Noncommutative Chern-Simons gauge and gravity theories and their geometric Seiberg-Witten map

We use a geometric generalization of the Seiberg-Witten map between noncommutative and commutative gauge theories to find the expansion of noncommutative Chern-Simons (CS) theory in any odd dimension $D$ and at first order in the noncommutativity parameter $\theta$. This expansion extends the classical CS theory with higher powers of the curvatures and their derivatives. A simple explanation of the equality between noncommutative and commutative CS actions in $D=1$ and $D=3$ is obtained. The $\theta$ dependent terms are present for $D\geq 5$ and give a higher derivative theory on commutative space reducing to classical CS theory for $\theta\to 0$. These terms depend on the field strength and not on the bare gauge potential. In particular, as for the Dirac-Born-Infeld action, these terms vanish in the slowly varying field strength approximation: in this case noncommutative and commutative CS actions coincide in any dimension. The Seiberg-Witten map on the $D=5$ noncommutative CS theory is explored in more detail, and we give its second order $\theta$-expansion for any gauge group. The example of extended $D=5$ CS gravity, where the gauge group is $SU(2,2)$, is treated explicitly.


Introduction and Summary
Noncommutative (NC) actions can be expanded order by order in the noncommutativity parameter θ and be interpreted as effective actions on commutative spacetimes, the noncommutativity leading to extra interaction terms, possibly capturing some quantum spacetime effect. These actions involve higher derivatives in the field strengths. For gauge theory actions one can expand not only the ⋆-products but also the noncommutative fields in terms of the commutative ones using the Seiberg-Witten (SW) map [1] . This allows to define NC gauge theories with any simple gauge group in arbitrary representations [2,3]. In the literature mainly first order in θ deformations have been studied. In [4], [5,6,7,8,9], extensions of Yang-Mills and gravity theories have been obtained at second order in the noncommutativity parameter θ starting from NC actions. The second order expansion is needed in the case of D = 4 gravity theories because the first order θ-correction vanishes 1 .
Some gauge theory actions have the remarkable property of being invariant under the SW map. This is notably the case for the CS action in 3 dimensions [13] and, if we consider slowly varying field strengths, for the Dirac-Born-Infeld theory in any dimension [1,14] . Noncommutative CS actions can be studied in any (odd) dimension [15], [16], [17] , [18]. In [18] noncommutativity is given by a Drinfeld twist defined by a set of commuting vector fields (so-called abelian Drinfeld twist), this noncommutativity including as a special case Moyal-Groenewold noncommutativity.
In this paper we apply the geometric Seiberg-Witten map [5] (i.e. the geometric generalization of the SW map that applies to Drinfeld twist noncommutativity) to the noncommutative Chern-Simons actions in any odd dimension studied in [18]. We obtain the correction to the classical action up to first order in the noncommutativity parameter θ. The correction is expressed in terms of the curvature R = dΩ − Ω ∧ Ω, of its contraction along the vector fields determining the noncommutativity and of its covariant derivative. These terms are covariant under gauge transformations and therefore the correction is truly gauge invariant (not just up to boundary terms). The construction of these extended commutative CS actions obtained by adding correction terms order by order in θ applies to any gauge group G. For slowly varying field strength we show that these correction terms vanish, so that, as for Dirac-Born-Infeld theory, also noncommutative and commutative CS theories coincide in any dimension in this approximation.
The variation of the NC Chern-Simons form under SW map has an intriguing structure. We find that the SW map relates the NC topological terms T r(R n ) and T r(R n+1 ) and therefore relates the NC CS forms in D and in D + 2 dimensions. In fact D-dimensional CS forms are mapped into double contractions of the (D + 2)dimensional CS forms, plus contractions of (D + 1)-forms, plus extra terms that are covariant under gauge transformations. Since in D dimensions (D + 2)-and (D + 1)-forms vanish, only the extra covariant terms are relevant in computing the SW map of the CS action. In D = 1 and D = 3 the extra covariant terms are absent, which easily explains why in these cases the SW map is trivial, as first observed by [13]. Our results confirm those of [19], where, using a different approach based on operator valued fields, a (generalized) NC CS action defined only in terms of covariant derivatives was shown to be nontrivial under the SW map in dimension D > 3. We sharpen the findings in [19] by explicitly computing and analyzing the extra terms in D > 3. Moreover we are not constrained to consider Moyal-Groenewold noncommutativity, and our commutative limit, never involving the inverse of θ µν , is well under control for θ → 0.
The first nontrivial θ dependence occurs in D = 5 NC CS theory. For this case we compute also the second order expansion in θ.
Next we specialize to NC CS gravity [18] where the gauge group is G = SU(2, 2), and explicitly compute its expansion to first order in θ in terms of component fields.
In this paper we focus on local properties of CS forms: in particular the connection Ω is always globally defined, and the underlying principal G-bundle is trivial.
It would be interesting to extend our analysis to the D=5 noncommutative Chern-Simons supergravity theory constructed in [18], invariant under the local action of the ⋆-supergroup U(2, 2|N) that includes N supersymmetries. In this case the SW map relates ⋆-supersymmetry to ordinary supersymmetry, so that the θ-correction terms of the SW expansion are separately invariant under ordinary supersymmetry. The result is an extended D = 5 CS supergravity with (locally) supersymmetric higher order terms. This work is in progress.
The paper is organized as follows. In Section 2 we recall a few facts on Chern-Simons forms and their noncommutative versions. In Section 3 we recall the geometric generalization of the Seiberg-Witten map. In Section 4 we compute the SW variation of the NC topological term T r(R n ), of the CS forms and of the CS actions. In Section 5 we apply these results to D = 5 CS and present the first and second order corrections in θ; finally we consider the case of D = 5 CS gravity. In Appendix A we give the derivation of the SW variation of the noncommutative topological term T r(R n ). In Appendix B we collect useful identities, and Appendix C contains a summary of D = 5 gamma matrix properties. The CS Lagrangian in (2n − 1)-dimensions is a (2n − 1)-form given in terms of the connection Ω and its exterior derivative dΩ, or equivalently its curvature 2-form R = dΩ − Ω ∧ Ω, by the following expressions (see e.g. [20,21]): where we have omitted writing explicitly the wedge product. For example: These expressions are obtained by solving the condition The CS form L (2n−1) CS contains (exterior products of) the Lie(G)-valued gauge potential one-form Ω and its exterior derivative. The trace T r is taken on some representation of the Lie algebra Lie(G).
Because of (2.6), the CS action on the boundary ∂M of a manifold M is related to a topological action in 2n dimensions via Stokes theorem: (2.7) Infinitesimal gauge transformations are defined by so that T r(R n ) is manifestly gauge invariant. Therefore also the CS action is gauge invariant under infinitesimal gauge transformations 2 .
Considering L (2n−1) CS as a function of Ω and R, a convenient formula for its gauge variation is (see for example ref. [18]) where j ε is a contraction acting selectively on Ω, i.e.

