We give a pedagogical introduction to the basic ideas and concepts of the Asymptotic Safety program in quantum Einstein gravity. Using the continuum approach based upon the effective average action, we summarize the state of the art of the field with a focus on the evidence supporting the existence of the non-trivial renormalization group fixed point at the heart of the construction. As an application, the multifractal structure of the emerging space-times is discussed in detail. In particular, we compare the continuum prediction for their spectral dimension with Monte Carlo data from the causal dynamical triangulation approach.

Quantum Einstein gravity

It is well-known that any 4-dimensional hyperkahler metric with two commuting Killing fields may be obtained explicitly, via the Gibbons-Hawking Ansatz, from a harmonic function invariant under a Killing field on • 3. In this paper, we find all selfdual Einstein metrics of nonzero scalar curvature with two commuting Killing fields. They are given explicitly in terms of a local eigenfunction of the Laplacian on the hyperbolic plane. We discuss the relation of this construction to a class of selfdual spaces found by Joyce, and some Einstein-Weyl spaces found by Ward, and then show that certain 'multipole' hyperbolic eigenfunctions yield explicit formulae for the quaternion-kahler quotients of • Pm—1 by an (m — 2)-torus studied by Galicki and Lawson. As a consequence we are able to place the well-known cohomogeneity one metrics, the quaternion-kahler quotients of • P2 (and noncompact analogues), and the more recently studied selfdual Einstein Hermitian metrics in a unified framework, and give new complete examples.

/pdf/selfdual-einstein-metrics-with-torus-symmetry-8omn2p3tns.pdf

Selfdual Einstein Metrics with Torus Symmetry

It is well known that any 4-dimensional hyperkahler metric with two commuting Killing fields may be obtained explicitly, via the Gibbons-Hawking Ansatz, from a harmonic function invariant under a Killing field on R^3. In this paper, we find all selfdual Einstein metrics of nonzero scalar curvature with two commuting Killing fields. They are given explicitly in terms of a local eigenfunction of the Laplacian on the hyperbolic plane. We discuss the relation of this construction to a class of selfdual spaces found by Joyce, and some Einstein-Weyl spaces found by Ward, and then show that certain `multipole' hyperbolic eigenfunctions yield explicit formulae for the quaternion-kahler quotients of HP(m-1) by an (m-2)-torus first studied by Galicki and Lawson. As a consequence we are able to place the well-known cohomogeneity one metrics, the quaternion-kahler quotients of HP(2) (and noncompact analogues), and the more recently studied selfdual Einstein hermitian metrics in a unified framework, and give new complete examples.

Selfdual Einstein metrics with torus symmetry

We give a brief review of the quantum Hall effect in higher dimensions and its relation to fuzzy spaces. For a quantum Hall system, the lowest Landau level dynamics is given by a one-dimensional matrix action whose large N limit produces an effective action describing the gauge interactions of a higher dimensional quantum Hall droplet. The bulk action is a Chern–Simons type term whose anomaly is exactly cancelled by the boundary action given in terms of a chiral, gauged Wess–Zumino–Witten theory suitably generalized to higher dimensions. We argue that the gauge fields in the Chern–Simons action can be understood as parametrizing the different ways in which the large N limit of the matrix theory is taken. The possible relevance of these ideas to fuzzy gravity is explained. Other applications are also briefly discussed.

https://iopscience.iop.org/article/10.1088/0305-4470/39/41/S05/pdf

Quantum Hall effect in higher dimensions, matrix models and fuzzy geometry

We determine the one-instanton corrections to the universal hypermultiplet moduli space coming both from Euclidean membranes and NS-fivebranes wrapping the cycles of a (rigid) Calabi-Yau threefold. These corrections are completely encoded by a single function characterizing a generic four-dimensional quaternion-Kahler metric without isometries. We give explicit solutions for this function describing all one-instanton corrections, including the fluctuations around the instanton to all orders in the string coupling constant. In the semi-classical limit these results are in perfect agreement with previous supergravity calculations.

https://dspace.library.uu.nl/bitstream/handle/1874/19495/vandoren_06_membrane.pdf?sequence=1

Membrane and fivebrane instantons from quaternionic geometry

U(1)×U(1) quaternionic metrics from harmonic superspace

Four-dimensional Hall mechanics as a particle on CP3

We construct, using harmonic superspace and the quaternionic quotient approach, a quaternionic-K\"ahler extension of the most general two centres hyper-K\"ahler metric. It possesses $U(1)\times U(1)$ isometry, contains as special cases the quaternionic-K\"ahler extensions of the Taub-NUT and Eguchi-Hanson metrics and exhibits an extra one-parameter freedom which disappears in the hyper-K\"ahler limit. Some emphasis is put on the relation between this class of quaternionic-K\"ahler metrics and self-dual Weyl solutions of the coupled Einstein-Maxwell equations. The relation between our explicit results and the recent general ansatz of Calderbank and Pedersen for quaternionic-K\"ahler metrics with $U(1)\times U(1)$ isometries is traced in detail.

Pierre-Yves Casteill

Papers

U(1)×U(1) quaternionic metrics from harmonic superspace

Four-dimensional Hall mechanics as a particle on CP3

U(1) x U(1) Quaternionic Metrics from Harmonic Superspace

Quaternionic extension of the double Taub-NUT metric

Dualised σ-models at the two-loop order