Semi-global Kuranishi charts and the definition of contact homology

We define the contact homology algebra for any contact manifold and show that it is an invariant of the contact manifold. More precisely, given a contact manifold $(M,\xi)$ and some auxiliary data $\mathcal{D}$, we define an algebra $HC(\mathcal{D})$. If $\mathcal{D}_1$ and $\mathcal{D}_2$ are two choices of auxiliary data for $(M,\xi)$, then $HC(\mathcal{D}_1)$ and $HC(\mathcal{D}_2)$ are isomorphic. We use a simplified version of Kuranishi perturbation theory, consisting of semi-global Kuranishi charts.Comment: 95 pages. Proposition 9.4.2 was removed, as its proof has a gap. The main theorem is not affecte

Compact complex surfaces with geometric structures related to split quaternions

We study the problem of existence of geometric structures on compact complex surfaces that are related to split quaternions. These structures, called para-hypercomplex, para-hyperhermitian and para-hyperk\"ahler are analogs of the hypercomplex, hyperhermitian and hyperk\"ahler structures in the definite case. We show that a compact oriented 4-manifold carries a para-hyperk\"ahler structure iff it has a metric of split signature together with two parallel, orthogonal and null vector fields. Every compact complex surface admiting a para-hyperhermitian structure has vanishing first Chern class and we show that, unlike the definite case, many of these surfaces carry infinite dimensional families of such structures. We provide also compact examples of complex surfaces with para-hyperhermitian structures which are not locally conformally para-hyperk\"ahler. Finally, we discuss the problem of non-existence of para-hyperhermitian structures on Inoue surfaces of type $S^0$ and provide a list of compact complex surfaces which could carry para-hypercomplex structures.Comment: final version, to appear in Nucl.Phys.

From F-theory to brane webs: Non-perturbative effects in type IIB String Theory

Tesis doctoral inédita leída en la Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Física Teórica. Fecha de lectura: 13-06-2016We analyse the flavour sector of SU(5) Grand Unified Theories in F–theory. Two classes of local models are formulated, one with enhancement to E6 where the masses of the up–type quarks are generated, and one with enhancement to either E7 or E8 where the masses for all fermions of the Standard Model are generated. A full rank 3 Yukawa matrix is attained only after the inclusion of non–perturbative effects in the compactification space. By performing a scan over the parameters defining the local models we check whether realistic masses for the fermions may be attained. Secondly we present two example of the appearance of linear equivalence between cycles in D–brane models. In the first case we show how linear equivalence is tied with kinetic mixing between open and closed string massless U(1)’s and discuss potential phenomenological implications for dark matter and unification of gauge couplings. Secondly we show how taking into account the coupling with closed string moduli some of the brane moduli may acquire a mass. We clarify the microscopic origin of this effect and its connection with linear equivalence of cycles, and finally match it with the 4d supergravity description. Finally we discuss the application of topological string techniques for the computation of the Nekrasov partition function for theories in the Higgs branch. We formulate a general algorithm for the computation of the Nekrasov partition function of the 5d TN theory in a generic point of the Higgs branch. Afterwards we present a generalisation of the topological vertex applicable to a wide class of non– toric varieties. In both cases we provide some explicit examples of the application of the new rules formulatedSe analiza el sector de sabor en teorías SU(5) de Gran Unificación en Teoría F. Se construyen dos clases de modelos locales, una con aumento del grupo gauge a E6 donde se generan las masas de los quarks de tipo up, y una con aumento del grupo gauge a E7 o E8 donde se generan las masas de todos los fermiones del Modelo Estándar. Solamente después de haver incluido efectos no perturbativos en el espacio de compactificación se consigue una matriz de Yukawa de rango 3. Haciendo una búsqueda en los valores de los parámetros que definen los modelos locales se comprueba si es posible conseguir masas realistas para los fermiones. En segundo lugar se presentan dos ejemplos de cómo la equivalencia lineal entre ciclos aparece en modelos de D–branas. En el primer caso se demuestra cómo la equivalencia lineal está conectada con la mezcla cinética entre U(1)’s sin masa de cuerda abierta y cerrada y se discuten implicaciones fenomenológicas para materia oscura y unificación de los acoplos de gauge. Después se demuestra cómo algunos de los módulos de brana reciben masa al tener en cuenta el acoplo con los módulos de cuerda cerrada. Se aclara el origen microscópico de este efecto y su conexión con la equivalencia lineal de ciclos, comparandolo por último con la descripción en supergravedad en 4d. Finalmente se discute la aplicación de técnicas de cuerda topológica para el cálculo de funciones de partición de Nekrasov para teoría en la rama de Higgs. Se formula un algoritmo general para el cálculo de la función de partición de Nekrasov de la teoría TN en 5d en un punto genérico de la rama de Higgs. Después se presenta una generalización del vértice topológico que se puede aplicar a una amplia clase de variedades no tóricas. En ambos casos se presentan algunos ejemplos de la aplicación de las nuevas reglas que se han formulad

