Sources of torsion in Poincarè gauge gravity

We study sources for torsion in Poincarè gauge theory in any dimension, signature, and spin. We find that symmetric kinetic terms for non-Yang–Mills bosonic fields of arbitrary rank drive torsion. Our detailed discussion of spin-3/2 Rarita–Schwinger fields shows that they source all independent parts of the torsion. We develop systematic notation for spin-(2k+1)/2 fields and find the spin tensor for arbitrary k in n ≥\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ge $$\end{document} 2k+1 dimensions. For k>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k>0$$\end{document} there is a novel direct coupling between torsion and spinor fields. We also cast the well-known gauge relation between the canonical and Belinfante–Rosenfield energy tensors in terms of different choices of independent variables.


Introduction
The development of Riemann-Cartan geometry using the Einstein-Hilbert action is known as the Einstein-Cartan-Sciama-Kibble (ECSK) model of gravity.Its long history begins with Cartan's generalization of Riemannian geometry [13,14,15,16].A few years later Einstein used torsionful geometry to discuss teleparallel model [17] though this theory is not cast in the same terms as general relativity.Originally, the evolving ECSK theory was the study of the metric variation of the Einstein-Hilbert action S EH [g] in a Riemann-Cartan geometry.The gauge theory approach was more fully developed starting with Utiyama and continuing with the work of Sciama and Kibble, taking its present form with the work of Ne'eman and Regge [2,3,4,7,8,9,10].A detailed review is given in [18].With the advent of modern gauge theory it has become natural to vary both metric and connection S EH [g, Γ] or both solder form and spin connection S EH [e, ω].
Basing gravity theory on the Einstein-Hilbert action with source fields, torsion is found to be nonpropagating and vanishing away from material sources.This is perhaps a benefit, since there is no direct experimental evidence of torsion, and limits on torsion coupling to matter are strong (see Donald E. Neville1 [19]).For this reason, much study of ECSK theory has focussed on showing that torsion does not persist in physical situations (e.g., [20]).It is natural that the seemingly pathological non-integrability, the anomolous effect on angular momentum, and in general the extreme success of general relativity should have this effect.Nonetheless, the study of ECSK theory has drawn considerable attention over the last century, including generalizations to propagating torsion [19,21,20,22,23].The latter have been criticized as incapable of simultaneous unitarity and normalizability [24].
On the other hand, a deeper understanding of geometry and general relativity is to be gained by fully exploring nearby theories.This is the goal of the present work: to describe broad classes of sources for torsion in Poincarè gauge theory.Our results hold in any dimension n and any signature (p, q).The exercise includes some important physical predictions, since some of the sources we discuss, notably the spin-3 2 Rarita-Schwinger field, are predicted by string and other supergravity theories.
In the next Section we present the basic properties of Poincarè gauge theory using Cartan methods.We include the structure equations, Bianchi identities, the solution for the spin connection in terms of the compatible connection and the contorsion, and the decomposition of the torsion into invariant parts.These results are geometrical.
The ECSK action is introduced in Section (3).The variation of the solder form makes the canonical energy tensor asymmetric.It is well-known that the choice of the local Lorentz gauge leads to the symmetric Belinfante-Rosenfield energy tensor.Alternatively, in the gravity case the same result is achieved using a diffeomorphism.Here we delve a little more deeply into this equivalence, discussing two gauge-equivalent but distinct methods of variation.We show that the difference between the Belinfante-Rosenfield and canonical energy tensors corresponds to a different choice of independent variables.For the first variation the action is taken as a functional of the solder form and the full spin connection, S [e a , ω a b ], in the spirit of Palatini but allowing torsion.The second variation uses the decomposition of the spin connection into a compatible piece and the contorsion tensor ω a b = α a b + C a b .This directly respects the Lorentz fiber structure of the bundle by varying only the Lorentz tensors-the solder form and the contorsion, while treating the compatible part of the spin connection as a functional of the solder form α a b = α a b (e a ).With matter, δS [e a , ω a b ] leads to the canonical energy while S [e a , C a b ] gives the symmetric Belinfante-Rosenfeld energy tensor.The fact that the two sets of independent variables are related by a local Lorentz tranformation recovers the known result.The field equations in the two cases are identical, modulo the constraints of Lorentz invariance.
The bulk of our investigation, presented in Section (4), concerns the effects of various types of fundamental fields on torsion.The exceptional cases of Klein-Gordon and Yang-Mills fields are treated first.The actions for these fields do not depend on the spin connection and therefore do not provide sources for torsion.Next, we study a class of bosonic fields of arbitrary spin with actions quadratic and symmetric in covariant derivatives.Except for scalars, these drive torsion.In Subsection (4.3) we derive the well-known axial current source for totally antisymmetric torsion arising from Dirac fields.We also check the effect of nonvanishing spin tensor in the limit of general relativity where the torsion vanishes.
The effect of the less thoroughly studied Rarita-Schwinger field on torsion is examined in Subsection (4.4).While the axial source for Dirac fields arises from the anticommutator of a γ-matrix with the spin connection, the Rarita-Schwinger field couples through a similar anticommutator but with the product of three γ-matrices.In addition, we find a new direct coupling of the spin- 3  2 field to torsion.Unlike the Dirac field with only an axial current source, the Rarita-Schwinger field drives all three independent pieces of the torsion.Except in dimensions 5, 7 and 9, spin- 3  2 fields have enough degrees of freedom to drive all components of the torsion independently.
Finally, we introduce new compact notation for spin-2k+1 2 spinor-valued p-form fields in Subsection (4).This enables us to write actions for arbitrary k and find the general form of the spin tensor.The physical properties appear to echo those of the Rarita-Schwinger field.
We conclude with a brief summary of our results.

