The Conformal Bootstrap
The Conformal Bootstrap
- Miguel Fernandes PaulosMiguel Fernandes PaulosEcole Normale Superieure, CNRS, Université Paris Sciences et Lettres, Sorbonne Universite
Conformal field theories (CFTs) have a wide range of experimental and theoretical applications. They describe classical and quantum critical phenomena, low (or high) energy limits of quantum field theories, and even quantum gravity via the Anti-de Sitter space/CFT correspondence (AdS/CFT). Most interesting, CFTs are strongly interacting and difficult to analyze. The Conformal Bootstrap program is an approach that exploits only basic consistency conditions of CFTs, such as unitarity, locality, and symmetry, encoded into a set of bootstrap equations. The hope is that such conditions might be strong enough to uniquely determine the full set of consistent theories. This philosophy was first used successfuly in the 1980s to analytically determine and classify large classes of critical phenomena in two spatial dimensions. Starting from 2008, major developments have allowed the exploration of CFTs in more general spacetime dimension. The key breakthrough was to realize that one could exploit methods from linear and semidefinite optimization theory to analyze the bootstrap equations and obtain strong, universal constraints on the space of CFTs.
The Conformal Bootstrap has led to a number of important results in the study of CFTs. One of the main outcomes consists of general bounds on the data defining a CFT, such as critical exponents and operator–product expansion coefficients. This has been done for a number of contexts, such as different space-time dimensions, global symmetry groups, and various amounts of supersymmetry. More remarkably, this approach not only leads to general results on the space of theories but is also powerful enough to give extremely precise determinations of the properties of specific models, such as the critical exponents of the critical 3d Ising and O(2) models. Finally the conformal-bootstrap program also includes the formal study and non-perturbative definition of CFTs and their observables. These include not only the study of Euclidean correlation functions but also a study of their properties in Lorentzian signature; the study of defects, interfaces, and boundary conditions; finite temperature; and connections to the AdS/CFT correspondence.
- Condensed Matter and Materials Physics
- Particles and Fields
The goal of this article is to review recent developments in what is known as the Conformal Bootstrap, the use of elementary properties that are axiomatically satisfied by Conformal Field Theories (CFTs), to constrain them and, in some special cases, determine them. Unlike what happens in the analysis of ordinary quantum field theories, bootstrap methods are intrinsically non-perturbative: there are no Feynman diagrams or other perturbative methods, and at every step there are finite, well-defined quantities and computations. The conformal bootstrap is a rapidly developing field; the focus here will be mainly on well-established methods and results, with brief comments on recent developments.
Conformal field theories are quantum field theories invariant under the action of the conformal group. This is the group of coordinate transformations that leave the spacetime metric invariant up to an overall, possibly local, change of scale. The history of conformal symmetry in physics is nicely reviewed in Kastrup (2008). It seems to have first made its appearance in theoretical physics in work by Lord Kelvin, which showed covariance of the Laplace equation under coordinate inversions. Later, Bateman and Cunningham showed that Maxwell’s equations also transform covariantly under conformal transformations. The development of conformal field theory began in the 1970s and is ongoing. One of the high points is the paper by Belavin, Polyakov, and Zamolodchikov in 1984 (Belavin et al., 1984), with the discovery of infinite-dimensional conformal symmetry in two dimensions and exact solutions of the so-called minimal models that describe a large set of critical phenomena. Many modern developments were then spurred by the discovery of the AdS/CFT correspondence in the late 1990s: the conjectural equivalence of four-dimensional supersymmetric Yang-Mills theory with gauge group (a supersymmetric cousin of quantum chromodynamics), and type IIB superstring theory in asymptotically spaces (Aharony et al., 2000; Maldacena, 1998). This led to broad interest in the general properties of conformal field theories ,in particular in space-time dimension greater than two where the conformal group is now finite dimensional. The conformal bootstrap as understood in this article began with the seminal paper of Rattazzi, Rychkov, Tonni, and Vichi (Rattazzi et al., 2008), which first showed that it is possible to obtain strong bounds on CFTs using only crossing symmetry and unitarity.
Some motivation for studying CFTs is provided by considering one of their key applications: the description of critical phenomena, that is, systems undergoing second-order phase transitions. One of the most famous experimental examples of such phenomena occurs in boiling water: more precisely, water at its critical point at a temperature and pressure . This is a special point on the water phase diagram, shown schematically in Figure 1, where the line of first-order phase transitions separating the liquid and vapor phases terminates. At this point critical opalescence is observed: experimentally it is found that water becomes opaque. This can be explained by the existence of density fluctuations in the fluid of all sizes (which allow for scattering of light even for optical wavelengths). The correlator of density fluctuations has the form
The quantity is what is known as a critical exponent. It is one of the fundamental quantities characterizing the transition. This characteristic power-law behavior is not limited to density correlators but rather is observed generically for many other observables, such as the specific heat. Power-laws imply the absence of a preferred scale in the system: it signals an emergent scaling invariance, or scaling symmetry.
