A Twistor Space Action for Yang-Mills Theory

We consider the twistor space ${\cal P}^6\cong{\mathbb R}^4{\times}{\mathbb C}P^1$ of ${\mathbb R}^4$ with a non-integrable almost complex structure ${\cal J}$ such that the canonical bundle of the almost complex manifold $({\cal P}^6, {\cal J})$ is trivial. It is shown that ${\cal J}$-holomorphic Chern-Simons theory on a real $(6|2)$-dimensional graded extension ${\cal P}^{6|2}$ of the twistor space ${\cal P}^6$ is equivalent to self-dual Yang-Mills theory on Euclidean space ${\mathbb R}^4$ with Lorentz invariant action. It is also shown that adding a local term to a Chern-Simons-type action on ${\cal P}^{6|2}$, one can extend it to a twistor action describing full Yang-Mills theory.


Introduction
Let M 4 be an oriented real four-manifold with a Riemannian metric and P (M 4 , SO(4)) the principal bundle of orthonormal frames over M 4 . The twistor space Tw(M 4 ) of M 4 can be defined as an associated bundle [1] Tw An almost complex structure J on M 4 is a global section of the bundle (1.2). Note that while a manifold M 4 admits in general no almost complex structure (e.g. four-sphere S 4 ), its twistor space Tw(M 4 ) can always be equipped with two natural almost complex structures. The first, J = J + , introduced in [1], is integrable if and only if the Weyl tensor of Riemannian metric on M 4 is self-dual, while the second, J = J − , introduced in [2], is non-integrable (and never integrable), i.e. the Nijenhuis tensor of J does not vanish.
Twistor space P 6 = Tw(R 4 ) ∼ = R 4 ×S 2 of R 4 with an almost complex structure J is a particular case of almost complex six-manifolds to be discussed in this paper. Twistor space (P 6 , J ) is a complex manifold P 3 C for integrable J and it is an almost complex manifold with an SU(3)structure and non-vanishing torsion for non-integrable J . Twistor literature focuses on complex twistor space P 3 C (see e.g. [3,4,5]) and very rarely on the non-integrable case (see e.g. [2,6,7]). The goal of twistor theory is to take some unconstraint analytic object on Tw(M 4 ) (e.g. Dolbeault cohomology classes) and transform them to objects on M 4 which will be constrained by some differential equations [3,4]. In particular, the self-dual Yang-Mills (SDYM) equations on Euclidean space R 4 can be described as field equations of holomorphic Chern-Simons theory defining holomorphic bundles on the complex twistor space P 3 C via the Penrose-Ward correspondence [3,4,5]. This correspondence can be extended to the non-integrable case (see e.g. [6,7]).
The field equations of J -holomorphic Chern-Simons (J -hCS) theory on (P 6 , J ) read F 0,2 = P 0,1 P 0,1 F = (dA + A ∧ A) 0,2 = 0 , where P 0,1 = 1 2 (Id + iJ ) is the projector onto (0,1)-part of one-forms, A is a connection one-form on a complex vector bundle E over (P 6 , J ) and F = dA + A ∧ A is the curvature of A. One can expect that equations (1.3) are obtained by variation of the action functional where Ω is a (3,0)-form w.r.t. J on (P 6 , J ), i.e. Ω is a global section of the canonical bundle of (P 6 , J ). However, the canonical bundle of P 3 C ∼ = CP 3 \ CP 1 is the non-trivial holomorphic line bundle O(−4) with the first Chern class -4. Hence, there is no non-singular holomorphic volume form Ω on P 3 C . Thus, the functional (1.4) is not defined on P 3 C .
The triviality of the canonical bundle can be restored if instead of P 3 C one considers the supertwistor space P 3|4 C ∼ = CP 3|4 \CP 1|4 with four holomorphic fermionic dimensions, each of type ΠO (1) bundle, where the operator Π inverts the Grassmann parity of fibre coordinates. The canonical bundle of P 3|4 C is trivial and hence there is a holomorphic volume form Ω on P 3|4 C . This fact was used by Witten for introducing twistor string theory and holomorphic Chern-Simons theory (hCS) on P 3|4 C [8]. The action of hCS theory on P 3|4 C can be written in the form (1.4) after substituting Ω instead of Ω and integrating over P 3|4 C . The field equations will be (1.3) with A 0,1 = P 0,1 A depending on four Grassmann variables taking values in the bundle ΠO(1) ⊗ C 4 over P 3 C . This hCS theory on P 3|4 C in turn is equivalent [8] to self-dual subsector of N =4 supersymmetric Yang-Mills theory on R 4 (see e.g. [9,10,11] for reviews and references) in the form of Chalmers and Siegel [12]. The N =4 SDYM equations can be truncated to the bosonic SDYM equations [12] and on the twistor level this was discussed e.g. in [13,14,15].
