Supersymmetry Constraints and String Theory on K3

Journal of High Energy Physics, Dec 2015

Abstract We study supervertices in six dimensional (2, 0) supergravity theories, and derive supersymmetry non-renormalization conditions on the 4- and 6-derivative four-point couplings of tensor multiplets. As an application, we obtain exact non-perturbative results of such effective couplings in type IIB string theory compactified on K3 surface, extending previous work on type II/heterotic duality. The weak coupling limit thereof, in particular, gives certain integrated four-point functions of half-BPS operators in the nonlinear sigma model on K3 surface, that depend nontrivially on the moduli, and capture worldsheet instanton contributions.

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Supersymmetry Constraints and String Theory on K3

JHE Supersymmetry Constraints and String Theory on K3 Ying-Hsuan Lin 0 2 Shu-Heng Shao 0 2 Yifan Wang 0 1 Xi Yin 0 2 0 Cambridge , MA 02138 U.S.A 1 Center for Theoretical Physics, Massachusetts Institute of Technology 2 Je erson Physical Laboratory, Harvard University We study supervertices in six dimensional (2; 0) supergravity theories, and derive supersymmetry non-renormalization conditions on the 4- and 6-derivative four-point couplings of tensor multiplets. As an application, we obtain exact non-perturbative results of such e ective couplings in type IIB string theory compacti ed on K3 surface, extending previous work on type II/heterotic duality. The weak coupling limit thereof, in particular, gives certain integrated four-point functions of half-BPS operators in the nonlinear sigma model on K3 surface, that depend nontrivially on the moduli, and capture worldsheet instanton contributions. Superstrings and Heterotic Strings; Scattering Amplitudes; Supergravity - HJEP12(05)4 Models 2.1 2.2 2.3 4.1 4.2 5.1 5.2 1 Introduction 2 Supervertices in 6d (2; 0) supergravity 6d (2; 0) Super-spinor-helicity formalism Supervertices for tensor multiplets Supervertices for supergravity and tensor multiplets 3 4 Di erential constraints on f (4) and f (6) couplings An example of f (4) and f (6) from type II/heterotic duality Type II/heterotic duality Heterotic string amplitudes and the di erential constraints 4.2.1 4.2.2 One-loop four-point amplitude Two-loop four-point amplitude 5 Implications of f (4) and f (6) for the K3 CFT Reduction to the K3 CFT moduli space A1 ALE limit 6 Discussions A Explicit check of the di erential constraints A.1 Four-derivative coupling f (4) A.2 Six-derivative coupling f (6) B Relation to 5d MSYM amplitudes B.1 Four-derivative coupling f (4) B.2 Six-derivative coupling f (6) We will focus on the 4 and 6-derivative couplings of tensor multiplets, of the schematic form fa(4bc)d( )HaHbHcHd and fa(6b;)cd( )D2(HaHb)HcHd; where stands for the massless scalar moduli elds, that parameterize the moduli space [11] M = O( 21;5)nSO(21; 5)=(SO(21) SO(5)); and we have omitted the contraction of the Lorentz indices on the self-dual tensor elds Ha in the tensor multiplets (not to be confused with the anti-self-dual tensor elds in the supergravity multiplet), a = 1; fa(6b;)cd( ). These equations are of the schematic form By consideration of the factorization of six-point superamplitudes through graviton and tensor poles, we derive second order di erential equations that constrain fa(4bc)d( ) and fi(j4k)` ! gIVIBK`s43 Aijk`('); one-loop and two-loop heterotic string amplitudes, with the results fa(4bc)d = fa(6b;)cd = IK JL + IL JK Z d 2 (yj ; ) ( ) ; Z Y d 2 IJ (det Im ) 2 10( ) (yj ; ) is the theta function of the (yj ; ) is an analogous genus two theta function. The precise expressions of these theta functions will be given later. The above two expressions depend on the embedding of the lattice into R21;5 through the theta functions, and the space of inequivalent embeddings is the same as the moduli space M (1.1) of the 6d (2; 0) supergravity. cusp form of SL(2; Z), and 10( ) is the weight 10 Igusa cusp form of Sp(4; Z). The result for the 4-derivative term f (4) has previously been obtained in [12]. ( ) = 24( ) is the weight 12 We will verify, through rather lengthy calculations, that (1.3) indeed obey second order di erential equations of the form (1.2), and x the precise numerical coe cients in these equations. While the expressions (1.3) for the coupling coe cients f (4) and f (6) are fully nonperturbative in type IIB string theory, the results are nontrivial even at string tree-level. For instance, in the limit of weak IIB string coupling gIIB, f (4) reduces to (1.1) (1.2) (1.3) (1.4) HJEP12(05)4 Z d2z 2 ; 20), the moduli of the 2d (4; 4) CFT given by the supersymmetric nonlinear sigma model on K3 (we will refer to this as the K3 CFT). From the point of view of the worldsheet CFT, we can express Aijk`(') as an integrated four-point function of marginal BPS operators of the K3 CFT, through the expansion j j iRR(z) are the weight ( 14 ; 14 ) RR sector superconformal primaries in the R-symmetry singlet, related to the NS-NS sector weight ( 12 ; 21 ), exactly marginal, superconformal primaries by spectral ow. The z-integral is de ned using Gamma function regularization, or equivalently, analytic continuation in s and t from the domain where the integral converges. While Aijk` gives the tree-level contribution to f (4), Bij;k` captures the tree-level contribution to f (6). Note that, in contrast to the Riemannian curvature of the Zamolodchikov metric [13], which is contained in a contact term of the four-point function [14], Aijk` and Bij;k` are determined by the non-local part of the four-point function and do not involve the contact term. Unlike the Zamolodchikov metric which has constant curvature on the moduli space of K3 (with the exception of orbifold type singularities), Aijk` and Bij;k` are nontrivial functions of the moduli. In particular, the latter coe cients blow up at the points of the moduli space where the CFT becomes singular, corresponding to the K3 surface developing an ADE type singularity, with zero B- eld ux through the exceptional divisors. We can give a simple formula for Aijk` in the case of A1 ALE target space, which may be viewed as a certain large volume limit of the K3. In this case, the indices i; j; k; ` only take a single value (denoted by 1), corresponding to a single multiplet that parameterizes the 4-dimensional moduli space MA1 = R 3 Z2 S1 : MA1 has two orbifold xed points by the Z2 quotient, one of which corresponds to the C2=Z2 free orbifold CFT, whereas the other corresponds to a singular CFT, singular in the sense of a continuous spectrum, that is described by the N = 4 A1 cigar CFT [15{17]. While the Zamolodchikov metric does not exhibit any distinct feature between these two points on the moduli space, the integrated four-point function A1111 does. The latter is a harmonic function on MA1 , is nite at the free orbifold point, but blows up at the A1 cigar point. When the A1 singularity is resolved, in the limit of large area of the exceptional divisor, we nd that A1111 receives a one-loop contribution in 0, plus worldsheet instanton contributions (5.21). The paper is organized as follows. In section 2 we set up the super-spinor-helicity formalism in 6d (2; 0) supergravity and classify the supervertices of low derivative orders. In section 3, we derive the di erential equation constraints on the four-point 4- and 6derivative coupling between the tensor multiplets based on the absence of certain six-point supervertices, with some model-independent constant coe cients yet to be determined. { 3 { In section 4, using type II/heterotic duality, we obtain the exact non-perturbative 4- and 6-derivative couplings in type IIB string theory on K3. We verify that these couplings indeed satisfy the di erential equations and x the constant coe cients in these equations. In section 5, we consider the weak coupling limit of the above results, which gives the integrated four-point function of BPS primaries in the K3 CFT, with an explicit dependence on the moduli space. We also consider the A1 ALE sigma model limit of the K3 CFT and study the 4-derivative couplings in that limit. Supervertices in 6d (2; 0) supergravity 6d (2; 0) Super-spinor-helicity formalism Following [18{20], we