itself, i.e.
Dζ a dσ = f (σ)ζ a , where f (σ) is the proportionality function. This yields the affine geodesic equation, One may change the parameterization from σ to u(σ) by writing and for a suitable choice u(σ) we can eliminate the right hand side of Eq. (3) to write the geodetic equation Here the parameter u is said to be an affine parameter with respect to the connection ∇ a Although the parameterization may have changed, the curves remain the same. Furthermore different connections, say∇ a and ∇ a can sometimes admit the same geodesics. If so, then ∇ and ∇ belong to the same projective equivalence class. Thomas showed how one can write a gauge theory over this projective symmetry [25,26]. We discuss this gauge theory presently.

B. Projectively equivalent paths
Consider a d-dimensional manifold M with coordinates x a where italic latin indices a, b, c, m, n, · · · = 0, 1, . . . , d − 1. Let∇ a be a connection on M where ζ a is geodetic, i.e.
Now consider another connection whose coefficients are defined as where v b is an arbitrary one form. The geodesic equation for this connection is then and where f (τ ) = 2v b dx b dτ . Since Eq. (8) can also be made geodetic by a suitable reparameterization of τ to u(τ ) both Eq. (6) and Eq. (8)admit the same geodesic curves. Eq. (7) is called a projective transformation and establishes the projective equivalence relation,Γ a bc ∼ Γ a bc .
In [25,26], Thomas presents a "gauge" theory of projectively equivalent connections that is projectively invariant and general coordinate invariant. This begins by defining the which is traceless by construction Π a ba = Π a ab = 0 (10) and invariant under a projective transformation, Eq. (7), for an arbitrary one form v a . Using the fundamental projective invariant Π a bc one can write a geodetic equation that is projectively invariant. However this equation is not covariant as Π a bc transforms as under a general coordinate transformation from x → x ′ (x) with J a b = ∂x ′a ∂x b , the Jacobian of the transformation. We will denote the inverse Jacobian asJ a b = ∂x a ∂x ′b . The last summand spoils the covariance and can be related to volume, as it involves the determinant of the Jacobian of the transformation J = det (J a b ). Thomas then constructs a line bundle over M which is a d + 1-dimensional manifold N referred to as the Thomas Cone [27] [28]. The coordinates on the Thomas Cone are (x 0 , x 1 , . . . , x d−1 , λ), where λ is denoted the volume coordinate. Since the volume coordinate, λ, takes values 0 < λ < ∞, N is called a cone.
The coordinates transform as From here on, we refer to transformations in Eq. (13) as TCN -transformations. Here, Greek indices are over N coordinates and take values α, β, µ, · · · = 0, 1, 2, . . . , d and italic latin indices are over coordinates on M and take values a, b, m, n, · · · = 0, 1, 2, . . . d − 1. We reserve the index λ and the upright letter d to refer to the volume coordinate x d = x λ = λ.
For every coordinate transformation on M there is a unique coordinate transformation on N .

C. Thomas projective connections
Thomas was able to find a connection on N that transforms as a connection by extending the fundamental projective invariant to a d + 1-dimensional projective connectionΠ α µν . It is defined as follows [25,29,30] where R ab is constructed from the equi-projective curvature "tensor" R m with an associated equi-projective Ricci "tensor" With this,Π α µν transforms as a connection under a TCN -transformation as so that one may construct a projective curvature tensor whose non-vanishing components are The projective Ricci tensor is defined as the trace of the projective curvature tensor and vanishes identically This construction is only a specific example of a projective connection but it laid the ground work for the more general setting we now present.

