• Asymptotic Safety
• Dilaton
• Gravity
• Quantum Gravity
• renormalization
• Unimodular

Nonrenormalizability of 4-dimensional Gravity is a well-known problem, which in recent years has been faced with many different attempts. In 1979, Steven Weinberg proposed an alternative method instead of the traditional perturbative renormalization approach: asymptotic safety. Asymptotic safety is a generalization of the notion of renormalizability in which an RG-, non-Gaussian, UV-fixed point can allow us to control the high energy behavior of a theory, even if it is not perturbatively renormalizable. Slightly above 2 dimensions, the theory can be analytically continued in dimensionality and we show that an UV fixed point exists, with a finite-dimensional critical surface, which can provide the theory with predictivity. Our computation is applied to the so-called "Unidmodular Dilaton Gravity", which is an equivalent formulation of Einstein-Hilbert one, but with a different implementation of the diffeomorphisms invariance. Unimodular theory seems to be the right framework in where to work because its 2-dimensional limit is continuous (unlike Einstein-Hilbert's) and reproduces the Liouville theory, which is the model considered to study the dynamics of 2-dimensional gravity. This model is studied in its "pure gravity" version and with a cosmological constant operator. Finally, following Weinberg's argument of "dimensional continuation", we try to discuss the implication of our results in 4 dimensions.