10.2 Virasoro algebra and conformal field theory

The Virasoro algebra extends the Witt algebra, adding a central element to create an infinite-dimensional Lie algebra. It's crucial in conformal field theory, describing symmetries of two-dimensional systems and classifying their properties through the central charge.

Representations of the Virasoro algebra, characterized by highest weight states, are key to understanding conformal field theories. The algebra's structure determines operator product expansions, correlation functions, and critical behavior in physical systems like the Ising model.

Virasoro algebra and its extension

Central extension of the Witt algebra

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• The Virasoro algebra is an infinite-dimensional Lie algebra that extends the Witt algebra, the Lie algebra of polynomial vector fields on the circle
• It is spanned by generators $L_n$ ($n \in \mathbb{Z}$) and a central element $c$, satisfying the commutation relations $[L_n, L_m] = (n - m)L_{n+m} + \frac{c}{12}(n^3 - n)\delta_{n,-m}$
• The central charge $c$ characterizes the representation of the Virasoro algebra and is crucial in classifying conformal field theories (CFTs)
• Setting $c = 0$ reduces the Virasoro algebra to the Witt algebra, isomorphic to the Lie algebra of one-dimensional conformal transformations

Regularization and the central charge

• The central extension arises from the need to regularize the product of two operators in quantum field theory
• This regularization leads to the appearance of the central charge $c$
• The central charge plays a vital role in determining the properties of the CFT, such as the conformal anomaly and the critical dimensions

Representation theory of Virasoro algebra

Highest weight representations and Verma modules

• Representations of the Virasoro algebra are characterized by the central charge $c$ and the highest weight $h$, the eigenvalue of $L_0$ on the highest weight state
• Highest weight representations are constructed by acting with negative modes $L_n$ ($n < 0$) on the highest weight state, generating a Verma module
• The Verma module may contain null states, which are both primary (annihilated by $L_n$, $n > 0$) and descendant (obtained by acting with $L_n$, $n < 0$ on a primary state)

Irreducible and unitary representations

• Irreducible representations are obtained by quotienting the Verma module by the submodule generated by the null states
• Unitary representations, physically relevant for CFTs, are characterized by the Kac determinant formula and exist only for specific values of $c$ and $h$
• The minimal models are a series of unitary representations with $c < 1$, which are of particular importance in CFT (Ising model, tricritical Ising model)

Virasoro algebra in conformal field theory

Symmetry algebra and the energy-momentum tensor

• The Virasoro algebra is the symmetry algebra of two-dimensional CFTs, which are invariant under conformal transformations
• The generators $L_n$ correspond to the modes of the energy-momentum tensor in a CFT, which generates conformal transformations
• The central charge $c$ is related to the conformal anomaly, measuring the breaking of conformal symmetry at the quantum level

Primary fields and operator product expansion

• Primary fields in a CFT are operators that transform covariantly under conformal transformations and are associated with highest weight representations of the Virasoro algebra
• The operator product expansion (OPE) in a CFT is determined by the commutation relations of the Virasoro algebra and the conformal dimensions of the primary fields
• The Virasoro algebra plays a key role in classifying CFTs through its unitary representations and the corresponding values of $c$ and $h$ (minimal models, Liouville CFT)

Virasoro algebra for two-dimensional CFTs

Conformal bootstrap and Kac-Moody algebras

• The conformal bootstrap approach uses Virasoro algebra constraints, such as crossing symmetry of four-point functions, to determine the spectrum and correlation functions of a CFT
• Kac-Moody algebras, extensions of the Virasoro algebra by additional currents, describe CFTs with extra symmetries (Wess-Zumino-Witten models, affine Lie algebras)

Partition functions and modular invariance

• The Virasoro algebra is used to compute the partition function of a CFT on the torus, encoding information about the spectrum and exhibiting modular invariance
• Modular invariance imposes strict constraints on the possible values of $c$ and $h$, leading to a classification of rational CFTs

Boundary conditions, defects, and the AdS/CFT correspondence

• The representation theory of the Virasoro algebra classifies possible boundary conditions and defects in a CFT, described by Cardy states and Ishibashi states, respectively
• The Virasoro algebra is essential in studying the AdS/CFT correspondence, relating gravity in anti-de Sitter space to a CFT on its boundary (string theory, holography)