#### \( \mathcal{N} \) = 4 superconformal bootstrap of the K3 CFT

Received: April
= 4 superconformal bootstrap of the K3 CFT
Ying-Hsuan Lin 0 1 2 5 6 7 8
Shu-Heng Shao 0 1 2 5 6 7 8
David Simmons-Du n 0 1 2 3 6 7 8
Yifan Wang 0 1 2 4 6 7 8
Xi Yin 0 1 2 5 6 7 8
0 77 Massachusetts Ave , Cambridge, MA 02139 , U.S.A
1 1 Einstein Drive , Princeton, NJ 08540 , U.S.A
2 17 Oxford Street, Cambridge, MA 02138 , U.S.A
3 School of Natural Sciences, Institute for Advanced Study
4 Center for Theoretical Physics, Massachusetts Institute of Technology
5 Je erson Physical Laboratory, Harvard University
6 Open Access , c The Authors
7 nd an exact equivalence
8 [79] A. Adams, N. Arkani-Hamed , S. Dubovsky, A. Nicolis and R. Rattazzi, Causality, analyticity
c = 6, corresponding to nonlinear sigma models on K3 surfaces, using the superconformal bootstrap. This is made possible through a surprising relation between the BPS N = 4 superconformal blocks with c = 6 and bosonic Virasoro conformal blocks with c = 28, and an exact result on the moduli dependence of a certain integrated BPS 4-point function. Nontrivial bounds on the non-BPS spectrum in the K3 CFT are obtained as functions of the CFT moduli, that interpolate between the free orbifold points and singular CFT points. We observe directly from the CFT perspective the signature of a continuous spectrum above a gap at the singular moduli, and nd numerically an upper bound on this gap that is saturated by the A1 N = 4 cigar CFT. We also derive an analytic upper bound on the rst nonzero eigenvalue of the scalar Laplacian on K3 in the large volume regime, that depends on the K3 moduli data. As two byproducts, we between a class of BPS N = 2 superconformal blocks and Virasoro conformal blocks in two dimensions, and an upper bound on the four-point functions of operators of su ciently low scaling dimension in three and four dimensional CFTs. ArXiv ePrint: 1511.04065
Conformal Field Theory; Extended Supersymmetry; Field Theories in Lower
1 Introduction
2 Review of N = 4 superconformal representation theory N = 4 superconformal blocks
N = 4 Ak 1 cigar CFT
Four-point function and the Ribault-Teschner relation
N = 2 superconformal blocks
5 The integrated four-point functions
6 Special loci on the K3 CFT moduli space
T 4=Z2 free orbifold
N = 4 Ak 1 cigar CFT
7 Bootstrap constraints on the K3 CFT spectrum: gap
Crossing equation for the BPS four-point function
The gap in the non-BPS spectrum as a function of A1111
Constraints on the OPE of two di erent 12 -BPS operators
8 Bootstrap constraints on the critical dimension b crt
A simple analytic bound on OPE coe cients and b crt
8.2 Improved analytic bounds on b crt
8.3 Numerical bounds on b crt b crt in 2, 3, and 4 spacetime dimensions b crt for the K3 CFT
9 The large volume limit
10 Summary and discussion
Parameterization of the K3 moduli
9.2 Bounding the rst nonzero eigenvalue of the scalar Laplacian on K3
B Conformal blocks under the q-map
C More on the integrated four-point function Aijk`
C.1 Conformal block expansion
crt and the divergence of the integrated four-point function A1111
The conformal bootstrap [1{3], the idea that a conformal eld theory can be determined
entirely based on (possibly extended) conformal symmetry, unitarity, and simple
assumptions about the spectrum, has proven to be remarkably powerful. Such methods have been
implemented analytically to solve two-dimensional rational CFTs [4{7], and later extended
to certain irrational CFTs [8{11]. The numerical approach to the conformal bootstrap has
been applied successfully to higher dimensional theories [12{29], as well as putting
nontrivial constraints on the spectrum of two-dimensional theories that have been previously
unattainable with analytic methods [30{32].
bootstrap. Our primary example1 is the supersymmetric nonlinear sigma model with the
K3 surface as its target space. We refer to this theory as the K3 CFT. The conformal
manifold and BPS spectrum of the K3 CFT has been well known [34{39]. Much less was
known about the non-BPS spectrum of the theory, except at special solvable points in the
moduli space [40{42], and in the vicinity of points where the CFT becomes singular [43{45].
To understand the non-BPS spectrum of the K3 CFT is the subject of this paper.
There are two essential technical ingredients that will enable us to bootstrap the K3
the sphere four-point bosonic Virasoro conformal block of central charge 28, with external
weights 1 and internal primary weight h + 1. This relation is observed by comparing
correlators in the bosonic Liouville theory, through the relation of Ribault and Teschner
that expresses SL(2) WZW model correlators in terms of Liouville correlators [46, 47].
We generalize the above argument to establish an exact equivalence between a class
blocks of c = 13 + 6k + 6=k in section 4.
The second ingredient is the exact moduli dependence of certain integrated four-point
functions Aijk` of 12 -BPS operators (corresponding to marginal deformations) in the K3
CFT. They are obtained from the weak coupling limit of the non-perturbatively exact
results on 4- and 6-derivative terms in the spacetime e ective action of type IIB string
theory compacti ed on the K3 surface [48, 49]. This allows us to encode the moduli of the
K3 CFT directly in terms of CFT data applicable in the bootstrap method, namely the
four-point function.
The numerical bootstrap then proceeds by analyzing the crossing equation, where
Zamolodchikov's recurrence relations [8, 50]. The reality condition on the OPE coe cients,
the ALF CFT [33], for which our bootstrap method also applies.
which follows from unitarity, leads to two kinds of bounds on the scaling dimension of
nonBPS operators, which we refer to as the gap dimension
gap and a critical dimension
gap is the scaling dimension of the lowest non-BPS primary that appear in the
OPE of a pair of 12 -BPS operators. b crt is de ned such that, roughly speaking, the OPE
coe cients of (and contributions to the four-point function from) the non-BPS primaries
at dimension
> b crt are bounded from above by those of the primaries of dimension
b crt. A consequence is that, when the four-point function diverges at special points on
the conformal manifold, the CFT either develops a continuum that contains b crt or some of
its OPE coe cients diverge. In the case when the OPE coe cients are bounded (which is
not always true as we will discuss in section 7.4), b crt provides an upper bound on the gap
below the continuum of the spectrum that is developed when the CFT becomes singular.
We will see that the numerical bounds on b crt and
gap are saturated by the free
move along the moduli space. The moduli dependence is encoded in the integrated
fourpoint function of 12 -BPS operators Aijk`, which has been determined as an exact function
of the moduli. Our results provide direct evidence for the emergence of a continuum in the
CFT spectrum, at the points on the conformal manifold where the K3 surface develops ADE
singularities, using purely CFT methods (as opposed to the knowledge of the spacetime
BPS spectrum of string theory [38, 43, 51]). Our bounds are also consistent with, but not
saturated by, the OPE of twist elds in the free orbifold CFT.
We further discuss analytic and numerical bounds on b crt in general CFTs in 2,3, and
4 dimensions. Using crossing equations, we derive a crude analytic bound b crt
is the scaling dimension of the external scalar operator. This bound on b crt is
then re ned numerically, and we observe that it meets at the unitarity bound for
in 3 dimensions and
. 2 in 4 dimensions, thus giving universal upper bounds on the
four-point functions for this range of external operator dimension.
In the large volume limit of the K3 target space, the spectrum of the CFT is captured
by the eigenvalues of the Laplacian on the K3. Using a positivity condition on the
qexpansion of conformal blocks and four-point functions [52, 53], we will derive an upper
bound on the gap in the spectrum, or equivalently on the rst nonzero eigenvalue of the
scalar Laplacian on the K3, that depends on the moduli and remains nontrivial in the large
volume limit. Namely, it scales with the volume V as V 2 and thereby provides a bound
on the rst nonzero eigenvalue of the scalar Laplacian on the K3.
