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Twistor Action for General Relativity, [58] A

Twistor Action for General Relativity, [58] A

arXiv:2104.07031v2 [hep-th] 28 Jul 2021 o l relvlapiue in amplitudes formulae tree-level to all rise for give Twistor strings ambitwistor amplitudes. many and of scattering heart in the at structures lie remarkable space twistor on morphic ino h ogsadn ogypolmo wso the- twistor of resolu- problem non-linear googly fully long-standing but the classical of tion a sector providing non-self-dual the GR, encodes of also action with Our space-times to metrics. complex spaces almost twistor on certain structures associates non-linear that tool Penrose’s [37] main of construction generalization Our novel a [36]. is in (without discussed GR constant) Euclidean for cosmological action chiral the to equivalent [35]. investigations Further roadblocks similar sector. encountered MHV equivalence the beyond manifest GR any with lacked but MHV [31–34], graviton amplitudes tree-level reproduce were encode constructively These to to able [31]. were expanded interactions 30] non-self-dual later [29, leading-order GR and self-dual constructed for and successfully actions 28] down twistor [5, broke Meanwhile, but conformal [26], [27]. multiplicity in graviton high out for at carried rules also MHV were finding descrip- scattering at action attempts effective Direct an lacked tion. it but [23–25], gravity n aers otete mltdsof amplitudes tree [22] in the formulated to was rise gravity gave for elu- more and much twistor be A to proven sive. has (GR) relativity general non- in 21]. scattering [20, gluon backgrounds on first trivial the results yielded all-multiplicity corre- also has other ever it numerous Recently, and [13–19]. 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Penrose e igti,w rv htortitraction twistor our that prove we this, sing α 2 ˙ M I HRLFRUAINO GR OF FORMULATION CHIRAL II. α β Γ S µ ˙ αβ st h eiaino computational of derivation the to as l o-iergaio osrcin we construction, graviton non-linear s αβ oia osata nato governing action an as constant logical ( [ α satisfying 0 = ˙ αβ e, nerbeams ope structures complex almost integrable ρ eafu-iesoa aiodwt Rieman- with manifold four-dimensional a be csa arnemlile n moe the imposes and multiplier Lagrange a as acts α ]= Γ] ) oeulteADsi oncinassociated connection ASD the equal to ˙ αβ where , g etc. , , ecniu ocl t“pc-ie.We “space-time”. it call to continue We . ,˙ 1, S nti ae tflosfo h structure the from follows it case, this In . Σ αβ SD α Z αβ d M [ s htteADsi oncini flat is connection spin ASD the that = e, ǫ 2 Σ = ntdKingdom United αβ κ Σ ]= Γ] = 0 ˙ αβ , stegaiainlculn.This coupling. gravitational the is ǫ r pnridcsand indices spinor are 1 γβ ˙ ǫ ( αβ αβ ∧ Z = ) (dΓ M ǫ α = ˙ δ β γ Γ α ˙ αβ e e αβ and α α α α ˙ + ˙ ∧ ∧ e ∗ κ β dΣ ǫ e 2 β α ˙ ˙ e β κ β Γ α , ˙ αβ α ˙ ǫ α α ˙ → γ ˙ . γ β o hsmetric, this for . ˙ h ∧ = κ λ fG,an GR, of 0 Γ δ γβ i γ α ˙ ˙ = ǫ Spinor . (3) ) αβ λ , α ǫ κ (1) (4) (2) α αβ ˙ α β ˙ , 2

III. EUCLIDEAN IV. OFF-SHELL NON-LINEAR GRAVITON

We start by recalling the twistor correspondence for a. Curved twistor spaces. Instead of working covari- Euclidean signature flat space (see [40–42] for a review). antly with the Atiyah-Hitchin-Singer almost complex The of R4 is PT = P3 \ P1. This is structure [40, 43] like in [5, 30, 35], we now build a new also the total space of the holomorphic vector bundle local model of twistor spaces for off-shell curved space- 1 A α˙ O(1) ⊕ O(1) → P . Let Z = (µ , λα) be homogeneous times. Penrose’s non-linear graviton [37, 44] will emerge twistor coordinates, with λα denoting coordinates on the as a corollary. base P1 and µα˙ up the fibers of O(1) ⊕ O(1). We endow Let PT be a manifold that is diffeomorphic to PT PT with a reality structure induced by the quaternionic (equivalently PS) and possesses an almost complex struc- A A α˙ conjugation: Z 7→ Zˆ = (ˆµ , λˆα) with ture with Dolbeault operator