Another important example that will be relevant is a ferromagnet. If a ferromagnet is heated to a certain critical temperature, known as the Curie point, it will continuously lose its magnetization properties. Remarkably, the critical exponents in this case are the same as for a fluid at the critical point! This is known as universality: at criticality, most microscopic details of the system become irrelevant and only basic features of the system such as symmetries and dimensionality survive. Heuristically, both universality and scale invariance can be understood due to the fact that, at a critical point, correlation lengths diverge. Furthermore, the discreteness of the system becomes unimportant, and hence variables such as the density and local magnetization become continuous fields. Combined with scale invariance, this means that conformal field theories are the appropriate theoretical objects to describe these systems.
2. Basics of Conformal Field Theories
2.1 Conformal Correlation Functions
2.1.1 Conformal Transformations
The crucial property of a conformal field theory, which gives it its name, is that it has a symmetry under conformal transformations (Ferrara et al., 1971, 1972, 1973, 1974, 1975; Mack & Salam, 1969; Polyakov, 1974). These transformations include rotations, translations, and rigid scaling transformations, which, are all naturally associated to a critical point. They also include so-called special conformal transformations. There is no a priori reason why critical phenomena should be expected to exhibit this extra symmetry, but in many cases they do (though not always), as first noticed by Polyakov (1970). Under some assumptions, it can, in fact, be rigorously shown that scale invariance implies conformal invariance in space-time dimension 2 and 4 (Dymarsky et al., 2015, 2016; Polchinski, 1988; Zamolodchikov, 1986).
A conformal transformation is a coordinate transformation that leaves the metric invariant up to a scale factor
The metric is assumed to be flat and have Euclidean signature:
One way of justifying this is that in applications to statistical systems time-independent correlations are often of interest: a three-dimensional Euclidean CFT can be used to describe the phase transition in a piece of ferromagnetic material. It is possible nevertheless to analytically continue observables such as correlation functions to Lorentzian signature, where, in fact, many of the properties of the CFT become clearer.
The full set of conformal transformations is given by the mappings:
Here are arbitrary -dimensional vectors and a real number and is an antisymmetric real matrix, so that is an orthogonal matrix. The acronym SCT stands for special conformal transformations. Finally, the infinitesimal generators are written as:
The non-trivial commutation relations are given by
There are a total of generators, which is the same as for the algebra , which is, in fact, the algebra described by these commutation relations.
For the most, part conformal transformations are straightforward to understand. Only special conformal transformations are more unusual. One way to think about them is as compositions of translations with inversions:
whose effect is shown in Figure 2. But this begs the question: Why then are inversions not included? Inversions are not continuously connected to the identity transformation, and so do not have an associated infinitesimal generator.
There is an important addendum to this discussion. In two space-time dimensions, the group of conformal transformations is actually infinite dimensional. This is because the metric can be written in complex coordinates
It is easy to see that any holomorphic transformation leaves the metric invariant up to an overall factor. This larger group of symmetries can be used to obtain a number of powerful results about 2D CFTs, and in particular exact solutions for a large set of theories (Belavin et al., 1984; Di Francesco et al., 1997). However, this article focuses only on those symmetries, mentioned previously, that are present for any space-time dimension and not just .
2.1.2 Correlators of Primary Operators
As usual in quantum mechanics, the basic quantities of interest in a CFT are correlators of local operators, or equivalently of expectation values of products of operators, in the ground state (vacuum) of the theory:
In applications to critical phenomena, these have interpretations as statistical averages over thermal fluctuations. Particularly important operators are those that transform covariantly under conformal transformations. For instance, scalar operators should satisfy:
Those operators that satisfy this transformation property are called primary. The number is called the conformal or scaling dimension of . It measures the mass dimension of the operators. Conformal dimensions are directly related to critical exponents in the context of critical phenomena. Derivatives of primaries are called descendant operators. A primary operator together with the infinite tower of descendant operators that can be obtained from it by acting with the derivative, that is, , spans a representation of the conformal algebra. The correlators of operators in any such multiplet are all related by the action of the symmetry algebra: for example, those of descendants are simply related to those of primaries by taking derivatives. Therefore, attention can always be restricted to understanding correlators of the primaries. These turn out to be quite constrained, as a consequence of the transformation law:
To see what this implies, consider first the two-point function. In this case:
This is an important result: in a CFT, two-point functions are diagonal and behave as power laws, with an exponent determined by the scaling dimension of the operator. Hence, the job of determining critical exponents becomes equivalent to determining the spectrum of operators in a CFT.
Moving on, consider the three-point functions, focusing on scalars again for simplicity. In this case it can be shown that they must take the form
The three-point function is, therefore, uniquely fixed up to an undetermined coefficient, , which is known as the OPE (operator product expansion) coefficient. The reason for this name will become clear later. It can be shown that for any set of operators with arbitrary quantum numbers, their three-point functions are kinematically fixed up to a finite number of such coefficients. A case that will be of interest later has three-point functions of two scalars and one traceless-symmetric tensor, in which case there is only one such coefficient.
