Conformal field theory overview

Identify the conformal algebra in d dimensions from the requirement of preserving angles.

Classify finite conformal transformations: translations, rotations, dilatations, and special conformal transformations.

Construct the conformal group as SO(d+1,1) and describe its relation to the Poincaré group.

Define primary operators and their descendants in terms of their transformation properties under the conformal algebra.

Compute the action of conformal generators on primary and descendant operators.

Establish the state-operator correspondence using radial quantization.

Apply Ward identities to constrain the form of correlation functions under conformal transformations.

Determine the universal form of two-point and three-point functions of primary operators.

Express four-point functions in terms of conformal cross-ratios.

Define the Operator Product Expansion (OPE) and relate it to the operator algebra.

Decompose correlation functions into conformal blocks, representing the contribution of an exchanged primary operator and its descendants.

Formulate the conformal bootstrap equations by imposing crossing symmetry on the OPE.

Relate scale invariance at critical points in statistical systems to the emergence of conformal symmetry.

Identify critical exponents with the scaling dimensions of operators in the corresponding CFT.

Describe the renormalization group (RG) flow between CFTs in the space of theories.

Classify operators as relevant, irrelevant, or marginal based on their scaling dimensions and their effect on RG flow.

Describe the principle of universality classes and their classification by CFTs.

Analyze the critical Ising model as a fixed point of the RG flow described by a specific 2D CFT.

Explain the origin of infinite-dimensional conformal symmetry in 2D via holomorphic factorization.

Construct the Virasoro algebra as the central extension of the 2D conformal algebra and interpret the central charge.

Construct Verma modules as highest-weight representations of the Virasoro algebra.

Identify null vectors in Verma modules using the Kac determinant formula.

Classify unitary minimal models based on their central charge and operator content using the Kac table.

Apply null state conditions to derive differential equations for correlation functions.

Compute the Operator Product Expansion of the stress-energy tensor with itself.

Calculate correlation functions in minimal models using null state decoupling.

Define the CFT partition function on a torus and describe its modular transformation properties.

Compute the partition function for a free boson on a torus and verify its modular invariance.

Apply Wick's theorem to compute correlation functions in free bosonic and fermionic CFTs.

Define characters of Virasoro representations and use modular invariance to constrain the operator spectrum of a rational CFT.

Construct affine Lie algebras (Kac-Moody algebras) as central extensions of loop algebras.

Define highest-weight representations of affine Lie algebras and the role of the level k.

Describe the Wess-Zumino-Witten (WZW) action and use it to derive the equations of motion and conserved currents.

Relate the stress tensor to the currents via the Sugawara construction and compute the central charge.

Identify the primary fields of a WZW model with integrable representations of the underlying affine Lie algebra.

State the Knizhnik-Zamolodchikov (KZ) equations for correlation functions in WZW models.

Define supercharges as operators and construct basic on-shell supersymmetry representations (supermultiplets).

Construct the superconformal algebra in 2D (N=1) and its Neveu-Schwarz (NS) and Ramond (R) sectors.

Construct the superconformal algebra in 4D (N=1) and identify its key subalgebras.

Define the BPS bound and the principle of representation shortening in superconformal theories.

Identify BPS operators in simple SCFTs by applying shortening conditions.

Describe spectral flow in 2D SCFTs as an automorphism of the algebra that relates the NS and R sectors.

Express four-point functions in d>2 in terms of a conformal partial wave expansion.

Apply crossing symmetry and unitarity constraints to derive the fundamental bootstrap equations.

Explain how unitarity constrains the spectrum of operator dimensions.

Formulate the conformal bootstrap as a numerical optimization problem to find bounds on CFT data.

Interpret exclusion plots for CFT data (dimensions and OPE coefficients) generated by numerical bootstrap methods.

Describe the key results of the numerical bootstrap for the 3D Ising and O(N) models.

Explain the large N expansion in matrix models and its role in simplifying QFT dynamics.

Describe the holographic principle and how large N theories can lead to classical gravity descriptions.

State the AdS/CFT correspondence, relating a specific large N gauge theory (N=4 SYM) to string theory in AdS space.

Map the global conformal group in d dimensions to the isometry group of AdS_{d+1}.

Use the operator-field correspondence to relate CFT operator dimensions to bulk field masses.

Calculate CFT two-point functions from the bulk perspective using bulk-to-boundary propagators.

Explain the role of superconformal symmetry in the precise formulation of the AdS/CFT correspondence.

Analyze the Polyakov action and fix the conformal gauge to describe the string worldsheet as a 2D CFT.

Identify the worldsheet CFT for the bosonic string and derive the critical dimension from Virasoro constraints.

Construct vertex operators for string states using the state-operator correspondence.

Quantize the bosonic string using the bc-ghost system and define physical states via BRST cohomology.

Compute a tree-level string scattering amplitude by evaluating a CFT correlation function of vertex operators.