Abstract
An easily readable introduction to the main concepts and techniques of conformal invariance is provided. Starting from the global scale-invariance at a critical point, it is argued, through the local conformal Ward identities, that under mild conditions an extension to a local form of scale-invariance, namely conformally invariance, is in general possible. In two space dimensions, the particular role of the infinite-dimensional Lie algebra of conformal transformations is outlined and the main concepts, namely those of a primary scaling operator, the conformal energy-momentum tensor, the Virasoro algebra and the central charge and the main facts of their unitary and/or minimal representations will be presented. Some simple applications for the explicit calculation of two-point functions will be given. The free boson will be used as a paradigmatic illustration and we shall close with an outline of surface critical phenomena and their description in terms of boundary conformal field-theory.
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Notes
- 1.
Unless explicitly stated otherwise, we shall choose units such that the Boltzmann constant k B = 1.
- 2.
For example, in the 2D Ising universality class, one has x σ = 1/8 and x ε = 1/2.
- 3.
In the literature, the alternative notation: \(h,\bar{h}\) for the conformal weights and \(\varDelta= h +\bar{h}\) for the scaling dimension, is also met with frequently.
- 4.
Although this algebra was first defined by É. Cartan in 1909, it is unfortunately often referred to as ‘Witt algebra’. Witt studied this algebra only much later (in the 1930s), over fields of characteristic p > 0, when the algebra is spanned by the ℓ n with − 1 ≤ n ≤ p − 2.
- 5.
With the normalisation ϕ ab = δ ab in (1.16), the coefficient \( {{C}_{{123}}} \) is universal and not arbitrary.
- 6.
Equation (1.25) borrows from the theory of elasticity, where T μν is called ‘stress-energy tensor’. This analogy is unlikely to be valid for theories with long-range interactions.
- 7.
To dissipate any belief that linear, massless dispersion relations would only belong to the fictitious worlds of the stringy theorist: exactly this kind of dispersion relation is actually realised in graphene, where the ‘carriers of the charge behave as (2+1)D ultra-relativistic particules without mass’ [32].
- 8.
The physical content of a mathematical classification has still to be established by external evidence. To quote a well-known example, the periodic system of the chemical elements follows from the representation theory of the rotation Lie group \(\mathfrak{so}(3)\). Still, that classification alone does not tell you that the 8th element keeps fires burning and allows vertebrates to breathe or that the 79th element has since prehistoric times attracted the greed of many.
- 9.
A good example of this is the Fermi theory of weak interactions, where the momentum-dependence of the propagators of the intermediate weak bosons W ± and Z of the unified electroweak theory can be neglected for energies ≪ M W, Z c 2≈80 [GeV].
- 10.
Note that in this expression the factor \((\operatorname{Im}\tau )^{-1/2}\) of (1.84) has disappeared.
- 11.
We do not require in this book the much more rich phenomenology of surface critical behaviour in d ≥ 3 dimensions, with its ‘extraordinary’ and ‘special’ transitions.
- 12.
One restricts here and in what follows to scaling operators which are scalars deep in the bulk, with \(\varDelta=\overline{\varDelta}=x/2\).
- 13.
It remains perfectly possible to describe by (1.134) a two-point function such as 〈σε〉 with a fixed non-vanishing magnetisation imposed at the surface and x σ ≠ x ε , which of course vanishes in the bulk.
References
Alcaraz, F.C., Grimm, U., Rittenberg, V.: The XXZ Heisenberg chain, conformal invariance and the operator content of c < 1 systems. Nucl. Phys. B 316, 735 (1989)
Alcaraz, F.C., Levine, E., Rittenberg, V.: Conformal invariance and its breaking in a stochastic model of a fluctuating interface. J. Stat. Mech., 08003 (2006)
Baake, M., Christe, P., Rittenberg, V.: Higher spin conserved currents in c = 1 conformally invariant systems. Nucl. Phys. B 300, 637 (1988)
Baxter, R.J.: Exactly Solved Models in Statistical Mechanics. Academic Press, London (1982)
Belavin, A.A., Polyakov, A.M., Zamolodchikov, A.B.: Infinite conformal symmetry in two-dimensional quantum field-theory. Nucl. Phys. B 241, 333 (1984)
Blumenhagen, R., Plauschinn, E.: Introduction to Conformal Field-Theory. Lecture Notes in Physics, vol. 779. Springer, Heidelberg (2009)
Boyer, T.H.: Conserved currents, renormalization and the Ward identity. Ann. of Phys. 44, 1 (1967)
Calabrese, P., Cardy, J.L.: Time-dependence of correlation functions following a quantum quench. Phys. Rev. Lett. 96, 136801 (2006)
