# Contents

## Idea

Recall that a TQFT is an FQFT defined on the $\left(\infty ,n\right)$-category of cobordisms whose morphisms are plain cobordisms and diffeomorphisms between these.

In a conformal quantum field theory the cobordisms are equipped with a conformal structure (a Riemannian metric structure modulo pointwise rescaling).

A conformal field theory (CFT) is accordingly a functor on such a richer category of conformal cobordisms.

See the discussion at FQFT for more details.

The conformally invariant quantum field theories have fields for whom the correlation functions have a specific behaviour accounting for the so called conformal dimension of the fields. This kind of constraints coming from conformal invariance, leads to constraints on the possible shapes of correlation functions; this has being formulated in 1971 by Polyakov as a bootstrap program: the conformal invariance should be sufficient to classify consistent conformal QFTs directly from the analysis of symmetries, rather than computing the Feynman integrals from possible actions; this avoids the usual problems with regularization and renormalization.

Precisely in 2-dimensions is the representation theory of the conformal group exceptionally interesting. In $d>2$, the global transformations, i.e. the elements of the conformal group, are given by the Poincare algebra, dilations and special conformal transformations. In $2$ dimensions, there are additionaly infinitely many local generators of conformal transformations whose commutation relations are given by the Virasoro algebra. This circumstance enables Belavin, Polyakov and Zamolodchikov to make a breakthrough in the bootstrap program in 1984 (what also helped a 1984-1985 revolution in string theory). There are well-developed tools for handling the theory locally (conformal nets, vertex operator algebras) and at least in the rational case there are complete classification results for the full theories (defined on cobordisms of all genera).

For this reason often in the literature the term “CFT” often implicitly refers to 2d CFT.

$2$-dimensional conformal field theories have two major applications:

• they describe critical phenomena on surfaces in condensed matter physics;

• they are the building blocks used in string theory

In the former application it is mostly the local behaviour of the CFT that is relevant. This is encoded in vertex operator algebras.

In the string theoretic applications the extension of the local theory to a full representation of the 2d conformal cobordism category is crucial. This extension is called solving the sewing constraints .

## Abstract

In the definition paragraph we will show how to define a conformal field theory using the axiomatic approach of Wightman resp. Osterwalder-Schrader. There are several approaches to axiomatically define conformal field theory, the said approach is not the most “popular” or “elegant” one. There are two reasons to consider the Wightman approach however: If one is already familiar with the Wightman approach, it helps to put conformal field theories into context. The second reason is that several notions often used by physicists can be easily and rigorously defined and explained using this approach. Therefore, it may serve as a bridge between mathematicians and physics literature.

## 2d CFT on the plane

### Definition in Terms of the Osterwalder-Schrader Axioms

A definition of conformal field theories can be formulated using the appropriate version of the Osterwalder--Schrader axioms, that is by a system of axioms of the correlation functions?. From the correlation functions it is then possible to deduce the existence of field operators in the sense of the Wightman axioms, that is fields aka field operators are operator valued distributions. Both the Hilbert space and the field operators are therefore not defined in the axioms, but reconstructed from the correlations functions defined in the first three axioms.

Let

${M}_{n}:=\left\{\left({z}_{1},...,{z}_{n}\right)\in {ℂ}^{n}:{z}_{i}\ne {z}_{j}\phantom{\rule{thickmathspace}{0ex}}\text{for}\phantom{\rule{thickmathspace}{0ex}}i\ne j\right\}$M_n := \{(z_1, ..., z_n) \in \mathbb{C}^n : z_i \neq z_j \; \text{for} \; i \neq j \}

be the space of configuration points. Let ${B}_{0}$ be a countable index set and $B:={\bigcup }_{n\in ℕ}{B}_{0}^{n}$.

