12,140 research outputs found
On the Convergence of Gromov-Witten Potentials and Givental's Formula
Let X be a smooth projective variety. The Gromov-Witten potentials of X are
generating functions for the Gromov-Witten invariants of X: they are formal
power series, sometimes in infinitely many variables, with Taylor coefficients
given by Gromov-Witten invariants of X. It is natural to ask whether these
formal power series converge. In this paper we describe and analyze various
notions of convergence for Gromov-Witten potentials. Using results of Givental
and Teleman, we show that if the quantum cohomology of X is analytic and
generically semisimple then the genus-g Gromov-Witten potential of X converges
for all g. We deduce convergence results for the all-genus Gromov-Witten
potentials of compact toric varieties, complete flag varieties, and certain
non-compact toric varieties.Comment: 38 pages, 1 figure, v2: corrected several error
Minimum Distances in Non-Trivial String Target Spaces
The idea of minimum distance, familiar from R 1/R duality when the string
target space is a circle, is analyzed for less trivial geometries. The
particular geometry studied is that of a blown-up quotient singularity within a
Calabi-Yau space and mirror symmetry is used to perform the analysis. It is
found that zero distances can appear but that in many cases this requires other
distances within the same target space to be infinite. In other cases zero
distances can occur without compensating infinite distances.Comment: 21 pages, IASSNS-HEP-94/1
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