Cordonier Tello, Fabrizio Bastián (2019): Modified gravitational backgrounds: Horndeski's theory and non-geometrical backgrounds in string theory. Dissertation, LMU München: Faculty of Physics
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### Abstract

This thesis addresses a two-part study of modified backgrounds in theories that generalize gravity. In the first part of this work we explore Horndeski's theory within Cartan's first order formalism, and study gravitational waves considering torsion to be non-vanishing. In part two we move on to string theory and study T-duality transformations for a particular non-linear sigma model containing open strings. This prompts the study of the backgrounds arising from such transformations. In part one we analyze Horndeski's Lagrangian in Cartan's first-order formalism. This formalism allows torsion to be non-zero, whereas in standard general relativity it is a vanishing quantity. Horndeski's Lagrangian is the most general Lagrangian in four dimensions featuring all possible interactions between a scalar field $\phi$ and gravity whose equations of motion are partial differential equations up to second order. This feature of such equations of motion prevents the existence of ghosts. Since this Lagrangian contains well-known modified theories of gravity as particular cases, we focus on the role of torsion and its impact at the linear perturbation regime. In order to make our analysis manageable, we cast Horndeski's Lagrangian in differential form language and we take the spin connection $\omega^{ab}$ and the vierbein $e^a$ to be independent of each other, following Cartan's formalism. We take the full Horndeski Lagrangian and compute the equations of motion for the scalar field, the spin connection and the vierbein. We argue that in order to recover the torsionless case and make contact with standard General Relativity, we have to impose a constraint via Lagrange multipliers. As a preparation for the analysis of the linear perturbation regime, we define several differential operators capable to discern spacetime torsion. These operators are capable to act covariantly on $p$-forms carrying Lorentz indices. In particular, we provide with a generalization of the Weitzenböck identity that includes torsion. Later on, we consider linear perturbations around a generic background for the vielbein, spin connection and scalar field and study Horndeski's Lagrangian under such perturbations. What we find is that non-minimal couplings and second derivatives of the scalar field are generic sources of torsion. This makes a contrast to what was known from the Einstein-Cartan-Sciama-Kibble framework, where torsion can be sourced only from fermions. In fact, we find that background torsion couples with the propagating metric degrees of freedom. This provides with a potential way to falsify torsion via gravitational waves. In part two we work inside the framework of string theory and we set to study T-duality transformations via Buscher's procedure for the open string. Such transformations lead to the study of interesting geometries which play an important role in string theory, as in moduli stabilization or in the construction of inflationary potentials. Such spaces are called non-geometric backgrounds. In this second part, we work out technical details which have been missing in the literature. These details regard the presence of D-branes and the effect of T-duality transformations on them. We study a non-linear sigma model for the open string with fields defined on the worldsheet of such open string $\Sigma$ and on its boundary $\partial\Sigma$. We take into account non-trivial topologies for this worldsheet and we present the appropriate boundary conditions for the open string. According to Buscher's procedure, we assume certain conditions for the background configuration and define additional fields on $\Sigma$ and $\partial\Sigma$ taking into account the presence of D-branes. We follow Buscher's procedure and perform T-duality transformations by gauging a worldsheet symmetry and intregrating-out worldsheet gauge fields. We reach in this way the dual configuration for the open and closed string sector. In particular, we find that the dual Kalb-Ramond field $B$ features a residual part which will play a major role when we discuss the dual configuration for the open string. To illustrate our formalism, we consider the standard configuration of the three-torus with $H$-flux and perform one, two and three collective T-duality transformations for different D-brane configurations. We read off the dual backgrounds for the closed and open string sector. We find the standard non-geometric backgrounds found in the literature, noting that such backgrounds can receive contributions coming from the dual open string sector. Regarding the dual open string sector, we study the boundary conditions of the open strings in the dual configuration and we find that they comply with the usual results in CFT. We study the global well-definedness of these D-branes on such dual backgrounds and we illustrate the application of the Freed-Witten anomaly cancelation condition for some of the examples presented.