Lectures in Quantum Gravity

Abstract

Formulating a quantum theory of gravity lies at the heart of fundamental theoretical physics. This collection of lecture notes encompasses a selection of topics that were covered in six mini-courses at the Nordita PhD school "Towards Quantum Gravity". The scope was to provide a coherent picture, from its foundation to forefront research, emphasizing connections between different areas. The lectures begin with perturbative quantum gravity and effective field theory. Subsequently, two ultraviolet-complete approaches are presented: asymptotically safe gravity and string theory. Finally, elements of quantum effects in black hole spacetimes are discussed.

arXiv commentLecture notes PhD school "Towards Quantum Gravity", Nordita Scientific Program "Quantum Gravity: from gravitational EFTs to UV complete approaches". 265 pages + references, 37 figures - V2: typos fixed and references added - V3: matches published version

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• Holographic Derivation of Entanglement Entropy from AdS/CFT
• The World as a Hologram
• String Theory and Noncommutative Geometry
• de Sitter Vacua in String Theory
• Microscopic Origin of the Bekenstein-Hawking Entropy
• M Theory As A Matrix Model: A Conjecture
• Planck 2018 results. X. Constraints on inflation
• Exact evolution equation for the effective potential
• String Theory Dynamics In Various Dimensions
• The galileon as a local modification of gravity
• Resummation of Massive Gravity
• The String Landscape, Black Holes and Gravity as the Weakest Force
• General relativity as an effective field theory: The leading quantum corrections
• On the Geometry of the String Landscape and the Swampland
• Generalized gravitational entropy
• Information in Black Hole Radiation
• Nonperturbative Evolution Equation for Quantum Gravity
• Supergravity description of field theories on curved manifolds and a no go theorem
• A Large-N Reduced Model as Superstring
• Generalization of the Fierz-Pauli Action
• The Stretched Horizon and Black Hole Complementarity
• The Exact Renormalisation Group and Approximate Solutions
• Causality, Analyticity and an IR Obstruction to UV Completion
• Heat kernel expansion: user's manual
• Replica Wormholes and the Entropy of Hawking Radiation
• The fuzzball proposal for black holes: an elementary review
• Symmetries and Strings in Field Theory and Gravity
• The Swampland: Introduction and Review
• Aspects of the Functional Renormalisation Group
• A Spin Entanglement Witness for Quantum Gravity
• Isolated and dynamical horizons and their applications
• On Renormalization Group Flows in Four Dimensions
• The entropy of bulk quantum fields and the entanglement wedge of an evaporating black hole
• Optimised Renormalisation Group Flows
• The Page curve of Hawking radiation from semiclassical geometry
• Soft Hair on Black Holes
• The Thermodynamics of Black Holes
• Gravitationally-induced entanglement between two massive particles is sufficient evidence of quantum effects in gravity
• Quantum Gravity in Everyday Life: General Relativity as an Effective Field Theory
• The entropy of Hawking radiation
• Matrix String Theory
• Logarithmic correction to the Bekenstein-Hawking entropy
• A Correspondence Principle for Black Holes and Strings
• Dimensional Reduction in Quantum Gravity
• Leading Quantum Correction to the Newtonian Potential
• Higgs inflation: consistency and generalisations
• The nonperturbative functional renormalization group and its applications
• Twenty Years of the Weyl Anomaly
• Causality Constraints on Corrections to the Graviton Three-Point Coupling
• Optimisation of the exact renormalisation group
• Strings in AdS_3 and the SL(2,R) WZW Model. Part 1: The Spectrum
• Perturbative Quantum Field Theory in the String-Inspired Formalism
• The String Landscape and the Swampland
• Open Strings
• Why is the Matrix Model Correct?
