Cut-touching linear functionals in the conformal bootstrap

Journal of High Energy Physics, Jun 2017

The modern conformal bootstrap program often employs the method of linear functionals to derive the numerical or analytical bounds on the CFT data. These functionals must have a crucial “swapping” property, allowing to swap infinite summation with the action of the functional in the conformal bootstrap sum rule. Swapping is easy to justify for the popular functionals involving finite sums of derivatives. However, it is far from obvious for “cut-touching” functionals, involving integration over regions where conformal block decomposition does not converge uniformly. Functionals of this type were recently considered by Mazáč in his work on analytic derivation of optimal bootstrap bounds. We derive general swapping criteria for the cut-touching functionals, and check in a few explicit examples that Mazáč’s functionals pass our criteria.

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Cut-touching linear functionals in the conformal bootstrap

Received: May Cut-touching linear functionals in the conformal Open Access 0 1 2 3 c The Authors. 0 1 2 3 0 CERN, Theoretical Physics Department 1 Sorbonne Universites, UPMC Univ. Paris 06, CNRS , 75005 Paris , France 2 Ecole Normale Superieure, PSL Research University 3 1211 Geneva 23 , Switzerland The modern conformal bootstrap program often employs the method of linear functionals to derive the numerical or analytical bounds on the CFT data. These functionals must have a crucial \swapping" property, allowing to swap in nite summation with the action of the functional in the conformal bootstrap sum rule. Swapping is easy to justify for the popular functionals involving nite sums of derivatives. However, it is far from obvious for \cut-touching" functionals, involving integration over regions where conformal block decomposition does not converge uniformly. Functionals of this type were recently considered by Mazac in his work on analytic derivation of optimal bootstrap bounds. We derive general swapping criteria for the cut-touching functionals, and check in a few explicit examples that Mazac's functionals pass our criteria. Conformal Field Theory; Conformal and W Symmetry 1 Introduction 2 3 4 Analytic continuation Functionals: general considerations Cut-touching functionals Relation to the work of Mazac A Toy counterexample B Spectra with accumulation points Introduction Any conformal eld theory (CFT) is characterized by the spectrum of local primary operators and by their operator product expansion (OPE) coe cients, called collectively the CFT data. This data has to satisfy a consistency conditions following from the OPE associativity. The program of constraining or solving the CFT data using the OPE associativity is known as the conformal bootstrap [1{4]. The conformal bootstrap equations are mathematically well de ned and can be studied on a computer [5]. This approach has led in the last 10 years to a wealth of rigorous numerical results about CFTs, many of which are currently out of reach of analytical methods. Following [5], the numerical conformal bootstrap analysis is usually formulated in terms of linear functionals, as we will now review (see also [6, 7]). Let us specialize to the case of a one-dimensional (1d) CFT, by which we mean here a theory of local operators in 1d whose correlation functions transform covariantly under the fractional linear transformations x ! conformal transformations in d dimensions. Consider in such a theory a four point (4pt) correlation function of a primary operator of scaling dimension . Conformal invariance constrains this correlator to have the form: h (x4) (x1) (x2) (x3)i = jx12j where xij = xi are assuming the operators to be cyclically ordered on the conformally compacti ed real axis (as appropriate in 1d). If we put the operators at 0; z; 1; 1 then we have1 h (1) (0) (z) (1)i = z 2 The function G(z) is initially de ned on the real interval 0 < z < 1 (its analytic continuation will be discussed below). Since we are dealing with a 4pt function of four identical operators, this function satis es on this interval the following crossing relation (F 's also depend on but we omit this dependence in the notation.) This equation can be used to put constraints on the allowed unitary CFT spectra [5]. The strategy is to look for a linear functional which satis es the conditions which is the simplest example of a conformal bootstrap equation. Near the endpoints of the interval, we have asymptotic behavior This is given by the unit operator contribution in the OPE , and is clearly consistent with the crossing relation. Furthermore, the function G(z) can be expanded into conformal blocks [8{10]:2 G(z) = (1 G(z) = G (z) = z 2F1( ; ; 2 ; z) : 1De ning as usual (1) = limz!1 jzj2 2For d > 1 conformal blocks depend also on the spin of the exchanged primary and on the second cross i are the scaling dimension of all primary operators appearing in the OPE pi are the squares of the OPE coe cients. We have 0 = 1, p0 = 1 corresponding to the unit operator. We will assume that our theory is unitary. In such theories all