adopt the convention for 6d spinor-helicity variables pAB = A B ; pAB 2 1 ABCDpCD = eA _ eB _ _ _ ; and de ne Grassmannian variables I and e_ I , where the lower and upper A; B are SO(5; 1) chiral and anti-chiral spinor indices respectively, ( ; _ ) are SU(2) SU(2) little group indices, and I = 1; 2 is an auxiliary index which may be identi ed with the spinor index of an SO(3) subgroup of the SO(5) R-symmetry group. Let us represent the 1-particle states in the (2; 0) tensor multiplet and the (2; 0) supergravity multiplet as polynomials in the Grassmannian variables I and ~ _ I . The 1particle states of the (2; 0) tensor multiplet transform in the following representations of the SU(2) SU(2) little group, (2.1) (2.2) (2.3) (2.4) (2.5) These 1-particle states can be represented collectively as a polynomial up to degree 4 in , but with no ~. In particular, the monomial to the self-dual two form (3; 1) and the monomials 1; I J I J IJ corresponds ; 4 correspond to the 5 scalars (1; 1). The 1-particle states of the (2; 0) supergravity multiplet, on the other hand, transform in the following representations of the SU(2) SU(2) little group, IJ corresponds to the graviton (3; 3) and the , and 4 correspond to the 5 anti-self-dual tensor elds (1; 3). { 4 { The 16 supercharges are represented on 1-particle states as They obey the supersymmetry algebra The 10 SO(5) R-symmetry generators are qAI = A I ; fqAI ; qBJ g = pAB IJ ; fq; qg = fq; qg = 0: J I when acting on 1-particle states. In an n-point scattering amplitude, we will associate to each particle spinor helicity variables iA ; ~iA _ and Grassmannian variables i I ; ~i _ I , with i = 1; ; n. Correspondingly we de ne the supercharges for each particle, qiAI = iA iI ; : The supercharges acting on the amplitude are represented by sums of the 1-particle representations QAI = X qiAI ; QAI = X qiAI ; and so are the R-symmetry generators i i X( i2)IJ ; J I : (2.6) (2.7) (2.8) (2.9) (2.10) (2.11) (2.12) (2.13) The solutions to the supersymmetry Ward identities can be expressed in terms of the super-spinor-helicity variables. If such expression is local in these variable, we call it a supervertex, otherwise it is a superamplitude. Among all the supervertices, the D-term type takes the form where 8(Q) = QA;I QAI , and P is a polynomial in the super-spinor-helicity variables i; ~i; i; ~i associated with the external particles labeled by i = 1; : : : ; n, that is Lorentz invariant and little group invariant. On the other hand, the F-term supervertices are of the form 8(Q)Q8P( i; ~i; i; ~i); 8(Q)F ( i; ~i; i; ~i); where F is a Lorentz invariant and little group invariant polynomial in the super-spinorhelicity variables that cannot be written in the D-term form [9, 21{24]. From momentum counting, we expect D-term supervertices in general to come at or above 8-derivative order. In the following subsections, we will focus on three- and four-point supervertices in the (2; 0) supergravity. We will start with supervertices involving tensor multiplets only, whose classi cation coincides with that of the (2; 0) SCFT on the tensor branch. We will then introduce couplings to the supergravity multiplet and classify the supervertices thereof. In particular, we will discover that the four-point D-term supervertices involving supergravitons do not appear until at 12-derivative order. { 5 { Among the four-point supervertices that only involve the (2; 0) tensor multiplets, the leading F-term ones arise at 4 and 6-derivative orders and take the form where QAI = Pi4=1 qiAI and 8(Q) = QA;I QAI . The coe cients f (4); f (6) are constant in s; t; u but functions of the moduli. Their dependence on the moduli is the main object of the current paper. The subscripts a; b; c; d label the 21 tensor multiplets. They contain the H4 and D2H4 couplings, respectively, where H denotes the self-dual three form eld strength in the 21 tensor multiplets. 8(Q)Q8P( i; ~i; i). For There are also four-point D-term supervertices of the form this expression to be non-vanishing, we need P 8to contain at least eight 's. On the other hand, by exchanging the order of 8(Q) and Q , we see that we cannot have more than eight 's in P because there are in total 4 4 's from the four 1-particle states. Hence the lowest derivative order D-term supervertices for tensor multiplets arise at 8-derivative (2.14) (2.15) (2.16) (2.17) order Pi<j i j 4 4 . though we could act Q on other little group singlets made out of eight i's, like for instance ( 12)IJ ( 22)IJ ( 32)KL( 42)KL, such expressions always turn out to be proportional to Next, we will show that three-point supervertices of tensor multiplets are absent. In general it is more intricate to write down the three-point supervertices due to the kinematic constraints,1 and we will work in a frame where the three momenta p1; p2; p3 lie in a null plane spanned by e0 + e1 (the 0 -direction) and e2 + ie3 (the 1 -direction). The null plane is equivalently speci ed by the linear operator, such that the spinor helicity variables associated with the momenta satisfy Nb = p1mp2n mn (p+)2 0 1 ; NbAB iB = 0; Nb AB eiB _ = 0: We write both the lower (chiral) and upper (anti-chiral) SO(5; 1) spinor index A as ( ) which represent spins on the 01 and 23 planes, while the spin in the 45 plane is xed by the 01 and 23 spins due to the chirality condition. For instance, we write iA and eiA _ as ei _ . By de nition, is0s1 (or eis_0s1 ) has charge s20 and s21 under the SO(1; 1)01 boost and SO(2)23 rotation in the 01 and 23 planes, respectively. Then by the chirality as i , 1As will be shown in this section, for any choice of the three momenta, two QAI 's and two QAI 's vanish. While this implies that 8(Q) = 0 (and hence the naive construction of the supervertices as in the fourpoint and higher-point cases does not apply), the supersymmetry Ward identities associated with the two vanishing QAI 's also become trivial, which means that the full factor of 8(Q) is not needed in a supervertex. { 6 { in the three orthogonal planes. Note that in the last two rows, we choose a frame where p1 is parallel to e0 + e1, and p2 is parallel to e2 + ie3. condition, is0s1 has charge ( 1) s0 +2s1 2 and ei s0s1 has charge \tiny group" that rotates the 45 plane. The momentum p+ has charge 12 under both the SO(1; 1)01 and SO(2)23, and is not charged under the SO(2)45. For clarity, these charges under the SO(2)45 are summarized in table 1. The constraint (2.17) implies that i = ei _ Q I and QI vanish identically. The expression = 0. Consequently the supercharges group charge following form is thus annihilated by all 16 supercharges QAI and QAI . Since (2.18) has SO(2)45 tiny 1, a general three-point supervertex for the tensor multiplets must take the Y I=1;2 I Q+ Q +Q++ I I 0 Q+ Q +Q++ I I A fabc( i; i); Symbol ++ / e++ + / e+ + / e + / e p+ p p + 1 + 2 1 2 1 2 1 2 1 2 1 2 1 0 SO(1; 1)01 12 / 12 / where fabc must be annihilated by Q up to terms proportional to Q, invariant with respect to the little groups, and have charge +1 under tiny group. By consideration of CPT conjugation,2 fabc cannot depend on i (otherwise the CPT conjugate expression would 2For an n-point (n 4) supervertex or superamplitude the CPT conjugate is For a three-point supervertex the CPT conjugate is V = 8(Q)F( i; i; ei; ei); i : n I=1;2 V = Y QI+ QI +QI++F( i; i; ei; ei); V = I=1;2 i : n { 7 { involve fewer than 6 's and cannot be proportional to Q6). Little group invariance then forces it to be a function of the momenta only. In particular, since all three momenta are SO(2)45 tiny group invariant, fabc would have to be tiny group invariant by itself which then forces it to vanish. Supervertices for supergravity and tensor multiplets We will now incorporate the coupling to the supergravity multiplet. Below to ease the notation, we will de ne qeAI eA _ e _ I ; e (q2)AB qeAI qeBJ IJ : (2.24) which is at 12-derivative order and contains the D4R4 coupling.3 tensor multiplet and two supergravity multiplets as external states We also have a four-point F-term supervertex at 8-derivative order that involves one e e 8(Q)(q12)AB(q22)CDp3AC p4BD; which contains the D2(R2H2) coupling.4 We can also obtain a 10-derivative F-term by multiplying the 8-derivative one (2.27) by s12, which contains the D4(R2H2) coupling. The lowest derivative order D-term is 8(Q)Q8 34 44(qe12)AB(q22)CDp3AC p4BD; e which is at 12-derivative order and contains the D6(R2H2) coupling. The fact that D term four-point supervertices involving the supergravity multiplet only start appearing at 12-derivative order is a special feature of (2; 0) supergravity, in contrast to the naive momentum counting that may suggest they occur at 8-derivative order (as in the case of maximally supersymmetric gauge theories, with sixteen supersymmetries). 