III. THOMAS-WHITEHEAD PROJECTIVE GEOMETRY
A. The general projective connection The original Thomas projective connection,Π α µν , can be generalized to a connectionΓ α βγ [29][30][31], where explicitlyΓ and where Here the connection Γ a bc is any representative member of the equivalence class [Γ a bc ] of projectively equivalent connections, related via Eq. 7, and α a is that chosen member's trace component. However, keep in mind that Π a bc exists in its own right in that it is traceless and transforms like a traceless part of an affine connection. Notice also that only the λ component for α µ appears in the projective connectionΓ µ αβ . On M, the transformation laws are In the above, D ab generalizes the work of Thomas and transforms in such a way thatΓ α βγ transforms as an affine connection on N . This is the origin of the diffeomorphism field D ab . In this construction, Υ is the fundamental vector on the Thomas cone and satisfies the compatibility relation∇ so Υ β satisfies the fundamental geodesic equation with unit proportionality For functions on N showing that Υ generates scaling in the λ direction. One-forms β α on N are uniquely defined by β a on M when β α Υ α = 1 and the Lie derivative with respect to Υ vanishes i.e, L Υ β ρ = 0, so that it is scale invariant. Under a TCN -transformation, Eq. (13), Υ α and the covariant derivative transform as Demanding thatΓ α µν transforms as an affine connectioñ and using the transformation laws of Υ α andΓ λ ab , one finds that D ab transforms under a coordinate transformation on M as where we define j a = ∂ a log |J| − 1 d+1 . One can show that the coordinate transformation law of D ab as stated by Eq. (35) is an action of the general linear group on the components of D. This property holds despite the presence of the coordinate-dependent object Π a bc in the transformation law [32]. This transformation law will become important later in the correspondence with coadjoint elements of the Virasoro algebra in one-dimension.
A general tensor on M with m-contravariant and n-covariant indices we express as In what follows we refer to (m, n)-tensor on M as objects that transform as under coordinate transformations. Similarly, we refer to objects as (m, n)-TC tensors on N that transform as under a TCN -transformation. This will allow us to build actions that are invariant with respect to TCN -transformations.

B. Geodetics revisited
Before discussing projective curvature relations, we now revisit geodesics and geodetics to illuminate the projective connection. Consider a geodetic on N associated with the vector field ζ α = dx α du . The parameter u is an affine parameter for∇ such that Separating the M coordinates from λ, we have the expressions Together, these equations are covariant and projectively invariant. Let us consider a reparameterization that can render Eq. (40) geodetic. In other words, does there exist a parameter τ that is affine with respect to the projective invariant Π a bc ? Let u → τ (u) so that This will eliminate the RHS of Eq. (40) and we can use this to eliminate λ in Eq. (41) with With this, one finds that the reparameterization is viable if where S(τ : u) is the Schwarzian derivative of τ with respect to u. so we see that the preferred class of parameters for Π a bc is preserved by projective transformations rather than affine transformations. This motivates the description of Π a bc as a projective connection. The inclusion of Π a bc in the TW connection, which incorporates the field D bc , allows us to apply techniques that are typically available for affine connections.

C. Projective geometry
One constructs the projective curvature tensor in the usual way from connections that transform as in Eq. (34). In terms of the connections, the curvature can be written explicitly as This transforms as a (1,3) TC tensor on N . Using Eq. (22) to expandΓ α µν we find the only non-vanishing components of the projective curvature tensor to be By contracting the first and third indices of the projective curvature tensor, we can write the projective Ricci tensor whose only non-vanishing components are R bd is the equi-projective Ricci tensor from Eq. (16). The expressions in Eq. (48) are precisely of the form seen in conformal geometry where W a bcd is the Weyl tensor, P db is the Schouten tensor, and C cab is the Cotton-York tensor. In the above, W a bcd is analogous to K a bcd in Eq. (48). If we consider the contraction of the projective curvature tensor with a volume one-form g µ , that transforms as Eq. (28) and is also invariant under projective transformations, we can form the projective Cotton-York ναβ . Then we can write where ∆ a ≡ g a − α a is a one form on M. K(g) ναβ is now explicitly seen as a (0,3)-TC tensor on N and K nab is a (0,3)-tensor on M. When we introduce a metric tensor g am on M in the next section, we will find that g µ = (g a , 1 λ ), where g a ≡ − 1 d+1 ∂ a log |g|, is a suitable volume one-form, Eq. (59). This also introduces the projective Schouten tensor [16] P ab , which is a (0,2)-tensor on M. The form ofΓ α µν in Eq. (22) allows for D ab to become dynamical as K α µαβ = 0, relaxing the Ricci flat condition in [25,26,31]. This allows us to extend the Einstein-Hilbert action to projective geometry as in [10,19].
If we choose a member of the equivalence class [Γ c ab ], then we may express Π c ab in terms of a specific connection and its associated trace α µ . With this, one may write P ab in terms of D ab as The above is a generalization of [10,19], where constant volume coordinates were used and Γ e bc was regarded as Levi-Civita so α a = 0. Then, in that case, D ab = P ab and is a tensor in the volume preserving coordinates. As stated above, P ab transforms as a tensor on M which we may call the projective Schouten tensor in analogy with conformal geometry.