• Geometric Classification of Conformal Anomalies in Arbitrary Dimensions
• Mission design of LiteBIRD
• Black Holes and Large N Species Solution to the Hierarchy Problem
• Asymptotic safety of gravity and the Higgs boson mass
• Quantum Gravitational Corrections to the Nonrelativistic Scattering Potential of Two Masses
• Lectures on the Swampland Program in String Compactifications
• Gauge Invariance in Field Theory and Statistical Physics in Operator Formalism
• Quantum Gravity from Causal Dynamical Triangulations: A Review
• Non-Trivial Ultraviolet Fixed Point in Quantum Gravity
• Introduction to the functional RG and applications to gauge theories
• A Short Review of Loop Quantum Gravity
• Stringy gravity, interacting tensionless strings and massless higher spins
• Agravity
• Black Hole Bound on the Number of Species and Quantum Gravity at LHC
• Flow Equations for N Point Functions and Bound States
• The scattering matrix approach for the quantum black hole, an overview
• Fractal Spacetime Structure in Asymptotically Safe Gravity
• Relative entropy and the Bekenstein bound
• Black Holes in Higher-Derivative Gravity
• Theorems on gravitational time delay and related issues
• Matter matters in asymptotically safe quantum gravity
• Quantum Foam and Topological Strings
• xTras: a field-theory inspired xAct package for Mathematica
• The EFT-Hedron
• Landau gauge Yang-Mills correlation functions
• New Test of the Gravitational $1/r^2$ Law at Separations down to 52 $\mu$m
• A Model for High Energy Scattering in Quantum Gravity
• Asymptotic safety guaranteed
• Asymptotically Safe Lorentzian Gravity
• Some remarks on high derivative quantum gravity
• Further evidence for asymptotic safety of quantum gravity
• xPerm: fast index canonicalization for tensor computer algebra
• Constraints on symmetry from holography
• Energy's and amplitudes' positivity
• xPert: Computer algebra for metric perturbation theory
• The causal set approach to quantum gravity
• Evidence for a Lattice Weak Gravity Conjecture
• The S-matrix Bootstrap I: QFT in AdS
• UV-Completion by Classicalization
• New positivity bounds from full crossing symmetry
• Renormalization group improved gravitational actions: a Brans-Dicke approach
• Positivity Bounds for Scalar Theories
• Information loss problem and a `black hole' model with a closed apparent horizon
• Weak Gravity Conjecture from Unitarity and Causality
• The Weak Gravity Conjecture in three dimensions
• Extremal Effective Field Theories
• The low energy expansion of the one-loop type II superstring amplitude
• Quadratic Gravity
• Beyond Amplitudes' Positivity and the Fate of Massive Gravity
• Asymptotic safety of gravity-matter systems
• Softness and Amplitudes' Positivity for Spinning Particles
• Is There Really a de Sitter/CFT Duality
• Global aspects of the space of 6D N = 1 supergravities
• The Weak Gravity Conjecture: A Review
• Local Quantum Gravity
• Gravitational Effective Field Theory Islands, Low-Spin Dominance, and the Four-Graviton Amplitude
• The Gravitational Two-Loop Counterterm is Asymptotically Safe
• Towards apparent convergence in asymptotically safe quantum gravity
• Ultraviolet Properties of N = 8 Supergravity at Five Loops
• Quantum Break-Time of de Sitter
• UV complete me: Positivity Bounds for Particles with Spin
• Top mass from asymptotic safety
• Asymptotically Safe Inflation
• Unitarity, stability and loops of unstable ghosts
• One force to rule them all: asymptotic safety of gravity with matter
• The Dark Dimension and the Swampland
• Quantum gravity on foliated spacetime - asymptotically safe and sound
• Free partition functions and an averaged holographic duality
• Asymptotic safety: a simple example
• A Critique of the Asymptotic Safety Program
• Large-N bounds on, and compositeness limit of, gauge and gravitational interactions
• Renormalization group fixed points of foliated gravity-matter systems