Despite the success of the supertwistor description of supersymmetric Yang-Mills theories, there was a desire to get a twistor description of pure bosonic SDYM theory. Recently, it was proposed by Costello to work with hCS theory on the bosonic twistor space P 3 C by allowing Ω in (1.4) to be meromorphic instead of holomorphic [16]. After choosing a meromorphic form Ω on P 3 C and imposing some boundary conditions on fields at poles of Ω, one can reduce the action (1.4) to the 4d action for SDYM theory as it was demonstrated in [16,17]. Depending on the gauge choice, the twistor action is reduced to the action for group-valued fields [18,19] or to the action for Liealgebra valued fields [20,21], both of which are well known in the literature. However, the choice of (3,0)-form Ω and of its singularities is not unique and different choices lead to a range of actions on R 4 , not all of which have equations of motion equivalent to the SDYM equations [17].
All the above-mentioned actions break Lorentz invariance. The actions [18]- [21] for the SDYM equations were discussed long time ago by Chalmers and Siegel in [12], where it was shown that these 4d actions at more than one loop generate diagrams that do not relate to quantum Yang-Mills theory. These flaws are absent for the Chalmers-Siegel 4d action which is a truncation (a limit of small coupling constant) of the standard Yang-Mills action. We want to obtain this 4d action in the framework of twistor approach. We show that this is possible by using a non-integrable almost complex structure J on the twistor space P 6 such that the canonical bundle becomes trivial and hence there exists a globally defined (3,0)-form Ω on (P 6 , J ) which can be used in (1.4).
The action [12] contains gauge field coupled with a propagating anti-self-dual auxiliary field Gαβ = ε αβ G αα,ββ withα,β = 1, 2. The field Gαβ corresponds to additional degrees of freedom parametrized by some cohomology groups on the complex twistor space P 3 C [22,11] and can be obtained from the component A 0,1 along CP 1 ֒→ P 3|4 C in hCS theory on the supertwistor space (see e.g. [11] and references therein). This Gαβ enters into the N =4 SDYM supermultiplet (f αβ , χ αi , φ ij ,χα i , Gαβ), where the fields have helicities (+1, + 1 2 , 0, − 1 2 , −1), i = 1, ..., 4. Truncations of the self-dual N =4 super-Yang-Mills to the case N < 4, including the bosonic case N =0, can be obtained by considering weighted projective supertwistor space [14,10] or exotic supertwistor space [15,9]. The approach similar to that in [14,15] can be used in the case of twistor space (P 6 , J ) with non-integrable almost complex structure J on P 6 . We will show that the 4d Chalmers-Siegel action [12] can be obtained from an action functional for J -hCS theory on a graded twistor space P 6|2 with two real fermionic directions, each parametrizing trivial real line bundle over (P 6 , J ). The Chern-Simons type action on P 6|2 is introduced by using globally defined form Ω = Ω∧dη 1 ∧dη 2 on P 6|2 , where Ω is a global section of the trivial canonical bundle of P 6 . Compo-nents of gauge potential A in this theory take values in the Grassmann algebra Λ(R 2 ) generated by two real scalars η 1 , η 2 . We also show that this action can be extended to a twistor action describing full Yang-Mills theory on R 4 after adding some local terms to J -hCS Lagrangian on the twistor space P 6|2 . of vertical and horizontal subbundles of T Q 6 . The space V q in q ∈ Q 6 is tangent to the fibre π −1 (π(q)) over x = π(q) ∈ M 4 of the projection π : Q 6 → M 4 . Recall that the fibre over x = π(q) is identified with S 2 x ∼ = SO(4)/U(2) and so it has a natural complex structure J v . Hence, we can define an almost complex structure J on Q 6 using the decomposition (2.1) by setting where J h is an almost complex structure equal in the point q ∈ Q 6 to the complex structure J h q on H q ∼ = T π(q) M 4 = T x M 4 . Thus, the twistor space Q 6 has a natural almost complex structure J .
