We describe local mirror symmetry from a mathematical point of view and make several A-model calculations using the e-print archive: http://xxx.lanl.gov/hep-th/9903053 496 T.-M. CHIANG, A. KLEMM, S.-T. YAU, AND E. ZASLOW mirror principle (localization). Our results agree with B-model computations from solutions of Picard-Fuchs differential equations constructed form the local geometry near a Fano surface within a Calabi-Yau manifold. We interpret the GromovWitten-type numbers from an enumerative point of view. We also describe the geometry of singular surfaces and show how the local invariants of singular surfaces agree with the smooth cases when they occur as complete intersections.

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We present an explicit method for translating between the linear sigma model and the spectral cover description of SU(r) stable bundles over an elliptically fibered Calabi-Yau manifold. We use this to investigate the 4-dimensional duality between (0,2) heterotic and F-theory compactifications. We indirectly find that much interesting heterotic information must be contained in the `spectral bundle' and in its dual description as a gauge theory on multiple F-theory 7-branes. A by-product of these efforts is a method for analyzing semistability and the splitting type of vector bundles over an elliptic curve given as the sheaf cohomology of a monad.Comment: 40 pages, no figures; minor cosmetic reorganization of section 4; reference [6] update

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