α˙ 1˙ 0˙ ¯ ¯ λˆα = (λ1, −λ0) , µˆ = (µ , −µ ) . (5) ∇ = ∂ + V. (13) The points xαα˙ ∈ R4 of flat space are in 1:1 correspon- We assume that, like PT, it has a fibration PT → P1 dence with projective lines in twistor space that are left that is at least smooth. This lets us use twistor coordi- invariant by the ˆ· conjugation: nates ZA as well as spinor bundle coordinates (x, λ) as PT αα˙ P1 α˙ αα˙ local coordinates on (when using the latter, we occa- x ←→ X ≃ : µ = x λα (6) sionally abuse notation and refer to PT by PS as well). ¯ that simultaneously satisfyµ ˆα˙ = xαα˙ λˆ . This correspon- In these coordinates, ∂ is the “background” Dolbeault α PS dence recovers R4 as the moduli space of such lines. operator on , If we let x vary, pullback to these real twistor lines ∂¯ =e ¯0 ∂¯ +¯eα˙ ∂¯ , (14) provides a diffeomorphism between PT and the projec- 0 α˙ PS R4 P1 tive spinor bundle of undotted = × 0,1 PS 1,0 αα˙ while V ∈ Ω ( ,TPS ) provides a finite deformation with coordinates (x , λα). It is useful to work directly on PS when building action principles. The (0, 1)-vector ˙ V ≡ V α˙ ∂ = e¯0 V α˙ +¯eβ V α˙ ∂ . (15) fields determining the twistor complex structure on PS α˙ 0 β˙ α˙  are spanned by α˙ α˙ Occasionally, we also set V0 ∂α˙ ≡ V0 and Vβ˙ ∂α˙ ≡ Vβ˙ . ¯ ˆ ∂ ¯ α For the deformation to be compatible with the fibration ∂0 = −hλ λi λα , ∂α˙ = λ ∂αα˙ , (7) PT → P1, we have taken V y e0 = 0. ∂λˆα In what follows, we will also need to assume that V αα˙ where ∂αα˙ ≡ ∂/∂x . Their dual (0, 1)-forms are is a hamiltonian vector field with respect to the Poisson

αα˙ bivector I given in (11). This leads to a zero-divergence Dλˆ λˆ dx α˙ e¯0 = , e¯α˙ = α , (8) condition on the (0, 1)-form valued components V : hλ λˆi2 hλ λˆi α˙ div V ≡ L∂α˙ V =0 . (16) where Dλˆ ≡ hλˆ dλˆi. We also list convenient bases of (1, 0)-vector fields and (1, 0)-forms: We will impose this as a constraint in our action, though one can also solve it in terms of a hamiltonian h [30]. ˆ ˆα λα ∂ λ ∂αα˙ In the deformed complex structure, the (0, 1)-vector ∂0 = , ∂α˙ = − , (9) ¯ ¯ hλ λˆi ∂λα hλ λˆi fields are spanned by ∂0 + V0, ∂α˙ + Vα˙ . The associated 0 α˙ αα˙ basis of (1, 0)-forms on PT is e = Dλ, e = λα dx , (10) 0 α˙ α˙ α˙ where Dλ ≡ hλ dλi is the canonical holomorphic top- e = Dλ, θ = e − V , (17) form on P1. In terms of these, we can equip PT with a as these annihilate the (0, 1)-vector fields. A computation holomorphic Poisson structure through the bivector produces the structure equations de0 = 0 and α˙ β˙ I = ǫ ∂α˙ ∧ ∂β˙ (11) α˙ 0 α˙ β˙ α˙ α˙ dθ = e ∧ L∂0 θ − θ ∧ L∂β˙ V − N , (18) whose symplectic leaves are the fibers of O(1) ⊕ O(1). In the computations below, we also use the fact that where the “torsion” N α˙ is found to be exterior derivatives of projective differential forms on PS α˙ 2 1 with homogeneity n in λ and 0 in λˆ receive corrections N ≡ N ∂α˙ = ∇¯ = ∂V¯ + [V, V ] . (19) α α 2 from the Chern connection on O(n) → P1: This is consistent with the Newlander-Nirenberg theorem hλˆ dλi ¯ 2 dPS ≡ d = dS + n ∧ , (12) that ∇ be the obstruction to the integrability of the hλ λˆi distribution of (0, 1)-vector fields. The almost complex structure is integrable precisely when N vanishes. where S = R4 × C2 is the non-projective spinor bundle. 3

b. Reconstruction of space-time. Much like R4, we Hence, assuming that (16) and (23) hold, we conclude 0 ∗ 1 can construct the space-time M associated to PT as a that e ∧ p Σ is globally holomorphic in λα ∈ P . By moduli space of rational curves. In homogeneous coordi- Liouville’s theorem on P1, we finally obtain a triplet of nates, the twistor lines are deformed into a 4-parameter 2-forms Σαβ =Σ(αβ) on M via family of pseudo-holomorphic degree 1 rational curves ∗ αβ 0 labeled by moduli yαα˙ : p Σ= λα λβ Σ (y) mod e . (26)

