9 research outputs found
Fourier Mukai Transforms and Applications to String Theory
We give an introductory review of Fourier-Mukai transforms and their
application to various aspects of moduli problems, string theory and mirror
symmetry. We develop the necessary mathematical background for Fourier-Mukai
transforms such as aspects of derived categories and integral functors as well
as their relative version which becomes important for making precise the notion
of fiberwise T-duality on elliptic Calabi-Yau threefolds. We discuss various
applications of the Fourier-Mukai transform to D-branes on Calabi-Yau manifolds
as well as homological mirror symmetry and the construction of vector bundles
for heterotic string theory.Comment: 52 pages. To appear in Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A
Mat. Minor changes, reference of conjecture in section 7.5 changed,
references update
Notes on fundamental algebraic supergeometry. Hilbert and Picard superschemes
These notes aim at providing a complete and systematic account of some
foundational aspects of algebraic supergeometry, namely, the extension to the
geometry of superschemes of many classical notions, techniques and results that
make up the general backbone of algebraic geometry, most of them originating
from Grothendieck's work. In particular, we extend to algebraic supergeometry
such notions as projective and proper morphisms, finiteness of the cohomology,
vector and projective bundles, cohomology base change, semicontinuity theorems,
relative duality, Castelnuovo-Mumford regularity, flattening, Hilbert and Quot
schemes, faithfully flat descent, quotient \'etale relations (notably, Picard
schemes), among others. Some results may be found elsewhere, and, in
particular, there is some overlap with a recent preprint by Moosavian and Zhou.
However, many techniques and constructions are presented here for the first
time, notably, a first development of Grothendieck relative duality for proper
morphisms of superschemes, the construction of the Hilbert superscheme in a
more general situation than the one already known (which in particular allows
one to treat the case of sub-superschemes of supergrassmannians), and a
rigorous construction of the Picard superscheme for a locally superprojective
morphism of noetherian superschemes with geometrically integral fibres.
Moreover, some of the proofs given here are new as well, even when restricted
to ordinary schemes. In a final section we construct a period map from an open
substack of the moduli of proper and smooth supercurves to the moduli stack of
principally polarized abelian superchemes.Comment: 86 pages. v2: 88 pages. Added a comparison of the superperiod map
with the D'Hoker-Phong superperiod matrix. Minor correction
Mirror symmetry on K3 surfaces via Fourier-Mukai transform
We use a relative Fourier-Mukai transform on elliptic K3 surfaces to
describe mirror symmetry. The action of this Fourier-Mukai transform on the
cohomology ring of reproduces relative T-duality and provides an
infinitesimal isometry of the moduli space of algebraic structures on
which, in view of the triviality of the quantum cohomology of K3 surfaces, can
be interpreted as mirror symmetry.Comment: 15 pages, AMS-LaTeX v1.2. Final version to appear in Commun. Math.
Phy