3This is also the only D-term supervertex of supergravity multiplet at the 12-derivative order. The e's anti-commute with the supercharges. Their only role is to form supergraviton states and they must be contracted with the e's to form little group singlets. 4The 6-derivative order supervertex that contains the R2H2 coupling appears to be absent. { 8 { (2.25) (2.26) (2.27) (2.28) 0 1 Y I=1;2 0 Three-point supervertices. Let us now discuss the three-point supervertices between the (2; 0) supergravity multiplet and the tensor multiplets. Below we will explicitly construct the 2-derivative supervertices and also argue for the absence of three-point supervertices at 4-derivative order and beyond. At 2-derivative order, the 3-supergraviton supervertex is given by I Q+ Q +Q++ I I A (qe12)AA0 (q22)BB0 (q32)CC0 PABCA0B0C0 ( i; i): e e (2.30) I The power of p+ is xed by the SO(1; 1)01 and SO(2)23 invariance, and this expression is also invariant under the SO(2)45 tiny group, thereby consistent with the full SO(1; 5) Lorentz symmetry. More generally, a cubic supervertex of the supergravity multiplet must be of the form PABCA0B0C0 must be annihilated by Q up to terms proportional to Q, invariant with respect to the little groups, and must have charge 2 under tiny group scaling. As we have argued for the 3-tensor supervertices in the previous subsection, by applying CPT conjugation and little group invariance, we conclude PABCA0B0C0 is a tiny group invariant that only depends the momenta. The tiny group invariance of the full amplitude then forces (AA0; BB0; CC0) to have a total of 4 's and 8 +'s, and then SO(1; 1)01 and SO(2)23 invariance forces PABCA0B0C0 to scale like (p+) 4, and we are back to the two-derivative cubic supervertex (2.29). This rules out any higher derivative cubic supervertices of the supergravity multiplet. Now let us consider the three-point supervertex for one supergravity and two tensor multiplets. We can further choose the lightcone coordinates to be aligned with the momenta of the rst and second particle, by demanding that p1 = p1+(e0 + e1), and p2 = p2+(e2 + ie3). This amounts to the restriction At two-derivative order, the gravity-tensor-tensor supervertex is 1 + = 0; + 2 = 0: 1 0 1 Y I=1;2 I Q+ Q +Q++ I I A (qe12)(+ ;+ ); where p1 labels the momentum of the supergraviton. At 4-derivative order and beyond, there do not appear to be three-point supervertices for the gravity-tensor-tensor coupling, using the same argument as above. Similarly one can argue that no gravity-gravity-tensor supervertex exists.5 5It appears that one can write down a 2-derivative order supervertex 1 (p1+)2p2+ I=1;2 Y QI+ QI +QI++ (q12)(+ ;+ )(q22)(++; +) + (1 $ 2): e e ! { 9 { (2.31) (2.32) (2.33) Supervertices ggg gtt ggt ttt gggg ggtt tttt 2 only 2 only absent absent D-terms: 12+ D-terms: 12+ D-terms: 8+ F-terms: 8 and possibly 12+ F-terms: 8, 10, and possibly 12+ F-terms: 4, 6, and possibly 8+ and tensor supermultiplets which include R and H, respectively. The derivative order includes the derivatives implicit in the elds. For example, D2(R2H2) is regarded as an 8-derivative supervertex (2 + 2 2 + 2 1). It is claimed in [25] that in type IIB string theory on K3, there is a CP-odd RH2 e ective coupling that arises at one-loop order, where here H refers to a mixture of the self-dual two-form in a tensor multiplet and the anti-self-dual two-form in the multiplet that also contains the dilaton . This would seem to correspond to a 4-derivative cubic supervertex. A more careful inspection of the 6d IIB cubic vertex of [25] shows that it in fact vanishes identically [26], which is consistent with our nding based on the super spinor helicity formalism. The classi cation of three-point and four-point supervertices given in this section is summarized in table 2. In particular, the three- and four-point supervertices are all invariant under the SO(5) R-symmetry (2.8). In other words, our classi cation implies that SO(5) breaking supervertices in (2; 0) supergravity can only start appearing at ve-point and higher. The simplest examples of such supervertices are 8(Q) at n-point with n > 4, which transform in the [n 4; 0] representations of the SO(5) R-symmetry [27]. 3 Di erential constraints on f (4) and f (6) couplings In this section, we shall deduce the general structures of the di erential constraints on f (4) and f (6) couplings due to supersymmetry, using superamplitude techniques [9, 10, 23]. The construction of the f (4) and f (6) supervertices in (2; 0) supergravity gives the on-shell supersymmetric completion of the H4 and D2H4 couplings. In particular, given their relatively low derivative orders, such supervertices must be of F-term type which are rather scarce and have been classi ed and explicitly constructed in the previous section. However, after restoring the full SO(1; 5) Lorentz invariance, the resulting expression cannot be a local supervertex. This can be seen by noting that the expression is SO(5) R-symmetry invariant, and there simply does not exist any 2-derivative three-point coupling that involves two elds from the gravity multiplet and multiplet states while the dotted lines stand for the supergravity multiplet states. The black circles represent the 4-derivative four-tensor-multiplet supervertex, and the trivalent vertices represent the 2-derivative supervertex involving one gravity and two tensor multiplets. As we shall see below, the absence of certain higher point supervertices of these derivative orders will lead to di erential constraints on the moduli dependence of the aforementioned couplings in the quantum e ective action of (2; 0) supergravity. For example, we can expand the supersymmetric f (4) coupling, in terms of the moduli elds, and obtain higher-point vertices. In particular, the resulting six-point '2H4 coupling in the singlet representation of SO(5) R-symmetry can be related to a symmetric double soft limit of the corresponding six-point superamplitude (at 4-derivative order) [10, 27]. The absence of SO(5) R-symmetry invariant six-point supervertices at 4-derivative order [27] means that this six-point '2H4 coupling from expanding f (4) cannot possibly have a local supersymmetric completion. Rather, it must be related to polar pieces of the superamplitude via supersymmetry; in other words, it is xed by the residues in all factorization channels. The '2H4 superamplitude can only factorize through the 4-derivative supervertex for tensor multiplets and 2-derivative cubic supervertices for two tensor and one graviton multiplets (see gure 1), giving rise to r(e rf)fa(4bc)d = U fa(4bc)d ef + V f((e4()abc d)f) + W fe(f4)(ab cd): Here a natural SO(21; 5) homogeneous vector bundle W over M arises as the quotient V V where R21;5 transforms as a vector under SO(21) SO(5). We de ne the covariant derivative rai, where a = 1; : : : ; 21 and i = 1; : : : ; 5, by the SO(21; 5) invariant connection on W that gives rise to the symmetric space structure of the scalar manifold M. Further imposing invariance under the SO(5) R-symmetry means we can focus on the SO(21) subbundle W. The coupling fa(4bc)d becomes a section of the symmetric product vector bundle , on which the second order di erential operator r(e rf) acts naturally. (3.1) (3.2) tensor multiplet states while the dotted lines stand for the supergravity multiplet states. The black and white circles represent the 4 and 6-derivative four-tensor-multiplet supervertices, respectively, and the trivalent vertices represent the 2-derivative supervertex involving one gravity and two tensor multiplets. For the f (6) coupling, recall that it is de ned as the coe cient in the superamplitude 8(Q)(fa(6b;)cds + fa(6c;)bdt + fa(6d);bcu): Due to the relation s + t + u = 0, there is an ambiguity in the de nition of fa(6b;)cd, where we can shift f (6) by a term that is totally symmetric in three of the four indices. We x this ambiguity by demanding that fa(6(b);cd) = 0, which makes f (6) a section of the V vector bundle. The corresponding D2('2H4) superamplitude can also factorize through two f (4) supervertices (see gure 2), and we end up with the following di erential constraint 2r(e rf)fab;cd = u1fab;cd ef + u2(fe(f6;)ab cd + fe(f6;)cd ab) + u02fe(f6;)(c(a b)d) (6) (6) + u3(fe(a6;)fb cd + fe(c6;)fd ab) + u03fe((6c);f(a b)d) 2 + u4(fe((6c);ab fd) + fe((6a);cd fb)) + u5(fe((6b);a)(c d)f + fe((6d);c)(a b)f ) + v1 (fg(a4)b(cfd()4e)fg +fg(c4d)(afb()4e)fg)+v2fegabfc(d4f)g +v3fe(g4a)(cfd()4b)fg +(e $ f ) : (4) (3.3) (3.4) As we shall argue in the next section, the constant coe cients in (3.1) and (3.4) can be xed using results from the type II/heterotic duality and heterotic string perturbation theory. 