Compactifications of moduli spaces inspired by mirror symmetry

We study moduli spaces of nonlinear sigma-models on Calabi-Yau manifolds, using the one-loop semiclassical approximation. The data being parameterized includes a choice of complex structure on the manifold, as well as some ``extra structure'' described by means of classes in H^2. The expectation that this moduli space is well-behaved in these ``extra structure'' directions leads us to formulate a simple and compelling conjecture about the action of the automorphism group on the K\"ahler cone. If true, it allows one to apply Looijenga's ``semi-toric'' technique to construct a partial compactification of the moduli space. We explore the implications which this construction has concerning the properties of the moduli space of complex structures on a ``mirror partner'' of the original Calabi-Yau manifold. We also discuss how a similarity which might have been noticed between certain work of Mumford and of Mori from the 1970's produces (with hindsight) evidence for mirror symmetry which was available in 1979. [The author is willing to mail hardcopy preprints upon request.]Comment: 25 pp., LaTeX 2.09 with AmS-Font

Through the Looking Glass

It is frequently possible to produce new Calabi-Yau threefolds from old ones by a process of allowing the complex structure to degenerate to a singular one, and then performing a resolution of singularities. (Some care is needed to ensure that the Calabi-Yau condition be preserved.) There has been speculation that all Calabi-Yau threefolds could be linked in this way, and considerable evidence has been amassed in this direction. We propose here a natural way to relate this construction to the string-theoretic phenomenon known as ``mirror symmetry.'' We formulate a conjecture which in principle could predict mirror partners for all Calabi-Yau threefolds, provided that all were indeed linked by the degeneration/resolution process. The conjecture produces new mirrors from old, and so requires some initial mirror manifold construction---such as Greene-Plesser orbifolding---as a starting point. (Lecture given at the CIRM conference, Trento, June 1994, and at the Workshop on Complex Geometry and Mirror Symmetry, Montr\'eal, March 1995.)Comment: latex2e, 22 pages with 1 figur

On tachyons, gauged linear sigma models, and flip transitions

We study systems of multiple localized closed string tachyons and the phenomena associated with their condensation, in C3/ZN nonsupersymmetric noncompact orbifold singularities using gauged linear sigma model constructions, following hep-th/0406039. Our study reveals close connections between the combinatorics of nonsupersymmetric flip transitions (between topologically distinct resolutions of the original singularity), the physics of tachyons of different degrees of relevance and the singularity structure of the corresponding residual endpoint geometries. This in turn can be used to study the stability of the phases of gauged linear sigma models and gain qualitative insight into the closed string tachyon potential.Comment: Latex, 38 pages, 7 eps figs. v2. minor modification

F-Theory and N=1 SCFTs in Four Dimensions

Using the F-theory realization, we identify a subclass of 6d (1,0) SCFTs whose compactification on a Riemann surface leads to N = 1 4d SCFTs where the moduli space of the Riemann surface is part of the moduli space of the theory. In particular we argue that for a special case of these theories (dual to M5 branes probing ADE singularities), we obtain 4d N = 1 theories whose space of marginal deformations is given by the moduli space of flat ADE connections on a Riemann surface.Comment: v2: added an exampl