1 α˙ αα˙ α˙ Y ≃ P : µ = x (y, λ) λα ≡ F (y, λ) . (20) Comparing this with (24) yields the existence of a matrix α˙ C Hβ˙ (x, λ) ∈ SL(2, ) of homogeneity 0 in λα such that Our space-time M is taken to be the moduli space of such curves that are invariant under ZA 7→ ZˆA. It exists and is ∗ α˙ β˙ αα˙ 0 p Hβ˙ θ = λα e (y) mod e , (27) generically four-dimensional for suitable data V [5, 45].  PT is then diffeomorphic to a patch of the projective for some 1-forms eαα˙ on M. In terms of these, we find P1 αα˙ αβ αα˙ β αα˙ spinor bundle M× →M coordinatized by (y , λα). Σ = e ∧ e α˙ . The e comprise a tetrad for our 1 As the curves are parametrized by λα ∈ P , we will sought after metric (1) on M.  ¯ 0 0 abuse notation by using {∂0, ∂0} and {e , e¯ } to also de- α˙ note the standard bases of vector fields and forms on Y . Hβ˙ (x, λ) provides a spin-frame on the bundle of dot- In local coordinates, this diffeomorphism can then be ex- ted spinors O⊕O→ PS. Due to being SL(2, C)-valued, α˙ β˙ γ˙ δ˙ γ˙ δ˙ pressed as a map it satisfies ǫ Hα˙ Hβ˙ = ǫ and drops out of invariant objects like Σ. It also solves the d-bar equation p : M× P1 → PT , (y, λ) 7→ (x(y, λ), λ) (21) γ˙ ¯ ¯ y ∗ γ˙ ∗ α˙ δ ∂ + ∂ p L∂ ˙ V p Hγ =0 , (28) β˙ 0 0 β ˙ satisfying the PDE for pseudo-holomorphic curves [46]:  found by acting with L ¯ on (27) and simplifying using ¯ y ∗ α˙ ∂0 ∂0 p θ =0 , (18), (23). Alternatively, we can take (28) as its definition α˙ ∗ α˙ (22) i.e., ∂¯0F = ∂¯0 y p V . when constructing the action. As of yet, eαα˙ do not satisfy any equations of motion. When V = 0, xαα˙ = yαα˙ . More generally, one can always The original non-linear graviton construction arises as a solve (22) for xαα˙ (y, λ) locally as it is an elliptic PDE [45]. corollary of proposition 1. We first use (16) and (18) to We now prove our new result that constructs an off- show that shell metric on M from the almost complex structure of α˙ 0 dΣ = −2 N ∧ θα˙ mod e . (29) PT . The main fact that will be useful for us is that this does not require complete integrability N = 0. This brings us to Proposition 1 (Off-shell ) Corollary 1.1 (Penrose [37]) The resulting metric on Every hamiltonian complex structure deformation V of M is self-dual Ricci-flat if and only if N α˙ =0. PT satisfying V y e0 =0 and Proof: Pulling back (29) to M× P1 using (26), we con- ∗ α˙ αβ α˙ ∂¯0 y p N = 0 (23) clude that dΣ (y) = 0 is equivalent to N = 0. As expected, SD vacuum space-times arise from integrable, gives rise to a metric on the associated space-time M. hamiltonian complex structure deformations.  Proof: We begin by introducing a (2, 0)-form of weight +2 in λα: V. TWISTOR ACTION FOR GRAVITY α˙ Σ := θ ∧ θα˙ . (24) Like its space-time counterpart (3), our proposal for When the almost complex structure is integrable, this the twistor action decomposes into an action for the SD gives Gindikin’s holomorphic symplectic form on the subsector, plus an interaction term encoding the non-self- fibers of PT → P1 [31, 47]. More generally, it follows dual excitations: from (18) and (22) that κ2 S[∇¯ ,B,C]= S [∇¯ ,B,C]+ S [∇¯ ,B] . (30) ∗ 0 ¯ y ∗ 0 SD 4 int L∂¯0 p (e ∧ Σ) = ∂0 p (div V ∧ e ∧ Σ) ∗ α˙ 0 ∗ + 2 (∂¯0 y p N ) ∧ e ∧ p θα˙ , (25) This depends on three fields: the almost complex struc- ture represented by the Dolbeault operator ∇¯ described ∗ 0 0 α˙ having noted that p e = e as p preserves λα. We ob- above, a (1, 1)-form B ≡ θ ∧ Bα˙ with coefficients 0,1 serve that the first term on the right vanishes due to V Bα˙ ∈ Ω (PS, O(−5)) acting as a Lagrange multiplier being divergence-free, whereas the second term can be imposing N ≡ ∇¯ 2 = 0 in the SD subsector, and a further made to vanish if and only if we assume (23), i.e., take Lagrange multiplier C ∈ Ω0,2(PS, O(−4)) for the zero- N α˙ to be trivial along the curves. divergence condition (16). In our correspondence, ∇¯ is 4

