Cuttouching linear functionals in the conformal bootstrap
Received: May
Cuttouching linear functionals in the conformal
Jiaxin Qiao 0 1 2 3 4
Slava Rychkov 0 1 2 3 4
0 Sorbonne Universites, UPMC Univ. Paris 06, CNRS , 75005 Paris , France
1 Ecole Normale Superieure, PSL Research University
2 Laboratoire de Physique Theorique, Departement de Physique de l'ENS
3 1211 Geneva 23 , Switzerland
4 CERN, Theoretical Physics Department
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 \cuttouching" 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 cuttouching functionals, and check in a few explicit examples that Mazac's functionals pass our criteria.
Conformal Field Theory; Conformal and W Symmetry

HJEP06(217)
1 Introduction 2 3 4
5
6
Analytic continuation
Functionals: general considerations
Cuttouching functionals
4.1
4.2
Finiteness
Swapping
Relation to the work of Mazac
Conclusions
A Toy counterexample
B Spectra with accumulation points
correlation functions transform covariantly under the fractional linear transformations x !
(ax + b)=(cx + d). These form the group SL(2; R), the 1d counterpart of the group of global
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
2
jx34j
2
G(z) ;
(1.1)
{ 1 {
where xij = xi
xj and z = x12x34=(x13x24) is the conformally invariant cross ratio. We
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
G(z) :
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
ratio z.
! : f 7! !(f )
!(F 0 ) > 0 ;
!(F i ) > 0
(i > 1)
{ 2 {
which is the simplest example of a conformal bootstrap equation. Near the endpoints of
the interval, we have asymptotic behavior
G(z) ! 1 (z ! 0) ;
G(z)
(1
1
z)2
(z ! 1) :
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
1De ning as usual (1) = limz!1 jzj2
(z).
2For d > 1 conformal blocks depend also on the spin of the exchanged primary and on the second cross
Here
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
G (z)
(1
z) 2
G (1
z) :
(1.6)
for all
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
which is impossible in view of (1.8) and of p0 = 1, pi > 0 in unitary CFTs. The putative
spectrum is thus ruled out.
The linear functionals used in [
5
] and in essentially all subsequent numerical work were
linear combinations of a nite number of derivatives at the midpoint z = 12 :
verges 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 \cuttouching" 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
d = 1, the standard we impose has a natural extension to d > 1, and we hope it will be
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 cuttouching 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
3This is also true for functionals in [11] using points away from z = 12 but within the region of uniform
convergence.
{ 3 {
the particular cases of lowlying
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.
2
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) =
(1 + p
1
z
z)2
;
z( ) =
4
(1 + )2
The cut z plane is thus mapped to the disk j j < 1.
Consider the series (1.5) transformed to the
Using hypergeometric function identities, we have [15]:
We will not actually need this exact formula but three properties of conformal blocks that
it implies:
and
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
G~ ( ) = (4 ) 2F1
; ;
+ ; 2 :
1
2
1
2
G~ (r) > 0
G~ (r) = O log
1
1
r
(0 6 r < 1) ;
(r ! 1) ;
jG~ (rei )j 6 G~ (r) :
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
= rei , r < 1, each term in the series is dominated by its
value at
= r. So the series converges in the disk j j < 1. The original series (1.5) then
converges in the cut z plane.
The above argument is a 1d adaptation of the general ddimensional argument
from [14]. The argument is robust and can be extended in several directions. For example,
{ 4 {
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
over
i >
satis es the uniform bound:
X
i>
pi G~ i ( ) 6 C
for all j j 6 1
:
(2.7)
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
and
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
argument around z = 1). In this way one can explore the full domain of analyticity of
.4 The subdisks j j 6 1
G(z), which is an in nitelysheeted 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
h ( 1) (
) ( ) (1)i :
(2.8)
For real
< 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.
{ 5 {
3
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:
Functions F 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
and j (1
z)j < 1
are satis ed (see gure 2).
We would like to consider linear functionals !(f ) which have the following two
properties:
P1. (Finiteness) !(F ) is nite for any
P2. (Swapping) For any possible 4pt function of an operator of dimension
, eq. (1.6)
implies that
the series converging in the usual sense.
It's important to emphasize that the functional should be de ned not just on the
functions F
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
shortly.
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 nitelysheeted Riemann surface. The original
crossing relation in the form (1.3) is then true everywhere on this surface with appropriate identi cation
z ! 1
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.