4 An example of f (4) and f (6) from type II/heterotic duality In section 3, we wrote down the di erential constraints (3.1) and (3.4) on the 4- and 6derivative four-point couplings f (4) and f (6) between the 21 tensor multiplets in 6d (2; 0) supergravity, with undetermined model-independent constant coe cients. To determine these coe cients, we can consider the speci c example of four-point scattering amplitudes in type IIB string theory on K3. In this section, we will relate the exact non-perturbative 4and 6-derivative couplings in type IIB on K3 to a certain limit of the one- and two-loop amplitudes in the T 5 compacti ed heterotic string theory, via a chain of string dualities. With explicit expressions for the heterotic amplitudes, we verify the di erential constraints (see appendix A for the detailed computations), and thereby determine the model-independent constant coe cients. 4.1 gA T 4 Mh We consider type IIB string theory on K3 rB. The 6d limit of interest corresponds to keeping gB SB1 with string coupling gB, and circle radius O(1) while sending rB ! 1. We shall work in units with type II string tension 0 = 1. By T-duality, we can equivalently look at type IIA string theory on K3 SA1 with string coupling gA = gB=rB rA and circle size rA = 1=rB. In terms of type IIA parameters, the 6d limit corresponds to rA ! 0. Now we use type IIA/heterotic duality to pass to heterotic string theory on S1 where the size of both T 4 and S1 are of order rA in type II string units. Since the heterotic string is dual to a wrapped NS5 brane on K3, their tensions satisfy the relation HJEP12(05)4 1=`h 1=gA 1=rA. The heterotic string coupling, on the other hand, can be xed by matching the 6d (or 5d) supergravity e ective couplings to be gh 1=rA Mh.6 Hence in the limit where the circle of K3 SB1 in the type IIB picture decompacti es rB ! 1, we have 1 g 2 A Mh8rA4 g 2 h gh Mh ! 1 (4.1) (4.2) in the dual T 5 compacti ed heterotic string theory. Under the duality, the 21 tensor multiplets of (2; 0) supergravity on SB1 are related to the 21 abelian vector multiplets of heterotic string on T 5. In particular the e ective action of the tensor multiplets in the (2; 0) supergravity is captured by that of the vector multiplets in heterotic string. Let us now focus on the four-point amplitude of abelian vector multiplets in heterotic string on T 5. As we shall see in the next subsection, apart from the tree-level contribution at 2-derivative order due to supergraviton exchange, the four-point amplitudes at 4-derivative and 6-derivative orders receive contributions up to one-loop and two-loop, respectively. Furthermore we will argue that, in the limit of interest gh Mh ! 1, these couplings in the e ective action are free from contributions at higher loop orders. can be written as Relative to the tree-level contribution to the 2-derivative amplitude f (2), the 4-derivative f (4) coupling in the 6d (2; 0) supergravity from type IIB on K3, which contains H4 f (4) f (2) lim ` 2 h ; (4.3) where ngh2n is the n-loop contribution. By using type I/heterotic duality and con rmed by two-loop computation in [28{31], it has been argued that the F 4 coupling in heterotic string is does not receive contributions beyond one-loop, namely n = 0 for n 2. Therefore, we expect that f (4) is completely captured by the one-loop contribution 1gh2`2h.7 Indeed, we 6One can also derive this using the equivalence between the IIA string and the wrapped heterotic NS5 brane on T 4. 7Note that the tree-level contribution 0`2h to the 4-derivative coupling vanishes in the limit of interest gh 1=`h ! 1. Similarly, the tree-level and one-loop contributions 0`4h and 1`4hgh2 to the 6-derivative coupling vanish in that limit. will see in section 4.2.1 that the heterotic one-loop amplitude (4.6) satis es the di erential constraint for the 4-derivative coupling (3.1) in 6d (2; 0) supergravity. Likewise, the 6-derivative f (6) coupling which contains D2H4 can be written as lim We will see in section 4.2.2 that the two-loop contribution (4.31) corresponding to the 2gh4`4h term alone satis es the di erential constraint (3.4) for the 6-derivative coupling in 6d (2; 0) supergravity. This strongly suggests that the D2F 4 does not receive higher than two-loop contributions in the T 5 compacti ed heterotic string theory, though we are not aware of a clear argument.8 HJEP12(05)4 Heterotic string amplitudes and the di erential constraints In this subsection, we compute the four-point amplitude of scalars in the abelian vector multiplets in ve dimensions, of heterotic string on T 5, or more precisely, heterotic string compacti ed on the Narain lattice 21;5. As explained in the previous subsection, the cove dimensions at genus one and genus two in heterotic string capture exactly the six dimensional e ective couplings f (4) and f (6) of type IIB string theory on K3, expanded in the string coupling constant including the instanton corrections. These results can be extracted by slightly modifying the 10d heterotic string amplitudes, computed by D'Hoker and Phong (see for instance (6.5) of [29] and (1.22) of [31]). Furthermore, we x the constant coe cients in the di erential constraints (3.1) and (3.4) by explicitly varying the heterotic amplitudes with respect to the moduli elds. 4.2.1 One-loop four-point amplitude The scalar factor in the 4-gauge boson amplitude at one-loop takes the form9 A1 = Z d 2 2 ( ) i=1 2 ( ; ) Y4 d2zi e 12 Pi<j sijG(zi;zj) * 4 Y jai (zi) i=1 + : (4.5) Here denotes the even unimodular lattice 21;5, ( ) = form of SL(2; Z), and is the theta function of the lattice ( )24 is the weight 12 cusp with modular weight ( 221 ; 52 ). ja stand for the current operators associated to the 5d Cartan gauge elds in the Narain lattice CFT and G(zi; zj ) is the scalar Green function on the torus. The zi integrals are performed over the torus and the over F1 which is the fundamental domain of SL(2; Z) 2 on H . Note that the integrand has total modular weight (2; 2), and hence the integral 8The consistency check with the di erential constraints still allows for the possibility of shifting the 4and 6-derivative coupling f (4) and f (6) by eigenfunctions of the covariant Hessian. However, we believe that f (4) and f (6) are given exactly by the low energy limit of the heterotic one- and two-loop contributions. The above possibility can in principle be ruled out by studying the limit to 6d (2; 0) SCFT, but we will not demonstrate it here. 9The summation over spin structures has been e ectively carried out already in this expression. is independent of the choice of the fundamental domain of SL(2; Z). To extract the F 4 coe cient, we can simply set sij to zero in the above scalar factor (4.5), and write fa(41a)2a3a4 ( A1jF 4 )a1a2a3a4 = Z d 2 2 4 = 5 2 2 ( ; ) Y4 d2zi * 4 ( ) i=1 Z d 2 2 5 2 2 2 Y jai (zi) i=1 ( ) ; + where y is a vector in R21, that lies in the positive subspace of the R21;5 in which the is embedded. We have rewritten the four-point function of the currents as a fourth derivative on the theta function (see, for example, [32]). The theta function is de ned as variations of the embedding on the lattice As we have argued previously, the f (4) coupling satis es a di erential constraint (3.1) on M due to supersymmetry. Given the explicit expression for f (4) (4.6) from the type