subsequent operators satisfy the unitarity bound i > 0. Also the OPE coe cients are real in unitary theories, implying pi > 0. We will also assume for simplicity that the spectrum of operators is discrete without accumulation points, so that there is a nite number of operators below any xed dimension. This assumption is not crucial and can be relaxed; see appendix B. Eqs. (1.3), (1.5) can be rewritten as a sum rule X piF i (z) = 0 ; F (z) = z 2 i in some putative spectrum. Applying ! to (1.6) and assuming we can swap the order of summation with the action of the functional (a nontrivial requirement since the number of operators in the OPE is always in nite), we have spectrum is thus ruled out. The linear functionals used in [5] and in essentially all subsequent numerical work were X pi !(F i ) = 0 ; !(f ) = As we will review below, for these functionals the above-mentioned crucial swapping assumption is very easy to justify, basically because the conformal block decomposition converges uniformly near z = 12 .3 Recently Mazac [12] introduced a new class of linear functionals. Unlike the functionals of [5], his functionals involve integrals of f over regions of the z space approaching the analyticity cuts where the conformal block decomposition seizes to converge. The use of such \cut-touching" functionals raises anew the problem of justifying swapping. His functionals are very interesting because, as explained in his paper, they lead to an analytic understanding of some optimal conformal bootstrap bounds previously conjectured by extrapolating numerical results. He veri ed the relevant conditions analogous to (1.8) in his paper. However, he has not discussed nor even mentioned swapping. This is an unfortunate gap in his otherwise beautiful analysis. Although some amends were made in the online presentation [13], we consider this issue not fully clari ed, and su ciently important to dedicate this short note to it. The functionals like in [12], or even more complicated ones, may well become widespread in the conformal bootstrap. Anticipating these developments, we will show here a minimal standard of rigor in dealing with such functionals. Following this standard is important to ensure that the results are technicalIy correct. While in this note we focus on the case followed there as well. We start in section 2 by discussing the analytic continuation of the 4pt function and of its conformal block decomposition into the plane of complex z. In section 3 we formalize the swapping property (along with the more obvious niteness) which the linear functionals used in the conformal bootstrap must have. For the usual functionals these properties are trivially veri ed. Then in section 4 we turn to the cut-touching functionals. We derive some general criteria guaranteeing that such functionals obey niteness and swapping. In section 5 we use our criteria to prove swapping for the functionals used in [12], at least for the particular cases of low-lying where [12] provides su cient details. The general case remains incomplete. Appendix A contains a simple counterexample, showing that taking swapping for granted and proceeding formally can lead to manifestly wrong results. Appendix B deals with the case when the operator spectrum is continuous or has accumulation points. Analytic continuation The function G(z), while originally de ned on the interval 0 < z < 1, allows an analytic continuation into the complex plane of z with cuts along ( 1; 0) and (1; +1) (\cut plane"). Analytic continuation is provided by the series (1.5). Clearly, the individual terms in the series are analytic functions in the cut plane. In addition, the series converges in the cut plane. To show this latter fact, it is convenient to work in the coordinate [14, 15]: (z) = z( ) = The cut z plane is thus mapped to the disk j j < 1. Consider the series (1.5) transformed to the G~( ) = G~( ) = G z( ) ; G~ ( ) = G Using hypergeometric function identities, we have [15]: We will not actually need this exact formula but three properties of conformal blocks that G~ ( ) = (4 ) 2F1 G~ (r) = O log (0 6 r < 1) ; The rst property is obvious, the second is a standard hypergeometric asymptotics. The last property follows by expanding the hypergeometric function in (2.3) in a power series and noticing that all coe cients are positive (if > 0 as demanded by unitarity). Now we can nish the convergence proof. For real = r, 0 < r < 1, the function G~( ) is nite, and all terms in its series (2.2) are positive, so the series does converge. On the other hand, for complex converges in the cut z plane. The above argument is a 1d adaptation of the general d-dimensional argument from [14]. The argument is robust and can be extended in several directions. For example, the z plane). the same argument shows that the convergence in any subdisk j j 6 1 is uniform. It is also easy to argue that the convergence in any such subdisk is exponentially fast [14]. For a precise formulation, let be any real number in the range 1 < 1. Then there is a constant C such that for any the tail of the series (2.2) corresponding to summing satis es the uniform bound: The r.h.s. of this