(3.1)
{ 6 {
\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
's):
HJEP06(217)
Z
S
d f (z) :
!(f ) = f (n)(z0) :
{ 7 {
X
<
p F (z) + F
(z) = 0 ;
F
(z)
X
>
p F (z) :
Now we can apply ! and get:
X
<
p !(F ) + !(F
) = 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 !,
!(F
) ! 0
(
! 1) :
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
measure d
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:
(3.2)
(3.3)
! 1,
(3.4)
(3.5)
(3.6)
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.
4
Cuttouching functionals
We will now consider a functional of the following form:
Z
z
!(f ) = Im
dz H(z)f (z) ;
where H(z) is a
xed analytic function in the upper halfplane. The function f (z) on
which the functional acts is also assumed analytic in the upper halfplane (in fact it will
be analytic in the cut plane). The contour
z starts at z = 1 and ends at z = +1, as
shown in
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 halfplane, touching its boundary
only at two points as shown.
As in [12], let us pass from the coordinate z to coordinate
x(z) =
z
z
1
;
z(x) =
1
1
x
:
The upper halfplane of z is mapped to the upper halfplane 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:
Z
x
!(f ) = Im
may have some singularities on the real axis
but we will assume it is analytic in the upper halfplane. The factor (1
x) 2
is factored
{ 8 {
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 halfplane.
out for future convenience, as in [12]. We will assume in our analysis that contour
x
approaches x = 0; 1 not tangentially to the real axis.
It's clear that for such functionals the proof of swapping given above for examples 1, 2
cannot be applied. The problem is that the convergence of the series (1.6) near z = 1; +1
(which map to x = 0; 1) is not uniform. To establish (3.4), we will need to understand how
behaves near these points. The condition for swapping, whatever it is, will depend in
a nontrivial way on
and on the asymptotics of h(x) near x = 0; 1. Our goal here will
be to work out this condition.
4.1
Finiteness
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) ;
G (1
z) = O(jxj ) :
For x = 1 + ,
! 0 we have z
1= . To estimate G (z) we pass to the
coordinate:
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= )
jG~ ( )j 6 G~ (j j) = O(log 1=j j) ;
(4.4)
(4.5)
(4.6)
(4.7)
(4.8)
Combining the above estimates for G 's we can estimate F . We have:
over the part of x near x = 1 :
~
G j (z)j = O(jxj 2
~
G j (1
z)j = O(1) :
j (z)j
const.pjxj
) ;
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
oscillating near endpoints x = 0; 1, which can be rendered
nite by cancellations.
4.2
Swapping
Let us split the contour x into three parts, two \end parts", one close to x = 0 and one
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
absolute value:
jF
z(x) j 6 jzj 2
jG
(z)j + j1
z 2
j
jG
(1
z)j :
Here G
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
z) with
1 + x (recall that the contour is not along the real axis) this
(1
z).
Now, for x ! 0, z
strategy gives us:
and
(4.13)
(4.14)
(4.15)
(4.16)
(4.17)
(4.19)
(4.20)
On the other hand, for x = 1 + , using the estimates on 's from the previous section we
nd:
G j (z)j ; G~ j (1
~
z)j = O(j j
2
)
We now combine these estimates on G~ to get estimates on F
. We have:
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
convergent:
Z
Z
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
is arbitrary,
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
possible (i.e. necessary) for h(x) which do not rapidly oscillate near the endpoints x = 0; 1.
This 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.
The cuttouching 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
HJEP06(217)
f z(x) ;
with h(x) = pn(x) a Legendre polynomial. The function f is assumed analytic in the cut
plane. He chooses the contour
to run as in gure 6, staying away from the point x = 0.
Conditions for the
niteness7 and swapping of these functionals can be examined exactly
as above. It's clear that only conditions at x = 1 need to be imposed. The
niteness
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) =
X anpn(x) :
n
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
here.
Then the functional has to have the swapping property. This was not actually discussed
in [12]. Near x = 1 we must have (4.20), which should arise as a result of cancelation
between various terms in the sum de ning h(x).8
Additional complications arise near x = 0. Namely, as a result of the in nite
summation, 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
1)
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.