It was shown in [1] that if the Weyl tensor of M 4 is self-dual then the almost complex structure (2.2) on Q 6 is integrable and (Q 6 , J int ) inherits the structure of a complex analytic 3-manifold Q 3 C . It was also shown in [2] that is an almost complex structure on Q 6 which is never inegrable. These structures differ in sign along M 4 .
Twistor corespondence. Let E be a rank k complex vector bundle over M 4 and A a connection one-form (gauge potential) on E with the curvature where * denotes the Hodge star operator, ε µνλσ is the completely antisymmetric tensor on M 4 with ε 1234 = 1 in the Riemannian metric ds 2 = δ µν e µ e ν for an orthonormal basis {e µ } on T * M 4 .
Bundles E with self-dual connections A are called self-dual. It was proven in [1] that the self-dual bundle E over self-dual manifold M 4 lifts to a holomorphic bundle E over the complex twistor space Q 3 C = (Tw(M 4 ), J int ) and E is holomorphically trivial on fibres CP 1 x of projections π : Q 6 → M 4 for each x ∈ M 4 . The bundle E = π * E is defined by the connection A = π * A such that its curvature F = dA + A∧A satisfies the equations (1.3) and F = π * F is the pull-back to 3) on Q 6 the manifold (Q 6 , J non ) is not complex. However, on (Q 6 , J non ) one can introduce bundles with J -holomorphic structure (pseudoholomorphic bundles) [23]. Let E be a complex rank k vector bundle over Q 6 endowed with a connection A. According to Bryant [23], a connection A on E is said to define a J -holomorphic structure if it has curvature F of type (1,1) w.r.t. J , i.e.
It is not difficult to show that twistor correspondence between solutions of SDYM equations (2.4) on M 4 and solutions of J -hCS equations (2.5) on the almost complex twistor space (Q 6 , J ) still persists (see e.g. [7]). This will be discussed in more details later for the case of flat Euclidean space M 4 = R 4 .

Twistor space of R 4
According to the definition ( but also over S 2 , with spaces R 4 as fibres. Almost complex structures J . In section 2 we described generic construction of an almost complex structure J on a twistor space Tw(M 4 ). Here, we give explicit form of J for the case Recall that a complex structure J on R 4 is a tensor J = (J ν µ ) such that J σ µ J ν σ = −δ ν µ . All constant complex structures on R 4 are parametrized by the two-sphere for s a ∈ R 3 , a, b = 1, 2, 3. One can choose generic J in the form are antisymmetric 't Hooft tensors, µ, ν = 1, ..., 4. Using the identities one can show that J 2 = −Id. Here, we consider R 4 as a space with the metric ds 2 Let {e α } represents an orthonormal coframe on S 2 , i.e.
for α, β = 1, 2. The canonical form of complex structure j on S 2 is and are almost complex structures on the twistor space P 6 of R 4 . Complex twistor space P 3 C = (P 6 , J ) with integrable almost complex structure J = J int has been studied a lot in the literature and in the following we will focuse on non-integrable almost complex structure J = J non .
Complex coordinates for J = J int . The two-sphere S 2 , global coordinates s a on which are used in (3.4), is conformally equivalent to R 2 . One can cover S 2 by two patches in which the metric on S 2 is conformally flat, On the intersection of two patches we have v α where α, β = 1, 2.
By using the complex structure (3.4) on R 4 , one can introduce a CP 1 -family of complex coordinates on R 4 given by formulae where The coordinates (3.15) together with (3.16) provide complex coordinates on P 6 given by w 1 + , w 2 + and w 3 + = λ + on U + = U + × R 4 ⊂ P 6 (3.17) and . On the intersection U + ∩ U − of patches U ± ⊂ P 6 these coordinates are related by formulae Hence, the transition functions relating w a + and w a − are holomorphic functions on U + ∩ U − , a = 1, 2, 3. This means that J int is an integrable almost complex structure and P 3 C = (P 6 , J int ) is a complex 3-manifold. From (3.16) -(3.18) it follows that the manifold P 3 C can be identified with the total space of the holomorphic vector bundle over CP 1 , with coordinates w α ± on fibres C 2 J over points J ∈ CP 1 parametrized by λ ± ⊂ U ± ⊂ CP 1 .