We summarize our results and discuss possible extensions of the current work in
the concluding section.
Various technical details are presented in the appendices. In
appendix A, we x the normalization of the integrated four-point function by comparing
with known results at the free orbifold point. In appendix B, we review the q-expansion of
the Virasoro conformal blocks and Zamolodchikov's recurrence relations. In appendix C,
we explain the subtle technical details on how to incorporate the integrated four-point
function Aijk` into the bootstrap equations, and also derive a bound on the integrated
cuss how the critical dimension b crt gives an upper bound on the gap below the continuum
when the integrated four-point function diverges at some points on the moduli space.
h = `, 0
h = k40 , 0
[Lm; Ln] = (m
n)Lm+n +
algebra SU(2)R and outer-automorphism SU(2)out is generated by a energy-momentum
tensor T , super-currents G A transforming as (2; 2) under SU(2)R
SU(2)out and the
SCA is captured by the commutation relations
[Lm; Gr A] =
fGr A; Gs Bg = 2
[Lm; J ni] =
[J mi; Gr A] =
[J mi; Jnj ] = i ijkJ mk+n + m
Review of N
The small N
acts as [54],
where ( i)
are the Pauli matrices and ( i)
= ( i)
= +
= +1. Here we
are focusing on the left-moving (holomorphic) part. The subscripts r; s take half-integer
values for the NS sector and integer values for the R sector.
2 Z=2. In particular, spectral ow with
2 Z + 12 connects the NS and R sectors.
` in the NS sector and h
(massless or short) representations and the non-BPS (massive or long) representations,
on both the left and right sides are called 12 -BPS; the operators which are BPS on one side
and non-BPS on the other are 14 -BPS. We should emphasize that our terminology of BPS
operators exclude the currents which will be lifted at generic moduli of the K3 CFT.
m= 1
chnho;`n-BPS(q; z; y) = qh Y
while the non-BPS NS sector character is
The character for the BPS representation in the NS sector is
(1+yzqm+ 12 )(1+y 1zqm+ 12 ) (1+yz 1qm+ 12 )(1+y 1z 1qm+ 12 )
m= 1
1+yzqn 12 1+y 1zqn 12 1+yz 1qn 12 1+y 1
q(k0+1)m2+(2`+1)m z2((k0+1)m+`)
where z and y are the fugacities for the third components of SU(2)R and SU(2)out,
respectively. The Ramond sector characters are related to the above by spectral ow.
labelled by Oi
forms on K3 (i = 1;
; 20). In particular, the weight- 12 BPS primaries Oi
to exactly marginal operators of the K3 CFT. Under spectral ow, the identity operator
whereas Oi
iRR. The K3 CFT also contains 14 BPS primaries of weight (s; 12 ) and ( 12 ; s), for integer
1.2 The weight (s; 12 ) 14 -BPS primaries have left SU(2)R spin 0 and right SU(2)R
in the (R,R) sector,
characters [35, 55, 56],
ZKNS3 = 20chB1 PS + ch0BPS
ch0non-BPS(90q + 462q2 + 1540q3 +
where the (s; 12 ) BPS primaries are counted by the character
90q + 462q2 + 1540q3 +
We assume the absence of currents at generic moduli of the K3 CFT, which may be justi ed
by conformal perturbation theory, so that the 14 BPS primaries are the only contributions
to the non-BPS character terms in the elliptic genus (2.5). While the currents (of general
spin) may appear at special points in the moduli space, they can be viewed as limits of
non-BPS operators and therefore do not a ect our bootstrap analysis.
We are interested in the four-point function of Oi
RR by spectral ow). Below
we will make a general argument, based on N
2Note that the 14 -BPS primaries are fermionic with half integer spin, and are themselves projected out
in the spectrum of the K3 SCFT. Rather, their integer spin (4; 4) SCA descendants comprise the true 14
BPS operators of the K3 SCFT.
`2 respectively can only contain superconformal primaries O
SU(2)R spin ` within the range j`1
`;m (and descendants of), with
1; `1 + `2 and m labels
the identity operator and non-BPS operators can appear. Consequently, only the identity
block and non-BPS blocks contribute to the four-point function of 12 -BPS primaries Oi .
We start with the 3-point function
m m1 m2 is an arbitrary word with J03 =
m2 under left SU(2)R and
identically. The main idea is to perform contour deformation a number of times to strip
or just the correlator of the superconformal primaries themselves which vanish due to
SU(2)R invariance.
Let us suppose ` does not belong to j`1
1; `1 + `2. By
inserting an appropriate number of J0 at x1 and x2 in (2.7), and redistributing them by
contour deformations, we can reduce the correlator (2.7) to
`1;`1 (x1) `22; `2 (x2)[W `2 `1 m
x2)G+nA 1=2 =
G+nA+1=2 +
G+A(z)(z
3We will focus on the holomorphic part in this argument.
4Similar contour arguments have been used in [57, 58] to argue that the three point functions of BPS
primaries are covariantly constant over the moduli space.
5Note that we do not have contributions when deforming the contour past in nity for n
We can immediately strip o
all Virasoro generators L n in W `2 `1 m by deforming
This will relate the original three-point
correlator to the derivatives of those without L n. Similarly, we can deform the contour of
x3) nJ 3(z) to move J 3 n on
x3) nJ +(z), we can replace its insertion by
x2)J +n =
and deforming the contour. Note that the second term in (2.9) has a vanishing contribution
function by G+nA+1=2 for n
when we deform the contour to encircle either
point function with J +n in W `2 `1 m is related to another with the operator replaced by
J +n+1 in W `2 `1 m. Repeating this procedure a number of times, we can be replace J +n
by J0+.5 Similarly we can substitute J n by J0 . By commuting J0i all the way to right,
we obtain a bunch of three point correlators of the form (2.8) with W `2 `1 m purely made
leftmost letter in W . As before for J +n, we can replace this insertion in the three-point
`1;`1 or `2; `2 , hence the original
threereduce the number of G Ar's in W `2 `1 m by two. Therefore we have reduced the correlator
to that of the form (2.8) with W `2 `1 m being either G Ar or removed completely. In the
former case, we can perform the replacement (2.10) and contour deformation again and
conclude the reduced three-point function vanishes. In the latter case, the resulting 3-point
correlator also vanishes due to SU(2)R invariance. This completes the argument.
For the purpose of bootstrapping the K3 CFT, we will need the sphere four-point
superR sector. The intermediate representation will be taken to be that of a non-BPS primary
tor BPS correlator of the form hO+(z)O (0)O+(1)O (1)i by Fh
of weight h (and necessarily SU(2)R spin 0). Let us denote the NS BPS primary by O
(exhibiting the left SU(2)R doublet index only), and the Ramond BPS primary by
N =4;NS(z) (see gure 1),
and the corresponding block with R sector external primaries, associated with a correlator
of the form h R(z) R(0) R(1) R(1)i, by Fh
N =4;R(z). The NS and R sector blocks are
decomposition of the BPS four-point function in the K3 CFT, because neither the 12 -BPS
nor the 14 -BPS operators appear in the OPE of a pair of 12 -BPS primaries, as demonstrated
in the previous section. The identity representation superconformal block, on the other
maries and internal non-BPS primary of weight h is identi ed with the bosonic Virasoro
weight h + 1 for the internal primary, through the relation
z) 21 FcV=ir28(1; 1; 1; 1; h + 1; z):
Here FcVir(h1; h2; h3; h4; h0; z) denotes the sphere four-point Virasoro conformal block with
central charge c, external weights hi, and internal weight h0.8;9
6One can apply a similar procedure if G+Ar is the leftmost letter in W .
7By a contour argument similar to the one in section 2, one can show there is only one independent
OPE coe cient between two BPS superconformal primaries.
superscript V ir for the bosonic Virasoro conformal blocks.
9A similar relation between superconformal blocks and non-SUSY blocks with shifted weights was found
in [29, 59{61] for SCFTs in d > 2.
N =4,NS(z) =
O−(0)
O−(∞)
cont =
and intermediate non-BPS primary of weight h.
