Puzzles on the Duality between Heterotic and Type IIA Strings

We discuss the possibility of the extension of the duality between the webs of heterotic string and the type IIA string to Calabi-Yau 3-folds with another K3 fiber by comparing the dual polyhedron of Calabi-Yau 3-folds given by Candelas, Perevalov and Rajesh.Comment: Latex, 12 pages, Replacement: solution to puzzles was added, i.e., double K3 fibration

Critical Strings from Noncritical Dimensions: A Framework for Mirrors of Rigid Vacau

The role in string theory of manifolds of complex dimension $D_{crit} + 2(Q-1)$ and positive first Chern class is described. In order to be useful for string theory, the first Chern class of these spaces has to satisfy a certain relation. Because of this condition the cohomology groups of such manifolds show a specific structure. A group that is particularly important is described by $(D_{crit} + Q-1, Q-1)$--forms because it is this group which contains the higher dimensional counterpart of the holomorphic $(D_{crit}, 0)$--form that figures so prominently in Calabi--Yau manifolds. It is shown that the higher dimensional manifolds do not, in general, have a unique counterpart of this holomorphic form of rank $D_{crit}$. It is also shown that these manifolds lead, in general, to a number of additional modes beyond the standard Calabi--Yau spectrum. This suggests that not only the dilaton but also the other massless string modes, such as the antisymmetric torsion field, might be relevant for a possible stringy interpretation.Comment: 7 pages, NSF-ITP-93-3

On Semi-Periods

The periods of the three-form on a Calabi-Yau manifold are found as solutions of the Picard-Fuchs equations; however, the toric varietal method leads to a generalized hypergeometric system of equations which has more solutions than just the periods. This same extended set of equations can be derived from symmetry considerations. Semi-periods are solutions of this extended system. They are obtained by integration of the three-form over chains; these chains can be used to construct cycles which, when integrated over, give periods. In simple examples we are able to obtain the complete set of solutions for the extended system. We also conjecture that a certain modification of the method will generate the full space of solutions in general.Comment: 18 pages, plain TeX. Revised derivation of $\Delta^*$ system of equations; version to appear in Nuclear Physics

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

A Stringy Test of the Fate of the Conifold

By studying string loop corrections to superpotential of type II strings compactified on Calabi-Yau threefolds we find a quantum stringy test and a confirmation of a recent proposal of Strominger on the fate of the conifold singularity. We also propose a connection between the spectrum of Bogomolnyi saturated solitons and one-loop string partition function of $N=2$ topological strings.Comment: 12 page

On the Connectedness of the Moduli Space of Calabi--Yau Manifolds

We show that the moduli space of all Calabi-Yau manifolds that can be realized as hypersurfaces described by a transverse polynomial in a four dimensional weighted projective space, is connected. This is achieved by exploiting techniques of toric geometry and the construction of Batyrev that relate Calabi-Yau manifolds to reflexive polyhedra. Taken together with the previously known fact that the moduli space of all CICY's is connected, and is moreover connected to the moduli space of the present class of Calabi-Yau manifolds (since the quintic threefold P_4[5] is both CICY and a hypersurface in a weighted P_4, this strongly suggests that the moduli space of all simply connected Calabi-Yau manifolds is connected. It is of interest that singular Calabi-Yau manifolds corresponding to the points in which the moduli spaces meet are often, for the present class, more singular than the conifolds that connect the moduli spaces of CICY's.Comment: 22 pages plain TeX, Tables and references adde

Exact Results for N=2 Compactifications of Heterotic Strings

We search for $N=2$, $d=4$ theories which can be realized both as heterotic string compactifications on $K_{3}\times T^{2}$ and as type II string compactifications on Calabi-Yau threefolds. In such cases, the exact non-perturbative superpotential of one string theory is given in terms of tree level computations in the other string theory. In particular we find concrete examples which provide the stringy realization of the results of Seiberg and Witten on N=2 Yang-Mills theory, corrected by gravitational/stringy effects. We also discuss some examples which shed light on how the moduli spaces of different N=2 heterotic vacua are connected.Comment: 30 pages, Expansions and Modifications on more potential dual pairs involving K3 fibrations. Version to appear in Nuclear Physics

On Periods for String Compactifications

Motivated by recent developments in the computation of periods for string compactifications with $c=9$, we develop a complementary method which also produces a convenient basis for related calculations. The models are realized as Calabi--Yau hypersurfaces in weighted projective spaces of dimension four or as Landau-Ginzburg vacua. The calculation reproduces known results and also allows a treatment of Landau--Ginzburg orbifolds with more than five fields.Comment: HUPAPP-93/6, IASSNS-HEP-93/80, UTTG-27-93. 21 pages,harvma

A Three-Generation Calabi-Yau Manifold with Small Hodge Numbers

We present a complete intersection Calabi-Yau manifold Y that has Euler number -72 and which admits free actions by two groups of automorphisms of order 12. These are the cyclic group Z_12 and the non-Abelian dicyclic group Dic_3. The quotient manifolds have chi=-6 and Hodge numbers (h^11,h^21)=(1,4). With the standard embedding of the spin connection in the gauge group, Y gives rise to an E_6 gauge theory with 3 chiral generations of particles. The gauge group may be broken further by means of the Hosotani mechanism combined with continuous deformation of the background gauge field. For the non-Abelian quotient we obtain a model with 3 generations with the gauge group broken to that of the standard model. Moreover there is a limit in which the quotients develop 3 conifold points. These singularities may be resolved simultaneously to give another manifold with (h^11,h^21)=(2,2) that lies right at the tip of the distribution of Calabi-Yau manifolds. This strongly suggests that there is a heterotic vacuum for this manifold that derives from the 3 generation model on the quotient of Y. The manifold Y may also be realised as a hypersurface in the toric variety. The symmetry group does not act torically, nevertheless we are able to identify the mirror of the quotient manifold by adapting the construction of Batyrev.Comment: PDFLaTeX. 50 pages, 9 figures, 7 table

Max Kreuzer's Contributions to the Study of Calabi-Yau Manifolds

This is a somewhat personal account of the contributions of Max Kreuzer to the study of Calabi-Yau manifolds and has been prepared as a contribution to the Memorial Volume: Strings, Gauge Fields, and the Geometry Behind - The Legacy of Maximilian Kreuzer, to be published by World Scientific.Comment: 11 pages, pdflatex with pdf figure