The Einstein-Weyl Equations, Scattering Maps, and Holomorphic Disks

Recent years have witnessed a resurgence of twistor methods in both mathematics and physics. On the physics side, this development has primarily been driven by Witten’s discovery [21] that strings in twistor space can be used to calculate Yang-Mills scattering amplitudes, resulting in concrete experimental predictions that can be tested using existing particle accelerators. Here, one of the main strands of thought [1] involves the introduction of open strings —– Riemann surfaces with non-empty boundary, where the boundary is constrained to lie on a specified submanifold, called a D-brane. On the mathematical side, there has been a parallel development. Penrose-type twistor correspondences study a component of moduli spaces of compact holomorphic curves C in a complex manifold Z; when Z is equipped with an anti-holomophic involution σ : Z → Z, the parameter space for those curves C ⊂ Z with σ(C) = C then often turns out to carry a natural geometrical structure which is the general solution of an interesting differential-geometric problem. By contrast, the new paradigm is instead to study the moduli spaces of holomorphic curves-with-boundary C, where ∂C 6= ∅ is constrained to lie in a totally real submanifold P ⊂ Z. It turns out that this framework is well adapted to the study of many natural problems in global differential geometry where solutions typically are of very low regularity. For example, Zoll metrics (and, more generally, Zoll projective structures) on surfaces turn out to arise [14] from holomorphic disks in CP2 with boundaries in a totally real RP →֒ CP2. Analogous results [15] describe split-signature self-dual conformal structures on 4-manifolds in terms of holomorphic disks in CP3 with boundary on a totally real RP →֒ CP3. For closely-related results on the Yang-Mills equations in split signature, see [16]. This article will show that these techniques also lead to definitive results concerning the Einstein-Weyl equations for a 3-dimensional Lorentzian space-time, thereby substantiating a claim made in [13]. Recall that the Einstein-Weyl equations, for a conformal class of metrics [g] and a compatible torson-free connection ▽, are precisely the requirement that the trace-free symmetric part of the Ricci tensor of ▽ should vanish; that is, the torsion-free connection ▽ is required to satisfy ▽g = α⊗ g for some 1-form α, and its curvature tensor R is required to satisfy