Poincare gauge theory
All results below hold in arbitrary dimension n = p + q and signature s = p − q.The group we gauge is then SO (p, q) or Spin (p, q) with the familiar spacetime case having p = 3, q = 1.

The structure of Riemann-Cartan geometry
We review the formal features of Poincarè gauge theory.All results below hold in arbitrary dimension n = p + q and signature s = p − q so while we continue to refer to the Poincarè group ISO (3, 1) and its Lorentz subgroup SO (3, 1) we actually work with P = ISO (p, q) or P = ISpin (p, q) with subgroups L = SO (p, q) or L = Spin (p, q) respectively.The local Lorentz arena for general relativity in n dimensions follows by setting q = 1.
The most relevant results of the Cartan construction are the 2-form expressions for the Riemann-Cartan curvature R a b and torsion T a in terms of the solder form e a and spin connection ω a b .
Each of these may be expanded in the orthonormal basis In a coordinate basis T a is given by any antisymmetric part of the connection.The Bianchi identities generalize to where the covariant exterior derivatives are given by The frame field e a is (p, q)-orthonormal, e a , e b = η ab = diag (1, . . ., 1, −1, . . ., −1), with the connection assumed to be metric compatible Since dη ab = 0, the spin connection is antisymmetric, ω ab = −ω ba .When the connection is assumed to be compatible with the metric, Eqs.( 1)-( 6) describe Riemann-Cartan geometry in the Cartan formalism.Note that the Cartan-Riemann curvature, R a b , differs from the Riemann curvature R a b by terms dependent on the torsion.When the torsion vanishes, T a = 0, the Riemann-Cartan curvature R a b reduces to the Riemann curvature R a b and Eqs.( 1) and ( 2) exactly reproduce the expressions for the connection and curvature of a general Riemannian geometry.At the same time, Eqs.( 5) and ( 6) reduce to the usual first and second Bianchi identities.
The structure equations, Eqs.( 1) and ( 2), allow us to derive explicit forms for the connection and curvature.From Eq.( 2), the result for the spin connection is where C a b is the contorsion, The decomposition of the connection is unique.Local Lorentz transformations transform α a b inhomogeneously in the familiar way while torsion and contorsion are tensors.The form of contorsion (9) in terms of torsion is unique and invertible.
We may recover the torsion by wedging and contracting with e b .
The torsion now enters the curvature through the connection.Expanding the Cartan-Riemann curvature of Eqs.(1) using Eq.( 8) and identifying the α-covariant derivative, This is the Riemann-Cartan curvature expressed in terms of the Riemann curvature and the contorsion.Note that the α-covariant derivative is compatible with the solder form, De a = de a − e b ∧ α a b = 0.If we contract with e b we recover the Bianchi identity.This happens because our solution for the connection automatically satisfies the integrability condition for the connection.
Given Eq.( 10) for the Cartan-Riemann curvature in terms of the Riemannian curvature and connection, we may also expand the generalized Bianchi identities of Eqs.( 5) and (6).The first Bianchi becomes Using De a = 0 and replacing C c b ∧ e b = T c leads to the Riemannian Bianchi e b ∧ R a b = 0. Similarly, expanding the derivative in the second Bianchi gives

to several cancellations and finally
DR a b = 0 so that the Cartan-Riemann Bianchi identities hold if and only if the Riemann Bianchi identities hold.The first Bianchi identity relates the triply antisymmetric part of the curvature tensor R a b to the exterior derivative of the torsion.Expanding both sides of Eq.( 5), antisymmetrizing and contracting leads shows that the antisymmetric part of the Ricci-Cartan tensor is simply minus the divergence (11) where we define In 2-dimensions T a bc includes only one of the two degrees of freedom of the torsion.Although the Ricci tensor of the Cartan-Riemann curvature acquires an antisymmetric part there remains only a single independent contraction.Because the curvature is a 2-form, and the spin connection is antisymmetric, the curvature still satisfies These results are geometric; a physical model follows when we posit an action functional.The action may depend on the bundle tensors e b , T a , R a b and the invariant tensors η ab and e ab...d .To this we may add source functionals built from any field representations of the fiber symmetry group L, including scalars, spinors, vector fields, etc.
Constraining the torsion to zero, specifying the Einstein-Hilbert form of action, and varying only the solder form, the q = 1 theory describes general relativity as a gauge theory in n-dimensions.We cannot vary the metric and connection independently because this can introduce nonzero sources for torsion, making the T a = 0 constraint inconsistent.
Dropping the torsion constraint while retaining the Einstein-Hilbert action gives the Einstein-Cartan-Sciama-Kibble (ECSK) theory of gravity in Riemann-Cartan geometry.The torsion is found to depend on the spin tensor, given by the connection variation of the source σ µ ab = δL δω ab µ .Without modifying the action to include dynamical torsion, the resulting torsion survives only within matter.