IV. COVARIANT METRIC TENSOR ON N
In projective geometry, a vector field χ on M may be lifted to a vector fieldχ on N by where κ a is some object that transforms as j a in Eq. (35), i.e.
under a general coordinate transformation on M. We write the components ofχ as Similarly, if a one form v on M can be related to a projective one formṽ viã It is clear thatχ αṽ α = χ a v a . A generic vector on N , which has components η ⊥ that are unrelated to vectors on M, may be written as The fundamental vector field Υ in Eq. (26) has no component parallel to M, for example.
We are interested in building an invariant action using the projective curvature. This will require a soldering metric which transforms as a tensor on N and which is projectively invariant. Taking a metric g ab on M, one may view this soldering metric as the local tensor product of two one-forms and write Here we have replaced κ a with g a ≡ − 1 d+1 ∂ a log |g| as it is naturally built from the metric degrees of freedom and does not introduce a connection. The constant λ 0 has units of length (like λ), and ensures that G µν remains dimensionless when g ab is dimensionless. Since G µν depends only on the spacetime metric g ab , it is indeed projectively invariant. One can check that G µν satisfies the transformation law when (x a , x λ ) → (y a , y λ ) = (y a , x λ |J| − 1 d+1 ). Furthermore, under this coordinate change the volume form on N remains invariant, i.e.
Here G(x a , x λ ) and G(y a , y λ ) are the metric determinants in the different coordinates. This follows since from Eq. (59), we see that where g is the determinant of g ab on M. Since , these terms exactly conspire in Eq. (61) to maintain the invariant volume on N . Again, this motivates why λ is called the volume coordinate. Lastly, the inverse of G µν is given by where g ab is the inverse of the spacetime metric g ab . This metric generalizes the work in [10,19], allowing TW gravity to be used in any coordinates. We can succinctly write the metric and its inverse as where we have defined g α ≡ (g a , 1 λ ). In TW gravity, the metric g ab , the projective invariant Π a bc , and the diffeomorphism field D ab will be treated as independent degrees of freedom in the spirit of the Palatini formalism [22].
V.γ µ ON N Now we seek theγ α matrices associated with the projective metric G µν given by Equation 59. The gamma matrices, γ m , on a d-dimensional spacetime are defined by where {·, ·} is the anti-commutator, g µν is the spacetime metric, N = 2 ⌊d/2⌋ , and I N is the Letγ µ be the gamma matrices for the metric G µν on N . These matrices satisfy as in Eq. (66). We will stay in even space-time dimensions. In this case, the gamma matrices γ µ for G µν will have the same dimension as the gamma matrices γ m for g mn .
Using the inverse of G µν , Eq. (63), we immediately must haveγ µ = γ µ if µ is a spacetime coordinate index, say m, and where γ m are the gamma matrices for the spacetime metric g mn . The remaining gamma matrix isγ λ . This matrix must satisfy Recall the chiral matrix γ 5 in four-dimensional spacetime. We will refer to it as γ d+1 in the general even dimensional case. It satisfies Comparing Eqs. (68) and (69) to Eqs. (70) and (71), we see that we should havẽ as the final gamma matrix for G µν . Explicitly, the chiral gamma matrix γ d+1 has the following construction in terms of the other gamma matrices in d-dimensions where a i = 0, . . . , d − 1 and ǫ is the totally antisymmetric Levi-Civita tensor on M. Specifically, for d = 4, the gamma matrices for G µν arẽ γ m = γ m when m = 0, 1, 2, 3 The fifth gamma matrix γ 5 is crucial in discussions about chirality, which we will see when we apply the TW connection to spinor fields. Eq. (74) shows that the volume bundle metric G µν explicitly builds in γ 5 . Thus, we will expect our dynamical theory for D mn to be chiral in nature when interacting with fermions.
Eqs. (72, 73, 74) also serve to further establish the relationship between the projective gauge field D mn and the notion of volume on M. Any Lagrangian for D mn will involve the metric G µν on N , which in turn can be constructed from gamma matrices. Eq. (72) says that one of these gamma matrices includes a rescaling of γ d+1 by λ, where γ d+1 is itself related to volume due to the presence of the epsilon tensor ǫ a 1 ...a d . The epsilon tensor is alternating in its indices and transforms as a tensor density that is used to construct volume forms on M. Therefore, we can again, view λ as a parameter which determines a rescaling of the volume element on M.

