92 research outputs found
Abelian Yang-Mills theory on Real tori and Theta divisors of Klein surfaces
The purpose of this paper is to compute determinant index bundles of certain
families of Real Dirac type operators on Klein surfaces as elements in the
corresponding Grothendieck group of Real line bundles in the sense of Atiyah.
On a Klein surface these determinant index bundles have a natural holomorphic
description as theta line bundles. In particular we compute the first
Stiefel-Whitney classes of the corresponding fixed point bundles on the real
part of the Picard torus. The computation of these classes is important,
because they control to a large extent the orientability of certain moduli
spaces in Real gauge theory and Real algebraic geometry.Comment: LaTeX, 44 pages, to appear in Comm. Math. Phy
Instanton bundles on Fano threefolds
We introduce the notion of an instanton bundle on a Fano threefold of index
2. For such bundles we give an analogue of a monadic description and discuss
the curve of jumping lines. The cases of threefolds of degree 5 and 4 are
considered in a greater detail.Comment: 31 page, to appear in CEJ
On complex surfaces diffeomorphic to rational surfaces
In this paper we prove that no complex surface of general type is
diffeomorphic to a rational surface, thereby completing the smooth
classification of rational surfaces and the proof of the Van de Ven conjecture
on the smooth invariance of Kodaira dimension.Comment: 34 pages, AMS-Te
Moduli of vortices and Grassmann manifolds
We use the framework of Quot schemes to give a novel description of the
moduli spaces of stable n-pairs, also interpreted as gauged vortices on a
closed Riemann surface with target Mat(r x n, C), where n >= r. We then show
that these moduli spaces embed canonically into certain Grassmann manifolds,
and thus obtain natural Kaehler metrics of Fubini-Study type; these spaces are
smooth at least in the local case r=n. For abelian local vortices we prove
that, if a certain "quantization" condition is satisfied, the embedding can be
chosen in such a way that the induced Fubini-Study structure realizes the
Kaehler class of the usual L^2 metric of gauged vortices.Comment: 22 pages, LaTeX. Final version: last section removed, typos
corrected, two references added; to appear in Commun. Math. Phy
Knot homology via derived categories of coherent sheaves II, sl(m) case
Using derived categories of equivariant coherent sheaves we construct a knot
homology theory which categorifies the quantum sl(m) knot polynomial. Our knot
homology naturally satisfies the categorified MOY relations and is
conjecturally isomorphic to Khovanov-Rozansky homology. Our construction is
motivated by the geometric Satake correspondence and is related to Manolescu's
by homological mirror symmetry.Comment: 51 pages, 9 figure
Cohomology of bundles on homological Hopf manifold
We discuss the properties of complex manifolds having rational homology of
including those constructed by Hopf, Kodaira and
Brieskorn-van de Ven. We extend certain previously known vanishing properties
of cohomology of bundles on such manifolds.As an application we consider
degeneration of Hodge-deRham spectral sequence in this non Kahler setting.Comment: To appear in Proceedings of 2007 conference on Several complex
variables and Complex Geometry. Xiamen. Chin
Hyperholomorpic connections on coherent sheaves and stability
Let be a hyperkaehler manifold, and a torsion-free and reflexive
coherent sheaf on . Assume that (outside of its singularities) admits a
connection with a curvature which is invariant under the standard SU(2)-action
on 2-forms. If the curvature is square-integrable, then is stable and its
singularities are hyperkaehler subvarieties in . Such sheaves (called
hyperholomorphic sheaves) are well understood. In the present paper, we study
sheaves admitting a connection with SU(2)-invariant curvature which is not
necessarily square-integrable. This situation arises often, for instance, when
one deals with higher direct images of holomorphic bundles. We show that such
sheaves are stable.Comment: 37 pages, version 11, reference updated, corrected many minor errors
and typos found by the refere
Chirality Change in String Theory
It is known that string theory compactifications leading to low energy
effective theories with different chiral matter content ({\it e.g.} different
numbers of standard model generations) are connected through phase transitions,
described by non-trivial quantum fixed point theories.
We point out that such compactifications are also connected on a purely
classical level, through transitions that can be described using standard
effective field theory. We illustrate this with examples, including some in
which the transition proceeds entirely through supersymmetric configurations.Comment: 50 pages, 2 figure
Heterotic Compactification, An Algorithmic Approach
We approach string phenomenology from the perspective of computational
algebraic geometry, by providing new and efficient techniques for proving
stability and calculating particle spectra in heterotic compactifications. This
is done in the context of complete intersection Calabi-Yau manifolds in a
single projective space where we classify positive monad bundles. Using a
combination of analytic methods and computer algebra we prove stability for all
such bundles and compute the complete particle spectrum, including gauge
singlets. In particular, we find that the number of anti-generations vanishes
for all our bundles and that the spectrum is manifestly moduli-dependent.Comment: 36 pages, Late
Moduli of ADHM Sheaves and Local Donaldson-Thomas Theory
The ADHM construction establishes a one-to-one correspondence between framed
torsion free sheaves on the projective plane and stable framed representations
of a quiver with relations in the category of complex vector spaces. This paper
studies the geometry of moduli spaces of representations of the same quiver
with relations in the abelian category of coherent sheaves on a smooth complex
projective curve . In particular it is proven that this moduli space is
virtually smooth and related byrelative Beilinson spectral sequence to the
curve counting construction via stable pairs of Pandharipande and Thomas. This
yields a new conjectural construction for the local Donaldson-Thomas theory of
curves as well as a natural higher rank generalization.Comment: 61 pages AMS Latex; v2: minor corrections, reference added; v3: some
proofs corrected using the GIT construction of the moduli space due to A.
Schmitt; main results unchanged; final version to appear in J. Geom. Phy
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