Decompostion of the torsion
We identify well-known invariant pieces of the torsion.The torsion includes a totally antisymmetric piece T abc e abcde e d ∧ e e in particular giving the well-known axial vector in 4-dimensions.There is also a single vectorial contraction, T b ba .
In components the decomposition is simply While the vector and pseudovector each have 4 degrees of freedom in 4-dimensions, the situation is very different in higher dimensions.In general the torsion has a total of n 2 (n−1) 2 degrees of freedom.Therefore, while the trace contains only n degrees of freedom for a fraction 2 n(n−1) ∼ 1 n 2 of the total, the antisymmetric part includes 1  3! n (n − 1) (n − 2) or roughly n−2 3n ∼ 1 3 .The residual tensor τ a includes the remaining 3n(n−1) ∼ 2 3 .Thus, the antisymmetric part is a major contributor in higher dimensions.

ECSK theory
The physical content of the Einstein-Cartan-Sciama-Kibble theory enters through use of the Einstein-Hilbert action in Riemann-Cartan geometry.q 1 e ε abcd .Now the ECSK action in n-dimensions is where with R ab the Riemann-Cartan curvature scalar, S matter has the general form for fields ξ A of any type, and S GHY is the Gibbons-Hawking-York surface term where K is the trace of the second fundamental form, ǫ = ±1 and h the induced metric on the boundary δM.
For the gravity action, we restrict attention to the Einstein-Hilbert form but with the Riemann-Cartan scalar curvature.Alternatives with propagating torsion are considered in [19,21,20,22], and with additional modification in [26].
The Gibbons-Hawking-York surface term [27,28,29,30] is necessary because fixing both δe a = 0 and δω a b = 0 on the boundary overdetermines the solution in the bulk.This can be seen from the conditions for the initial value problem-specifying the metric and the intrinsic curvature of an initial Cauchy surface is enough to propagate a unique solution as the time evolution.It is straightforward to check that adding the Gibbons-Hawking-York surface term resolves the issue, while leaving the expected field equations in the bulk.Having checked this, our considerations below focus on the gravity and matter terms of the action.
The form of the field equations (and the physical content for some nonstandard sources) also depends on the choice of independent variables.While only the metric was varied in the original formulation in Riemann-Cartan geometry, the gauge theory approach leads naturally to a Palatini variation, S [e a , ω a b ].With this change and certain sources, it becomes impossible to set the torsion to zero since varying the spin connection in some matter actions gives nonvanishing sources for torsion.We explore the nature of these sources for a variety of types of field in Section 4, which contains our principal results.
Before studying sources for torsion, we examine the dependence of the field equations on the choice of independent variables.It is immediate that the asymmetry of the solder form means that the Einstein tensor and canonical energy tensor may acquire antisymmetric parts [25].Belinfante [31] and Rosenfeld [32] showed that a symmetric energy tensor may be formed by adding a combination of spin currents to the canonical energy.The resulting Belinfante-Rosenfeld energy tensor remains divergence-free.It is well-known that the difference between the canonical and symmetric energy tensors can be traced to a change of local Lorentz gauge, or in Riemannian geometry to a diffeomorphism.
Perhaps not so widely recognized is the relationship between the two forms of energy tensor and the choice of independent variables.In the remainder of this Section, we show first that the Palatini variation δS [e a , ω a b ] = 0 leads to the asymmetric form of the Ricci tensor and the canonical energy tensor.Next, we show that the antisymmetric part of the Einstein equation is identically satisfied as a consequence of Lorentz invariance.Finally, we show that the alternative variation δS [e a , C a b ]-allowed by the decomposition of the connection in Eq.( 8)-leads to a symmetric combination of the Ricci and spin tensors for the geometry and to the symmetric Belinfante-Rosenfeld energy tensor for matter sources.The symmetrizing terms arise because the compatible part of the connection must be treated as a functional of the solder form,

Palatini variation
Variation of the Riemann-Cartan curvature with respect to the spin connection leads to the ω ab -covariant derivative of the variation, δR ab = D δω ab = d δω ab − δω eb ∧ω a e −(δω ae )∧ω b e .To avoid ambiguities for surface terms, we integrate only the exterior derivative by parts, using Lorentz invariance of the Levi-Civita tensor to redistribute the spin connections in the remaining terms.
Varying the solder form and connection independently then leads to The asymmetric Einstein tensor is sourced by the asymmetric canonical energy tensor while the torsion is sourced by the spin tensor The resulting form of the field equations displays a pleasing completeness.The two Casimir operators of the Poincarè group show the invariance of mass and spin, and the corresponding energy and spin tensors provide the sources for curvature and torsion, respectively.
In the absence of sources T c ab = 0 implies vanishing torsion for all n > 2, and therefore vanishing contorsion, C a c = 0. Using Eq.( 10) to separate the usual Einstein tensor from the contorsion contributions and setting C a c = 0 reduces R a b to the Riemann curvature R a b and therefore to the usual Einstein equation of Riemannian geometry, R ab − 1 2 η ab R = 0. Therefore vacuum Poincarè gauge theory reproduces vacuum general relativity.However, the theories differ when matter fields other than Yang-Mills or Klein-Gordon type are included.

