Mirror symmetry and self-duality equations

Abstract

A central problem of mirror symmetry is to understand Calabi-Yau moduli spaces at large complex structure. The basic techniques for this problem have been developped by Griffiths, Deligne and Schmid. The moduli space for complex structures is mapped into Griffiths' classifying space for Hodge structures, and the image is approximated by a nilpotent orbit up to exponentially small terms. Schmid described the geometry of the nilpotent orbit by equations which later were found by the speaker in the context of magnetic monopoles. The nilpotent orbit case corresponds to the special case of a maximally degenerate spectral curve.Keywords: large complex structure, moduli, conformal field theory. Boundaries of the M-theory moduli spaceThere are two limits in which non-perturbative string theory (alias M-theory) is accessible to calculations, namely small spacetime curvature and small string coupling. In the first case, in particular for strings on the product of a flat spacetime with a compact internal space of large radius, one obtains in the limit a conventional quantum field theory on an Einstein spacetime. The behaviour at small string coupling, on the other hand, is described by a superconformal theory on the string worldsheet. If the present indications of a low Higgs mass will hold true, one even can hope for low energy supersymmetry and consequently for extended supersymmetry in the worldsheet theory. Together with T-and S-duality, the two limits might give sufficient control over the boundaries of the (hypothetical) M-theory moduli space to get a handle on more realistic situations. At present, even the understanding of the boundaries is far from complete. Even in the favorable case of extended supersymmetry, the conformal field theories are only solvable for special parameters, and there is still much to be learned from the case where the two limits are performed together. But at least a complete understanding of the geometry and metric of their moduli spaces seems feasible. This means that it will be much easier to classify the theories then to describe them in detail. Moreover, for the special physical quantities described by meromorphic functions the description of the moduli space takes us very close to an explicit computation, since for a conventional vacuum one expects compactifiable spaces. Classical and quantum moduli spacesThe moduli space M of a conformal theory with extended supersymmetry is closely related to certain moduli spaces of classical geometrical structures. In the limiting case when both the string coupling and the spacetime curvature tend to zero, it is clear that the conformal theory is given by a non-linear sigma model on a Calabi-Yau manifold X, such that M can be projected to the moduli space M (X) of complex structures of the underlying differentiable manifold. For IIA theories with (exactly) N = (2, 2) supersymmetry, in particular for central charges c = 3 and c = 9, one also obtains a projection to the moduli space of the mirror manifold X , and M = M(X) × M(X ), at least...