Finally, consider the four-point correlator of scalar operators, which for simplicity is taken to be identical. In this case the correlator has to take the form
where is now an arbitrary function of two independent conformal cross-ratios:
It can be easily checked that such conformal cross-ratios are left invariant under conformal transformations. They appear only when there are at least four independent points, which explains their absence in two- and three-point functions. This logic generalizes. An arbitrary correlation function is always given by an overall kinematic prefactor that carries the non-trivial conformal transformation properties of the operators, and which can be written in several ways, times a function of a finite set of independent cross-ratios.
2.2 The Operator–Product Expansion
Perhaps the most important additional property of a CFT as compared to an ordinary quantum field theory is the existence of an operator–product expansion with a finite radius of convergence. This is a consequence of the state–operator correspondence discussed here.
2.2.1 Radial Quantization
Starting from a CFT originally defined on flat Euclidean space, there is a natural way to define the theory in the same space with a different metric, as long as they are related by an overall Weyl transformation:
The correlators of the theory on Weyl related metrics are defined by setting:
Specific CFT examples with Lagrangian descriptions can often be conformally coupled to the background metric, both sides independently computed and agreement checked. In general however, this should be thought of as a definition of the theory in a different space. There is one important exception to this rule. Under a Weyl transformation, the stress tensor does not in general transform as in the prior. This is because, in general, the Weyl symmetry of the CFT is anomalous in even dimensions. The effect of the anomaly is very mild, modifying only correlation functions involving the stress tensor.
The most important application of this is the “plane”-cylinder map (“plane” means flat -dimensional Euclidean space). The flat space metric can be written in radial coordinates,
where is the volume element of the dimensional sphere. Introducing gives
where the metric on the cylinder has been introduced. The mapping is shown pictorially in Figure 3. It follows that
with , and vectors satisfying . Notice how the radius on the plane has become the time coordinate on the cylinder. In particular, the Hamiltonian on the cylinder is nothing but the dilatation operator on the plane. Hence, quantizing the theory on the cylinder amounts to studying the spectrum of the dilatation operator on the plane: this is called “radial quantization.” On the plane, fixed “time” slices correspond to spheres where the states of the theory are defined.
2.2.2 State–Operator Correspondence
Consider an operator inserted at the origin in the plane (which is on the cylinder). This defines a state . On the cylinder the state is defined by sending an operator to and multiplying it by a prefactor,
In any case,
The conclusion is that operators with definite scaling dimension (these can be primary or descendants) create states with definite energy on the cylinder, so really . This is called the state–operator correspondence:
State–operator correspondence: to each primary or descendant operator corresponds an eigenstate of the CFT Hamiltonian on the cylinder.
The full Hilbert space of a CFT decomposes into representations of the conformal algebra. Such representations can be built by starting with a “lowest weight” state, the primary, and acting with as many as desired in all possible configurations. The number of is sometimes called the level of the descendant. Any such state is of the form where the dots represent indices. The action of the rotation generators commutes with dilatations, so it just mixes descendants of the same level. All such states are energy eigenstates on the cylinder, with various kinds of angular dependence.
2.2.3 The Operator–Product Expansion
The Hilbert space of a CFT has been constructed through radial quantization. Conformal primaries and their descendants map one to one with the eigenstates of the cylinder Hamiltonian; hence, knowing all scaling dimensions (and other quantum numbers) of the primaries in the CFT amounts to solving the spectral problem on the cylinder. Since energy eigenstates form a complete basis of the Hilbert space, then any state can be decomposed into this basis. Consider local operators inserted in some region, and some large-enough sphere containing these insertions. Radial quantization tells us that these insertions generate some state on the sphere. Because any state can be decomposed into the basis of eigenstates of dilatations, and because each eigenstate is associated to a primary or descendant operator, the conclusion is the:
Operator–product expansion: An arbitrary product of CFT operators can be expanded into a sum of primary and descendant operators.
The expansion is written with operators at the center of a sphere containing the initial product of operators. Furthermore, the expansion has a finite radius of convergence, which is the radius of the largest such sphere that does not intersect any other operator insertions.
It is clear that it is sufficient to determine the OPE for just two operators, because if there are more, they can always be iterated:
The (infinite) sum runs over all conformal primaries in the theory and their descendants, indicated schematically by acting on the primary (in reality each has an index, such indices being contracted into indices in the and the operator if it has spin). It is important to emphasize that this is not a formal expression but can actually be inserted into correlation functions, where it will converge exponentially fast (Pappadopulo et al., 2012).
The fact that both three- and two-point functions of primaries are uniquely fixed by conformal invariance determined the OPE. By inserting this expansion into a three-point function and comparing with the exact expression, it is possible to fix all coefficients in terms of the three-point couplings appearing in those three-point functions, which are therefore known as OPE coefficients. In particular, contributions of descendant operators to the OPE are completely fixed by symmetry in terms of those of the corresponding primary operator. It is therefore common to represent an OPE schematically as
It is important to realize that if somehow successful in determining the spectrum of a CFT, that is, the set of all scaling dimensions and other quantum numbers of primary operators, and their OPE coefficients, then an arbitrary correlation function can be reduced by repeatedly using the OPE to a sum of known two-point functions. This is summarized in the concept of:
CFT data: the quantum numbers of all primary operators in the theory as well as their three-point couplings or OPE coefficients.
