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

Journal of High Energy Physics, May 2017

We study two-dimensional (4, 4) superconformal field theories of central charge 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 \( \mathcal{N} \) = 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 find numerically an upper bound on this gap that is saturated by the A 1 \( \mathcal{N} \) = 4 cigar CFT. We also derive an analytic upper bound on the first nonzero eigenvalue of the scalar Laplacian on K3 in the large volume regime, that depends on the K3 moduli data. As two byproducts, we find an exact equivalence between a class of BPS \( \mathcal{N} \) = 2 superconformal blocks and Virasoro conformal blocks in two dimensions, and an upper bound on the four-point functions of operators of sufficiently low scaling dimension in three and four dimensional CFTs.

A PDF file should load here. If you do not see its contents the file may be temporarily unavailable at the journal website or you do not have a PDF plug-in installed and enabled in your browser.

Alternatively, you can download the file locally and open with any standalone PDF reader:

https://link.springer.com/content/pdf/10.1007%2FJHEP05%282017%29126.pdf

\( \mathcal{N} \) = 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). Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. Fiz 66 (1974) 23 [INSPIRE]. conformally covariant operator product expansion, Annals Phys. 76 (1973) 161 [INSPIRE]. Two-Dimensions, Nucl. Phys. B 247 (1984) 83 [INSPIRE]. [INSPIRE]. [7] P. Bouwknegt and K. Schoutens, W symmetry in conformal eld theory, Phys. Rept. 223 [8] A.B. Zamolodchikov and A.B. Zamolodchikov, Structure constants and conformal bootstrap in Liouville eld theory, Nucl. Phys. B 477 (1996) 577 [hep-th/9506136] [INSPIRE]. Nucl. Phys. B 546 (1999) 390 [hep-th/9712256] [INSPIRE]. Phys. B 571 (2000) 555 [hep-th/9906215] [INSPIRE]. [hep-th/0104158] [INSPIRE]. 4D CFT, JHEP 12 (2008) 031 [arXiv:0807.0004] [INSPIRE]. Phys. Rev. D 80 (2009) 045006 [arXiv:0905.2211] [INSPIRE]. Theories, JHEP 05 (2011) 017 [arXiv:1009.2087] [INSPIRE]. (2012) 110 [arXiv:1109.5176] [INSPIRE]. [arXiv:1203.6064] [INSPIRE]. (2014) 091 [arXiv:1307.6856] [INSPIRE]. Lett. 111 (2013) 071601 [arXiv:1304.1803] [INSPIRE]. bootstrap, JHEP 03 (2016) 183 [arXiv:1412.7541] [INSPIRE]. the 3d Ising Model with the Conformal Bootstrap II. c-Minimization and Precise Critical Exponents, J. Stat. Phys. 157 (2014) 869 [arXiv:1403.4545] [INSPIRE]. dimensions, JHEP 09 (2014) 143 [arXiv:1406.4814] [INSPIRE]. 3d Superconformal Bootstrap, JHEP 03 (2015) 130 [arXiv:1412.0334] [INSPIRE]. Phys. Rev. D 91 (2015) 086014 [arXiv:1412.7746] [INSPIRE]. Dimensions, arXiv:1412.6549 [INSPIRE]. Symmetries and the Conformal Bootstrap, JHEP 01 (2016) 110 [arXiv:1507.04424] [INSPIRE]. 3D Fermions, JHEP 03 (2016) 120 [arXiv:1508.00012] [INSPIRE]. [28] C. Beem, M. Lemos, L. Rastelli and B.C. van Rees, The (2; 0) superconformal bootstrap, [INSPIRE]. [arXiv:1510.03866] [INSPIRE]. [arXiv:0902.2790] [INSPIRE]. Math. Phys. 324 (2013) 107 [arXiv:1209.4649] [INSPIRE]. (2015) 124 [arXiv:1506.08407] [INSPIRE]. linear -models, JHEP 02 (2015) 110 [arXiv:1406.6342] [INSPIRE]. B 303 (1988) 286 [INSPIRE]. Compacti cation on Manifolds with SU(N ) Holonomy, Nucl. Phys. B 315 (1989) 193 Int. J. Mod. Phys. A 6 (1991) 1749 [INSPIRE]. [hep-th/9912067] [INSPIRE]. Models, Nucl. Phys. B 296 (1988) 757 [INSPIRE]. Spaces, arXiv:0803.0377 [INSPIRE]. 