We will discuss an explicit check of (3.2) on the z-expansion coe cients of the
conformal block in section 4.
N = 4 Ak 1 cigar CFT
product of N = 2 coset SCFTs [43, 62, 63],
SL(2)k=U(1)
SU(2)k=U(1):
The N = 4 Ak 1 cigar theory has 4(k
1) normalizable weight ( 12 ; 1 ) BPS primaries,
cor2
responding to 4(k
1) exactly marginal deformations,11 and a continuum of delta function
normalizable non-BPS primaries above the gap
in the scaling dimension. Later when we consider a sector of primaries with nonzero
R-charges, the continuum develops above a gap of larger value and there may also be
discrete, normalizable non-BPS primaries below the gap. The continuum states are in
correspondence with those of the supersymmetric SU(2)k
CFT, where R is a linear
Four-point function and the Ribault-Teschner relation
Let us recall the computation of the sphere four-point function of the BPS primaries in
the Ak 1 cigar CFT, studied in [47]. The weight ( 14 ; 14 ) 12 -BPS RR sector primaries lie in
Note that ` + 1 is also the charge with respect to a Zfk symmetry that acts on the twisted
sectors, and is conserved modulo k. They can be constructed from SL(2) and SU(2) coset
primaries as either
VR+;` = V 2` ; `+22 ;2`+22
2 V ` ` `
VR;` = V `
coset primary Vjs;ml; (z) is
j(j + 1) + (m + )
j0(j0 + 1)
We have the identi cation VR;` = VR+;k 2 `.
The correlator of interest is
DVR+;`(z; z)VR+;`(0)VR;`(1)VR;`(1) ;
E
of the correlator was determined in [47], using Ribault and Teschner's relation [46] between
the bosonic SL(2) WZW and Liouville correlators. The result is of the form (see (3.37)
and (3.39) of [47])12
F Vir(h1; h2; h3; h4; hP ; z)j2:
Here F Vir(h1;
N is a normalization constant. Q is the background charge of a corresponding bosonic
Liou
They are related to k (labeling the Ak 1 cigar theory) and ` (labeling the BPS primaries) by
Q = b + ;
1 =
2 =
h1 = h2 =
h3 = h4 =
(` + 2)(2k
(k + ` + 2)(k
b2 =
3 =
4 =
12Note that the identity block does not show up in the cigar CFT four-point function because the identity
operator is non-normalizable. This can also be understood from the normalization when compared with
the K3 CFT discussed in section 6.2.
Note that the Liouville background charge Q is not the same as the background of the
intermediate continuous state in the Liouville theory is
hP =
P =
0 Qi3=1 (2 i)
C( 1; 2; 3) is the structure constant of Liouville theory [8, 64],
C( 1; 2; 3) = e
where ~ =
(b2)b2 2b2 is the dual cosmological constant to
with (x) = (x)= (1
central charge
c = 1 + 6Q2 = 28;
( ) =
2(Q=2jb; b 1)2
2(xja1; a2) is the Barnes double Gamma function [65].
m=b and
The integration contour in (3.10) is the standard one if i lie on the line Q2 +iR. We need
to analytically continue i to the real values given above. In doing so, the integral may pick
up residues from poles in the Liouville structure constants. These residue contributions, if
present, correspond to discrete intermediate state contributions [66]. We will have more to
say about these discrete intermediate state contributions to the four-point function (3.10)
( ) has zeroes at
in the N = 4 Ak 1 cigar CFT in section 7.4.
the cigar CFT is simply given by one bosonic linear dilaton R , with background charge p1 ,
CFT are in one-to-one correspondence with exponential operators in the bosonic part of
the asymptotic linear dilaton CFT, of the form
V = e2 ; with
Importantly, these non-BPS primaries are labeled by the same quantum number, a real
number P , as the intermediate Liouville primaries in (3.10).
The result (3.10) that expresses the BPS four-point function in terms of Virasoro
conformal blocks labeled by the Liouville primaries V then strongly suggests that in the
sition (3.10) in terms of Virasoro conformal blocks. Here, the Virasoro block is that of
intermediate Liouville primary with weight hP to the corresponding N
= 4 non-BPS
description, would be constructed from an SL(2) primary of spin13
with conformal weight
On the other hand, by the relation of Ribault and Teschner (see also (3.17) of [47]), the
intermediate Liouville primary in (3.10) is labeled by the exponent P given by
j =
h =
P =
hP =
hP = h + 1:
Using (3.12), we obtain the weight of the intermediate Liouville primary in terms of P
This leads us to identify the relation between the Virasoro primary weight hP and the
normalization in the z ! 0 limit, we then deduce the relation (3.2).
m [67], with m = 0; 1;
. By a similar contour argument as in section 2, only the U(1)R
chiral operator + and anti-chiral operator
,15 hence the claim.
13The p2 is introduced to match with the convention in (3.10).
14We thank Sarah Harrison for a discussion on this issue.
primaries with U(1)R charge q1 and q2 can only contain a primary (and descendants of) with U(1)R charge
0 and q1 + q2
0 and q1 + q2
0. In particular when we consider the OPE
of one chiral and one antichiral primaries with opposite U(1)R charges, only the U(1)R neutral primaries
(and descendants) can appear.
Fq,−q,q,−q|h
N =2,c= 3(kk+2) ,NS(z) =
q =
; ` = 0; 1;
chiral/anti-chiral primaries
U(1)R neutral non-BPS primary of weight h.
of weight j2qj and U(1)R charge
q =
`+k2 , and intermediate
More generally, one can extract the chiral-anti-chiral NS superconformal block (see
gure 2) of a general N
= 2 SCA with central charge c = 3(k+2) from the N
k
= 2
one can show that the c = 3(k+2)
anti-chiral operators of weight j2qj and U(1)R charge q, q, q, q, with16
q =
` = 0; 1;
and the internal U(1)R neutral non-BPS primary with weight h, is related to the bosonic
N =2; c= 3(kk+2) ; NS
(z) = z
(`+2)(k ` 2)
(`+2)(3k 2` 4)
FcV=ir13+6k+ k6
hq; h q; hq; h q; h +
hq =
(` + 2)(2k
h q =
`)(k + ` + 2)
the intermediate weight hP = h + k+2 comes from the di erence between Q2=4 and 1=4k,
4
that (4.3) (and therefore (3.2) as a special case) holds up to level 4 superconformal
descendants with various values of q in (4.2). We expect (4.3) to hold for (anti)chiral primaries
in ` and k.
charge from the cigar CFT will be presented elsewhere.
16Under spectral ow, the NS sector chiral primaries are mapped to R sector ground states with R-charges
= 2
The integrated four-point functions
In this section we discuss the integrated four-point function of 12 -BPS operators, whose
exact moduli dependence will be later incorporated into the bootstrap equations (see
section 7.3 and appendix C). The integrated sphere four-point functions Aijkl and Bij;kl
are de ned as [49]17
iRR(z; z) jRR(0) kRR(1) `RR(1)
+ Aijk` + Bij;k`s + Bik;j`t + Bi`;jku + O(s2; t2; u2);
where iRR are the RR sector 12 -BPS primaries of weight ( 14 ; 14 ) that are related to NS-NS 12
BPS primaries Oi
by spectral ow, and the variables s; t; u are subject to the constraint
(kl), and under the exchange (ij) $ (kl). Furthermore, Bij;kl is subject to the constraint
amplitude [68, 69].
The rst term in (5.1) is related to the tree-level amplitude of tensor multiplets in type
IIB string theory compacti ed by K3 at two-derivative order. In particular, it captures
the Riemannian curvature of the Zamolodchikov metric on the K3 CFT moduli space.
