arXiv research

We define $(p,q)$ hermitian geometry as the target space geometry of the two dimensional $(p,q)$ supersymmetric sigma model. This includes generalised Kähler geometry for $(2,2)$, generalised hyperkähler geometry for $(4,2)$, strong Kähler with torsion geometry for $(2,1)$ and strong hyperkähler with torsion geometry f…

Maps strong Kähler with torsion to Generalised Complex Geometry.

Generalised contact structures are studied from the point of view of reduced generalised complex structures, naturally incorporating non-coorientable structures as non-trivial fibering. The infinitesimal symmetries are described in detail, with a geometric description given in terms of gerbes. As an application of the …

Complex geometry and symplectic geometry are mirrors in string theory. The recently developed generalised complex geometry interpolates between the two of them. On the other hand, the classical and quantum mechanics of a finite number of degrees of freedom are respectively described by a symplectic structure and a comp…

The paper applies generalised geometry to semi-Riemannian immersions and hypersurfaces.

The paper explores linear generalised complex structures over vector bundles.

Researchers create a Kähler structure on complex projective plane using elliptic functions.

Constructs brane current algebras from QP-manifolds, generalizing string currents.

In this paper we define the analogue of Calabi--Yau geometry for generic $D=4$, $\mathcal{N}=2$ flux backgrounds in type II supergravity and M-theory. We show that solutions of the Killing spinor equations are in one-to-one correspondence with integrable, globally defined structures in $E_{7(7)}\times\mathbb{R}^+$ gene…

A family of new algebraic Poisson varieties will be constructed, generalising the complex character varieties of Riemann surfaces. Then the well-known (Poisson) mapping class group actions on the character varieties will be generalised.

The paper generalizes Cartan Geometry using Polacek and Siegel's approach.

Study of string theory flux compactifications using generalized geometry.

Y-algebroids generalize symmetry structures for string/M-theory compactifications.

We construct a Kahler structure (which we call a generalised Kahler cone) on an open subset of the cone of a strongly pseudo-convex CR manifold endowed with a 1-parameter family of compatible Sasaki structures. We determine those generalised Kahler cones which are Bochner-flat and we study their local geometry. We prov…

Study topological G₂ and Spin(7) strings at 1-loop using double complexes.

This paper attempts to define a generalisation of the standard Einstein condition (in conformal/metric geometry) to any parabolic geometry. To do so, it shows that any preserved involution $σ$ of the adjoint bundle $\mc{A}$ gives rise, given certain algebraic conditions, to a unique preferred affine connection $\nabla$…

Heterotic backgrounds described using generalised geometry, preserving minimal supersymmetry.

Unified framework for exceptional and generalised geometry, and Poisson-Lie duality.

Extends nonlinear theory of distributional geometry.

We examine how generalised geometries can be associated with a labelled Dynkin diagram built around a gravity line. We present a series of new generalised geometries based on the groups $\mathit{Spin}(d,d)\times\mathbb{R}^+$ for which the generalised tangent space transforms in a spinor representation of the group. In …

The correspondence between 2-parameter families of oriented lines in ${\Bbb{R}}^3$ and surfaces in $T{\Bbb{P}}^1$ is studied, and the geometric properties of the lines are related to the complex geometry of the surface. Congruences generated by global sections of $T{\Bbb{P}}^1$ are investigated and a number of theorems…

Develops a new global theory of generalised functions on manifolds.

Study of real and quaternionic Lie algebroid connections on manifolds.

We generalise the notion of contact manifold by allowing the contact distribution to have codimension two. There are special features in dimension six. In particular, we show that the complex structure on a three-dimensional complex contact manifold is determined solely by the underlying contact distribution.

In the context of generalised geometry we investigate reductions to $SU(m)\times SU(m)$ together with an integrability condition which in dimension 6 describes the geometry of type II supergravity compactifications.

Study new ECS structures in 5D Minkowski compactifications of M-theory.

Unified description of string and brane worldvolumes using auto-parallel vector fields.