II/heterotic duality, we can now proceed to x the constant coe cients in (3.1). The above expression for the 4-derivative term f (4) has previously been determined in [12]. In the following we will explicitly parametrize the coset moduli space M and write down the covariant Hessian r(e rf). We will work in a trivialization of the SO(21) vector bundle V . In particular, we shall identify the coordinates on the base manifold M with Let eI be a set of lattice basis vectors, I = 1; ; 26, with the pairing eI eJ = IJ given by the even unimodular quadratic form of 21;5. Let P+ and P be the linear projection operator onto the positive and negative subspace, R21 and R5, respectively, of R21;5. We can write P+eI as (eILa)a=1; ;21, and P eI as (eIRi)i=1; ;5. are then y We expand the lattice vectors into components y = yeI eI . The left components of y a = yeI eILa. The requirement that y lies in the positive subspace means that y y = yaya stays invariant, y itself must vary, and so does ` y = `I eILaya. yeI eIRi = 0. This constraint implies that, under a variation of the lattice embedding, while From 21 X eILaeJLa a=1 5 i=1 X eIRieJRi = IJ ; eILa up to an SO(5) rotation. eIRi by itself is subject to the constraint we see that (eILa; eIRi) is the inverse matrix of IJ (eJLa; eJRi). Note that eIRi are speci ed by This constraint leaves 26 5 15 independent components of eIRi. The SO(5) rotation of the negative subspace further removes 10 degrees of freedom from eIRi, leaving the 21 5 = IJ eIRieJRj = ij : (4.6) (4.7) (4.8) (4.9) 105 moduli of the lattice embedding which give rise to a parametrization of the scalar manifold M.10 Now consider variation of the lattice embedding, eIa ! eILa + eILa; L eIi ! eIRi + eIi: R R subject to the constraints IJ eIL(a eJb) = O(( e)2); L IJ eILaeJRi + eILa eJi = O(( e)2); R IJ eIR(i eJj) = O(( e)2); R L L e(Ia eJ)a e(Ii eJ)i = O(( e)2): R R Let f (eIRi) be a scalar function on the moduli space M of the embedding of 21;5. We can expand f (eIRi + eIRi) = f (e) + f Ii(e) eIRi + f IJij (e) eIRi eJRj + O(( e)3): f Ii and f IJij are subject to shift ambiguities f Ii ! f Ii + f IJij ! f IJij + IJ eJRj gij ; 2 1 IJ gij + IK eRKkhJKijk + JK eRKkhIKijk for arbitrary symmetric gij and hIJijk, due to the constraints on eIRi. We can x these ambiguities by demanding eIRj f Ii = 0; eIRkf IJij = 0: This can be achieved, for instance, by shifting f Ii with IJ eJRj gij , for some gij . We can then de ne feai = eILaf Ii; feabij = eILaeJLbf IJij + 1 2 abeIR(if Ij): Note that these are invariant under the shift (4.15) and hence give rise to well-de ned di erential operators on the moduli space M. and we can therefore write the covariant Hessian of fa(4bc)d (4.6) as This construction can be straightforwardly generalized to non-scalar functions on M, In appendix A.1, we explicitly compute f~a(4bc)d;efii and nd 10Consider the symmetric matrix MIJ de ned by We have MIJ = eILaeJLa + eIRieJRi = 2eILaeJLa IJ = IJ + 2eIRieJRi: MIK KLMLJ = eILa abeJLb + eIRi( ij)eJRj = IJ : The symmetric matrix M , subject to the constraint M M = , can be used to parameterize the coset SO(21; 5)=(SO(21) SO(5)). (4.12) (4.13) (4.14) (4.15) (4.16) (4.17) (4.18) (4.19) (4.10) (4.11) !I (z) are a basis of holomorphic one-form on the genus two Riemann surface normalized such that !J = IJ ; !J = IJ ; where the cycles I and J have intersection numbers ( I ; J ) = IJ , ( I ; J ) = ( I ; J ) = genus two Riemann surface. G(zi; zj) is the Green's function on the genus two Riemann surface. Again, the contribution to D2F 4 coe cient is simply extracted as t u Z 3 Q To proceed, we need to compute the four-point correlation function of TIa = d2zja(z)!I (z) = ja(z)dz; on the genus two Riemann surface . This allows us to express the correlators of TIa in terms of the theta function (see for instance [32]), which allows us to x the constant coe cients in (3.1) to be 3 2 ; U = V = 2; The scalar factor in the two-loop heterotic amplitude takes the form [31] A2 = Z Q where YS is given by (4.20) (4.22) (4.23) (4.24) (4.25) (4.27) I I 5 Z I I ( ; ) Z I I where ( ; ) = X (yj ; ); (4.26) Thus, we can simplify the result to Next, we would like to verify that the coe cient functions f (6) extracted from A2 obeys the di erential constraint (3.4) on the moduli space SO(5; 21)=(SO(5) SO(21)), and also x the precise coe cients thereof. In principle, it should be possible to show that f (6) is f (6) plus the integral of a total derivative on the moduli space F2 of the genus two Riemann surface , which reduces to a boundary contribution where is pinched into two genus one surfaces. However, this calculation is somewhat messy so instead we will x the coe cients of for (f (4))2 by comparison to similar di erential constraints on the tensor branch of the 6d (2; 0) SCFT. (yj ; ): (4.28) We can write A2jD2F 4 as HJEP12(05)4 ( A2jD2F 4 )a1a2a3a4 = fa(61a)2;a3a4 s12 + fa(61a)3;a2a4 s13 + fa(61a)4;a2a3 s14 ; However, the de nition (4.29) of fa(61a)2;a3a4 is ambiguous because s12 + s13 + s14 = 0. We fa(61()a2;a3a4) = 0: ( A1A3 A2A4 + A1A4 A2A3 ) Z Q I J d (yj ; ) y=0 : (4.29) (4.30) (4.31) (4.32) (4.33) (4.34) (4.35) x this ambiguity by imposing Explicitly, fa(61a)2a3a4 is given by, fa(61a)2;a3a4 enjoys the symmetry The condition (4.30) gives rise to constraints between the coe cients in (3.4), fa(61a)2;a3a4 = fa(62a)1;a3a4 = fa(61a)2;a4a3 = fa(63a)4;a1a2 : u0 2 u2 + 2 = 0; u3 + 3 = 0: u0 2 We therefore end up with 5 coe cients u1; u2; u3; u4; u5 to determine for the terms proportional to f (6) on the r.h.s. of the di erential equation (3.4). We determine the ui's by explicit computation of the covariant Hessian in appendix A.2 and nd u1 = On the other hand, to determine the 2 independent coe cients v1; v2 for the (f (4))2 terms in (3.4), we shall take advantage of the following di erential constraint on the 6derivative four-point term in the tensor branch e ective action of (2; 0) SCFT [35] Fg(a4b)(cFd)efg +Fg(c4d)(aFb)efg +w2Fe(g4a)bFc(d4f)g +w3Fe(g4a)(cFd)bfg +(e $ f ) : (4) (4) (4) Here F (6) and F (4) are 6-derivative and 4-derivative four-point couplings of tensor multiplets in the (2; 0) SCFT, which are related to the supergravity couplings f (6) and f (4) by taking the large volume limit of K3 and zooming in on an ADE singularity that gives rise to the particular 6d SCFT.11 The 4- and 6-derivative terms on the tensor branch of the 6d (2; 0) SCFT can be in turn computed by the one- and two-loop amplitudes in 5d maximal SYM on its Coulomb branch as discussed in [27]. Explicitly, we have (see appendix B.1 and B.2 for details) A1jF 4 ! 210 123 F (4); A2jD2F 4 ! 215 9F (6): In [35], the coe cients in (4.35) are xed to be12 Hence we x the rest of the constants in (3.4) to be In summary, the 4- and 6-derivative couplings satisfy the following di erential (4.36) (4.37) (4.38) (4.39) HJEP12(05)4 + fe((6c);ab fd + fe((6a);cd fb) + (e $ f ) : 3 1 Implications of f (4) and f (6) for the K3 CFT As alluded to in the introduction, since spacetime supersymmetry imposes di erential constraints on the four-point string perturbative amplitudes which involve, in particular, integrated correlation functions of exactly marginal operators in the internal K3 CFT, we will be able to derive nontrivial consequences for the K3 CFT itself. As an illustration, we will see how the resulting moduli dependence of the f (4) and f (6) couplings at tree-level can pinpoint the singular points on the moduli space of the K3 CFT which the Zamolodchikov metric does not detect (since the moduli space is a symmetric space). Below we will rst explain how to extract the data relevant for K3 CFT from string treelevel amplitudes and general features thereof. We will then demonstrate their implications, 11The contribution due to supergraviton exchange on the r.h.s. of (3.4) is absent in (4.35) due to this decoupling limit. A similar reduction of the genus one and two amplitudes in the type II string theory to supergravity amplitudes was considered in [34]. 