• Information Preservation and Weather Forecasting for Black Holes
• Generalized Parametrization Dependence in Quantum Gravity
• Consistency of the Standard Model Effective Field Theory
• M-theory observables for cosmological space-times
• Black Hole Formation and Classicalization in Ultra-Planckian 2 -> N Scattering
• Minimal conditions for the existence of a Hawking-like flux
• Quantum Gravity effects near the null black hole singularity
• Fakeons And Lee-Wick Models
• An asymptotically safe solution to the U(1) triviality problem
• The gravitational S-matrix
• Quantum-gravity effects on a Higgs-Yukawa model
• Form Factors in Asymptotic Safety: conceptual ideas and computational toolbox
• On the quantum field theory of the gravitational interactions
• Causality as an emergent macroscopic phenomenon: The Lee-Wick O(N) model
• The Invar Tensor Package
• Non-minimal coupling in Higgs-Yukawa model with asymptotically safe gravity
• Massive Galileon Positivity Bounds
• Upper bound on the Abelian gauge coupling from asymptotic safety
• Automorphic properties of low energy string amplitudes in various dimensions
• Curvature dependence of quantum gravity
• Causality constraints on corrections to Einstein gravity
• Superrenormalizable gauge and gravitational theories
• Where is String Theory?
• Observations of Hawking radiation: the Page curve and baby universes
• Hawking-like radiation from evolving black holes and compact horizonless objects
• Positivity bounds on vector boson scattering at the LHC
• Strings from Massive Higher Spins: The Asymptotic Uniqueness of the Veneziano Amplitude
• Constraining de Sitter Space in String Theory
• Nonviolent nonlocality
• Asymptotic safety in an interacting system of gravity and scalar matter
• Bootstrapping QCD: the Lake, the Peninsula and the Kink
• String universality in ten dimensions
• Quantum radiation from an evaporating non-singular black hole
• Quantum Gravity, Fakeons And Microcausality
• Cobordism Conjecture, Anomalies, and the String Lamppost Principle
• How perturbative is quantum gravity?
• The Universal RG Machine
• Convex Geometry Perspective to the (Standard Model) Effective Field Theory Space
• Higgs scalar potential in asymptotically safe quantum gravity
• Scale of Quantum Gravity
• Quantum Horizons of the Standard Model Landscape
• Quantum mechanics of 4-derivative theories
• Branes and the Swampland
• The Ultraviolet Behavior Of Quantum Gravity
• Positivity Bounds and the Massless Spin-2 Pole
• Positive Signs in Massive Gravity
• Asymptotic safety of higher derivative quantum gravity non-minimally coupled with a matter system
• Into the EFThedron and UV constraints from IR consistency
• Effective universality in quantum gravity
• Zooming in on fermions and quantum gravity
• Non-perturbative unitarity and fictitious ghosts in quantum gravity
• QED positivity bounds
• On the running of the gravitational constant
• Gravitational positivity bounds
• The arrow of causality and quantum gravity
• Asymptotically safe inflation from quadratic gravity
• Renormalisation group flows for gauge theories in axial gauges
• Bare Action and Regularized Functional Integral of Asymptotically Safe Quantum Gravity
• Quantum Gravitational Corrections to the Entropy of a Schwarzschild Black Hole
• Finite Quantum Gravity Amplitudes -- no strings attached
• From renormalization group flows to cosmology
• Inflationary Perturbations in No-Scale Theories
• Correlation functions on a curved background
• Nonminimal hints for asymptotic safety
• Lattice Quantum Gravity and Asymptotic Safety
• Non-supersymmetric heterotic model building
• Black Hole Gravitational Waves in the Effective Field Theory of Gravity
• Partition functions of the tensionless string
• Field Parametrization Dependence in Asymptotically Safe Quantum Gravity
• Effective Field Theory of Gravity to All Orders
• Positivity Constraints for Pseudo-linear Massive Spin-2 and Vector Galileons