bound becomes exponentially small for large are mapped onto the subregions of the z plane shown in gure 1. In any such subregion the series (1.5) converges uniformly and exponentially fast. Also notice that the cut through ( 1; 0) is present in the above argument only because the factors z in the conformal blocks have this cut. The convergence is not spoiled by the presence of this cut. In fact the argument proves that the function G(z) can be analytically extended through this cut, and one can circle around the origin through a second, third etc sheet. The same is of course true for the cut (1; +1) because the function G(z) is crossing symmetric, eq. (1.3) (or because we can equivalently run the G(z), which is an in nitely-sheeted Riemann surface if work we will stay on the rst sheet. is an irrational number. In this A comment is in order concerning the origin of the positivity property of the conformal blocks and of their power series coe cients, which played an important role in the above proof. In terms of the 4pt unction, passing to the coordinate corresponds to mapping it conformally to the con guration < 1, the new con guration is re ection positive. This explains why all terms in the power series expansion of G~( ) have to be positive [14]. 4One can also put = 1 at the cost of making the constant C grow as a power of [14], but we will not need this sharper estimate here. Functionals: general considerations So let us go back to the crossing relation (1.6) satis ed by a 4pt function of some 1d CFT. Based on the discussion of the previous section, the following facts are true: are analytic in the cut plane. The series converges in the cut plane.5 The convergence is uniform in the subregions where both conditions j (z)j < 1 are satis ed (see gure 2). We would like to consider linear functionals !(f ) which have the following two prop P1. (Finiteness) !(F ) is nite for any P2. (Swapping) For any possible 4pt function of an operator of dimension , eq. (1.6) the series converging in the usual sense. X pi !(F i ) = 0 ; It's important to emphasize that the functional should be de ned not just on the and on their nite linear combinations, but on a wider class of functions.6 This class should at the very least include the functions F (z) which will be introduced In practice, the functional !(f ) will be given by some sort of integral or a combination of derivatives and property P1 should be relatively easy to check, especially given that 5The following comment may be useful. As mentioned in the previous section, the 4pt function can be analytically extended through the branch cuts to an in nitely-sheeted Riemann surface. The original crossing relation in the form (1.3) is then true everywhere on this surface with appropriate identi cation z, not just on the rst sheet. However, it has to be emphasized that the crossing relation in the series form (1.6) really stops converging on the branch cuts. This precludes any straightforward use of the series relation beyond the rst sheet. 6In this respect the notation of eqs. (2.8) and (2.22) of [12] is confusing, while that in [13] is OK. the conformal blocks in 1d are known explicitly. Property P2 is more tricky. It can be \derived" by applying functional !(f ) to both sides of (1.6). However this is formal since it requires interchanging the action of the functional with in nite summation. Sometimes this formal argument is easy to justify, sometimes more work is needed. We will see examples in a second. Assuming that P1 holds, the strategy to establish P2 is as follows. Split (1.6) into two parts (we switch from summing over i to summing over the discrete set of occurring !(f ) = !(f ) = f (n)(z0) : (z) = 0 ; Now we can apply ! and get: ) = 0 : Notice that here we interchanged the functional with a nite summation, which is always a legal operation. Furthermore, the function F goes to zero in the cut plane as uniformly so in the regions shown in gure 2. So we may expect that, under wide conditions on the functional !, If we can show this rigorously, then (3.1) follows and we are done. This is what it takes to justify the formal argument. Let us consider two examples where (3.4) is immediate. Example 1. Suppose the functional ! is given by an integral over some integrable mea whose support S is a bounded set, which is fully contained in the cut plane and does not touch the cuts (see gure 3): Then (3.4) follows trivially from the uniform convergence of (1.6) on S. Example 2. Suppose the functional !(f ) is a derivative of a nite order n at a point z0 lying strictly inside the cut plane: This example can be reduced to the previous one, by representing the derivative via Cauchy's formula as a contour integral over a circle fully contained in the cut plane. Clearly, a nite linear combination of derivatives will do as well. The functionals (1.10) used in the numerical bootstrap belong to this class. The simplicity of veri cation of (3.1) in this case explains why it was left implicit in the literature. For example, the authors of ref. [14] carefully established the convergence of the conformal block decomposition in the cut plane and stated that this puts the numerical conformal bootstrap results on \mathematically solid ground". What they had in mind was a kind of the above argument. Cut-touching functionals We will now consider a functional of the following form: !