12The factor 2 in the numerator comes from the relative normalization between the lattice vectors and the 5d scalars. In particular, the mass square of the W -boson is m2 = 2 `2R. in a particular slice of the K3 CFT moduli space where the K3 CFT is approximated by the supersymmetric nonlinear sigma model on A1 ALE space. In principle, we expect to arrive at the same set of constraints from the K3 CFT worldsheet Ward identities with spin elds associated to the Ramond sector ground states which appear in the spacetime supercharge. The same set of constraints is expected to hold for all c = 6 (4; 4) SCFTs.13 We will leave this generalization to future work. Reduction to the K3 CFT moduli space The string theory amplitudes we have obtained in the previous section are exact results which can be regarded as sections of certain SO(21) vector bundles over the full moduli elds, and the rest 80 NSNS scalars describe the moduli space SO(5). Among the 105 moduli, one comes from the IIB Globally the K3 CFT has moduli space [36, 37] MK3 = O( 20;4)nSO(20; 4)=SO(20) SO(4); parametrized by the scalars inside 20 of the 21 tensor multiplet, 'i from the 6d perspective. From the worldsheet CFT point of view, 'i with i = 1; 2 : : : ; 20, are associated with the BPS superconformal primaries that are doublets of the two SU(2) current algebras. We will restrict the full string four-point amplitude obtained from the type II/heterotic duality to these 20 tensor multiplets, and expand in the limit of small gIIB.14 In this limit, the theta function of the 21;5 lattice can be approximated by the product of the theta function of the 20;4 lattice and that of the 1;1 lattice whose integral basis has the following embedding in R1;1, u = (r0; r0); v = with r0 ! 1 in the limit. Since at genus one we have in this limit 1 2r0 ; 1;1 ( ) 1 2r0 ; r0 and at genus two 1;1 ( ) (5.1) (5.2) (5.3) r02, (5.4) (5.5) full AH4 r0ArHed4('i); AfDu2llH4 r02ArDed2H4 ('i): Now on the other hand, working with the canonically normalized elds (Einstein frame) which involves rescaling the string frame metric and B- elds by G ! M6 2G ; B ! M6 2B where M6 = (VK3=gI2IB`s8)1=4 is the 6d Planck scale, we know that the four-point coupling f (4) must scale as M62 and f (6) as M64. From this we conclude that r0 M62. 13The condition c = 6 is used in writing down the spacetime supercharges (in combination with the spacetime part of the worldsheet CFT) hence making connection to the spacetime supersymmetry constraints on the integrated CFT correlation functions. 14Note that by doing so we break the SO(5) R symmetry to SU(2) SU(2). The di erential constraint on A full in the perturbation expansion implies a similar red. Focusing on the scalar component of the superamplitude, we have the constraint on A following derivative expansion k ' ` ) s t u = s2 ij k` + ik j` + i` jk + Aijk` + Bij;k`s + Bik;j`t + Bi`;jku + O(s2) : where the rst two terms come from the supergraviton exchange, while Aijkl and Bij;kl are obtained from the tree-level limit of the f (4) and f (6) couplings respectively. The coe cients Aijk` and Bij;k` for the N = 4 AK 1 cigar CFT, which is the ZK orbifold of the supersymmetric SU(2)K =U(1) SL(2)K =U(1) coset CFT, are studied in [38]. On the other hand, Ared can be evaluated directly from IIB tree-level perturbation theory. The K3 CFT admits a small (4; 4) superconformal algebra, that contains left and right moving SU(2) R-current algebra at level k = 1 [39]. Focusing on the left moving part, the super-Virasoro primaries are labeled by its weight h and SU(2) spin `. The BPS super-Virasoro primaries in the (NS,NS) sector consist of the identity operator (h = ` = h = ` = 0), and 20 others labeled by Oi with h = ` = h = ` = 1=2 which correspond to the 20 (1; 1) harmonic forms in the K3 sigma model.15 The BPS primaries Oi exactly marginal primaries of the K3 CFT, corresponding to the moduli elds 'i . Under spectral ow, the identity operator is mapped to a unique h = h = 1=4, ` = ` = 1=2 ground are the state O0 in the (R,R) sector, whereas the weight-1=2 BPS super-Virasoro primaries give rise to h = h = 1=4, ` = ` = 0 (R,R) sector ground states labeled by i operators for the 6d massless elds all involve these 21 BPS super-Virasoro primaries and RR [39]. The vertex their spectral owed partners. More explicitly, the vertex operators in the NSNS sector are Here eik X comes from the R1;5 part of the worldsheet CFT. The associated 1-particle states transform under the SU(2) SU(2) little group as e e eik X eik X 1; Oi ; i = 1; ; 20; The 80 scalars in the second line of (5.7) are denoted by 'i . On the other hand, the vertex operators in the RR sector are e e =2 =2S _ S _ eik X =2 =2S S eik X O0 ; iRR; i = 1; ; 20: 15Here Oi are BPS superconformal primaries of the N = (4; 4) superconformal algebra. With respect to an N = (2; 2) superconformal subalgebra, Oi Oi + is an anti-chiral primary on the left and a chiral primary on the right. ++ is a chiral primary both on the left and the right, whereas 4h=h= 14 20 1h=h= 14 6d multiplet supergravity + tensor 20 tensors K3. The 1 and 4 denote the trivial and ` = ` = 12 representations of the worldsheet SU(2) R-symmetry, and the arrows represent the spectral ow. The chiralities of the spin elds are dictated by the IIB GSO projection in RR sector, which depends on the SU(2) R-charge of the vertex operator.16 The associated 1-particle SU(2) little group. (5.8) and (5.11) together give the 1-particles states in the (2,0) supergravity multiplet and the 21 tensor multiplets. See table 3 for summary. The four-scalar amplitude of 'i in tree-level string theory is given by k ' ` ) = Z d2z D 2 G 1 G 12 Oi 2 ++eik1 X (z) G 1 G 12 Oj 2 ++eik2 X (0) Ok eik3 X (1) O` eik4 X (1) account in the above. correlation function vanishes identically 2 2 i where G(z) is the N = 1 super-Virasoro current, which acts on both Oi and eik X .17 We have put two vertex operators in the ( 1; 1) picture and the other two in the (0; 0) picture to add up to the total picture number ( 2; 2) for the tree-level string scattering amplitude. The correlator of the superconformal ghosts have already been taken into By deforming the contour of G 1 = H dw G (w), it is easy to see that the following D G 1 G 12 Oi++(z) G 1 G 12 Oj++(0) Ok (1) O` (1) 2 2 E = G 1 G 12 Oi++(z) G 1 G 12 Oj++(0) Ok (1) O` (1) : 2 2 16The IIB GSO projection in the RR sector is [17, 38] FL + 2JL3 1 2 2 2Z; FR + 2JR3 1 2 2 2Z where FL;R are the left and right worldsheet fermion numbers in R1;5, and JL3;R denote the left and right SU(2) Cartan R-charges of the internal K3 CFT. 17The sigma model on R1;5 current G(z) is the sum of N = 2 super-Virasoro currents G+(z) + G (z), and G of the N = 4 super-Virasoro currents. The U(1) charge of the N = 2 algebra coincides with the J3 charge are each a combination K3 has N = 2 worldsheet supersymmetry. The N = 1 super-Virasoro of the N = 4. (5.11) E (5.12) (5.13) (5.10) Therefore in (5.12) we can take G 1 ; G 1 to act on eik X only, which gives 2 2 k ' ` ) = s2 Z d2z 2 j j z s 2 1 j zj t DOi++(z)Oj++(0)Ok (1)O` (1)E : Thus, comparing with (5.6), we obtain the relation Z d2z 2 j j z s 2 1 j zj t DOi++(z)Oj++(0)Ok (1)O` (1) E s t u = partners iRR, all channels, Z d2z 2 From the CFT perspective, the polar terms in t and u are simply due to the appearance of the identity operator in the OPE of O ++ with O , while Aijk` and Bijk` capture information about all intermediate primaries in the conformal block decomposition of the four-point function of the marginal operators. It is then natural to expect this relation to hold for exactly marginal operators in any c = 6 (4; 4) SCFT. Furthermore, we expect Aijk` and Bijk` to obey the same kind of di erential equations as f (4) and f (6), for any Using the relation between the correlation function of Oi and their spectral owed j j ij k` + ik j` + i` jk + Aijk` + Bij;k`s + Bik;j`t + Bi`;jku + O(s2; t2; u2): (5.14) (5.15) (5.17) (5.18) DOi++(z)Oj++(0)Ok (1)O` (1)E = z j j j 1 z j iRR(z) jRR(0) kRR(1) `RR(1) ; (5.16) we can put (5.15) into an equivalent form, where the crossing symmetries are manifest in To illustrate the power of the relation (5.15), we consider the A1 ALE limit where we zoom in on and resolve an A1 singularity. In other words, we focus on a slice near the boundary of the full moduli space MK3, where the K3 CFT is reduced to a sigma model on A1 ALE space, which is related to the sigma model on C2=Z2 by exactly marginal deformations [37, 40, 41]. The slice of interest is parametrized by the normalizable exactly marginal deformations of the orbifold CFT C2=Z2, which is simply the moduli space of the A1 SCFT18 MA1 = R 3 Z2 S1 ; 18In [ 42 ], the moduli space of the non-linear sigma model on a general hyperkahler manifold is discussed. For the A1 ALE space sigma model, the moduli space metric is at because we have scaled the Zamolodchikov metric by an in nite volume factor of the target space. 