VI. THE VIRASORO ALGEBRA AND PROJECTIVE GEOMETRY
Here we will review three ways in which there is a correspondence between the projective connection's reduction to one dimension and the coadjoint elements of the Virasoro algebra.
The Virasoro algebra [4,33,34] may be regarded as the centrally extended algebra of vector fields in one dimension. Let (ξ, a) and (η, b) denote centrally extended vector fields in one dimension where a and b are elements in the center. Then the Lie algebra of these centrally extended vector fields is given through the commutator where ξ • η is defined via Here we explicitly expose the valence of the one dimensional vectors. The symbol ((ξ, η)) 0 is called the Gelfand-Fuchs two-cocycle [35] and is defined explicitly as where g ab is a one-dimensional metric. Eqs. (77) and (78) demonstrate an invariant pairing between ξ and η ′′′ . The Gelfand-Fuchs two-cocycle is an example of an invariant pairing between a vector and a quadratic differential B In the Gelfand-Fuchs two-cocycle, the pairing is between a vector ξ and a one-cocycle of η, where this one-cocycle is a projective transformation [4,5] that has mapped the vector field η into a quadratic differential. Explicitly, The invariant pairing in Eq. (79) follows if the action of another centrally extended algebra element, say (η, d), leaves the pairing invariant, i.e.
Then, a more general invariant two-cocycle relative to the centrally extended coadjoint element B = (B, c) can be written as One sees that the Gelfand-Fuchs case lives in the pure gauge sector, i.e. B = (0, c), of the space of coadjoint elements. It was also observed [5] that this action is the same as the action of the space of Sturm-Liouville operators on vector fields. Thus there is a correspondence where on the left side (B, c) is identified with a centrally extended coadjoint element of the Virasoro algebra and on the right side is a Sturm-Liouville operator with weight c and B(x) as the Sturm-Liouville potential.

A. Correspondence through the transformation laws
Here, we show how the relation between a coadjoint element of the Virasoro algebra and the Sturm-Liouville operator is reconciled by Thomas-Whitehead projective connections.
We will evaluate the connection in one-dimension where one can construct a Laplacian even though curvature is unavailable.
Consider the transformation of the diffeomorphism field D ab in one dimension. One can show that in one dimension, Eq. (35), i.e.
reduces to [32] under an infinitesimal coordinate transformation. We may let D = qD where q is an arbitrary constant. Then or equivalently Choosing q = 1 2c , we see a correspondence between the one-dimensional Thomas projective connection and the coadjoint element in Eq. (82). This improves the argument made in [10].

B. Correspondence through two-cocycles
The covariant metric allows us to improve upon another correspondence between the projective connection and coadjoint elements discussed in [10]. We consider a projective 2-cocycle on N for a path C as where σ parameterizes the path. The vector ζ µ ≡ dx µ dσ defines the path C. Here, the coordinates on N are x α = (x, λ). We choose the vector fields as ξ β = (ξ b , −λξ a g a ) and η β = (η b , −λη a g a ). Consider a path given by a fixed value λ = λ 0 along the vector ζ µ λ 0 = ( dx dσ , 0). The metric used to construct the projective Laplacian is the one-dimensional version of Eq. (63). Setting the metric to a constant g 11 and the components of the vector fields to ξ 1 and η 1 , respectively and keeping in mind that Π a bc = 0 in one dimension, one finds that Comparing this to Eq. (83), we make the observation that the projective connection and the coadjoint element (B, q) are in correspondence through which recovers Eq. (83) for q = c 2π .

