###### Abstract

We derive a Lorentzian OPE inversion formula for the principal series of . Unlike the standard Lorentzian inversion formula in higher dimensions, the formula described here only applies to fully crossing-symmetric four-point functions and makes crossing symmetry manifest. In particular, inverting a single conformal block in the crossed channel returns the coefficient function of the crossing-symmetric sum of Witten exchange diagrams in AdS, including the direct-channel exchange. The inversion kernel exhibits poles at the double-trace scaling dimensions, whose contributions must cancel out in a generic solution to crossing. In this way the inversion formula leads to a derivation of the Polyakov bootstrap for . The residues of the inversion kernel at the double-trace dimensions give rise to analytic bootstrap functionals discussed in recent literature, thus providing an alternative explanation for their existence. We also use the formula to give a general proof that the coefficient function of the principal series is meromorphic in the entire complex plane with poles only at the expected locations.

A Crossing-Symmetric OPE Inversion Formula

Dalimil Mazáč,

[1cm] C. N. Yang Institute for Theoretical Physics, Stony Brook University

Stony Brook, NY 11794, USA

Simons Center for Geometry and Physics, Stony Brook University

Stony Brook, NY 11794, USA

###### Contents

## 1 Introduction

The conformal bootstrap is the idea of solving for the dynamics of conformal field theories starting from the basic principles of conformal invariance, unitarity and associativity of the operator product expansion. This approach has received widespread attention ever since the numerical studies of [1, 2] demonstrated its unexpected constraining power in more than two dimensions.^{1}^{1}1See [3, 4, 5] for early formulations of the conformal bootstrap and [6] for a review of recent developments. In parallel, an analytic approach to the conformal bootstrap has been developed, initially based on the expansion of the bootstrap equations near a pair of light-cones [7, 8, 9, 10].

More recently, the results of the analytic conformal bootstrap have been unified and extended through the so-called Lorentzian inversion formula [11].^{2}^{2}2See [12, 13] for more details on and generalizations of the original Lorentzian inversion formula. The formula exploits complex analyticity of the four-point function to extract its OPE decomposition from the double commutator, also called double discontinuity and denoted .^{3}^{3}3For identical external scalar primaries, is defined by , where are the coordinate vectors. The cross-ratios are defined by , with . The notation and conventions of this note follow closely those of [12]. More precisely, the formula computes the coefficient function of the decomposition of the four-point function into a complete set of conformal partial waves labelled by their dimension and spin . The OPE decomposition of can be read off from the poles and residues of . The Lorentzian inversion formula expresses as an integral of the double discontinuity times an appropriate inversion kernel over a Lorentzian spacetime diamond .

One of the most important virtues of the Lorentzian inversion formula is that it allows one to compute the OPE data exchanged in a given channel in terms of the OPE data in the crossed channels. This in turn leads to systematic expansions of the OPE data at large spin, and large scaling dimension [14]. Furthermore, the contributions of crossed-channel operators at mean-field double-trace scaling dimensions are suppressed by the formula, allowing for a recursive determination of the OPE data in perturbation theory around the generalized free field [15, 16, 17, 18, 19, 20].

The inversion formula of [11] is valid as long as the spacetime dimension is greater than one. One may be interested in having also an analogous formula which assumes only the minimal conformal symmetry, namely the global conformal symmetry of a line, corresponding to the algebra . One reason to look for such formula is that there exist intrinsically one-dimensional conformal-invariant systems, such as line defects in higher-D CFTs [21, 22, 23, 24, 25, 26] or SYK-like models [27, 28], to which the standard Lorentzian inversion formula in does not apply. Furthermore, one should keep in mind that every higher-D CFT is in particular also a 1D CFT since its correlators can be restricted to a line and satisfy all the axioms of the conformal bootstrap. Finally, 1D CFTs provide a simpler but still constraining setting for testing ideas about higher-D conformal bootstrap.

One needs to face some obvious challenges when trying to derive a Lorentzian inversion formula for . First, it may not be immediately clear what the distinction is between the meaning of “Euclidean” and “Lorentzian” in one dimension. Second, the existence of the Lorentzian formula in is closely tied to the fact that the CFT data are analytic in spin. This property may seem mysterious from the point of view for the simple reason that there is no spin in one dimension. In fact, these two points have already been addressed in [12] and [28]. To explain their resolution of the above puzzles and how the present work fits in the existing literature, let us first quickly review the basic kinematics of 1D CFTs.

