Related papers: Observations on the Darboux coordinates for rigid …

In previous work, we have considered Hamiltonians associated with 3 dimensional conformally flat spaces, possessing 2, 3 and 4 dimensional isometry algebras. Previously our Hamiltonians have represented free motion, but here we consider the…

Motivated by understanding the limiting case of a certain systolic inequality we study compact Riemannian manifolds having all harmonic 1-forms of constant length. We give complete characterizations as far as K\"ahler and hyperbolic…

For every $2n\times 2n$ real positive definite matrix $A,$ there exists a real symplectic matrix $M$ such that $M^TAM=\diag(D,D),$ where $D$ is the $n\times n$ positive diagonal matrix with diagonal entries $d_1(A)\le \cdots\le d_n(A).$ The…

We reconsider the (rational) Calogero-Moser system from the point of view of bi-Hamiltonian geometry. By using geometrical tools of the latter, we explicitly construct set(s) of spectral canonical coordinates, that is, complete sets of…

The moduli space of SL(2) flat connections on a punctured Riemann surface with the fixed conjugacy classes of the monodromies around the punctures is endowed with a system of holomorphic Darboux coordinates, in which the generating function…

Given a compact symplectic toric manifold $(M,\omega, \mathbb{T})$, we identify a class $DGK_{\omega}^{\mathbb{T}}(M)$ of $\mathbb{T}$-invariant generalized K\"ahler structures for which a generalisation the Abreu-Guillemin theory of toric…

We study the moduli space of flat maximal space-like embeddings in $\mathbb{H}^{2,2}$ from various aspects. We first describe the associated Codazzi tensors to the embedding in the general setting, and then, we introduce a family of…

Harmonic maps from S^2 to S^2 are all weakly conformal, and so are represented by rational maps. This paper presents a study of the L^2 metric gamma on M_n, the space of degree n harmonic maps S^2 -> S^2, or equivalently, the space of…

In this paper we use a diffeo-geometric framework based on manifolds that are locally modeled on "convenient" vector spaces to study the geometry of some infinite dimensional spaces. Given a finite dimensional symplectic manifold…

We develop a real-analytic framework, called perplex analysis, in which the complex, split-complex, and dual numbers arise as members of a single four-parameter family of two-dimensional commutative real algebras. Within this unified…

We introduce a natural symplectic structure on the moduli space of quadratic differentials with simple zeros and describe its Darboux coordinate systems in terms of so-called homological coordinates. We then show that this structure…

Let $(M,h)$ be a Hermitian manifold and $\psi$ a smooth weight function on $M$. The $\partial$-complex on weighted Bergman spaces $A^2_{(p,0)}(M,h, e^{-\psi})$ of holomorphic $(p,0)$-forms was recently studied in [[10] and [9]. It was shown…

The aim of these lecture notes is to give a pedagogical introduction to the subject of non-holomorphic deformations of special geometry. This subject was first introduced in the context of N=2 BPS black holes, but has a wider range of…

We construct a local characteristic map to a symplectic manifold M via certain cohomology groups of Hamiltonian vector fields. For each p in M, the Leibniz cohomology of the Hamiltonian vector fields on R^{2n} maps to the Leibniz cohomology…

We investigate a new 8-dimensional Riemannian geometry defined by a generic closed and coclosed 3-form with stabiliser PSU(3), and which arises as a critical point of Hitchin's variational principle. We give a Riemannian characterisation of…

The geometry that is defined by the scalars in couplings of Einstein-Maxwell theories in N=2 supergravity in 4 dimensions is denoted as special Kaehler geometry. There are several equivalent definitions, the most elegant ones involve the…

In these lectures I review the general structure of electric--magnetic duality rotations in every even space--time dimension. In four dimensions, which is my main concern, I discuss the general issue of symplectic covariance and how it…

For a finite dimensional symplectic manifold $(M,\omega)$ with a symplectic form $\omega$, corresponding loop space ($LM=C^\infty(S^1,M)$) admits a weak symplectic form $\Omega^\omega$. We prove that the loop space over $\mbr^n$ admits…

We apply the techniques of special Kaehler geometry to investigate AdS_4 vacua of general N=2 gauged supergravities underlying flux compactifications of type II theories. We formulate the scalar potential and its extremization conditions in…

In this article we study multisymplectic geometry, i.e., the geometry of manifolds with a non-degenerate, closed differential form. First we describe the transition from Lagrangian to Hamiltonian classical field theories, and then we…