(f ) = Im dz H(z)f (z) ; where H(z) is a xed analytic function in the upper half-plane. The function f (z) on which the functional acts is also assumed analytic in the upper half-plane (in fact it will be analytic in the cut plane). The contour z starts at z = 1 and ends at z = +1, as gure 4. Of course since the functions are analytic we may deform the contour. For example, we may want to make it run along the cut. Such contour deformations may be useful in actual explicit calculations, but for the proof of properties P1, P2 it will be convenient to keep the contour in the bulk of the upper half-plane, touching its boundary only at two points as shown. As in [12], let us pass from the coordinate z to coordinate x(z) = z(x) = The upper half-plane of z is mapped to the upper half-plane of x with points 0; 1; 1 and the contour mapped as in gure 5. It is equivalent but more convenient to analyze the functional in terms of the x coordinate: !(f ) = Im The function h(x) = H z(x) z0(x)(1 may have some singularities on the real axis but we will assume it is analytic in the upper half-plane. The factor (1 transformation from z to x. The function h(x) and the functions f z(x) on which the functional is evaluated will be analytic in the upper half-plane. out for future convenience, as in [12]. We will assume in our analysis that contour It's clear that for such functionals the proof of swapping given above for examples 1, 2 behaves near these points. The condition for swapping, whatever it is, will depend in a nontrivial way on be to work out this condition. we have z To check niteness, we need to estimate how F z(x) behaves near x = 0; 1. For x ! 0 G (z) = O(log 1=jxj) ; z) = O(jxj ) : For x = 1 + , 1= . To estimate G (z) we pass to the Since we are assuming that is not parallel to the real axis we have both For the crossed channel we have: Using the estimate (2.6), we have and analogously for G~ ( 0). = (z) ( 1= ) 0 = (1 (1= ) j j; j 0j = 1 jG~ ( )j 6 G~ (j j) = O(log 1=j j) ; Combining the above estimates for G 's we can estimate F . We have: z(x) = O(log 1=jxj) + O(jxj ) = O(jxj 2 where we used > 0. Further x = 1 + : z(x) = z2 = O G (z) + O G (1 z) = O(log 1=j j) : Finiteness will hold if the following integrals along x involving these bounds are absolutely convergent: over the part of x near x = 0 ; over the part of x near x = 1 : z)j = O(1) : Notice that these are su cient conditions for niteness. However, it's easy to see that these conditions are also necessary, unless one considers functionals with h(x) rapidly nite by cancellations. close to x = 1, and the \bulk part". As ! 1, the function F z(x) goes to zero uniformly (and exponentially fast) on the bulk part. So that part of the integral can be made arbitrarily small by choosing a su ciently large On the end parts, we will estimate F as follows. First of all we bound all terms by is the tail of the conformal block decomposition, de ned as F in (3.2) but summing over G . Using (2.6), we can estimate these tails by the whole function G evaluated at the absolute value of the The second estimate can be understood for example by estimating the 4pt function using the OPEs ( ) (1) and ( ( 1) in (2.8). Alternatively it just follows from the second of the asymptotics (1.4). There is also an analogous estimate for G 1 + x (recall that the contour is not along the real axis) this strategy gives us: z)j = O(j j We now combine these estimates on G~ to get estimates on F . We have: x = 1 + : (1 x) 2 z(x) = O(jxj 2 z(x) = z2 = O G (1 z) = O(j j It's crucial for what follows that the r.h.s. of these estimates does not depend on Suppose now that the following integrals of h(x) against these bounds are absolutely over the part of x near x = 0 ; over the part of x near x = 1 : Then we claim that the swapping property holds. To show (3.4) we argue as follows. Pick any > 0. Take the end parts of the contour su ciently short so that those parts of the integral, for any , are smaller in absolute value than . This is possible by the conditions (4.19), (4.20). The bulk part of (3.4) tends to zero as that the large ! 1, since the integrand uniformly converges to zero there. We conclude limit of (3.4), in absolute value, is smaller than . Since the limit is zero. This completes the proof. Conditions (4.19), (4.20) are su cient conditions for swapping. Moreover we believe that, just like the niteness conditions (4.11), (4.12), these conditions are also best posThis may not be totally obvious from our proof. For example, one might think that the estimates (4.13) and (4.14) are too crude. However, we think that in general there cannot be a full cancellation between the terms whose absolute values are summed in the r.h.s. of (4.13). As for (4.14), we only use this estimate near the endpoints of the contour, where it's basically best possible. Notice that while (4.19) is identical to (4.11), the other condition is stricter than (4.12). So, the fact that