⋆-Exterior products from abelian Drinfeld twists
The preceding discussion is based on algebraic manipulations, and relies on the (graded) cyclicity of T r. As such, it can be exported immediately to the noncommutative setting, provided we ensure that cyclicity holds. The noncommutativity we consider here is controlled by an abelian twist, and amounts to a deformation of the exterior product: where θ AB is a constant antisymmetric matrix, and ℓ A are Lie derivatives along commuting vector fields X A . The product is associative due to [X A , . If the vector fields X A are chosen to coincide with the partial derivatives ∂ µ , and if τ , τ ′ are 0-forms, then τ ⋆ τ ′ reduces to the well-known Moyal-Groenewold product [22]. A short review on twisted differential geometry can be found for example in [23]. The deformed exterior product differs from the undeformed one by a total Lie derivative, indeed since [ℓ A , ℓ B ] = 0 we can write where for brevity we have renamed the summation Q A 1 . In particular we have without boundary or if the forms τ and τ ′ have suitable boundary conditions. The equality (2.13) implies that the integral of ⋆-wedge products of exterior forms has the usual graded cyclic property τ If we now consider forms T , T ′ that are Lie algebra valued, the trace of the ∧ ⋆ -product of forms is still graded cyclic up to total Lie derivative terms: (2.14) and for suitable boundary conditions the integral of the trace has the graded cyclic property.