C. Correspondence through gauge invariant action
Using the action in [10], we write the invariant projective Einstein-Hilbert terms as where we have used the projective Schouten tensor to write this in terms of the Riemann scalar curvature for familiarity. In two-dimensions, the Einstein-Hilbert term is the Gauss-Bonnet topological invariant. The Polyakov metric has constant volume and D ab and P ab are equivalent. Evaluating this on the Polyakov metric in two dimensions gives the coupling to the coadjoint element Again, the importance of this is to show dimensional universality of the interaction term in the Polyakov action as |G| K has meaning in any dimension. Thus, D ab is to the Virasoro algebra of one-dimensional centrally extended vector fields as the Yang-Mills gauge field A a is to affine Lie algebras in one-dimension. Furthermore, the projective curvature K α µνβ can be used to build dynamical theories for D ab just as the gauge curvature F ab can provide dynamics for the gauge fields related to external gauge symmetries.
Similarly, the frame fields denotedẽ µ α will be associated with the metric G µν on N , and the indices range over all dimensions, including λ for the curved coordinates on N We may also use the frame fields to write the components of the Dirac matrices in curved spacetime coordinatesγ For the metric G µν given by Eq. (59), the frame fields are listed as follows: 20 The inverse frame field components are then given by: e a m = e a m e 5 m = λ 0 g m e a λ = 0 LetΓ µ νρ be the components of the TW connection, and call∇ µ the corresponding covariant derivative operator that acts only on the curved indices as opposed to flat indices. Definẽ We use the geometric objectω of Eq. (98) to define a new spin covariant derivativẽ which recognizes tensorial objects, such as the vector V α , written in flat coordinates. We can take the full covariant derivative of a geometric object with curved and flat spacetime indices by using the ordinary connection coefficientsΓ µ νρ for curved indices and the spin connection coefficientsω µ αν for flat indices. From now on, we will denote this full covariant derivative operator by∇ µ . By construction, the frame fields are covariantly constant, satisfying Then for any vector V µ , we have∇ We simply need to plug these coefficients and the frame fields into Eq. (98) to get the TW spin coefficients that we desire. For example, if a, b = 5 and µ = λ (which aligns with our chosen index conventions), we havẽ and since η a5 = 0 andẽ c λ = 0, this reduces tõ where ω abm (without a tilde) is the coefficient of the spin connection for the underlying spacetime connection Γ n pm . Thus, we see how the TW spin connection coefficientsω abm are offset from the spacetime spin connection coefficients ω abm for a, b = 5. Below, we present the full list of independent TW spin connection coefficients: In Eqs. (104) and (105), we have explicitly written out Π ν ρµ in terms of a member of the equivalence class, Γ a bc and its trace α a . This allows us to see the relationship to the spin connection on M. It is clear thatω abρ is not anti-symmetric in a and b.

B. The spinor connection
Let Ψ(x) andΨ(x) be a spinor field and its Pauli adjoint, respectively representing a fermion and its anti-partner on the manifold N . Then the covariant derivative acting on the spinor is∇ Similarly, The spin connection in Eqs. (107) and (108) have in general both symmetric and antisymmetric components in their flat indices a, b. This is because the connectionΓ µ νρ on N is not a metric compatible connection, since∇ cannot be made metric compatible. The enveloping algebra of the gamma matrices is thus where the Sigma matrices generate the local SO(4, 1) Lorentz algebra on the Thomas cone, i.e.
Theω [αβ]µ therefore correspond to gauge fields for the local Lorentz transformation, while theω (αβ)µ generate a translation on the fermions to their tensor densities. Let us write Ω µ = Ω S µ + Ω A µ , such that the symmetric component is and the expected SO(4,1) connection is In Eq. (112), the space-time component of this Abelian connection is Γ a ac ≡ Γ c . In differential geometry, such a term appears in the presence of weighted spinors [36] that transform relative to an unweighted spinor φ as in four-dimensions. The spinor ψ vw is said to have weight (vI 4 + wγ 5 ). For these weighted spinors, the spin connection is augmented to be [36] Ω m → Ω m + (vI 4 + wγ 5 )Γ m .
We use this to define spinor representations ( 1 2 integer spin) on the Thomas Cone. First, we remark that on the Thomas Cone γ 5 is an invariant tensor since , we can expect weighted spinor representations on the Thomas Cone to be .
And forΨ,/ ω abmγ aγbγm +ω abλγ aγbγλ +ω 5bmγ 5γbγm +ω 55λγ 5γ5γλ . (120) Evaluating these with the coefficients from Eq. (105) yields whilst/ i λ 0 D rm − ∂ m g r + Γ n rm g n + α m g r + g m α r g mr Here, ∇ m (without a tilde) is the spinor covariant derivative operator associated with the space-time connection Γ m nr . Using this decomposition, we write Eq. (121) as where we've defined B m and Ξ as Ξ ≡ λ 0 (D rm − ∂ m g r + Γ n rm g n + α m g r + g m α r ) g mr The TW Dirac Lagrangian Density that yields the Dirac equation for a mass M and a chiral mass M χ , may be written explicitly in covariant and self conjugate form as The last term arises because the metric and covariant derivative operator are not compatible does not vanish. The commutator term is precisely where the field D mn resides. We can rewrite this so that the field equations on Ψ (orΨ) are explicit if we integrate by parts the derivative term onΨ. Then The total space-time derivative may be eliminated on the boundary. However the total λ derivative will in general be finite and could contribute to the field equations. Let us examine this term more carefully. One sees that From the spinor projective representations in Eqs. (116) and (117), this will vanish when w = 0, eliminating any chiral density terms. We also observe that the term, g mΨ γ m Ψ, would vanish if the coordinates were gauge fixed so that g m = 0 (constant volume Had we wished to add a Yang-Mills potential to the action, we would have a term where a chiral mass term M A = 1 λ 0 is induced. This follows since the corresponding projective one-form for the matrix valued potential is where 1 is in the center of the algebra. Then, using Eq. (72), we have the result In the Lagrangian density, Eq. (134), we have left terms with explicit λ dependence of Ψ. Following Eqs. (116) and (117), along with the requirement that the action be a scalar, where v is the density weight which determines precisely how Ψ will transform under λ → λ ′ .
In the TW Dirac Lagrangian L T W D of Eq. (134), this representation of Ψ will only affect the terms so that the TW Dirac Lagrangian can be reduced to a Lagrangian on ψ with v a weight parameter A special choice of the weight v = − 1 2 eliminates the induced chiral mass term (in the absence of gauge fields) and also eliminates the metric density contribution in the coupling to iφγ m φ.
The only λ dependence is in the overall coefficient λ 0 λ |g|. As we will discuss in Section VIII, we may write ℓ ≡ λ/λ 0 to be a dimensionless scale. By writing |G| = 1 where ℓ i and ℓ f are original and final length scales. With this, we can make a field redefinition of the fermions φ and define ψ = φ λ 0 log(ℓ f /ℓ i ) so that the fermions ψ have the dimensions of four-dimensional fermions. The four-dimensional TW Dirac action becomes We see that λ 0 still sets the chiral scale due to its presence in the last two summands of the action.
In the discussion following Eq. (74), we noted that we should expect a dynamical theory of D mn to be sensitive to chirality of fermions. This expectation is realized by the TW Dirac Lagrangian, Eq. (134), due to the presence of γ 5 . The theory is therefore chiral in this sense. We remark that one can still eliminate d degrees of freedom by using a coordinate gauge choice. For example, we could set g a = 0 (constant volume gauge for the metric), α a = 0 (constant volume for the connection) or even g a = α a (compatibility of condition) in Eq. (141). However, no gauge choice will eliminate the D ab fermion interaction.

