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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 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 to argue that it should appear in nature. The basic idea behind its plausibility is that local scale invariant theories have their currents given by $$T_{\mu \nu} \xi^\nu$$ where $$\xi^\nu$$ is a Killing vector and $$T_{\mu \nu}$$ is a conserved operator (the stress-tensor) of dimension exactly $$d$$. For the associated symmetries to include scale but not conformal transformations, the trace $$T_\mu^\mu$$ has to be a non-zero total derivative implying that there is a non-conserved operator of dimension exactly $$d - 1$$.

Under some assumptions it is possible to completely rule out this type of non-renormalization and hence prove that scale invariance implies conformal invariance in a quantum field theory, for example in unitary compact conformal field theories in two dimensions.

While it is possible for a quantum field theory to be scale invariant but not conformally invariant, examples are rare. 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 $$ PSL_2(\mathbb{C}) $$, which is finite-dimensional. On the other hand, infinitesimal conformal transformations form the infinite-dimensional Witt algebra: 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 $$\xi(z)$$ yield the infinity of Killing vector fields $$z^n\partial_z$$.

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 $$ PSL_2(\mathbb{C}) $$. This turns out to be unique to non-unitary theories; an example is the biharmonic scalar. This property should be viewed as even more special than scale without conformal invariance as it requires $$T_\mu^\mu$$ to be a total second derivative.

Global conformal symmetry in two dimensions is a special case of conformal symmetry in higher dimensions, and is studied with the same techniques. This is done not only in theories that have global but not local conformal symmetry, but also in theories that do have local conformal symmetry, for the purpose of testing techniques or ideas from higher-dimensional CFT. In particular, numerical bootstrap techniques can be tested by applying them to minimal models, and comparing the results with the known analytic results that follow from local conformal symmetry.

Conformal field theories with a Virasoro symmetry algebra
In a conformally invariant two-dimensional quantum theory, the Witt algebra of infinitesimal conformal transformations has to be centrally extended. The quantum symmetry algebra is therefore the Virasoro algebra, which depends on a number called the central charge. This central extension can also be understood in terms of a conformal anomaly.

It was shown by Alexander Zamolodchikov that there exists a function 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.

In addition to being centrally extended, the symmetry algebra of a conformally invariant quantum theory has to be complexified, resulting in two copies of the Virasoro algebra. In Euclidean CFT, these copies are called holomorphic and antiholomorphic. In Lorentzian CFT, they are called left-moving and right moving. Both copies have the same central charge.

The space of states of a theory is a representation of the product of the two Virasoro algebras. This space is a Hilbert space if the theory is unitary. This space may contain a vacuum state, or in statistical mechanics, a thermal state. Unless the central charge vanishes, there cannot exist a state that leaves the entire infinite dimensional conformal symmetry unbroken. The best we can have is a state that is invariant under the generators $$L_{n\geq -1}$$ of the Virasoro algebra, whose basis is $$(L_n)_{n\in\mathbb{Z}}$$. This contains the generators $$L_{-1},L_0,L_1$$ of the global conformal transformations. The rest of the conformal group is spontaneously broken.

Definition and Jacobian
For a given spacetime and metric, a conformal transformation is a transformation that preserves angles. We will focus on conformal transformations of the flat $$d$$-dimensional Euclidean space $$\mathbb{R}^d$$ or of the Minkowski space $$\mathbb{R}^{1,d-1}$$.

If $$x\to f(x)$$ is a conformal transformation, the Jacobian $$J^\mu_\nu(x) = \frac{\partial f^\mu(x)}{\partial x^\nu} $$ is of the form

J^\mu_\nu(x) = \Omega(x) R^\mu_\nu(x), $$ where $$\Omega(x)$$ is the scale factor, and $$R^\mu_\nu(x)$$ is a rotation (i.e. an orthogonal matrix) or Lorentz transformation.

Conformal group
The conformal group is locally isomorphic to $$SO(1, d + 1)$$ (Euclidean) or $$SO(2,d)$$ (Minkowski). This includes translations, rotations (Euclidean) or Lorentz transformations (Minkowski), and dilations i.e. scale transformations

x^\mu \to \lambda x^\mu. $$

This also includes special conformal transformations. For any translation $$T_a(x) = x + a$$, there is a special conformal transformation

S_a = I \circ T_a \circ I, $$ where $$ I $$ is the inversion such that

I\left(x^\mu\right) = \frac{x^\mu}{x^2}. $$

In the sphere $$S^d = \mathbb{R}^d \cup \{\infty\}$$, the inversion exchanges $$0$$ with $$\infty$$. Translations leave $$\infty$$ fixed, while special conformal transformations leave $$0$$ fixed.

Conformal algebra
The commutation relations of the corresponding Lie algebra are

 * $$\begin{align}[]

[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}, \end{align}$$

where $$P$$ generate translations, $$D$$ generates dilations, $$K_\mu$$ generate special conformal transformations, and $$M_{\mu\nu}$$ generate rotations or Lorentz transformations. The tensor $$\eta_{\mu\nu}$$ is the flat metric.

Global issues in Minkowski space
In Minkowski space, the conformal group does not preserve causality. Observables such as correlation functions are invariant under the conformal algebra, but not under the conformal group. As shown by Lüscher and Mack, it is possible to restore the invariance under the conformal group by extending the flat Minkowski space into a Lorentzian cylinder. The original Minkowski space is conformally equivalent to a region of the cylinder called a Poincaré patch. In the cylinder, global conformal transformations do not violate causality: instead, they can move points outside the Poincaré patch.

Correlation functions and conformal bootstrap
In the conformal bootstrap approach, a conformal field theory is a set of correlation functions that obey a number of axioms.

The $$n$$-point correlation function $$\left\langle O_1(x_1)\cdots O_n(x_n)\right\rangle $$ is a function of the positions $$x_i$$ and other parameters of the fields $$O_1,\dots ,O_n$$. In the bootstrap approach, the fields themselves make sense only in the context of correlation functions, and may be viewed as efficient notations for writing axioms for correlation functions. Correlation functions depend linearly on fields, in particular $$ \partial_{x_1} \left\langle O_1(x_1)\cdots \right\rangle = \left\langle \partial_{x_1}O_1(x_1)\cdots \right\rangle $$.

We focus on CFT on the Euclidean space $$\mathbb{R}^d$$. In this case, correlation functions are Schwinger functions. They are defined for $$x_i\neq x_j$$, and do not depend on the order of the fields. In Minkowski space, correlation functions are Wightman functions. They can depend on the order of the fields, as fields commute only if they are spacelike separated. A Euclidean CFT can be related to a Minkowskian CFT by Wick rotation, for example thanks to the Osterwalder-Schrader theorem. In such cases, Minkowskian correlation functions are obtained from Euclidean correlation functions by an analytic continuation that depends on the order of the fields.

Behaviour under conformal transformations
Any conformal transformation $$x\to f(x)$$ acts linearly on fields $$O(x) \to \pi_f(O)(x)$$, such that $$f\to \pi_f$$ is a representation of the conformal group, and correlation functions are invariant:

\left\langle\pi_f(O_1)(x_1)\cdots \pi_f(O_n)(x_n) \right\rangle = \left\langle O_1(x_1)\cdots O_n(x_n)\right\rangle. $$ Primary fields are fields that transform into themselves via $$\pi_f$$. The behaviour of a primary field is characterized by a number $$\Delta$$ called its conformal dimension, and a representation $$\rho$$ of the rotation or Lorentz group. For a primary field, we then have

\pi_f(O)(x) = \Omega(x')^{-\Delta} \rho(R(x')) O(x'), \quad \text{where}\ x'=f^{-1}(x). $$ Here $$\Omega(x)$$ and $$R(x)$$ are the scale factor and rotation that are associated to the conformal transformation $$f$$. The representation $$\rho$$ is trivial in the case of scalar fields, which transform as $$ \pi_f(O)(x) = \Omega(x')^{-\Delta} O(x') $$ . For vector fields, the representation $$\rho$$ is the fundamental representation, and we would have $$ \pi_f(O_\mu)(x) = \Omega(x')^{-\Delta} R_\mu^\nu(x') O_\nu(x') $$.

