Conformal field theory in two dimensions/Bootstrap approach

For a history of the conformal bootstrap approach, see Ref.

An axiomatic approach

Fields and correlation functions

In conformal field theory, the observables are correlation functions. An n-point correlation function is written as

${\Big \langle }{\mathcal {O}}_{1}(x_{1}){\mathcal {O}}_{2}(x_{2})\cdots {\mathcal {O}}_{n}(x_{n}){\Big \rangle }$

where ${\mathcal {O}}_{i}(x_{i})$ is a field at position $x_{i}\in \mathbb {R} ^{d}$. Correlation functions are defined by the axioms that they obey, and fields can be thought of as convenient notations for writing these axioms.

Here we will state a few basic axioms. Really nontrivial axioms, such as the existence of operator product expansions, will come later.

1. Linearity: correlation functions depend linarly on fields: ${\Big \langle }\left(\lambda {\mathcal {O}}(x)+\lambda '{\mathcal {O}}'(x)\right)\cdots {\Big \rangle }=\lambda {\Big \langle }{\mathcal {O}}(x)\cdots {\Big \rangle }+\lambda '{\Big \langle }{\mathcal {O}}'(x)\cdots {\Big \rangle }\quad ,\quad {\Bigg \langle }{\frac {\partial }{\partial x^{\mu }}}{\mathcal {O}}(x)\cdots {\Bigg \rangle }={\frac {\partial }{\partial x^{\mu }}}{\Big \langle }{\mathcal {O}}(x)\cdots {\Big \rangle }$
2. Permutation invariance: the ordering of fields does not matter.
3. State-field correspondence: the space of fields is an $x$-independent vector space, also called the space of states or the spectrum. This means that the field indices take values in an $x$-independent set $I$, and the space of fields is ${\text{Span}}\left({\mathcal {O}}_{i}\right)_{i\in I}$.

It is natural to assume that the space of fields does not depend on the position, but why do we identify it with the space of states? This is motivated by scale invariance. In a quantum theory, states live on a constant time slice of spacetime. If we take Euclidean time to be a radial coordinate around $x$, then states live on spheres centered at $x$. By scale invariance, we do not lose information by sending the radius of the spheres to zero. So it is reasonable to assume that states correspond to objects that are localized at $x$, i.e. fields.

Comparison with the functional integral approach

In the functional integral approach, correlation functions are constructed as

${\Big \langle }{\mathcal {O}}_{1}(x_{1})\cdots {\mathcal {O}}_{n}(x_{n}){\Big \rangle }=\int D\phi \ e^{-S[\phi ]}{\mathcal {O}}_{1}[\phi ](x_{1})\cdots {\mathcal {O}}_{n}[\phi ](x_{n})$

where

1. The field $\phi$ is a function on $\mathbb {R} ^{d}$, with values in some space $X$.
2. The action $S$ is a functional, for example $S[\phi ]=\int _{\mathbb {R} ^{d}}dx\ \left(\partial _{\mu }\phi \partial ^{\mu }\phi +\cdots \right)$. (Here the field takes values in $X=\mathbb {R}$.)
3. ${\mathcal {O}}_{i}$ is a functional, built from $\phi (x_{i})$ and its derivatives at $x_{i}$. For example, ${\mathcal {O}}_{p,\mu }[\phi ](x)=e^{p\phi (x)}\partial _{\mu }\phi (x)$.
4. The integration measure $D\phi$ on the space of functions $\mathbb {R} ^{d}\to X$ cannot always be constructed rigorously. To some extent it is defined by axioms, so that the functional integral approach is partly axiomatic and not purely constructive.
5. The zero-point function $\int D\phi \ e^{-S[\phi ]}$ is called the partition function.

In contrast to the bootstrap approach, symmetries are not assumed, but derived from the functional integral. For example, linearity and permutation invariance of correlation functions follow from the functional integral. It can happen that a symmetry of the action is not a symmetry of the integration measure: we say that we have an anomaly.

Conformal symmetry, operator product expansions, and crossing symmetry

This section is adapted from Ref.