By repeated use of the OPE, knowledge of the CFT data suffices to compute any correlation function in the theory.
2.3 Conformal Blocks and the Crossing Equation
2.3.1 Conformal Blocks
The OPE can be used to understand conformal four-point functions by applying it twice to end up with a double sum over two-point functions. Because these must be diagonal, the double sum reduces to a single one, running over contributions of primary operators and their descendants. Beause the contributions of descendant operators are fixed by symmetry in terms of those of primaries, they should be resumed to obtain an expression that depends only on the dynamical data. This resummation of the OPE leads to what is known as a conformal block.
Conformal blocks depend on the detailed quantum numbers both of the external operators appearing in the four-point function, as well asthose of the exchanged conformal primary. The main complication occurs in allowing for non-scalar operators as external operators. The allowed spin-representations that can appear in intermediate channels are then much larger and space-time-dimension dependent. There can also be more than one kind of block for the same exchanged representation.
In the simplest case where the external primaries are scalars, the possible exchanged states must transform as traceless symmetric tensors, which are labeled by a single spin quantum number, , that is, the OPE takes the form
At this point, an important remark is that demanding that all states in the CFT Hilbert space have positive norm leads to the following constraints on quantum numbers, known as unitarity bounds:
The four-point function can now be written in terms of conformal blocks. Using the OPE on the pairs of operators 1,2 and 3,4 leads to an expression
that involves only the OPE coefficients, the quantum numbers of primary operators, and the kinematically determined conformal blocks. In this simple case, exact expressions for the blocks in even space-time dimensions can be given. In particular, for the interesting cases and give
Here, the Dolan-Osborn variables
have been introduced and
Note that with in for Euclidean kinematics. A great number of interesting properties, identities, and relations about conformal blocks were worked out in the papers of Dolan and Osborn (Dolan & Osborn, 2001, 2004, 2012). One key insight is that conformal blocks are eigenfunctions of the Casimir operator of the conformal algebra. This has led to a number of applications and extensions, for example, Isachenkov and Schomerus (2016) and Costa et al. (2011). Conformal blocks cannot always be written down in closed form, but by now there are methods for computing them numerically to great accuracy in any space-time dimension. Usually this is done by determining rapidly convergent power-series expansions for the conformal blocks. One particularly convenient method is based on an idea of Zamolodchikov (Zamolodchikov, 1984) originally used for the special case (where the conformal group is infinite). In this method, conformal blocks are treated as analytic functions of the exchanged scaling dimension. By studying poles and residues in this variable, it is then possible to set up a rapidly convergent recursion relation for an expansion of the blocks around (Kos et al., 2014).
2.3.2 Crossing Symmetry
The OPE allows a four-point function to be represented as a sum of conformal blocks. However, this can be done in several ways: either in operator pairs (14) and (23), or (13) and (24). The resulting expressions for the correlator would look very different: the operators appearing in the OPE will not necessarily be the same, the OPE coefficients are different, and even the blocks have a different form. That any two such representations can lead to the same four-point function is a very strong constraint on the CFT data. As an example, consider the four-point function of identical scalar operators of dimension . Equating OPE expansions in the (12) and (14) channels leads to the following crossing equation:
where and the contribution of the identity operator with , which always appear in the OPE of two identical operators (the identity satisfies ), has been separated out. To a large extent, the conformal bootstrap is the analysis of the constraints that follow from this kind of equation, and in particular how exactly they constrain the allowed sets of CFT data. The equation is shown pictorially in Figure 4.
In general, the OPE should be associative: that is, the order in which products of operators are taken should not matter. Depending on this order, apparently very different representations of the same product of operators may result. Demanding all such representations agree is known as crossing symmetry. It can be checked that crossing symmetry for all four-point functions is enough to ensure it for all higher order correlation functions. Furthermore, four-point functions impose relatively simple quadratic constraints on the OPE coefficients and involve simpler expressions than at higher points, which has led to the vast majority of efforts in the bootstrap to focus on this case.
3. Numerical Bootstrap
This section shows how to obtain constraints on the CFT data by borrowing techniques from optimization theory to analyse crossing equations. A striking application of these methods is presented to determine the properties of the CFT describing the 3D Ising model.
3.1 Numerical Techniques
3.1.1 Basic Picture: Bounds
Even in this simplest of cases, the crossing equation for the four-point function, that is, equation (31), is formidable: it involves a continuously infinite set of constraints on a continuously infinite set of variables. What does this equation imply about the allowed sets of ?
The key idea is to think about the crossing equation as a linear equation, with the interpreted as vectors (Rattazzi et al., 2008). In fact, they can literally be turned into vectors, by considering just a finite dimensional set of constraints. Usually this is done by straightforward Taylor expansion around the crossing symmetric point . For instance:
The crossing equation then expresses linear dependence between the identity and the contributions of other operators in the OPE
If the vectors have linearly independent components generically any choice of vectors in the sum in (33) will form a basis of , and so it would seem that this crossing equation can always be solved trivially. However, the coefficients in that equation must be non-negative. This is a consequence of unitarity of the CFT, which guarantees that the OPE coefficients of real operators can always be chosen real, if their two-point functions are normalized as in (11). The set of all non-negative linear combinations of vectors forms a cone, and it might just be that the vector lies outside the cone.
