11.5 Conformal field theory

Conformal field theory (CFT) is a powerful framework for studying quantum systems with scale invariance. It's crucial in understanding critical phenomena and provides insights into von Neumann algebras, particularly in operator algebras and quantum statistical mechanics.

CFTs use conformal symmetry to constrain correlation functions, simplifying calculations. The Virasoro algebra, central to 2D CFTs, extends conformal symmetry and connects to representation theory. Primary and descendant fields form the building blocks of CFTs, linking to highest weight states in Hilbert space.

Fundamentals of conformal field theory

• Conformal field theory (CFT) serves as a powerful framework for studying quantum field theories with scale invariance and additional symmetries
• CFTs play a crucial role in understanding critical phenomena and phase transitions in statistical mechanics
• The study of CFTs provides valuable insights into the structure of von Neumann algebras, particularly in the context of operator algebras and quantum statistical mechanics

Conformal symmetry principles

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• Conformal transformations preserve angles locally while allowing for scale changes
• Include translations, rotations, dilations, and special conformal transformations
• Generate an infinite-dimensional algebra in two dimensions, leading to powerful constraints on correlation functions
• Conformal invariance imposes strict restrictions on the form of correlation functions, simplifying many calculations

Virasoro algebra basics

• Infinite-dimensional algebra of conformal generators in two-dimensional CFTs
• Central extension of the Witt algebra, characterized by the central charge c
• Commutation relations: $[L_m, L_n] = (m-n)L_{m+n} + \frac{c}{12}m(m^2-1)\delta_{m+n,0}$
• Generators $L_n$ correspond to modes of the energy-momentum tensor
• Positive and negative modes create and annihilate excitations respectively

Primary fields vs descendant fields

• Primary fields transform covariantly under conformal transformations
• Characterized by their conformal dimension and spin
• Descendant fields obtained by acting with Virasoro generators on primary fields
• Form conformal families or Verma modules
• Operator-state correspondence relates primary fields to highest weight states in the Hilbert space

Correlation functions

• Correlation functions in CFTs encode the fundamental physical observables of the theory
• Their structure is highly constrained by conformal symmetry, leading to powerful predictive capabilities
• The study of correlation functions in CFTs provides insights into the algebraic structure of von Neumann algebras, particularly in the context of operator product expansions and fusion rules

Two-point correlators

• Fully determined by conformal symmetry up to a normalization constant
• Take the form: $\langle \phi_1(x_1) \phi_2(x_2) \rangle = \frac{C_{12}}{|x_1 - x_2|^{2h}}$
• $h$ represents the conformal dimension of the fields
• Vanish unless the two fields have the same scaling dimension
• Serve as a fundamental building block for more complex correlators

Three-point correlators

• Also fixed by conformal symmetry up to a constant
• General form: $\langle \phi_1(x_1) \phi_2(x_2) \phi_3(x_3) \rangle = \frac{C_{123}}{|x_1 - x_2|^{h_1 + h_2 - h_3} |x_2 - x_3|^{h_2 + h_3 - h_1} |x_3 - x_1|^{h_3 + h_1 - h_2}}$
• $C_{123}$ known as the structure constant or OPE coefficient
• Determine the operator product expansion (OPE) of the theory
• Play a crucial role in the conformal bootstrap approach

Operator product expansion

• Allows expressing the product of two local operators as a sum over other local operators
• Takes the form: $\phi_i(x) \phi_j(0) = \sum_k C_{ijk} |x|^{h_k - h_i - h_j} \phi_k(0) + \text{descendants}$
• Convergent expansion within correlation functions
• Encodes the algebraic structure of the CFT
• Provides a powerful tool for computing higher-point correlation functions

Conformal bootstrap method

• Non-perturbative approach to solving conformal field theories
• Exploits consistency conditions imposed by conformal symmetry and unitarity
• Connects to the theory of von Neumann algebras through the study of operator algebras and their representations

Crossing symmetry constraints

• Arise from different ways of applying the OPE to four-point functions
• Lead to functional equations for conformal blocks and OPE coefficients
• Provide powerful constraints on the spectrum and OPE coefficients of CFTs
• Can be formulated as linear or semidefinite programming problems

Conformal blocks

• Represent the contribution of a primary operator and its descendants to a four-point function
• Depend only on the conformal dimensions and spins of the external and exchanged operators
• Can be computed recursively or through differential equations (Casimir equation)
• Play a central role in the conformal bootstrap program
• Connect to representation theory of the Virasoro algebra