Moreover Aijkl and Bij;kl can be identi ed as the tree level amplitudes of tensor multiplets
in the 6d (2; 0) supergravity at 4- and 6-derivative orders respectively. They can be obtained
from the weak coupling limit of the exact results for the 4- and 6-derivative order tensor
e ective couplings determined in [48, 49]. In this paper, we will make use of
Aijk` =
16 2 @yi@yj @yk@y` y=0 F
where F is the fundamental domain of PSL(2; Z) acting on the upper half plane,
even unimodular lattice
20;4 embedded in R20;4, which parameterizes the moduli of the
K3 CFT, and the theta function
is de ned to be
Here `L and `R are the projection of the lattice vector ` onto the positive subspace R20 and
y is an auxiliary vector in the R20, whose components are in correspondence with the 20
BPS multiplets of the K3 CFT. Note that in (5.2), the integral is modular invariant only
The expression (5.2) is obtained from the weak coupling limit of (1.3) in [49] (by
21;5 =
1;1, and taking a limit on the
1;1). The normalization can
17More precisely, this integral is de ned by analytic continuation in s; t from the region where it converges.
free orbifold CFT, as shown in appendix A. There is an analogous formula for Bij;kl as an
integral of ratios of modular forms over the moduli space of a genus two Riemann surface.
If we assume that all non-BPS primaries have scaling dimension above a gap
can derive an inequality between the integrated four-point function A1111 of a single 12 -BPS
1, and the four-point function f (z; z) itself evaluated at a given cross ratio, say
z = 12 , of the form (see appendix C.1)
3A0 + M ( ) [f (1=2)
that f ( 12 ) diverges in the singular CFT limits.
Here A0 and f0 are constants, and M ( ) is a function of
that goes like 1=
! 0 limit. Since A1111 is known as an exact function of the moduli, this inequality will
provide a lower bound on f ( 12 ) over the moduli space. In particular, it can be used to show
Special loci on the K3 CFT moduli space
Some loci on the moduli space of the K3 CFT are more familiar to us, such as near the
free orbifold points19 and where ADE singularities develop. This section reviews certain
properties of the K3 CFT at these special points, that will allow us to check the consistency
of our bootstrap results in section 7. In fact, some of the examples we discuss here will
saturate the bounds from bootstrap analysis.
T 4=Z2 free orbifold
There is a locus on the K3 CFT moduli space that corresponds to the Z2 free orbifold of
a rectangular T 4 of radii (R1; R2; R3; R4). Let us rst consider the twisted sector ground
state in the RR sector (z; z), associated with one of the Z2 xed points. Its OPE with
itself will receive contributions from all states in the untwisted sector with even winding
The four-point function of (z; z) is [71, 72]
f (z; z) = jz(1
where20 q(z) = exp(i
(z)), (z) = iF (1
and the lattice
z)=F (z), F (z) = 2F1( 21 ; 21 ; 1jz) = [ 3(q(z))]2,
2Ri. Note again
that the untwisted sector operators with odd winding numbers are absent in (6.1) due to the
selection rule in the orbifold theory [70]. The map z ! q(z) is due to Zamolodchikov [8, 50]
and is explained further in appendix B. The range of this q-map is shown in gure 3.
18Note that the assumption of a nonzero gap holds in the singular CFT limits where the K3 develops
ADE type singularities, but obviously fails in the large volume limit.
20Our convention for 3(q) is 3(q) = P
n2Z qn2 , with q = ei .
of the q-map (B.2). The regions D1, D2 and D3 each contains two fundamental domains of the S3
crossing symmetry group. See appendix B.
f (1=2; 1=2) =
jF (1=2)j4 i=1
=Ri2 )j
and section 7.3, we will compare the twisted sector four-point function with our bootstrap
bounds on the gap in the spectrum.
Next let us consider the four-point function of untwisted sector operators. The NS
sector 12 -BPS operators in the untwisted sector can be built from the free fermions
which satisfy the OPE
N = 4 descendant of identity or the current
superconformal primary.
= 12
e B AB. Its four-point function is,
From the bilinears of
A we have either the SU(2)R current
B AB which is an
which is a weight (1; 0) non-BPS
Consider a single 12 -BPS operator in the untwisted sector of the free orbifold theory
f (z; z) = hO
(1)i =
In the OPE between O
e D : of weight (1,1). This will show up as a special example in section 7.2
and section 7.3 when we study the bootstrap constraint on the gap in the spectrum. Note
, the lowest non-identity primary is
be checked explicitly from (5.1) and (C.1).
More generally, we can consider two 12 -BPS operators 1
in the untwisted
with hP =
; h1 = h2 =
(` + 2)(2k
with det(M ) = 0.
with any other 12 -BPS primary
where MAB and M AB are some independent general 2
2 complex matrices. Below we
will show that if the identity block is absent in the OPE of a 12 -BPS primary i
untwisted sector with itself, the (1; 0) non-BPS primary must appear in the OPE of i
in the untwisted sector if the identity block appears
there. The OPE coe cient of the identity block in the
1 1 OPE is proportional to
det(M ), whereas that in the 1 2 OPE is proportional to AB CDMAC M BD. Therefore,
but not in the 1 2 OPE. If the (1; 0) primary is absent in the 1 2 channel, we require
AB with a nonzero proportionality constant. This is in contradiction
In this case, the lowest primary in the i i OPE would be a (1; 1) non-BPS primary
which combines the holomorphic (1; 0) primary with its antiholomorphic counterpart. In
other words, if the i i channel does not contain identity whereas the i j channel contains
gap = 1 in the i j channel and
gap = 2 in the i i channel. As we will see
in subsection 7.4, if we take j to be the complex conjugate of i, this corresponds to a
special kink on the boundary of the numerical bound for the h
i correlator.
N = 4 Ak 1 cigar CFT
2 we use the identi cation VR;` = VR+;k 2 `.
on its continuous spectrum and divergent OPE coe cients.
We will consider the RR sector 12 -BPS primaries VR+;` and VR;` ((3.5) and (3.6)) [49,
73, 74] with Zfk charge (` + 1). Here ` ranges from 0 to b k 2 2 c. For ` between b k 2 2 c + 1
Continuum in the cigar CFT.
As already mentioned in (3.4), in the OPE between
VR+;` and VR;`, there is a continuum of delta function normalizable non-BPS primaries above
Here we have adopted the notation that will be used in subsection 7.4 where we denote
Let us move on to the lowest weight operator that lies at the bottom of the continuum
the two parts by hsl and hsu, respectively.
hsl can be determined by studying the four-point function (3.10) together with the
cont =
of the latter is given by (3.7) to be 41k + 18 . Hence,
where we have used (3.11) and hP =
state in the continuum. Recall that Q = p
P ) with P = Q=2 for the lowest dimension
k + p1k is the background charge of the
corresponding bosonic Liouville theory in the Ribault-Teschner relation. Writing the four-point
perconformal block with intermediate state being the bottom state in the continuum and
hsl =
(` + 2)(2k
between two V ` ` `
su;(1;1), whose holomorphic weight is given by (3.8),
Adding hsl and hsu together, we obtain the lowest scaling dimension
of the OPE channel between VR+;` and VR+;`,
cont in the continuum
hsu =
cont = 2(hsl + hsu) =
As we will show below, in addition to the continuum, there are generally discrete states
contributing to the four-point function (3.10) of the cigar CFT with divergent structure
constant when normalized properly.
Discrete Non-BPS primaries.
As mentioned in section 3, the discrete state
contributions come from the poles in the Liouville structure constants C( 1; 2
; P ) when we
analytically continue the external states, labeled by their exponents
their actual values on the real line given in (3.11) [66]. The relevant factor in the Liouville
structure constant is
P ) in the denominator of (3.13),21 where
x =
x =
The argument of ( 1 + 2
P ) is deformed from Q=2 + iR to `p+2
k + p1 , the question of identifying the poles is equivalent to asking whether
that Q = p
structure constant C( 3; 4; Q2
gives the same set of poles.
Q) in (3.13) will give other discrete states with the same weights. The
iP ) yields an identical analysis with ` replaced by k
2 `, and hence
contains any of the poles in (6.11). It is not hard to see that the only possible poles in (6.11)
that lie in the above interval are
x =
n = 0; 1;
Note that k
4 for these poles to contribute.22 These poles occur at
or, in other words,
+ iP =
P = i p
The imaginary shift of the momentum shifts the scaling dimension of the discrete non-BPS
primary of question from the continuum gap by the amount of 2P 2, to
+ 2P 2 = 2(n + 1)
2(n + 1)(2 + 2` + n)
The lowest scaling dimension
discrete of such a discrete state (with divergent structure
discrete = 2
K3 CFT is
for generic cross ratio.
