Quantum Gravity via Causal Dynamical Triangulations

"Causal Dynamical Triangulations" (CDT) represent a lattice regularization of the sum over spacetime histories, providing us with a non-perturbative formulation of quantum gravity. The ultraviolet fixed points of the lattice theory can be used to define a continuum quantum field theory, potentially making contact with quantum gravity defined via asymptotic safety. We describe the formalism of CDT, its phase diagram, and the quantum geometries emerging from it. We also argue that the formalism should be able to describe a more general class of quantum-gravitational models of Horava-Lifshitz type.Comment: To appear in "Handbook of Spacetime", Springer Verlag. 31 page

CDT---an Entropic Theory of Quantum Gravity

In these lectures we describe how a theory of quantum gravity may be constructed in terms of a lattice formulation based on so-called causal dynamical triangulations (CDT). We discuss how the continuum limit can be obtained and how to define and measure diffeomorphism-invariant correlators. In four dimensions, which has our main interest, the lattice theory has an infrared limit which can be identified with de Sitter spacetime. We explain why this infrared property of the quantum spacetime is nontrivial and due to "entropic" effects encoded in the nonperturbative path integral measure. This makes the appearance of the de Sitter universe an example of true emergence of classicality from microscopic quantum laws. We also discuss nontrivial aspects of the UV behaviour, and show how to investigate quantum fluctuations around the emergent background geometry. Finally, we consider the connection to the asymptotic safety scenario, and derive from it a new, conjectured scaling relation in CDT quantum gravity.Comment: 49 pages, many figures. Lectures presented at the "School on Non-Perturbative Methods in Quantum Field Theory" and the "Workshop on Continuum and Lattice Approaches to Quantum Gravity", Sussex, September 15th-19th 2008 . To appear as a contribution to a Springer Lecture Notes in Physics boo

Wilson loops in CDT quantum gravity

By explicit construction, we show that one can in a simple way introduce and measure gravitational holonomies and Wilson loops in lattice formulations of nonperturbative quantum gravity based on (Causal) Dynamical Triangulations. We use this set-up to investigate a class of Wilson line observables associated with the world line of a point particle coupled to quantum gravity, and deduce from their expectation values that the underlying holonomies cover the group manifold of SO(4) uniforml

Renormalization Group Flow in CDT

We perform a first investigation of the coupling constant flow of the nonperturbative lattice model of four-dimensional quantum gravity given in terms of Causal Dynamical Triangulations (CDT). After explaining how standard concepts of lattice field theory can be adapted to the case of this background-independent theory, we define a notion of "lines of constant physics" in coupling constant space in terms of certain semiclassical properties of the dynamically generated quantum universe. Determining flow lines with the help of Monte Carlo simulations, we find that the second-order phase transition line present in this theory can be interpreted as a UV phase transition line if we allow for an anisotropic scaling of space and time.Comment: Typos corrected, 21 page

Mapping social reward and punishment processing in the human brain:A voxel-based meta-analysis of neuroimaging findings using the social incentive delay task

Social rewards or punishments motivate human learning and behaviour, and alterations in the brain circuits involved in the processing of these stimuli have been linked with several neuropsychiatric disorders. However, questions still remain about the exact neural substrates implicated in social reward and punishment processing. Here, we conducted four Anisotropic Effect Size Signed Differential Mapping voxel-based meta-analyses of fMRI studies investigating the neural correlates of the anticipation and receipt of social rewards and punishments using the Social Incentive Delay task. We found that the anticipation of both social rewards and social punishment avoidance recruits a wide network of areas including the basal ganglia, the midbrain, the dorsal anterior cingulate cortex, the supplementary motor area, the anterior insula, the occipital gyrus and other frontal, temporal, parietal and cerebellar regions not captured in previous coordinate-based meta-analysis. We identified decreases in the BOLD signal during the anticipation of both social reward and punishment avoidance in regions of the default-mode network that were missed in individual studies likely due to a lack of power. Receipt of social rewards engaged a robust network of brain regions including the ventromedial frontal and orbitofrontal cortices, the anterior cingulate cortex, the amygdala, the hippocampus, the occipital cortex and the brainstem, but not the basal ganglia. Receipt of social punishments increased the BOLD signal in the orbitofrontal cortex, superior and inferior frontal gyri, lateral occipital cortex and the insula. In contrast to the receipt of social rewards, we also observed a decrease in the BOLD signal in the basal ganglia in response to the receipt of social punishments. Our results provide a better understanding of the brain circuitry involved in the processing of social rewards and punishment. Furthermore, they can inform hypotheses regarding brain areas where disruption in activity may be associated with dysfunctional social incentive processing during diseas

Gaugino Condensation and Nonperturbative Superpotentials in Flux Compactifications

There are two known sources of nonperturbative superpotentials for K\"ahler moduli in type IIB orientifolds, or F-theory compactifications on Calabi-Yau fourfolds, with flux: Euclidean brane instantons and low-energy dynamics in D7 brane gauge theories. The first class of effects, Euclidean D3 branes which lift in M-theory to M5 branes wrapping divisors of arithmetic genus 1 in the fourfold, is relatively well understood. The second class has been less explored. In this paper, we consider the explicit example of F-theory on $K3 \times K3$ with flux. The fluxes lift the D7 brane matter fields, and stabilize stacks of D7 branes at loci of enhanced gauge symmetry. The resulting theories exhibit gaugino condensation, and generate a nonperturbative superpotential for K\"ahler moduli. We describe how the relevant geometries in general contain cycles of arithmetic genus $\chi \geq 1$ (and how $\chi > 1$ divisors can contribute to the superpotential, in the presence of flux). This second class of effects is likely to be important in finding even larger classes of models where the KKLT mechanism of moduli stabilization can be realized. We also address various claims about the situation for IIB models with a single K\"ahler modulus.Comment: 24 pages, harvmac, no figures, references adde