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The current understanding of theoretical physics tells us that there exists a unique, nonperturbative quantum theory living in 11D spacetime (M-theory), from which five 10D superstring theories arise as perturbative limits Finding the explicit form of this M-theory is one of the greatest theoretical challenges of the twenty first century. In this thesis, we shed the light on some important aspects, vacuum energy, moduli stabilization and gaugino condensates in the framework of 5D heterotic M-theory. The central question we are trying to answer in this, thesis is: what is the mechanism for radion stabilization?. To answer this question we calculate the total bulk vacuum energy, which is the difference between the twisted and untwisted fermion vacuum energies. in both flat and curved spaces.

To this date, experimental searches for exotics (fractionally charged states) have yielded negative results, thus imposing strong limitations on their existence at low energies. In this work we argue that a possible scenario that could explain the experimental data is that such states are simply not present at the low energy limit. This thesis presents the phenomenology of the first string model which does not have any exotic states at the massless level, and where only the top quark Yukawa coupling exists at the tri-linear level superpotential. More specifically we present its spectrum, cubic superpotential and a viable semi-realistic phenomenological scenario which is supported by a specific set of F - and D-fl.at solutions. The discovery of this model is the result of the statististical exploration of a class of heterotic Pati-Salam vacua, out of which we managed to extract three generation exophobic models with the required representations needed to induce spontaneous breaking to the Standard Model. We also exhibit the derivation of the analytic formulae that permitted the exact identification of several properties of a string vacuum, and thus allowed its distinction among supersymmetric vacua that share the same gauge group.

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This thesis is dedicated to the study of K-theoretical properties of D-branes and Ramond-Ramond fields. We construct abelian groups which define a homology theory on the category CW-complexes, and prove that this homology theory is equivalent to the bordism 3n of KO-homology, the dual theory to KO-theory. We construct an isomorphism between our geometric representation and the SLUdlytic representation of KO-homology, which induces a natural equivalence of homology functors. We apply this framework to describe mathematical properties of D-branes in type I String theory.

In this thesis, we discuss time-dependent solutions of the low-energy limit of heterotic M-theory. In particular, we are interested in cases where the solution undergoes a dynamical transition from a five- to a four-dimensional description. We give explicit examples of such cases, and show how to match the properties of the different regimes. We begin by discussing how the five- and four-dimensional actions arise in heterotic M-theory, and why reduction differs from that of a standard Kaluza-Klein compacti- fication. The truncated actions relevant for our cosmological solutions are presented, and the process of compactification on a non-trivial vacuum is explained in detail. We derive various cosmological solutions, both of the heterotic M-theory action and a more phenomenologically motivated action relevant for inflation, and discuss the conditions necessary for these solutions to possess a four-dimensional limit. We pay particular attention to the behaviour of these solutions in the four-dimensional regime. VVe give a brief introduction to inflation, explaining why this epoch could potentially provide observable signatures of Horava- Witten cosmology. Having derived solutions with an (A)dS5 bulk and dS4 branes, we consider the slow-roll conditions that must be imposed if these backgrounds are to be upgraded to inflationary models. We show that the structure of the five-dimensional space places strong restrictions on a bulk scalar field. An example of a five-dimensional inflationary background which becomes four-dimensional as slow-roll progresses is found. Finally, we consider the problem of stabilization of the four-dimensional moduli during inflation. Specifically, in the context of the weakly coupled heterotic string we are interested in whether the dilaton can naturally be stabilized near its present value at the time of inflation. We demonstrate a fairly generic mechanism that ensures the gauge coupling at the unification scale was at the approximately the same value during inflation as today. The consequences of this for inflationary model-building are discussed.

The AdS5 ×S5 superstring action is constructed by the Green-Schwarz formalism. For quantization it is necessary to eliminate unphysical degrees of freedom from the action by solving the Virasoro constraints and fixing the fermionic kappa-symmetry, which can be achieved by the Pohlmeyer reduction preserving the two-dimensional Lorentz invariance and the integrability. The resulting system is a gauged Wess-Zumino-Witten (gWZW) model deformed with a certain integrable potential and two-dimensional fermions. This thesis explores the quantum relation between the AdS5 × S5 superstring theory and the deformed gWZW model by evaluating the reduced theory quantum partition functions for respective classical string configurations. To understand the quantum relation between the original string theory and the reduced theory, the one-loop computation in the reduced theory is first studied for homogeneous and inhomogeneous string configurations localized in subspaces. For these classical backgrounds we demonstrate that the reduced theory partition function is exactly the same as the string theory one, then they are equivalent at one-loop level. Next we investigate the two-loop relation between the original string theory the reduced theory. The two-loop computation in the reduced theory is performed by considering the long folded string localized in AdS3. We show that the nontrivial finite terms of the two-loop partition functions of the two theories match, exhibiting the same patterns of the bosonic contributions and fermionic contributions. This is a strong indication that the AdS5 × S5 GS string and its reduced form are closely related at the quantum level.

The Pohlmeyer reduction of the AdS5 x S5 superstring is a fermionic generalization of the relationship between the O(3) sigma model and the sine-Gordon model. In the reduction procedure the Virasoro constraints are solved and the resulting reduced theory is Lorentz-invariant, integrable and classically equivalent to the superstring sigma model. Its action is that of a gauged WZW model plus an integrable potential coupled to fermions. Furthermore, the theory is UV-finite and conjectured to be related to the superstring at a quantum level. This thesis begins with a review of the Pohlmeyer reduction, concentrating on its rôle in string theory. The main focus of the thesis is an investigation into the S-matrix of the Pohlmeyer-reduced AdS5 x S5 superstring. Expanding around the trivial vacuum, a local quartic action is constructed for the 8 + 8 (bosonic and fermionic) massive asymptotic degrees of freedom. The resulting perturbative S-matrix has the same tensorial structure and group factorization property as the light-cone gauge-fixed superstring S-matrix. However, it does not satisfy the Yang-Baxter equation. As a possible resolution it is proposed to consider a particular limit of the quantum-deformed (psu(2|2)xR3)-invariant R-matrix of Beisert and Koroteev. The exact form of the corresponding S-matrix is constructed and possible relations to the perturbative computation are explored. The on-shell symmetry of the quantum-deformed S-matrix may be interpreted as a quantum-deformed N = 8 two-dimensional supersymmetry. After describing the representation theory of Uq(psu(2|2)xR2) and the pole structure of the deformed S-matrix the bootstrap programme is used to construct the S-matrix elements for the bound states. The thesis concludes with a discussion of the current status of the Pohlmeyer reduction and open questions.