VIII. GAUGE INVARIANT TW ACTION
The TW Action was introduced in [10] in order to give dynamics to the diffeomorphism field. There, the correspondence with the coadjoint orbits of the Virasoro algebra was determined in the background of the gauged fixed 2D metric of Polyakov [21] that had constant volume. Similarly in [19], the interest was to study the diffeomorphism field as a primeval source for dark energy in a Friedman-Lemaitre-Robertson-Walker background in constant volume coordinates. As we have just seen in the Dirac action, writing the TW action in a gauge invariant form reveals physically interesting structure. From [10] the TW dynamical action is: where the projective Einstein-Hilbert action is and the projective Gauss-Bonnet action is We remark that both terms are generalized Gauss-Bonnet terms and one could presumably continue adding generalized Gauss-Bonnet terms for higher interaction without compromising causality in the metric field equations [23]. Recall that the components of the TW curvature tensor K α βγρ are given by where again for any affine connection Γ a bc . The non-zero components of the TW Ricci tensor K αβ are Then the projective Gauss-Bonnet action S P GB may be decomposed as and we have defined the Gauss-Bonnet operator as and for convenience, in terms of the metric on M, B bbgḡrr aā = g aā g bb g gḡ g rr − 4δ g a δḡāg bb g rr + δ g a δḡāg br gbr .
Finally, we can write the full dynamical action as S = S P EH + S P GB1 + S P GB2 + S P GB3 This form of the action is convenient for computing field equations. The curvature components K a bcd and K λ bcd carry all of the Π a bc and D bc (equivalently Γ a bc and P bc ) dependence, while the metric tensor g ab appears elsewhere in each part of the action, including in the Gauss-Bonnet operator B.
To illustrate explicit general coordinate invariance, it is also possible to decompose the action as Eqs. (146) and (148) demonstrate that K a bcd , K ab , and K are tensors on the spacetime manifold M. Furthermore, we introduce K bcd as the following rank-three tensor on M where ∇ a is the covariant derivative operator associated with the spacetime connection Γ a bc . Since g a and α a have the same coordinate transformation law, we see that K bcd is indeed a tensor on M. This demonstrates that the action is a scalar as well as projectively invariant.
Owing to Eq. (154), all the λ dependence appears as overall coefficients. We will use the interpretation of the coupling constants as in [19] to write them in terms of scale dependent quantities. Let ℓ ≡ λ/λ 0 be a dimensionless scale. Since only dλ 1 λ appears in the overall coupling, we again write |G| = 1 ℓ |g|. Then by integrating over ℓ, we can rewrite the action in terms of coupling constants that have familiar interpretations Thus a natural scaling of the gravitational coupling constant κ 0 and angular momentum parameter J 0 occurs as we move from one length scale to another. In this way, projective geometry has a potential renormalization group interpretation. This link is under further investigation. The characteristic projective length scale (inverse mass scale) is set by λ 0 .
With this, we can rewrite the TW action as where K bcd has indices raised by the inverse metric on M