014 [hep-th/0502048] [INSPIRE]. Num. Theor. Phys. 6 (2012) 1 [arXiv:1106.4315] [INSPIRE]. symmetry, JHEP 02 (2014) 022 [arXiv:1309.4127] [INSPIRE]. Phys. B 463 (1996) 55 [hep-th/9511164] [INSPIRE]. theories, JHEP 01 (2005) 027 [hep-th/0411041] [INSPIRE]. Unreasonable E ectiveness of 6D SYM), JHEP 12 (2014) 176 [arXiv:1407.7511] [INSPIRE]. [49] Y.-H. Lin, S.-H. Shao, Y. Wang and X. Yin, Supersymmetry Constraints and String Theory [50] A.B. Zamolodchikov, Conformal symmetry in two-dimensions: an explicit recurrence formula for the conformal partial wave amplitude, Commun. Math. Phys. 96 (1984) 419 [INSPIRE]. Algebras in Two-Dimensions, Phys. Lett. B 184 (1987) 191 [INSPIRE]. Phys. Lett. B 200 (1988) 315 [INSPIRE]. and the attractor mechanism, JHEP 03 (2009) 030 [arXiv:0809.0507] [INSPIRE]. primary 3-point functions, JHEP 07 (2012) 137 [arXiv:1203.1036] [INSPIRE]. Covariant Approaches to Superconformal Blocks, JHEP 08 (2014) 129 [arXiv:1402.1167] [INSPIRE]. general scalar operators, JHEP 08 (2014) 049 [arXiv:1404.5300] [INSPIRE]. Model, JHEP 11 (2014) 109 [arXiv:1406.4858] [INSPIRE]. [INSPIRE]. and an IR obstruction to UV completion, JHEP 10 (2006) 014 [hep-th/0602178] [INSPIRE]. eld theories, Int. J. Mod. Phys. A 9 (1994) 3007 [hep-th/9304135] [INSPIRE]. as mirror symmetry, JHEP 08 (2001) 045 [hep-th/0104202] [INSPIRE]. [1] A. Polyakov , Nonhamiltonian approach to conformal quantum eld theory, Zh . Eksp. Teor. [2] S. Ferrara , A.F. Grillo and R. Gatto , Tensor representations of conformal algebra and [3] G. Mack , Duality in quantum eld theory, Nucl . Phys . B 118 ( 1977 ) 445 [INSPIRE]. [4] A.A. Belavin , A.M. Polyakov and A.B. Zamolodchikov , In nite Conformal Symmetry in Two-Dimensional Quantum Field Theory, Nucl . Phys . B 241 ( 1984 ) 333 [INSPIRE]. [5] V.G. Knizhnik and A.B. Zamolodchikov , Current Algebra and Wess-Zumino Model in [6] D. Gepner and E. Witten , String Theory on Group Manifolds , Nucl. Phys . B 278 ( 1986 ) 493 [10] J. Teschner , Operator product expansion and factorization in the H3+-W ZN W model, Nucl . [11] J. Teschner , Liouville theory revisited, Class . Quant. Grav. 18 ( 2001 ) R153 [12] R. Rattazzi , V.S. Rychkov , E. Tonni and A. Vichi , Bounding scalar operator dimensions in [13] V.S. Rychkov and A. Vichi , Universal Constraints on Conformal Operator Dimensions , [14] D. Poland and D. Simmons-Du n, Bounds on 4D Conformal and Superconformal Field [15] D. Poland , D. Simmons-Du n and A. Vichi , Carving Out the Space of 4D CFTs , JHEP 05 [16] S. El-Showk , M.F. Paulos , D. Poland , S. Rychkov , D. Simmons-Du n and A. Vichi , Solving the 3D Ising Model with the Conformal Bootstrap , Phys. Rev . D 86 ( 2012 ) 025022 [17] F. Kos , D. Poland and D. Simmons-Du n, Bootstrapping the O(N ) vector models , JHEP 06 [18] C. Beem , L. Rastelli and B.C. van Rees , The N = 4 Superconformal Bootstrap , Phys. Rev. [19] C. Beem , M. Lemos , P. Liendo , L. Rastelli and B.C. van Rees , The N = 2 superconformal [20] S. El-Showk , M.F. Paulos , D. Poland , S. Rychkov , D. Simmons-Du n and A. Vichi , Solving [21] S.M. Chester , J. Lee , S.S. Pufu and R. Yacoby , The N = 8 superconformal bootstrap in three [22] S.M. Chester , J. Lee , S.S. Pufu and R. Yacoby , Exact Correlators of BPS Operators from the [23] S.M. Chester , S.S. Pufu and R. Yacoby , Bootstrapping O (N ) vector models in 4 < d < 6, [24] J.