Abstract:
We examine the stabilization of the two typical moduli, the length $\rho$ of the eleventh segment and the volume $V$ of the internal six manifold, in compactified heterotic $M$-theory. It is shown that, under certain conditions, the phenomenologically favored vacuum expectation values of $\rho$ and $V$ can be obtained by the combined effects of multi-gaugino condensations on the hidden wall and the membrane instantons wrapping the three cycle of the internal six manifold.

Abstract:
We reconsider the ingredients of moduli stabilization in heterotic M-theory. On this line we close a gap in the literature deriving the Kaehler potential dependence on vector bundle moduli and charged matter. Crucial in this derivation is our superspace formulation of 5d heterotic M-theory taking into account the Bianchi identities modified by brane terms. Likewise, we obtain the Fayet-Iliopolous terms due to brane localised anomalous U(1)'s. After assembling perturbative and non-perturbative contributions to the superpotential, we study supersymmetric (adS) vacua. It is found that the susy condition decouples the bundle moduli from the geometric moduli. We show that M-theory supersymmetric vacua without five-branes can be found, albeit not at phenomenologically interesting values of the geometric moduli. This result is fairly independent of the choice of vector bundle at the observable brane.

Abstract:
Type II compactifications with varying string coupling can be described elegantly in F-theory/M-theory as compactifications on U - manifolds. Using a similar approach to describe Super Yang-Mills with a varying coupling constant, we argue that at generic points in Narain moduli space, the $E_8 \times E_8$ Heterotic string compactified on $T^2$ is described in M(atrix) theory by N=4 SYM in 3+1 dimensions with base $S^1 \times CP^1$ and a holomorphically varying coupling constant. The $CP^1$ is best described as the base of an elliptic K3 whose fibre is the complexified coupling constant of the Super Yang-Mills theory leading to manifest U-duality. We also consider the cases of the Heterotic string on $S^1$ and $T^3$. The twisted sector seems to (almost) naturally appear at precisely those points where enhancement of gauge symmetry is expected and need not be postulated. A unifying picture emerges in which the U-manifolds which describe type II orientifolds (dual to the Heterotic string) as M- or F- theory compactifications play a crucial role in Heterotic M(atrix) theory compactifications.

Abstract:
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.

Abstract:
In this talk we discuss a few relevant aspects of heterotic M-theory. These are the stabilization of the two relevant moduli (the length of the eleventh segment (pi rho) and the volume of the internal six manifold (V)) in models where supersymmetry is broken by multiple gaugino condensation and non-perturbative corrections to the Kahler potential; the existence of almost flat directions in the scalar potential; the possibility of lifting them, and their role in constructing a viable model of inflation. Finally, we review the status of the moduli problem within these models. T

Abstract:
We discuss the moduli space approximation for heterotic M-theory, both for the minimal case of two boundary branes only, and when a bulk brane is included. The resulting effective actions may be used to describe the cosmological dynamics in the regime where the branes are moving slowly, away from singularities. We make use of the recently derived colliding branes solution to determine the global structure of moduli space, finding a boundary at which the trajectories undergo a hard wall reflection. This has important consequences for the allowed moduli space trajectories, and for the behaviour of cosmological perturbations in the model.

Abstract:
We study the effective low energy supergravity of the strongly coupled heterotic string compactified on a Calabi-Yau 3-fold with generic E8 x E8 gauge bundle. We focus on the effective potential for the chiral scalars. The effective superpotential can be studied using the dual 11-dimensional M-theory background involving insertions of M5 branes along an interval. In such backgrounds, in some regions of moduli space, the leading nonperturbative contributions are due to open membrane instantons. These instantons lead to both attractive and repulsive forces between the 5-branes and the orientifold ``M9-branes,'' depending on the region of moduli space. The resulting dynamics on moduli space include a strong coupling dual to the Dine-Seiberg instability, in which the interval grows. We discuss conditions under which the 5-branes are attracted to the wall and comment on the relevance of these results to the study of chirality-changing phase transitions in heterotic M-theory.

Abstract:
We discuss several aspects of the cosmological evolution of moduli fields in heterotic string/M-theory scenarios. In particular we study the equations of motion of both the dilaton and overall modulus of these theories in the presence of an expanding Universe and under different assumptions. First we analyse the impact of their couplings to matter fields, which turns out to be negligible in the string and M-theory scenarios. Then we examine in detail the possibility of scaling in M-theory, i.e. how the moduli would evolve naturally to their minima instead of rolling past them in the presence of a dominating background. In this case we find interesting and positive results, and we compare them to the analogous situation in the heterotic string.

Abstract:
A formalism for calculating the open supermembrane contribution to the non-perturbative 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 non-perturbative 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 non-vanishing.

Abstract:
Brane actions with chiral bosons present special challenges. Recent progress in the description of the two main examples -- the M theory five-brane and the heterotic string -- is described. Also, double dimensional reduction of the M theory five-brane on K3 is shown to give the heterotic string.