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 to condensed matter physics, statistical mechanics, quantum statistical mechanics, and string theory. Statistical and condensed matter systems are indeed often conformally invariant at their thermodynamic or quantum critical points.
Scale invariance vs conformal invariance
In quantum field theory, scale invariance is a common and natural symmetry, because any fixed point of the renormalization group is by definition scale invariant. Conformal symmetry is stronger than scale invariance, and one needs additional assumptions For this reason, the terms are often used interchangeably in the context of quantum field theory.
Two dimensions vs higher dimensions
The number of independent conformal transformations is infinite in two dimensions, and finite in higher dimensions. This makes conformal symmetry much more constraining in two dimensions. All conformal field theories share the ideas and techniques of the conformal bootstrap. But the resulting equations are more powerful in two dimensions, where they are sometimes exactly solvable (for example in the case of minimal models), in contrast to higher dimensions, where numerical approaches dominate.
The development of conformal field theory has been earlier and deeper in the two-dimensional case, in particular after the 1983 article by Belavin, Polyakov and Zamolodchikov.
The term conformal field theory has sometimes been used with the meaning of two-dimensional conformal field theory, as in the title of a 1997 textbook.
Higher-dimensional conformal field theories have become more popular with the AdS/CFT correspondence in the late 1990s, and the development of numerical conformal bootstrap techniques in the 2000s.
Global vs local conformal symmetry in two dimensions
The global conformal group of the Riemann sphere is the group of Möbius transformations , which is finite-dimensional.
On the other hand, infinitesimal conformal transformations form the infinite-dimensional Witt algebra: the conformal Killing equations in two dimensions, <math>\partial_\mu \xi_\nu + \partial_\nu \xi_\mu = \partial \cdot\xi \eta_{\mu \nu},~</math> reduce to just the Cauchy-Riemann equations, , the infinity of modes of arbitrary analytic coordinate transformations <math>\xi(z)</math> yield the infinity of Killing vector fields .
Strictly speaking, it is possible for a two-dimensional conformal field theory to be local (in the sense of possessing a stress-tensor) while still only exhibiting invariance under the global . This turns out to be unique to non-unitary theories; an example is the biharmonic scalar.
: <math>
\left\langle O(x_1)O(x_2) \right\rangle = \frac{1}{|x_1-x_2|^{2\Delta,
</math>
where we choose the normalization of the field such that the constant coefficient, which is not determined by conformal symmetry, is one. Similarly, two-point functions of non-scalar primary fields are determined up to a coefficient, which can be set to one. In the case of a symmetric traceless tensor of rank , the two-point function is
: <math> \left\langle O_{\mu_1,\dots,\mu_\ell}(x_1) O_{\nu_1,\dots,\nu_\ell}(x_2)\right\rangle = \frac{\prod_{i=1}^\ell I_{\mu_i,\nu_i}(x_1-x_2) - \text{traces{|x_1-x_2|^{2\Delta,
</math>
where the tensor <math>I_{\mu,\nu}(x)</math> is defined as
: <math>
I_{\mu,\nu}(x) = \eta_{\mu\nu} - \frac{2x_\mu x_\nu}{x^2}.
</math>
The three-point function of three scalar primary fields is
: <math>
\left\langle O_{1}(x_1)O_{2}(x_2)O_{3}(x_3)\right\rangle = \frac{C_{123{|x_{12}|^{\Delta_1+\Delta_2-\Delta_3}|x_{13}|^{\Delta_1+\Delta_3-\Delta_2} |x_{23}|^{\Delta_2+\Delta_3-\Delta_1,
</math>
where , and <math>C_{123}</math> is a three-point structure constant. With primary fields that are not necessarily scalars, conformal symmetry allows a finite number of tensor structures, and there is a structure constant for each tensor structure. In the case of two scalar fields and a symmetric traceless tensor of rank , there is only one tensor structure, and the three-point function is
: <math>
\left\langle O_{1}(x_1)O_{2}(x_2)O_{\mu_1,\dots,\mu_\ell}(x_3)\right\rangle
= \frac{C_{123}\left(\prod_{i=1}^\ell V_{\mu_i}-\text{traces}\right)}{|x_{12}|^{\Delta_1+\Delta_2-\Delta_3}|x_{13}|^{\Delta_1+\Delta_3-\Delta_2} |x_{23}|^{\Delta_2+\Delta_3-\Delta_1,
</math>
where we introduce the vector
: <math>
V_\mu = \frac{x_{13}^\mu x_{23}^2 - x_{23}^\mu x_{13}^2}{|x_{12}||x_{13}||x_{23}|}.
</math>
Four-point functions of scalar primary fields are determined up to arbitrary functions <math>g(u,v)</math> of the two cross-ratios
: <math>
u = \frac{x_{12}^2 x_{34}^2}{x_{13}^2 x_{24}^2} \ , \ v = \frac{x_{14}^2 x_{23}^2}{x_{13}^2 x_{24}^2}.
</math>
The four-point function is then and in dimension <math>4 - \epsilon</math>
When N is large, the O(N) model can be solved perturbatively in a 1/N expansion by means of the Hubbard–Stratonovich transformation. In particular, the <math>N \to \infty</math> limit of the critical O(N) model is well-understood.
The conformal data of the critical O(N) model are functions of N and of the dimension, on which many results are known.
Further reading
- Martin Schottenloher, A Mathematical Introduction to Conformal Field Theory, Springer-Verlag, Berlin, Heidelberg, 1997. , 2nd edition 2008, .
External links
- CFT Zoo