TASI Lectures on the Conformal Bootstrap

Abstract

These notes are from courses given at TASI and the Advanced Strings School in summer 2015. Starting from principles of quantum field theory and the assumption of a traceless stress tensor, we develop the basics of conformal field theory, including conformal Ward identities, radial quantization, reflection positivity, the operator product expansion, and conformal blocks. We end with an introduction to numerical bootstrap methods, focusing on the 2d and 3d Ising models.

65 pages + appendices and references

Cited by in corpus (69)

• EPFL Lectures on Conformal Field Theory in D>= 3 Dimensions
• A Mellin space approach to the conformal bootstrap
• Analytic Bounds and Emergence of ${\text{AdS}}_2$ Physics from the Conformal Bootstrap
• Non-perturbative methodologies for low-dimensional strongly-correlated systems: From non-abelian bosonization to truncated spectrum methods
• The Epsilon Expansion Meets Semiclassics
• Soft Bootstrap and Supersymmetry
• Anomalies and Bounds on Charged Operators
• The Inflationary Wavefunction from Analyticity and Factorization
• Crossing Symmetry in Alpha Space
• Universality at large transverse spin in defect CFT
• TASI lectures on AdS/CFT
• Bulk Phase Shift, CFT Regge Limit and Einstein Gravity
• Evolution operators in conformal field theories and conformal mappings: the entanglement Hamiltonian, the sine-square deformation, and others
• The critical O(N) CFT: Methods and conformal data
• Momentum space approach to crossing symmetric CFT correlators
• The BMS Bootstrap
• Analytic Euclidean Bootstrap
• ETH and Modular Invariance of 2D CFTs
• Colour-dressed hexagon tessellations for correlation functions and non-planar corrections
• Integrability and Conformal Bootstrap: One Dimensional Defect CFT
• Bootstrapping Mixed Correlators in the Five Dimensional Critical O(N) Models
• Dimensional Reduction for Conformal Blocks
• Conformal correlators as simplex integrals in momentum space
• Feynman diagrams and the large charge expansion in $3-\epsilon$ dimensions
• Decoding a Three-Dimensional Conformal Manifold
• Mass Generation, the Cosmological Constant Problem, Conformal Symmetry, and the Higgs Boson
• Carving out the Space of Open-String S-matrix
• Floquet conformal field theories with generally deformed Hamiltonians
• String Field Theory -- A Modern Introduction
• The nuts and bolts of the BMS Bootstrap
• Momentum space conformal three-point functions of conserved currents and a general spinning operator
• Notes on flat-space limit of AdS/CFT
• Anomalous Symmetries End at the Boundary
• Navigator Function for the Conformal Bootstrap
• The Cosmological Bootstrap: Spinning Correlators from Symmetries and Factorization
• On the Positive Geometry of Conformal Field Theory
• Anomalous dimensions at finite conformal spin from OPE inversion
• Degenerate Operators and the $1/c$ Expansion: Lorentzian Resummations, High Order Computations, and Super-Virasoro Blocks
• Helicity basis for three-dimensional conformal field theory
• Dispersion Formulas in QFTs, CFTs, and Holography
• Upper bound on the mass anomalous dimension in many-flavor gauge theories: a conformal bootstrap approach
• The unitary subsector of generalized minimal models
• Transition between algebraic and ${\mathbb{Z}}_2$ quantum spin liquids at large $N$
• Bulk Locality in AdS/CFT Correspondence
• On Local and Integrated Stress-Tensor Commutators
• Conformal Correlation functions in four dimensions from Quaternionic Lauricella system
• Superconformal Blocks for Mixed 1/2-BPS Correlators with $SU(2)$ R-symmetry
• Conformal partial waves in momentum space
• Exact Diagonalization, Matrix Product States and Conformal Perturbation Theory Study of a 3D Ising Fuzzy Sphere Model
• 2d small N=4 Long-multiplet superconformal block
• The Analytic Bootstrap in Fermionic CFTs
• The four-point correlation function of the energy-momentum tensor in the free conformal field theory of a scalar field
• More on the operator-state map in non-relativistic CFTs
• Determinant Formula for Parabolic Verma Modules of Lie Superalgebras
• RG flows on $S^d$ and Hamiltonian truncation
• Classification of Convergent OPE Channels for Lorentzian CFT Four-Point Functions
• Casimir Forces in CFT with Defects and Boundaries
• Mellin Bootstrap for Scalars in Generic Dimension
• Numerical estimation of structure constants in the 3d Ising CFT through Markov chain UV sampler
• Quantum Groups as Global Symmetries II. Coulomb Gas Construction
• CFTs on curved spaces
• Lectures on chiral algebras of $\mathcal{N}⩾2$ superconformal field theories
• Simple Bulk Reconstruction in AdS/CFT Correspondence
• Bootstrapping mixed correlators in the 2D Ising model
• Stability of $SU(N_c)$ QCD3 from the $\epsilon$-Expansion
• Operator mixing, UV asymptotics of nonplanar/planar $2$-point correlators, and nonperturbative large-$N$ expansion of QCD-like theories
• Virasoro conformal bootstrap with $c>1$
• Crossing, Modular Averages and $N\leftrightarrow k$ in WZW Models
• PCT Theorem, Wightman Axioms and Conformal Bootstrap