4,879 research outputs found
Veech surfaces and simple closed curves
We study the SL(2,R)-infimal lengths of simple closed curves on
half-translation surfaces. Our main result is a characterization of Veech
surfaces in terms of these lengths. We also revisit the "no small virtual
triangles" theorem of Smillie and Weiss and establish the following dichotomy:
the virtual triangle area spectrum of a half-translation surface either has a
gap above zero or is dense in a neighborhood of zero. These results make use of
the auxiliary polygon associated to a curve on a half-translation surface, as
introduced by Tang and Webb.Comment: 12 pages. v2: added proof of continuity of infimal length functions
on quadratic differential space; 16 pages, one figure; to appear in Israel J.
Mat
Local mirror symmetry of curves: Yukawa couplings and genus 1
We continue our study of equivariant local mirror symmetry of curves, i.e.
mirror symmetry for X_k=O(k)+O(-2-k) over P^1 with torus action
(lambda_1,lambda_2) on the bundle. For the antidiagonal action
lambda_1=-lambda_2, we find closed formulas for the mirror map and a rational B
model Yukawa coupling for all k. Moreover, we give a simple closed form for the
B model genus 1 Gromov-Witten potential. For the diagonal action
lambda_1=lambda_2, we argue that the mirror symmetry computation is equivalent
to that of the projective bundle P(O+O(k)+O(-2-k)) over P^1. Finally, we
outline the computation of equivariant Gromov-Witten invariants for A_n
singularities and toric tree examples via mirror symmetry.Comment: 20 pages, no figures; v2: added details on connection to
hep-th/0606120; v3: corrected triple intersection number, which gives sleek
formula for Yukawa coupling
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