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Calabi yau manifold5/17/2023 ![]() Symplectic manifolds are smooth even dimensional manifolds admitting a symplectic struc- ture: a non-degenerate closed 2-form. This essay presents some aspects of the Gromov-Witten theory from the point of view of symplectic topology. Mirror pairs are also more general than exactly soluble models that have hitherto been discussed since we solve the theory for all points of the moduli space. The burden of the present work is that a mirror pair of Calabi-Yau manifolds is an exactly soluble superconformal theory, at least as far as the massless sector is concerned. This sum over instantons points to a deep connection between the modular group and the rational curves of the Calabi-Yau manifold. In particular there are no “sub-instanton” corrections. ![]() It is also shown that the quantum corrections to the coupling decompose into a sum over instanton contributions and moreover that this sum converges. ![]() The moduli space is found to be subject to the action of a modular group which, among other operations, exchanges large and small values of the radius, though the action on the radius is not as simple as. In this way we obtain all the sigma model corrections to the Yukawa couplings and moduli space metric for the original manifold. We compute the prepotentials and the geometry of the moduli spaces for a Calabi-Yau manifold and its mirror. ![]()
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