[en] A procedure for computing the dimensions of the moduli spaces of reducible, holomorphic vector bundles on elliptically fibered Calabi-Yau threefolds X is presented. This procedure is applied to poly-stable rank n+m bundles of the form V + pi* M, where V is a stable vector bundle with structure group SU(n) on X and M is a stable vector bundle with structure group SU(m) on the base surface B of X. Such bundles arise from small instanton transitions involving five-branes wrapped on fibers of the elliptic fibration. The structure and physical meaning of these transitions are discussed. (author)

[en] A formalism for calculating the open supermembrane contribution to the nonperturbative superpotential of moduli in heterotic M-theory is presented. This is explicitly applied to the Calabi-Yau (1,1)-moduli and the separation modulus of the end-of-the-world BPS three-branes, whose nonperturbative superpotential is computed. The role of gauge bundles on the boundaries of the open supermembranes is discussed in detail, and a topological criterion presented for the associated superpotential to be nonvanishing

[en] Structure group SU(4) gauge vacua of both weakly and strongly coupled heterotic superstring theory compactified on torus-fibered Calabi-Yau threefolds Z with Z₂ x Z₂ fundamental group are presented. This is accomplished by constructing invariant, stable, holomorphic rank four vector bundles on the simply connected cover of Z. Such bundles can descend either to Hermite-Yang-Mills instantons on Z or to twisted gauge fields satisfying the Hermite-Yang-Mills equation corrected by a non-trivial flat B-field. It is shown that large families of such instantons satisfy the constraints imposed by particle physics phenomenology. The discrete parameter spaces of those families are presented, as well as a lower bound on the dimension of the continuous moduli of any such vacuum. In conjunction with Z₂ x Z₂ Wilson lines, these SU(4) gauge vacua can lead to standard-like models at low energy with an additional U(1)_B-L symmetry. This U(1)_B-L symmetry is very helpful in naturally suppressing nucleon decay. (author)

[en] We construct smooth Calabi-Yau threefolds Z, torus-fibered over a dP₉ base, with fundamental group π₁=Z₂xZ₂. To do this, the structure of rational elliptic surfaces is studied and it is shown that a restricted subset of such surfaces admit at least a Z₂xZ₂ group of automorphisms. One then constructs Calabi-Yau threefolds X as the fiber product of two such dP₉ surfaces,demonstrating that the involutions on the surfaces lift to a freely acting Z₂xZ₂ group of automorphisms on X. The threefolds Z are then obtained as the quotient Z=X/(Z₂xZ₂). These Calabi-Yau spaces Z admit stable, holomorphic SU(4) vector bundles which, in conjunction with Z₂xZ₂ Wilson lines, lead to standard-like models of particle physics with naturally suppressed nucleon decay. (author)

[en] Torus-fibered Calabi-Yau three folds Z, with base dP₉ and fundamental group π ₁(Z) = Z₂ x Z₂, are reviewed. It is shown that Z = X/(Z₂ x Z₂), where X = BX_P1 B^' are elliptically fibered Calabi-Yau three folds that admit a freely acting Z₂ x Z₂ automorphisms group. B and B^' are rational elliptic surfaces, each with a Z₂ x Z₂ group of automorphisms. It is shown that the Z₂ x Z₂ invariant classes of curves of each surface have four generators which produce, via the fiber product, seven Z₂ x Z₂ invariant generators in H₄(X,Z). All invariant homology classes are computed explicitly. These descend to produce a rank seven homology group H₄(Z,Z) on Z. The existence of these homology classes on Z is essential to the construction of anomaly free, three family standard-like models with suppressed nucleon decay in both weakly and strongly coupled heterotic superstring theory. (author)

[en] Expressions for the number of moduli of arbitrary SU(n) vector bundles constructed via Fourier-Mukai transforms of spectral data over Calabi-Yau threefolds are derived and presented. This is done within the context of simply connected, elliptic Calabi-Yau threefolds with base F_r, but the methods have wider applicability. The condition for a vector bundle to possess the minimal number of moduli for fixed r and n is discussed and an explicit formula for the minimal number of moduli is presented. In addition, transition moduli for small instanton phase transitions involving non-positive spectral covers are defined, enumerated and given a geometrical interpretation

[en] We develop numerical methods for approximating Ricci flat metrics on Calabi-Yau hypersurfaces in projective spaces. Our approach is based on finding balanced metrics and builds on recent theoretical work by Donaldson. We illustrate our methods in detail for a one parameter family of quintics. We also suggest several ways to extend our results

[en] We provide the methods to compute the complete massless spectra of a class of recently introduced supersymmetric E₈ x E₈ heterotic string models which invoke vector bundles with U(N) structure group on simply connected Calabi-Yau manifolds and which yield flipped SU(5) and MSSM string vacua of potential phenomenological interest. We apply Leray spectral sequences in order to derive the localisation of the cohomology groups Hⁱ(X, V_a x V_b), Hⁱ(X, Λ²V) and Hⁱ(X, S²V) for vector bundles defined via Fourier-Mukai transforms on elliptically fibered Calabi-Yau manifolds. By the method of bundle extensions we define a stable U(4) vector bundle leading to the first flipped SU(5) model with just three generations, i.e. without any vector-like matter. Along the way, we propose the notion of λ-stability for heterotic bundles

[en] Techniques are presented for computing the cohomology of stable, holomorphic vector bundles over elliptically fibered Calabi-Yau threefolds. These cohomology groups explicitly determine the spectrum of the low energy, four-dimensional theory. Generic points in vector bundle moduli space manifest an identical spectrum. However, it is shown that on subsets of moduli space of co-dimension one or higher, the spectrum can abruptly jump to many different values. Both analytic and numerical data illustrating this phenomenon are presented. This result opens the possibility of tunneling or phase transitions between different particle spectra in the same heterotic compactification. In the course of this discussion, a classification of SU(5) GUT theories within a specific context is presented. (author)

[en] A formalism for determining the massless spectrum of a class of realistic heterotic string vacua is presented. These vacua, which consist of SU(5) holomorphic bundles on torus-fibered Calabi-Yau threefolds with fundamental group Z₂, lead to low energy theories with standard model gauge group (SU(3)_C x SU(2)_L x U(1)_Y)/Z₆ and three families of quarks and leptons. A methodology for determining the sheaf cohomology of these bundles and the representation of Z₂ on each cohomology group is given. Combining these results with the action of a Z₂ Wilson line, we compute, tabulate and discuss the massless spectrum