0 0 ⊥ “dual” to the space-time metric while B will map to the where Bα˙ |Y is in the span of e , e¯ and Bα˙ in the span of αα˙ ∗ ∗ ASD field Γαβ of (3). dy . Next note that due to (22), ∂¯0 y p Ω = 0, i.e., p Ω The SD action takes the form has noe ¯0-component. So, the integral in (34) can only be 0 ∗ saturated by thee ¯ -component of p Bα˙ . Consequently, S = Ω ∧ B ∧ N α˙ +Ω ∧ C ∧ div θ , (31) B⊥ drops out of S . It will only be present in S . SD Z α˙ α˙ int SD PT We begin the reduction by integrating out C. This having used the canonical (3, 0)-form on PT , imposes (33) as a constraint. In the remaining SD action, we perform a “change of integration variables” using the Ω := Dλ ∧ Σ , (32) diffeomorphism p. This yields

β˙ ∗ of weight +4 in λα. Since L∂ ˙ e = 0, the equation of α˙ α SSD = p (Ω ∧ Bα˙ ∧ N ) motion of C in coordinates is ZM×P1 (36) ∗ ⊥ ∗ div θ ≡ L θα˙ = −div V =0 . (33) = p Ω ∧ B + B ∧ p N α˙ , ∂α˙ Z α˙ Y α˙ M×Y  (31) is in fact the standard twistor action of self-dual con- ¯ y ∗ ¯ y ⊥ formal gravity [5], now adapted to the fibration PT → P1 having used (35). Again, since ∂0 p Ω=0= ∂0 Bα˙ , and augmented with the zero-divergence constraint. the second term in this integral only retains the compo- ∗ α˙ ⊥ On the other hand, the non-self-dual interactions are nent of p N alonge ¯0. Thus, integrating out Bα˙ from ¯ y ∗ α˙ captured by the theory imposes ∂0 p N = 0 as a constraint. By proposition 1, this allows us to construct a space- 2 time tetrad eαα˙ as in (27). With this in hand, we can use ˙ ˙ ∗ β1β2 α˙ i α˙ Sint = h12i ǫ pi (Bi α˙ i Hi ˙ ∧ Ωi) (34) Ω = Dλ ∧ θ ∧ θ to recast the rest of the SD action as Z βi α˙ i^=1 PS×MPS ˙ S = −2 Dλ ∧ p∗θα˙ ∧ B ∧ p∗(N β ∧ θ ) αα˙ SD Z α˙ Y β˙ with the integral being over points (y , λ1 α, λ2 α) of the M×Y 1 2 PS M PS P fiberwise product × ≃ M× ( ) . We have ∗ α˙ ∗ = Dλ ∧ p θ ∧ Bα˙ Y ∧ p dΣ (37) abbreviated hλ1 λ2i ≡ h12i. The remaining ingredients ZM× α˙ α˙ Y are Bi ≡ B(x, λi), Ωi ≡ Ω(x, λi), Hi β˙ ≡ H β˙ (x, λi), ∗ ∗ α˙ αβ so that p B ≡ B (x(y, λ ), λ ), etc. This interaction is = Dλ ∧ p θ ∧ Bα ∧ λα λβ dΣ (y) . i i i i i Z ˙ Y non-local on twistor space but reduces to the expected M×Y interaction term on space-time. To get the first line, we have used 2 θα˙ ∧ θ = δα˙ θγ˙ ∧ θ . As written, our twistor action is only partially covari- β˙ β˙ γ˙ The second line follows from (29), while the third line ant as it singles out the coordinate λα along the base of the fibration PT → P1. It is only invariant under is a consequence of (26). Hence, defining the space-time diffeomorphisms preserving the fibration and the Pois- field Γαβ(y) as the Penrose transform of B [31], son bivector in (11), as these also preserve the zero- ∗ divergence constraint. Nevertheless, this is enough to Γαβ(y) := Dλ ∧ λα λβ p B Z give rise to the covariant action (3) on space-time. Y (38) = Dλ ∧ λ λ p∗θα˙ ∧ B , Z α β α˙ Y Y VI. EQUIVALENCE WITH GR the SD action reduces to (4). Recall the diffeomorphism p : M× P1 → PS ≃ PT . (ii) Using (38), we can also prove the equivalence of We now prove that we can compactify the twistor action the interaction term in (3) with the non-SD action (34): (30) along P1 to obtain the complete action (3) of GR as 2 αβ γ the effective theory on M. κ Σ ∧ Γα ∧ Γγβ ZM Proposition 2 On performing the Penrose transform, 2 αα˙ β = κ h12i λ1 α e ∧ λ2 β e α˙ (i) the SD twistor action (31) reduces to the SD space- ZPS×MPS time action (4). ∗ β˙ ∗ γ˙ ∧ Dλ1 ∧ p1 θ1 ∧ B1 β˙ ∧ Dλ2 ∧ p2 θ2 ∧ B2γ ˙ (ii) The non-SD interaction term (34) reduces to the κ2   = S , (39) interaction term in the space-time action (3). 4 int Proof: (i) Given a V , we will integrate out the compo- α˙ α˙ with θi ≡ θ (x, λi), etc. To get the second equality, we αα˙ nents of B orthogonal to its pseudo-holomorphic curves. have substituted for λi α e using (27).  Using {e0, e¯0, dyαα˙ } as a basis of T ∗(M× P1), we split (38) is invariant under B 7→ B + ∇¯ χ for smooth (1, 0)- ∗ ⊥ α˙ 1,0 PT p Bα˙ = Bα˙ Y + Bα˙ , (35) forms χ = χα˙ (x, λ) θ ∈ Ω ( , O(−4)). This gives