We consider the four-point function of identical primaries in a unitary, parity-invariant 1D CFT. It can be written as

(1.1) |

where are positions on the line and plays an important role in various contexts, including the present note. If the 1D four-point function arises by restricting a higher-D four-point function to collinear configurations, we find . The four-point function can be expanded in a complete set of conformal partial waves using the Euclidean inversion formula. In , this complete set consists only of the *principal series* with and . The complete set in includes both the principal series and the *discrete series* .^{4}^{4}4Here and in the rest of this note, stands for *positive* integers and for positive real numbers. Correspondingly, the four-point function is described by a pair of coefficient functions: for the principal series and for the discrete series. is analogous to from in the sense that primary operators in the OPE with generic scaling dimensions translate into poles of on the positive real axis. As we review in the main text, the main role of the discrete series is to cancel spurious contributions of the principal series.^{5}^{5}5In special circumstances, the discrete series may capture the OPE data of physical operators, see [29]. The Euclidean inversion formula expresses the coefficient functions as integrals of over the real line as follows is the only cross-ratio of four points. A priori, the cross-ratio ranges over all real numbers but the analytic continuation to complex

(1.2) |

where are the conformal partial waves. A priori, these formulas only compute the coefficient functions for restricted to the principal and discrete series respectively and thus do not provide their analytic continuation to the complex plane. Consequently, does not hold in general.

We are interested in a Lorentzian, rather than Euclidean, inversion formula for and . Following [12], we will take this to mean a formula which extracts the coefficient functions from the double discontinuity of , rather than from the value of the Euclidean correlator. Such definition ensures that the 1D formula replicates the usefulness of its higher-D cousin for the analytic bootstrap. In 1D, the double discontinuity is defined by

(1.3) |

where and are the analytic continuations of from to above and below the branch cut. When arises from a correlator by restricting to collinear configuration, then the above definition agrees with the standard double discontinuity restricted to . Furthermore, this definition also agrees with the thermal expectation value of the double commutator , where and are evolved in Lorentzian time. The terms and correspond to the out-of-time-order contributions in the double commutator. The limit of these terms can be used to diagnose chaos and is analogous to the Regge limit of [30, 31, 32, 28].

The authors of [12] derived the following Lorentzian inversion formula for the discrete series of

(1.4) |

where is the 1D conformal block. This formula provides an analytic continuation of away from positive even integers and thereby answers the second point raised above: the analogue of analyticity in spin for is analyticity in the label of the discrete series. This is indeed needed to make the correlator bounded in the Regge limit [28].

The main result of this note is an analogous formula for the coefficient function of the principal series. Such formula is clearly needed for many interesting applications since , and not , carries information about the spectrum in a generic OPE. The formula takes the form

(1.5) |

where is an appropriate inversion kernel. We will fix by demanding compatibility of (1.5) with the Euclidean inversion formula. For the formula to be valid, must be bounded in the Regge limit.

While (1.5) is similar to the standard higher-D inversion formula, there are some key differences between the two. Most notably, the present formula only works for four-point functions of identical external operators. At the practical level, this is because the contour-deformation argument relating (1.5) to the Euclidean inversion formula is only valid provided is fully Bose- or Fermi-symmetric.^{6}^{6}6On the other hand, Caron-Huot’s formula works for four-point functions of arbitrary sets of external operators and indeed Bose symmetry plays no role in its derivation. Correspondingly, we will have one formula for identical bosons and one for identical fermions. The two cases are almost identical, but the bosonic one has some additional subtleties, which we suppress in the introduction. In the fermionic case, the rest of the introduction applies without any amendment.

At first sight, the requirement of Bose/Fermi symmetry may seem like a limitation, but it implies that (1.5) leads to an interesting reformulation of the crossing equation, as we explain in the next few paragraphs.