• Dispersion Relation Bounds for pi pi Scattering
• The Emergence Proposal in Quantum Gravity and the Species Scale
• Extremal Kerr black holes as amplifiers of new physics
• IR/UV Mixing, Towers of Species and Swampland Conjectures
• Gravitational interaction to one loop in effective quantum gravity
• Planck-Suppressed Operators
• On Classical Stability with Broken Supersymmetry
• Flavor Constraints from Unitarity and Analyticity
• On sigma model RG flow, "central charge" action and Perelman's entropy
• Entropy Bound and Unitarity of Scattering Amplitudes
• Crossing Symmetric Spinning S-matrix Bootstrap: EFT bounds
• EPFL Lectures on General Relativity as a Quantum Field Theory
• Direct Production of Lightest Regge Resonances
• On the fate of singularities and horizons in higher derivative gravity
• On the non-local heat kernel expansion
• Scattering amplitudes in super-renormalizable gravity
• Connections and geodesics in the space of metrics
• Natural Selection Rules: New Positivity Bounds for Massive Spinning Particles
• High-energy gravitational scattering and black hole resonances
• Baby Universes, Holography, and the Swampland
• Stringy AdS3 from the Worldsheet
• Solving functional flow equations with pseudo-spectral methods
• Solutions to the reconstruction problem in asymptotic safety
• Cobordism Classes and the Swampland
• Fakeons, Microcausality And The Classical Limit Of Quantum Gravity
• Lorentz invariance violations in the interplay of quantum gravity with matter
• Asymmetric Orbifolds, Noncommutative Geometry and Type I String Vacua
• Lorentz symmetry is relevant
• Compactness of Brane Moduli and the String Lamppost Principle in $d>6$
• Holography and the KKLT Scenario
• Covariant Perturbation Theory (IV). Third Order in the Curvature
• Essential Quantum Einstein Gravity
• Massive Higher Spins: Effective Theory and Consistency
• Inflation in an effective gravitational model & asymptotic safety
• Massless Positivity in Graviton Exchange
• Cosmological bounds on the field content of asymptotically safe gravity-matter models
• White Paper and Roadmap for Quantum Gravity Phenomenology in the Multi-Messenger Era
• On the physical mechanism underlying Asymptotic Safety
• de Sitter in non-supersymmetric string theories: no-go theorems and brane-worlds
• Positivity Constraints on Interacting Spin-2 Fields
• A Bound on Massive Higher Spin Particles
• Asymptotic Safety, Emergence and Minimal Length
• Predictions of quantum gravity in inflationary cosmology: effects of the Weyl-squared term
• The End of a Black Hole's Evaporation -- Part I
• Brane annihilation in non-supersymmetric strings
• Causal Effective Field Theories
• Asymptotic safety, string theory and the weak gravity conjecture
• Elastic positivity vs extremal positivity bounds in SMEFT: a case study in transversal electroweak gauge-boson scatterings
• A Quadrillion Standard Models from F-theory
• Characterizing black hole metrics in quadratic gravity
• Wald's gravitational entropy for ghost-free, infinite derivative theories of Gravity
• Gravitational Regge bounds
• Two-Point String Amplitudes
• On Unitarity of Tree-Level String Amplitudes
• Bootstrapping Massive Quantum Field Theories
• The Asymptotically Safe Standard Model: From quantum gravity to dynamical chiral symmetry breaking
• More asymptotic safety guaranteed
• On Calculation of $2+\veps$ RG Functions in the Gross Neveu Model from Large $N$ Expansions of Critical Exponents
• Primordial Entropy Production and Lambda-driven Inflation from Quantum Einstein Gravity
• Causality Violation, Gravitational Shockwaves and UV Completion
• Positivity in electron-positron scattering: testing the axiomatic quantum field theory principles and probing the existence of UV states
• Essential renormalisation group
• Topological Superconductors on Superstring Worldsheets
• Thermality of the Hawking flux
• Veneziano Variations: How Unique are String Amplitudes?