Solutions to the crossing equation are known to exist, so this will not be the case unless some restrictions are made. One of the simplest ideas is to impose a gap in the spectrum. For instance, if the dimension of first scalar operator in the OPE is larger than some value , the equation has solutions if there are OPE coefficients
Remarkably there are no such coefficients if is large enough! Finding the maximal such is known as the gap maximization problem. This maximal value is referred to as a function of by . It implies that:
in the OPE of two identical fields, there must exist a non-identity scalar operator whose dimension is no larger than .
In other words, there is an upper bound on the scaling dimension of the leading operator in the OPE. This bound, given by can only improve as increases. This again is a key insight: even though the full crossing equation is very complicated, it can nevertheless be truncated and interesting rigorous bounds that any CFT must satisfy can still be extracted. These bounds can be systematically improved by considering larger truncations.
3.1.2 Functionals: Linear and Semidefinite Optimization
How can bounds be computed in practice? The idea is that if the identity does not lie inside the cone spanned by the other crossing vectors, then it is possible to find a hyperplane that separates the latter from the former. If such a hyperplane can be found, there can be no solution to crossing. This is shown in Figure 5. A hyperplane is simply a linear functional . The feasibility problem can be formalized:
Feasibility problem (single correlator)
The existence of such a linear functional rules out solutions to the crossing equation where the exchanged operators have dimensions and spins lying in set :
For instance, this set was chosen to consist of all scalar operators with , and spinning operators consistent with unitarity, . Clearly if a functional can be found for some , it can also be found for any other by merely setting some components to zero. In this way the feasible region is non-increasing as is increased.
Problem (35) is an example of a semi-infinite linear program. A linear program is an optimization problem with a linear objective function subject to linear constraints. In the present case, the objective function is simply zero because the aim is merely to find a functional consistent with the constraints. There are similar problems such as maximizing values of OPE coefficients for which this objective is non-zero (Rattazzi et al., 2011). In any case, similar problems have been studied extensively and there exist efficient algorithms for solving them for any (Hettich & Kortanek, 1993; Lopez & Still, 2007; Reemtsen & Rückmann, 1998).
Including the constraints arising from crossing symmetry for a finite set of correlation functions involving more than one kind of operator (Kos et al., 2014) poses a problem, because for a typical correlator the OPE will not involve squares of OPE coefficients. Positivity is therefore lost, and it seems like the method described here fails. However, this can be circumvemted by focusing on the constraints of crossing for a set of correlations functions involving fields in a finite set. This will lead to a finite set of crossing equations, labeled by an index , each of which presents a quadratic constraint on the OPE coefficients. These can be written:
Here, for each , is some known matrix is computed in terms of conformal blocks, while the vector contains the OPE coefficients between the operator and all possible pairs of external fields appearing in the set of correlators under consideration,
A feasibility problem can be formulated in this case too. A set of linear functionals (which can be taken to consist of various derivatives at the crossing symmetric point) and this problem posed:
Feasibility problem (multiple correlators)
To clarify, notice that each functional acts componentwise in the entries of each matrix , so that the net outcome of acting with the functional is still a matrix. Positive semidefinite conditions on this matrix are required, for example, if for all . If these can be satisfied then there is no solution with because
The feasibility problem (39) is an example of what is known as a semidefinite program: an optimization problem (in this case the objective function is trivial) subject to positive semidefiniteness constraints. These can again be solved using standard algorithms such as interior point methods (Alizadeh, 1993).
3.1.3 Extremal Functionals and Spectrum Extraction
In the feasibility problem, for each truncation order of the crossing equation there is a sharp boundary between the regions where a functional can be found—the excluded region—and where one cannot—the allowed region. In the excluded region, there can be no solutions to crossing. As for the allowed region, although it cannot be ruled out that some particular spectrum will not be excluded by increasing , at least for fixed there are infinitely many approximate solutions to crossing, that is, solutions to the truncated set of crossing equations. At the boundary between the two regions, something very special happens: there exists both a certain functional (called the extremal functional) and a uniquely determined solution to the truncated crossing equations. This is possible because the extremal functional annihilates the identity and at most a finite set of crossing vectors , which are precisely the vectors solving the truncated crossing equations (El-Showk & Paulos, 2013, 2018). One of these must necessarily be the scalar operator of dimension , that is,
and the extremal functional, called , satisfies
where is zero unless lies on the boundary of (i.e., the allowed set of CFT data). The derivative conditions are necessary for positivity of the functional in the vicinity of its zeros. It follows that at the boundary of the allowed region, the functional action can be simply plotted and its zeros determined to obtain a spectrum that solves the truncated crossing equations. Increasing the size of the truncation causes the spectrum to slowly stabilize, starting from operators with small scaling dimension. If a certain CFT of interest sits at the boundary of the allowed region, this method provides a way of obtaining a large set of the corresponding CFT data.