Constraints on Four Dimensional Effective Field Theories From String and F-theory

This thesis is a study of string theory compactifications to four dimensions and the constraints the Effective Field theories must exhibit, exploring both the closed and open sectors. In the former case, we focus on axion monodromy scenarios and the impact the backreaction of the energy density induced by the vev of an axion has on its field excursions. For all the cases studied, we find that the backreaction is small up to a critical value, and the proper field distance is flux independent and at most logarithmic in the axion vev. We then move to the open sector, where we use the framework of F-theory. We first explore the relation between the spectra arising from F-theory GUTs and those coming from a decomposition of the adjoint of E_8 to SU(5)x U(1)^n. We find that extending the latter spectrum with new SU(5)-singlet fields, and classifying all possible ways of breaking the Abelian factors, all the spectra coming from smooth elliptic fibration constructed in the literature fit in our classification. We then explore generic properties of the spectra arising when breaking SU(5) to the Standard Model gauge group while retaining some anomaly properties. We finish by a study of F-theory compactications on a singular elliptic fibration via Matrix Factorisation, and find the charged spectrum of two non-Abelian examples

The geometry and physics of F-theory compactifications

In this PhD thesis we study the structure of gauge and gravitational anomalies in effective theories obtained by compactfication of F-theory on Calabi-Yau manifolds. In particular, we study the continuous local anomalies in 2D N = (0, 2) effective theories from elliptically fibered Calabi-Yau five-fold compactifications and discrete gauge anomalies in 6D N = (1, 0) theories from F-theory on genus-one fibrations of Calabi-Yau three-folds. Certain anomalies associated with these symmetries, induced at 1-loop in perturbative theories, can be cancelled by a corresponding generalized Green-Schwarz mechanism operating at the level of chiral fields in the effective theories. We derive closed expressions for types Green-Schwarz mechanisms in F-theory compactifications, as well as the gravitational and gauge anomalies. These expressions in both cases involve topological invariants of the underlying fibrations of Calabi-Yau manifolds. Cancellation of these anomalies in the effective theories predicts intricate topological identities which must hold on every corresponding Calabi-Yau manifold. Some of the identities we find on elliptic 5-folds are related in an intriguing way to previously studied topological identities governing the structure of local anomalies for continuous symmetry in 6D N = (1, 0) and 4D N = 1 theories obtained from F-theory

Gauge Fluxes in F-theory Compactifications

In this thesis, we study the geometry and physics of gauge fluxes in F-theory compactifications to four dimensions. Motivated by the phenomenological requirement of chiral matter in realistic model building scenarios, we develop methods for a systematic analysis of primary vertical G4-fluxes on torus-fibred Calabi-Yau fourfolds. In particular, we extend the well-known description of fluxes on elliptic fibrations with sections to the more general set-up of genus-one fibrations with multi-sections. The latter are known to give rise to discrete abelian symmetries in F-theory. We test our proposal for constructing fluxes in such geometries on an explicit model with SU(5)xZ2 symmetry, which is connected to an ordinary elliptic fibration with SU(5)xU(1) symmetry by a conifold transition. With our methods we systematically verify anomaly cancellation and tadpole matching in both models. Along the way, we find a novel way of understanding anomaly cancellation in 4D F-theory in purely geometric terms. This observation is further strengthened by a similar analysis of an SU(3)xSU(2)xU(1)xU(1) model. The obvious connection of this particular model with the Standard Model is then investigated in a more phenomenologically motivated survey. There, we will first provide possible matchings of the geometric spectrum with the Standard Model states, which highlights the role of the additional U(1) factor as a selection rule. In a second step, we then utilise our novel methods on flux computations to set up a search algorithm for semi-realistic chiral spectra in our Standard- Model-like fibrations over specific base manifolds B. As a demonstration, we scan over three choices P3, Bl1P3 and Bl2P3 for the base. As a result we find a consistent flux that gives the chiral Standard Model spectrum with a vector-like triplet exotic, which may be lifted by a Higgs mechanism