Lorentz invariance and the symmetry of the energy tensor
Now consider Noether's theorem applied to the action.Each term in the total action must be Lorentz invariant, and this implies conditions on the fields.We express these conditions for a general action. .This means that under an infinitesimal gauge transformation we must include changes in both the solder form and the spin connection.
Neglecting the GHY surface term the action is Application of the Noether theorem to each term is straightforward.Following a general variation, we impose the field equations and restrict the variation to the symmetry.
For the matter action we immediately have and with ε ab = −ε ba otherwise arbitrary this fixes the antisymmetric part of the canonical energy tensor to equal the divergence of the spin tensor.
The calculation for the gravity action is somewhat more involved.The steps result in a vanishing divergence The final term vanishes while the first two resolve to With the arbitrariness of ε ab and the definition T a bc = T a bc − δ a b T e ec + δ a c T e eb this becomes which is in exact agreement with the trace of the Bianchi identity, Eq.( 11).
This fully resolves the difference between asymmetric and symmetric energy tensors.Combining the requirements for Lorentz invariance, Eqs.( 19) and (20), with the field equations (17) shows that the symmetric part of the Einstein tensor equals the symmetric part of the energy source, with torsion sourced by the spin tensor as before Using the Lorentz conditions the antisymmetric part of the Einstein equation now becomes ab and this is now automatically satisfied as the divergence of the torsion field equation 22.

The Belinfante-Rosenfeld energy and the choice of independent variables
In 1940 Belinfante [31] and Rosenfeld [32] showed how to modify the canonical energy with the addition of divergences of the spin tensor to produce a simultaneously symmetric and conserved form of the energy.The Belinfante-Rosenfeld modification is related to the canonical energy by Lorentz transformation.Here we derive the Belinfante-Rosenfeld energy directly by a judicious choice of independent variables.Equation (18) shows how Lorentz invariance involves both the solder form and the spin connection.This means that Lorentz transformations will mix the field equations following from the Palatini form of variation.We can change this using the decomposition of the spin connection into compatible α a b and contorsion C a b pieces, Eq.( 8).This combines both solder form and connection variations because compatibility requires α a b to be varied as a functional of the solder form.This is sufficient to subsume the Lorentz conditions ( 19) and ( 20 The Einstein-Hilbert variation leads to the replacement of the Einstein tensor G ab = R bc − 1 2 Rη bc by while the canonical energy is replaced by the Belinfante-Rosenfeld energy tensor The added terms enforce the symmetry of each side.Antisymmetrizing the gravitational terms by Lorentz invariance (20) and for T BR bc we have again directly expressing the Lorentz invariance found in Eq.( 19).
If we write the alternative form of the curvature field equation in symmetric and antisymmetric pieces we see that the added terms in the symmetrized part vanish by the torsion field equation while each of the two antisymmetric combinations vanishes by Lorentz invariance, Eqs.( 20) and ( 19) respectively.The two variations δS [e a , ω a b ] = 0 and δS [e a , C a b ] = 0 are therefore equivalent.We therefore see that the Belinfante-Rosenfeld tensor arises directly as the gravitational source when the independent variables are chosen as the compatible connection and the contorsion, and the proof of its symmetry follows from Lorentz invariance.Conservation of energy follows as usual in general relativity, from general coordinate invariance.

Sources for torsion
We now come to our central results, the study of various sources for torsion.Before considering fields with nonvanishing spin tensor, we note some classes for with σ a bc = 0. Fields other than these exceptional types generically drive torsion.

Exceptional cases
There are two important exceptional cases-Klein-Gordon fields and Yang-Mills fields.
Appropriately for a scalar field, there is no spin tensor.This holds true for internal multiplets of scalar fields φ i as well.

Yang-Mills fields
Yang-Mills fields comprise the second important class of exceptions.Let i, j, . . .index the generators of an internal Lie symmetry g ∈ G, that is, the fiber symmetry of a principal fiber bundle.Then the connection satisfies the Maurer-Cartan equation, dA i = − 1 2 c i jk A j ∧A k where c i jk are the structure constants.Curving the bundle the field strength is independent of the spacetime connection and the corresponding action has vanishing spin density.The result also holds for p-form electromagnetism [33] and the Proca field [34].These observations mean that the Higgs and Yang-Mills fields of the standard model do not drive torsion.

Bosonic matter sources
The currents of generic bosonic sources have nonvanishing spin tensors.We consider source fields of arbitrary integer spin Θ a...b having quadratic kinetic energies.
When the kinetic term of the fields is symmetric in derivatives we have where we assume Q AB = Q BA is independent of the connection, though it may depend on the metric.The field equations (17) or the reduced equations (21,22) hold without modification.We need only find the relevant variations of the matter actions.
For these fields the solder form variation only enters through the metric variation as η ab (δe µ a e ν b + e µ a δe ν b ) = δg µν since Therefore the energy tensor takes the usual symmetric form plus any (symmetric) dependence on Q AB .The spin tensor is therefore This has the form of a current density.
From Lorentz invariance Eq.( 19) and the symmetry of the energy tensor T [ab] = 0 we immediately have conservation of the spin tensor We conclude that for the types of bosonic action considered the Poincarè gauge equations take the form Coupling such higher spin fields to other sources may lead to failure of causality or other pathologies.For example, for a vector field with Q ab = η ab the kinetic action is simply so the energy tensor has the usual form and the current density is simply The torsion remains nonpropagating and vanishes whenever the source field Θ b vanishes.