Suppose we start from a four-point function of identical primaries in a unitary theory and apply (1.5) to it. Just like in , can be computed by expanding in the t-channel. The first step is then to understand the coefficient function obtained by applying the Lorentzian inversion formula to a single t-channel conformal block of general dimension , which we denote as follows

(1.6) |

We will argue in the main text that describes the crossing-symmetric sum of exchange Witten diagrams in in the s-, t- and u-channel with exchanged dimension . We call this crossing-symmetric object the *Polyakov block* for reasons that will become clear soon. The Polyakov block with exchanged dimension will be denoted . The s-channel OPE decomposition of contains the single-trace conformal block of dimension with unit coefficient, as well as an infinite tower of double-trace contributions with dimensions or in the bosonic and fermionic case respectively, where . In summary, using (1.5) to invert a single block in the crossed channel returns a manifestly crossing-symmetric object. This is in contrast with what happens when using the standard Lorentzian inversion formula, as described in Section 7 of this note.

Having understood the contribution of an individual crossed-channel block to , we will argue that one can commute the integral in (1.5) and the t-channel OPE applied to . This means can be expanded using the coefficient functions of the Polyakov blocks as follows

(1.7) |

where the sum runs over primaries in the OPE and are the OPE coefficients. The sum converges (absolutely and uniformly in any compact set) in the entire complex -plane away from poles of the individual terms in the sum. Since are meromorphic functions of , it follows that is also meromorphic, with poles only at locations of poles of the individual terms. The same expansion then holds also at the level of the correlator

(1.8) |

On the other hand, we know can be expanded in the s-channel conformal blocks as follows

(1.9) |

For the last two equations to be compatible, the double-trace contributions to the Polyakov blocks must cancel out when the sum over in (1.8) has been performed. This leads to an infinite set of sum rules on the OPE data, with two independent sum rules for every double-trace operator. This is precisely the idea behind Polyakov’s approach to the conformal bootstrap [4] recently revisited and refined using Mellin-space techniques in [33, 34, 35].

These sum rules were recently derived and studied in a closely related work [36]. Among other things, reference [36] demonstrated not only that (1.8) holds for every unitary solution of crossing, but also the stronger claim that the totality of sum rules arising from the equivalence of (1.8) and (1.9) is in fact completely equivalent to the standard crossing equation

(1.10) |

One goal of the present note is to offer an alternative perspective on the same core idea from the point of view of the Lorentzian inversion formula.

In [36], the relevant sum rules were derived by applying suitable linear functionals to the crossing equation (1.10), building on the constructions of [37, 38]. The functionals themselves are interesting because they are examples of optimal (or extremal) functionals of the numerical bootstrap [39]. For example, they can be used to show rigorously that the gap above identity in the OPE in a unitary CFT (in any ) is at most , where is a scalar primary or a component of a spinning primary. The bound becomes optimal if only the minimal (1D) conformal symmetry is assumed. The most important properties of the functionals is that when acting on the conformal block (minus the crossed-channel conformal block) of dimension , they are positive from a certain onwards and exhibit double zeros for at the double-trace values. The functionals of [37, 38, 36] are constructed as contour integrals against suitable holomorphic kernels in the complex -plane. The kernels are constrained by an intricate functional equation which guarantees the above properties.

In this note, we will derive this construction starting from the Lorentzian inversion formula (1.5). The Lorentzian inversion kernel has double poles for at the double-trace dimensions. These poles precisely reproduce the double-trace contributions to the Polyakov block in formula (1.6). It turns out that the coefficients of the simple and double poles of at the double traces are precisely the kernels used to define the functionals of [37, 38, 36]. The contour integral prescription for the functionals of these works is nothing but a way to reconstruct the double discontinuity while staying on the first sheet in the variable. The intricate functional equation satisfied by the functional kernels is a consequence of the equation satisfied by which guarantees that the Lorentzian and Euclidean inversion formulas give the same answer for . The functionals exhibit positivity and double zeros at the double traces because the double discontinuity of conformal blocks in the crossed channel has these properties. In this way, the inversion formula of this note unifies all the functionals considered in [37, 38, 36] into a single object.

### Outline and summary of results

The rest of this note is structured as follows. In Section 2, we review 1D kinematics and the expansion of a four-point function into a complete set of conformal partial waves provided by the Euclidean inversion formula.