The correlation functions are a family $\left({G}_{i}{\right)}_{i\in B}$ of continuous and polynomially bound functions

${G}_{{i}_{1},...,{i}_{n}}:{M}_{n}\to ℂ$G_{i_1, ..., i_n}: M_n \to \mathbb{C}
###### Axiom 1

locality

For all indexes $\left({i}_{1},...,{i}_{n}\right)$, configuration points $\left({z}_{1},...,{z}_{n}\right)$ and permutations $\pi :\left\{1,...,n\right\}\to \left\{1,...,n\right\}$ one has

${G}_{\left({i}_{1},...,{i}_{n}\right)}\left({z}_{1},...,{z}_{n}\right)={G}_{\left(\pi \left({i}_{1}\right),...,\pi \left({i}_{n}\right)\right)}\left(\pi \left({z}_{1}\right),...,\pi \left({z}_{n}\right)\right)$G_{(i_1,..., i_n)}(z_1,..., z_n) = G_{(\pi(i_1),..., \pi(i_n))}(\pi(z_1),..., \pi(z_n))

In this context, covariance is meant with respect to the Euclidean group? $E={E}_{2}$.

###### Axiom 2

covariance

For every index $i\in {B}_{0}$ there are conformal weights ${h}_{i},\overline{{h}_{i}}\in ℝ$ ($\overline{{h}_{i}}$ is completly independent from ${h}_{i}$, it is not the complex conjugate, this notation is widly used in the physics literature so we use it here, too) such that for $n\ge 1$, $w\in E$ and ${w}_{i}:=w\left({z}_{i}\right)$, ${h}_{j}:={h}_{{i}_{j}}$ one has

${G}_{\left({i}_{1},...,{i}_{n}\right)}\left({z}_{1},\overline{{z}_{i}},...,{z}_{n},\overline{{z}_{n}}\right)=\prod \left(\frac{\mathrm{dw}}{\mathrm{dz}}\left({z}_{j}\right){\right)}^{{h}_{j}}\left(\overline{\frac{\mathrm{dw}}{\mathrm{dz}}\left({z}_{j}\right)}{\right)}^{\overline{{h}_{j}}}{G}_{\left({i}_{1},...,{i}_{n}\right)}\left({w}_{1},\overline{{w}_{1}}...,{w}_{n},\overline{{w}_{n}}\right)$G_{(i_1,..., i_n)}(z_1, \overline{z_i}, ..., z_n, \overline{z_n}) = \prod (\frac{dw}{dz} (z_j))^{h_j} (\overline{\frac{dw}{dz}(z_j)})^{\overline{h_j}} G_{(i_1,..., i_n)}(w_1, \overline{w_1}..., w_n, \overline{w_n})

The ${s}_{i}:={h}_{i}-\overline{{h}_{i}}$ are called conformal spin (for the index $i$) and ${d}_{i}:={h}_{i}+\overline{{h}_{i}}$ is the scaling dimension. As an axiom 2.2 we assume that all conformal spins and scaling dimensions are integers.

For the next axiom, which is about reflection positivity, we need some notation:

We write $z\in ℂ$ as $z=t+\mathrm{iy}$ and identify $y$ as the space coordinate and $t$ as (imaginary) time. So, time reflection $\theta$ is simply:

$\theta :ℂ\to ℂ$\theta: \mathbb{C} \to \mathbb{C}
$\theta :z=t+\mathrm{iy}↦-t+\mathrm{iy}=-\overline{z}$\theta: z = t + iy \mapsto -t + iy = - \overline{z}

Let $𝒮\left({ℂ}^{n}\right)$ be the space of Schwartz functions? and

${M}_{n}^{+}:=\left\{\left({z}_{1},...,{z}_{n}\right)\in {M}_{n}:Re\left({z}_{i}\right)>0\right\}$M_n^+ := \{ (z_1, ..., z_n) \in M_n: \operatorname{Re}(z_i) \gt 0 \}
${𝒮}_{0}^{+}:=ℂ$\mathcal{S}_0^+ := \mathbb{C}
${𝒮}_{n}^{+}:=\left\{f\in 𝒮\left({ℂ}^{n}\right):Supp\left(f\right)\subset {M}_{n}^{+}\right\}$\mathcal{S}_n^+ := \{f \in \mathcal{S}(\mathbb{C}^n): \operatorname{Supp}(f) \subset M_n^+ \}

Finally let $\underline{{𝒮}^{+}}$ the space of all sequences $\underline{f}=\left({f}_{i}{\right)}_{i\in B}$ with ${f}_{i}\in {𝒮}_{n}^{+}$ fpr $i\in {B}_{0}^{n}$ with only finitely many entries $\ne 0$.