Conformal covariance of correlation functions

Let $x\mapsto x'=f(x)$ be a conformal transformation. We may require that correlation functions are invariant, but this would be too restrictive. For example, correlation functions of scalar fields are invariant under rotations, but correlation functions that involve vector fields or more general tensor fields transform non-trivially under rotations. We therefore allow fields to transform, writing this as ${\mathcal {O}}\mapsto {\mathcal {O}}'$. Correlation functions are not invariant, but covariant:

${\Bigg \langle }\prod _{i=1}^{n}{\mathcal {O}}'_{i}(x'_{i}){\Bigg \rangle }={\Bigg \langle }\prod _{i=1}^{n}{\mathcal {O}}_{i}(x_{i}){\Bigg \rangle }$

A class of fields that play an important role in CFT are called scalar primary fields. They are invariant under rotations, but not necessarily under dilations. A scalar primary field is characterized by a conformal dimension $\Delta \in \mathbb {C}$, and transforms as

${\mathcal {O}}'(x')=\Omega (x)^{\Delta }{\mathcal {O}}(x')$

where $\Omega (x)$ is the scale factor associated to the conformal transformation $x\mapsto x'$. There exist fields whose behaviour is more complicated: for example, derivatives of scalar primary fields.

Conformal covariance is a strong constraint on correlation functions. In particular, it determines 2-point and 3-point functions of scalar primary fields. 2-point functions are nonzero only if the 2 fields have the same conformal dimension,

${\Big \langle }{\mathcal {O}}_{1}(x_{1}){\mathcal {O}}_{2}(x_{2}){\Big \rangle }=\delta _{\Delta _{1},\Delta _{2}}|x_{1}-x_{2}|^{-2\Delta _{1}}$

where the $x_{i}$-independent prefactor is normalized to one. For 3-point functions, we have

${\Big \langle }{\mathcal {O}}_{1}(x_{1}){\mathcal {O}}_{2}(x_{2}){\mathcal {O}}_{3}(x_{3}){\Big \rangle }=C_{123}|x_{1}-x_{2}|^{\Delta _{3}-\Delta _{1}-\Delta _{2}}|x_{1}-x_{3}|^{\Delta _{2}-\Delta _{1}-\Delta _{3}}|x_{2}-x_{3}|^{\Delta _{1}-\Delta _{2}-\Delta _{3}}$

where the $x_{i}$-independent prefactor $C_{123}$ is called a 3-point structure constant.

The conformal dimensions of primary fields, together with the 3-point structure constants, are called the CFT data of a given conformal field theory. These data determine not only 2-point and 3-point functions, but actually all correlation functions in flat space. This is because of operator product expansions, which allow us to reduce higher correlation functions to 2-point and 3-point functions.

Operator product expansions

The axiom that OPEs exist states that a product of fields is a linear combinations of fields:

${\mathcal {O}}_{1}(x_{1}){\mathcal {O}}_{2}(x_{2})=\sum _{k}f_{12k}(x_{1},x_{2},y){\mathcal {O}}_{k}(y)$

This expansion is supposed to be valid in any correlation functions, with coefficients $f_{12k}(x_{1},x_{2},y)$ that do not depend on the correlation function, provided there are no other fields too close to $x_{1},x_{2},y$. However, the region of convergence depends on the correlation function: namely

$|x_{1}-y|,|x_{2}-y|<\min _{i\neq 1,2}|x_{i}-y|$

The point $y$ may be chosen arbitrarily: different values are related by Taylor expanding ${\mathcal {O}}_{k}(y)$. (Derivatives of fields are also fields, and appear in the OPE.) A usual choice is $y=x_{2}$. Conformal symmetry constraints OPEs, in the same way as it constrains correlation functions. As a result, OPE coefficients take the form

$f_{12k}(x_{1},x_{2},y)=C_{12k}{\hat {f}}_{12k}(x_{1},x_{2},y)$

where $C_{12k}$ are 3-point structure constants, and ${\hat {f}}_{12k}(x_{1},x_{2},y)$ is completely determined by the behaviour of ${\mathcal {O}}_{1},{\mathcal {O}}_{2},{\mathcal {O}}_{k}$ under conformal transformations, and can be computed explicitly. For example, if one of these fields is a scalar primary, then ${\hat {f}}_{12k}(x_{1},x_{2},y)$ is a function of its conformal dimension.