Dirac fields with torsion
It is well-known that the Dirac field provides a source for torsion (among the earliest references see, e.g., [35,36,37,38,39,18,40]).The flat space Dirac action takes the same form in any dimension where ✁ ∂ = γ a e µ a ∂ µ .The principal difference in dimension n is that the spinors are representations of Spin (p, q) and therefore elements of a 2 [ n 2 ] -dimensional complex vector space while the γ a satisfy the Clifford algebra relations γ a , γ b = −2η ab 1 where η ab is the (p, q) metric.
However, in a curved space the spin connection introduces an additional term.The covariant derivative of a spinor is given by where h is Hermitian h † = h and reality of a vector v a = ψ † hγ a ψ under Spin (p, q) requires It follows that σ ab † h = −hσ ab .While h is generally taken to be γ 0 in spacetime, h transforms as a 0 2 spin tensor while γ 0 transforms as a 1 1 spin tensor so that h = γ 0 can hold only in a fixed basis.There exist satisfactory choices for h in any dimension or signature (see below).The solder form components e µ a connect the orthonormal basis of the Clifford algebra to the coordinate basis for the covariant derivative, γ a e µ a D µ .The conjugate action now differs, so we take the manifestly real combination showing that the connection now couples to a triple of Dirac matrices − i 2 ω bca γ a , σ bc = −2iω bca γ [a γ b γ c] .This form is valid in any dimension.In 4-or 5-dimensions the triple antisymmetrization may be shortened using γ 5 .The action is now where The simple form for the anticommutator turns out to be a low-dimensional accident.In the Appendix we show that the general form for the anticommutator Γ a1a2...a k , σ bc depends on both Γ a1a2...a k+1 and Γ a1a2...a k−1 with the second form absent for the Dirac k = 1 case.Here we define Γ a1a2...a k ≡ γ [a1 γ a2 . . .γ a k ] .This includes the particular cases Γ = 1 and σ ab = γ a , γ b for the Spin (p, q) generators.For k < n 2 we may write Γ a1a2...a k in terms of γ 5 ≡ i m Γ a1...an and Γ a1a2...a n−k , where i m is chosen so that γ † 5 = γ 5 .

Spinor metric
The Clifford relation for the gamma matrices is γ a , γ b = −2η ab with η ab = diag (−1, . . ., −1, 1, . . ., 1).Here the γ-matrices are numbered γ 1 . . .γ q γ q+1 . . .γ q+p and we take the first q matrices hermitian.Then for a, b ≤ q the γs satisfy the timelike Clifford relation The final p γs must be antihermitian to give hermiticities of σ ab appropriate for generating both rotations and boosts.We seek a spinor metric h such that both the spinor inner product and the n-vector v a ≡ ψ † hγ a ψ are real.These immediately imply To satisfy the second condition we take h proportional to the product of all timelike γs, h = λγ 1 . . .γ q .This insures that γ a † h = (−1) q−1 hγ a with the same sign for all γ a .Then hermiticity requires λ = i q(q−1) 2 . This is all we need for q odd.When q is even we include an additional factor of γ 5 where γ 5 = i p+ n(n−1) 2 γ 1 . . .γ n .In this case we must also include an additional i q .Therefore we define Adopting the usual notation, we may now let ψ = ψ † h for spinors in any dimension.We note that γ 5 h = (−1) q hγ 5

The general relativity limit
We wish to examine general relativity with coupled Dirac sources.This source still has a spin density, despite the absence of torsion, and it is necessary to determine whether this puts a constraint on the Dirac field.With vanishing torsion the connection is compatible, ω bc µ → α bc µ though the action must still be made real by adding the conjugate.From the curvature field equation Eq. (25) with Although there is nonvanishing spin density there is no second field equation.There is now an antisymmetric part to the Einstein equation.
This is exactly the part that vanishes by Lorentz symmetry.The Einstein equation therefore reduces to the symmetric expression where the spin tensor is the antisymmetric current Because this is totally antisymmetric, σ bac + σ cab = 0 and we recover the Einstein equation with the usual symmetrized energy tensor and no additional coupling.
Therefore, despite nonvanishing spin tensor, Dirac fields make only the expected contribution to the field equation of general relativity with no additional constraint.

Rarita-Schwinger
The spin-3 2 Rarita-Schwinger field [41] is known to give rise to acausal behavior when coupled to other fields [42].This problem is overcome when a spin- 3  2 field representing the gravitino is coupled supersymmetrically.Therefore, we first examine the 11-dimensional supergravity Lagrangian.

11-d Supergravity
Here the basic Lagrangian includes the scalar curvature R, the spin-3 2 Majorana gravitino field ψ α , and a complex 4-form field built from a 3-form potential as F = dA.The covariant derivative has connection ω a b and γ µ = e µ a γ a .This starting Lagrangian is augmented by ψ α -F coupling terms and a Chern-Simons term required to enforce the supersymmetry ( [43,44,46,47]).The result is the Lagrangian for 11D supergravity, first found by Cremmer, Julia and Scherk [46].
Since we are primarily interested in sources for torsion, we will only need the kinetic term for the Rarita-Schwinger field.While is it possible that supergravity theories-which exist only in certain dimensions-are the only consistent formulation of spin-3 2 fields, there may be alternative couplings that allow them.For this reason, we will consider the original Rarita-Schwinger kinetic term in arbitrary dimension as a source for torsion, omitting additional couplings.