3 corresponds to the Kahler and complex structure deformations associated with the exceptional divisor of the C2=Z2, and the S1 is parameterized by the integral of the B eld on the exceptional divisor. This Z2 can be understood from the fact that the SO(3) rotation of the asymptotic geometry of the circle bration of the Eguchi-Hanson geometry that exchanges the two points of degenerate ber e ectively also ips the orientation of the P1 hence re ects the B- eld ux. The two orbifold singularities on the moduli space corresponds to the free orbifold point and the singular CFT point where a linear dilaton throat develops. The distinction between these two points on the moduli space is not detected by the Zamolodchikov metric, but should be detected by f (4) restricted to the single tensor multiplet corresponding to this exceptional divisor (or rather A1111).19 Since the overall volume of the CFT target space is in nite, A1111 is a harmonic function on the moduli space.20 Near the singular CFT point, A1111 goes like 1=j'~j2, where '~ is a local Euclidean coordinate on the moduli space, as in the case of the A1 DSLST at treelevel (either (2; 0) or (1; 1)) [38, 43]. At the free orbifold point, on the other hand, the four-point function of marginal operators are perfectly non-singular, and A1111 should be nite. This together with the harmonicity and R-symmetry determines A1111 to be (up to an overall coe cient) ; where R is the radius of the S1 of the moduli space.21 It is easy to identify from (5.19) that, '~ = (0; 0; 0; 0) is the singular CFT point, and '~ = (0; 0; 0; R) is the free orbifold point, since A1111 is non-singular at the latter point, and the Z2 symmetry is clearly preserved. Let us de ne r2 = Pi3=1 'i2, '4 = R + y. Then near the free orbifold point, r; y are small, we have A1111 = 1 4R2 + 3y2 r 2 48R4 + O(r4; y4; r2y2): One should be able to con rm this using conformal perturbation theory [ 44 ]. In the large ' regime, where the CFT is described by a nonlinear sigma model on T CP1, performing Poisson summation on (5.19), we can write A1111 as the expansion A1111 = 1 2Rr 1 n=1 1 + X( )ne n(r+iy) R + X( )ne n(r iy) # R : 1 n=1 Since r scales like the area of the CP1, the leading 1=r contribution should come from one-loop order in 0 perturbation theory. The e nr=R corrections, on the other hand, are expected to come from worldsheet instanton e ects. Moreover, the phase e iny=R indicates that there are contributions from both holomorphic and anti-holomorphic worldsheet 19Note that f (6) vanishes in this case because there is only one tensor multiplet involved. 20The contribution from supergraviton exchange on the r.h.s. of (3.1) is suppressed in this limit. 21The S 1 parameterized by the B- eld ux through the exceptional divisor P1 is of constant size along the R3. This is because the marginal primary operator associated with the normalizable harmonic 2-form on the ALE space with unit integral on the P1 also has a normalized two-point function. (5.19) (5.20) (5.21) instantons. In other words, our exact result based on supersymmetry constraints gives the striking prediction that in 0 perturbation theory, A1111 which is related to the fourpoint function of exactly marginal operators of the A1 SCFT, receives only one-loop plus worldsheet instanton contributions. It would be interesting to understand if a similar worldsheet instanton expansion applies for the K3 CFT at string [45{48]. In particular, the N = 4 topological string amplitudes are written as integrals over the fundamental domain F1 and also satisfy certain di erential equations on the moduli space [48]. 6 The main result of this paper is the exact non-perturbative coupling of tensor multiplets at 4 and 6-derivative orders in type IIB string theory compacti ed on K3, fa(4bc)d( ) and fa(6b;)cd( ), and the di erential equations they obey on the 105-dimensional moduli space. In the weak coupling limit (tree-level string theory), as described in section 5, they reduce to (up to a factor involving the IIB string coupling) the functions Aijk`(') and Bij;k`(') on the 80-dimensional moduli space of the K3 CFT. Aijk` and Bij;k` are integrated four-point functions of 12 -BPS operators in the K3 CFT on the sphere. Unlike the Zamolodchikov metric or its curvature [14], Aijk` and Bij;k` do not receive contribution from contact terms, and depend nontrivially on the moduli. In particular, these functions diverge at the points in the moduli space where the CFT develops a continuous spectrum (corresponding to ADE type singularities on the K3 surface, with no B- eld through the exceptional divisors [37]). This allows us to pinpoint the location on the moduli space using CFT data alone (as opposed to, say, BPS spectrum of string theory), and makes it possible to study the K3 CFT through the superconformal bootstrap [49{52] (e.g. constraining the non-BPS spectrum of the CFT) at any given point on its moduli space. This is currently under investigation [53]. In the full type IIB string theory on K3, at the ADE points on the moduli space, there are new strongly interacting massless degrees of freedom, characterized by the 6d (2; 0) superconformal theory at low energies. Near these points, the components of fa(4bc)d( ) and fa(6b;)cd( ) associated with the moduli that resolve the singularities are precisely the H4 and D2H4 couplings on the tensor branch of the (2; 0) SCFT, studied in [27, 54]. Note that this is di erent from the ALE space limit discussed in section 5.2, which was restricted to the weak string coupling regime. As pointed out in section 2, there are F-term supervertices involving the supergraviton in 6d (2; 0) supergravity theories as well, including one that corresponds to a coupling of the schematic form fR( )R4 + . It appears that a six-point supervertex involving 4 supergravitons and 2 tensor multiplets in the SO(5)R singlet does not exist at this derivative order (namely 8), and so by the same reasoning as section 3, we expect that fR( ) obeys a second order di erential equation with respect to the moduli, whose form is determined by the factorization structure of the six-point superamplitude of 4 supergravitons and 2 tensor multiplets. One complication here is the potential mixing of the coe cients of R4, D2(R2H2), and D4H4, in the di erential constraining equations. In particular, D4H4 is a D-term, and by itself is not subject to such constraining equations. We leave a detailed analysis of the supersymmetry constraints on the higher derivative supergraviton couplings in (2; 0) supergravity to future work. One can similarly classify the supervertices in the 6d (1,1) supergravity theory and derive di erential constraints for the higher derivative couplings. In this case however, the string coupling lies in the 6d supergraviton multiplet rather than the vector multiplets, and its dependence is not controlled by the same type of di erential equations considered in this paper. Finally, one may wonder whether our exact results for integrated correlators in the HJEP12(05)4 K3 CFT can be extended to 2d (4; 4) SCFTs with c = 6k for k > 1, such as the D1-D5 CFT [55, 56]. While this is conceivable, the arguments used in this paper are based on the spacetime supersymmetry of the string theory and cannot be applied directly to the k > 1 case. In the CFT language, our constraints can be recast as Ward identities involving insertions of spin elds, and we have implicitly used the property that the spin elds of the c = 6 (4; 4) SCFT transform in a doublet of the SU(2)R symmetry. It would be interesting to understand whether there are analogous Ward identities in the c = 6k (4; 4) SCFTs, where the spin elds carry SU(2)R spin j = k2 .22 Acknowledgments We would like to thank Clay Cordova, Thomas Dumitrescu, Hirosi Ooguri, David SimmonsDu n, Cumrun Vafa for discussions. We would like to thank the Quantum Gravity Foundations program at