Conformal field theory
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A conformal field theory (CFT) is a quantum field theory (or statistical mechanics model at the critical point) that is invariant under conformal transformations. Conformal field theory is often studied in two dimensions where there is an infinite-dimensional group of local conformal transformations, described by the holomorphic functions. Conformal field theory has important applications in string theory, statistical mechanics, and condensed matter physics.
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[edit] Scale invariance vs. conformal invariance
While it is possible for a quantum field theory to be scale invariant but not conformally-invariant, examples are rare [1]. For this reason, the terms are often used interchangeably in the context of quantum field theory, even though the conformal symmetry is much larger.
In some particular cases it is possible to prove that scale invariance implies conformal invariance in a quantum field theory, for example in unitary compact conformal field theories in two dimensions.
[edit] Two-dimensional conformal field theory
Two-dimensional CFTs are (in some way) invariant under an infinite-dimensional symmetry group. For example, consider a CFT on the Riemann sphere. It has the Möbius transformations as the conformal group, which is isomorphic to (the finite-dimensional) PSL(2,C). However, the infinitesimal conformal transformations form an infinite-dimensional algebra, called the Witt algebra and only the primary fields (or chiral fields) are invariant with respect to the full infinitesimal conformal group.
In most conformal field theories, a conformal anomaly, also known as a Weyl anomaly, arises in the quantum theory. This results in the appearance of a nontrivial central charge, and the Witt algebra is modified to become the Virasoro algebra.
This symmetry makes it possible to classify two-dimensional CFTs much more precisely than in higher dimensions. In particular, it is possible to relate the spectrum of primary operators in a theory to the value of the central charge, c. The Hilbert space of physical states is a unitary module of the Virasoro algebra corresponding to a fixed value of c. Stability requires that the energy spectrum of the Hamiltonian be nonnegative. The modules of interest are the highest weight modules of the Virasoro algebra.
A chiral field is a holomorphic field W(z) which transforms as
and
Similarly for an antichiral field. Δ is the conformal weight of the chiral field W.
Furthermore, it was shown by Zamolodchikov that there exists a function, C, which decreases monotonically under the renormalization group flow of a two-dimensional quantum field theory, and is equal to the central charge for a two-dimensional conformal field theory. This is known as the Zamolodchikov C-theorem, and tells us that renormalization group flow in two dimensions is irreversible.
[edit] Conformal algebra
- [Pμ,Pν] = 0 , [D,Kμ] = − Kμ,
- [D,Pμ] = Pμ , [Kμ,Kν] = 0,
- [Kμ,Pν] = ημνD − iMμν,
[edit] See also
- AdS/CFT correspondence
- Operator product expansion
- Vertex operator algebra
- WZW model
- critical point
- primary field
- superconformal algebra
[edit] References
- ^ Six dimensional noncritical self-dual string theory is an example of a scale invariant theory which isn't conformally-invariant. Basically, conformal invariance means that the theory is independent of the Weyl connection.
- Martin Schottenloher, A Mathematical Introduction to Conformal Field Theory, Springer-Verlag, Berlin Heidelberg, 1997. ISBN3-540-61753-1.
- Paul Ginsparg, Applied Conformal Field Theory. arXiv:hep-th/9108028.
- P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory, Springer-Verlag, New York, 1997. ISBN 0-387-94785-X.
- A.A. Belavin, A.M. Polyakov, A.B. Zamolodchikov, Infinite conformal symmetry in two-dimensional quantum field theory, Nucl. Phys. B241 (1984) 333–380.
- A.B Zamolodchikov, ``Irreversibility' Of The Flux Of The Renormalization Group In A 2-D Field Theory, JETP Lett.43:730-732,1986 [1] (Russian version).