In Section 3, we discuss the Lorentzian inversion formula for the principal series and explain how the inversion kernel is constrained by compatibility with the Euclidean formula.

We find explicit formulas for the Lorentzian inversion kernel in Section 4. This includes closed formulas for the bosonic kernel when and the fermionic kernel when . Furthermore, we find the series expansion of the kernel around for general .

Section 5 explains that the Lorentzian inverse of a single conformal block in the t-channel is the coefficient function of a fully crossing-symmetric sum of exchange Witten diagrams in , including the s-channel exchange.

The implications of the last observation are analyzed in Section 6. We prove that of a crossing-symmetric four-point function in a unitary theory can be expanded in the coefficient functions of crossing-symmetrized exchange Witten diagrams. We explain why this implies is meromorphic in the entire complex plane with poles only at the expected locations. Furthermore, we explain how consistency with the usual OPE leads to infinitely many sum rules on the CFT data. Finally, we demonstrate that optimal functionals of the numerical bootstrap arise from residues of the Lorentzian inversion kernel at the double-trace locations.

We conclude with a discussion and open questions in Section 7.

## 2 Kinematics and the Euclidean inversion formula

### 2.1 The four-point function

In this note, we will focus on the four-point function of identical operators in a conformal field theory, denoted . Let us restrict the four operators to lie on a straight line in the Euclidean space and let denote the coordinate along the line. There is a conformal symmetry acting along this line. We will take to be a primary operator of dimension with respect to this . Thus can be for example a scalar primary operator or a component of a spinning operator in a CFT, or simply a primary operator of a 1D CFT. In the rest of the note, we will distinguish the two cases where has bosonic and fermionic statistics. Let us focus on the bosonic case first. The two-point function then reads

(2.1) |

where . Symmetry under implies that the four-point function can be written as

(2.2) |

where is the cross-ratio

(2.3) |

We can use the conformal symmetry and a permutation of labels 1 and 3 if necessary to set , and . is then equal to and thus ranges over all real numbers. When arises by restricting a four-point function to a line, is obtained by restricting the full four-point function to .

Since the four-point function is singular at coincident points , it is useful to define , and as the functions to which reduces in the three disconnected regions

(2.4) |

The functions can be analytically continued to complex values of , but in general are not analytic continuations of each other. Instead, they can be related by Bose symmetry of the four-point function. The symmetry under determines in terms of and the symmetry under determines in terms of as follows

(2.5) | ||||

Clearly determines the whole four-point function. In addition, symmetry under implies must satisfy the crossing relation

(2.6) |

It will be convenient to define the function

(2.7) |

for which crossing symmetry becomes .

The four-point function can be expanded using the s-channel OPE . Since we are only assuming symmetry, the appropriate conformal blocks are the blocks

(2.8) |

It will also be useful to define the conformal block for negative

(2.9) |

The s-channel OPE reads

(2.10) |

where the sum runs over the primary operators appearing in the OPE and is the appropriate OPE coefficient. A priori, the expansion holds for but a standard argument using the coordinate [40, 41] shows that the sum converges also for complex away from . The conformal blocks have a power-law branch cut at implying has branch cuts for and . Let us also note that thanks to an asymptotic bound on OPE coefficients [42, 43], the convergence of the s-channel OPE is uniform in any compact region of not containing , implying is holomorphic away from the branch cuts and .

We will also need the knowledge of how behaves for large . In order to avoid the branch cuts, we should take the limit in the upper half-plane^{7}^{7}7The lower half-plane being related by crossing . as , and . This limit of the four-point function is precisely the Regge limit of the u-channel, as explained in detail in Section 2 of [36]. It is also the same limit that can be used to diagnose chaos from out-of-time-order correlators [30, 31, 32]. Since all channels are equivalent for identical external operators, we will simply refer to this limit as the Regge limit. We will also take to mean approaching in any direction in the upper (or lower) half-plane. Four-point functions in unitary theories satisfy a boundedness condition in the Regge limit. To see that, one can work in the coordinate and use positivity of the s-channel expansion together with the fact that the t-channel is dominated by the identity operator [41]. The result is that

(2.11) |

Note that the bound can not be improved in general since of the generalized free field approaches a constant in the Regge limit

(2.12) |

as . For technical reasons, it will sometimes be useful to consider functions which are better-behaved than just bounded in the Regge limit. Thus, let us define to be *super-bounded* if

(2.13) |

for .