• Asymmetric Orientifolds, Brane Supersymmetry Breaking and Non-BPS Branes
• Asymptotic Safety: Swampland or Wonderland?
• Gravitational Causality and the Self-Stress of Photons
• Causal vs. Analytic constraints on anomalous quartic gauge couplings
• The spatial Functional Renormalization Group and Hadamard states on cosmological spacetimes
• Fine Tuning, Sequestering, and the Swampland
• Massive Gravity is not Positive
• Generalized Veneziano and Virasoro amplitudes
• Minimal Black Holes and Species Thermodynamics
• Shedding black hole light on the emergent string conjecture
• Stringy Dynamics from an Amplitudes Bootstrap
• String Tension between de Sitter vacua and Curvature Corrections
• Graviton-Mediated Scattering Amplitudes from the Quantum Effective Action
• A note on the representations of $\text{SO}(1,d+1)$
• Taming the Landscape of Effective Theories
• The covariant perturbative string spectrum
• Bespoke Dual Resonance
• The Stringy S-matrix Bootstrap: Maximal Spin and Superpolynomial Softness
• Interpretation of quantum mechanics with indefinite norm
• Lectures on the string landscape and the Swampland
• Fixed Points of Quantum Gravity and the Dimensionality of the UV Critical Surface
• Higgs vacuum (in)stability during inflation: the dangerous relevance of de Sitter departure and Planck-suppressed operators
• Positivity Constraints on Interacting Pseudo-Linear Spin-2 Fields
• Einstein gravity from the $\mathbb{N}=4$ spinning particle
• Effective theory for quantum gravity
• Functional renormalization group and 2PI effective action formalism
• Where is M-theory in the space of scattering amplitudes?
• Black Holes in Asymptotically Safe Gravity
• UV properties of Galileons: Spectral Densities
• Multiparticle Factorization and the Rigidity of String Theory
• Formation and evaporation of quantum black holes from the decoupling mechanism in quantum gravity
• UV graviton scattering and positivity bounds from IR dispersion relations
• Lorentz-violating vs ghost gravitons: the example of Weyl gravity
• Properties Of The Classical Action Of Quantum Gravity
• Graviton propagator, renormalization scale and black-hole like states
• Generalized elastic positivity bounds on interacting massive spin-2 theories
• Cosmological ${\alpha }^{\prime }$-corrections from the functional renormalization group
• Sudden breakdown of effective field theory near cool Kerr-Newman black holes
• UV-Complete Gravity Amplitudes and the Triple Product
• BICEP/Keck data and Quadratic Gravity
• Inflation near a metastable de Sitter vacuum from moduli stabilisation
• Non-commutative Geometry from Perturbative Quantum Gravity
• Entropy Production during Asymptotically Safe Inflation
• Properties of infinite product amplitudes: Veneziano, Virasoro, and Coon
• Functional Renormalisation for $f(R_{\mu \nu \rho \sigma })$ Quantum Gravity
• A Worldline Theory for Supergravity
• Form Factors in Quantum Gravity - contrasting non-local, ghost-free gravity and Asymptotic Safety
• Species scale, worldsheet CFTs and emergent geometry
• Aspects of perturbative unitarity
• On the Origin of Species Thermodynamics and the Black Hole - Tower Correspondence
• Bootstrapping the $a$-anomaly in $4d$ QFTs
• Evaluating one-loop string amplitudes
• Topological modular forms and the absence of all heterotic global anomalies
• An algebraic QFT approach to the Wetterich equation on Lorentzian manifolds
• Stable Vacua for Tachyonic Strings
• Bootstrap Principle for the Spectrum and Scattering of Strings
• UV Completion on the Worldline
• Off-diagonal heat-kernel expansion and its application to fields with differential constraints
• Finite energy sum rules for gravitational Regge amplitudes
• Bounds on photon scattering
• Asymptotically Safe Non-Minimal Inflation