The normalization of structure constants.
We now argue these discrete non-BPS
operators, when viewed as a limit of those in the K3 CFT (that is described by the cigar CFT
near a singularity), have divergent structure constants with the external 12 -BPS primaries.
Let us rst clarify the normalization of operators in the cigar CFT versus in the K3
CFT. In comparing the cigar CFT correlators to the K3 CFT correlators, there is a
divergent normalization factor involving the length L of the cigar. That is, let V be some
operator in the cigar CFT, then an n-point function hV V
V i in the cigar CFT of order
i goes like L, which diverges in the in nite L limit,
The discrete non-BPS states discussed above contribute to the four-point
function (3.10) by an amount that is a
nite fraction of the continuum contribution, and
both diverge in the singular cigar CFT limit. Consequently, these discrete states in the
OPE of two 12 -BPS operators
RR have divergent structure coe cients in this limit.
point function is cancelled by poles from other factors in the Liouville structure constant. In any case, the
potential discrete state lies at the bottom of the continuum and therefore does not a ect the distinction
discrete with
f (z; z) =
Ch2L;hR FhRL (z)FhRR (z);
z) 21 FcV=ir28(1; 1; 1; 1; h + 1; z);
0 =
0 =
and FcVir(h1; h2; h3; h4; h; z) is the sphere four-point conformal block of the Virasoro algebra
of central charge c. Crossing symmetry relates the decomposition in the z ! 0 channel to
that in the z ! 1 channel
This is equivalent to the statement that
Bootstrap constraints on the K3 CFT spectrum: gap
Crossing equation for the BPS four-point function
Let us consider the four-point function f (z; z)
h RR(z; z) RR(0) RR(1) RR(1)i of
identical R sector ground states (the four-point function in the NS sector is related by spectral
ow). Decomposed into c = 6 N
for all possible linear functionals
[12]. In particular, we can pick our basis of linear
functionals to consist of derivatives evaluated at the crossing symmetric point
m;n[H ;s(z; z)] trivially vanishes for m + n even, we want to consider functionals
that are linear combinations of m;n for m+n odd. Restricting to this subset of functionals,
the crossing equation becomes
m;n = @zm@zn z=1=2 :
0 =
FhRL (z)FhRR (z) :
where for convenience we de ne
Using the crossing equation, we will constrain the spectrum of intermediate primaries
appearing in the
RR RR OPE, by
nding functionals that have certain positivity
properties. In particular, we will be interested in bounding the gap in the non-BPS spectrum,
as well as the lowest scaling dimension in the continuum of the spectrum in the singular
scalar as a function of
blocks), 3 dimensions, and 4 dimensions. The blue line shows the analytic bound p
in 2 dimensions (using global conformal
The red bounds are computed numerically with derivative order 12; 20; 28, with the darkest line
and strongest bound corresponding to derivative order 28. For
. 1 in 3d and
. 2 in 4d, the
red bounds meet at the unitary bounds, thus giving universal OPE bounds in this range of
K3 CFT. Our results show rigorously that
crt in the K3 CFT must lie below 0:29321, at
every point on the moduli space. By extrapolating to in nite order, we nd that b crt is
saturated, within numerical error, by the A1 cigar whose continuum lies above
crt = 1=4.
As in section 7.4, we can consider a correlator h RR RR RR RRi for two di erent
RRsector 12 -BPS operators that are complex conjugate of each other, and bound the divergent
operator of the lowest scaling dimension in the
RR channels. We
x ( b crt; b crt), and search for nonzero functionals ~ that satisfy
If such a functional exists, then
Figure 9 shows the allowed region of ( crt; crt) obtained at various derivative orders.
crt, the bound on
crt cannot be worse than the single correlator bound
crt . 0:25. For
crt . 1:5, extrapolating to in nite order gives bounds on
close to the single correlator bound. For
crt & 1:5, the bound on
crt decreases until it
reaches 0 at
Derivative order d
quadratic t
A1 cigar
0.02 0.04 0.06 0.08 0.10 0.12 0.14
derivative order is increased, as well as the extrapolation to in nite order using a quadratic t.
Also shown is the value of crt for the A1 cigar.
■ A1⊕A1 and Ak-1 (k⩾3) cigar
Single correlator bound
lowest scaling dimension in the respective OPEs, at derivative orders ranging from 8 to 20. At
in nite order, the bound cannot be worse than the single correlator bound 0.25 indicated by the
dashed line. We also
crt is bounded above by 2, beyond which
crt = 0. The square
dots indicate the values for the A1
A1 (at (1=4; 1=4)) and Ak 1 (k
3) cigar theories.
Ak 1 cigar CFT. Let us comment on where the Ak 1 cigar CFTs analyzed in section 6.2
gure 9. For the cigar CFT, we take
VR+;` and VR;` ((3.5) and (3.6)). The continua of the Ak 1 cigar CFT in
RR start at
cont = (k
1)2=2k and
cont = 1=2k, respectively (see (6.10)
and (6.6)). For k
4, there are discrete state contributions to the four-point function in
RR to be RR sector 12 -BPS primaries
the channel RR
RR starting at
their OPE coe cients are divergent when compared with a generic K3 CFT. Since
de ned as the lowest scaling dimension such that either a continuous spectrum appears or
the structure constants of some states in the discrete spectrum diverge, we have
crt = min
if k = 2; 3;
in the OPE channel between VR+;` and VR+;` in the Ak 1 cigar CFT. On the other hand, in the
OPE channel between VR+;` and VR;`,
crt =
cont = 1=2k as in (6.6). We would like to
emphasize that the presence of these R-charge non-singlet discrete states below the continuum
is crucial for the consistency with the bootstrap bound derived from the crossing equations.
can be realized at an A1
point on the moduli space, and the other black dots at Ak 1 points with k
asymptote to (2; 0) at large k.32
The large volume limit
In this section we consider the gap in the OPE of 21 -BPS operators in the large volume
regime of the K3 CFT. Based on unitarity constraints on the superconformal block
decomposition of the BPS 4-point function (but without making direct use of the crossing
equation), we will derive an upper bound on the gap, which remains nontrivial in the
large volume regime, and leads to an interesting inequality that relates the rst nonzero
eigenvalue of the scalar Laplacian on the K3 to an integral constructed from a harmonic
2-form, and data of the lattice
19;3 that parameterize the K3 moduli. The eigenvalues of
the Laplacian on K3 can be studied using the explicit numerical metric in [84, 85].
Parameterization of the K3 moduli
The quantum moduli space of the K3 CFT can be parameterized by the embedding of the
20;4 into R20;4, or equivalently, the choice of a positive 4-dimensional hyperplane
in the span of 20;4. Let us write
19;3, with the
19;3 identi ed with the
cohomology lattice H2(K3; Z) [86]. Let u; v be a pair of null basis vectors of the 1;1, with
2 = v
2 = 0, u v = 1. Let
with the hyperkahler structure of the K3 surface, normalized so that i
j = ij . We will
denote by B the cohomology class of a at B- eld, and by V the volume of the K3 surface
(more precisely it is (2 )4 times the volume in units of 02). An orthonormal basis of the
32The minimal resolution of an ADE singularity of rank
exceptional divisors which are dual to
self-dual elements of H1;1(K3), thus
19. In particular, the K3 surface can develop an Ak singularity
19. However our bound on b crt is insensitive to the identity superconformal block contribution,
and applies to noncompact theories as well, such as nonlinear sigma model on ALE spaces [83] and the
N = 4 cigar CFTs.