###### Axiom 3

reflection positivity

There is a map

$*:{B}_{0}\to {B}_{0}$*: B_0 \to B_0

with ${*}^{2}={\mathrm{id}}_{{B}_{0}}$ that extends to a map

$*:B\to B$*: B \to B
$*:i↦{i}^{*}$*: i \mapsto i^*

so that

1. ${G}_{i}\left(z\right)={G}_{{i}^{*}}\left(\theta \left(z\right)\right)={G}_{{i}^{*}}\left(-\overline{z}\right)$

2. $⟨\underline{f},\underline{f}⟩\ge 0$ for all $\underline{f}\in \underline{{𝒮}^{+}}$

The scalar product $⟨\underline{f},\underline{f}⟩$ is defined as

$\sum _{i,j\in B,m,n\in ℕ}{\int }_{{M}_{n+m}}{G}_{{i}^{*}j}\left(\theta \left({z}_{1}\right),...,\theta \left({z}_{n}\right),{w}_{1},...,{w}_{n}\right)\overline{{f}_{i}\left(z\right)}{f}_{j}\left(w\right){d}^{n}z{d}^{m}w$\sum_{i, j \in B, m, n \in \mathbb{N} } \int_{M_{n+m}} G_{i^* j} (\theta(z_1), ...,\theta(z_n), w_1, ..., w_n) \overline{f_i(z)} f_j(w) d^n z d^m w
###### Proposition

Both a Hilbert space and the field operators of the theory can be (re-) constructed from axioms 1, 2 and 3.

TODO: Details

###### Axiom 4

scaling covariance

The correlation functions satisfy the covariance condition also for dilatations $w\left(z\right)={e}^{t}\left(t\right),t\in ℝ$.

Explicitly, the last condition says that

${G}_{i}\left({z}_{1},...,{z}_{n}\right)=\left({e}^{t}{\right)}^{{h}_{1}+\overline{{h}_{1}}+...+{h}_{n}+\overline{{h}_{n}}}{G}_{i}\left({e}^{t}\left({z}_{1}\right),...,{e}^{t}\left({z}_{n}\right)\right)$G_i(z_1, ..., z_n) = (e^t)^{h_1 + \overline{h_1} +...+ h_n + \overline{h_n}} G_i(e^t(z_1), ..., e^t(z_n))

Given axioms 1-4, the 2-point functions can be fully classified, see below.

###### Axiom 5

existence of the energy-momentum tensor

There are four fields ${T}_{\mu ,\nu },\mu .\nu \in \left\{0,1\right\}$ with the following properties:

Symmetry:

${T}_{\mu ,\nu }={T}_{\nu ,\mu },\phantom{\rule{thickmathspace}{0ex}}{T}_{\mu ,\nu }\left(z{\right)}^{*}={T}_{\nu ,\mu }\left(\theta \left(z\right)\right)$T_{\mu, \nu} = T_{\nu, \mu}, \; T_{\mu, \nu}(z)^* = T_{\nu, \mu}(\theta (z))
${\partial }_{t}{T}_{\mu ,0}+{\partial }_{y}{T}_{\mu ,1}=0$\partial_t T_{\mu, 0} + \partial_y T_{\mu, 1} = 0

${T}_{\mu ,\nu }$ has scaling dimension 2 and the conformal spin $s$ is restricted by

$s\left({T}_{00}-{T}_{11}±2i\phantom{\rule{thickmathspace}{0ex}}{T}_{01}\right)=±2$s(T_{00} - T_{11} \pm 2i \; T_{01}) = \pm 2

The energy momentum tensor allows us to define two densly defined operators ${L}_{-n},\overline{{L}_{-n}}$ that both define unitary representations of the Virasoro algebra.

