The inverse twistor transform for ten-dimensional supergravity: what data on the twistor space of supersymmetric light-like lines characterize a supergravity background?

Witten's 1986 twistor-like transform established a one-to-one correspondence between ten-dimensional supersymmetric Yang–Mills fields and holomorphic bundles on the complex manifold N of supersymmetric light-like lines, and showed that any N=1, d=10 supergravity background determines such an N, with each superspace point corresponding to a quadric Q in N whose family reconstructs spacetime up to a conformal metric. The catch is that a generic N with such a quadric family only yields a superspace satisfying a weak integrability condition (the first torsion relation, eq. (60) of the source), not the full supergravity constraints. The open problem is the inverse transform: specify the additional geometric data on N — candidates include the closed three-form $H=dB$ (the field strength of the two-form gauge field) and compatibility conditions analogous to the self-dual four-dimensional construction — that are necessary and sufficient for the reconstructed superspace to be a genuine ten-dimensional N=1 supergravity solution, or prove that no such characterization exists. Later work solved the linearized equations by twistor methods (Galperin–Howe–Townsend 1993), characterized supergravity backgrounds through pure spinor superspace and worldsheet consistency (Berkovits–Howe 2002; Adamo–Casali–Skinner 2015), and built twistorial ambitwistor-string descriptions of ten- and eleven-dimensional amplitudes (Geyer–Mason 2020), but no twistor-space characterization of full nonlinear ten-dimensional supergravity has been established, and no obstruction is known.

Background

Twistor theory replaces space-time geometry by the geometry of the space of light rays. In four dimensions, Penrose's twistor space parametrizes null geodesics, and classic correspondences relate massless fields to holomorphic data on twistor space: massless linear field equations are solved by twistor cohomology classes, and (anti-)self-dual gauge fields are identified with holomorphic vector bundles on twistor space (the four-dimensional twistor transform of Penrose and its higher-dimensional pure-spinor generalizations are described concisely in Berkovits & Cherkis 2004). In 1986, Witten constructed a ten-dimensional analogue of this transform for supersymmetric field theory, motivated — in the words of the paper's abstract — by the possibility that "the twistor transform of ten-dimensional supersymmetric field theory is the proper starting point for understanding the geometrical meaning of superstring theory".

Ten-dimensional N=1 supersymmetric Yang–Mills theory is compactly described in superspace: one enlarges complexified ten-dimensional space-time with fermionic coordinates $\theta^\alpha$ forming a 16-component Majorana–Weyl spinor — a real spinor of definite chirality, which exists in ten dimensions — (a superspace of dimension (10,16)) and imposes the single constraint $F_{\alpha\beta}=0$ on the spinor components of the gauge field strength; the superspace Bianchi identities then imply the full ten-dimensional Yang–Mills equations. Witten showed that this constraint is equivalent to the gauge connection being trivial along every supersymmetric light-like line L, a (1,8)-dimensional locus generalizing an ordinary null line (the orbit of a classical massless superparticle), and deduced a twistor transform: solutions on a region U correspond one-to-one to holomorphic vector bundles E over the complex manifold N of such lines whose restriction to each quadric $Q(p)\subset N$ — the set of lines through a superspace point $p\in U$ — is holomorphically trivial.

For N=1 supergravity in ten dimensions the analogous construction is only partially successful. The theory (graviton, dilaton, two-form gauge field B with three-form field strength $H=dB$, and their fermionic partners) is encoded by torsion constraints on the curved superspace vielbein $e^m,e^\alpha$,

$$\mathscr{D}e^{m}=T^{m}{}_{np}\,e^{p}e^{n}+\Gamma^{m}{}_{\alpha\beta}\,e^{\alpha}e^{\beta},$$ $$\mathscr{D}e^{\alpha}=T^{\alpha}{}_{np}\,e^{p}e^{n}+\phi^{\epsilon\alpha}\Gamma_{m\beta\epsilon}\,e^{\beta}e^{m},$$

together with the Bianchi identities, including $dH=0$; here $\mathscr{D}$ is the Lorentz-covariant exterior derivative, T the torsion, and $\phi^{\epsilon\alpha}$ an auxiliary spinor field fixed by the constraints. Witten showed that these constraints imply the existence of a twistor space N — the complex manifold of dimension (17,8) parametrizing massless superparticle orbits, in which each superspace point corresponds to a quadric $Q\subset N$ — and that the family of quadrics reconstructs the space-time M with a natural conformal metric. However, a generic complex manifold N carrying such a family of quadrics yields a reconstructed superspace satisfying only a weaker integrability condition — the first torsion relation above (equation (60) of the paper), which is the condition for the local fermionic (kappa-)symmetry of the massless superparticle — rather than the full supergravity constraints. The paper explicitly leaves open what additional data on N (the closed three-form H, and further compatibility conditions analogous to those needed in the self-dual four-dimensional gravity construction — Penrose's "nonlinear graviton", which encodes a self-dual (chiral) gravitational field in the complex structure of a deformed region of twistor space (Penrose 1976)) would make the transform invertible, conjectures that no analogous twistor interpretation may exist for eleven-dimensional supergravity, and asks whether the whole construction generalizes to N=2 (type II) ten-dimensional supergravity and to string-theoretic twistor spaces. The central question is thus the ten-dimensional analogue of asking for a twistor characterization of full — not merely self-dual — gravity.

Modern context: the pure spinor formalism introduces a bosonic spinor $\lambda^\alpha$ obeying the pure spinor constraint $\lambda\gamma^{m}\lambda=0$, a projectivized version of the same null geometry, and is a direct descendant of this program; so is the ambitwistor string, a chiral, infinite-tension string whose target is the space of complex null geodesics (ambitwistor space). These descendants characterize ten-dimensional supergravity through worldsheet consistency conditions and scattering equations rather than through geometric data on N, which is why the original inverse-transform question has survived.

Problem Statement

Establish the inverse twistor transform for ten-dimensional N=1 supergravity. Let N be the complex (17,8)-dimensional manifold parametrizing supersymmetric light-like lines, so that any ten-dimensional N=1 supergravity solution — a curved (10,16) superspace satisfying the standard torsion constraints (the two torsion relations on the vielbein $e^m,e^\alpha$ quoted in the Background, together with the Bianchi identities including $dH=0$ for the closed three-form $H=dB$) — determines on N a family of quadrics Q, one quadric per superspace point, from which the superspace M is reconstructed up to a natural conformal metric; a general complex manifold N with such a quadric family reconstructs a superspace satisfying only the weaker first torsion relation (equation (60) of Witten 1986), the integrability condition for the superparticle's local fermionic symmetry. Determine what additional geometric data on N — the candidates proposed in the source are a closed three-form related to H and further compatibility conditions on the quadric family, in analogy with the self-dual four-dimensional construction — are necessary and sufficient for the reconstructed superspace to satisfy the full ten-dimensional supergravity torsion constraints, and prove the characterization in both directions: every supergravity solution induces data of the stated kind on its twistor space, and every N carrying such data satisfying the stated conditions arises from a supergravity solution, with the local or global scope of the claim stated explicitly. Alternatively, prove an obstruction: that no data of any natural class on such an N characterizes full ten-dimensional supergravity, in which case the twistor program fails in this formulation. Such an obstruction must cover at least the classes named in this statement — the closed three-form on N compatible with the quadric family proposed in the source, together with the further compatibility conditions named with it — and may not rest on eliminating a single candidate alone.

References

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