We present a general formalism for incorporating the string corrections in generalised geometry, which necessitates the extension of the generalised tangent bundle. Not only are such extensions obstructed, string symmetries and the existence of a well-defined effective action require a precise choice of the (generalise…

Motivated by strong desire to understand the natural geometry of moduli spaces of hyperbolic monopoles, we introduce and study a new type of geometry: pluricomplex geometry. It is a generalisation of hypercomplex geometry: we still have a 2-sphere of complex structures, but they no longer behave like unit imaginary qua…

We reinterpret the generalised Lie derivative of M-theory $E_6$ generalised geometry as hamiltonian flow on a graded symplectic supermanifold. The hamiltonian acts as the nilpotent derivative of the tensor hierarchy of exceptional field theory. This construction is an M-theory analogue of the Courant algebroid and reve…

Survey and generalization of implosion and contraction in symplectic and hyperkähler geometry.

A twist construction for manifolds with torus action is described generalising certain T-duality examples and constructions in hypercomplex geometry. It is applied to complex, SKT, hypercomplex and HKT manifolds to construct compact simply-connected examples. In particular, we find hypercomplex manifolds that admit no …

The aim of this paper is to describe the geometry of conformal structures in Lorentzian signature, which admit a lightlike conformal Killing vector field whose corresponding adjoint tractor acts as complex structure on the standard tractor bundle of conformal geometry. Key to the treatment of this problem is CR-geometr…

Extended supersymmetry leads to Yano F structures on a manifold.

Constructs a unique Levi-Civita connection for generalised metrics.

The paper develops harmonic theory on vector bundles with singular metrics and extends results from complex geometry.

CDC-FM improves generative model quality-generalization tradeoff by regularizing with geometry-aware noise.

We introduce the notion of a special complex manifold: a complex manifold (M,J) with a flat torsionfree connection \nabla such that (\nabla J) is symmetric. A special symplectic manifold is then defined as a special complex manifold together with a \nabla-parallel symplectic form ω. This generalises Freed's definition …

We consider links that are alternating on surfaces embedded in a compact 3-manifold. We show that under mild restrictions, the complement of the link decomposes into simpler pieces, generalising the polyhedral decomposition of alternating links of Menasco. We use this to prove various facts about the hyperbolic geometr…

We study cubical sets without degeneracies, which we call square sets. These sets arise naturally in a number of settings and they have a beautiful intrinsic geometry; in particular a square set C has an infinite family of associated square sets J^i(C), i=1,2,..., which we call James complexes. There are mock bundle pr…

In recent work N. Hitchin introduced the concept of "generalised geometry". The key feature of generalised structures is that that they can be acted on by both diffeomorphisms and 2-forms, the so-called $B$-fields. In this lecture, we give a basic introduction and explain some of the fundamental ideas. Further, we disc…

Researchers redefine spinor field derivatives in generalized geometry.

In this article, we study a generalisation of the Seiberg-Witten equations, replacing the spinor representation with a hyperKahler manifold equipped with certain symmetries. Central to this is the construction of a (non-linear) Dirac operator acting on the sections of the non-linear fibre-bundle. For hyperKahler manifo…

We show that generalised geometry gives a unified description of maximally supersymmetric consistent truncations of ten- and eleven-dimensional supergravity. In all cases the reduction manifold admits a "generalised parallelisation" with a frame algebra with constant coefficients. The consistent truncation then arises …

We discuss the structure of "exceptional generalised geometry" (EGG), an extension of Hitchin's generalised geometry that provides a unified geometrical description of backgrounds in eleven-dimensional supergravity. On a d-dimensional background, as first described by Hull, the action of the generalised geometrical O(d…

We define intrinsic torsion in generalised geometry and use it to introduce a new notion of generalised special holonomy. We then consider generic warped supersymmetric flux compactifications of M theory and Type II of the form $\mathbb{R}^{D-1,1}\times M$. Using the language of $E_{d(d)}\times\mathbb{R}^+$ generalised…

Geometry-aware noise improves model generalization on complex manifolds.

In this paper, we study the geometry of surfaces with the generalised simple lift property. This work generalises previous results by Bernstein and Tinaglia, and it is motivated by the fact that leaves of a minimal lamination obtained as a limit of a sequence of properly embedded minimal disks satisfy the generalised s…