A primary field that is characterized by the conformal dimension $$\Delta$$ and representation $$\rho$$ behaves as a highest-weight vector in an induced representation of the conformal group from the subgroup generated by dilations and rotations. In particular, the conformal dimension $$ \Delta$$ characterizes a representation of the subgroup of dilations. In two dimensions, the fact that this induced representation is a Verma module appears throughout the literature. For higher-dimensional CFTs (in which the maximally compact subalgebra is larger than the Cartan subalgebra), it has recently been appreciated that this representation is a parabolic or generalized Verma module.

Derivatives (of any order) of primary fields are called descendant fields. Their behaviour under conformal transformations is more complicated. For example, if $$O$$ is a primary field, then $$\pi_f(\partial_\mu O)(x) = \partial_\mu\left(\pi_f(O)(x)\right)$$ is a linear combination of $$ \partial_\mu O$$ and $$O$$. Correlation functions of descendant fields can be deduced from correlation functions of primary fields. However, even in the common case where all fields are either primaries or descendants thereof, descendant fields play an important role, because conformal blocks and operator product expansions involve sums over all descendant fields.

The collection of all primary fields $$O_p$$, characterized by their scaling dimensions $$\Delta_p$$ and the representations $$\rho_p$$, is called the spectrum of the theory.

Dependence on field positions
The invariance of correlation functions under conformal transformations severely constrain their dependence on field positions. In the case of two- and three-point functions, that dependence is determined up to finitely many constant coefficients. Higher-point functions have more freedom, and are only determined up to functions of conformally invariant combinations of the positions.

The two-point function of two primary fields vanishes if their conformal dimensions differ.

\Delta_1\neq \Delta_2 \implies \left\langle O_{1}(x_1)O_{2}(x_2)\right\rangle= 0. $$ If the dilation operator is diagonalizable (i.e. if the theory is not logarithmic), there exists a basis of primary fields such that two-point functions are diagonal, i.e. $$ i\neq j\implies \left\langle O_i O_j\right\rangle = 0$$. In this case, the two-point function of a scalar primary field is

\left\langle O(x_1)O(x_2) \right\rangle = \frac{1}{|x_1-x_2|^{2\Delta}}, $$ where we choose the normalization of the field such that the constant coefficient, which is not determined by conformal symmetry, is one. Similarly, two-point functions of non-scalar primary fields are determined up to a coefficient, which can be set to one. In the case of a symmetric traceless tensor of rank $$\ell$$, the two-point function is
 * $$ \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}},

$$ where the tensor $$I_{\mu,\nu}(x)$$ is defined as

I_{\mu,\nu}(x) = \eta_{\mu\nu} - \frac{2x_\mu x_\nu}{x^2}. $$

The three-point function of three scalar primary fields is

\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}}, $$ where $$x_{ij}=x_i-x_j$$, and $$C_{123}$$ is a three-point structure constant. With primary fields that are not necessarily scalars, conformal symmetry allows a finite number of tensor structures, and there is a structure constant for each tensor structure. In the case of two scalar fields and a symmetric traceless tensor of rank $$\ell$$, there is only one tensor structure, and the three-point function is

\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}}, $$ where we introduce the vector

V_\mu = \frac{x_{13}^\mu x_{23}^2 - x_{23}^\mu x_{13}^2}{|x_{12}||x_{13}||x_{23}|}. $$

Four-point functions of scalar primary fields are determined up to arbitrary functions $$g(u,v)$$ of the two cross-ratios

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}. $$ The four-point function is then

\left\langle \prod_{i=1}^4O_i(x_i)\right\rangle = \frac{\left(\frac{|x_{24}|}{|x_{14}|}\right)^{\Delta_1-\Delta_2} \left(\frac{|x_{14}|}{|x_{13}|}\right)^{\Delta_3-\Delta_4}}{|x_{12}|^{\Delta_1+\Delta_2} |x_{34}|^{\Delta_3+\Delta_4}}g(u,v). $$

Operator product expansion
The operator product expansion (OPE) is more powerful in conformal field theory than in more general quantum field theories. This is because in conformal field theory, the operator product expansion's radius of convergence is finite (i.e. it is not zero). Provided the positions $$x_1,x_2$$ of two fields are close enough, the operator product expansion rewrites the product of these two fields as a linear combination of fields at a given point, which can be chosen as $$ x_2$$ for technical convenience.

The operator product expansion of two fields takes the form

O_1(x_1)O_2(x_2) = \sum_k c_{12k}(x_1-x_2) O_k(x_2), $$

where $$c_{12k}(x)$$ is some coefficient function, and the sum in principle runs over all fields in the theory. (Equivalently, by the state-field correspondence, the sum runs over all states in the space of states.) Some fields may actually be absent, in particular due to constraints from symmetry: conformal symmetry, or extra symmetries.

If all fields are primary or descendant, the sum over fields can be reduced to a sum over primaries, by rewriting the contributions of any descendant in terms of the contribution of the corresponding primary:

O_1(x_1)O_2(x_2) = \sum_p C_{12p}P_p(x_1-x_2,\partial_{x_2}) O_p(x_2), $$

where the fields $$O_p$$ are all primary, and $$C_{12p}$$ is the three-point structure constant (which for this reason is also called OPE coefficient). The differential operator $$ P_p(x_1-x_2,\partial_{x_2})$$ is an infinite series in derivatives, which is determined by conformal symmetry and therefore in principle known.

Viewing the OPE as a relation between correlation functions shows that the OPE must be associative. Furthermore, if the space is Euclidean, the OPE must be commutative, because correlation functions do not depend on the order of the fields, i.e. $$ O_1(x_1)O_2(x_2) = O_2(x_2)O_1(x_1)$$.

The existence of the operator product expansion is a fundamental axiom of the conformal bootstrap. However, it is generally not necessary to compute operator product expansions and in particular the differential operators $$ P_p(x_1-x_2,\partial_{x_2})$$. Rather, it is the decomposition of correlation functions into structure constants and conformal blocks that is needed. The OPE can in principle be used for computing conformal blocks, but in practice there are more efficient methods.

Conformal blocks and crossing symmetry
Using the OPE $$O_1(x_1)O_2(x_2)$$, a four-point function can be written as a combination of three-point structure constants and s-channel conformal blocks,

\left\langle \prod_{i=1}^4 O_i(x_i) \right\rangle = \sum_p C_{12p}C_{p34} G_p^{(s)}(x_i). $$ The conformal block $$G_p^{(s)}(x_i)$$ is the sum of the contributions of the primary field $$O_p$$ and its descendants. It depends on the fields $$O_i$$ and their positions. If the three-point functions $$\left\langle O_1O_2O_p\right\rangle$$ or $$\left\langle O_3O_4O_p\right\rangle$$ involve several independent tensor structures, the structure constants and conformal blocks depend on these tensor structures, and the primary field $$O_p$$ contributes several independent blocks. Conformal blocks are determined by conformal symmetry, and known in principle. To compute them, there are recursion relations and integrable techniques.

