Scaling Behavior in String Theory

In Calabi--Yau compactifications of the heterotic string there exist quantities which are universal in the sense that they are present in every Calabi--Yau string vacuum. It is shown that such universal characteristics provide numerical information, in the form of scaling exponents, about the space of ground states in string theory. The focus is on two physical quantities. The first is the Yukawa coupling of a particular antigeneration, induced in four dimensions by virtue of supersymmetry. The second is the partition function of the topological sector of the theory, evaluated on the genus one worldsheet, a quantity relevant for quantum mirror symmetry and threshold corrections. It is shown that both these quantities exhibit scaling behavior with respect to a new scaling variable and that a scaling relation exists between them as well.Comment: 10pp, 4 eps figures (essential

Complex Multiplication of Exactly Solvable Calabi-Yau Varieties

We propose a conceptual framework that leads to an abstract characterization for the exact solvability of Calabi-Yau varieties in terms of abelian varieties with complex multiplication. The abelian manifolds are derived from the cohomology of the Calabi-Yau manifold, and the conformal field theoretic quantities of the underlying string emerge from the number theoretic structure induced on the varieties by the complex multiplication symmetry. The geometric structure that provides a conceptual interpretation of the relation between geometry and the conformal field theory is discrete, and turns out to be given by the torsion points on the abelian varieties.Comment: 44 page

Conifold Transitions and Mirror Symmetries

Recent work initiated by Strominger has lead to a consistent physical interpretation of certain types of transitions between different string vacua. These transitions, discovered several years ago, involve singular conifold configurations which connect distinct Calabi-Yau manifolds. In this paper we discuss a number of aspects of conifold transitions pertinent to both worldsheet and spacetime mirror symmetry. It is shown that the mirror transform based on fractional transformations allows an extension of the mirror map to conifold boundary points of the moduli space of weighted Calabi-Yau manifolds. The conifold points encountered in the mirror context are not amenable to an analysis via the original splitting constructions. We describe the first examples of such nonsplitting conifold transitions, which turn out to connect the known web of Calabi-Yau spaces to new regions of the collective moduli space. We then generalize the splitting conifold transition to weighted manifolds and describe a class of connections between the webs of ordinary and weighted projective Calabi-Yau spaces. Combining these two constructions we find evidence for a dual analog of conifold transitions in heterotic N$=$2 compactifications on K3$\times$T$^2$ and in particular describe the first conifold transition of a Calabi-Yau manifold whose heterotic dual has been identified by Kachru and Vafa. We furthermore present a special type of conifold transition which, when applied to certain classes of Calabi-Yau K3 fibrations, preserves the fiber structure.Comment: 23 page

On the Geometric Interpretation of N = 2 Superconformal Theories

We clarify certain important issues relevant for the geometric interpretation of a large class of N = 2 superconformal theories. By fully exploiting the phase structure of these theories (discovered in earlier works) we are able to clearly identify their geometric content. One application is to present a simple and natural resolution to the question of what constitutes the mirror of a rigid Calabi-Yau manifold. We also discuss some other models with unusual phase diagrams that highlight some subtle features regarding the geometric content of conformal theories.Comment: 25 pages, note adde

Raising the unification scale in supersymmetry

In the minimal supersymmetric standard model, the three gauge couplings appear to unify at a mass scale near $2 \times 10^{16}$ GeV. We investigate the possibility that intermediate scale particle thresholds modify the running couplings so as to increase the unification scale. By requiring consistency of this scenario, we derive some constraints on the particle content and locations of the intermediate thresholds. There are remarkably few acceptable solutions with a single cleanly defined intermediate scale far below the unification scale.Comment: 22 pages, macros included. One figure, available at ftp://ftp.phys.ufl.edu/incoming/rais.ep

Arithmetic Spacetime Geometry from String Theory

An arithmetic framework to string compactification is described. The approach is exemplified by formulating a strategy that allows to construct geometric compactifications from exactly solvable theories at $c=3$. It is shown that the conformal field theoretic characters can be derived from the geometry of spacetime, and that the geometry is uniquely determined by the two-dimensional field theory on the world sheet. The modular forms that appear in these constructions admit complex multiplication, and allow an interpretation as generalized McKay-Thompson series associated to the Mathieu and Conway groups. This leads to a string motivated notion of arithmetic moonshine.Comment: 36 page

Heterotic/Type II Duality in D=4 and String/String Duality

We discuss the structure of heterotic/type II duality in four dimensions as a consequence of string-string duality in six dimensions. We emphasize the new features in four dimensions which go beyond the six dimensional vacuum structure and pertain to the way particular K3 fibers can be embedded in Calabi-Yau threefolds. Our focus is on hypersurfaces as well as complete intersections of codimension two which arise via conifold transitions.Comment: 6 pages, espcrc