Consider now the case of identical fermions . The two-point function has an extra ordering sign

(2.14) |

The four-point function is defined analogously to the bosonic case

(2.15) |

The functions are defined exactly as in (2.4). Symmetry under the permutations of the external operators again fixes and in terms of . The transposition introduces an ordering sign on both sides of (2.15), which thus cancel and give

(2.16) |

On the other hand, the transposition leads to an extra sign compared to the bosonic case

(2.17) |

We will again define . Just like in the bosonic case, satisfies crossing symmetry and boundedness in the Regge limit (the latter whenever the theory is unitary).

### 2.2 Review of the Euclidean inversion formula

The Plancherel theorem for allows us to expand into a complete set of eigenfunctions of the s-channel Casimir. Both in the bosonic and fermionic case, is invariant under for . Since the s-channel Casimir respects the same symmetry, we can restrict to eigenfunctions invariant under this symmetry. Moreover, the eigenfunctions need to satisfy a boundary condition at to ensure the Casimir operator is self-adjoint, see for example Section 3.2.2 of [28] for more details. The relevant eigenfunctions are called conformal partial waves and can be written as the following conformal integral^{8}^{8}8The integral representation converges for . For other values of , can be defined by an analytic continuation from this region.

(2.18) |

For , the conformal partial waves are a linear combination of a conformal block and its shadow

(2.19) |

where

(2.20) |

For and , the conformal partial waves are determined from as follows

(2.21) | ||||

The invariant inner product on gives the following inner product of functions of

(2.22) |

The set of conformal partial waves which is orthogonal and complete with respect to this inner product consists of the principal series with and the discrete series . Note that on the discrete series, the second term in (2.19) vanishes and we find

(2.23) |

The scalar products among the complete set are

(2.24) | ||||

where

(2.25) |

Let us define the following coefficient functions as overlaps of the four-point function with the principal and discrete series conformal partial waves

(2.26) | ||||

The four-point function can then be expanded in the complete set as follows

(2.27) |

For , we can use (2.19) and to write this as

(2.28) |

If is normalizable with respect to the inner product (2.22), is holomorphic in some neighbourhood of the principal series, and is finite for . Note that it may not always be true that is the analytic continuation of from the principal series to since the integral defining may not converge along the path of the analytic continuation.

The s-channel OPE is recovered by closing the contour to the right, so that terms of the OPE come from poles of and the terms of the discrete series sum. Concretely, we can define

(2.29) |

so that the presence of in the OPE translates to a simple pole of at with residue . There is no reason for the OPE to always contain operators with scaling dimensions exactly at even integers, so how should we think of the contribution of the discrete series? The answer comes from noting that has simple zeros on the discrete series, leading to potentially unphysical poles of . When there is no physical operator at , we have . This guarantees that the residue of the integral in (2.28) coming from the zero of at precisely cancels the corresponding term of the sum over the discrete series. On the other hand, if there is a physical operator precisely at , we should have so that the principal series integral and the discrete series sum combine to a non-vanishing contribution at .

We have seen that the formulas (2.26) extract the OPE data from the Euclidean four-point function and for this reason are known as the Euclidean inversion formulas. To see some concrete examples, we can consider the four-point functions of the generalized free boson (GFB) and fermion (GFF). In the bosonic case, the four-point function is

(2.30) |

The inversion integral (2.26) is most easily done by using the integral representation of the conformal partial waves (2.18). The final result is^{9}^{9}9Strictly speaking, to compute one needs to remove the non-normalizable contribution of the identity and work with instead.

(2.31) |

is essentially the simplest meromorphic function with the right poles and residues which respects the shadow symmetry. The four-point function of the generalized free fermion of dimension is

(2.32) |

Note that in spite of the fermionic statistics, we have and we can use the same set of conformal partial waves as in the bosonic case. The coefficient function reads

(2.33) |

Before closing this section, let us note that if we are using the Euclidean inversion formula to extract the OPE data of a four-point function restricted to , we can be agnostic about the statistics of the external operators. Indeed, suppose we are given for satisfying and we want to find coefficient functions , such that (2.28) holds. In order to compute the inversion integrals (2.26), we need to extend to a function defined for all , such that . One way to do this is to pretend the external operators are identical bosons, which gives

(2.34) |

Another option is to pretend they are identical fermions^{10}^{10}10From now on, we will call the external dimension also in the fermionic case to simplify notation.