Using the OPE $$O_1(x_1)O_4(x_4)$$ or $$O_1(x_1)O_3(x_3)$$, the same four-point function is written in terms of t-channel conformal blocks or u-channel conformal blocks,

\left\langle \prod_{i=1}^4 O_i(x_i) \right\rangle = \sum_p C_{14p}C_{p23} G_p^{(t)}(x_i) =\sum_p C_{13p}C_{p24} G_p^{(u)}(x_i). $$ The equality of the s-, t- and u-channel decompositions is called crossing symmetry: a constraint on the spectrum of primary fields, and on the three-point structure constants.

Conformal blocks obey the same conformal symmetry constraints as four-point functions. In particular, s-channel conformal blocks can be written in terms of functions $$g_p^{(s)}(u,v)$$ of the cross-ratios. While the OPE $$O_1(x_1)O_2(x_2)$$ only converges if $$|x_{12}|<\min(|x_{23}|,|x_{24}|)$$, conformal blocks can be analytically continued to all (non pairwise coinciding) values of the positions. In Euclidean space, conformal blocks are single-valued real-analytic functions of the positions except when the four points $$x_i$$ lie on a circle but in a singly-transposed cyclic order [1324], and only in these exceptional cases does the decomposition into conformal blocks not converge.

A conformal field theory in flat Euclidean space $$\mathbb{R}^d$$ is thus defined by its spectrum $$\{(\Delta_p,\rho_p)\}$$ and OPE coefficients (or three-point structure constants) $$\{C_{pp'p''}\}$$, satisfying the constraint that all four-point functions are crossing-symmetric. From the spectrum and OPE coefficients (collectively referred to as the CFT data), correlation functions of arbitrary order can be computed.

Unitarity
A conformal field theory is unitary if its space of states has a positive definite scalar product such that the dilation operator is self-adjoint. Then the scalar product endows the space of states with the structure of a Hilbert space.

In Euclidean conformal field theories, unitarity is equivalent to reflection positivity of correlation functions: one of the Osterwalder-Schrader axioms.

Unitarity implies that the conformal dimensions of primary fields are real and bounded from below. The lower bound depends on the spacetime dimension $$d$$, and on the representation of the rotation or Lorentz group in which the primary field transforms. For scalar fields, the unitarity bound is

\Delta \geq \frac12(d-2). $$

In a unitary theory, three-point structure constants must be real, which in turn implies that four-point functions obey certain inequalities. Powerful numerical bootstrap methods are based on exploiting these inequalities.

Compactness
A conformal field theory is compact if it obeys three conditions:
 * All conformal dimensions are real.
 * For any $$\Delta\in\mathbb{R}$$ there are finitely many states whose dimensions are less than $$\Delta$$.
 * There is a unique state with the dimension $$\Delta =0$$, and it is the vacuum state, i.e. the corresponding field is the identity field.

(The identity field is the field whose insertion into correlation functions does not modify them, i.e. $$ \left\langle I(x)\cdots \right\rangle = \left\langle \cdots \right\rangle $$.) The name comes from the fact that if a 2D conformal field theory is also a sigma model, it will satisfy these conditions if and only if its target space is compact.

It is believed that all unitary conformal field theories are compact in dimension $$d>2$$. Without unitarity, on the other hand, it is possible to find CFTs in dimension four and in dimension $$4 - \epsilon$$ that have a continuous spectrum. And in dimension two, Liouville theory is unitary but not compact.

Extra symmetries
A conformal field theory may have extra symmetries in addition to conformal symmetry. For example, the Ising model has a $$\mathbb{Z}_2$$ symmetry, and superconformal field theories have supersymmetry.

Mean field theory
A generalized free field is a field whose correlation functions are deduced from its two-point function by Wick's theorem. For instance, if $$\phi$$ is a scalar primary field of dimension $$\Delta$$, its four-point function reads

\left\langle \prod_{i=1}^4\phi(x_i) \right\rangle = \frac{1}{|x_{12}|^{2\Delta}|x_{34}|^{2\Delta}} + \frac{1}{|x_{13}|^{2\Delta}|x_{24}|^{2\Delta}} + \frac{1}{|x_{14}|^{2\Delta}|x_{23}|^{2\Delta}}. $$ For instance, if $$\phi_1,\phi_2$$ are two scalar primary fields such that $$\langle \phi_1\phi_2\rangle=0$$ (which is the case in particular if $$\Delta_1\neq\Delta_2$$), we have the four-point function

\Big\langle \phi_1(x_1)\phi_1(x_2)\phi_2(x_3)\phi_2(x_4)\Big\rangle = \frac{1}{|x_{12}|^{2\Delta_1}|x_{34}|^{2\Delta_2}}. $$ Mean field theory is a generic name for conformal field theories that are built from generalized free fields. For example, a mean field theory can be built from one scalar primary field $$\phi$$. Then this theory contains $$\phi$$, its descendant fields, and the fields that appear in the OPE $$\phi \phi$$. The primary fields that appear in $$\phi \phi$$ can be determined by decomposing the four-point function $$\langle\phi\phi\phi\phi\rangle$$ in conformal blocks: their conformal dimensions belong to $$2\Delta+2\mathbb{N}$$: in mean field theory, the conformal dimension is conserved modulo integers.

Similarly, it is possible to construct mean field theories starting from a field with non-trivial Lorentz spin. For example, the 4d Maxwell theory (in the absence of charged matter fields) is a mean field theory built out of an antisymmetric tensor field $$F_{\mu \nu}$$ with scaling dimension $$\Delta = 2$$.

Mean field theories have a Lagrangian description in terms of a quadratic action involving Laplacian raised to an arbitrary real power (which determines the scaling dimension of the field). For a generic scaling dimension, the power of the Laplacian is non-integer. The corresponding mean field theory is then non-local (e.g. it does not have a conserved stress tensor operator).

Critical Ising model
The critical Ising model is the critical point of the Ising model on a hypercubic lattice in two or three dimensions. It has a $$\mathbb{Z}_2$$ global symmetry, corresponding to flipping all spins. The two-dimensional critical Ising model includes the $$\mathcal{M}(4,3)$$ Virasoro minimal model, which can be solved exactly. There is no Ising CFT in $$d \geq 4$$ dimensions.

Critical Potts model
The critical Potts model with $$q=2,3,4,\cdots$$ colors is a unitary CFT that is invariant under the permutation group $$S_q$$. It is a generalization of the critical Ising model, which corresponds to $$q=2$$. The critical Potts model exists in a range of dimensions depending on $$q$$.

The critical Potts model may be constructed as the continuum limit of the Potts model on d-dimensional hypercubic lattice. In the Fortuin-Kasteleyn reformulation in terms of clusters, the Potts model can be defined for $$q\in\mathbb{C}$$, but it is not unitary if $$q$$ is not integer.

Critical O(N) model
The critical O(N) model is a CFT invariant under the orthogonal group. For any integer $$N$$, it exists as an interacting, unitary and compact CFT in $$d=3$$ dimensions (and for $$N=1$$ also in two dimensions). It is a generalization of the critical Ising model, which corresponds to the O(N) CFT at $$N=1$$.

The O(N) CFT can be constructed as the continuum limit of a lattice model with spins that are N-vectors, discussed here.

Alternatively, the critical $$O(N)$$ model can be constructed as the $$\varepsilon \to 1$$ limit of Wilson-Fisher fixed point in $$d=4-\varepsilon$$ dimensions. At $$\varepsilon = 0$$, the Wilson-Fisher fixed point becomes the tensor product of $$N$$ free scalars with dimension $$\Delta = 1$$. For $$0 < \varepsilon < 1$$ the model in question is non-unitary.

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 $$N \to \infty$$ limit of the critical O(N) model is well-understood.

Conformal gauge theories
Some conformal field theories in three and four dimensions admit a Lagrangian description in the form of a gauge theory, either abelian or non-abelian. Examples of such CFTs are conformal QED with sufficiently many charged fields in $$d=3$$ or the Banks-Zaks fixed point in $$d=4$$.

