Abstract
We define the notion of mirror of a Calabi–Yau manifold with a stable bundle in the context of type II strings in terms of supersymmetric cycles on the mirror. This allows us to relate the variation of Hodge structure for cohomologies arising from the bundle to the counting of holomorphic maps of Riemann surfaces with boundary on the mirror side. Moreover it opens up the possibility of studying bundles on Calabi–Yau manifolds in terms of supersymmetric cycles on the mirror.
Highlights
Mirror symmetry, which was conjectured as a generalization of R → 1/R duality for Calabi-Yau compactifications [1][2] has played a major role in our understanding of dynamical issues in string theory
At the level of string perturbation theory, the concrete examples of mirror pairs [3] was shown to result in a deeper understanding of sigma model instantons [4]
Non-perturbatively mirror symmetry plays a key role in the geometric engineering approach to constructing quantum field theories
Summary
Mirror symmetry, which was conjectured as a generalization of R → 1/R duality for Calabi-Yau compactifications [1][2] has played a major role in our understanding of dynamical issues in string theory. The purpose of this note is to extend the mirror conjecture to situations including bundles over CalabiYau. The basic idea of this extended conjecture was motivated by the topological open and closed strings on Calabi-Yau threefolds [10][11] and was discussed in a preliminary form in [12]. Bundles over Calabi-Yau 3-folds have phenomenological applications for constructing N = 1, d = 4 vacua for heterotic string. In this context they lead to (2, 0) sigma model on the worldsheet and one could attempt to construct a mirror in this context. Some of the constructions are most natural for the Calabi-Yau threefold which will be the main focus of this paper, though generalizations are straight-forward for any n-fold Calabi-Yau and some of our general discussion is in that context
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