5 an additional gauge symmetry of the twistor action. The ple. A preliminary study of the MHV vertices originating invariance of Sint holds by construction, while that of from (34) has already been performed by using them to SSD follows from symmetries of the SD compute on-shell graviton MHV amplitudes in flat space α˙ twistor action [5] and because ∇¯ Ω ∧ χα˙ N ∝ div V = 0. [31, 34] as well as in non-trivial classes of self-dual space- This extra gauge symmetry is expected to be a crucial times [20]. It would be very interesting to see if the meth- ingredient for the derivation of gravitational MHV rules, ods of these works can be applied to the off-shell MHV as it allows for useful gauge choices [7, 8, 31, 32]. vertices and propagator of our action. It should also be possible to adapt our twistor action to the case of a non-vanishing as VII. DISCUSSION well as to non-trivial amounts of . This could give insights into MHV rules for graviton scattering We have proposed a new twistor action for gravity. in (A)dS4, possibly making contact with the worldsheet Proposition 2 shows that it is equivalent to Plebanski’s formulae in [49]. Our action may also find analogues on formulation of Euclidean . Solutions to the spinor bundles of Lorentzian or split-signature space- its field equations should yield twistor spaces for generic times [41, 50], in other dimensions [51], holographic back- vacuum space-times that are not necessarily self-dual, grounds [52], and integrable systems [50, 53]. while its perturbation theory should provide new prin- It would also be interesting to explore the recently dis- ciples for the computation of graviton amplitudes. covered twistorial origins of double copy [54, 55] at the Previous attempts like [35] built covariant twistor ac- level of twistor actions. This could help extend the anal- tions by means of the Atiyah-Hitchin-Singer almost com- ysis of the kinematic algebra in [56] to the non-self-dual plex structure, but were unable to reduce them to GR sector. Further points of interest include building connec- by compactification on P1. Unlike them, we have built tions with other contemporary work on reformulations of an analogue of Kodaira-Spencer gravity [48] by taking GR. See for instance [57] for graviton scattering using the the almost complex structure of PT to be a deforma- chiral action on space-time, and [58] for possible implica- tion of the complex structure of the “flat” background tions for double copy. It would also be worthwhile to try PT. We also assumed the existence of a smooth fibra- developing links with inherently quantum formulations tion PT → P1, as this appropriately reduced the degrees like loop [59]. of freedom in compatible deformations V . An important next step would be to complete this into a fully covariant, Acknowledgments: It is a pleasure to thank Tim background-independent formalism, along with studying Adamo, Roland Bittleston, Lionel Mason and David its quantum consistency. Skinner for helpful discussions and comments on the A primary motivation for our construction is to derive draft. AS is supported by a Mathematical Institute Stu- robust gravitational MHV rules from an action princi- dentship, Oxford.

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