[Paper Review] Applied Conformal Field Theory

Paul Ginsparg|ArXiv.org|Nov 11, 1988

Algebraic structures and combinatorial modelsMathematics105 references451 citations

TL;DR

This seminal 1988 Les Houches lecture series by Paul Ginsparg provides a comprehensive, pedagogical introduction to two-dimensional conformal field theory (CFT), emphasizing its applications to critical statistical mechanics and string theory. It systematically develops the Virasoro algebra, central charge, highest weight representations, and modular invariance, deriving fusion rules via modular S-matrix duality and establishing connections to coset constructions and W-algebras, with explicit realizations of the critical Ising model via free fermions and bosonization.

ABSTRACT

These lectures consisted of an elementary introduction to conformal field theory, with some applications to statistical mechanical systems, and fewer to string theory. Contents: 1. Conformal theories in d dimensions 2. Conformal theories in 2 dimensions 3. The central charge and the Virasoro algebra 4. Kac determinant and unitarity 5. Identication of m = 3 with the critical Ising model 6. Free bosons and fermions 7. Free fermions on a torus 8. Free bosons on a torus 9. Affine Kac-Moody algebras and coset constructions 10. Advanced applications

Motivation & Objective

• To provide a self-contained, accessible introduction to two-dimensional conformal field theory for a mixed audience of experts and beginners.
• To clarify the role of the central charge and the Virasoro algebra in classifying CFTs and their physical realizations.
• To demonstrate how modular invariance and the modular S-matrix encode fusion rules and character decompositions in rational CFTs.
• To connect abstract CFT structures to concrete physical models, particularly the critical Ising model and free fermion/boson systems.
• To lay the groundwork for advanced topics such as coset constructions, W-algebras, and the A-D-E classification of modular invariants.

Proposed method

• Derives the conformal algebra and Ward identities in 2D using radial quantization and mode expansions of primary fields.
• Constructs the Virasoro algebra via mode expansions of the stress-energy tensor and identifies highest weight states and descendants.
• Introduces the Kac determinant to analyze unitarity and classify allowed representations, particularly for c < 1 minimal models.
• Uses modular invariance of torus partition functions to classify consistent CFTs and derive fusion rules via the S-matrix.
• Applies the coset construction to build new CFTs from affine Kac-Moody algebras, with explicit examples like SU(2)k and SU(3)1×SU(3)1/SU(3)2.
• Demonstrates that modular invariant partition functions are diagonal in the largest chiral algebra, with fusion rules determined by the S-matrix via S-diagonalization.

Experimental results

Research questions

• RQ1 How do the constraints of conformal invariance in 2D lead to the structure of the Virasoro algebra and its central charge?
• RQ2 What is the role of the Kac determinant in determining unitarity and classifying allowed representations in 2D CFT?
• RQ3 How can modular invariance of the torus partition function be used to classify consistent CFTs and extract fusion rules?
• RQ4 In what way do coset constructions and W-algebras provide alternative realizations of known CFTs like the critical Ising model?
• RQ5 How does the modular S-matrix relate to the fusion rules of primary fields in rational CFTs?

Key findings

• The critical Ising model is realized as a free fermion theory with c = 1/2, and its partition function is modular invariant on the torus.
• Fusion rules for c < 1 minimal models are derived via the S-matrix diagonalization of the fusion algebra, confirming agreement with differential equation methods.
• The partition function (9.56) for the spin-3 W algebra is diagonal in the larger chiral algebra, with characters χ′₀ = χ₀ + χ₃, and corresponds to the coset SU(3)₁×SU(3)₁/SU(3)₂ with c = 4/5.
• The modular S-matrix Sij diagonalizes the fusion rules, yielding the formula Nijk = Σₙ Sjn Sin S†ₙₖ / S₀ₙ, which reproduces known SU(2)k fusion rules.
• The N = 2 superconformal discrete series and SU(2)k parafermion models coincide at m = k + 2, with the latter realized via a free boson and parafermions.
• Orbifolds such as S¹/Z₂ and SU(2)₃/U(1) provide alternative realizations of the same CFTs, including the critical Ising model with c = 1/2.