Spinor notation. The rotation group SO(4) of space R 4 is locally isomorphic to the group SU(2)×SU(2), where both groups SU(2) have two-dimensional fundamental (spinor) representations µ = (µ α ) and λ = (λα) . Obviously, λ + = λ −1 − if λ1 = 0 and λ2 = 0. Isomorphism SO(4) ∼ =SU(2)×SU(2) allows also to introduce spinor notation for complex coordinates on R 4 by formula (x αα ) = x 11 x 12 From (3.26) it follows that x 11 =x 22 and where the overbar denotes complex conjugation. By using (3.26), one can rewrite (3.16) and (3.20) as follows w α Vector fields and one-forms. On the twistor space (P 6 , J ) with J from (3.10) we have the natural basis ∂ ∂z a ± for the space of (1,0) vector fields. On the intersection we have (3.30) Using formulae (3.28), we can express these vector fields in terms of coordinates (x α1 , λ ± ) and their complex conjugates according to where we have used together with the convention ε12 = −ε21 = −1, which implies εαβεβγ = δγα. Thus, the vector fields can be chosen as a basis of vector fields of type (1,0) on U ± ⊂ P 6 in the coordinates (x αα , λ ± ,λ ± ). Complex conjugate of (3.33) provide us with the vector fields which form a basis of vector fields of type (0,1) on U ± ⊂ P 6 .
It is easy to check that the basis of (1,0)-and (0,1)-forms on U ± , which are dual to the vector fields (3.33) and (3.34), is given by forms One can easily verify that Geometry of (P 6 , J ). We consider the twistor space (P 6 , J ) with J from (3.10) and coordinates {z a ± } on U ± ⊂ P 6 given by (3.20)- (3.22). In the following we often omit the signs ± in coordinates, vector fields, one-forms etc. by considering all formulae on the patch U + ⊂ P 6 .
By direct calculations we obtain that nonzero commutators of vector fields (3.33) and (3.34 where we used the formulae ∂ λ (γλα) = γ 2λα and ∂λ(γλα) = −γ 2 λα . To prove integrability of an almost complex structure J one has to show that commutators of vector fields of type (0,1) w.r.t. J will again be vector fields of type (0,1) [24]. From (3.37) we see that this is not the case and therefore J is not integrable. For one-forms (3.35) we have and complex conjugate formulae. The first terms in (3.40) define a torsionful connection on P 6 with values in u(1) ⊂ su(3) and the last terms define the Nijenhuis tensor (torsion) with non-vanishing components plus their complex conjugate N1 23 = N2 31 = γ −1 . From (3.40) we again see that (P 6 , J ) is not a complex manifold but the total space (3.23) of the anti-holomorphic bundle over CP 1 . Furthermore, from (3.40) we see that (P 6 , J ) has an SU(3)-structure and the globally defined (3,0)-form Ω with Hence, the canonical bundle of (P 6 , J ) is trivial. From (3.40) it follows that i.e. the real part of Ω is not closed. For the volume form on P 6 we have Twistor correspondence. To conclude this section we describe a twistor correspondence between the SDYM model on R 4 and J -hCS theory on (P 6 , J ).

Twistor actions for Yang-Mills theory
Graded twistor space P 6|2 . Recall that on P 6 there are globally defined (3,0)-form Ω given by (3.42) and its complex conjugate (0,3)-formΩ. Hence, the J -hCS action functional (1.4) is well defined on (P 6 , J ). However, if we substitute (3.49) into (1.4) then we obtain S = 0 since (0,3)-part of Chern-Simons form CS(A) on (P 6 , J ) vanishes if A 3 =Ā 3 = 0. To obtain a nontrivial Lagrangian, one can perform a gauge transformation, which will give some non-vanishing terms 2 as it was done in [16,17]. We will not follow this path here because this way we can at best get the actions [18]- [21] which have various limitations in comparison with the Chalmers-Siegel action [12].
The action [12] cannot be obtained without introducing additional degrees of freedom since it contains an extra propagating field Gαβ. One of the possibilities for introducing additional fields is to consider vector bundles E over P 6 that are not trivial after restriction to CP 1 ֒→ P 6 [25]. Another possibility is to consider a graded extension of the twistor space (P 6 , J ) similar to the cases considered by Wolf [10,14] and Sämann [9,15] for the complex twistor space P 3 C . We will use the second option and introduce a graded twistor space P 6|2 .
