Wilson loops in nonperturbative quantum gravity

A bstract We study how meaningful physical predictions can arise in nonperturbative quantum gravity in a closed Lorentzian universe. In such settings, recent developments suggest that the quantum gravitational Hilbert space is one-dimensional and real for each α -sector, as induced by spacetime wormholes. This appears to obstruct the conventional quantum-mechanical prescription of assigning probabilities via projection onto a basis of states. While previous approaches have introduced external observers or augmented the theory to resolve this issue, we argue that quantum gravity itself contains all the necessary ingredients to make physical predictions. We demonstrate that the emergence of classical observables and probabilistic outcomes can be understood as a consequence of partial observability: physical observers access only a subsystem of the universe. Tracing out the inaccessible degrees of freedom yields reduced density matrices that encode classical information, with uncertainties exponentially suppressed by the environment’s entropy. We develop this perspective using both the Lorentzian path integral and operator formalisms and support it with a simple microscopic model. Our results show that quantum gravity in a closed universe naturally gives rise to meaningful, robust predictions without recourse to external constructs.

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We formulate the string field theory in zero-dimensional target space corresponding to the two-dimensional quantum gravity theory defined through Causal Dynamical Triangulations. This third quantization of the quantum gravity theory allows us in principle to calculate the transition amplitudes of processes in which the topology of space changes in time, and to include non-trivial topologies of space-time. We formulate the corresponding Dyson-Schwinger equations and illustrate how they can be solved iteratively.

This chapter is the first chapter of the book dedicated to the study of the combinatorics of various quantum gravity approaches. After a brief introductory section to quantum gravity, we shortly mention the main candidates for a quantum theory of gravity: string theory, loop quantum gravity, and group field theory (GFT), causal dynamical triangulations, matrix models. The next sections introduce some GFT models such as the Boulatov model, the colourable and the multi-orientable model. The saddle point method for some specific GFT Feynman integrals is presented in the fifth section. Finally, some algebraic combinatorics results are presented: definition of an appropriate Conne–Kreimer Hopf algebra describing the combinatorics of the renormalization of a certain tensor GFT model (the so-called Ben Geloun–Rivasseau model) and the use of its Hochschild cohomology for the study of the combinatorial Dyson–Schwinger equation of this specific model.

The conference Conceptual and Technical Challenges for Quantum Gravity at Sapienza University of Rome, from 8 to 12 September 2014, has provided a beautiful opportunity for an encounterbetween different approaches and different perspectives on the quantum-gravity problem. It contributedto a higher level of shared knowledge among the quantum-gravity communities pursuing each specific research program.There were plenary talks on many different approaches, including in particular string theory, loop quantum gravity, spacetime noncommutativity, causal dynamical triangulations, asymptotic safety and causal sets. Contributions from the perspective of philosophy of science were also welcomed.In addition several parallel sessions were organized. The present volume collects contributions from the Noncommutative Geometry and Quantum Gravity parallel session4, with additional invited contributions from specialists in the field.Noncommutative geometry in its many incarnations appears at the crossroad of many researches in theoretical and mathematical physics:• from models of quantum space-time (with or without breaking of Lorentz symmetry) to loop gravity and string theory,• from early considerations on UV-divergencies in quantum field theory to recent models of gauge theories on noncommutative spacetime,• from Connes description of the standard model of elementary particles to recent Pati-Salam like extensions.This volume provides an overview of these various topics, interesting for the specialist as well as accessibleto the newcomer. 4partially funded by CNRS PEPS /PTI ‘‘Metric aspect of noncommutative geometry: from Monge to Higgs’’

We review some recent results from the causal dynamical triangulation (CDT) approach to quantum gravity. We review recent observations of dimensional reduction at a number of previously undetermined points in the parameter space of CDT, and discuss their possible relevance to the asymptotic safety scenario. We also present an updated phase diagram of CDT, discussing properties of a newly discovered phase and its possible relation to a signature change of the metric.

We study the supersymmetric circular Wilson loops of N=4 super Yang-Mills in large representations of the gauge group. In particular, we obtain the spectral curves of the matrix model which captures the expectation value of the loops. These spectral curves are then proven to be precisely the hyperelliptic surfaces that characterize the bubbling solutions dual to the Wilson loops, thus yielding an example of a geometry emerging from an eigenvalue distribution. We finally discuss the Wilson loop expectation value from the matrix model and from supergravity.

Shaken, but not stirred—Potts model coupled to quantum gravity

One of the problems which must be overcome before a nonperturbative theory of quantum gravity can be formulated is the unboundedness of the Euclidean action. The author presents a brief outline of the density of states reconstruction method of analysing data from Monte Carlo simulations and then proposes a way to use this method to overcome the unboundedness problem.