The Rarita-Schwinger equation
In flat 4-dimensional space the uncoupled Rarita-Schwinger equation may be written as with real action In curved spacetime, generalizing to the covariant derivative ∂ α ψ β → D α ψ β where we must explicitly make it real.As with the Dirac field, the extra terms give an anticommutator.Noticing that we have and therefore, taking the adjoint and rearranging The explicit torsion coupling here is surprising, and forces us to be clear about the independent variables.We may set T a = de Before carrying out the variation we develop Rarita-Schwinger action in higher dimensions.

The Rarita-Schwinger action in arbitrary dimension
To explore higher dimensions we introduce some general notation.Clearly we will need the Hodge dual, but it yields a more systematic result if we combine the dual with the gamma matrices. Define: In particular, Γ 0 is just the volume form Φ.
It is not hard to check that the Dirac case may be written as by expanding the forms.
To rewrite the Rarita-Schwinger action in arbitrary dimensions we replace the volume form and set m ψµ (−1) q 1 2 σ ρσ 1 2 e αβµν e αβρσ ψ ν 1 4! e def g e d ∧ e e ∧ e f ∧ e g This allows us to eliminate the 4-dimensional Levi-Civita tensor by reducing the Levi-Civita pairs (−1) q 4! ǫ µκρν e def g and (−1) q 4! e αβµν e def g , to combine a solder form with each spinor.Then σ ρσ e ρσde e d ∧ e e ∧ ψ Now set to write the action as By using the Hodge dual in Γ 2 and Γ 3 we have eliminated the specific reference to dimension.Equation ( 31) is the Rarita-Schwinger action in flat (p, q)-space.

Rarita-Schwinger in curved spaces
To generalize Eq.( 31) we now replace the exterior derivative with the covariant exterior derivative keeping the action real by taking Therefore, the direct torsion-Rarita-Schwinger coupling will occur in higher dimensions as well.

The Rarita-Schwinger spin tensor
Varying the action with respect to the spin connection or contorsion Expanding the forms, setting δω a b = A a bc e c , and collecting the basis into volume forms this becomes , σ ba ψ e Φ so antisymmetrizing on ab and expanding the anticommutator as ] ψ e + i η ac η bd − η bc η ad ψd γ e ψ e +i η ae η bc − η ac η be ψd γ d ψ e + i η ad η be − η ae η bd ψd γ c ψ e the spin tensor is After using the torsion equation, the source for the Einstein tensor is always the symmetrized canonical tensor (21) but the torsion is now driven by much more than the axial current.We next use the full spin tensor , Eq.( 33), to compute the source for each indepdendent part of the torsion.Since the reduced field equation shows that κ 2 T c ab = −σ c ab it suffices to find the trace, totally antisymmetric, and traceless, mixed symmetry parts of σ cab .The corresponding parts of the torsion are proportional to these.
First, the trace of the spin tensor reduces to a simple vector current.
For the antisymmetric part there is no change in the totally antisymmetric piece i ψd γ [a γ b γ c γ d γ e] ψ e .Of the last three terms involving metrics, the first two vanish while the antisymmetrization of the third gives i η ad η be − η ae η bd ψd γ c ψ e The remaining terms require the abc antisymmetrization of ψe γ [e γ b γ c] ψ a and ψb γ [a γ e γ c] ψ e .This is complicated by the existing antisymmetry of ebc.Write these out in detail and collecting terms we find with the full contribution to σ [abc] being i 2 times these.Combining everything, the source for the totally antisymmetric part of the torsion is containing 1-, 3-, and 5-gamma currents.The traceless, mixed symmetry part σcab is found by subtracting the trace and antisymmetric pieces.
The result is The traceless, mixed symmetry piece therefore depends on 1-and 3-gamma currents.Therefore, while the Dirac field produces only an axial vector source for torsion, the Rarita-Schwinger field provides a source for each independent piece.Moreover, since a spin- 3  2 field in n-dimensions has n × 2 [ n 2 ]+1 degrees of freedom while the torsion has 1  2 n 2 (n − 1), generic solutions may be expected to produce generic torsion except in dimensions n = 5, 7 or 9.

Higher spin fermions
We have seen that the vacuum Dirac (k = 0) and Rarita-Schwinger (k = 1) actions for spin-2k+1 2 may be written as The pattern seen here generalizes immediately to higher fermionic spins in any dimension n ≥ 2k + 1, with the flat space kinetic term depending on Γ 2k+1 and the mass term depending on Γ 2k for spin 2k+1 2 fields.Including the covariant derivative then adds torsion and anticommutator couplings.
For the generalized Γs it is useful to normalize to avoid overall signs.Setting hΓ k = hΓ k † introduces a factor of (−1) k , but including the fields the adjoint of the combination ψ ∧ Γ 2k+1 ∧ idψ introduces an additional factor of (−1) k .We therefore require no phase factor and can conveniently define for all integers m.