On the space P 6 we consider the space Gr P 6 of locally defined functions (a sheaf) with values in the Grassmann algebra Λ(R 2 ). A manifold P 6 with the sheaf Gr P 6 is a graded manifold P 6|2 = (P 6 , Gr P 6 ) [26,27] that can be viewed as the trivial bundle P 6 × Λ 1 (R 2 ) → P 6 . Tangent spaces of P 6|2 are defined by the even vector fields (3.33), (3.34) together with the odd vector fields commuting with the even vector fields on P 6 . Respectively, the space of differential forms on P 6|2 has the local basis {E a ,Ē a , dη i } with commutation relations where {E a ,Ē a } are given in (3.35).
Chern-Simons type theory on P 6|2 . Let E be a trivial rank k complex vector bundle over P 6|2 and A a connection one-form on E. We choose the connection A depending on all coordinates on P 6|2 and having no components along the Grassmann directions. The curvature F of such A is is the fermionic part of d.
Field equations on P 6|2 . Having given necessary ingredients, we may now consider J -hCS field equations (4.17). These equations on the patchÛ where " " denotes the interior product of vector fields with differential forms. Here we used components of A in the expansion As usual in the twistor approach, we work in a gauge in whichĀ 3 = 0 but the bosonic part ofĀ 3 is zero. Note that in general the gauge potential A in (4.18) and (4.19) can be expanded in the odd coordinates η i as For simplicity and more clarity we first consider the truncated case ψ i = 0 and discuss the case ψ i = 0 afterwards.
Remark. The connection (4.20) on the vector bundle E over P 6|2 ∼ = P 6 × Λ 1 (R 2 ) takes values in the Lie algebra g of a semi-simple Lie group G. Note that maps from the space Λ 1 (R 2 ) in (4.5) to the group G form a supergroup super-T G [30], where T G = G ⋉ g is the semi-direct product of G and g. That is why the field A in (4.20) can be considered as a connection on a super-T G bundle E ′ over the bosonic twistor space P 6 . This kind of correspondence was found by Witten when studying Chern-Simons theories on 3-manifolds [30].
Substituting (4.21) into (4.18), we obtain the equations showing that G α(αβγ) are composite fields describing no independent degrees of freedom. Other nontrivial equations following from (4.18) after substituting (4.21) read Hence, B αα is a tangent vector at A αα to the solution space of the SDYM equations. It is a secondary field (a symmetry) depending on A αα and for simplicity we neglect it by choosing B αα = 0. The rest equations (4.24) and (4.26) are the Chalmers-Siegel equations describing the self-dual gauge potential A αα and the anti-self-dual field G αα,ββ = ε αβ Gαβ propagating in the self-dual background.
The action functional associated with (4.24) and (4.26) is given by with fαβ given by (3.52). This action can be obtained from (4.12) after splitting, into ordinary bosonic and even nilpotent parts, using the formula 3 and integrating over the nilpotent coordinate η and over CP 1 ֒→ P 6|2 .
Extra terms. As we mentioned earlier, the general expansion (4.20) of connection A in odd coordinates η i contains fermionic fields ψ i (x αα ) which we consider now. Expansion (4.20) can be written in components as A α = γ λαA αα (η 1 , η 2 ) and where A αα (η 1 , η 2 ) = A αα + η i (ψ i αα + γλβλγψ i α(αβγ) ) + η 1 η 2 (B αα + γλβλγG α(αβγ) ) , (4.32) ForĀ α andĀ 3 we havē Full Yang-Mills. So far, we have shown that the Chalmers-Siegel action (4.28) for SDYM theory can be obtained from the Chern-Simons type action (4.12) on the graded twistor space P 6|2 . It is known that the action (4.28) is a limit of the full Yang-Mills action for small coupling constant g YM . Namely, let us modify the action (4.28) by adding the term   with the coupling constant g YM = ε, plus the topological term. Therefore, for obtaining the Yang-Mills action (4.42) we should derive the term (4.38) from the twistor space.

Conclusions
In this paper we considered graded twistor space P 6|2 with a non-integrable almost complex structure J and J -holomorphic Chern-Simons theory on P 6|2 . It was shown that under some assumptions this theory is equivalent to self-dual Yang-Mills theory on R 4 . In our discussion we tried to be close to the consideration of the papers [14,15], where N < 4 SDYM theories were derived from holomorphic Chern-Simons theory on complex supertwistor spaces. We have also shown that the full Yang-Mills action in R 4 can be obtained from a twistor action on P 6|2 with a locally defined Lagrangian. We did not pursue the goal of studying all these tasks in full generality. We wanted to show the principal possibility of obtaining actions for Yang-Mills and its self-dual subsector from a twistor action. Examining all aspects of the model requires additional efforts.