Conformal field theory

A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations. In two dimensions, there is an infinite-dimensional algebra of local conformal transformations, and conformal field theories can sometimes be exactly solved or classified.

Conformal field theory has important applications^[1] to string theory, statistical mechanics, and condensed matter physics. Statistical and condensed matter systems are indeed often conformally invariant at their thermodynamic or quantum critical points.

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.^[2] For this reason, the terms are often used interchangeably in the context of quantum field theory, even though the scale symmetry group is smaller.

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.

Dimensional considerations

Two dimensions

There are two versions of 2D CFT: 1) Euclidean, and 2) Lorentzian. The former applies to statistical mechanics, and the latter to quantum field theory. The two versions are related by a Wick rotation.

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^[3] form an infinite-dimensional algebra, called the Witt algebra, but this infinity of conformal transformations do not have global inverses on ℂ. Only the primary fields (or chiral fields) are invariant with respect to this full infinitesimal conformal group. Its generators are indexed by integers $n$ ,

L_n=\oint_{z=0} \frac{dz}{2\pi i } z^{n+1} T_{zz}~,

where $T zz$ is the holomorphic part of the non-trace piece of the energy momentum tensor of the theory. E.g., for a free scalar field,

T_{zz}= \tfrac{1}{2} (\partial_z \phi)^2 ~.

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 extended to the Virasoro algebra.

In Euclidean CFT, one has both a holomorphic and an antiholomorphic copy of the Virasoro algebra. In Lorentzian CFT, one has a left-moving and a right moving copy of the Virasoro algebra (spacetime is a cylinder, with space being a circle, and time a line).

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

L_n W(z)=-z^{n+1} \frac{\partial}{\partial z} W(z) - (n+1)\Delta z^n W(z)

and

\bar L_n W(z)=0~.

Analogously, mutatis mutandis, for an antichiral field. $Δ$ is called the conformal weight of the chiral field $W$ .

Furthermore, it was shown by Alexander 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.

Frequently, we are not just interested in the operators, but we are also interested in the vacuum state, or in statistical mechanics, the thermal state. Unless c=0, there can't possibly be any state which leaves the entire infinite dimensional conformal symmetry unbroken. The best we can come up with is a state which is invariant under L_-1, L₀, L₁, L_i, $i > 1$ . This contains the Möbius subgroup. The rest of the conformal group is spontaneously broken.

Two-dimensional conformal field theories play an important role in statistical mechanics, where they describe critical points of many lattice models.

More than two dimensions

In d > 2 dimensions, the conformal group is isomorphic to $SO(d+1, 1)$ in Euclidean signature, or $SO(d, 2)$ in Minkowski space.

Higher-dimensional conformal field theories are prominent in the AdS/CFT correspondence, in which a gravitational theory in anti-de Sitter space (AdS) is equivalent to a conformal field theory on the AdS boundary. Notable examples are d=4 N = 4 supersymmetric Yang–Mills theory, which is dual to Type IIB string theory on AdS₅ x S⁵, and d=3 N=6 super-Chern–Simons theory, which is dual to M-theory on AdS₄ x S⁷. (The prefix "super" denotes supersymmetry, N denotes the degree of extended supersymmetry possessed by the theory, and d the number of space-time dimensions on the boundary.)

Conformal symmetry

Conformal symmetry is a symmetry under scale invariance and under the special conformal transformations having the following relations.

[P_\mu,P_\nu]=0,

[D,K_\mu]=-K_\mu,

[D,P_\mu]=P_\mu,

[K_\mu,K_\nu]=0,

[K_\mu,P_\nu]=\eta_{\mu\nu}D-iM_{\mu\nu},

where $P$ generates translations, $D$ generates scaling transformations as a scalar and $K_\mu$ generates the special conformal transformations as a covariant vector under Lorentz transformation.

Main articles: Conformal symmetry and Conformal Killing equation

References

↑ Paul Ginsparg (1989), Applied Conformal Field Theory. arXiv:hep-th/9108028. Published in Ecole d'Eté de Physique Théorique: Champs, cordes et phénomènes critiques/Fields, strings and critical phenomena (Les Houches), ed. by E. Brézin and J. Zinn-Justin, Elsevier Science Publishers B.V.
↑ One physical example is the theory of elasticity in two and three dimensions (also known as the theory of a vector field without gauge invariance). See Riva V, Cardy J (2005). "Scale and conformal invariance in field theory: a physical counterexample". Phys. Lett. B 622: 339–342. arXiv:hep-th/0504197. Bibcode:2005PhLB..622..339R. doi:10.1016/j.physletb.2005.07.010.
↑ Since the conformal Killing equations in two dimensions, $\partial_\mu \xi_\nu + \partial_\nu \xi_\mu =\partial \cdot\xi \eta_{\mu \nu},~$ reduce to just the Cauchy-Riemann equations, $\partial_{\bar{z}} \xi(z)=0=\partial_z \xi (\bar{z})$ , the infinity of modes of arbitrary analytic coordinate transformations ξ(z) yield the infinity of Killing vector fields $z n \partial z$ .

Martin Schottenloher, A Mathematical Introduction to Conformal Field Theory, Springer-Verlag, Berlin, Heidelberg, 1997. ISBN 3-540-61753-1, 2nd edition 2008, ISBN 978-3-540-68625-5.
P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory, Springer-Verlag, New York, 1997. ISBN 0-387-94785-X.
Conformal Field Theory page in String Theory Wiki lists books and reviews.
Slava Rychkov, Lectures on Conformal Field Theory in D≥ 3 Dimensions, 2012

Statistical mechanics

Statistical ensembles	Microcanonical Canonical Grand canonical Isothermal–isobaric Isoenthalpic–isobaric ensemble Open

Statistical thermodynamics	Characteristic state functions

Partition functions	Translational Vibrational Rotational

Equations of state	Dieterici Van der Waals Ideal gas law Birch–Murnaghan

Entropy	Sackur–Tetrode equation Tsallis entropy Von Neumann entropy

Particle statistics	Maxwell–Boltzmann statistics Fermi–Dirac statistics Bose–Einstein statistics

Statistical field theory	Conformal field theory Osterwalder–Schrader axioms

Quantum statistical mechanics	Density matrix Gibbs measure Partition function Phase space formulation of quantum mechanics Slater determinant

See also	Probability distribution Elementary particles

Quantum field theories

Standard	Chern–Simons model Chiral model Kondo model Conformal field theory Noncommutative quantum field theory Non-linear sigma model Quartic interaction Quantum chromodynamics Quantum electrodynamics Quantum Yang–Mills theory Schwinger model sine-Gordon Standard Model String theory Toda field theory Topological quantum field theory Wess–Zumino model Wess–Zumino–Witten model Yang–Mills theory Yang–Mills–Higgs model Yukawa model Thirring–Wess model

Four-fermion interactions	Thirring model Gross–Neveu model Nambu–Jona-Lasinio model

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