4-dimensional positive hyperplane is [86]
+ = Pi3=1(
2 H2(K3; Z) '
i) i. We have
We can now write the theta function
E0 =
Ei =
; i = 1; 2; 3:
e0 =
e = B
W u + W ;
= 1;
` = nu + mv + ;
` ` =
`2L + `2R =
`2R = (` E0)2 + X(` Ei)2
= (
B + n + m V
Now an orthonormal basis of the 20-dimensional negative subspace can be constructed as
2 span( 19;3) are a set of orthonormal vectors that are orthogonal to
correspond to a basis of anti-self-dual harmonic 2-forms on the K3 surface.
A general lattice vector of 20;4 can be written as
+ be the self-dual projection of , or equivalently,
4-point function of BPS operators (5.2) associated with deformations of 19;3 (as opposed
to the overall volume modulus, parameterizing the embedding of 1;1) becomes
V Z
Note that this result does not apply to the integrated 4-point function of the BPS operator
associated to the volume modulus, which in fact vanishes in the large volume limit.
Bounding the rst nonzero eigenvalue of the scalar Laplacian on K3
Let us write the four-point function of a given 12 -BPS, weight ( 14 ; 14 ) operator in the RR
RR, which is related to a weight ( 12 ; 12 ) NS-NS primary by spectral ow, as
RR(z; z) RR(0) RR(1) RR(1) = f (z; z):
16 2 @y4 y=0 F
f (z; z) + 6 ln
where according to our claim (3.2)
of central charge c. We can write
FhR(z) = (z(1
where the function gh(q) takes the form
f (z; z) = jF0R(z)j2 +
Ch2L;hR FhRL (z)FhRR (z);
z) 21 FcV=ir28(1; 1; 1; 1; h + 1; z);
Vir(h1; h2; h3; h4; h; z) is the sphere four-point conformal block of the Virasoro algebra
f (z; z) admits a conformal block decomposition (in the z ! 0 channel) of the form
In the rst approximation, we have dropped nite contributions that are unimportant in
the large volume limit, where A diverges like V 2 , while
0 goes to zero like V 2 . Here
gh(q) = qh 16 X1 anqn;
Positivity of the an follows from re ection positivity of the theory on the pillowcase [53].
In particular, we learn that FhR(z) obeys the inequality
q , 0 < z < 1 and 0 < q < 1.
In the large volume limit, A is dominated by the contribution from light non-BPS
0 in the spectrum of non-BPS (scalar) primaries. We can write in this limit
F R (z )
is a cuto
on the operator dimension that can be made small but nite, and (
small positive number. Taking
to zero after taking the large volume limit, we derive the
bound (which holds only in the large volume limit)
One might be attempted to take z to be small, but f (z ) diverges in the small z limit.
3(q 1 )4(q 1 ) 31 f (1=2)
jF0R(1=2)j2 ;
where q 1
q(z = 12
K3 surface, in units of 0.33
four-point function f ( 12 ) remains nite in the in nite volume limit. In this limit, we can
0 =
malized such that V 1 R
pg!ij !ij = 1. Let O!
corresponding moduli deformation. We have for instance O!
in the large volume limit. The 4-point function of the corresponding
be the BPS primary associated with the
where !2
1 Z
f!(1=2)
V R d2
@y!4 y!=0
z = 12 is
where e! is the unit vector in R20 associated with the deformation O!.
The upper bound (9.17) was derived by consideration of the 4-point function of a single
12 -BPS primary O!, and applies to the gap in the OPE of O! with itself. We see that in the
large volume limit, a light scalar non-BPS operator must appear in such an OPE, provided
that ! is not proportional to the Kahler form, so that A scales like
V . As noted earlier, if
we take ! to be the Kahler form J itself, the corresponding BPS operator OJ would have
an integrated 4-point function A that vanishes in the large volume limit instead, and we
cannot deduce the existence of a light operator in the OPE of OJ with itself.
33It is known [87, 88] that
where d is the diameter of the K3. The compatibility with our large volume bound then demands an
Summary and discussion
Let us summarize the main results of this paper.
2. We derived a lower bound on the four-point function of a 12 -BPS primary by the
integrated four-point function A1111, assuming the existence of a gap in the spectrum.
We also determined Aijkl as an exact function of the K3 CFT moduli (parameterized
by the embedding of the lattice 20;4).
3. We found an upper bound on the lowest dimension non-BPS primary appearing
in the OPE of two identical 12 -BPS primaries, as a function of the BPS four-point
function on the moduli space of the K3 CFT). Both vary monotonously from 2 to
14 , and interpolate between the untwisted sector of the free orbifold CFT and the A1
cigar CFT. It is also observed that A1111 must be non-negative from the bootstrap
constraints (see gure 6), which is consistent with the superluminal bound on the H4
coe cient in the 6d (2,0) supergravity coming from IIB string theory compacti ed
4. Bounding the contribution to the BPS four-point function by contributions from
non-BPS primaries of scaling dimension below b crt, and assuming the boundedness
of the OPE coe cients, we deduce that a continuum in the spectrum develops near
the ADE singular points on the K3 CFT moduli space, and
nd numerically that
b crt agrees with the gap below the continuum in the A1 cigar CFT, namely 14 .
5. We explored the possibility of the appearance of either a continuum or divergent
contribution from discrete non-BPS operators in the OPE of two distinct 12 -BPS
operators, near a singular point of the moduli space where the BPS four-point
function diverges (beyond the A1 case). The bootstrap bounds we found are consistent
appearance of discrete non-BPS primaries in the OPE below the continuum gap.
6. For general CFTs in 2,3,4 spacetime dimensions, we derived a crude analytic bound
was observed (see
is the scaling dimension of the external scalar operator. It
gure 8) from the stronger numerical bounds on b crt that they
meet at the unitarity bounds for
. 1 in 3 spacetime dimensions and
in 4 spacetime dimensions, thus providing universal upper bounds on the four-point
functions for this range of external operator dimension.
7. Independently of the crossing equation, but using nonetheless unitarity and exact
results of the integrated BPS four-point function, we derived in the large volume
regime a bound that is meaningful in classical geometry, namely an upper bound on
the rst nonzero eigenvalue of the scalar Laplacian on K3 surface, that depends on
the moduli of Einstein metrics on K3 (parameterized by the embedding of the lattice
19;3) and an integral constructed out of a harmonic 2-form on the K3.
While we have exhibited some of the powers of the crossing equation based on the
BPS spectrum and OPEs in the K3 CFT over the entire moduli space. We would like to
understand to what extent our bootstrap bounds can be saturated, away from free orbifold
and cigar points in the moduli space. In particular, it would be interesting to compare
with results from conformal perturbation theory.
Apart from a few basic vanishing results, the OPEs of the 14 -BPS primaries remain
largely unexplored. Neither have we investigated the torus correlation functions, which
should provide further constraints on the non-BPS spectrum. Note that there are certain
integrated torus four-point functions, analogous to Aijkl and Bij;kl, that can be determined
as exact functions of the moduli, by expanding the result of [49] perturbatively in the type
IIB string coupling.
There are a number of important generalizations of our bootstrap analysis that will be
left to future work. One of them is to derive bootstrap bounds on the non-BPS spectrum
of (2; 2) superconformal theories, with input from the known chiral ring relations. To do
so, we will need to extend the results of section 4 to ones that express a more general set
central charge and shifted weights). These relations can be extracted from BPS correlators
be presented in detail elsewhere.
Another generalization would be to extend our analysis to (4; 4) superconformal
the2, and use it to understand the
appearance of a continuous spectrum in the D1-D5 CFT at various singular points on its
block and bosonic Virasoro blocks, to the k0
2 case. This is currently under investigation.
Finally, our numerical bounds on b crt seem to allow for the possibility of having an
arbitrarily large four-point function when
& 1 in 3 spacetime dimensions and
4 spacetime dimensions. We are not aware of an example of such a CFT. It is conceivable
that such a CFT will be ruled out by unitarity constraints from other correlation functions,
but this remains to be seen.