IX. THE COVARIANT FIELD EQUATIONS
In the spirit of Palatini [22], we will treat the metric tensor g ab andΓ α βγ as independent degrees of freedom. This fits the framework of TW gravity, since the TW connection is to be thought of as a connection over the space of equivalence classes of connections and is not naturally tied to a particular metric. The metric G µν serves only to maintain general coordinate invariance on N , just as D ab exists in order to make the connection∇ µ covariant.
The covariant derivative is a projective invariant that is constructed only from projectively invariant quantities such as Π a bc and λ. However, as one sees in Eq. (22), the only degrees of freedom that are allowed to fluctuate are D ab and Π a bc . Therefore we will only need the field equations for Π a bc , D ab , and g ab . We note that λ does not fluctuate and only sets the volume scale.

A. Equations of motion for Π a bc
In order to simplify the computation of the field equations, we will use F to denote an object with the correct valence to form a scalar with another given object. For example, we might write an expression such as K a bcd F , where we would understand that F is an object with components F bcd a such that F forms a scalar upon tensor multiplication with K a bcd .
With this, we compute the field equations for Π a bc as: and These two variations lead to the full equations of motion for Π a bc that are associated with the appropriate object F . We have where the striked-out terms vanish because Π a bc as well as its variation δΠ a bc are traceless. The remaining contribution to the field equations would vanish if Π a bc were the traceless Levi-Civita connection of the metric g ab , consistent with the original Palatini equations [22]. The next contributions are |g|K a bcd B nbmced la − B nbecmd la +λ 2 0 g l g a g nb g ec g md − λ 2 0 g l g a g nb g mc g ed + |g|K a bcd Π n ef B f bmced la − B f becmd la +λ 2 0 g l g a g f b g ec g md − λ 2 0 g l g a g f b g mc g ed +Π e f l B nbf cmd ea − B nbmcf d ea +λ 2 0 g e g a g nb g mc g f d − λ 2 0 g e g a g nb g f c g md δΠ l mn (162) λ K λ f gh g l g bf g mg g dh Π n db + g a g nf g cg g mh Π a cl + |g|K a bcd g a g bm g cg g dn D gl δΠ l mn .
By definingK bgr a = KābḡrG bbgḡrr aā and g β = (g b , where the sums are restricted to M coordinates, the field equations for Π l mn may be written as Here,∇ a is the derivative operator with respect to the fundamental projective invariant Π l mn . We note that if the connection were chosen to be compatible with the metric g ab , then in the language of Tractor Calculus [18], Eq. (166) would imply that the projective curvature is Yang-Mills [16].

B. Equations of motion for D bc
To find the field equations for D bc , we proceed in the same manner as we did for Π a bc . The contributions are of the form: and Again, by assigning the appropriate object F to each term we have: δS P GB1 = 2J 0 c d d x |g|K e f gh B qf cgph ce − B qf pgch ce +2λ 2 0 g c g e g qf g pg g ch δD pq (170) λ K λ f gh g a g qf g ag g ph + ∂ g |g|K a bcd g a g bq g cp g dg + |g|K a bcd g a g bf g cp g dg Π q f g δD pq .
Then the variation with respect to δD pq yields Note the partial derivatives of K λ bcd , which make the field equations second-order differential equations in D bc .

C. Equations of motion for g bc
Finally, we will find the field equations for the spacetime metric tensor g bc . These equations will define the energy-momentum tensor Θ pq from the variation of the action with respect to the inverse metric g pq , i.e.
One recognizes that this implies the Einstein equations with κ 0 = 8πG c 4 , and where the energy-momentum tensor is defined in the usual way, i.e.
With this in mind, we will be able to extract the energy-momentum tensor by subtracting out the Einstein tensor from the action. First, let us consider the Einstein-Hilbert part of the action S P EH . It has the variation We recognize that the variation of S P EH has the form of the Einstein field equations, with K a bcd taking the place of R a bcd . This is expected. For the S P GB1 term, we first find Putting this all together defines the TW energy-momentum tensor Θ TW ij as where K bcd = g a K a bcd + 1 λ K λ bcd . This demonstrates that the energy-momentum tensor is indeed a tensor on M. Since we have used the Gauss-Bonnet action to describe dynamics for D ab , the field equations are second-order differential equations in g ab .