Noncommutative CS forms
We define noncommutative Chern-Simons actions by replacing ∧-products with ∧ ⋆products in the commutative Chern-Simons action. This procedure is unique if we integrate over manifolds without boundary or if the fields are properly behaving at the boundary; it is not unique for CS forms because of the cyclic ordering ambiguities (2.14), that are however irrelevant in the present paper. We denote by L gives T r(R n ) is algebraic, and relies only on the Leibniz rule property of the exterior derivative and on the graded cyclicity of the trace. Since the exterior derivative satisfies the Leibniz rule also in the noncommutative case, and the graded cyclicity of the trace holds up to total Lie derivatives, we can conclude that the noncommutative Chern-Simons form satisfies the relation where Q (2n)C is due to cyclic reorderings and is a sum of wedge products of Lie derivatives of connections and of their exterior derivatives. We note that Q (2n)C is local in the noncommutative connection, in the sense that expanding the ∧ ⋆ -product, for any finite order in θ there is a finite number of Lie or exterior derivatives.
In the noncommutative case the gauge group G usually has to be extended, because ⋆-commutators in general do not close in the original Lie algebra Lie(G). For example with the second term nonvanishing because the ∧ ⋆ -product is not antisymmetric. We therefore consider Lie algebras with representations T a that close under the usual matrix product (i.e. under commutators and anticommutators). It is easy to prove the invariance of the noncommutative Chern-Simons action under infinitesimal ⋆-gauge transformations defined by: for suitable boundary conditions. This is so because in the ⋆-deformed case the variation formulae (2.9, 2.10) still hold true under integration.
For example the D = 5 ⋆-Chern-Simons action reads and is invariant under the ⋆-gauge variations (2.17).

The Seiberg-Witten map
In the framework of Moyal deformed gauge theories, Seiberg and Witten showed how to relate noncommutative fields (that transform under deformed gauge transformations) to ordinary fields, called also classical fields, transforming with the usual gauge variation laws. The Seiberg-Witten map expresses the NC fields as functions of the ordinary fields in such a way that usual gauge variations on the latter induce ⋆-gauge variations on the former. The map is nonlinear, and is determined order by order in the noncommutativity parameter θ.
Under this map, a NC action can be re-expressed in terms of classical fields. The result is invariant under usual gauge variations (since the NC action is invariant under ⋆-gauge variations), and can be written as the classical action plus higher order θ corrections, each of which is separately gauge invariant under usual gauge variations (because usual gauge variations do not involve θ). This map provides therefore an interesting mechanism to generate extensions of usual commutative actions, with interaction terms that depend on θ (for gravity actions see [5,6,7,8,9]).
Denoting by Ω the NC gauge field, and by ε the NC gauge parameter, the Seiberg-Witten map relates Ω to the ordinary Ω, and ε to the ordinary ε and to Ω so as to satisfy: Thus the dependence of the NC gauge field on the ordinary gauge field is determined by requiring that ordinary gauge variations of Ω inside Ω(Ω) produce the noncommutative gauge variation of Ω. The condition (3.1) is satisfied if the following differential equations in the noncommutativity parameter θ AB hold [1,5]: where: • Ω A , R A are defined as the contraction i A along the tangent vector X A of the exterior forms Ω, R, i.e.
The differential equations (3.4)-(3.5) hold for any abelian twist defined by arbitrary commuting vector fields X A [5]. They reduce to the usual Seiberg-Witten differential equations [1] in the case of a Moyal-Groenewold twist, i.e. when X A → ∂ µ .
We can solve these differential equations order by order in θ by expanding Ω and ε in power series of θ, so that (the factor i 2 is inserted for ease of later notation) , and similarly for ε. For example up to first order in θ from (3.4) and (3.5) we immediately find In general Ω is a power series in θ of sums of products of commutative connections, also contracted and differentiated (e.g. Ω A , i A dΩ, ℓ A ℓ B Ω, etc.). Again we say that Ω is local in the commutative connection because for every power of θ only a finite number of exterior derivatives appears. It follows that in this framework noncommutative Lagrangians are power series in θ of commutative Lagrangians that are local in the connection Ω.

The SW variation of NC Chern-Simons forms
In the following we omit the hat denoting noncommutative fields, the ⋆ and ∧ ⋆ products, and simply write { , }, [ , ] for the ⋆-anticommutator and the ⋆-commutator

SW variation of T r(R n )
An equivalent expression for the SW variation of the connection 1-form is The "fat" Lie derivative IL B is defined by IL B ≡ ℓ B + L B where L B is the covariant Lie derivative along the tangent vector X B ; it acts on every field P as In fact the covariant Lie derivative L B can be written in Cartan form: where D is the covariant derivative: DP = dP − [Ω, P ] for P even form, DP = dP − {Ω, P } for P odd form. In particular DR = 0 (Bianchi identity) follows from the definition of R.
The SW variation of the connection implies the following variation for the curvature 2-form R = dΩ − Ω ∧ Ω (an easy derivation uses equation (A.4) with P = Q = Ω), From this formula and iterated use of (A.4) the SW variation of the trace of R n can be proven to be (see Appendix A): where the (2n − 1)-form U AB is given by . The precise expression (see Appendix A for details) of the 2n-form Q C AB , local in the NC connection, will not be relevant in the following.

