Nuclear Calabi-Yau space

In this note a Calabi-Yau manifold already found for⁹Li will be shown to carry an Euler number of six if Yang-Mills symmetry is broken. Not only does this specify the correct number of generations of quarks and leptons, but peaks on the manifold are associated with the lowest eigenvalues of aCP-invariant Dirac spin operatorC _[A].     

Complete intersection Calabi-Yau manifolds

An investigation is made of a class of Calabi-Yau spaces for which the manifold may be represented as a complete intersection of polynomials in a product of projective spaces. There are at least a hundred topologically distinct manifolds in this class. All manifolds in this class have negative Euler numbers which lie in the range of −200 ⩽ χ ⩽0.     

Crystal Melting and Toric Calabi-Yau Manifolds

We construct a statistical model of crystal melting to count BPS bound states of D0 and D2 branes on a single D6 brane wrapping an arbitrary toric Calabi-Yau threefold. The three-dimensional crystalline structure is determined by the quiver diagram and the brane tiling which characterize the low energy effective theory of D branes. The crystal is composed of atoms of different colors, each of which corresponds to a node of the quiver diagram, and the chemical bond is dictated by the arrows of the quiver diagram. BPS states are constructed by removing atoms from the crystal. This generalizes the earlier results on the BPS state counting to an arbitrary non-compact toric Calabi-Yau manifold. We point out that a proper understanding of the relation between the topological string theory and the crystal melting involves the wall crossing in the Donaldson-Thomas theory.     

Calabi-Yau 4-folds and toric fibrations

We present a general scheme for identifying fibrations in the framework of toric geometry and provide a large list of weights for Calabi-Yau 4-folds. We find 914 164 weights with degree d ≤ 150 whose maximal Newton polyhedra are reflexive and 525 572 weights with degree d ≤ 4000 that give rise to weighted projective spaces such that the polynomial defining a hypersurface of trivial canonical class is transversal. We compute all Hodge numbers, using Batyrev's formulas (derived by toric methods) for the first and Vafa's formulas (obtained by counting of Ramond ground states in N = 2 LG models) for the latter class, checking their consistency for the 109 308 weights in the overlap. Fibrations of k-folds, including the elliptic case, manifest themselves in the N lattice in the following simple way: The polyhedron corresponding to the fiber is a subpolyhedron of that corresponding to the k-fold, whereas the fan determining the base is a linear projection of the fan corresponding to the k-fold.     

Polynomial deformations and cohomology of Calabi-Yau manifolds

We study the method of polynomial deformations that is used in the physics literature to determine the Hodge numbers of Calabi-Yau manifolds as well as the related Yukawa couplings. We show that the argument generally presented in the literature in support of these computations is seriously misleading, give a correct proof which applies to all the cases we found in the literature, and present examples which show that the method is not universally valid. We present a general analysis which applies to all Calabi-Yau manifolds embedded as complete intersections in products of complex projective spaces, yields sufficient conditions for the validity of the polynomial deformation method, and provides an alternative computation of all the Hodge numbers in many cases in which the polynomial method fails.     

Cosmological solutions with Calabi-Yau compactification

Assuming a Calabi-Yau compactification, cosmological solutions are presented in ten-dimensional, N=1 Yang-Mills supergravity theory with the curvature squared term (R²_μνϱσ −4R_μν² + R²). In a vacuum state, Kasner-type soluti ons exist as well as (four-dimensional Minkoswki space-time)×(a Calabi-Yau space). In the later stage of the universe the (four-dimensional Friedmann universe)×(a constant Calabi-Yau space) is realized asymptotically like an attractor. This solution is asymptotically stable against small perturbations.     

Noncommutative Resolutions of ADE Fibered Calabi-Yau Threefolds

In this paper we construct noncommutative resolutions of a certain class of Calabi-Yau threefolds studied by Cachazo et al. (Geometric transitions and N = 1 quiver theories. , 2001). The threefolds under consideration are fibered over a complex plane with the fibers being deformed Kleinian singularities. The construction is in terms of a noncommutative algebra introduced by Ginzburg (Calabi-Yau algebras. , 2006) which we call the “N = 1 ADE quiver algebra”.     