###### Proposition

Lüscher-Mack

T is holomorphic. Therefore, the operators

${L}_{-n}:=\frac{1}{2\pi i}{\oint }_{\mid \zeta \mid =1}\frac{T\left(\zeta \right)}{{\zeta }^{n+1}}d\zeta$L_{-n} := \frac{1}{2 \pi i} \oint_{| \zeta | = 1} \frac{T(\zeta)}{\zeta^{n+1}} d\zeta

and

$\overline{{L}_{-n}}:=\frac{1}{2\pi i}{\oint }_{\mid \zeta \mid =1}\frac{\overline{T}\left(\zeta \right)}{{\zeta }^{n+1}}d\zeta$\overline{L_{-n}} := \frac{1}{2 \pi i} \oint_{| \zeta | = 1} \frac{\overline{T}(\zeta)}{\zeta^{n+1}} d\zeta

are well defined and satisfy the commutation relations of two commuting Virasoro algebras with the same central charge.

###### Definition

primary field

A conformal field ${\psi }_{i},i\in {B}_{0}$, is called a primary field if

$\left[{L}_{n},{\psi }_{i}\right]={z}^{n+1}{\partial }_{z}{\psi }_{i}\left(z\right)+{h}_{i}\left(n+1\right){z}^{n}{\psi }_{i}\left(z\right)$[L_n, \psi_i] = z^{n+1} \partial_z \psi_i(z) + h_i (n+1) z^n \psi_i(z)

and likewise for $\overline{z}$.

So, primary fields are the fields whose correlation functions are covariant “infinitesimally” with respect to all holomorphic functions, the “infinitesimal symmetry” expressed in the definition.

The fields are operator valued distributions and cannot be multiplied in general. The possibility of multiplication of fields evaluated at different points and some control of the singularities of this product are part of the axioms of conformal field theory:

###### Pseudodefinition

operator product expansion

An operator product expansion (OPE) for a family of fields means that there is for all fields and all ${z}_{1}\ne {z}_{2}$ a relation of the form

${\psi }_{i}\left({z}_{1}\right){\psi }_{j}\left({z}_{2}\right)\sim \sum _{k\in {B}_{0}}{C}_{\mathrm{ijk}}\left({z}_{1}-{z}_{2}{\right)}^{{h}_{k}-{h}_{i}-{h}_{j}}{\psi }_{k}\left({z}_{2}\right)$\psi_i(z_1) \psi_j(z_2) \sim \sum_{k \in B_0} C_{ijk} (z_1 - z_2)^{h_k - h_i - h_j} \psi_k(z_2)

Here $\sim$ means modulo regular functions.

A rigorous interpretation of an OPE would interpret the given relation as a relation of e.g. matrix elements or vacuum expectation values.

An OPE is called associative if the expansion of a product of more than two fields does not depend on the order of the expansion of the products of two factors. Since the OPE has no interpretation as defining products of operators, or more generally the product in a ring, the notion of associativity does not refer to the associativity of a product in a ring, as the term may suggest.

###### Axiom 6

existence of associative operator product expansion

The primary fields have an associative OPE.

Many formal calculations of physicists in CFT involving OPE can be justified by using vertex operator algebras, so that vertex operator algebras have become a standard way to formulate CFT.

### Properties

###### Proposition

Any 2-point function satisfying axioms 1-4 has the following form:

${G}_{\mathrm{ij}}\left({z}_{1},{z}_{2}\right)={C}_{\mathrm{ij}}\left({z}_{1}-{z}_{2}{\right)}^{-\left({h}_{i}+{h}_{j}\right)}\left(\overline{{z}_{1}}-\overline{{z}_{2}}{\right)}^{-\left(\overline{{h}_{i}}+\overline{{h}_{j}}\right)}$G_{ij}(z_1, z_2) = C_{ij} (z_1 - z_2)^{-(h_i + h_j)} (\overline{z_1} - \overline{z_2})^{-(\overline{h_i} + \overline{h_j})}

with some constant ${C}_{\mathrm{ij}}\in ℂ$.

## Conformal anomaly

The Liouville cocycle appears when one moves from genuine representations of 2-dimensional conformal cobordisms to projective representations. The obstruction for such a projective representation to be a genuine representation is precisely given by the central charge $c$; when $c\ne 0$, one says that the conformal field theory has a conformal anomaly.

## CFT in AQFT language

In AQFT conformal field theory is modeled in terms of local nets that are conformal nets.