Acknowledgments
We would like to thank Chris Beem, Clay Cordova, Thomas Dumitrescu, Matthew
Headrick, Christoph Keller, Petr Kravchuk, Sarah Harrison, Juan Maldacena, Hirosi Ooguri,
Nati Seiberg, Steve Shenker, Cumrun Vafa, Shing-Tung Yau, and Alexander Zhiboedov
for discussions. We would like to thank the workshop \From Scattering Amplitudes to
the Conformal Bootstrap" at Aspen Center for Physics, the Simons Summer Workshop in
Mathematics and Physics 2015, and the workshop Amplitudes in Asia 2015, for hospitality
during the course of this work. DSD is supported by DOE grant DE-SC0009988 and a
William D. Loughlin Membership at the Institute for Advanced Study. YW is supported
in part by the U.S. Department of Energy under grant Contract Number DE-SC00012567.
XY is supported by a Simons Investigator Award from the Simons Foundation, and in part
by DOE grant DE-FG02-91ER40654.
In this appendix we compare the proposed exact formula for the integrated four-point
free orbifold CFT. The twist elds of the latter are associated with the 16 Z2 xed points
on the T 4. We will focus on the case where i; j; k; l label the same Z2 xed point (denote
A1111 = 6 2
Here is related to the cross ratio z by the mapping
e4;4 is the Narain lattice associated with the T 4 with all radii rescaled by p2. The factor 6
comes from the integration over the fundamental domain of (2), which consists of 6 copies
= iF (1 z)=F (z), F = 2F1( 21 ; 12 ; 1; z).
to a divergent integral, which is regularized by analytic continuation in the Mandelstam
variables s; t; u and then dropping the polar terms in the s; t; u ! 0 limit as before.
We will take the original T 4 (before orbifolding) to be a rectangular torus with radii
Ri, i = 1;
; 4. To compare (A.1) with our exact formula for Aijkl as a function of the
K3 moduli, we need to construct the lattice embedding
R20;4 that corresponds to
R16. Let (ui; vi)
wR the projection of a vector w 2 R20;4 in the positive and negative subspaces, R20 and
R4 respectively. We can write juiLj = juiRj =
q 20h R1h , jviLj = jviRj = q 210h Rih. Note that,
i
importantly, Rih are not to be identi ed with Ri. Rather, they are related by (see (2.5),
(2.6) and footnote 2 of [90])
Let Ai be the following vectors in the R16,
Ri =
2R1hR2hR3hR4h
A1 =
A2 =
A3 =
A4 =
(1; 1; 1; 1; 1; 1; 1; 1; 0; 0; 0; 0; 0; 0; 0; 0);
(1; 1; 1; 1; 0; 0; 0; 0; 1; 1; 1; 1; 0; 0; 0; 0);
(1; 1; 0; 0; 1; 1; 0; 0; 1; 1; 0; 0; 1; 1; 0; 0);
(1; 0; 1; 0; 1; 0; 1; 0; 1; 0; 1; 0; 1; 0; 1; 0):
the following generators
Note that Ai Aj = 1 + ij . Let
16 be the root plus chiral spinor weight lattice of
SO(32) embedded in the R16, generated by the root vectors (0;
; 0; 1; 1; 0;
) with even number of minuses. Now
20;4 can be constructed as the span of
X Ai Aj
One can verify that this lattice is indeed even and unimodular.34
In the large Ri limit, we can approximate the theta function of 20;4 as
Note that the -dependence of
evaluate the integral
16 is entirely through the factor e 2 2 y2 . We can then
Qi4=1 Rih
16 (yj ; ) =
R1R2R3R4
e 2 2 y2 factor dropped, due to the 2 !
2d 1d 2. In the last line, the holomorphic function e 16 (yj ) is
16 with the
Furthermore, only the y4 term is kept in the Laurent expansion in y, and in particular
the constant term 1 in the lattice sum in e 16 does not contribute. The only contribution
comes from the terms of order q in e 16 (yj ), giving
1 limit taken in going to the boundary of F .
In particular,
@y14 y=0 F
60 = 210 5
The factor 60 comes from the sum of (Ea e^1)6 for all root vectors Ea of so(32), with
e^1 = (1; 0;
34This lattice can also be used to describe the compacti cation of SO(32) heterotic string on a rectangular
T 4 with radii Rih and Wilson line turned on. This can be seen from the large Rih limit, where ui and vi are
approximations to primitive lattice vectors. Note that in the opposite limit, say small R1h, u21 and 2v1 are
approximations to primitive lattice vectors. This means that the T-dual E8
E8 heterotic string lives on a
Note that in the large radii limit, the four-point function of twist elds at a given cross
ratio in the free orbifold CFT diverges like the volume, as is Aijkl.35 Comparison with (A.1)
then xes the overall normalization of Aijk` as a function of moduli to be that of (5.2).
Conformal blocks under the q-map
complex moduli
of the T 2 is related to the cross ratio z of the four punctures by a map
F (z) = 2F1(1=2; 1=2; 1jz):
lies in the upper half plane, the \nome" de ned as
c = 1 + 6Q2;
Q = b + ;
hm;n =
m;n =
then H( i2; hjq) satis es Zamolodchikov's recurrence relation [8, 50]
H( i2; hjq) = 1 +
qmnRm;n(f ig) H( i2; hm;n + mnjq);
exp(i (z))
has the property that its value lies inside the unit disk. We shall simply refer to this map
z ! q(z) as the q-map. The q-map has a branch cut at (1; 1); the value of q(z) covered
by one branch is shown in gure B.2, and crossing to other branches brings us outside this
eye-shaped region. Also shown are the regions D1, D2, D3 de ned by
If we de ne
each of which contains two fundamental domains of the S3 crossing symmetry group.
The holomorphic Virasoro block for a four-point function hO1(z)O2(0)O3(1)O4(1)i
with central charge c, external weights hi, and intermediate weight h has the following
and Rm;n(f ig) are given by
Rm;n(f ig) = 2
where hm;n are the conformal weights of degenerate representations of the Virasoro algebra,
35This is to be contrasted with the large volume limit of a smooth K3, where the four-point function of
BPS operators remain
nite at generic cross ratio, while Aijkl diverges like the square root of volume.
The product of (r; s) is taken over
and the product of (k; `) is taken over
r =
s =
n + 1; n + 3;
k =
` =
n + 1; n + 2;
1, the prefactor multiplying H( i2; hjq) gives the large h asymptotics of
the conformal block. The superconformal block FhR(z) which is related to the Virasoro
conformal block via (3.2) also has the same large h asymptotics.
More on the integrated four-point function Aijk`
The purpose of this appendix is the explain how knowing the value of the integrated
fourpoint function Aijk` can improve the bootstrap bounds on the spectrum. We rst explain
the problem with naively incorporating Aijk` into semide nite programming, and then
discuss two solutions. The
rst way is to cleverly use crossing symmetry to choose an
appropriate region over which to integrate the conformal blocks. The second way is to use
A1111 indirectly by bounding it above by the four-point function evaluated at the crossing
Conformal block expansion
We can write the integrated four-point function Aijk` as
Aijk` = lim
iRR(z; z) jRR(0) kRR(1) `RR(1)
+ 2 ln ( ij k` + ik j` + i` jk) :
In expressing the four-point function of the 12 -BPS operators in terms of conformal blocks,
we would like the divergence in the z-integral to appear in the identity conformal block
alone, so that the regularization can be performed on the identity block contribution alone.
This can be achieved by dividing the integral over the z-plane into the contributions from
three regions D1, D2 and D3 de ned in (B.3). Note that regions D2 and D3 can be mapped
from D1 by z 7! 1=z and z 7! 1=(1
z), respectively. We have
Aijk` =
where the constant C0 is given by
+ (j $ k) + (j $ `) ;
non-BPS O
CijOCk`OFhRL (z)FhRR (z)
C0 = lim
Now the integral in the domain D1 can be performed term by term in the summation over
superconformal blocks. De ne the constant A0 and the function A(hL; hR) by
A0 can also be obtained as a limit of A( ; 0) by
We can now write
Aijk` =
A0 =
A( ; s) =
A0 = lim
CijOCk`OA( ; s) + (j $ k) + (j $ `):
Let us examine this equation for identical external operators
0 = (3A0
A1111) + 3
It takes the same form as the equations corresponding to acting linear functionals
0 =
m;n(H0(z; z)) +
Clearly, if we can nd a set of coe cients a and am;n such that
A1111) + X am;n m;n[H0(z; z)] > 0;
3aA( ; s) + X am;n m;n[H ;s(z; z)] > 0 for
are satis ed, then the gap in the non-BPS spectrum
in order to be consistent with the positivity of C121O.