Numerical bootstrap techniques

• Involve discretizing the crossing equations and applying semidefinite programming
• Allow for rigorous bounds on scaling dimensions and OPE coefficients
• Have led to high-precision determinations of critical exponents in various models
• Require efficient methods for computing conformal blocks and their derivatives
• Connect to computational aspects of operator algebras and von Neumann algebras

Central charge

• Fundamental parameter characterizing conformal field theories
• Appears in various contexts, including the Virasoro algebra and the trace anomaly
• Plays a crucial role in the classification of von Neumann algebras arising from conformal field theories

Virasoro central charge

• Appears in the central extension term of the Virasoro algebra
• Quantifies the number of degrees of freedom in the CFT
• Determines the transformation properties of the energy-momentum tensor
• Takes rational values for minimal models and irrational values for more general CFTs
• Relates to the coefficient of the two-point function of the energy-momentum tensor

c-theorem

• States that the central charge decreases along renormalization group flows
• Proved by A.B. Zamolodchikov for two-dimensional theories
• Generalizes to higher dimensions as the a-theorem and F-theorem
• Provides a measure of the number of degrees of freedom at different energy scales
• Has important implications for the classification of quantum field theories

Minimal models

• Rational conformal field theories with finite number of primary fields
• Characterized by central charge $c = 1 - \frac{6}{m(m+1)}$ for $m \geq 2$
• Include important models such as the Ising model $(m=3)$ and tricritical Ising model $(m=4)$
• Possess a finite number of irreducible representations of the Virasoro algebra
• Serve as building blocks for more complex conformal field theories

Conformal anomaly

• Describes the breaking of conformal symmetry at the quantum level
• Manifests in various forms, including the trace anomaly and Weyl anomaly
• Provides important connections between conformal field theory and the geometry of spacetime

Trace anomaly

• Non-vanishing trace of the energy-momentum tensor in curved spacetime
• Proportional to the central charge in two dimensions
• Takes the form $\langle T^\mu_\mu \rangle = \frac{c}{24\pi} R$ in 2D, where R is the Ricci scalar
• Generalizes to higher dimensions with additional geometric terms
• Reflects the response of the theory to changes in the background geometry

Weyl anomaly

• Describes the change in the effective action under Weyl transformations
• Related to the trace anomaly but defined for arbitrary background metrics
• Encoded in the Polyakov action in two dimensions
• Plays a crucial role in string theory and AdS/CFT correspondence
• Connects conformal field theory to the geometry of the space on which it is defined

Holographic interpretation

• Central charge related to the AdS radius in AdS/CFT correspondence
• Weyl anomaly of the boundary CFT matches gravitational anomalies in the bulk
• Provides a geometric interpretation of the c-theorem in terms of the holographic RG flow
• Connects conformal anomalies to properties of black holes in the dual gravitational theory
• Offers insights into the relationship between quantum information and geometry

Vertex operator algebras

• Algebraic structures that formalize the operator content of two-dimensional conformal field theories
• Provide a rigorous mathematical framework for studying CFTs
• Connect conformal field theory to various areas of mathematics, including representation theory and algebraic geometry

Definition and properties

• Infinite-dimensional graded vector space V with a vacuum vector and a translation operator
• Equipped with a vertex operator map $Y: V \otimes V \rightarrow V((z))$
• Satisfy locality and associativity axioms
• Include the Virasoro algebra as a subalgebra
• Provide a mathematical formalization of the operator-state correspondence in CFT

Modules and representations

• Generalize the notion of modules for Lie algebras to the vertex operator algebra setting
• Classified by their conformal weights and character formulas
• Include irreducible, indecomposable, and logarithmic modules
• Connect to the representation theory of the Virasoro algebra and affine Lie algebras
• Play a crucial role in understanding the structure of rational and irrational CFTs

Moonshine phenomena

• Unexpected connections between finite simple groups and modular functions
• Most famous example: Monster moonshine relating the Monster group to the j-function
• Generalizations include Mathieu moonshine and umbral moonshine
• Involve the study of vertex operator algebras with special symmetries
• Provide deep connections between number theory, group theory, and conformal field theory

Applications in von Neumann algebras

• Conformal field theory provides a rich source of examples and constructions in the theory of von Neumann algebras
• The algebraic structure of CFTs naturally leads to operator algebras with interesting properties
• Studying CFTs in the context of von Neumann algebras offers new perspectives on both subjects

Conformal nets

• Algebraic approach to conformal field theory using von Neumann algebras
• Associate local algebras of observables to intervals on the circle
• Satisfy axioms of isotony, locality, Möbius covariance, and positivity of energy
• Provide a rigorous framework for studying CFTs in the operator algebraic setting
• Allow for the construction of subfactors and the study of their properties