Continuous phase transitions
Continuous phase transitions (critical points) of classical statistical physics systems with D spatial dimensions are often described by Euclidean conformal field theories. A necessary condition for this to happen is that the critical point should be invariant under spatial rotations and translations. However this condition is not sufficient: some exceptional critical points are described by scale invariant but not conformally invariant theories. If the classical statistical physics system is reflection positive, the corresponding Euclidean CFT describing its critical point will be unitary.

Continuous quantum phase transitions in condensed matter systems with D spatial dimensions may be described by Lorentzian D+1 dimensional conformal field theories (related by Wick rotation to Euclidean CFTs in D+1 dimensions). Apart from translation and rotation invariance, an additional necessary condition for this to happen is that the dynamical critical exponent z should be equal to 1. CFTs describing such quantum phase transitions (in absence of quenched disorder) are always unitary.

String theory
World-sheet description of string theory involves a two-dimensional CFT coupled to dynamical two-dimensional quantum gravity (or supergravity, in case of superstring theory). Consistency of string theory models imposes constraints on the central charge of this CFT, which should be c=26 in bosonic string theory and c=10 in superstring theory. Coordinates of the spacetime in which string theory lives correspond to bosonic fields of this CFT.

AdS/CFT correspondence
Conformal field theories play a prominent role 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 × S5, and d = 3, N = 6 super-Chern–Simons theory, which is dual to M-theory on AdS4 × 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.)

title: Conformal field theory


text:
Conformal group

In mathematics, the conformal group of an inner product space is the group of transformations from the space to itself that preserve angles. More formally, it is the group of transformations that preserve the conformal geometry of the space.

Several specific conformal groups are particularly important:
 * The conformal orthogonal group. If V is a vector space with a quadratic form Q, then the conformal orthogonal group CO(V, Q) is the group of linear transformations T of V for which there exists a scalar λ such that for all x in V
 * $$Q(Tx) = \lambda^2 Q(x)$$
 * For a definite quadratic form, the conformal orthogonal group is equal to the orthogonal group times the group of dilations.

 * The conformal group of the sphere is generated by the inversions in circles. This group is also known as the Möbius group.
 * In Euclidean space En, n > 2, the conformal group is generated by inversions in hyperspheres.
 * In a pseudo-Euclidean space Ep,q, the conformal group is Conf(p, q) ≃ O(p + 1, q + 1) / Z2.

All conformal groups are Lie groups.

Angle analysis
In Euclidean geometry one can expect the standard circular angle to be characteristic, but in pseudo-Euclidean space there is also the hyperbolic angle. In the study of special relativity the various frames of reference, for varying velocity with respect to a rest frame, are related by rapidity, a hyperbolic angle. One way to describe a Lorentz boost is as a hyperbolic rotation which preserves the differential angle between rapidities. Thus, they are conformal transformations with respect to the hyperbolic angle.

A method to generate an appropriate conformal group is to mimic the steps of the Möbius group as the conformal group of the ordinary complex plane. Pseudo-Euclidean geometry is supported by alternative complex planes where points are split-complex numbers or dual numbers. Just as the Möbius group requires the Riemann sphere, a compact space, for a complete description, so the alternative complex planes require compactification for complete description of conformal mapping. Nevertheless, the conformal group in each case is given by linear fractional transformations on the appropriate plane.

Mathematical definition
Given a (Pseudo-)Riemannian manifold $$M$$ with conformal class $$[g]$$, the conformal group $$\text{Conf}(M)$$ is the group of conformal maps from $$M$$ to itself.

More concretely, this is the group of angle-preserving smooth maps from $$M$$ to itself. However, when the signature of $$[g]$$ is not definite, the 'angle' is a hyper-angle which is potentially infinite.

For Pseudo-Euclidean space, the definition is slightly different. $$\text{Conf}(p,q)$$ is the conformal group of the manifold arising from conformal compactification of the pseudo-Euclidean space $$\mathbf{E}^{p, q}$$ (sometimes identified with $$\mathbb{R}^{p,q}$$ after a choice of orthonormal basis). This conformal compactification can be defined using $$S^p\times S^q$$, considered as a submanifold of null points in $$\mathbb{R}^{p+1, q+1}$$ by the inclusion $$(\mathbf{x}, \mathbf{t})\mapsto X = (\mathbf{x}, \mathbf{t})$$ (where $$X$$ is considered as a single spacetime vector). The conformal compactification is then $$S^p\times S^q$$ with 'antipodal points' identified. This happens by projectivising the space $$\mathbb{R}^{p+1,q+1}$$. If $$N^{p,q}$$ is the conformal compactification, then $$\text{Conf}(p,q) := \text{Conf}(N^{p,q})$$. In particular, this group includes inversion of $$\mathbb{R}^{p,q}$$, which is not a map from $$\mathbb{R}^{p,q}$$ to itself as it maps the origin to infinity, and maps infinity to the origin.

Conf(p,q)
For Pseudo-Euclidean space $$\mathbb{R}^{p,q}$$, the Lie algebra of the conformal group is given by the basis $$\{M_{\mu\nu}, P_\mu, K_\mu, D\}$$ with the following commutation relations:

and with all other brackets vanishing. Here $$\eta_{\mu\nu}$$ is the Minkowski metric.

In fact, this Lie algebra is isomorphic to the Lie algebra of the Lorentz group with one more space and one more time dimension, that is, $$\mathfrak{conf}(p,q) \cong \mathfrak{so}(p+1, q+1)$$. It can be easily checked that the dimensions agree. To exhibit an explicit isomorphism, define

It can then be shown that the generators $$J_{ab}$$ with $$a, b = -1, 0, \cdots, n = p+q$$ obey the Lorentz algebra relations with metric $$\tilde \eta_{ab} = \operatorname{diag}(-1, +1, -1, \cdots, -1, +1, \cdots, +1)$$.

Conformal group in two spacetime dimensions
For two-dimensional Euclidean space or one-plus-one dimensional spacetime, the space of conformal symmetries is much larger. In physics it is sometimes said the conformal group is infinite-dimensional, but this is not quite correct as while the Lie algebra of local symmetries is infinite dimensional, these do not necessarily extend to a Lie group of well-defined global symmetries.

For spacetime dimension $$n > 2$$, the local conformal symmetries all extend to global symmetries. For $$n = 2$$ Euclidean space, after changing to a complex coordinate $$z = x + iy$$ local conformal symmetries are described by the infinite dimensional space of vector fields of the form

Hence the local conformal symmetries of 2d Euclidean space is the infinite-dimensional Witt algebra.

Conformal group of spacetime
In 1908, Harry Bateman and Ebenezer Cunningham, two young researchers at University of Liverpool, broached the idea of a conformal group of spacetime They argued that the kinematics groups are perforce conformal as they preserve the quadratic form of spacetime and are akin to orthogonal transformations, though with respect to an isotropic quadratic form. The liberties of an electromagnetic field are not confined to kinematic motions, but rather are required only to be locally proportional to a transformation preserving the quadratic form. Harry Bateman's paper in 1910 studied the Jacobian matrix of a transformation that preserves the light cone and showed it had the conformal property (proportional to a form preserver). Bateman and Cunningham showed that this conformal group is "the largest group of transformations leaving Maxwell’s equations structurally invariant." The conformal group of spacetime has been denoted $C(1,3)$

Isaak Yaglom has contributed to the mathematics of spacetime conformal transformations in split-complex and dual numbers. Since split-complex numbers and dual numbers form rings, not fields, the linear fractional transformations require a projective line over a ring to be bijective mappings.

