arXiv subjects

James Gundry

Publications and source records attributed to James Gundry.

5 recordsLinked to original sources

arXiv2018

Jumps, folds, and singularities of Kodaira moduli spaces

For any integer $k$ we construct an explicit example of a twistor space which contains a one--parameter family of jumping rational curves, where the normal bundle changes from $O(1)+O(1)$ to $O(k)+O(2-k)$. For $k>3$ the resulting anti--self--dual Ricci-flat manifold is a Zariski cone in the space of holomorphic sections of $O(k)$. In the case $k=2$ we recover the canonical example of Hitchin's folded hyper-Kahler manifold, where the jumping lines form a three--parameter family. We show that in this case there exist normalisable solutions to the Schrodinger equation which extend through the fold.

math.DG↗

arXiv2017

Kodaira Families and Newton-Cartan Structures with Torsion

We describe the induced geometry on several classes of Kodaira moduli spaces of rational curves in twistor spaces. By constructing connections and frames on the moduli spaces we build and review twistor theories pertaining to relativistic and non-relativistic geometries. Focussing on the cases of three- and five-dimensional moduli spaces we establish novel twistor theories of Newton-Cartan spacetimes. We generalise a canonical class of connections on Kodaira moduli spaces to encompass torsion and prove that in three dimensions deformations of the twistor space's holomorphic structure induce Newton-Cartan structures with torsion. In five dimensions the canonical connections contain generalised Coriolis forces, and here also we consider the introduction of torsion by deformation. Non-relativistic limits are exhibited as jumping phenomena of normal bundles. We consider jumping phenomena in Newtonian twistor theory, as well as exhibiting via twistor theory that three-dimensional torsional Newton-Cartan geometry generically exists on the jumping hypersurfaces of Gibbons-Hawking manifolds.

math.DG↗

arXiv2016

Higher Symmetries of the Schrödinger Operator in Newton-Cartan Geometry

We establish several relationships between the non-relativistic conformal symmetries of Newton-Cartan geometry and the Schrödinger equation. In particular we discuss the algebra $\mathfrak{sch}(d)$ of vector fields conformally-preserving a flat Newton-Cartan spacetime, and we prove that its curved generalisation generates the symmetry group of the covariant Schrödinger equation coupled to a Newtonian potential and generalised Coriolis force. We provide intrinsic Newton-Cartan definitions of Killing tensors and conformal Schrödinger-Killing tensors, and we discuss their respective links to conserved quantities and to the higher symmetries of the Schrödinger equation. Finally we consider the role of conformal symmetries in Newtonian twistor theory, where the infinite-dimensional algebra of holomorphic vector fields on twistor space corresponds to the symmetry algebra $\mathfrak{cnc}(3)$ on the Newton-Cartan spacetime.

math-ph↗

arXiv2015

Non-relativistic twistor theory and Newton--Cartan geometry

We develop a non-relativistic twistor theory, in which Newton--Cartan structures of Newtonian gravity correspond to complex three-manifolds with a four-parameter family of rational curves with normal bundle ${\mathcal O}\oplus{\mathcal O}(2)$. We show that the Newton--Cartan space-times are unstable under the general Kodaira deformation of the twistor complex structure. The Newton--Cartan connections can nevertheless be reconstructed from Merkulov's generalisation of the Kodaira map augmented by a choice of a holomorphic line bundle over the twistor space trivial on twistor lines. The Coriolis force may be incorporated by holomorphic vector bundles, which in general are non--trivial on twistor lines. The resulting geometries agree with non--relativistic limits of anti-self-dual gravitational instantons.

hep-th↗

arXiv2014

Gauge Theories and Dessins d'Enfants: Beyond the Torus

Dessin d'Enfants on elliptic curves are a powerful way of encoding doubly-periodic brane tilings, and thus, of four-dimensional supersymmetric gauge theories whose vacuum moduli space is toric, providing an interesting interplay between physics, geometry, combinatorics and number theory. We discuss and provide a partial classification of the situation in genera other than one by computing explicit Belyi pairs associated to the gauge theories. Important also is the role of the Igusa and Shioda invariants that generalise the elliptic $j$-invariant.

hep-th↗