Spin 2k+1 2 fields
To start, we take the flat space Spin 2k+1 2 action to be after taking the conjugate and expanding the forms explicitly to check that S 0 k is real.Notice that ψ ∧ dψ is a (2k + 1)-form and therefore S 0 k exists only for n ≥ 2k + 1.This makes Rarita-Schwinger the maximal case in 4-dimensional spacetime.Then, replacing d ⇒ D using Eq.( 34) and symmetrizing, the gravitationally coupled Spin 2k+1 2 action is As with the Rarita-Schwinger case, we find the real part of the torsion and σ ab parts.For the torsion terms while the σ ab terms still give an anticommutator Therefore, the action for gravitationally coupled Spin 2k+1 2 fields is The spin tensor always contains the anticommutator, which always brings in couplings involving Γ 2k−1 and Γ 2k+3 only (see the Appendix).The Dirac field has k = 0, so only the Γ 3 term is possible, while for Rarita-Schwinger fields with k = 1 we see both Γ 1 and Γ 5 .
There are also direct torsion couplings of the form so the Spin 2k+1 2 field may emit and absorb torsion.This is absent from Dirac interactions because there is no vector index on ψ, but does show up in the Rarita-Schwinger spin tensor.If the action includes a dynamical torsion term this constitutes a new interaction unless there is a consistent interpretation of torsion in terms of known interactions.
The spin tensor is given by a simple variation, followed by reducing the basis forms to a volume form.The result is The anticommutator is a linear combination of Γ 2k−1 , Γ 2k+3 (See Appendix 5) so together with the torsion contribution we have the original and both adjacent couplings Γ 2k−1 , Γ 2k+1 , Γ 2k+3 .It is extremely likely that, like the Rarita-Schwinger field, higher spin fermions drive all invariant parts of the torsion.

Low dimensions
We note that the Spin 2k+1 2 action of Eq.( 36) may be present as long as n ≥ 2k + 1.In particular we see Dirac fields in 2-dimensions, while both Dirac and Rarita-Schwinger fields may be present in 3-dimensions.We look briefly at these two low dimensional cases.
but as we have seen, Yang-Mills sources do not source torsion.For normal physical fields we therefore expect the usual conformal solutions.If we do have a non-standard source for torsion, the action is where T a = T a12 .

Three dimensions
In 3-dimensional spacetime general relativity gives vanishing Weyl curvature and is therefore conformally flat [49] and exactly soluble [51].When a cosmological constant is included solutions become more general [50] and if negative permit the BTZ black hole [54].Coupled to Maxwell theory still other solutions become possible [53].Alternatively, torsion has been included in models with topological gravity [52].There has been considerable study of all of these models.
In Poincarè gauge gravity torsion modifications of the curvature tensor extend it beyond its one conformal degree of freedom, and will also permit nontrivial solutions.Dirac fields add one additional degree of freedom, but we find that Rarita-Schwinger fields are also possible in 3-dimensions and yield all 9 degrees of freedom of fully general torsion.We briefly explore the Dirac and Rarita-Schwinger contributions.
With Dirac matrices γ Varying the spin connection, the torsion is determined by Even if the scalar ψψ is constant, the C c b ∧ C a c term in the curvature contributes an effective cosmological constant, allowing solutions in 3-dim Poincarè gravity as general as those in [50,54], though the energy tensor of the Dirac field must also be included.Nonconstant values will allow more general solutions.
In 3-dimensions the spin density for spin-3 2 fields reduces Eq.( 33) to This may be written in terms of two currents: as simply Given the 12 degrees of freedom of ψ a , this appears sufficiently general to drive all components of the torsion.

Conclusions
We implemented Poincarè gauging in arbitrary dimension n and signature (p, q) using Cartan's methods.The principal fields are the curvature and torsion 2-forms, given in terms of the solder form and local Lorentz spin connection.The inclusion of torsion produces a Riemann-Cartan geometry rather than Riemannian.We displayed the Bianchi identities and showed that the Riemann-Cartan identities hold if and only if the Riemannian Bianchi identities hold.
Replicating familiar results, we reproduced general relativity in Riemannian geometry by setting the torsion to zero and varying only the metric.The resulting Riemannian geometry is known to be consistent and metric variation leads to a symmetric energy tensor.
We examined sources for the ECSK theory, that is, the gravity theory in Riemann-Cartan geometry found by using the Einstein-Hilbert form of the action with the Einstein-Cartan curvature tensor.The vacuum theory agrees with general relativity even when both the solder form and connection are varied independently, but there are frequently nonvanishing matter sources for both the Einstein tensor and the torsion.
The first issue we dealt with in depth was the choice of independent variables.The spin connection is the sum of the solder-form-compatible connection and the contorsion tensor ω a b = α a b + C a b .We compared and constrasted the resulting two allowed sets of independent variables: the solder form and spin connection (e a , ω a b ) on the one hand and the solder form and the contorsion tensor (e a , C a b ) on the other.When choosing the latter pair the compatible part of the spin connection α a b must be treated through its dependence on the solder form.We demonstrated explicitly how the two choices of independent variable differ in their relationship to the Lorentz fibers of the Riemann-Cartan space.
Changing independent variables changes the energy tensor.We showed that the difference between these two choices leads to the difference between the (asymmetric) canonical energy tensor and the (symmetric) Belinfante-Rosenfield energy tensor.When the field equations are combined both methods yield the same reduced system.
Our main contribution was a more thorough analysis of sources for torsion.Many, perhaps most, of the research on ECSK theory or its generalizations to include dynamical torsion have restricted attention to Dirac fields as sources.This yields a single axial current and totally antisymmetric torsion.This amounts to only n of the 1  2 n 2 (n − 1) degrees of freedom of the torsion.We took the opposite approach, considering fields of all spin.Only scalar and Yang-Mills fields fail to determine nonvanishing torsion.In addition to these we looked at symmetric bosonic kinetic forms and found all to provide sources for torsion.We studied Dirac and Rarita-Schwinger fields in greater depth.After reproducing the well-known result for Dirac fields, we developed formalism to describe the spin-3 2 Rarita-Schwinger field in arbitrary dimension.Surprisingly, in addition to dependence on the anticommutator of three gammas with the spin generator, γ [a γ b γ c] , σ de , there is a direct coupling to torsion, ψ a T a .Continuing, we showed that Rarita-Schwinger fields drive all three independent parts of the torsion: the trace, the totally antisymmetric part, and the traceless, mixed-symmetry residual.Except in dimensions 5, 7, and 9 the Rarita-Schwinger field has enough degrees of freedom to produce generic torsion.
Finally, we looked at higher and lower dimensional cases, generalizing to spin-2k+1 2 sources for all k.Like the Rarita-Schwinger case, these have direct torsion couplings in addition to an anticommutator γ [a1 γ a2 . . .γ a 2k+1 ] , σ de , and appear to drive all components of the torsion.Specializing to 3-dimensions, the we find the geometric structure substantially enhanced if Rarita-Schwinger fields are present.The essential feature here is that the anticommutator coupling between σ de and Γ 2k+1 always leads to a linear combination of Γ 2k+3 and Γ 2k−1 and only these.

