Invariant Homology on Standard Model Manifolds

Torus-fibered Calabi-Yau threefolds Z, with base dP_9 and fundamental group pi_1(Z)=Z_2 X Z_2, are reviewed. It is shown that Z=X/(Z_2 X Z_2), where X=B X_{P_1} B' are elliptically fibered Calabi-Yau threefolds that admit a freely acting Z_2 X Z_2 automorphism group. B and B' are rational elliptic surfaces, each with a Z_2 X Z_2 group of automorphisms. It is shown that the Z_2 X Z_2 invariant classes of curves of each surface have four generators which produce, via the fiber product, seven Z_2 X Z_2 invariant generators in H_4(X,Z). All invariant homology classes are computed explicitly. These descend to produce a rank seven homology group H_4(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.Comment: 57 pages, 13 figure

Torus-Fibered Calabi-Yau Threefolds with Non-Trivial Fundamental Group

We construct smooth Calabi-Yau threefolds Z, torus-fibered over a dP_9 base, with fundamental group Z_2 X Z_2. 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 Z_2 X Z_2 group of automorphisms. One then constructs Calabi-Yau threefolds X as the fiber product of two such dP_9 surfaces, demonstrating that the involutions on the surfaces lift to a freely acting Z_2 X Z_2 group of automorphisms on X. The threefolds Z are then obtained as the quotient Z=X/(Z_2 X Z_2). These Calabi-Yau spaces Z admit stable, holomorphic SU(4) vector bundles which, in conjunction with Z_2 X Z_2 Wilson lines, lead to standard-like models of particle physics with naturally suppressed nucleon decay.Comment: 60 pages, 13 figures, Typos correcte

The Spectra of Heterotic Standard Model Vacua

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 Z_2, lead to low energy theories with standard model gauge group (SU(3)_C x SU(2)_L x U(1)_Y)/Z_6 and three families of quarks and leptons. A methodology for determining the sheaf cohomology of these bundles and the representation of Z_2 on each cohomology group is given. Combining these results with the action of a Z_2 Wilson line, we compute, tabulate and discuss the massless spectrum.Comment: 41+1pp, 2 fig

SU(4) Instantons on Calabi-Yau Threefolds with Z_2 x Z_2 Fundamental Group

Structure group SU(4) gauge vacua of both weakly and strongly coupled heterotic superstring theory compactified on torus-fibered Calabi-Yau threefolds Z with Z_2 x Z_2 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 Z_2 x Z_2 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.Comment: 68 pages, no figure

Vector Bundle Moduli and Small Instanton Transitions

We give the general presciption for calculating the moduli of irreducible, stable SU(n) holomorphic vector bundles with positive spectral covers over elliptically fibered Calabi-Yau threefolds. Explicit results are presented for Hirzebruch base surfaces B=F_r. The transition moduli that are produced by chirality changing small instanton phase transitions are defined and specifically enumerated. The origin of these moduli, as the deformations of the spectral cover restricted to the ``lift'' of the horizontal curve of the M5-brane, is discussed. We present an alternative description of the transition moduli as the sections of rank n holomorphic vector bundles over the M5-brane curve and give explicit examples. Vector bundle moduli appear as gauge singlet scalar fields in the effective low-energy actions of heterotic superstrings and heterotic M-theory.Comment: 52 pages, LATEX, corrected typo

The Particle Spectrum of Heterotic Compactifications

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.Comment: 77 pages, 3 figure

Vector Bundle Moduli Superpotentials in Heterotic Superstrings and M-Theory

The non-perturbative superpotential generated by a heterotic superstring wrapped once around a genus-zero holomorphic curve is proportional to the Pfaffian involving the determinant of a Dirac operator on this curve. We show that the space of zero modes of this Dirac operator is the kernel of a linear mapping that is dependent on the associated vector bundle moduli. By explicitly computing the determinant of this map, one can deduce whether or not the dimension of the space of zero modes vanishes. It is shown that this information is sufficient to completely determine the Pfaffian and, hence, the non-perturbative superpotential as explicit holomorphic functions of the vector bundle moduli. This method is illustrated by a number of non-trivial examples.Comment: 81 pages, LaTeX, corrected typo

The Dynamics of Small Instanton Phase Transitions

The small instanton transition of a five-brane colliding with one end of the S1/Z2 interval in heterotic M-theory is discussed, with emphasis on the transition moduli, their potential function and the associated non-perturbative superpotential. Using numerical methods, the equations of motion of these moduli coupled to an expanding Friedmann-Robertson-Walker spacetime are solved including non-perturbative interactions. It is shown that the five-brane collides with the end of the interval at a small instanton. However, the moduli then continue to evolve to an isolated minimum of the potential, where they are trapped by gravitational damping. The torsion free sheaf at the small instanton is ``smoothed out'' into a vector bundle at the isolated minimum, thus dynamically completing the small instanton phase transition. Radiative damping at the origin of moduli space is discussed and shown to be insufficient to trap the moduli at the small instanton point.Comment: LaTeX, 23 pages, 7 figures; minor corrections, references adde

The Moduli of Reducible Vector Bundles

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.Comment: 33+1 page

Moduli Dependent mu-Terms in a Heterotic Standard Model

In this paper, we present a formalism for computing the non-vanishing Higgs mu-terms in a heterotic standard model. This is accomplished by calculating the cubic product of the cohomology groups associated with the vector bundle moduli (phi), Higgs (H) and Higgs conjugate (Hbar) superfields. This leads to terms proportional to phi H Hbar in the low energy superpotential which, for non-zero moduli expectation values, generate moduli dependent mu-terms of the form H Hbar. It is found that these interactions are subject to two very restrictive selection rules, each arising from a Leray spectral sequence, which greatly reduce the number of moduli that can couple to Higgs-Higgs conjugate fields. We apply our formalism to a specific heterotic standard model vacuum. The non-vanishing cubic interactions phi H Hbar are explicitly computed in this context and shown to contain only four of the nineteen vector bundle moduli.Comment: 23 pages, LaTe