It has been traditional since the work of Ludwik Silberstein in 1914 to use the ring of biquaternions to represent the Lorentz group. For the spacetime conformal group, it is sufficient to consider linear fractional transformations on the projective line over that ring. Elements of the spacetime conformal group were called spherical wave transformations by Bateman. The particulars of the spacetime quadratic form study have been absorbed into Lie sphere geometry.

Commenting on the continued interest shown in physical science, A. O. Barut wrote in 1985, "One of the prime reasons for the interest in the conformal group is that it is perhaps the most important of the larger groups containing the Poincaré group."

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Conformal group


text:
Gary Soto

Gary Anthony Soto (born April 12, 1952) is an American poet, novelist, and memoirist.

Life and career
Soto was born to Mexican-American parents Manuel (1910–1957) and Angie Soto (1924-). In his youth, he worked in the fields of the San Joaquin Valley. Soto's father died in 1957, when he was five years old. As his family had to struggle to find work, he had little time or encouragement in his studies. Soto notes that in spite of his early academic record, while at high school he found an interest in poetry through writers such as Ernest Hemingway, John Steinbeck, Jules Verne, Robert Frost and Thornton Wilder.

Soto attended Fresno City College and California State University, Fresno, where he earned his B.A. degree in English in 1974, studying with poet Philip Levine. He did graduate work in poetry writing at the University of California, Irvine, where he was the first Mexican-American to earn a M.F.A. in 1976. He states that he wanted to become a writer in college after discovering the novelist Gabriel García Márquez and the contemporary poets Edward Field, W. S. Merwin, Charles Simic, James Wright and Pablo Neruda, whom he calls "the master of them all."

Soto taught at University of California, Berkeley and at University of California, Riverside, where he was a Distinguished Professor.

Soto was a 'Young People's Ambassador' for the United Farm Workers of America, introducing young people to the organization's work and goals. Soto became the sponsor for the Pattonville High School Spanish National Honor Society in 2009.

Soto lives in northern California, dividing his time between Berkeley and Fresno, but is no longer teaching.

Work
Soto's poetry focuses on daily experiences, often reflecting on his life as a Mexican American. Regarding his relationship with the Mexican-American community, Soto commented "as a writer, my duty is not to make people perfect, particularly Mexican Americans. I’m not a cheerleader. I’m one who provides portraits of people in the rush of life."

Soto writes novels, plays and memoirs, and has edited several literary anthologies. His story "The No-Guitar Blues" was made into a film, and he produced another film based on his book "The Pool Party." He is a prolific writer of children's books.

About his work Joyce Carol Oates noted "Gary Soto's poems are fast, funny, heartening, and achingly believable, like Polaroid love letters, or snatches of music heard out of a passing car; patches of beauty like patches of sunlight; the very pulse of a life."

Awards and honors
Soto's first collection of poems, The Elements of San Joaquin, won the United States Award of the International Poetry Forum in 1976 prior to its publication in the Pitt Poetry Series in 1977. The New York Times Book Review also honored the book by reprinting six of the poems. In 1985, his memoir Living Up the Street received the Before Columbus Foundation's American Book Award.

In 1993, Soto received the Andrew Carnegie Medal for Film Excellence from the Association for Library Service to Children for his production work on the film The Pool Party. In 1999, Soto received the Hispanic Heritage Award for Literature, the Author-Illustrator Civil Rights Award from the National Education Association, and the PEN Center West Book Award for Petty Crimes.

Other honors include the "Discovery"/The Nation Prize, the Bess Hokin Prize and the Levinson Award from Poetry. He has received The California Library Association's John and Patricia Beatty Award (twice), a Recognition of Merit from the Claremont Graduate School for Baseball in April, the Silver Medal from the Commonwealth Club of California, and the Tomás Rivera Prize.

The library at Winchell Elementary School in Fresno was named after Soto.

In 2011, the Old Administration Building at Fresno City College became the permanent home of the Gary Soto Literary Museum.

In 2014, Soto received the Phoenix Award for his 1994 children's book Jesse. The award committee stated: "Jesse is both a coming-of-age story of one Mexican-American boy with a poetic sensibility and the story of a community and a country at a difficult time—facing poverty and prejudice and war, problems we are still facing today. Jesse offers an unembellished slice of life in Vietnam-era Fresno, California."

Poetry collections

 * Downtime (Gunpowder Press, 2023)
 * Meatballs for the People: Proverbs to Chew On (Red Hen Press, 2017)
 * Partly Cloudy: Poems of love and longing (Harcourt, 2009)
 * A Simple Plan (Chronicle Books, 2007)
 * One Kind of Faith (Chronicle Books, 2003)
 * A Natural Man (Chronicle Books, 1999)
 * Junior College (1997)
 * New and selected poems (Chronicle Books, 1995) National Book Award finalist
 * Canto Familiar/Familiar Song (1994)
 * Neighborhood Odes (1992)
 * Home Course in Religion (1991)
 * Saturday at the Canal  (1991)
 * Who Will Know Us? (1990)
 * Black Hair (1985)
 * Where Sparrows Work Hard (1981)
 * The Tale of Sunlight (1978)
 * The Elements of San Joaquin (1977)
 * ''Waiting at the curb: Lynwood California (1967)
 * ''Waiting at the curb: Lynwood California (1967)

Young adult/children's books

 * Baseball in April (1990)
 * A Fire in My Hands (1991)
 * Taking Sides (1991)
 * Pacific Crossing (1992), sequel to Taking Sides added by DaeQuan Jones
 * Too Many Tamales (1992)
 * The Skirt (1992)
 * The Pool Party (1993)
 * Local News (1993)
 * Jesse (1994)
 * 7th grade (1995)
 * Crazy Weekend (1994)
 * Boys at Work (1995)
 * Summer On Wheels (1995)
 * Canto Familiar (1995)
 * Buried Onions (1997)
 * The Cat's Meow (1997)
 * Jessie De La Cruz: A Profile of a United Farm Worker (2000)
 * Fearless Fernie (2002)
 * If the Shoe Fits (2002)
 * Marisol (2005)
 * When Dad Came Back (2011), ebook

Chato
Beginning in 1995 with Chato's Kitchen (Chato y su cena), Soto released a series of children's picture books in Spanish and English about a real, cool cat (gato), a low rider from the barrio of East Los Angeles. They were illustrated by Susan Guevara, and the second one Chato and the Party Animals (Chato y los amigos pachangueros.) (2000) won the Pura Belpre Medal for best illustration in 2002. The series continued with Chato Goes Cruisin' (2004) and Chato's Day of Dead (2006).

Anthologies as editor

 * Entrance: Four Latino Poets (1976)
 * California Childhood (1988)
 * Pieces of Heart (1993)
 * Afterlife(1999)

Memoir

 * Why I Don't Write Children's Literature (2015)
 * What Poets Are Like: Up and Down with the Writing Life (2013)
 * Living Up the Street (1985), American Book Award
 * Small Faces (1986)
 * Lesser Evils: Ten Quartets (1988)
 * A Summer Life (1990)
 * The Effects of Knut Hamsun on a Fresno Boy (2001)
 * The Jacket (1983)

Plays

 * Novio Boy: A play (2006)

Film

 * The No-Guitar Blues (1991) based on a story from Baseball in April
 * The Bike (1991, director)
 * The Pool Party (producer, 1992) Carnegie Medal for Excellence in Children's Video (1993)
 * Novio Boy (1994, director)

title: Gary Soto


text:
Pietermaritzburg

Pietermaritzburg (uMgungundlovu) is the capital and second-largest city in the province of KwaZulu-Natal, South Africa after Durban. It was named in 1838 and is currently governed by the Msunduzi Local Municipality. The town was named in Zulu after King Dingane's royal homestead uMgungundlovu. Pietermaritzburg is popularly called Maritzburg in Afrikaans and is often informally abbreviated to PMB. It is a regionally important industrial hub, producing aluminium, timber and dairy products, as well as the main economic hub of uMgungundlovu District Municipality. The public sector is a major employer in the city due to local, district and provincial government offices located here.

