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.
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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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