Generalised asymptotic equivalence for extensive and non-extensive entropies

We extend the Hanel and Thurner asymptotic analysis to both extensive and non-extensive entropies on the basis of a wide class of entropic forms. The procedure is known to be capable to classify multiple entropy measures in terms of their defining equivalence classes. Those are determined by a pair of scaling exponents taking into account a large number of microstates as for the thermodynamical limit. Yet, a generalisation to this formulation makes it possible to establish an entropic connection between Markovian and non-Markovian statistical systems through a set of fundamental entropies $S_{\pm}$, which have been studied in other contexts and exhibit, among their attributes, two interesting aspects: They behave as additive for a large number of degrees of freedom while they are substantially non-additive for a small number of them. Furthermore, an ample amount of special entropy measures, either additive or non-additive, are contained in such asymptotic classification. Under this scheme we analyse the equivalence classes of Tsallis, Sharma-Mittal and R\'enyi entropies and study their features in the thermodynamic limit as well as the correspondences among them.Comment: 6 pages, 2 figure

Heterotic Mini-landscape in blow-up

Localization properties of fields in compact extra dimensions are crucial ingredients for string model building, particularly in the framework of orbifold compactifications. Realistic models often require a slight deviation from the orbifold point, that can be analyzed using field theoretic methods considering (singlet) fields with nontrivial vacuum expectation values. Some of these fields correspond to blow-up modes that represent the resolution of orbifold singularities. Improving on previous analyses we give here an explicit example of the blow-up of a model from the heterotic Mini-landscape. An exact identification of the blow-up modes at various fixed points and fixed tori with orbifold twisted fields is given. We match the massless spectra and identify the blow-up modes as non-universal axions of compactified string theory. We stress the important role of the Green-Schwarz anomaly polynomial for the description of the resolution of orbifold singularities.Comment: 34 pages, 5 figure

Matching the heterotic string on orbifolds and their resolutions

We study the symmetry breaking mechanism under which a 6d orbifold compactification of the 10d heterotic string turns into a smooth Calabi--Yau compactification. This process is naturally required to preserve N=1 supersymmetry in 4d. The cause is the existence of a Fayet--Iliopoulos D--term generated by the anomalous U(1) gauge--symmetry on the orbifold theory. An orbifold is constructed by modding out a symmetry from a toroidal 6d lattice, resulting in an almost everywhere flat variety with the exception of fixed sets under the symmetry action. Those sets are fixed points and fixed tori, and the states localized at them are the so called twisted states. D--flatness leads to a vacuum with non--zero expectation values of twisted scalars. Those scalars play the role of blow--up modes: their vevs deform the local geometry and smooth out the singularities. We study Calabi--Yau manifolds obtained by blowing--up (resolving) the singularities using toric geometry. We analyze the massless spectrum and the anomaly cancellation on the deformed orbifold and in the resolution obtaining a perfect map. On the orbifold we can compute the full particle spectrum and the interactions using the CFT world--sheet description of the heterotic string. To compactify on the resolution, as the metric is unknown, we have to start with the 10d N=1 supergravity and super Yang--Mills effective theory and perform dimensional reduction. In the thesis we first review the 10d heterotic string, the heterotic supergravity, the orbifold and Calabi--Yau compactifications and the toric geometry techniques required for resolving orbifolds. We perform an study of potential orbifold 4d discrete symmetries in factorizable and non--factorizable orbifolds arising from the torus lattice automorphisms. We then come to our focus, which is the orbifold--resolution transition in two compact orbifold models with the Minimal Supersymmetric Standard Model Physics. First, we study the T6/Z7 orbifold and its resolution. This orbifold contains all the ingredients of realistic models. It is simpler because it is prime and has therefore only fixed points and no orbifold brother models. We find the field redefinitions that identify the orbifold and blow--up massless spectrum. A local index theorem is crucial in this process. We study then the Green--Schwarz anomaly cancellation mechanism after dimensional reduction on the resolution and from the massless spectrum and the field redefinitions on the orbifold. We find that both results perfectly agree. This determines the blow--up modes as non--universal axions on the Calabi--Yau manifold. After these encouraging results we study now a more involved case, this is the T6/Z6II orbifold and its resolution. As the orbifold is non--prime there are fixed tori. This makes the identification of the blow--up modes and the search of the field redefinitions more difficult. We overcome that difficulty exploring in a Mini--landscape of phenomenologically promising orbifold models to select a suitable one, and on it we are able to identify the blow--up modes. We find perfect agreement of the massless spectrum, including the orbifold generated mass terms. To culminate, we study in detail the anomaly cancellation mechanism. We find here as in the Z7 case that the blow--up modes play the role of the resolution non--universal axions. Our work interplays between string theory consistency, which expresses itself through 10d anomaly cancellation, and the physics and the geometry of supersymmetric space--time vacua

Landscaping with fluxes and the E8 Yukawa Point in F-theory

Integrality in the Hodge theory of Calabi-Yau fourfolds is essential to find the vacuum structure and the anomaly cancellation mechanism of four dimensional F-theory compactifications. We use the Griffiths-Frobenius geometry and homological mirror symmetry to fix the integral monodromy basis in the primitive horizontal subspace of Calabi-Yau fourfolds. The Gamma class and supersymmetric localization calculations in the 2d gauged linear sigma model on the hemisphere are used to check and extend this method. The result allows us to study the superpotential and the Weil-Petersson metric and an associated tt* structure over the full complex moduli space of compact fourfolds for the first time. We show that integral fluxes can drive the theory to N=1 supersymmetric vacua at orbifold points and argue that fluxes can be chosen that fix the complex moduli of F-theory compactifications at gauge enhancements including such with U(1) factors. Given the mechanism it is natural to start with the most generic complex structure families of elliptic Calabi-Yau 4-fold fibrations over a given base. We classify these families in toric ambient spaces and among them the ones with heterotic duals. The method also applies to the creating of matter and Yukawa structures in F-theory. We construct two SU(5) models in F-theory with a Yukawa point that have a point on the base with an $E_8$-type singularity on the fiber and explore their embeddings in the global models. The explicit resolution of the singularity introduce a higher dimensional fiber and leads to novel features.Comment: 150 page