Since the seminal paper by Belavin, Polyakov and Zamolodhikov, Conformal Field Theory has been a golden mine of beautiful theoretical ideas of far-reaching consequences.
Besides the classification of the universality classes of two-dimensional critical phenomena, it has promoted the understanding of massive/massless integrable models and the nature of
renormalization group flows. It has also enlightened the role of supersymmetry both in string theory and statistical physics, in model like the Tricritical Ising Model or the Gaussian Model, for istance. The link between the two subjects were established in a series of papers in collaboration with G. Sotkov and M. Staniskov, where it was shown that the Ramond sector of superstring theory can be fully characterized in terms of the spin field of the Ising model. It is also worth remind that Calabi-Yau spaces can be also be constructed in terms of minimal conformal models of statistical physics, as shown originally by D. Gepner.
The basic mathematical structure of Conformal Field Theory, i.e. the one relative to fusion rules, Verlinde algebra, modular transformations and braiding matrices, has proved to be one the key ingredients of non-abelian anyons, i.e. the excitations used in topological quantum computation.
Its boundary version, introduced originally by J.L. Cardy in order to analyze surface phase transitions, has found lately an enormous spectrum of applications, in fields so diverse as quantum transport, entanglement and quantum quenches.
Relevant Publications
• G. Mussardo, G. Sotkov, M. Staniskov, N=2 Superconformal Minimal Models, Int. Journ. Mod. Phys. A 4 1135 (1989).
• G. Mussardo, G. Sotkov, M. Staniskov, Fusion Rules, Four-Point Functions and Discrete Symmetries of N=2 Superconformal Models, Phys. Lett. B 218, 191 (1989).
• G. Mussardo, G. Sotkov, M. Staniskov, Finite Structure of the Supersymmetric Operator Product Expansion Algebra, Nucl. Phys. B 305, 69 (1988).
• G. Mussardo, G. Sotkov, M. Staniskov, Ramond Sector of the Supersymmetric Minimal Models, Phys. Lett. B 195, 397 (1987).
Main References
• A.A. Belavin, A. M. Polyakov, A.L. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys.B 241 333-380 (1984).
• Al. B. Zamolodchikov, A.L. Zamolodchikov, Conformal field theory and critical phenomena in two-dimensional systems, Sov. Scie. Rev. A Physics, 10 (1989), 269.
• J.L. Cardy, Boundary Conditions, Fusion Rules and the Verlinde formula, Nucl. Phys. B 328 , 521 (1989).
• P. Di Francesco, P. Mathieu, D. Senechal, Conformal Field Theory, Springers 1997.
• C. Itzykson, H. Saleur, J.B. Zuber, Conformal Invariance and Applications to Statistical Mechanics, World Scientific (Singapore), 1988.
• J. Cardy, Conformal Field Theory and Statistical Mechanics, Les Houches Summer School: Session 89: Exacts Methods in Low-Dimensional Statistical Physics and Quantum Computing, Les Houches, 2008, arXiv:0807.3472
Conformal Field Theory
“A physical law must possess mathematical beauty”
-- P.A.M. Dirac