Using OPEs, we can compute any $n$-point function in terms of CFT data. To do this, we reduce the $n$-point function to a combination of $(n-1)$-point function, and so on, until we reach 2-point functions.

In a quantum field theory that is not a CFT, there may exist OPEs, but their radiuses of convergence are typically zero. And in the absence of conformal symmetry, OPE coefficients are not so simple.

Crossing symmetry

OPEs reduce correlation functions to CFT data. But is the reduction unique? And how do we determine CFT data? These two questions are related: the reduction is not unique, and comparing different reductions constrains CFT data.

In the case of a 4-point function, there are 3 different reductions, called s-channel, t-channel and u-channel, corresponding to the OPEs ${\mathcal {O}}_{1}{\mathcal {O}}_{2}$, ${\mathcal {O}}_{1}{\mathcal {O}}_{4}$, ${\mathcal {O}}_{1}{\mathcal {O}}_{3}$. The equality of these reductions is a constraint on CFT data called crossing symmetry. Crossing symmetry reads

$\sum _{k}C_{12k}C_{k34}{\mathcal {G}}_{k}^{(s)}=\sum _{k}C_{41k}C_{k23}{\mathcal {G}}_{k}^{(t)}=\sum _{k}C_{13k}C_{24k}{\mathcal {G}}_{k}^{(u)}$

where ${\mathcal {G}}_{k}^{(x)}$ are conformal blocks: functions of the positions $x_{1},x_{2},x_{3},x_{4}$, which also depend on the fields ${\mathcal {O}}_{1},{\mathcal {O}}_{2},{\mathcal {O}}_{3},{\mathcal {O_{4}}},{\mathcal {O}}_{k}$, and which are known in principle. In crossing symmetry equations, the unknowns are therefore CFT data. While symmetries (including conformal symmetry) give rise to linear equations for correlation functions called Ward identities, crossing symmetry is quadratic in the 3-point structure constants.

Schematically, the 3 channels are depicted as follows:

Fields and the conformal algebra

Let $x'=f(x)$ be a conformal transformation, i.e. $f^{*}g=\Omega ^{2}g$. This means that locally, our conformal transformation is an isometry modulo a scale factor,

${\frac {\partial x'^{\mu }}{\partial x^{\nu }}}=\Omega (x)\Lambda _{\nu }^{\mu }(x)\quad {\text{with}}\quad g_{\rho \sigma }\Lambda _{\mu }^{\nu }\Lambda _{\nu }^{\sigma }=g_{\mu \nu }$

Let us specialize to a flat Euclidean or Minkowskian metric $g_{\mu \nu }=\eta _{\mu \nu }$, then the isometry $\Lambda$ is a rotation.

The conformal algebra has generators $M_{\mu \nu }$ for rotations, $D$ for dilations, $P_{\mu }$ for translations, and $K_{\mu }$ for special conformal transformations. The respective numbers of generators are

${\frac {d(d-1)}{2}}+1+d+d={\frac {(d+1)(d+2)}{2}}$

The nonzero commutation relations are

${\begin{aligned}{}[M_{\mu \nu },M_{\rho \sigma }]&=\eta _{\nu \rho }M_{\mu \sigma }-\eta _{\mu \rho }M_{\nu \sigma }+\eta _{\nu \sigma }M_{\rho \mu }-\eta _{\mu \sigma }M_{\rho \nu }\\{}[M_{\mu \nu },P_{\rho }]&=\eta _{\nu \rho }P_{\mu }-\eta _{\mu \sigma }P_{\nu }\\{}[M_{\mu \nu },K_{\rho }]&=\eta _{\nu \rho }K_{\mu }-\eta _{\mu \sigma }K_{\nu }\\{}[D,P_{\mu }]&=P_{\mu }\\{}[D,K_{\mu }]&=-K_{\mu }\\{}[K_{\mu },P_{\nu }]&=2\eta _{\mu \nu }D-2M_{\mu \nu }\end{aligned}}$