Calabrese, P., Cardy, J.L.: Entanglement and correlation functions following a local quench: a conformal field theory approach. J. Stat. Mech., 10004 (2007)
Cardy, J.L.: Conformal invariance and surface critical behaviour. Nucl. Phys. B 240, 514 (1984)
Cardy, J.L.: Conformal invariance. In: Domb, C., Lebowitz, J.L. (eds.) Phase Transitions and Critical Phenomena, vol. 11. Academic Press, London (1986)
Cardy, J.L.: Effect of boundary conditions on the operator content of two-dimensional conformally invariant theories. Nucl. Phys. B 275, 200 (1986)
Cardy, J.L.: Boundary conditions, fusion rules and the Verlinde formula. Nucl. Phys. B 324, 581 (1989)
Cardy, J.L.: Conformal invariance and statistical mechanics. In: Brézin, E., Zinn-Justin, J. (eds.) Fields, Strings and Critical Phenomena, Les Houches XLIX. North-Holland, Amsterdam (1990)
Cardy, J.L.: Scaling and Renormalization in Statistical Physics. Cambridge University Press, Cambridge (1996)
Cardy, J.L.: Boundary conformal field theory. In: Encyclopedia of Mathematical Physics. Elsevier, Amsterdam (2006)
Cardy, J.L.: Conformal field theory and statistical mechanics. In: Exact Methods in Low-Dimensional Statistical Physics and Quantum Computing, Les Houches XLIX. North-Holland, Amsterdam (2008)
Chatelain, C., Berche, B.: Tests of conformal invariance in randomness-induced second-order phase transitions. Phys. Rev. E 58, 6899 (1998)
di Francesco, P., Mathieu, P., Sénéchal, D.: Conformal Field-Theory. Springer, Heidelberg (1997)
Diehl, H.W.: Field-theoretical approach to critical phenomena at surfaces. In: Domb, C., Lebowitz, J.L. (eds.) Phase Transitions and Critical Phenomena, vol. 10. Academic Press, London (1987)
Drewitz, A., Leidl, R., Burkhardt, T.W., Diehl, H.W.: Surface critical behaviour of binary alloys and antiferromagnets: dependence of the universality class on surface orientation. Phys. Rev. Lett. 78, 1090 (1997)
Drouffe, J.-M., Itzykson, C.: Statistical Field-Theory, vol. 2. Cambridge University Press, Cambridge (1988)
Fisher, M.E.: Scaling, universality and renormalisation group theory. In: Hahne, F.J.W. (ed.) Critical Phenomena. Lecture Notes in Physics, vol. 186, pp. 1–13. Springer, Heidelberg (1983)
Gaudin, M.: La Fonction d’onde de Bethe. Masson, Paris (1983)
Grandati, Y.: Éléments d’introduction à l’invariance conforme. Ann. Physique 17, 159 (1992)
Henkel, M.: Phase Transitions and Conformal Invariance. Springer, Heidelberg (1999)
Henkel, M., Patkós, A.: Critical exponents of defective Ising models and the U(1) Kac-Moody-Virasoro algebras. Nucl. Phys. B 285, 29 (1987)
Henkel, M., Pleimling, M.: Non-equilibrium Phase Transitions, vol. 2. Ageing and Dynamical Scaling Far from Equilibrium. Springer, Heidelberg (2010)
Henkel, M., Hinrichsen, H., Lübeck, S.: Non-equilibrium Phase Transitions, vol. 1. Absorbing Phase Transitions. Springer, Heidelberg (2009)
Itzykson, C., Drouffe, J.-M.: Théorie Statistique des Champs, vol. 2. InterÉditions/CNRS, Paris (1989)
Karevski, D., Henkel, M.: Finite-size effects in layered magnetic systems. Phys. Rev. B 55, 6429 (1995)
Neto, A.H.C., Guinea, F., Peres, N.M.R., Novosolev, K.S., Geim, A.K.: The electronic properties of graphene. Rev. Mod. Phys. 81, 109 (2009)
Oshikawa, M., Affleck, I.: Boundary conformal field theory approach to the two-dimensional critical Ising model with a defect line. Nucl. Phys. B 495, 533 (1997)
Petkova, V., Zuber, J.-B.: Conformal boundary conditions and what they teach us. In: Horváth, Z., Palla, L. (eds.) Non-perturbative Quantum Field Theoretic Methods and Their Applications. World Scientific, Singapore (2001)
Pleimling, M.: Critical phenomena at perfect and non-perfect surfaces. J. Phys. A, Math. Gen. 37, 79 (2004)
Pleimling, M., Selke, W.: Ising cubes with enhanced surface couplings. Phys. Rev. E 61, 933 (2000)
Polchinski, J.: Scale and conformal invariance in quantum field theory. Nucl. Phys. B 303, 226 (1988)
Riva, V., Cardy, J.L.: Scale and conformal invariance in field theory: a physical counterexample. Phys. Lett. B 622, 339 (2005)
Schottenloher, M.: A Mathematical Introduction to Conformal Field-Theory. Lecture Notes in Physics, vol. 759. Springer, Heidelberg (2008)
Yeomans, J.M.: Statistical Mechanics of Phase Transitions. Oxford University Press, Oxford (1992)
Zuber, J.-B.: An introduction of conformal field-theory. Acta Phys. Pol. B 26, 1785 (1995)
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Henkel, M., Karevski, D. (2012). A Short Introduction to Conformal Invariance. In: Henkel, M., Karevski, D. (eds) Conformal Invariance: an Introduction to Loops, Interfaces and Stochastic Loewner Evolution. Lecture Notes in Physics, vol 853. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-27934-8_1
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