Topics in gravity

We study questions relating to quantum gravity. The first part of this dissertation studies the degrees of freedom of non-commutative field theories. This class of theories presents an interesting playground for studying the UV-IR connection, a property assumed to hold in any theory of quantum gravity. To study these degrees of freedom, we submit the theories to a heat bath. We study a variety of theories with very different UV behavior. In all cases, we find indications in the non-planar sector for a reduced number of degrees of freedom at temperatures high with regard to the non-commutativity scale. Furthermore, the free energy in this sector exhibits a form which is suggestive of winding states in the spectrum. The starting point of the second part are recent astronomical observations that suggest that the universe is accelerating. Quintessence space-times have been proposed as alternative to asymptotic de Sitter spaces in order to accommodate these observations while bypassing the conceptual difficulties the latter pose for string theory. We argue that generic quintessence models that accommodate the present day acceleration tend to accelerate eternally. This implies that they exhibit horizons and hence pose the same problems for string theory as asymptotic de Sitter spaces. In the final part of this dissertation, we use open string mirror symmetry to calculate Ooguri-Vafa disk invariants of non-linear sigma models using the mirror Landau-Ginzburg theories. The target spaces we consider are the Calabi-Yau spaces obtained as the total space of the anticanonical line bundle of multiple blowups of the toric varieties P 2 and F2. Checking the integrality of the invariants requires calculating quantum corrections to the boundary variables. We show that these can be completely determined by using discrete symmetries of the superpotential of the mirror theory.