(5.1)
(5.2)
contour in gure 6 is no longer appropriate, and has to be modi ed. In fact, the behavior
of his h(x) near x = 0 can be described by the formula
HJEP06(217)
h(x) = h1(x) + h2(x) ;
where h1(x) is analytic near x = 0, while h2(x) has a cut along x < 0. The total functional
can then be de ned as a sum of three terms
1
2 i
Z
1
Z
2
Z
3
h(x)
f z(x) dx ;
(5.4)
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
on h2(x), while h1(x) does not have to satisfy any condition near x = 0. This contour
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
is integer
or hal nteger:
Using the properties of Legendre polynomials, this implies that
X anpn(x)
(
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.
(5.3)
(5.5)
(5.6)
(5.7)
(5.8)
(5.9)
and others into our notation) that near x = 0 we can decompose h~(x) as
Consider rst
2 N. In this case Mazac says in section 5.2 (we translate his eq. (5.18)
where h~1(x) is analytic near x = 0, while h~2(x) has a cut along x < 0 and satis es
h~(x) = h~1(x) + h~2(x) ;
h~2(x) = O(x2
log x)
(5.10)
(5.11)
(5.12)
Then by antisymmetry (5.8) we have the behavior near x = 1:
In section 5.3 he uses the freedom to add c(x) to set the behavior of
h~1(1
x) + c(x) = O (x
1)2
:
He conjectures that it's always possible although he only checked it up to
= 5. If so, we
can de ne this functional as in (5.3), (5.4) with h1(x) = h~1(x)+c(x) and h2(x) = h~2(x) and
have the conditions for the niteness and swapping satis ed near both x = 0 and x = 1.
Consider next
particular values
2 N
2
1 , discussed in [12], section 5.4 and appendix A. For the
= 1=2; 3=2; 5=2 he provides explicit h~(x), see his (4.29), (A.12){(A.14).
In these three cases we checked that near x = 0 one has decomposition (5.10) with the
nonanalytic part satisfying (5.11). The behavior near x = 1 is given by symmetry (5.9):
h~(x) = h~1(1
x) + h~2(1
x) :
h~1(1
x) + c(x) = O (x
1)2
:
He says that he's able, at least for
6 9=2, to use the freedom of adding c(x) to set:
If that's the case then the conditions for the niteness and swapping are indeed satis ed,
just as for
2 N.
The bottom line is that in the cases of lowlying
, 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
general
. This is beyond the scope of our work.
6
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 nonexplicit 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 Nonperturbative bootstrap).
A
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.
HJEP06(217)
Consider the following functions on the real interval 0 < t < 1:
0(t) = 1 ;
n(t) = (n + 1)tn
ntn 1
(n = 1; 2 : : :) :
The series of these functions sums to zero:
1
X
n=0
n(t) = 0
(0 < t < 1) :
Indeed, it was designed so that the subsequent terms cancel telescopically, so that the
partial sums
(A.1)
(A.2)
(A.3)
(A.4)
(A.5)
(A.6)
(A.7)
(A.8)
Let us check this in a couple of examples. If we take w(t) = 1
t, then things work nicely:
1
X In = 0 ;
n=0
I0 =
1
2
;
In =
1
n + 2
0
1
n + 1
In =
dt w(t) n(t) :
(n = 1; 2 : : :) ;
I0 = 1 ;
In = 0
(n > 0) ;
N
X
n=0
n(t) = (N + 1)tN
! 0
(0 < t < 1) :
Now consider formally integrating the series against some function w(t):
N (t) =
Z
1
X
n=N+1
dt w(t) N (t) ! 0
(N ! 1) :
n(t) =
(N + 1)tN :
and the series in (A.4) does converge to zero. On the other hand, for w(t) = 1 we have
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
integral
0
dP ( ) G (z) ;
associated with a monotonically increasing function P ( ), P (0) = 0. Convergence of this
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 RiemannStieltjes (RS) sums:
Z
0
N 1
corresponding to ner and ner subdivisions of the interval [0;
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
well.
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:
;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
(B.1)
(B.2)
(B.3)
(B.4)
! 1.
HJEP06(217)
N 1
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
!(F RS;N ) ! 0
(N ! 1 ;
xed) :
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 cuttouching functionals, a little
RS
;N goes
thought has to be given to what happens near the points x = 0; 1. Since this error term
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 (CCBY 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:
jGRS;N (z)j 6 const 1 + log
1
1
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:
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(B.7)
(B.8)
(B.9)
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HJEP06(217)
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