The city has many schools and tertiary education institutions, including a campus of the University of KwaZulu-Natal. It had a population of 228,549 in 1991; the current population is estimated at over 600,000 residents (including neighbouring townships) and has one of the largest populations of Indian South Africans in South Africa.

Pietermaritzburg is an emerging metropolitan area.

History
The city was occupied by Voortrekkers, in April 1838 following the murder of Piet Retief and his seventy-strong party at the Zulu Capital, Mgungundlovu (6 February 1838), when seeking land to settle around Port Natal (The Natal-Land Treaty), and from where the reprisal Wenkommando departed (November 1838) to defeat Dingane at the Battle of Blood River (16 December 1838). Pietermaritzburg is approximately 160 km direct from the Zulu capital, Mgungundlovu. Prior to the battle (9 December 1838) a vow was taken that if the Boers were granted victory over the Zulu, then a sacred church would be built which still stands today, The Church of the Vow. It was Jan Gerritze Bantjes, Secretary General to Andries Pretorius, Commander of the campaign who arranged the financing of the church by the Boer towns folk which by then had taken a low priority after the war. The town grew rapidly to become the capital of the short-lived Boer Republic or Natalia. Britain took over Pietermaritzburg in 1843 and it became the seat of the Natal Colony's administration with the first lieutenant-governor, Martin West, making it his home. Fort Napier, named after the governor of the Cape Colony, Sir George Thomas Napier, was built to house a garrison. In 1893, Natal received responsibility for its own government, and an assembly building was built along with the city hall.

On 7 June 1893, while the young Mahatma Gandhi was on his way to Pretoria, a white man objected to Gandhi's presence in a first-class carriage. Despite Gandhi having a first-class ticket, he was ordered by the conductor to move to the van compartment at the end of the train: he refused, and he was removed from the train at Pietermaritzburg. Shivering through the winter night in the waiting room of the station, Gandhi made the momentous decision to stay on in South Africa and fight the racial discrimination against Indians there. Out of that struggle emerged his unique version of nonviolent resistance, Satyagraha. Today, a bronze statue of Gandhi stands in Church Street, in the city center.

In 1910, when the Union of South Africa was formed, Natal became a province of the Union, and Pietermaritzburg became one of the capitals of the country. During apartheid, the city was segregated into various sections. 90% of the Indian population was moved to the suburb of Northdale while most of its Zulu inhabitants were moved to the neighbouring township of Edendale and white inhabitants were moved out of those areas.

Name
There exist two interpretations about the origin of the city's name. One is that it was named after Piet Retief (Pieter Mauritz Retief) and Gerrit Maritz (Gerhardus Marthinus Maritz), two Voortrekker leaders. The other is that it was originally named after Piet Retief's full name alone. In this interpretation the original name was "Pieter Mauritz' Burg", later transliterated to the current name. In 1938, the city announced officially that the second element Maritz should also honour Gert Maritz.

In fact, neither Retief nor Maritz ever reached Pietermaritzburg. Retief was killed by Dingane, successor to Shaka, king of the Zulus. Maritz died of illness on 23 September 1838 near the present-day town of Estcourt, some eighty kilometres northwest of Pietermaritzburg, after the battle with the Zulus at Bloukranz.

At the time of the rise of the Zulu Empire, the site that was to become Pietermaritzburg was called uMgungundlovu. This is popularly translated from the Zulu as "Place of the Elephant", although it could also be translated to mean "The elephant wins". Umgungundlovu is thus thought to be the site of some Zulu king's victory since "Elephant" (Indlovu) is a name traditionally taken by the Zulu monarch. Legend has it that Shaka had his warriors hunt elephant there to sell the ivory to English traders at Durban (then called Port Natal). Today, the town is still called by its Voortrekker name, although the municipality of which it is part bears the Zulu name.

Education
The University of Natal was founded in 1910. In 1922 the University extended to Durban. The two campuses were incorporated into the University of Natal in March 1949. It became a major voice in the struggle against apartheid and was one of the first universities in the country to provide education to African students. It became the University of KwaZulu-Natal on 1 January 2004.

Other historical events

 * The first newspaper in Natal, the Natal Witness (now known as The Witness), was published in 1846.
 * The 46 hectare Botanic Gardens were created in 1872 by the Botanic Society of Natal.
 * The city hall, which is the largest red-brick building in the Southern Hemisphere, was destroyed by a fire in 1895, but was rebuilt in 1901. It houses the largest pipe organ built by the Sheffield organ building company, Brindley & Foster.
 * The British built a concentration camp here during the Second Boer War to house Boer women and children.
 * In 1919 communist activist David Ivon Jones was arrested for distributed an anti-segregation leaflet he co-authored in Pietermaritzburg. His trial became the first major court case against communism in South African history.
 * During the Second World War, Italian prisoners of war were housed in Pietermaritzburg. During their stay, they built a church, which remains standing as a heritage site today.
 * In 1962, Nelson Mandela was arrested in the nearby town of Howick to the north of Pietermaritzburg. The arrest marked the beginning of Mandela's 27 years of imprisonment. A small monument has been erected at the location of his arrest. Immediately after his arrest Mandela was taken to the Old Prison in Pietermaritzburg. After a night in the prison, he was taken to Magistrate J. Buys's office in the old Magistrates Court Building in Commercial Road (now Chief Albert Luthuli Road), and was remanded for trial in Johannesburg.

Capital status
Pietermaritzburg was the capital of the Colony of Natal until 1910, when the Union of South Africa was formed, and Natal became a province of the Union. Prior to 1994, Pietermaritzburg was the capital of Natal Province. Following the first post-apartheid elections in South Africa, as a result of which the Inkatha Freedom Party won a majority in the KwaZulu-Natal provincial government, Pietermaritzburg shared its status as capital of the (then newly created) province of KwaZulu-Natal with Ulundi. Pietermaritzburg became the legislative capital of the new province, while Ulundi became the administrative capital. The IFP, being strongly Zulu nationalist, desired that Ulundi, the capital of the Zulu Kingdom at the time of its fall to the British in the Anglo-Zulu War, be the post-apartheid capital of the province. Ulundi had also been the capital of the bantustan KwaZulu, which makes up a portion of modern KwaZulu-Natal. However, Ulundi severely lacked the infrastructure to be an effective seat of government, and the African National Congress (ANC) and the Democratic Party, the two other strong political parties in the province, among others, called for Pietermaritzburg alone to be the capital. The debate came to an end when the ANC came to power in the province in 2004, and named Pietermaritzburg the sole capital of KwaZulu-Natal. This has resulted in the relocation of several government offices to Pietermaritzburg, an action that has generally been welcomed as a positive development for the region. Since 2004, progress such as the modernisation of several buildings in the city centre and a proliferation of retail and housing developments in the suburbs are results of recent investment in the city by both the public and private sectors.

Economy
Until the late-1990s, the region was renowned for the production of high quality textile, clothing and footwear products. An example of the latter included the production of Doc Marten shoes. However, these industries have declined in the area due to lower production costs in Asia. Extensive timber plantations and numerous citrus farms surround the city, contributing a significant share of the city's output.

The Liberty Group has made major investments in several phases in the region since 2002 with the development of the Liberty Midlands Mall (the area's largest shopping centre by gross lettable area and its most prestigious) and Stay Easy hotel. Hulett's Aluminium and Willowton cooking oil contributes a substantial part of the region's industrial output.

Pietermaritzburg is also the city where major South African pizza franchise Debonairs Pizza was founded.