the functional is nite on each F does not yet guarantee swapping. We would like to nish this section with the following comment. The problem of justifying the swap of integration and summation is of course standard in mathematics. One powerful result is Lebesgue's dominated convergence theorem. There are several reasons why we chose not to appeal to it in our exposition, but to deduce everything from scratch. First, Lebesgue's theorem is very general (it deals with an almost everywhere convergent sequence of measurable functions), and it's not good practice to shoot sparrows with a cannon. Second, if we did appeal to this theorem, we could eliminate but the paragraph following the conditions (4.19), (4.20). The estimates (4.18) would still have to be derived (\dominated convergence"), and this is what constitutes anyway the bulk of our argument. Finally, we believe that there is an added value in seeing what actually goes into the proof. Relation to the work of Mazac The cut-touching functionals from the previous section are closely related to the functionals constructed in [12], with the purpose to give an analytic proof of a certain optimal bootstrap bound involving operators of dimension 2 N=2. Let us review this connection in detail. Mazac begins by considering a family of basis functionals of the form !(f ) = plane. He chooses the contour Conditions for the niteness7 and swapping of these functionals can be examined exactly condition (4.12) is satis ed. On the other hand, the swapping condition (4.20) is not satis ed, because pn(1) 6= 0. That's not a problem because he does not act with the basis functionals themselves on the sum rule. Instead he considers their linear combinations, corresponding to h(x) = The coe cients an have to be chosen so that several conditions are satis ed. First of all, since his goal is to prove an optimal bootstrap bound, the functional has to be extremal, which means that it has to satisfy certain positivity conditions closely related to (1.8). These conditions have been discussed in detail in his work and we will not discuss them Then the functional has to have the swapping property. This was not actually discussed between various terms in the sum de ning h(x).8 tion, the function h(x) develops a cut over the negative real axis x < 0. For this reason the 7Ref. [12] actually works out !(F ) for all functionals in closed form. So their niteness is not in doubt. We will still discuss niteness for completeness, but our focus is on justifying swapping. 8The talk [13] (29m30s) cites the condition h(x) = O (x as needed \for the functional to be de ned on in nite sums of blocks bounded at in nity". This is not far from our condition (4.20), although a bit stronger than necessary. We emphasize however that the functional has to be not just \de ned", but has to satisfy eq. (3.4) from which the swapping property follows. contour in gure 6 is no longer appropriate, and has to be modi ed. In fact, the behavior h(x) = h1(x) + h2(x) ; can then be de ned as a sum of three terms !(f ) = f z(x) dx ; where the three parts of the contour are chosen as in gure 7. The niteness and swapping conditions for this functional are (4.12) and (4.20) imposed on h(x) and (4.19) imposed prescription is equivalent to the one discussed in [12] below eq. (5.19). After this introduction, let's see how the functionals of [12] fare with respect to all these conditions. To be more precise, his functionals correspond to h(x) = h~(x) + c(x) ; where h~(x) is a sum as in (5.2) with summation over even/odd n depending if or hal nteger: Using the properties of Legendre polynomials, this implies that h~(x) = h~(x) = 2 N) ; 2 N x) = x) = h~(x) 2 N On the other hand, c(x) is a nite degree polynomial which can be used to make the total h(x) vanish at x = 1 su ciently fast. Consider rst 2 N. In this case Mazac says in section 5.2 (we translate his eq. (5.18) h~(x) = h~1(x) + h~2(x) ; h~2(x) = O(x2 In section 5.3 he uses the freedom to add c(x) to set the behavior of h~(x) = x) + c(x) = O (x He conjectures that it's always possible although he only checked it up to = 5. If so, we Consider next 1 , discussed in [12], section 5.4 and appendix A. For the h~(x) = h~1(1 x) + c(x) = O (x He says that he's able, at least for If that's the case then the conditions for the niteness and swapping are indeed satis ed, just as for The bottom line is that in the cases of low-lying , where ref. [12] provides su cient information, we are able to apply our criteria and to prove swapping. A more detailed understanding and an extension of his argument would be needed to establish this for . This is beyond the scope of our work. Conclusions Conformal eld theories are both physically relevant and mathematically well de ned. They satisfy precise axioms, which can be used to derive rigorous bounds separating the possible from the impossible. These bounds are usually argued by contradiction, employing the method of linear functionals. The non-explicit character of such arguments requires special care, otherwise one risks to throw out the baby with the bathwater. In this note we proposed a blueprint which needs to be followed