Solving the optimization procedures outlined here may actually be more complex than implied. One of the issues is that the number of constraints involved in these problems is actually continuously infinite. This problem has been dealt with in various ways.
The most obvious solution, and the first to have been considered, is to simply discretize the set . Considering a large grid of reduces the number of constraints to a finite set. Whether refining the grid leads to any changes in the result can be checked a posteriori.
A second solution is to approximate the crossing vectors and their derivatives as rational functions of involving polynomials of some sufficiently large rank . In this case, it is possible to recast positivity of the functional action for all as a positive semidefiniteness problem on a large matrix (whose rank depends on the degree ). Notice this is not the same positive semidefinitess that arises in the feasibility problem involving multiple correlation functions, because these still have a continuously infinite set of constraints.
Finally, it is possible to implement versions of standard algorithms such as the Simplex Method adapted to a continous set of constraints. The key point in these algorithms is to have a sufficiently efficient way of searching for local and global minima of a functional action during the optimization procedure.
A very important peculiarity of the bootstrap problem is that owing to the exponentially fast convergence of the OPE, obtaining robust solutions of the optimization problems requires usage of arbitrary precision algorithms. At the time of writing this has led to two custom-made public codes for the conformal bootstrap (Landry & Simmons-Duffin, 2019; Paulos, 2014; Simmons-Duffin, 2015).
3.2 Bootstrapping the 3D Ising Model
Perhaps the most famous application of numerical bootstrap methods has been to determine the properties of the 3D Ising CFT. This is the CFT that is believed to describe the critical point of the Ising model on a 3D cubic lattice (Delamotte et al., 2016; Meneses et al., 2019). This lies in the same universality class as binary mixtures, the opalescence point in fluids, the Curie transition in uniaxial ferromagnets, and many others.
The Ising model is a simple model for a ferromagnet and consists of a -dimensional lattice of (classical) spins with nearest neighbor interactions, with Hamiltonian:
For this model has a phase transition at a finite critical temperature . The spin-spin correlator typically decays exponentially with some length scale, called the correlation length . But, at the critical temperature diverges:
and the spin-spin correlator now becomes a power law
The low-lying operators in the Ising CFT are then , the spin field, which couples to the external magnetic fields, and the energy operator , which couples to temperature. One way to define the 3D Ising CFT is as the hypothetically unique 3D CFT with a symmetry and two relevant operators (relevant operators correspond to the knobs an experimentalist has to tune to reach the critical point, in this case temperature and magnetic field in the case of a ferromagnet). The 3D Ising model is easily simulated using Monte-Carlo algorithms, showing a second-order phase transition at some critical temperature. It is then possible to measure the scaling dimensions , for example, Hasenbusch (2010), which are related to the critical exponents by
What can the conformal bootstrap say about this theory? Consider the four-point function of an operator, denoted . Its scaling dimension is unknown. Its OPE is assumed to take the form
In other words, is said not to appear in its own OPE, which is the same as saying that there is a symmetry under which is odd, and the first scalar and -even operator in the OPE is called , whose dimension is also unknown. The scaling dimension of cannot be arbitrarily large: there is a maximal allowed gap which can be found by constructing suitable functionals. The bound is obtained by expanding the crossing equation to some high order in derivatives and constructing suitable functionals numerically. Beyond some expansion order there is no visible change in the bound.
Figure 6 shows this maximal allowed gap as a function of the dimension of the field for 2D (Rattazzi et al., 2008; Rychkov & Vichi, 2009) and 3D CFTs (El-Showk et al., 2012, 2014a). (Note that the bootstrap methods are essentially agnostic about space-time dimension: the only difference lies in the computation of conformal blocks, which can be done for any desired value of dimension, even fractional (El-Showk et al., 2014b).) Both plots have a visible “kink” or inflection point. A major clue to the significance of this point is given by the fact that, in 2D, this kink occurs very accurately for
In other words, the kink seems to match with the known scaling dimensions of the 2D Ising CFT. This CFT describes the critical point of the 2D Ising model and can be solved analytically. The natural conjecture then is that the kink appearing in the bound for 3D CFTs corresponds to the 3D Ising model. This is corroborated by the fact that it closely coincides with Monte Carlo determinations of these quantities (Hasenbusch, 2010).
This result includes only the constraints arising from the crossing equation for the four-point function . It seems natural, however, to also include those constraints arising from correlation functions involving the field . In doing so assumptions consistent with what is known about the 3D Ising CFT can be made, namely that and should be the only two relevant operators in the theory. The OPEs take the form
Under these assumptions, the feasibility problem can be solved by setting appropriate gaps in each OPE. The results of the feasibility problem are shown in Figure 7.
Strikingly, the allowed region has shrunk to a very small “island” (there is also a “mainland” further to the right which is not shown) (Kos et al., 2014, 2016). This means that the OPEs described previously are consistent with crossing symmetry and unitarity only if and lie inside this tiny region. This region is compatible with, and significantly improves on, Monte Carlo determinations of the critical exponents. At the time of writing, the best determinations for these come from the bootstrap and give
Because the Ising CFT saturates a bound, it is possible to use the extremal functional method to obtain the corresponding CFT data. This has been used to obtain detailed determinations of the 3D Ising spectrum, including scaling dimensions of around 100 operators and their OPE coefficients to high accuracy (Simmons-Duffin, 2017).