Coat of arms
The Pietermaritzburg borough council assumed a coat of arms in 1861. The shield depicted an elephant standing on grass, and a cross of five stars was placed above the shield. The motto was Umgungunhlovu. It is unclear what the original colours were, but by 1910 the shield had been coloured blue. By 1931, the council had approved new artwork in which the stars were placed on a radiant sun. The arms were registered with the Natal Provincial Administration in November 1950. Many early renditions of the coat of arms, visible on older public building and wrought iron lampposts, features an elephant which is clearly an Asian elephant rather than an African elephant. More recent versions reflect an African elephant.

The final version of the arms was granted by the College of Arms in May 1961. It was registered at the Bureau of Heraldry in May 1973. The blazon was: Per fess Azure and Vert, over all an elephant statant Or, tusked Argent (i.e. the shield was divided horizontally into blue over green, and displayed a gold elephant with silver tusks). The crest was changed to a blue sun displaying gold and silver stars, and two black wildebeest were added as supporters. Each had a shield on its shoulder, the dexter supporter's shield displaying the Union Jack and the sinister supporter's the flag of the Natalia republic.

Road
Pietermaritzburg is on the N3 highway, the primary route between the harbour city of Durban, some 90 km away, and the Pretoria-Witwatersrand-Vereeniging conurbation.

The R33 connects Pietermaritzburg with Greytown and Paulpietersburg, while the R56 connects Pietermaritzburg with Ixopo and Kokstad.

Air
The city is served by Pietermaritzburg Airport, which has regular scheduled services to OR Tambo International Airport in Johannesburg. There are 4 flights to Johannesburg and 4 return flights daily, with a reduced number of weekend flights. It once operated flights to Cape Town International Airport in Cape Town – this route was terminated in 2019.

Rail
Pietermaritzburg Railway Station is served by long distance trains on the Durban-Johannesburg and Durban-Cape Town routes of Shosholoza Meyl.

It has been proposed that the Metrorail commuter rail system be expanded from Cato Ridge to Pietermaritzburg.

Bus
The Pietermaritzburg Municipality historically operated a tram service, which was closed down in the 1940s, and a bus service, which was closed down in the 1980s.

Nowadays regular daily bus services connect Pietermaritzburg to other major cities in South Africa. The bus station is located on Burger Street, opposite the McDonalds Centre, and it serves major bus companies. Greyhound and Intercape are the most reliable and they operate several round-trips from Durban to Johannesburg daily, where Pietermaritzburg is the second to last stop before Durban.

Taxis
Pietermaritzburg has two types of taxi services: metered taxis and minibus taxis. Unlike in many cities, metered taxis are not allowed to drive around the city to solicit fares and instead must be called and ordered to a specific location. A number of companies service Pietermaritzburg and surrounding areas. These taxis can also be called upon for airport transfers, point to point pickups and shuttles.

Minibus taxis are the standard mode of transport for the majority of the population who cannot afford private vehicles.

Integrated Rapid Public Transport (IRPT)
A bus rapid transit system is currently being developed in Pietermaritzburg.

The initial phase would see the development of a west to east corridor from Edendale to Northdale. The BRT route will traverse the CBD along Church Street.

Geography

Climate
Pietermaritzburg has a dry-winter humid subtropical climate (Köppen climate classification: Cwa). Summers are warm and occasionally hot, with frequent rainfall. Winters are dry with high diurnal temperature variation, with light air frosts being possible.

Sport

 * Pietermaritzburg is home to the oldest football (soccer) club of South Africa and the African continent: Savages FC PMB, founded in 1882.
 * The Comrades Marathon takes place annually in June between Pietermaritzburg and Durban. It has been run since 1921 and attracts thousands of entrants. The start of the race alternates between the two cities.
 * Between December 1953 and November 1981, Pietermaritzburg had an international standard motor racing circuit located on the outskirts of the city. The Roy Hesketh Circuit measured 1.803 mi. The circuit was named after South African driver Roy Hesketh. During its period of operation it hosted rounds of the South African National Drivers Championship, the Springbok Series  and national Formula Atlantic races. The circuit was also like a second home to Mike Hailwood. The track was known for hosting the Easter races as well – a festival of racing over three days. The expansion of the town of Pietermaritzburg eventually led to the redevelopment of the site as a residential and business zone after racing ceased at the end of 1981. The section from Henry's Knee to the top of Beacon still exists, and is undergoing protection from further development as an important piece of Pietermaritzburg's history. From 1948 until 1953 races were held at the 3.4 mi Alexandra Park Street Circuit on streets within the town of Pietermaritzburg.
 * Former Springbok and World Cup Winner Joel Stransky was born in Pietermaritzburg on 16 July 1967, he matriculated from Maritzburg College.
 * Former South African Cricketer Jonty Rhodes was born in Pietermaritzburg on 27 July 1969. He also attended Merchiston Prep school and later matriculated from Maritzburg College.
 * The English cricketer Kevin Pietersen was born in Pietermaritzburg on 27 June 1980, as was the Springbok cricketer Cuan McCarthy.
 * The yearly Amashovashova is a 106 km road cycling classic race held since 1986 which starts in Pietermartizburg and finishes in Durban. It is normally held in October.
 * South African cricketer David Miller was born in Pietermaritzburg on 10 June 1989.
 * In 2010, the BMX Racing World Championship took place in the city of Pietermaritzburg between 15 July and 1 August.
 * In January there is an annual canoe race, the Dusi Canoe Marathon, from Pietermaritzburg to Durban. The route follows the Msunduzi River into the Mgeni River, through the Valley of a Thousand Hills into the Inanda Dam and from here to the mouth of the Mgeni River.
 * The Midmar Mile is one of the largest open-water swimming events in the world; taking place at Midmar Dam, north of Pietermaritzburg in February every year, it attracts over 16,000 swimmers from around the world.
 * The Harry Gwala Stadium is a Multi-purpose stadium, and is mostly used for football matches, it is the home ground of Premier Soccer League club Maritzburg United.
 * The Pietermaritzburg Msunduzi Athletics Stadium is currently being developed.
 * The Pietermaritzburg Oval is considered one of the most picturesque cricket grounds in South Africa, and it hosted two matches during the 2003 Cricket World Cup.
 * Pietermaritzburg cricket ground is notable as one of the two grounds used regularly for first-class cricket that have a tree within the boundary (the other is St Lawrence Ground in Canterbury, Kent).
 * Woodburn Stadium is currently used for most club rugby in Pietermaritzburg.
 * The Kershaw Park Tennis Complex is a tennis stadium.
 * The above facilities form the centre of the Pietermaritzburg sports precinct.

Tourism
Some of the area's tourist attractions include; the KwaZulu-Natal Museum, City Hall, Colonial Buildings, Imperial Hotel, Comrades House and SANBI Botanical Gardens.

Attractions in the surrounding areas include; Albert Falls Nature Reserve, Howick Falls, Midmar Public Nature Reserve, Queens Elizabeth Park and World's View.

Schools

 * Alexandra High School
 * Carter High School (South Africa)
 * Cordwalles Preparatory School
 * Epworth School
 * Hilton college
 * MichaelHouse
 * Maritzburg College
 * Merchiston Preparatory School
 * Pietermaritzburg Girls' High School
 * Russell High School
 * St. Charles College, Pietermaritzburg
 * St. John's Diocesan School for Girls
 * St. Nicholas Diocesan School
 * The Wykeham Collegiate

Tertiary institutions
Pietermaritzburg has a well-developed higher system of public universities. Pietermaritzburg is served by two public universities, University of KwaZulu-Natal and Durban University of Technology. There are also many private and public colleges operating in the city, such as Varsity College.

Civil society
Pietermaritzburg is home to a number of prominent civil society organisations including the Abahlali baseMjondolo (shackdwellers) movement, GroundWork, CINDI, PACSA, and the KwaZulu Natal Christian Council.

