11,889 research outputs found
On the spectrum of the twisted Dolbeault Laplacian over K\"ahler manifolds
We use Dirac operator techniques to a establish sharp lower bound for the
first eigenvalue of the Dolbeault Laplacian twisted by Hermitian-Einstein
connections on vector bundles of negative degree over compact K\"ahler
manifolds.Comment: 14 pages. Completely revised: estimates corrected and shown to be
shar
Spectral curves and Nahm transform for doubly-periodic instantons
We explore the role played by the spectral curves associated with Higgs pairs
in the context of the Nahm transform of doubly-periodic instantons defined in
"Construction of doubly-periodic instantons" (math.DG/9909069) and "Nahm
transform for doubly-periodic instantons" (math.DG/9910120). More precisely, we
show how to construct a triple consisting of an algebraic curve plus a line
bundle with connection over it from a doubly-periodic instanton, and that these
coincide with the Hitchin's spectral data associated with the Nahm transformed
Higgs bundle.Comment: Completely reformulated, more differential-geometric approach. 12
page
Construction of doubly-periodic instantons
We construct finite energy instanton connection on which are periodic
in two directions via an analogue of the Nahm transform for certain singular
solutions of Hitchin's equations defined over a 2-torus.Comment: Final version to appear on CM
Trihyperkahler reduction and instanton bundles on CP^3
A trisymplectic structure on a complex 2n-manifold is a triple of holomorphic
symplectic forms such that any linear combination of these forms has constant
rank 2n, n or 0, and degenerate forms in belong to a non-degenerate
quadric hypersurface. We show that a trisymplectic manifold is equipped with a
holomorphic 3-web and the Chern connection of this 3-web is holomorphic,
torsion-free, and preserves the three symplectic forms. We construct a
trisymplectic structure on the moduli of regular rational curves in the twistor
space of a hyperkaehler manifold, and define a trisymplectic reduction of a
trisymplectic manifold, which is a complexified form of a hyperkaehler
reduction. We prove that the trisymplectic reduction in the space of regular
rational curves on the twistor space of a hyperkaehler manifold M is compatible
with the hyperkaehler reduction on M.
As an application of these geometric ideas, we consider the ADHM construction
of instantons and show that the moduli space of rank r, charge c framed
instanton bundles on CP^3 is a smooth, connected, trisymplectic manifold of
complex dimension 4rc. In particular, it follows that the moduli space of rank
2, charge c instanton bundles on CP^3 is a smooth complex manifold dimension
8c-3, thus settling a 30-year old conjecture.Comment: 42 pages, v. 3.2, changes in section 3.1: the notion of trisymplectic
structure stated differently, Clifford algebra action introduce
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