Mirror symmetry

Abstract: We compute the prepotentials and the geometry of the moduli spaces for a Calabi-Yau manifold and its mirror. In this way we obtain all the sigma model corrections to the Yukawa couplings and moduli space metric for the original manifold. 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 R → 1 R . It is also shown that the quantum corrections to the coupling decompose into a sum over instanton contributions and moreover that this sum converges. In particular there are no “sub-instanton” corrections. This sum over instantons points to a deep connection between the modular group and the rational curves of the Calabi-Yau manifold. 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. 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.

Abstract: It is argued that every Calabi-Yau manifold X with a mirror Y admits a family of supersymmetric toroidal 3-cycles. Moreover the moduli space of such cycles together with their flat connections is precisely the space Y . The mirror transformation is equivalent to T -duality on the 3-cycles. The geometry of moduli space is addressed in a general framework. Several examples are discussed.

Abstract: We prove mirror symmetry for supersymmetric sigma models on Kahler manifolds in 1+1 dimensions The proof involves establishing the equivalence of the gauged linear sigma model, embedded in a theory with an enlarged gauge symmetry, with a Landau-Ginzburg theory of Toda type Standard R -> 1/R duality and dynamical generation of superpotential by vortices are crucial in the derivation This provides not only a proof of mirror symmetry in the case of (local and global) Calabi-Yau manifolds, but also for sigma models on manifolds with positive first Chern class, including deformations of the action by holomorphic isometries

Abstract: Mirror symmetry (MS) was discovered several years ago in string theory as a duality between families of 3-dimensional Calabi-Yau manifolds (more precisely, complex algebraic manifolds possessing holomorphic volume elements without zeros). The name comes from the symmetry among Hodge numbers. For dual Calabi-Yau manifolds V, W of dimension n (not necessarily equal to 3) one has $$\dim {H^p}(V,{\Omega ^q}) = \dim {H^{n - p}}(W,{\Omega ^q}).$$ .

Abstract: We propose an explanation via string theory of the correspondence between the Coulomb branch of certain three-dimensional supersymmetric gauge theories and certain moduli spaces of magnetic monopoles. The same construction also gives an explanation, via SL(2, Z) duality of Type IIB superstrings, of the recently discovered “mirror symmetry” in three dimensions. New phase transitions in three dimensions as well as new infrared fixed points and even new coupling constants not present in the known Lagrangians are predicted from the string theory construction. An important role in the construction is played by a novel aspect of brane dynamics in which a third brane is created when two branes cross.