Subfactors and Jones index

• Subfactors arising from conformal field theories often have interesting properties
• Jones index measures the "size" of a subfactor and takes discrete values for rational CFTs
• Connected to the statistical dimension of superselection sectors in algebraic quantum field theory
• Provide a bridge between conformal field theory and the theory of operator algebras
• Allow for the classification of certain classes of conformal field theories

Modular invariance

• Constraint on the partition function of a CFT on a torus
• Reflects the invariance under large diffeomorphisms of the torus
• Leads to strong constraints on the spectrum of primary fields in rational CFTs
• Connected to the theory of modular forms and number theory
• Plays a crucial role in the classification of rational conformal field theories

Conformal field theory in physics

• CFTs describe critical phenomena in statistical mechanics and condensed matter physics
• Serve as the worldsheet theory in string theory, crucial for understanding the theory's spectrum and interactions
• Provide a framework for studying quantum field theories at their fixed points, offering insights into the renormalization group

String theory connections

• Worldsheet theory of the string is a two-dimensional CFT
• Central charge c = 26 for bosonic strings, c = 15 for superstrings
• Vertex operators in string theory correspond to primary fields in the worldsheet CFT
• Modular invariance of the partition function ensures consistency of the string spectrum
• Spacetime physics emerges from the properties of the worldsheet CFT

Condensed matter applications

• Describe critical points of phase transitions (Ising model, XY model)
• Quantum Hall effect described by Chern-Simons theory, which is related to certain CFTs
• Appear in the study of one-dimensional quantum systems (Luttinger liquids)
• Topological phases of matter often have low-energy descriptions in terms of topological CFTs
• Provide tools for studying entanglement entropy in many-body systems

AdS/CFT correspondence

• Relates conformal field theories to theories of quantum gravity in anti-de Sitter space
• Central charge of the CFT related to the AdS radius in the bulk
• Correlation functions in the CFT mapped to Witten diagrams in AdS
• Renormalization group flow in the CFT corresponds to radial direction in AdS
• Provides insights into strongly coupled quantum field theories and quantum gravity

Computational aspects

• Efficient computation of conformal blocks, fusion rules, and character formulas is crucial for practical applications of CFT
• Numerical methods play an important role in the conformal bootstrap program
• Computational techniques in CFT often have counterparts in the theory of von Neumann algebras

Conformal blocks calculation

• Casimir differential equation provides a method for computing conformal blocks
• Recursion relations allow for efficient numerical evaluation
• Series expansions in the cross-ratio provide analytic approximations
• Zamolodchikov recursion relation gives a powerful method for computing blocks in 2D
• Efficient computation of blocks and their derivatives crucial for numerical bootstrap

Fusion rules

• Determine which primary fields appear in the operator product expansion of two given primaries
• Can be computed using the Verlinde formula in rational CFTs
• Related to the representation theory of the chiral algebra (Virasoro algebra or extensions)
• Play a crucial role in determining the consistency of a CFT
• Connect to the theory of tensor categories in mathematics

Character formulas

• Encode the spectrum of states in a given representation of the chiral algebra
• Take the form of q-series with interesting modular properties
• Can be computed using the Kac determinant formula for Verma modules
• Play a crucial role in the study of modular invariance and partition functions
• Connect CFT to the theory of modular forms and number theory

Advanced topics

• These topics represent current areas of research in conformal field theory
• They often involve generalizations or extensions of the standard CFT framework
• Studying these advanced topics can provide new insights into the structure of von Neumann algebras

Supersymmetric CFTs

• Incorporate fermionic degrees of freedom and supersymmetry
• Include the superconformal algebra, extending the Virasoro algebra
• Play a crucial role in superstring theory and AdS/CFT correspondence
• Exhibit enhanced analytical properties due to supersymmetry
• Provide examples of CFTs with extended chiral algebras

Boundary CFTs

• Study CFTs in the presence of boundaries or interfaces
• Introduce boundary conditions and boundary operators
• Important for describing D-branes in string theory
• Relate to the study of quantum impurity problems in condensed matter
• Connect to the theory of subfactors in von Neumann algebras

Logarithmic CFTs

• Exhibit logarithmic singularities in correlation functions
• Involve indecomposable but reducible representations of the Virasoro algebra
• Describe critical points in disordered systems and percolation
• Require generalizations of standard CFT techniques
• Provide examples of non-semisimple tensor categories in mathematics