Calabi-Yau hypersurfaces in products of semi-ample surfaces

We study Calabi-Yau manifolds that are embedded as hypersurfaces in products of semi-ample complex surfaces. We classify the deformation classes of the latter and thereby achieve a classification of the Calabi-Yau manifolds that are constructed in this way. Complementing the results in the existing literature, we obtain the complete Hodge diamond for all Calabi-Yau hypersurfaces in products of semi-ample surfaces.     

Neutral Calabi-Yau Structures on Kodaira Manifolds

We construct neutral Calabi-Yau metrics and hypersymplectic structures on some Kodaira manifolds. Our structures are symmetric with respect to the central tori.Partially supported by The European Contract HPRN-CT-2000-00101, MURST and GNSAGA (Indam) of ItalyPartially supported by The European Contract HPRN-CT-2000-00101Partially supported by NSF     

D-branes on Calabi-Yau spaces and their mirrors

We study the boundary states of D-branes wrapped around supersymmetric cycles in a general Calabi-Yau manifold. In particular, we show how the geometric data on the cycles are encoded in the boundary states. As an application, we analyze how the mirror symmetry transforms D-branes, and we verify that it is consistent with the conjectured periodicity and the monodromy of the Ramond-Ramond field configuration on a Calabi-Yau manifold. This also enables us to study open string worldsheet instanton corrections and relate them to closed string instanton counting. The cases when the mirror symmetry is realized as T-duality are also discussed.     

Calabi-Yau manifolds of some special forms

Let M=M ₁×...×M _m be a product of Kähler C-spaces with second Betti numbers b ₂(M _i )=1 (1im). The work establishes that the complete intersections X of M produce a finite number of N-dimensional Calabi-Yau manifolds. Moreover, if b ₄(M _i )=1, then the complete intersections with vanishing first Pontrjagin classes are finitely many, as well.On the other hand, we consider hypersurfaces of weighted projective spaces and give an explicit formula for their Euler characteristics. As in the previous case, it turns out that only a finite number of these are Calabi-Yau manifolds.     

Enumerative Geometry of Calabi-Yau 4-Folds

Gromov-Witten theory is used to define an enumerative geometry of curves in Calabi-Yau 4-folds. The main technique is to find exact solutions to moving multiple cover integrals. The resulting invariants are analogous to the BPS counts of Gopakumar and Vafa for Calabi-Yau 3-folds. We conjecture the 4-fold invariants to be integers and expect a sheaf theoretic explanation. Several local Calabi-Yau 4-folds are solved exactly. Compact cases, including the sextic Calabi-Yau in , are also studied. A complete solution of the Gromov-Witten theory of the sextic is conjecturally obtained by the holomorphic anomaly equation.     

D-brane superpotentials and Ooguri-Vafa invariants of compact Calabi-Yau threefolds

We calculate the D-brane superpotentials for two compact Calabi-Yau manifolds X14(1,1,2,3,7) and X8(1, 1, 1, 2, 3) which are of non-Fermat type in the type II string theory. By constructing the open mirror symmetry,we also compute the Ooguri-Vafa invariants, which are related to the open Gromov-Witten invariants.     

Discrete symmetries and the complex structure of Calabi-Yau manifolds

We show how the discrete symmetries, which may be present after Calabi-Yau compactification for specific choices of the complex structure, extend to the h_2,1 moduli — the scalar fields whose vacuum expectation values determine the complex structure. This allows us to determine much about the coupling of the moduli and hence the energetically favoured complex structure. The discrete symmetry transformation properties of the moduli are worked out in detail for a three-generation Calabi-Yau model and it is shown how minimization of the effective potential involving these fields selects the complex structure which leaves unbroken a set of discrete symmetries. The phenomenological implications of these symmetries are briefly discussed.     

Connecting moduli spaces of Calabi-Yau threefolds

We demonstrate that many families of Calabi-Yau threefolds consist generically of small resolutions of nodal forms in other families and, in fact, that a large class of families is connected by this relation. Our result resonates with a conjecture of Reid that Calabi-Yau threefolds may have a universal moduli space even though they are of different homotopy types. Such ideas tie quite naturally to alluring prospects of unifying (super)string models.Supported by the Robert A. Welch Foundation and NSF Grants: PHY 8503890 and PHY 8605978On leave from Ruder Bokovi Institute, Bijenika 54, YU-41000 Zagreb, Croatia, Yugoslavia