## Full versus chiral CFT

some discussion of full vs. chiral CFT goes here… then:

The following result establishes which pairs of vertex operator algebras can appear as the left and right chiral parts of a full field algebra in the sense of Huang, Lepowski, Kong, etc. (see vertex operator algebra).

###### Definition

Two modular tensor categories are said to have the same Witt class if there exist two spherical categories $S$ and $T$ such that we have an equivalence of ribbon categories

$C⊠𝒵\left(S\right)\simeq D⊠𝒵\left(T\right)$C \boxtimes \mathcal{Z}(S) \simeq D \boxtimes \mathcal{Z}(T)

Here $𝒵\left(S\right)$ is the category whose objects are pairs $\left(Z,z\right)$ consisting of an object $Z\in S$ and a natural isomorphism ${z}_{X}:Z\otimes X\stackrel{\simeq }{\to }X\otimes Z$, such that for al objects $X,Y\in S$ we have

$\begin{array}{ccccc}Z\otimes X\otimes Y& & \stackrel{{z}_{X\otimes Y}}{\to }& & X\otimes Y\otimes Z\\ & {}_{{z}_{x}\otimes \mathrm{Id}}↘& & {↗}_{\mathrm{Id}\otimes {z}_{y}^{-1}}\\ & & X\otimes Z\otimes Y\end{array}$\array{ Z \otimes X \otimes Y &&\stackrel{z_{X \otimes Y}}{\to}&& X \otimes Y \otimes Z \\ & {}_{\mathllap{z_x \otimes Id}}\searrow && \nearrow_{Id \otimes \mathrlap{z_y^{-1}}} \\ && X \otimes Z \otimes Y }

This becomes a monoidal category itself by setting

$\left(Z,z\right)\otimes \left(W,w\right):=\left(Z\otimes W,z\otimes w\right)\phantom{\rule{thinmathspace}{0ex}}.$(Z,z) \otimes (W,w) := (Z \otimes W, z \otimes w) \,.
###### Theorem

(Michael Müger)

If $S$ is a spherical category then $𝒵\left(S\right)$ is a modular tensor category.

###### Theorem

(Michael Müger)

If $C$ is a modular tensor category then there is an equivalence of ribbon categories

$𝒵\stackrel{\simeq }{←}C⊠\overline{C}$\mathcal{Z} \stackrel{\simeq}{\leftarrow} C \boxtimes \bar C
###### Theorem

Two vertex operator algebras $V$ may appear as the left and right chiral halfs of a full conformal field theory precisely if their modular tensor categories of representations have the same Witt class.

## References

The first comprehensive physics textbook on CFT was maybe

• Philippe Di Francesco,Pierre Mathieu,David Sénéchal, Conformal field theory, Springer 1997

• Yu Nakayama, A lecture note on scale invariance vs conformal invariance (arXiv:1302.0884)

For a survey of perspectives in CFT with an eye towards string theory see various contributions in

### CFT on the plane

A textbook directed at mathematicians and explaining the “classical” concept of a CFT, i.e. without reference to categories, is this:

• Martin Schottenloher: A mathematical introduction to conformal field theory (2nd edition, ZMATH entry)

A survey of some of the basic manipulationsin chiral CFT, motivated from a path integral perspective is at

### CFT on complex curves/surfaces of arbitrary genus

For chiral part see

### Formulaton by conformal nets

For further references see conformal net.

### Formulation by functors on conformal cobordisms

The idea that CFT should have an axiomatization as FQFT – representations of cobordism categories for cobordisms equipped with conformal structure – was promoted in

• The definition of conformal field theory , in Differential geometrical methods in theoretical physics (Como, 1987), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., 250, Kluwer Acad. Publ., Dordrecht, (1988), 165-171

Two-dimensional conformal field theories and modular functors , in Proceedings of the IXth International Congress on Mathematical Physics , Swansea, 1988, Hilger, Bristol (1989) 22-37.

The definition of conformal field theory , preprint, 1988; also in: Topology, geometry and quantum field theory , ed. U. Tillmann, London Math. Soc. Lect. Note Ser., Vol. 308. Cambridge University Press, Cambridge (2004) 421-577.