2,417 research outputs found
Families of short cycles on Riemannian surfaces
Let be a closed Riemannian surface of genus . We construct a family of
1-cycles on that represents a non-trivial element of the k'th homology
group of the space of cycles and such that the mass of each cycle is bounded
above by . This result is optimal
up to a multiplicative constant.Comment: 16 pages, 3 figures. Exposition improved, to appear in Duke
Mathematical Journa
Pair Correlation Function of Wilson Loops
We give a path integral prescription for the pair correlation function of
Wilson loops lying in the worldvolume of Dbranes in the bosonic open and closed
string theory. The results can be applied both in ordinary flat spacetime in
the critical dimension d or in the presence of a generic background for the
Liouville field. We compute the potential between heavy nonrelativistic sources
in an abelian gauge theory in relative collinear motion with velocity v =
tanh(u), probing length scales down to r_min^2 = 2 \pi \alpha' u. We predict a
universal -(d-2)/r static interaction at short distances. We show that the
velocity dependent corrections to the short distance potential in the bosonic
string take the form of an infinite power series in the dimensionless variables
z = r_min^2/r^2, uz/\pi, and u^2.Comment: 16 pages, 1 figure, Revtex. Corrected factor of two in potential.
Some changes in discussio
Amoebas of algebraic varieties and tropical geometry
This survey consists of two parts. Part 1 is devoted to amoebas. These are
images of algebraic subvarieties in the complex torus under the logarithmic
moment map. The amoebas have essentially piecewise-linear shape if viewed at
large. Furthermore, they degenerate to certain piecewise-linear objects called
tropical varieties whose behavior is governed by algebraic geometry over the
so-called tropical semifield. Geometric aspects of tropical algebraic geometry
are the content of Part 2. We pay special attention to tropical curves. Both
parts also include relevant applications of the theories. Part 1 of this survey
is a revised and updated version of an earlier prepreint of 2001.Comment: 40 pages, 15 figures, a survey for the volume "Different faces in
Geometry
Extensions of Johnson's and Morita's homomorphisms that map to finitely generated abelian groups
We extend each higher Johnson homomorphism to a crossed homomorphism from the
automorphism group of a finite-rank free group to a finite-rank abelian group.
We also extend each Morita homomorphism to a crossed homomorphism from the
mapping class group of once-bounded surface to a finite-rank abelian group.
This improves on the author's previous results [Algebr. Geom. Topol. 7
(2007):1297-1326]. To prove the first result, we express the higher Johnson
homomorphisms as coboundary maps in group cohomology. Our methods involve the
topology of nilpotent homogeneous spaces and lattices in nilpotent Lie
algebras. In particular, we develop a notion of the "polynomial straightening"
of a singular homology chain in a nilpotent homogeneous space.Comment: 34 page
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