SW variation of T r(R n ) on a 2n-dimensional manifold M
If the forms are defined on a 2n-dimensional manifold M, T r(R n ) has top degree and its SW variation (4.4) simplifies since R n+1 = 0 being a (2n + 2)-form. Moreover, writing ℓ C = i C d + d i C and observing that dQ C AB = 0 because it is a (2n + 1)-form, we obtain the SW variation of the top form T r(R n ): Let's comment on the nontrivial information in this formula. The exactness of δ θ T r(R n ) is a trivial consequence of considering R n a top form. We write δ θ T r(R n ) = dη and compare this expression with the SW variation of (2.15) that when R n is a top form reads δ θ T r(R n ) = dδ θ L (2n−1) CS ⋆ − di C δ θ Q (2n) C . Recalling the differential equation (3.4) we see that η is local in the NC connection (i.e., that expanding the ∧ ⋆ -product, for any finite order in θ we have a finite number of Lie or exterior derivatives in the NC connection).
The nontrivial information in (4.6) is that η is the sum U AB + i C Q C AB where the second term is a contraction of a 2n-form (local in the NC connection), and the first term is expressed only in terms of products of curvatures, their contractions and covariant derivatives, i.e., in terms of only gauge covariant fields.

SW variation of L
where the Lie derivative term on the right hand side comes from cyclic reorderings (in the commutative limit Q (2n)C = 0 since the trace in that case is cyclic). Using this relation in (4.4) yields the SW variation of dL For forms living in a 2n-dimensional manifold M, this becomes where we used the identity ℓ C = i C d + d i C and the vanishing of forms of degree higher than 2n. Equivalently on M we have for some (2n − 1)-form ϕ written in terms of the connection, of exterior derivatives and of contraction operators along the noncommutative directions 3 . We now consider a (2n − 1)-dimensional submanifold N of M and choose commuting vector fields {X A } on M that restrict to vector fields on N. In this case L (2n−1) CS ⋆ is a top form on N, and Q C AB = δ θ Q (2n) C = 0 being 2n-forms on the (2n−1)dimensional manifold N. The SW variation of the CS action on a manifold N with no boundary or with fields that have appropriate boundary conditions is therefore where in the last equality we have evaluated the contraction operator, integrated by parts and cyclically reordered the addends.
This variation is zero for n = 1, 2 so that the first nonvanishing SW variation of a Chern-Simons action occurs for n = 3. In particular in three dimensions the SW expansion of the noncommutative Chern-Simons action equals the commutative Chern-Simons action; this result, for Moyal-Groenewold noncommutativity, was obtained in [13] 4 .
In higher dimensions the variation is nonvanishing, and is expressed in terms of the gauge covariant quantities R, R A and their covariant derivatives.

Slowly varying fields and invariance of NC CS action under SW map
In [1] (Section 4.1) it is shown that for slowly varying field strength the noncommutative and commutative Dirac-Born-Infeld actions coincide (up to a redefinition of the coupling constant and of the metric). In our geometric framework, where the noncommutativity is given by the vector fields {X A }, we can consider field strengths that are slowly varying just along the noncommutative directions. The gauge covariant formulation of the slowly varying field strenght condition is L A R ∼ 0. In this case the noncommutative and commutative CS actions coincide. Indeed (4.10)).
This result holds in particular in the U(1) case where the slowly varying field strenght condition on commutative spacetime reads ℓ A R commutative ∼ 0. For nondegenerate Moyal-Groenewold noncommutativity this is equivalent to ∂ µ R commutative νσ ∼ 0 that is the condition considered in [1].

Extended CS actions from NC CS actions
Consider the Taylor series expansion of a NC CS action in powers of θ (the θ dependence is due to the ⋆-product and to the SW map), . The right hand side is a higher derivative action on commutative spacetime. It is an extension, with θ corrections, of the commutative CS action L (2n−1) CS . The result of the previous section gives the first order θ-correction to the commutative CS theory, so that the action of the extended CS theory reads We notice that this action is well defined for any gauge group G, and that it has the same (off shell) degrees of freedom as the usual CS action. Like in modified gravity theories the θ correction is just a further interaction term among the fields.
Note In Section 2 we had to consider NC CS actions with fields valued in a Lie algebra representation T a closed under the matrix product rather than under commutators (recall (2.16)). This in general is a severe restriction on the gauge group G (typically requiring G = U(N)). Here however, the SW map relates the noncomutative fields corresponding to products of generators T a T b . . . to the classical gauge fields of any gauge group G [2,3]. Thus the SW map allows to define NC CS actions for any gauge group.

