MIRROR SYMMETRY AND DERIVED CATEGORIES

Mirror symmetry is a correspondence between symplectic and complex geometry developed to understand formulae for the numbers of rational curves on some 3-dimensional Calabi-Yau manifolds, discovered by physi-cists. These formulas involve Hodge theory on mirror dual Calabi-Yau manifolds. On the other hand, the problem of counting rational curves in Calabi Yau manifolds has been embedded in the general context of the theory of Gromov-Witten invariants. A physicist’s Calabi-Yau is a compact complex manifold with a trivial canonical class equipped with a Kähler class [w] and a B-field (an element of H2(X,R)/H2(X,Z). It is believed that to any Calabi-Yau we can as-sociate an N = 2 SCFT which depends on these geometric data. Mirror symmetry from a physicist point of view says that two different physicist’s Calabi-Yau manifolds (X,w) and (X,w′) give rise to a pair of N = 2 su-perconformal field theories SCFT’s related by a mirror morphism. A mirror morphism of N = 2 SCFT’s is an isomorphism of the underlying N = 1 SCFT’s which acts on the N = 2 super-Virasoro algebra as a mirror involu-tion. In this case (X,w) and (X ′, w′) are mirror to each other. Witten’s approach to mirror symmetry associates to any CY two topo-logical field theories TFT’s, the A and B models. The A model isolates the sympletic aspects of the CY and the B model the complex ones. Evenmore the state space of the A model is the Rham cohomology H∗,∗(X) and the B-model is the Dolbeault cohomology H∗(∧∗T 1,0X). The algebra structure in the A−case is the obviuos one and in the B−case, is a deformation of H∗(X), the quatum cohomology ring of X. M. Kontsevich presented the idea of homological mirror symmetry which states that there should be an equivalence of categories behind mir