(2.35) |

Both are perfectly consistent choices which lead to *different* coefficient functions and . and encode the same OPE data pertaining to the original correlator and thus must have the same residues at the physical poles. Therefore, their difference must be a meromorphic function with poles only at (and their shadow locations) since these poles (but not their shadows) cancel against zeros of in (2.28) and thus do not contribute to the OPE. The main result of this note are alternative formulas which extract and from , to which we turn now.

## 3 The Lorentzian inversion formula

### 3.1 The general form

For many applications, it is essential to have an alternative formula for the coefficient functions and , known as the Lorentzian inversion formula [11] (see also [12, 13]). The input of this formula is the double discontinuity of defined by

(3.1) |

When is obtained by restricting a higher-D four-point function to , then this definition agrees with the standard higher-D double discontinuity restricted to .

All s-channel conformal partial waves are annihilated by the double discontinuity. Crucially for many applications, the double discontinuity also annihilates t-channel double-trace conformal blocks and their derivatives with respect to . Which dimensions are counted as double-trace depends on whether we choose the bosonic or fermionic extension of the correlator, i.e. equations (2.34) or (2.35). In the bosonic case, the contribution of a t-channel conformal block of dimension to and is

(3.2) | ||||

which means its double discontinuity takes the form

(3.3) |

where the subscript on reminds us which extension of the correlator we choose. We see that the bosonic double discontinuity is non-negative for and exhibits double zeros at the bosonic double-trace dimensions

(3.4) |

The contribution of a t-channel conformal block in the fermionic case is

(3.5) | ||||

leading to the following double discontinuity

(3.6) |

this time exhibiting double zeros at the fermionic double-trace dimensions

(3.7) |

The authors of [12] found the following Lorentzian inversion formula for the discrete series coefficient function

(3.8) |

This formula applies to any physical four-point function satisfying , and in particular both the bosonic and fermionic extension of a crossing-symmetric . (3.8) provides a particular analytic continuation of to . As discussed in [12], this analytic continuation is holomorphic for and therefore in general can not agree with the principal series function .

Our goal in this section will be to derive a similar formula for the principal series coefficient function . More precisely, we will find Lorentzian inversion formulas for , , i.e. the coefficient functions corresponding to the bosonic and fermionic extensions of . The formulas take the form

(3.9) | ||||

Here and are appropriate inversion kernels and we took out a factor of 2 for future convenience. Unlike in higher dimensions or for the discrete series, we will find that are *not* eigenfunctions of the s-channel Casimir. In fact, they depend very non-trivially on the external dimension . Another unusual feature of the formula is that it has the crossing symmetry under built in, in the sense that it only holds for respecting this symmetry. Furthermore, the output of the formula is a coefficient function which manifestly leads to a crossing-symmetric correlator.^{11}^{11}11The higher-D inversion formula does not (and should not) always return crossing-symmetric OPE data. Indeed, if are scalar primaries of equal dimension, the exact same higher-D inversion formula applies to the correlators and . The latter is in general not symmetric under the crossing transformation. Our formula for the principal series only applies to fully symmetric correlators such as .

### 3.2 Constraining the inversion kernels

We will fix and by demanding that the Euclidean and Lorentzian inversion formulas (2.26) and (3.9) are compatible. Let us start from the Euclidean formula which we first split into integrals over the three regions

(3.10) |

Recall that and are related to by (2.34) and (2.35) respectively in the bosonic and fermionic case. Similarly, and are related to through (2.21). Let us plug these relations into (3.10) and change variables to bring all integrations to . We arrive at

(3.11) |

Here and in the following the upper sign applies for and the lower for . Recall that is crossing symmetric but the square bracket in the last formula is not, so let us symmetrize it to get a formula manifesting the full crossing symmetry^{12}^{12}12The partial waves satisfy , or equivalently