D = 5 CS form to second order in θ
To evaluate the second order variation of L we need the SW variation of R B and DR A . The first one is easily obtained by applying the contraction operator i B to the SW variation of the curvature 2-form R, eq. (4.3). The second one is obtained by summing the SW variation of dR A to the SW variation of {Ω, R A }, that is evaluated using (4.1) and (A.4). The result is Next with the help of (A.4) we compute δ θ DR A R B and finally using (B.3) we obtain 5 The expansion at second order in power series of θ of the D = 5 noncommutative CS action (2.19) is then given by where L

Extended D = 5 CS gravity to first order in θ
CS gravities and supergravities [25,26,27,28] present interesting alternatives to standard (super)gravities in odd dimensions. Indeed CS gravities are a particular case of Lovelock gravities [29], with at most second order equations for the metric. Moreover the gauge (super)group contains the anti-de Sitter (super)algebra, so that the theory is translation invariant and does not have dimensionful coupling constants. One can use group contraction to recover the (super)Poincaré algebra, but retrieving the Einstein-Hilbert term in this limit is problematic. There are however techniques (S-expansion method [30]) to recover Poincaré gravity from CS gravity with a particular "expanded" gauge algebra. We study here the example of D = 5 Chern-Simons AdS pure gravity. The commutative SU(2, 2) connection and curvature are given by all indices a, b, ... running on five values. The D = 5 gamma matrices γ a , together with their commutators γ ab ≡ 1 2 [γ a , γ b ], close on the D = 5 AdS algebra SU(2, 2) ≈ SO (2,4). The SU(2, 2) connection contains both the vielbein V a and the spin connection ω ab , and correspondingly the SU(2, 2) curvature contains both the AdS curvature R ab and the torsion R a . After applying the SW map to the D = 5 noncommutative CS action (2.19), and using the expression for the first order correction in (5.6), we obtain the extended D = 5 AdS gravity action: where D is the SO(1, 4) Lorentz covariant derivative (with connection ω ab ).

A The SW variation of T r(R n )
We first recall some formulas for the variation of a ∧ ⋆ -product of fields [6].
Lemma 1 Let P, Q be arbitrary exterior forms. Then This result easily follows from the previous lemma and from the ∧ ⋆ -product varia- We can now apply formula (A.4) to δ θ R n written as δ θ (RR n−1 ). Recalling the SW variation of R given in (4.3), and defining (R n−1 ) ′ (A.5) one finds leads to the recursive relation . This recursive relation is easily seen to be solved by Using this expression, the Leibniz rule for IL B and the identity IL B Ω A = R BA , we can rewrite the SW variation of T r(R n ) as where in the third line we have used cyclic reorderings to simplify the first line; the effect of these reorderings is the addition of a total Lie derivative term ℓ CQ C AB that can be explicitly computed. The last line is the rewriting of the second line using the Leibniz rule for D: We next use the Leibniz rule for the contraction operator in the form and then cyclic reorder the first and second terms in the last line: they drastically simplify to just two addends (up to a total Lie derivative absorbed in the term ℓ CQ C AB ), and we obtain To derive the last equality we observe that up to cyclic reorderings (absorbed in the ℓ CQ C AB term): • the covariant derivative can be replaced by the exterior derivative, • the first addend in δθ AB d T r( n−1 i=1 R i−1 DR A (R n−i ) B ), i.e. δθ AB d T r(DR A (R n−1 ) B ) vanishes. 6 6 one proves that up to cyclic reorderings Tr DR A (R n−1 ) B is a total derivative, and therefore its exterior derivative vanishes (since d 2 = 0). In conclusion the SW variation of T r(R n ) is given by where the sum T r(ℓ B {Ω A , R n } + ℓ CQ C AB ) has been renamed ℓ C Q C AB .

Cartan formulae
The usual Cartan calculus formulae simplify if we consider commuting vector fields X A , and read Other useful identities are (cf. also [6]): where L A P = ℓ A P − [Ω A , P ], IL A ≡ ℓ A + L A and R A ≡ i A R, R AB ≡ i B i A R.