In this thesis we present two geometric theories designed to extend general relativity. It can be seen as one of the aims of such theories to model the observed accelerating expansion of the universe as a gravitational phenomenon, or to provide a mathematical structure for the formulation of quantum field theories on curved spacetimes and quantum gravity. This thesis splits into two parts: In the first part we consider multimetric gravity theories containing N > 1 standard model copies which interact only gravitationally and repel each other in the Newtonian limit. The dynamics of each of the standard model copies is governed by its own metric tensor. We show that the antisymmetric case, in which the mutual repulsion between the different matter sectors is of equal strength compared to the attractive gravitational force within each sector, is prohibited by a no-go theorem for N = 2. We further show that this theorem does not hold for N > 2 by explicitly constructing an antisymmetric multimetric repulsive gravity theory. We then examine several properties of this theory. Most notably, we derive a simple cosmological model and show that the accelerating expansion of the late universe can indeed be explained by the mutual repulsion between the different matter sectors. We further present a simple model for structure formation and show that our model leads to the formation of filament-like structures and voids. Finally, we show that multimetric repulsive gravity is compatible with high-precision solar system data using the parametrized post-Newtonian formalism. In the second part of the thesis we propose a mathematical model of quantum spacetime as an infinite-dimensional manifold locally homeomorphic to an appropriate Schwartz space. This extends and unifies both the standard function space construction of quantum mechanics and the differentiable manifold structure of classical spacetime. In this picture we demonstrate that classical spacetime emerges as a finite-dimensional manifold through the topological identification of all quantum points with identical position expectation value. We speculate on the possible relevance of this geometry to quantum field theory and gravity.

It is shown that one can formulate Euclidean two-dimensional quantum gravity as the scaling limit of an ordinary statistical system. Scaling relations can be derived and the critical exponents have simple geometric interpretations. In addition it is possible to calculate reparametrization-invariant correlation functions as functions of the geodesic distance. The definitions for pure gravity generalize to matter systems coupled to two-dimensional quantum gravity. As long as the central charge c of the matter field is less than 1, it is still possible to solve the coupled system explicitly. For mean-field theory is reliable, and again we can solve the mean-field equations using the discretized approach. Finally, the discretized approach makes sense even in higher-dimensional spacetime. Although analytic solutions are still missing in the higher-dimensional case, numerical studies reveal an interesting structure and allow the identification of a fixed point where we can hope to define a genuine non-perturbative theory of four-dimensional quantum gravity.

We investigate the supersymmetric Wilson loops in $d=3$ $\mathcal{N}=4$ super Chern-Simons-matter theory obtained from non-chiral orbifold of ABJM theory. We work in both Minkowski spacetime and Euclidean space, and we construct 1/4 and 1/2 BPS Wilson loops. We also provide a complete proof that the difference between 1/4 and 1/2 Wilson loops is $Q$-exact with $Q$ being some supercharge that is preserved by both the 1/4 and 1/2 Wilson loops. This plays an important role in applying the localization techniques to compute the vacuum expectation values of Wilson loops. We also study the M-theory dual of the 1/2 BPS circular Wilson loop.

We introduce a new methodology to characterize properties of quantum spacetime in a strongly quantum-fluctuating regime, using tools from topological data analysis. Starting from a microscopic quantum geometry, generated nonperturbatively in terms of dynamical triangulations (DT), we compute the Betti numbers of a sequence of coarse-grained versions of the geometry as a function of the coarse-graining scale, yielding a characteristic “topological finger print”. We successfully implement this methodology in Lorentzian and Euclidean 2D quantum gravity, defined via lattice quantum gravity based on causal and Euclidean DT, yielding different results. Effective topology also enables us to formulate necessary conditions for the recovery of spacetime symmetries in a classical limit.

Partonic distributions for large x and renormalization of Wilson loop

We study Wilson loops in N=4 SYM theory which are non-constant in the scalar (S5) directions and open string solutions associated with them in the context of AdS/CFT correspondence. An interplay between Minkowskian and Euclidean pictures turns out to be non-trivial for time-dependent Wilson loops. We find that in the S5-rotating case there appears to be no direct open-string duals for the Minkowskian Wilson loops, and their expectation values should be obtained by analytic continuation from the Euclidean-space results. In the Euclidean case, we determine the dependence of the ``quark - anti-quark'' potential on the rotation parameter, both at weak coupling (i.e. in the 1-loop perturbative SYM theory) and at strong coupling (i.e. in the classical string theory in AdS5 x S5).