611 research outputs found
Deformation quantization of Poisson manifolds, I
I prove that every finite-dimensional Poisson manifold X admits a canonical
deformation quantization. Informally, it means that the set of equivalence
classes of associative algebras close to the algebra of functions on X is in
one-to-one correspondence with the set of equivalence classes of Poisson
structures on X modulo diffeomorphisms. In fact, a more general statement is
proven ("Formality conjecture"), relating the Lie superalgebra of polyvector
fields on X and the Hochschild complex of the algebra of functions on X.
Coefficients in explicit formulas for the deformed product can be interpreted
as correlators in a topological open string theory, although I do not use
explicitly the language of functional integrals. One of corollaries is a
justification of the orbit method in the representation theory.Comment: plain TeX and epsf.tex, 46 pages, 24 figure
Plato's cave and differential forms
In the 1970s and again in the 1990s, Gromov gave a number of theorems and
conjectures motivated by the notion that the real homotopy theory of compact
manifolds and simplicial complexes influences the geometry of maps between
them. The main technical result of this paper supports this intuition: we show
that maps of differential algebras are closely shadowed, in a technical sense,
by maps between the corresponding spaces. As a concrete application, we prove
the following conjecture of Gromov: if and are finite complexes with
simply connected, then there are constants and such that
any two homotopic -Lipschitz maps have a -Lipschitz homotopy (and
if one of the maps is a constant, can be taken to be .) We hope that it
will lead more generally to a better understanding of the space of maps from
to in this setting.Comment: 39 pages, 1 figure; comments welcome! This is the final version to be
published in Geometry & Topolog
F-theory, Geometric Engineering and N=1 Dualities
We consider geometric engineering of N=1 supersymmetric QFTs with matter in
terms of a local model for compactification of F-theory on Calabi-Yau fourfold.
By bringing 3-branes near 7-branes we engineer N=1 supersymmetric
gauge theory with flavors in the fundamental. We identify the Higgs
branch of this system with the instanton moduli space on the compact four
dimensional space of the 7-brane worldvolume. Moreover we show that the
Euclidean 3-branes wrapped around the compact part of the 7-brane worldvolume
can generate superpotential for as well as lead to quantum
corrections to the moduli space for . Finally we argue that Seiberg's
duality for N=1 supersymmetric QCD may be mapped to T-duality exchanging
7-branes with 3-branes.Comment: 15 page
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