Notable residents

 * Don Armand, rugby player (born 1988)
 * Neil Adcock, Springbok cricketer and radio cricket commentator
 * Arthur Bartman, (b.1972 - d.2019), Premier Soccer League goalkeeper.
 * Dale Benkenstein, former cricketer
 * Kork Ballington, World 250cc and 350cc Motorcycle Champion
 * Melissa Carlton, Paralympic swimmer who represented Australia
 * Amod Cassimjee, one of the earliest known Indian settlers
 * Pat Cilliers, rugby player (born 1987)
 * Brendon Dedekind, swimmer (born 1976)
 * Bathabile Dlamini, leader of the African National Congress Women's League and Minister of Women in the Presidency, and Minister of Social Development (born 1962)
 * Jon Ekerold, world 350cc motorcycle champion
 * Sarel Erwee, South Africa national cricket team player. Attended St. Charles College, Pietermaritzburg.
 * Brett Evans, (born 8 March 1982) South African Football (soccer) player attended Merchiston Preparatory School and Maritzburg College and played for Maritzburg City as an amateur
 * Adrian Furnham (born 1953), British-based organisational and applied psychologist and academic
 * Hayden Griffin (1943–2013), British stage designer
 * Tim Groenewald, Derbyshire and Warwickshire Cricketer (born 1984)
 * Archie Gumede, leader of the United Democratic Front (South Africa) and Member of the Parliament of South Africa
 * Adam Habib, vice-chancellor and principal of the University of the Witwatersrand (born 1965)
 * Jonathan Handley, singer-songwriter (born 1954), originally from Springs, founder of The Radio Rats who in 1979 had a hit single "ZX Dan" on Radio 5 (now 5FM)
 * Bessie Head, writer, was born in Pietermaritzburg in 1937.
 * Roy Hesketh, (1915–1944) racing driver and South African Air Force pilot.
 * Butch James, Springbok and rugby player attended Maritzburg College from 1994 to 1996. He played for Colleges' 3rd team.
 * Stratford Johns, (22 September 1925 – 29 January 2002) British stage, film and television actor. Left for Britain in 1948.
 * Peter Johnstone, Park Villa FC (born 1970), left PMB in 2004
 * Craig Joubert, international rugby union referee (born 8 November 1977). Attended Merchiston Preparatory School and Maritzburg College
 * Jesse Kriel, Springbok rugby player (born 1994). Attended Merchiston Preparatory School and Maritzburg College
 * Charlie Llewellyn, cricketer (born 1876), first non-White Test cricketer for South Africa.
 * Mbulelo Mabizela, South African national football team player (born 1981)
 * Nduduzo Makhathini, jazz musician
 * Clinton Marius, writer, performer (born 1966)
 * Thuso Mbedu, actress (born 1991). Attended Pietermaritzburg Girls' High School
 * Jomu Mbili, cricketer (born 1981)
 * Cuan McCarthy, fast-bowling The Proteas cricketer 1929–2000
 * Phyllis McCarthy, authority on and breeder of Rhodesian Ridgebacks
 * Ntuthuko Mchunu, (born 1999), rugby union player. Attended Piet Retief Primary School & Maritzburg College.
 * Cathcart William Methven (1849–1925), painter, engineer and architect
 * David Miller, cricketer (born 1989). Attended Maritzburg College.
 * Greg Minnaar, professional downhill mountain bike racer. He attended Carter High School.
 * Emmanuel Mkhize, cricketer (born 1989)
 * Zweli Mkhize, politician, former Kwazulu-Natal premier, national minister and South African current minister of health (born 2 February 1956). Born and raised in Willowfountain. Lived in Ashburton, Pietermaritzburg
 * Bryce Moon, (born 1986) South Africa national football team (soccer) player was born in Pietermaritzburg and played for Pirates in his youth
 * Ryan Moon, South Africa national football team player (born 15 September 1996). Attended Maritzburg College
 * Shaun Morgan, lead singer of Seether, former resident of Pietermaritzburg, attended both Merchiston Preparatory School and Maritzburg College while he lived there.
 * Peggy Mullany, South African water-colorist. Born (27 August 1916) in Pietermaritzburg and attended Wykeham School before it became The Wykeham Collegiate.
 * Nina Myskow, British journalist and television personality. Born in Pietermaritzburg and attended The Wykeham Collegiate, before moving to Scotland with her family aged 15.
 * Siphesihle Ndlovu, Premier Soccer League and South Africa national football team player.
 * Nkosingiphile Ngcobo, Premier Soccer League
 * Blade Nzimande (born 1958), South African national minister and president of the South African Communist Party. He attended Georgetown High School, Edendale & the University of Natal.
 * Alan Paton, author of Cry the Beloved Country, was born and schooled (Maritzburg College) in Pietermaritzburg.
 * Vyvyan Pearse (1891–1956), cricketer
 * Kevin Pietersen, (born 1980), cricketer for the England National Cricket Team. Attended Maritzburg College
 * Graeme Pope-Ellis, canoeist. Attended Alexandra High School.
 * Brian Raubenheimer, racing diver
 * Rowland Raw (1884–1915), cricketer
 * Jo-anne Reyneke, (born 1988), actress. Attended Russell High School (Pietermaritzburg)
 * Jonty Rhodes, national cricketer (born 1969)
 * Matthew Sates, (born 2003), Olympic games swimmer. Attended St. Charles College, Pietermaritzburg
 * Lunga Shabalala, (born 1989), TV presenter, TV personality and actor. He attended Maritzburg College.
 * Bandile Shandu, (born 1995), Premier Soccer League player. Attended Maritzburg College.
 * Tom Sharpe, novelist, who described the city as "half the size of a New York cemetery and twice as dead".
 * Peter Leslie Smith, auxiliary bishop for the Archdiocese of Portland in Oregon (born 1958)
 * Guy Spier, value investor (born 4 February 1966 in Grey's hospital).
 * Dale Stewart, bassist of Seether, former resident of Pietemaritzburg.
 * Joel Stransky, Springbok rugby player (born 1967). Born in Pietermaritzburg then attended Maritzburg College and University of Natal
 * Darian Townsend, Olympic swimmer. Attended Merchiston Preparatory School and Maritzburg College
 * Kevin Volans, composer (born in Pietermaritzburg, 26 July, 1949)
 * Lance Woolridge, rally driver
 * Philani Zulu, Premier Soccer League player
 * Dumisani Zuma, Premier Soccer League player
 * Lwandiswa Zuma South African first-class cricketer (born 11 July 1996) attended Maritzburg College

Various

 * Built in 1900, the City Hall was then the largest all-brick building in the southern hemisphere. It was declared a national monument in 1969.
 * At 14 metres high, the statue Pegasus adorning the entrance of the Golden Horse Casino was the largest statue of a horse in the world. However, as of July 2010, the statue has been demolished due to internal structural damage.

title: Pietermaritzburg


text:
Talk:Hunstanton

Untitled
Fixed the general sloppiness of the page... Aronjjxi (talk) 18:51, 28 January 2008 (UTC)

Need a how to get there section, especially local buses to king's lynn. —Preceding unsigned comment added by Radchenk (talk • contribs) 10:18, 21 July 2008 (UTC)

External links modified
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Pier used in Movie Barnacle Bill
Barnacle Bill (1957) (starring Alec Guinness) was shot at Hunstanton Pier, which was demolished on screen as part of the plot. It is alleged the pier was built in 1895.

https://en.wikipedia.org/wiki/Barnacle_Bill_(1957_film)#Production — Preceding unsigned comment added by 60.242.247.177 (talk) 03:14, 18 August 2018 (UTC)

title: Talk:Hunstanton