• Effective action and black hole solutions in asymptotically safe quantum gravity
• Running couplings and unitarity in a 4-derivative scalar field theory
• Snowmass White Paper: the Double Copy and its Applications
• Anomaly-free scale symmetry and gravity
• Exact quantum conformal symmetry, its spontaneous breakdown, and gravitational Weyl anomaly
• A new quantization principle from a minimally non time-ordered product
• Bootstrap and Amplitudes: A Hike in the Landscape of Quantum Field Theory
• Extension of positivity bounds to non-local theories: IR obstructions to Lorentz invariant UV completions
• Positivity Bounds on Massive Vectors
• Emergence of $R^4$-terms in M-theory
• On effective models of regular black holes inspired by higher-derivative and nonlocal gravity
• Proving the Weak Gravity Conjecture in Perturbative String Theory, Part I: The Bosonic String
• Asymptotic freedom in (3+1)-dimensional projectable Horava gravity: connecting ultraviolet to infrared
• Configuration space for quantum gravity in a locally regularized path integral
• Uniqueness Criteria for the Virasoro-Shapiro Amplitude
• (Non)-projective bounds on gravitational EFT
• Hint to Supersymmetry from GR Vacuum
• On amplitudes, resonances and the ultraviolet completion of gravity
• On the Running of Gauge Couplings in String Theory
• Noncommutativity and Physics: A non-technical review
• Tachyons and Misaligned Supersymmetry in Closed String Vacua
• Fully stabilized Minkowski vacua in the $2^6$ Landau-Ginzburg model
• Tadpole conjecture in non-geometric backgrounds
• Infinite Distances and Factorization
• Making sense of ghosts
• Anomalies and the Green-Schwarz Mechanism
• Snowmass White Paper: S-matrix Bootstrap
• Lorentzian contours for tree-level string amplitudes
• Conformal Invariance of (0,2) Sigma Models on Calabi-Yau Manifolds
• The Unruh vacuum and the in-vacuum in the Reissner-Nordström spacetime
• Asymptotic safety, quantum gravity, and the swampland: a conceptual assessment
• A Lorentzian renormalisation group equation for gauge theories
• Safe essential scalar-tensor theories
• Background independence and field redefinitions in quantum gravity
• Angular momentum content of a virtual graviton
• A Relation between One-Loop Amplitudes of Closed and Open Strings (One-Loop KLT Relation)
• Swamplandish Unification of the Dark Sector
• Dust collapse and horizon formation in Quadratic Gravity
• High energy scattering and string/black hole transition
• The disinformation problem for black holes (pop version)
• Power spectrum oscillations from Planck-suppressed operators in effective field theory motivated monodromy inflation
• Stringy Forces in the Black Hole Interior
• Free Fermionic Constructions of Heterotic Strings
• All Loop Scattering As A Counting Problem
• Trustworthy de Sitter compactifications of string theory: a comprehensive review
• Application of positivity bounds in asymptotically safe gravity
• Les Houches lectures on non-perturbative topological strings
• On the asymptotic density of states in solvable models of strings
• New Non-Supersymmetric Tachyon-Free Strings
• Physics-informed renormalisation group flows
• Flowing fields and optimal RG-flows
• Robustness of the derivative expansion in Asymptotic Safety
• Scale and Conformal Invariance in 2d Sigma Models, with an Application to N=4 Supersymmetry
• "Observables" in de Sitter Quantum Gravity: in Perturbation Theory and Beyond
• Cartographing gravity-mediated scattering amplitudes: scalars and photons
• Analyticity Constraints and Trace Anomaly Matching
• Do ${\Lambda }_{CC}$ and $G$ run?
• Asymptotic Safety within on-shell perturbation theory
• Graviton loops and negativity
• Running EFT-hedron with null constraints at loop level

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