244,547 research outputs found
Instantons, Monopoles and Toric HyperKaehler Manifolds
In this paper, the metric on the moduli space of the k=1 SU(n) periodic
instanton -or caloron- with arbitrary gauge holonomy at spatial infinity is
explicitly constructed. The metric is toric hyperKaehler and of the form
conjectured by Lee and Yi. The torus coordinates describe the residual
U(1)^{n-1} gauge invariance and the temporal position of the caloron and can
also be viewed as the phases of n monopoles that constitute the caloron. The
(1,1,..,1) monopole is obtained as a limit of the caloron. The calculation is
performed on the space of Nahm data, which is justified by proving the
isometric property of the Nahm construction for the cases considered. An
alternative construction using the hyperKaehler quotient is also presented. The
effect of massless monopoles is briefly discussed.Comment: 30 pages, latex2
On deformation of Poisson manifolds of hydrodynamic type
We study a class of deformations of infinite-dimensional Poisson manifolds of
hydrodynamic type which are of interest in the theory of Frobenius manifolds.
We prove two results. First, we show that the second cohomology group of these
manifolds, in the Poisson-Lichnerowicz cohomology, is ``essentially'' trivial.
Then, we prove a conjecture of B. Dubrovin about the triviality of homogeneous
formal deformations of the above manifolds.Comment: LaTeX file, 24 page
The map between conformal hypercomplex/hyper-Kaehler and quaternionic(-Kaehler) geometry
We review the general properties of target spaces of hypermultiplets, which
are quaternionic-like manifolds, and discuss the relations between these
manifolds and their symmetry generators. We explicitly construct a one-to-one
map between conformal hypercomplex manifolds (i.e. those that have a closed
homothetic Killing vector) and quaternionic manifolds of one quaternionic
dimension less. An important role is played by '\xi-transformations', relating
complex structures on conformal hypercomplex manifolds and connections on
quaternionic manifolds. In this map, the subclass of conformal hyper-Kaehler
manifolds is mapped to quaternionic-Kaehler manifolds. We relate the curvatures
of the corresponding manifolds and furthermore map the symmetries of these
manifolds to each other.Comment: 54 pages, 2 figures; v2: small corrections, version to be published
in CMP; v3: changes of statement on (3.5
Instantons and Killing spinors
We investigate instantons on manifolds with Killing spinors and their cones.
Examples of manifolds with Killing spinors include nearly Kaehler 6-manifolds,
nearly parallel G_2-manifolds in dimension 7, Sasaki-Einstein manifolds, and
3-Sasakian manifolds. We construct a connection on the tangent bundle over
these manifolds which solves the instanton equation, and also show that the
instanton equation implies the Yang-Mills equation, despite the presence of
torsion. We then construct instantons on the cones over these manifolds, and
lift them to solutions of heterotic supergravity. Amongst our solutions are new
instantons on even-dimensional Euclidean spaces, as well as the well-known
BPST, quaternionic and octonionic instantons.Comment: 40 pages, 2 figures v2: author email addresses and affiliations adde
A generalization of manifolds with corners
In conventional Differential Geometry one studies manifolds, locally modelled
on , manifolds with boundary, locally modelled on
, and manifolds with corners, locally
modelled on . They form categories . Manifolds with corners have
boundaries , also manifolds with corners, with .
We introduce a new notion of 'manifolds with generalized corners', or
'manifolds with g-corners', extending manifolds with corners, which form a
category with . Manifolds with g-corners are locally modelled on
for a weakly toric monoid,
where for .
Most differential geometry of manifolds with corners extends nicely to
manifolds with g-corners, including well-behaved boundaries . In
some ways manifolds with g-corners have better properties than manifolds with
corners; in particular, transverse fibre products in exist under
much weaker conditions than in .
This paper was motivated by future applications in symplectic geometry, in
which some moduli spaces of -holomorphic curves can be manifolds or
Kuranishi spaces with g-corners (see the author arXiv:1409.6908) rather than
ordinary corners.
Our manifolds with g-corners are related to the 'interior binomial varieties'
of Kottke and Melrose in arXiv:1107.3320 (see also Kottke arXiv:1509.03874),
and to the 'positive log differentiable spaces' of Gillam and Molcho in
arXiv:1507.06752.Comment: 97 pages, LaTeX. (v3) final version, to appear in Advances in
Mathematic
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