X. GEODESIC DEVIATION
To complete this study of the gauge covariant field equations and gauge invariant action we examine the geodesic deviation equations on the Thomas Cone and their image on the manifold M. Not only does geodesic deviation have importance in tidal forces, it can also provide a mechanism to study radiative degrees of freedom in D ab . Here, we will examine the modification to geodesic deviation that results from the presence of the projective gauge field D bc . A review of geodesic deviation and its derivation in general relativity can be found in textbooks such as [37].

A. The geodesic deviation equation
Let M be the spacetime manifold equipped with a metric g ab . Recall the geodesic equation for any connection Γ a bc on M where τ is some parameter. Here f (τ ) = 0 if and only if τ is an affine parameter for Γ a bc . In the presence of a gravitational field where the connection Γ a bc is compatible with the metric, freely moving objects will travel along geodesics specified by Eq. (183).
Consider the space of geodesics x a (s, τ ), where for each fixed value s = s 0 , we have that x a (s 0 , τ ) is a geodesic with affine parameter τ . This gives us a one-parameter family of geodesics which allows us to examine geodesics that are close to each other. The geodesic tangent vector T a (s, τ ) and geodesic deviation vector X a (s, τ ) are given by Eq. (184) leads to an immediate relation between derivatives of T a and X a ∂X a ∂τ = ∂T a ∂s .
For a vector field V a on M, the intrinsic derivative of V a along a curve x a (τ ) is given by Using Eq. (186), we can find an acceleration by taking the second intrinsic derivative of a vector field. If we do this with the geodesic deviation vector X a (s, τ ) with respect to τ , we find Eq. (187) can be simplified since x a (s, τ ) is a geodesic curve for all fixed s. Due to this fact, we know that Expanding Eq. (188) and rearranging terms yields Using Eq. (189), we eliminate ∂ 2 T a ∂s ∂τ from Eq. (187) and find D 2 X a ∂τ 2 = (∂ c Γ a db − ∂ d Γ a cb + Γ a ce Γ e db − Γ a be Γ e cb ) T c T This is the geodesic deviation equation. Note we did not use metric compatibility to arrive at this expression. The full Riemann curvature tensor R a bcd appears in the geodesic deviation equation, including the Weyl term which does not usually appear in Einstein field equations.
Gravitational radiation can influence geodesic deviation directly making it a useful observational tool. We will now explore the projective modifications of the geodesic deviation equation and insights on how the diffeomorphism field may be observed.

B. Projective geodesic deviation
We turn our attention to the diffeomorphism field which we also may consider as the projective gauge field D bc . To compute the resulting geodesic deviation on the spacetime manifold M for a general connection, we first must find the geodesic deviation of the TW connection on N , and project this deviation down onto M.
From Eq. (190), the geodesic deviation X α (τ ) of the TW connection on N is given by where the Greek indices range over all coordinates on N . Now, as in Eq. (58) let X α = (X a , −λX a g a + X 5 ) define the projective geodesic deviation vector. We have included a perpendicular component as physical vectors such as X α = Ψγ α Ψ might arise. However, for simplicity we will ignore the X 5 component in this discussion. We have used g a defined via a metric on N so as not to spoil the projective covariance of the equation. Let us first consider the geodesic deviation X a where a is a spacetime manifold coordinate specifically (not λ). Since the only non-vanishing components of K α βσρ are the components K λ bcd and K a bcd , then Eq. (191) reduces for α = a to Here R a bcd is the Riemann curvature tensor for a connection Γ a bc which is not necessarily compatible with the metric defining g a . Now the parameter τ is an affine parameter for the TW connection on N , not for the Γ a bc connection on M. If we make a change of parameterization τ → u so that u is an affine parameter for the spacetime manifold connection, we get using Eq. (42) here show precisely how any Dirac fermion will interact with the diffeomorphism field and how chiral masses become manifest due to a volume scale. These gravitationally induced chiral masses are affected by the dimension of the manifold, the number of gauge fields and the spinor's tensor density.
The use of geodesics extends far beyond gravitational theories and these results may be of value in fluid dynamics, optimization, other gauge theories and even quantum computing.
Several projects applying the general TW theory presented in this paper are currently underway including the quantization of the fully covariant TW theory, sourcing of cosmological inflation, constraints imposed by affects on gravitational radiation, and applications to the understanding of dark matter.