The conformal algebra acts on fields. For any field, we assume

$P_{\mu }{\mathcal {O}}(x)={\frac {\partial }{\partial x^{\mu }}}{\mathcal {O}}(x)$

The action of the other generators depend on the field. The definition of a scalar primary field of dimension $\Delta$ is equivalent to

$D{\mathcal {O}}=\Delta {\mathcal {O}}\quad ,\quad K_{\mu }{\mathcal {O}}=0\quad ,\quad M_{\mu \nu }{\mathcal {O}}=0$

More generally, a primary field of dimension $\Delta$ obeys the first two equations, but may transform in a nontrivial representation of rotation algebra, for instance the vector representation. Together with its all its derivatives, a primary field generates a representation of the conformal algebra. Primary fields, and the representations they generate, play an important role in CFT. In quite a few interesting CFTs, all fields are primaries or derivatives thereof.

Additional assumptions

Any set of correlation functions that obeys the axioms that we have seen constitutes, by definition, a conformal field theory. The space of CFTs is vast: to say anything interesting, we have to focus on a subspace or even on one specific CFT, by making more assumptions.

Symmetry assumptions

Many interesting CFTs have not only conformal symmetry, but also other symmetries. Just like it is a representation of the conformal algebra, the space of states is a representation of the extra symmetry groups or algebras.

This can be motivated by applications: for example, in the Ising model, the spins take values $\sigma \in \{-1,+1\}$, and the model is invariant under $\sigma \to -\sigma$. We therefore expect the corresponding CFT to have a $\mathbb {Z} _{2}$ symmetry, which commutes with conformal symmetry. This allows us to classify fields into $\mathbb {Z} _{2}$-even and $\mathbb {Z} _{2}$-odd fields, and OPEs respect this symmetry. (For example, the OPE of a field with itself only yields $\mathbb {Z} _{2}$-even fields.) Other statistical models have larger symmetry groups, such as the w:orthogonal group or the w:symmetric group, and the corresponding CFTs also have these symmetries.

Applications to string theory also lead to CFTs with global symmetries. In the worldsheet approach, string theory in a spacetime $Z$ is a 2d CFT on a worldsheet $M$, which describes the dynamics of embeddings $X:M\to Z$. Symmetries of the target space $Z$ give rise to symmetries of that CFT. Examples include w:Wess-Zumino-Witten models, for which $Z$ is a Lie group manifold.

In CFT on the sphere, the conformal group is the Möbius group $PSL_{2}(\mathbb {C} )$. However, local conformal symmetry gives rise to the Virasoro algebra, which is infinite-dimensional. There are CFTs with only global conformal symmetry, and CFT with local conformal symmetry. The latter are in principle much rarer, but they appear much more often in applications, and are much better studied. There are assumptions under which global conformal symmetry implies local conformal symmetry, just like scale invariance implies conformal invariance in many cases. Then there exist larger algebras that include the Virasoro algebra, generically called w:W-algebras. 2d CFTs can be constructed based on such algebras.

Assumptions on the spectrum

Consider the operator product expansion

${\mathcal {O}}_{1}(x_{1}){\mathcal {O}}_{2}(x_{2})=\sum _{k}C_{12k}|x_{1}-x_{2}|^{\Delta _{k}-\Delta _{1}-\Delta _{2}}{\mathcal {O}}_{k}(x_{2})$

where the dependence of OPE coefficients on positions is determined by scale and translation invariance, and for simplicity we write only primary fields. For the sum over $k$ to converge near $x_{1}=x_{2}$, we need the scaling dimensions (or conformal dimensions) $\Delta _{k}$ to be bounded from below. Therefore, we assume that the spectrum of conformal dimensions is bounded from below. Physically, this corresponds to the energies being bounded from below, which is a stability requirement in quantum mechanics. After this consistency requirement, let us discuss additional assumptions.