Define a volume form
as the Hodge dual of unity, Φ = * 1 = 1 n! e ab...c e a ∧ e b ∧ . . .∧ e c .It follows that * Φ = (−1) q in signature (p, q) and e a ∧ e b ∧ . . .∧ e c n terms = (−1) q e ab...c Φ where e ab...c is the Levi-Civita tensor.Let ε ab...c be the totally antisymmetric symbol with ε 12...n = 1 and e = det e a µ = |g|, so that e 12...n = eε 12...n and e 12...n = (−1) Under local Lorentz transformation Λ, both the solder form and spin connection change.The change in the spin connection is given by the usual gauge form ω = ΛωΛ −1 − dΛΛ −1 .In detail, for an infinitesimal gauge transformation Λ a b = δ a b + ε a b where ε ab = −ε ba this implies a change in the spin connection δ Λ ω a b = −Dε a b At the same time the solder form transforms as a Lorentz tensor, δ Λ e a = ε a b e b − 2)! ˆD −Dε ab ∧ e c ∧ . . .∧ e d e abc...d which requires the Ricci identity for D 2 ε ab and two integrations by parts.0 = − κ (n − 2)! ˆεef δ a e R b f ∧ e c ∧ . . .∧ e d e abc...d − κ 2 (n − 3)! ˆ ε ab DT c ∧ e d ∧ . . .∧ e e e abcd...e + κ 2 (n − 4)! ˆ ε ab T c ∧ T d ∧ e e ∧ . . .∧ e f e abcde...f ) derived above under the solder form variation.Now vary S [e a , C a b ].The contorsion variation follows immediately by nothing that the ω a b , α a b and C a b variations are related by ω a b = α a b + C a b so that δω a bc e b ∧ e c = δα a bc e b ∧ e c C constant = δC a bc e b ∧ e c α constant Therefore, variation of the contorsion gives the same result as the original Palatini variation.may also arrive at this by expanding the curvature and varying the DC a b − C c b ∧ C a c terms from Eq.(10) directly.The solder form variation is now more involved.The general form is δ e S = Palatini results, i.e., the Einstein tensor κ R ab − 1 2 Rη ab and the canonical energy T C ab .The second term introduces three derivatives to each side of the equation since δα a b = 1 2 (δ a d δ c b − η bd η ac ) D c δe d − e µ c η gh e g D d δe h µ − e α c D δe d α

4. 3 . 2 2 2 T
Energy tensor and spin density from the Dirac equation From the action (30) the energy tensor and spin current are immediate.Since the Dirac Lagrangian is proportional to the Dirac equation, there is no contribution from the volume form.Therefore the source for the Einstein tensor is δL δe b µ e c µ η ca = −iα ψγ a e µ b ← → ∂ µ ψ + 2iαη ac ω deb ψΓ cde ψ giving the curvature equation (17) the form κ R ab − 1 Rη ab = −iα ψγ (a e µ b) ← → D µ ψ + 2iαω de(b η a)c ψΓ cde ψ with 2iαω de(b η a)c ψΓ cde ψ becoming the axial current αω cd (a ε b)cde ψγ e γ 5 ψ in 4-dimensions.The spin density is σ cab ≡ δL δω abc = −iα ψΓ abc ψ so the torsion is given by κ cab = iα ψΓ abc ψ This is the axial current in 4-dimensions.Many studies of torsion in ECSK and generalizations to propagating torsion are restricted to this totally antisymmetric form of T cab .
a − e b ∧ ω a b and vary (e a , ω a b ) or we may write ω a b = α a b (e c ) + C a b and write the torsion in terms of the contorsion T a = C a b ∧ e b , then vary (e a , C a b ).We choose the latter course, since this respects the Lorentz fiber symmetry and yields the Belinfante-Rosenfield tensor as source.For the spin tensor it makes no difference because δ ω T a = −e b ∧ δω a b δ C T a = −e b ∧ δC a b