Twistor action for general relativity Atul Sharma1 1The Mathematical Institute, University of Oxford, OX2 6GG, United Kingdom∗ (Dated: July 29, 2021) We reformulate Euclidean general relativity without cosmological constant as an action governing the complex structure of twistor space. Extending Penrose’s non-linear graviton construction, we find a correspondence between twistor spaces with partially integrable almost complex structures and four-dimensional space-times with off-shell metrics. Using this, we prove that our twistor action reduces to Plebanski’s action for general relativity via the Penrose transform. This should lead to new insights into the geometry of graviton scattering as well as to the derivation of computational tools like gravitational MHV rules. I. INTRODUCTION ory [38]. This represents a significant step toward the construction of twistor spaces for non-self-dual solutions Dualities relating space-time field theories and holo- of Einstein’s equations. Furthermore, it paves a clear morphic theories on twistor space lie at the heart of many way for the derivation of an MHV formalism for gravity remarkable structures in scattering amplitudes. Twistor by means of its perturbative expansion. and ambitwistor strings give rise to worldsheet formulae for all tree-level amplitudes in N = 4 super-Yang-Mills [1–4]. The gauge theory twistor action was originally II. CHIRAL FORMULATION OF GR discovered as an effective action of twistor strings and proved to be equivalent to the Yang-Mills action up to a Let M be a four-dimensional manifold with Rieman- topological θ-term [5, 6]. This led to constructive proofs nian metric g. We continue to call it “space-time”. We of the Parke-Taylor formula, the maximally helicity vi- can introduce a (complex) null tetrad eαα˙ for this metric, olating (MHV) diagram formalism [7–9], the amplitude- 2 αα˙ ββ˙ Wilson loop duality [10–12], and numerous other corre- ds = ǫαβ ǫα˙ β˙ e e , (1) spondences [13–19]. Recently, it has also yielded the first ˙ ˙ ever all-multiplicity results on gluon scattering in non- where α = 0, 1,α ˙ = 0, 1 are spinor indices and ǫαβ, ǫα˙ β˙ trivial backgrounds [20, 21]. are Levi-Civita symbols. Spinor indices are raised using αβ α˙ β˙ αβ α α˙ β˙ α˙ On the other hand, the long-sought twistor action for ǫ , ǫ satisfying ǫ ǫγβ = δγ and ǫ ǫγ˙ β˙ = δγ˙ . Spinor α general relativity (GR) has proven to be much more elu- contractions are conventionally denoted by hλκi = λ κα, α˙ sive. A twistor string for gravity was formulated in [22] [µρ]= µ ρα˙ , etc. and gave rise to the tree amplitudes of N = 8 super- The anti-self-dual (ASD) 2-forms are spanned by gravity [23–25], but it lacked an effective action descrip- αβ (αβ) αα˙ β tion. Direct attempts at finding MHV rules for graviton Σ =Σ = e ∧ e α˙ . (2) scattering were also carried out in [26], but broke down at high multiplicity [27]. Meanwhile, twistor actions for Using these, we work with a version of Plebanski’s chiral conformal gravity [5, 28] and self-dual GR [29, 30] were action for GR espoused in [36]: successfully constructed and later expanded to encode αβ 2 γ leading-order non-self-dual interactions [31]. These were S[e, Γ] = Σ ∧ (dΓαβ + κ Γα ∧ Γγβ) (3) ZM arXiv:2104.07031v2 [hep-th] 28 Jul 2021 able to constructively reproduce tree-level graviton MHV amplitudes [31–34], but lacked any manifest equivalence given in terms of the tetrad and auxiliary 1-form fields with GR beyond the MHV sector. Further investigations Γαβ =Γ(αβ), where κ is the gravitational coupling. This encountered similar roadblocks [35]. chiral action is equivalent to the Einstein-Hilbert action In this letter, we present a new twistor action that is up to a topological term. The equation of motion of Γαβ 2 equivalent to the chiral action for Euclidean GR (without sets κ Γαβ to equal the ASD spin connection associated cosmological constant) discussed in [36]. Our main tool to g. The tetrad’s equation of motion then implies Ricci- is a novel generalization of Penrose’s non-linear graviton flatness. In the self-dual (SD) limit κ → 0 of GR, an construction [37] that associates certain almost complex integration by parts reduces this action to [39] structures on twistor spaces to space-times with off-shell metrics. Our action also encodes the non-self-dual sector αβ SSD[e, Γ] = Γαβ ∧ dΣ . (4) of GR, providing a classical but fully non-linear resolu- ZM tion of the long-standing googly problem of twistor the- Here, Γαβ acts as a Lagrange multiplier and imposes the closure of Σαβ. In this case, it follows from the structure equation for Σαβ that the ASD spin connection is flat ∗ [email protected] and the space-time is self-dual vacuum. 2 III. EUCLIDEAN TWISTOR THEORY 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 twistor space 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 spinors = × 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.