We may assume that there exists a vacuum state, equivalently an identity field. This is a primary field ${\mathcal {O}}_{0}$ of conformal dimension zero, with a trivial OPE ${\mathcal {O}}_{0}(x_{1}){\mathcal {O}}_{k}(x_{2})={\mathcal {O}}_{k}(x_{2})$. We can insert this trivial field in correlation functions, without modifying them. The non-empty statement is that this field appears in OPEs of other fields.

A CFT is called local if it has an energy-momentum tensor (also called stress-energy tensor): a primary field of conformal spin 2 and conformal dimension d, which is symmetric $T^{\mu ,\nu }=T^{\nu ,\mu }$ and conserved $\partial _{\mu }T^{\mu ,\nu }=0$. The motivation for assuming our CFT is local is w:Noether's theorem, which applies if the theory has a functional integral formulation, with an action $S=\int d^{d}x\ {\mathcal {L}}$. Then the energy-momentum tensor is the Noether current associated to spacetime translations, $T^{\mu ,\nu }=-{\frac {g^{\mu ,\nu }}{\sqrt {\det g}}}{\mathcal {L}}+\cdots$. Modulo changes of coordinates, spacetime translations are equivalent to changes of the metric, leading to the expression $T^{\mu ,\nu }=-{\frac {2}{\sqrt {\det g}}}{\frac {\delta S}{\delta g_{\mu ,\nu }}}$. The conformal dimension of the energy-momentum tensor can be inferred from dimensional analysis, assuming $[x]=-1,[g]=0$. By conformal invariance we have $[S]=0$ so that $[{\mathcal {L}}]=[T]=d$. For more details, see ref.

Until now we have mentioned general assumptions, which are satisfied by many CFTs. We could make much stronger assumptions, for example that the CFT is rational: the space of states decomposes into finitely many irreducible representations of the conformal algebra. Non-trivial rational CFTs are known only in $d=2$, and they can be classified: they are called minimal models.

When studying a given CFT, facts that are known from other approaches can be taken as assumptions in the bootstrap approach. In the case of the critical Ising model in d=3, we may assume that there is only one $\mathbb {Z} _{2}$-even primary field of conformal spin 0 that is relevant, i.e. whose conformal dimension obeys $0<\Delta <3$.

In d=2, we may assume modular invariance of the torus partition function: this assumption gives strong constraints on the spectrum, including the w:Cardy formula. This assumption follows from the more general assumption that the CFT is consistent on the torus, and not only on the sphere. The more general assumption also leads to constraints on structure constants.

Unitarity

Quantum theories predict probabilities of events, which are positive numbers. In quantum field theory, unitarity is the constraint that certain correlation functions are positive, so they can be interpreted as w:probability densities.

If we start with a statistical lattice model defined by a positive definite Hamiltonian, such as the Ising model, we should obtain a unitary CFT in the critical limit. However, other statistical lattice models give rise to non-unitary CFTs. For example, the w:random cluster model, which has a non-Hamiltonian definition in terms of statistical sums over bond configurations.

In string theory, unitarity of the worldsheet theory is roughly related to the target space having a positive definite metric. However, string theory is not defined by the sole metric but also involves the dilaton and antisymmetric fields. Unitarity can also depend on the values of these fields.

In the Osterwalder-Schrader axioms for quantum field theory on $\mathbb {R} ^{d}$, unitarity appears as reflection positivity: given a reflection $\theta$ with respect to a hyperplane, such as $\theta (x^{1},x^{2},\dots ,x^{d})=(-x^{1},x^{2},\dots ,x^{d})$ and n fields ${\mathcal {O}}_{1},\dots ,{\mathcal {O}}_{n}$, we have

${\Big \langle }{\mathcal {O}}_{1}(x_{1}){\mathcal {O}}_{1}(\theta (x_{1}))\dots {\mathcal {O}}_{n}(x_{n}){\mathcal {O}}_{n}(\theta (x_{n})){\Big \rangle }\geq 0$

This condition is quite complicated. In practice, necessary conditions for unitarity of a CFT are:

1. The space of states has a w:Hilbert space structure, with an inner product such that the dilation operator is self-conjugate.
2. Two-point functions are positive and three-point functions are real.