to guarantee that this does not happen. As an application, we checked that the functionals recently constructed in [12] can be safely used in the conformal bootstrap. Acknowledgments JQ is grateful to the CERN Theoretical Physics Department for hospitality. SR is supported by the National Centre of Competence in Research SwissMAP funded by the Swiss National Science Foundation, and by the Simons Foundation grant 488655 (Simons collaboration on the Non-perturbative bootstrap). Toy counterexample Mathematics textbooks are full of examples when one cannot swap integration with summation. We give one here so that you don't have to go look for it yourself. The example is based on simple power series expansions. However, the mechanism is general, and one should beware of falling into similar traps when working with conformal block expansions. Consider the following functions on the real interval 0 < t < 1: 0(t) = 1 ; n(t) = (n + 1)tn (n = 1; 2 : : :) : The series of these functions sums to zero: n(t) = 0 (0 < t < 1) : Indeed, it was designed so that the subsequent terms cancel telescopically, so that the t, then things work nicely: X In = 0 ; I0 = In = Now consider formally integrating the series against some function w(t): N (t) = n(t) = (N + 1)tN : in manifest contradiction with (A.4). To understand this \paradox", consider the tails of the series (A.2): To swap integration and summation, we must have a condition analogous to (3.4): This condition is satis ed for w(t) = 1 t but not for w(t) = 1. Spectra with accumulation points In the main text we made an assumption that the spectrum of operators appearing in the conformal block decomposition (1.5) is discrete without accumulation points. However, there exist 2d and 1d CFTs with continuous spectrum, such as the Liouville theory and its associated boundary CFTs (although in d > 2 there are no known examples showing such behavior). Here we will show that our main results remain unchanged if the spectrum is continuous or has accumulation points. In such a general situation, eq. (1.5) should be replaced by an inde nite Stieltjes G(z) = integral is understood in two steps. First one de nes the integral for a nite upper limit: This is de ned as the N ! 1 limit of the Riemann-Stieltjes (RS) sums: dP ( ) G (z) : corresponding to ner and ner subdivisions of the interval [0; 0 = 0 < N = Let z vary over a region where j (z)j < 1 uniformly continuously on . For such z, the functions G (z) depend ]. This is enough to guarantee that the RS sums have a uniform limit. Since the individual RS sums are analytic, their limit (B.2) is analytic as The second step is to de ne the integral in (B.1) as the limit of (B.2) as Since (B.2) monotonically grows with for 0 < z < 1, the limit does exist on this interval. Then one argues as in section 2, using the property (2.6) of conformal blocks, that the convergence as ! 1 is uniform in the regions j (z)j < 1 . This shows that the function G(z) is analytic in the cut complex plane, just as before. By the given argument, we have the following approximation of G(z) by nite sums of conformal blocks with two error terms: G(z) = The rst error term G ;N (z) is the di erence between (B.2) and the RS sum. This is to be compared with the situation in the main text, where we had only the rst error term. (z) is the di erence between (B.1) and (B.2), while the second error ) + !(F RS;N ) = 0 : When we take the limit N ! 1 and then ! 1, this will become the desired equation dP ( )!(F ) = 0 ; provided that we can show (3.4) (which is done exactly as before) and, in addition, that This extra condition is obvious for the simple functionals (3.5), (3.6) since F uniformly to zero in the relevant region of z. For the cut-touching functionals, a little RS satis es the same crude bound (B.6) as the conformal blocks, one can recycle the estimates from section 4.1. Conditions (4.11), (4.12) are then su cient to guarantee (B.9). The conclusion of this discussion is that the su cient conditions for the niteness and swapping derived in the main text remain valid when the spectrum is continuous or discrete with accumulation points. Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. The rst error term has the same properties as before: it goes uniformly to zero with , and it can be uniformly in bounded by the full ! 1 in the region j (z)j < 1 4pt function, as in eq. (4.14). On the other hand, as discussed above, the second error term can be made uniformly small in the same region j (z)j < 1 , by taking N ! 1 (for any xed ). Outside of this region we can use a crude upper bound: Here const may depend on but is independent of N . This bound follows from the fact that each individual conformal block satis es such a bound. Now we are in a position to repeat the analysis of section 3. Eq. (3.3) is replaced by: two-dimensional quantum Advanced Study Institute in Elementary Particle Physics: New Frontiers in Fields and [8] F.A. Dolan and H. 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Jiaxin Qiao, Slava Rychkov. Cut-touching linear functionals in the conformal bootstrap, Journal of High Energy Physics, 2017, 1-19, DOI: 10.1007/JHEP06(2017)076