There is a similar version of this story for models. In this case the correlators of the fundamental charged field and eventually the lowest-lying singlet and antisymmetric tensor operators are bootstrapped. It is then possible to arrive at a picture containing an “archipelago” of isolated points corresponding to critical models for varying (Chester et al., 2020a, 2020b; Kos et al., 2015, 2016).
4. Analytic Methods
This section describes some analytic methods for studying the consequences of crossing symmetry. This study is made difficult by the lack of non-perturbatively defined, exactly solvable CFTs in space-time dimension . However, there is one very special case. This CFT is the generalized free field theory (El-Showk & Papadodimas, 2012). Its correlation functions can all be obtained from Wick contractions of an elementary field , as in free theory, but with a non-trivial scaling dimension . For instance, the four-point function can be written:
or using the OPE:
with and known coefficient . Because finding any solution to crossing at all is non-trivial, it can well be speculated whether knowledge of this particular solution can help in constraining others. For instance, it is expected that, “on average,” the spectra of CFTs at high scaling and dimension and spin should be rather similar and indeed perhaps similar to the prior solution. The methods described here are in a sense a way of making this idea precise.
4.1 Lightcone Bootstrap and the Inversion Formula
The crossing equation lends itself to an analysis in the lightcone limit. This is a limit that must be taken after continuation to Lorentzian signature. At the level of the four-point function this amounts to considering the cross-ratios to be real and independent variables (instead of complex conjugate as in the Euclidean section). The lightcone limit is then . In this limit operators become null separated, so that, for example, but does not. The OPE is then dominated by operators with low twist, defined as . Indeed, in this limit conformal blocks can be approximated as
Because of the unitarity bound, for , which implies that for the lowest twist operator is always the identity. In the lightcone limit, taking also
The point now is that the diverges logarithmically as . Hence, to reproduce the power-law divergence on the left-hand side requires an infinite number of operators with increasingly large spins, and twists tending to . The leading correction to this result comes from including the subleading twist operator (with twist ) on the left-hand side, which gives a term with . This can be reproduced on the right-hand side by including a small anomalous dimension for the tower of operators at large spin of the form
Starting from Fitzpatrick et al. (2013) and Komargodski and Zhiboedov (2013), this sort of argument was refined and generalized to study conformal correlators in perturbation theory analytically without resorting to Lagrangian methods in a variety of contexts (Alday, 2017a, 2017b; Alday & Bissi, 2017; Alday & Zhiboedov, 2017; Alday et al., 2015a, 2015b, 2018, 2020; Kaviraj et al., 2015a, 2015b). This line of reasoning culminated in the discovery of the Lorentzian inversion formula (Caron-Huot, 2017) (see also Simmons-Duffin et al., 2018). This formula establishes analyticity of the OPE data in spin, so that operators in a CFT correlator lie in analytic curves in complex angular momentum known as Regge trajectories, in analogy with particle physics.
The understanding of this formula begins with the Euclidean inversion formula. This is merely the decomposition of a CFT correlator into a complete basis of eigenfunctions of the conformal Casimir:
The OPE function has poles in at the position of primaries in the correlator, so that closing the contour on the right yields the usual conformal block decomposition (up to several subtleties not discussed here). The formula may be inverted as
with some known normalization factor. This formula makes sense only for integer, because otherwise the eigenfunctions are not single-valued. The trick now is to deform the contour of integration to Lorentzian kinematics, so that the analytic continuation to complex is possible. The result is the Lorentzian inversion formula:
The important point is that this formula holds for general complex as long as . This means that the poles of , which give information on operators in the correlator, lie in families analytic in spin, as promised:
where is the twist of the -th Regge trajectory.
The formula is defined in terms of the double discontinuity of the correlation function, which in the CFT context plays the role of the imaginary part of the scattering amplitude in ordinary QFT. It is essentially given by the difference between the correlator in Euclidean kinematics and its analytic continuation to a regime where operators are pairwise timelike separated. Using the OPE gives, for instance
The sine-squared factor annihilates the spectrum of generalized free theory. This means that the inversion formula is very useful when considering weakly coupled CFT, because it can be used to derive the CFT data order by order in perturbation theory. This is completely analogous to similar methods in scattering amplitudes, where unitarity relates lower loop results to higher loops.
4.2 Analytic Functionals, Dispersive Sum Rules, and the Polyakov Bootstrap
The numerical bootstrap methods were based on applying a finite dimensional set of functionals to the crossing equation. The action of each such functional leads to a sum rule on the OPE data:
An exact solution to crossing such as the generalized free-field example is known to have to satisfy any such sum rule. However, this is quite non-trivial to check for the usual functionals given by derivatives or point-wise evaluation. It could well be wondered whether special functionals can be concocted that would make it obvious that the generalized free field is a solution.
