arXiv subjects

Timothy Moy

Publications and source records attributed to Timothy Moy.

5 recordsLinked to original sources

arXiv2026

Conformally Einstein anti-self-dual spaces and their generalisations

We study the existence of Einstein scales and compatible metrics in Grassmannian geometry. As a special case, this includes the conformal-to-Einstein problem for anti-self-dual (ASD) conformal structures. In the ASD setting, given a genericity condition algebraic in the Weyl curvature, we obtain necessary and sufficient conditions for the existence of a local Einstein scale that refine those previously appearing in the literature. We also prove that in Riemannian signature, a non-Ricci-flat ASD-K\"ahler metric is locally conformally Einstein if and only if it has a holomorphic Killing vector field with anti-self-dual derivative proportional to the Ricci spinor. Next, we consider some generalisations of the ASD conformal-to-Einstein problem for $(2n,2)$-Grassmannian structures. When $n > 1$, given an analogous genericity condition, we obtain necessary and sufficient conditions for the existence of a local Einstein scale. Using tractor calculus, we obtain algebraic obstructions to the existence of compatible metrics and show that the unique submaximally symmetric model of the geometry has the submaximal number of linearly independent compatible metrics.

math.DG↗

arXiv2026

Heavenly equations in de Sitter space

We demonstrate that all anti-self-dual Einstein metrics with non--zero cosmological constant $\Lambda$ locally arise from solutions of a single second order PDE introduced by Lipstein and Nagy. We show how this equation fits into the heavenly formalism of Pleba\'nski, and establish a Lax pair. Finally we show how Pleba\'nski's second heavenly equation arises in the limit as $\Lambda\rightarrow 0$.

hep-th↗

arXiv2025

Joyce structures from quadratic differentials on the sphere

Motivated by known examples of Joyce structures on spaces of meromorphic quadratic differentials, we consider the isomonodromic deformations of particular second-order linear ODEs with rational potential. We show the infinitesimal isomonodromic deformations are the kernel of a closed $2$-form arising from the intersection pairing of an algebraic curve defined by the potential. This observation enables us to construct Joyce structures on a class of moduli spaces of meromorphic quadratic differentials on the Riemann sphere, and provides a new, geometric description of the hyper-K\"ahler structures of previously computed examples. We focus on the case of moduli of quadratic differentials with poles of odd orders, where we obtain a complex hyper-K\"ahler metric with homothetic symmetry. We also include an example corresponding to the moduli space of quadratic differentials with four simple poles, which is a version of the classical isomonodromy problem that leads to the Painlev\'{e} VI equation.

math.DG↗

arXiv2024

Heavenly metrics, hyper-Lagrangians and Joyce structures

In \cite{B3}, Bridgeland defined a geometric structure, named a Joyce structure, conjectured to exist on the space $M$ of stability conditions of a $CY_3$ triangulated category. Given a non-degeneracy assumption, a feature of this structure is a complex hyper-K\"ahler metric with homothetic symmetry on the total space $X = TM$ of the holomorphic tangent bundle. \par Generalising the isomonodromy calculation which leads to the $A_2$ Joyce structure in \cite{BM}, we obtain an explicit expression for a hyper-K\"ahler metric with homothetic symmetry via construction of the isomonodromic flows of a Schr\"odinger equation with deformed polynomial oscillator potential of odd degree $2n+1$. The metric is defined on a total space $X$ of complex dimension $4n$ and fibres over a $2n$--dimensional manifold $M$ which can be identified with the unfolding of the $A_{2n}$-singularity. The hyper-K\"ahler structure is shown to be compatible with the natural symplectic structure on $M$ in the sense of admitting an \textit{affine symplectic fibration} as defined in \cite{BS}. \par Separately, using the additional conditions imposed by a Joyce structure, we consider reductions of Pleba\'nski's heavenly equations that govern the hyper-K\"ahler condition. We introduce the notion of a \textit{projectable hyper-Lagrangian} foliation and show that in dimension four such a foliation of $X$ leads to a linearisation of the heavenly equation. The hyper-K\"ahler metrics constructed here are shown to admit such a foliation.

math.DG↗

arXiv2022

Spinors in Five-Dimensional Contact Geometry

We use classical (Penrose) two-component spinors to set up the differential geometry of two parabolic contact structures in five dimensions, namely $G_2$ contact geometry and Legendrean contact geometry. The key players in these two geometries are invariantly defined directional derivatives defined only in the contact directions. We explain how to define them and their usage in constructing basic invariants such as the harmonic curvature, the obstruction to being locally flat from the parabolic viewpoint. As an application, we calculate the invariant torsion of the $G_2$ contact structure on the configuration space of a flying saucer (always a five-dimensional contact manifold).

math.DG↗