This immediately implies that conformal dimensions are real, since they are eigenvalues of the dilation operator. But this also leads to lower bounds on conformal dimensions of primary operators. For example, for a scalar operator i.e. a primary operator of conformal spin 0, the bound is $\Delta \geq {\frac {1}{2}}(d-2)$. For a traceless symmetric tensor of spin $\ell \geq 1$ the bound is $\Delta \geq \ell +d-2$, which is saturated by the energy-momentum tensor.

By systematically exploiting unitarity, together with crossing symmetry, it is possible to deduce inequalities for the conformal dimensions of fields, and for their structure constants. This can amount to determining these data with a good precision. For example, as of November 2024, the latest bootstrap inequalities for the conformal dimension of the spin field in the 3d Ising model are

${\underline {0.518148}}782<\Delta _{\sigma }<{\underline {0.518148}}830$

This is about 3 decimals better than directly computing $\Delta _{\sigma }$ in the lattice model by the w:Monte-Carlo method.

Exercises

• B1. Show that if the identity field appears in the OPE of two fields, then these two fields have the same conformal dimension.
• B2. Compute the 2-point and 3-point functions of scalar primary fields.
• B3. Complete the crossing relation for a 4-point function, by adding the u-channel decomposition, which is deduced from the OPE ${\mathcal {O}}_{1}{\mathcal {O}}_{3}$. If for all 4-point functions the s-channel and t-channel decompositions coincide, does this imply that the u-channel decomposition agrees as well?
• B4. Write the generators of the conformal algebra as differential operators, starting with $P_{\mu }={\frac {\partial }{\partial x^{\mu }}}$ and $D=x^{\mu }{\frac {\partial }{\partial x^{\mu }}}$. Deduce their commutation relations.
• B5. From reflection positivity, deduce that 2-point functions are positive and 3-point functions are real.

BOAP: Action of conformal generators on scalar primary fields

Consider a scalar primary field at $x=0$. Under a conformal transformation that fixes $x=0$, this field transforms as ${\mathcal {O}}(0)\to \Omega (0)^{-\Delta }{\mathcal {O}}(0)$. We would like to deduce the action of the dilations and special conformal generators on this field.

1. Compute the factor $\Omega (x)$ for dilations and special conformal transformations. Deduce $\Omega (0)$.
2. Consider an infinitesimal dilation by a factor $\lambda =1+\epsilon$. Show that our primary field transforms as ${\mathcal {O}}(0)\to {\mathcal {O}}(0)+\epsilon D\cdot {\mathcal {O}}(0)+O(\epsilon ^{2})$ where $D$ is the dilation generator of the conformal algebra, and deduce $D\cdot {\mathcal {O}}(0)$.
3. Similarly, compute how our field transforms under an infinitesimal special conformal transformation, and deduce $K_{\mu }\cdot {\mathcal {O}}(0)$.

BOSC: Conformal invariance of free scalar fields

1. Consider a massless free scalar field on $\mathbb {R} ^{d}$, of dimension 0. Is the action scale invariant? Is it conformally invariant?
2. If the action is conformally invariant, this means that the classical theory is conformally invariant. Is the quantum theory conformally invariant?
3. If the action is not conformally invariant, modify it so that it becomes conformally invariant, while remaining quadratic. (Hint: use non-integer powers of the w:Laplacian.) The resulting theory is called mean field theory.

BOUB: Unitarity bounds on conformal dimensions

We assume that there is an inner product such that the dilation operator is self-conjugate $D^{\dagger }=D$. We also assume that the inner product is compatible with the Lie algebra structure of the conformal algebra, in the sense that $[A,B]^{\dagger }=-[B,A]^{\dagger }$.

1. Compute the conjugates of all the generators of the conformal algebra.
2. Starting with a scalar primary state $v$ of conformal dimension $\Delta$, and assuming $(v,v)=1$, compute the squared norm $(P_{\mu }v,P_{\mu }v)$ of the descendant state $P_{\mu }v$.
3. Assuming unitarity, deduce a bound on $\Delta$.
4. Is this bound valid in the case of the vacuum state?