Despite the additional freedom of a, this naive incorporation of A1111 does not improve
the bound, for the following reasons. As explained at the end of appendix B, the
holomorphic superconformal block FhR(z) asymptotes to (16q(z))h at large h. This means that for
any spin s, the integrated block A( ; s) at large
is dominated by the integration near
the maximal value of jq(z)j in the domain D1, which is at (see gure 3),
gap must be bounded above by b gap,
and therefore has the asymptotic behavior
or q(z ) =
( 1)s=2(16e 23 ) ;
In comparison,
, whose sign oscillates with s. Thus positivity at large
cannot be improved despite specifying A1111.
forces a = 0, and b gap
One may wonder if we can choose a di erent region (that also consists of two
fundamental domains of the S3 crossing group) to integrate in, so that the leading large
behavior of the integrated block is (16e
) , same as
m;n(H ;s(z; z)). This is not
possible, because z
= 12
23 i at most exchange with each other under crossing. However, we
can integrate over a larger region D0 whose maximum jqj value is on the real axis (to avoid
the sign oscillation), and map the extra region D0 n D1 that needs to be subtracted o via
crossing to a region E inside D1. We thus have an equation for A1111 related to the naive
conformal block expansion by the replacement of D1 ! D0 n E as the integration region.
We are free to choose D0, but in the end the bootstrap bound should not be sensitive
to the choice. Let D0 be symmetric under q !
q and q ! q, so that it su ces to specify
D0 in the rst quadrant in the q-plane, or equivalently within the strip 0
-plane (recall q(z) = ei (z)). In this strip, the region D1 is bounded below by j j = 1. A
p
choice of D0 is the region bounded below by the lower arc of j
The corresponding region E is then the part of D1 that satis es j + 1j
21 j = 23 , with qmax = e p2 .
To perform semide nite programming e ciently, it is desirable to factor out certain
positive factors, including the exponential dependence on
, and just work with
polynomials. Our strategy is to factor out (16e
) , and approximate (16e ) A( ; s) by a
rational function in
, that works well up to a value beyond which A( ; s) is completely
dominated by the asymptotic (16qmax)
factor. We further demand a > 0, and that the
rational approximation be strictly bounded above by the actual value, so that the bound
can only be stronger as we improve the rational approximation to work well in a larger
range of .
An alternative is to use A1111 indirectly by bounding A1111 above by the four-point function
has the same large
asymptotics (16e
m;n(H0(z; z)), and the sign does not
in detail in section 7.2. This section is devoted to proving the inequality between A1111
and f (1=2).
function f (z; z) in the form (see (7.8) of [53])
with (z)
z)) 31 3(q) 2. The functions gh(q) take the form
where, importantly, the coe cients an are non-negative.
f (z; z) = j (z)j2 X
ghL (q)ghR (q);
gh(q) = qh 16 X anqn;
A1111 = 3A0 + 3
= 3A0 + 3
(hL;hR)6=(0;0)
ghL (q)ghR (q)
(hL;hR)6=(0;0)
= 3A0 + M ( ) f
maximal value of jq(z)j, x
0:0265799. On the r.h.s. of the inequality, f0 is a constant de ned by
f0 =
and the function M ( ) is given by
For a general complex cross ratio z, let x be the real value between 0 and 1 such that
q(x) = jq(z)j. De ne r = minfx; 1
f (x) =
We now make the assumption that the non-BPS operators have scaling dimensions above
a nonzero gap
. As before, we can write the integrated four-point function A1111 as 3
while regularizing the integral of the identity block contribution, in the form
Note that M ( ) goes like
! 0 limit, with lim
crt and the divergence of the integrated four-point function A1111
Recall that
crt de ned in section 8 is the lowest scaling dimension at which either a
continuous spectrum develops or the structure constant diverges, as the CFT is deformed
to a singular point in its moduli space. In this appendix we will describe how to use crossing
symmetry to bootstrap an upper bound on
crt that is universal across the moduli space. In
M ( ) = 3
3 Z
z)] = 0:
We will consider functionals L acting G (z; z) with the following properties,
for some b crt. Note that b crt depends on the choice of the functional L. The signi cance
of b crt is that it implies the structure constants above b crt are bounded by those below,
C2 L( ) =
Assuming that the integrated four-point function A1111 diverges at some points on the
moduli space, we will show that for any choice of the functional L, we always have
particular we will show that if the integrated four-point function A1111 diverges somewhere
on the moduli space, then b crt
crt with b crt de ned in section 8.
; 20). Let us consider in particular the four-point
function of the same operator, say, 1RR,
h 1RR(z; z) 1RR(0) 1RR(1) 1RR(1)i =
X C2 F (z; z) ;
where we did not write out the sum over the spin explicitly but it will not a ect the
argument signi cantly. The conformal block has the following asymptotic growth
The crossing equation takes the following expression,
In this way we can bootstrap an upper bound on
crt by scanning through a large class of
functionals L.
associated b crt <
To prove our goal (D.6), we assume that there exists a functional L such that the
crt, and show that it leads to contradiction. By assumption the density
of the spectrum is bounded and the structure constants are nite for
hence the r.h.s. of (D.5) is nite,
C2 L( ) =
In the following we will try to bound the integrated four-point function
A1111 = lim
roughly by P
butions. On the other hand, we know A1111 diverges, for example, at the cigar CFT points,
> b crt C2 L( ), which is nite by assumption, plus some other nite
contriand hence the contradiction.
Let us now ll in the details of the proof. As discussed in appendix C, in the expression
for A1111, we break the integral on the z-plane into three di erent regions D1; D2; D3 (B.3)
that are mapped to each other under z ! 1
z and z ! 1=z. Since the four-point function
is crossing symmetric, we can focus on region D1 alone. This has the advantage that the
divergence in the z-integral only shows up in the identity block. We will cut a small disk
We start by noting a bound on the conformal blocks. The functionals L we consider
where q 1
and 0 such that
if jq(z)j < jq 12 j. We can always tune 0 to be arbitrarily small by taking c0 to be large.
Note however that strictly at
= b crt, we have L( b crt) = 0.
For z in region I and jq(z)j < jq 12 j, from (D.10) we have
In particular it is true for 0 < z < 1=2.
Next we want to argue that (D.11) is true for z in the whole region I. First we note
that we can write the four-point function as an expansion in z; z with non-negative coe
cients [52, 53]. By the Cauchy-Schwarz inequality, we have
f (jzj; jzj):
Note that jzj 2 [ 21 ; 1] for z in region I but jq(z)j > jq 12 j. Next, by crossing symmetry, we
have f (jzj; jzj) = f (1
where we have used the fact that 1
Hence the bound (D.11) is true for all z in region I. We can therefore bound the integrated
jzj 2 [ 0; 21 ] if z is in region I with jq(z)j > jq 12 j.
four-point function as
region I fjzj= 0g
For integration inside the disk, we have to regularize the contribution from the identity
fjzj= 0g
reg. = c2 + c3
where we have assumed there is a gap in the spectrum. c2 is a moduli-independent constant
coming from the regularized identity block contribution.
Let us inspect every term in (D.14) and (D.15). First we tune 0 such that b crt + 0 is
crt, possibly at the price of having larger c0. After doing so, terms involving sums
nite by our assumption that the density of the spectrum is
bounded and the structure constants are nite for this range of
. On the other hand, for
terms involving sum of
above b crt + 0, they are both of the form
which is bounded from above by the l.h.s. of (D.7). Hence the l.h.s. of (D.14) and (D.15)
are both bounded. It follows that A1111 < 1 under the assumption that b crt <
which is a contradiction, say, at the cigar point. Thus we have proved our goal (D.6).
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