Such functionals do exist. They were first constructed in a simplified setting in the absence of spin, that is, for CFTs (Mazac, 2017, 2019; Mazáč & Paulos, 2019a, 2019b; Paulos, 2020a). For higher dimensional CFTs there are also proposed basis of functionals with this property. (Carmi & Caron-Huot, 2020; Caron-Huot et al., 2021; Gopakumar et al., 2021; Mazáč et al., 2019; Penedones et al., 2020; Sinha & Zahed, 2021).
The result is that there exist functionals that, essentially, satisfy the duality conditions
Furthermore, such bases of functionals are complete, in that they fully capture all constraints encoded in the crossing equation. The duality conditions imply the functionals will have double zeros at the position of generalized free-field operators. The sum rules associated to such functionals are called dispersive. In fact, such sum rules basically amount to the double discontinuity of the CFT correlator convolved with an appropriate kernel, and completeness can be translated into applicability of the Lorentzian inversion formula to any CFT.
The sum rules implied by these functionals are
These equations are a reformulation of the constraints of crossing symmetry, and therefore must hold for any CFT. As promised, the sum rules implied by these functionals are essentially trivially solved by the generalized free-field solution:
where these are the same coefficients as in (52).
The fact that functionals annihilate the identity has to be checked, whereas the second set of equations amounts to a determination of what the OPE data should be. Of course it agrees with a direct determination of this data by other methods.
The formulation of the crossing constraints in terms of such functionals has many advantages. They immediately lead to strong exact and sometimes optimal bounds on the CFT data (they are optimal because they are saturated by the generalized free-field CFT). Furthermore, since they are trivially solved by generalized free fields, in a sense these functionals already know about the correct asymptotics of any given CFT, and hence allow these asymptotics to be decoupled from the “low energy” (i.e., scaling dimension and spin) physics of interest. In practice, this means that, where it has been checked, numerical bootstrap methods seem to be vastly more efficient using such functionals than traditional derivative methods (Paulos, 2020b; Paulos & Zan, 2020).
Finally, this approach connects to an old idea of Polyakov that has been revived in recent years, namely (Dey et al., 2017; Gopakumar et al., 2017a, 2017b; Gopakumar & Sinha, 2018; Mazáč & Paulos, 2019b; Polyakov, 1974; Sen & Sinha, 2016; Sinha & Zahed, 2021), the conformal block expansion for a correlator manifests the OPE and unitarity. However, it obscures crossing. Can an expansion be found that does the reverse, such that crossing is manifest? It turns out this is possible. Any CFT correlator admits an expansion of the form:
where the Polyakov blocks are crossing symmetric functions. They themselves admit an OPE expansion of the form
The “Polyakov bootstrap” then amounts to replacing the constraints of crossing symmetry, which are automatic in (65), by those of validity of the OPE, which can work only if the “fictitious” operators present in each Polyakov block drop out in the sum over states
The resemblance to (63) is not accidental. In fact, it is possible to show that in order for the Polyakov bootstrap to be well-defined, must be directly related to the functional actions and .
This article offers a brief glimpse into the conformal bootstrap. This field is essentially a set of analytic, numerical, and computational tools to extract information from crossing equations. There are a number of obstacles to be overcome. First, kinematics must be understood: writing down the relevant set of independent crossing equations and the allowed sets of exchanged operators, which can be non-trivial when considering correlation functions with spinning fields, and/or of fields charged under a global symmetry. The relevant conformal blocks appearing in these equations must then be computed efficiently for numerical applications. Finally, the crossing equations themselves do not yield their constraints on the CFT data easily, and it is important to define efficient functional bases for doing so. There exists room for progress in all these fronts.
The bootstrap results show that crossing symmetry and positivity, from unitarity, are very powerful constraints on the CFT data. In two dimensions, it is known that these constraints are not sufficient: modular invariance, which equates the theory defined on equivalent torii, provides further constraints. Similar constraints also exist in higher dimensions, but unlike in two dimensions, it is hard to formulate them as conditions on the CFT data. In general, there could be other consistency conditions from placing CFTs on other manifolds, and understanding these is an open problem.
The bootstrap methods have mostly been applied to constrain correlation functions of local CFT operators. However, the bootstrap philosophy can also be applied in other contexts. An important set of examples are given by 2D CFT partition functions, obtained by placing the theory on the torus. The partition functions depend on the modulus of the torus and must be modular invariant, that is, invariant under transformations that lead the torus unchanged. This leads to an equation very similar to the one analyzed here, with characters of the Virasoro algebra replacing conformal blocks, and degeneracies of states replacing OPE coefficients; it is possible to obtain bounds on these quantities using bootstrap techniques (Hellerman, 2011). Crossing equations have also been written down for CFTs in the presence of defects, interfaces, and at finite temperature, for example (Billò et al., 2016; Gliozzi et al., 2015; Iliesiu et al., 2018; Liendo et al., 2013). The common difficulty with all these cases is lack of positivity in the OPE. Different methods are required to make progress, but what those are is not yet clear at the time of writing. See, however, Gliozzi (2013) and El-Showk and Paulos (2018) for some possibilities.
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