- B. Bae and S.-J. Rey , Conformal Bootstrap Approach to O(N ) Fixed Points in Five [25] S.M. Chester , S. Giombi , L.V. Iliesiu , I.R. Klebanov , S.S. Pufu and R. Yacoby , Accidental [26] L. Iliesiu , F. Kos , D. Poland , S.S. Pufu , D. Simmons-Du n and R. Yacoby, Bootstrapping [27] F. Kos , D. Poland , D. Simmons-Du n and A. Vichi , Bootstrapping the O(N) Archipelago, [29] M. Lemos and P. Liendo , Bootstrapping N = 2 chiral correlators , JHEP 01 ( 2016 ) 025 [30] S. Hellerman , A Universal Inequality for CFT and Quantum Gravity , JHEP 08 ( 2011 ) 130 [31] C.A. Keller and H. Ooguri , Modular Constraints on Calabi-Yau Compacti cations , Commun. [32] M.- A. Fiset and J. Walcher , Bounding the Heat Trace of a Calabi-Yau Manifold , JHEP 09 [33] J.A. Harvey , S. Lee and S. Murthy , Elliptic genera of ALE and ALF manifolds from gauged [34] N. Seiberg , Observations on the Moduli Space of Superconformal Field Theories, Nucl. Phys. [35] T. Eguchi , H. Ooguri , A. Taormina and S.-K. Yang , Superconformal Algebras and String [36] S. Cecotti , N = 2 Landau-Ginzburg versus Calabi-Yau -models: Nonperturbative aspects , [37] S. Cecotti and C. Vafa , Topological antitopological fusion, Nucl . Phys . B 367 ( 1991 ) 359 [38] P.S. Aspinwall and D.R. Morrison , String theory on K3 surfaces, hep-th/9404151 [INSPIRE]. [39] W. Nahm and K. Wendland , A Hiker's guide to K3: Aspects of N = ( 4 ; 4) superconformal eld theory with central charge c = 6, Commun . Math. Phys. 216 ( 2001 ) 85 [41] M.R. Gaberdiel , S. Hohenegger and R. Volpato , Symmetries of K3 -models, Commun. [42] M.R. Gaberdiel , A. Taormina , R. Volpato and K. Wendland , A K3 -model with Z82 : M20 [43] H. Ooguri and C. Vafa , Two-dimensional black hole and singularities of CY manifolds, Nucl . [44] T. Eguchi and Y. Sugawara , Conifold type singularities, N = 2 Liouville and SL (2 : R)=U(1) [45] T. Eguchi , Y. Sugawara and A. Taormina , Modular Forms and Elliptic Genera for ALE [46] S. Ribault and J. Teschner , H3 +-W ZN W correlators from Liouville theory, JHEP 06 ( 2005 ) [47] C.-M. Chang , Y.-H. Lin , S.-H. Shao , Y. Wang and X. Yin , Little String Amplitudes (and the [48] E. Kiritsis , N.A. Obers and B. Pioline , Heterotic/type-II triality and instantons on K3 , [51] P.S. Aspinwall , Enhanced gauge symmetries and K3 surfaces, Phys . Lett . B 357 ( 1995 ) 329 [52] T. Hartman , S. Jain and S. Kundu , Causality Constraints in Conformal Field Theory , JHEP [53] J. Maldacena , D. Simmons-Du n and A. Zhiboedov , Looking for a bulk point , JHEP 01 [54] A. Schwimmer and N. Seiberg , Comments on the N = 2, N = 3, N = 4 Superconformal [55] T. Eguchi and A. Taormina , Unitary Representations of N = 4 Superconformal Algebra , [56] T. Eguchi and A. Taormina , Character Formulas for the N = 4 Superconformal Algebra , [57] J. de Boer , J. Manschot , K. Papadodimas and E. Verlinde , The chiral ring of AdS3=CF T2 [58 ] M. Baggio , J. de Boer and K. Papadodimas , A non-renormalization theorem for chiral [59] A.L. Fitzpatrick , J. Kaplan , Z.U. Khandker , D. Li , D. Poland and D. Simmons-Du n, [60] Z.U. Khandker , D. Li , D. Poland and D. Simmons-Du n , N = 1 superconformal blocks for [61] N. Bobev , S. El-Showk , D. Mazac and M.F. Paulos , Bootstrapping SCFTs with Four [62] D. Kutasov , Orbifolds and solitons, Phys. Lett . B 383 ( 1996 ) 48 [hep-th /9512145] [INSPIRE]. [63] A. Giveon and D. Kutasov , Little string theory in a double scaling limit , JHEP 10 ( 1999 ) [64] J. Teschner , On the Liouville three point function , Phys. Lett. B 363 (1995) 65 [65] E.W. Barnes , The theory of the double gamma function , Phil. Trans. Roy. Soc. Lond. A 196 [66] J.M. Maldacena and H. Ooguri , Strings in AdS3 and the SL(2; R) WZW model . Part 3. Correlation functions , Phys. Rev. D 65 ( 2002 ) 106006 [hep-th /0111180] [INSPIRE]. [67] T. Eguchi and A. Taormina , On the Unitary Representations of N = 2 and N = 4 [68] N. Berkovits and C. Vafa , N=4 topological strings, Nucl . Phys . B 433 ( 1995 ) 123 [69] I. Antoniadis , S. Hohenegger and K.S. Narain , N = 4 Topological Amplitudes and String E ective Action, Nucl . Phys . B 771 ( 2007 ) 40 [hep-th /0610258] [INSPIRE]. [70] R. Dijkgraaf , E.P. Verlinde and H.L. Verlinde , C = 1 Conformal Field Theories on Riemann [71] L.J. Dixon , D. Friedan , E.J. Martinec and S.H. Shenker , The Conformal Field Theory of Orbifolds, Nucl . Phys . B 282 ( 1987 ) 13 [INSPIRE]. [72] D. Gluck , Y. Oz and T. Sakai , N = 2 strings on orbifolds , JHEP 08 ( 2005 ) 008 [73] O. Aharony , B. Fiol , D. Kutasov and D.A. Sahakyan , Little string theory and heterotic/type-II duality, Nucl . Phys . B 679 ( 2004 ) 3 [hep -th/0310197] [INSPIRE]. [74] O. Aharony , A. Giveon and D. Kutasov , LSZ in LST, Nucl. Phys . B 691 ( 2004 ) 3 [75] F. Kos , D. Poland and D. Simmons-Du n, Bootstrapping Mixed Correlators in the 3D Ising [76] D. Simmons-Du n, A Semide nite Program Solver for the Conformal Bootstrap , JHEP 06 [77] D. Pappadopulo , S. Rychkov , J. Espin and R. Rattazzi , OPE Convergence in Conformal [78] F. Caracciolo , A. Castedo Echeverri , B. von Harling and M. Serone , Bounds on OPE Coe cients in 4D Conformal Field Theories , JHEP 10 ( 2014 ) 020 [arXiv:1406.7845] [80] F. Caracciolo and V.S. Rychkov , Rigorous Limits on the Interaction Strength in Quantum [81] M. Hogervorst and S. Rychkov , Radial Coordinates for Conformal Blocks, Phys. Rev. D 87 [82] H. Kim , P. Kravchuk and H. Ooguri , Re ections on Conformal Spectra , JHEP 04 ( 2016 ) 184 [83] D. Anselmi , M. Billo , P. Fre , L. Girardello and A. Za aroni , ALE manifolds and conformal [84] M.R. Douglas , R.L. Karp , S. Lukic and R. Reinbacher , Numerical Calabi -Yau metrics, [85] M. Headrick and A. Nassar , Energy functionals for Calabi-Yau metrics, Adv . Theor. Math. [86] P.S. Aspinwall , K3 surfaces and string duality , hep-th/ 9611137 [INSPIRE]. [87] S.-Y. Cheng , Eigenfunctions and eigenvalues of laplacian , Proc. Sympos. Pure Math. 27 [88] P. Li and S.-T. Yau , Estimates of eigenvalues of a compact riemannian manifold , Proc. [89] K. Hori and A. Kapustin , Duality of the fermionic 2-D black hole and N = 2 Liouville theory [90] O. Bergman and M.R. Gaberdiel , NonBPS states in heterotic type IIA duality , JHEP 03


This is a preview of a remote PDF: https://link.springer.com/content/pdf/10.1007%2FJHEP05%282017%29126.pdf

Ying-Hsuan Lin, Shu-Heng Shao, David Simmons-Duffin, Yifan Wang, Xi Yin. \( \mathcal{N} \) = 4 superconformal bootstrap of the K3 CFT, Journal of High Energy Physics, 2017, 126, DOI: 10.1007/JHEP05(2017)126