Kavli Institute for Theoretical Physics, the workshop \From Scattering Amplitudes to the Conformal Bootstrap" at Aspen Center for Physics, and the Simons Summer Workshop in Mathematics and Physics 2015 for hospitality during the course of this work. SHS is supported by a Kao Fellowship at Harvard University. YW is supported in part by the U.S. Department of Energy under grant Contract Number DE-SC00012567. XY is supported by a Sloan Fellowship and a Simons Investigator Award from the Simons Foundation. A A.1 Explicit check of the di erential constraints Four-derivative coupling f (4) In this appendix we will explicitly show that the 4-derivative term coe cient fa(4bc)d between the 21 tensor multiplets satis es the following di erential equation and determine the 22In the case of the c = 12 (4; 4) SCFT, say described by the nonlinear sigma model on a hyperKahler 4-fold, one may compactify type IIB string theory to 2d, which generally leads to a (6; 0) supergravity theory in two dimension [57, 58], and examine the 4-derivative F-term coupling of moduli elds in this theory. However, we are not able to derive di erential constraining equations on these couplings based on soft limits of superamplitudes, due to the existence of local supervertices for the relevant six-point couplings at the same derivative order, in contrast to the 6d (2; 0) supergravity theory. coe cients U; V; W , re rf fa(4bc)d = U fa(4bc)d ef + V f((e4()abc d)f) + W fe(f4)(ab cd) : Let us rst decompose the 4-derivative coe cient fa(4bc)d into the fa(4bc)d = Aabcd + (abBcd) + (ab cd)C where Aabcd and Bcd are symmetric and traceless. The covariant Hessian r(a rb) of these tensors can be expressed, through a set of relations similar to (4.14) and (4.16), in the form The di erential constraints (A.1) can be expressed as X Aeabcd;efii; 5 i=1 X Beab;cdii; 5 X Ceabii: i=1 5 i=1 e;i c;i X Aeabcd;eeii = aAabcd; X Beab;ccii = bBab; X Ceaaii = cC; a;i 1 ef X Aeabcd;ggii = u (e(aAf)cde) + v e(a fbBcd) traces; X Beab;eeii = xAabcd + y (c(aBd)b) + z c(a b)dC traces; `2 `2 F We will relate the coe cients a; b; c; u; v; x; y; z; w to U; V; W later. To start with, let us determine the constant in the di erential equation for the scalar function C. From (4.6) and (A.2), we rst write C as C = 4! Z d 2 Z d 2 F F 1 2 2 1 2 2 ( ) ( ) X q `2` e 2 2`I eIRieJRi`J X q `2` " 2 3 21 2 `I `J eILaeJLa + + (`I `J eILaeJLa)2 2# e 2 2`I eIRieJRi`J ; where we have used `I `J eILaeJLa = ` ` + `I `J eIRieJRi. After integration by part, we have 16 4 Z d ( ) " X q `2` (` `)2 `2 11 2 ` ` + 33 # 2 2 2 e 2 2`I eIRieJRi`J : (A.1) (A.2) (A.3) (A.4) (A.5) (A.6) Under the variation eR ! eR + eR, the rst and second order variations of C are given by CIi eIRi + C IJij R R eIi eJj Z 2 F ( ) X `2 ` ` q 2 (` `)2 + 2 ` ` ` ` Z d 2 F ( ) 1 2 e 2 2`IeIRieJRi`J n 2 ` ` q 2 2 8 22 3 ` ` 2 ab`I `J eIRieJRj + 4 2 1 2 + ` ` We can now compute the Laplacian of C, 21 X ra raC = Ceaaii X a;i Z d 2 F ( ) n 1 2 X a;i Ceaaii = 25 2 C: This xes the constant c in (A.4) to be c = 25=2. where the hatted indices are taken to be symmetric traceless combinations. R eIi eJj R o : (A.7) ` ` ` ` 2 4 4 1 2 + 2 4 3 161 (` `)2 + e 2 2`IeIRieJRi`J : ` ` 2 2 1 " Similarly we can write Aabcd = (2 i)4 Bab = d F F d 2 1 2 2 ( ) 1 2 2 ( ) X `2 X `2 ` ` q 2 e # ` ` q 2 e ` ` # `I `J eILa^eJL^b; r(c rd)Bab = B~ab;cdii `I `J eILa^eJ^b L 24`M `N eLMceNd + cd L + a^c ^bd ` ` + `I `J eILa^eJ(d c)^b L ` ` + ` ` q 2 e ` ` r(a rb)C = Cea^^bii 704 4 Z 2 161 F ` ` q 2 e " ` ` where we have used ( xing the SO(21) freedom) eJa = eJRieaLI R L eIi + eLI R a R eIi eJi eJRj ejRM eLN a R R eMk eNk + : : : : After a somewhat tedious but straightforward calculation, we obtain all the di erential equations in (A.4), The covariant Hessians of Aabcd, Bab, and C can be computed straightforwardly to be r(e rf)Aabcd = Aeabcd;efii = 16 4 Z 2 ` ` q 2 e " 2`I `J `M `N eILa^eJL^beMc^eN(e f)d^ L L + 2 3 2 `I `J eILa^eJ^b c^e d^f L ` ` q 2 e 2 2`I eIRieJRi`J `I `J `M `N eILa^eJL^beMc^ Nd^ L L e ; `2 6 2 1 2 2 ( ) 4 X `2 1 2 1 2 # # ; 18 2 # 6 2 `I `J eILa^eJL^b; (A.12) (A.13) F 4 F 24 ef 2 25 2 2 ( ) X `2 Z F 1 2 2 ( ) 2 1 2 2 ( ) X `2 1 2 67 2 17 2 Aabcd; Bab; Aeabcd;eeii = Bab;ccii = X e;i X i X c;i X i X i 144 25 550 483 Cea^^bii Bab; 71 25 A eabcd;e^f^ii 2Aa^^bc^(e f)d^ Ba^^b c^(e f)d^ trace in (ef ); (A.14) B eab;c^d^ii = Aabcd + Ba^(c d)^b + 2C a^(c d)^b trace in (cd); where the hatted indices are taken to be symmetric and traceless. Together with (A.10), we have thus determined all the coe cients in (A.4), a = x = 67 2 144 25 b = y = 17 2 ; ; 71 25 c = z = 2; 25 2 ; u = w = 2; 550 483 : v = 1; (A.15) Determination of U; V; W . With the above 9 coe cients determined, we now arrange them into the form (A.1) and determine U; V; W . Let us start by inspecting the trace part in (ef ) of (A.1), r2fa(4bc)d = (21U + V )fa(4bc)d + W ef fe(f4)(ab cd): Noting that ef fe(f4)ab = 265 Bab + 233 abC, we obtain the rst three equations in (A.4), 25 6 23 3 W )Bab; W )C: V 2 e(aBbc d)f) Aefab + W Bef : trace in (ef), (abcd); 29 12 V + 25 9 W B(e(a b)f) (e(a b)f)C trace in (ef), (ab); (A.16) (A.17) (A.18) (A.19) Next, the traceless part in (ef ) of (A.1) can be written as r(e rf) r(e rf) 1 21 ef r 1 21 ef r 2 Aabcd = V A(e(abc d)f) + 2 Bab = r(e rf) 1 21 ef r 2 C = r2Aabcd = (21U + V )Aabcd; r2Bab = (21U + V + r2C = (21U + V + 6 25 + 1 161 3 2 25 6 V + V 2 25 6 25 6 25 9 W W 575 18 V + Matching (A.17) and (A.18) with (A.4), we nd the 9 coe cients a; b; c; u; v; x; y; z; w are indeed determined by U; V; W , which are U = V = 2; Six-derivative coupling f (6) In this appendix we will show that the 6-derivative term between the 21 tensor multiplets f (6) de ned in (4.29) satis es the following di erential equation, r(e rf)fa(61a)2;a3a4 = u1fa(61a)2;a3a4 ef + u2 fe(f6;)a1a2 a3a4 + fe(f6;)a3a4 a1a2 + u4 fe((6a)3;a1a2 fa4) + fe((6a)1;a3a4 fa2) + u5 fe((6a)2;a1)(a3 a4)f + fe((6a)4;a3)(a1 a2)f ; modulo terms of the schematic form (f (4))2. In the following the symmetrization on the indices (ef ) is always understood if not explicitly written. We have already taken the condition (4.30) and its consequence (4.33) into account. In the following we will use an abbreviated notation to simplify the notations, eIa e~Ii = eIRi, and MAB Im From (4.31), we can write fa(61a)2a3a4 as G H d 6 3 + 8 3M 1 MAB`AI `BJ (e2)a1a2;a3a4;IJ ( a1a2 a3a4 a1(a3 a4)a2 ) ; exp i AB`A B 2 MAB`AI `BJ e~Iie~Ji HJEP12(05)4 eILa, (A.21) (A.22) (A.23) (A.24) 1 1 2 1 2 1 2 1 They can be computed straightforwardly to be GIKJi;ab = e~IieaK eJb + e~JiebK eIa; + e~Iie~Jj eaM ebN ; EII1iI2I3I4;a1a2a3a4 = e~I1ieIa1 eI2a2 eI3a3 eI4a4 + 3 more; FIM1Ii2NI3jI4;a1a2a3a4 = e(aM1 IN1) e~I1ke~(kM eaN1) eI2a2 eI3a3 eI4a4 ij + 3 more HIMJ;iaNb j = e(M N) a I e~Ike~(kM eN) eJb ij + a b e(M N) J e~Jke~(kM eN) eIa ij b (e2)a1a2;a3a4;IJ := a3a4 eIa1 eJa2 a1a2 eIa3 eJa4 with symmetrization on the (IJ ) indices. 2 a2a3 eIa4 eJa1 + 2 a1a4 eIa2 eJa3 + 2 a2a4 eIa1 eJa3 + 2 a1a3 eIa2 eJa4 ; 1 Recall that under the variation e~ ! e~ + e~, eIa transforms as, up to second order, eIa = e~IieaJ e~Ji + e(M N) a I e~Ij e~(jM eN) a e~Mi e~Ni; where the M; N indices are raised by MN . We will de ne the tensor G; H; E; F as (eIaeJb) = GIKJi;ab e~Ki + HIMJ;iaNb j e~Mi e~Nj ; (eI1a1 eI2a2 eI3a3 eI4a4 ) = EII1iI2I3I4;a1a2a3a4 e~Ii + FIM1Ii2NI3jI4;a1a2a3a4 e~Mi e~Nj : 1 2 + e~I1ie~I2j eaM1 eaN2 eI3a3 eI4a4 + 5 more: (A.25) f (6) a1a2;a3a4= f (6) Ii e~Ii + f (6) MiNj e~Mi e~Nj a1a2;a3a4 n h Q G H d 6 GH 1 X exp i AB` A ` B 32 4( 2e4`4)a1a2;(a3a4) + 8 3M 1 MAB`AI `BJ (e2)a1a2;a3a4;IJ +4 2M 1 4 MAB`AI `BJ e~Ii e~Ji 2 MAB`AI `BJ ij + 8 2MABMCD`AI `BM `CJ `DN e~Mie~Nj e~Ii e~Jj + 32 4 +8 3M AB CD`AI1 `CI2 `BI3 `DI4hEIi I1I2I3I4;a1a2(a3a4) e~Ii +F MiNj I1I2I3I4;a1a2(a3a4) e~Mi e~Nj 1(MEF `EI `F J ) h e~Ki + HIMJ;iaN1aj2 e~Mi e~Nj + 5 more AB CD`AI1 `CI2 `BI3 `DI4 EMi I1I2I3I4;a1a2(a3a4) 1 MEF `EI `F J a3a4 GIMJ;ia1a2+5 more 4 MAB`AK `BN e~Kj e~Mi e~Nj i HJEP12(05)4 o (A.26) (A.27) where we have de ned X f a1a2;a3a4;ef ii = ( 2e4`4)a1a2;a3a4 = AB CD`AI `BJ `CM `DN eIa1 eMa2 eJa3 eN a4 : Note that ( 2e4`4)a1a2;(a3a4) = ( 2e4`4)(a1a2);(a3a4) = ( 2e4`4)(a3a4);(a1a2). The second derivative of f (6) is then given by X eIeeJf f (6) IJii + e~Iif (6) Ii 2 Q G H d 6 GH 1 X `1;`22 exp i AB ` A ` B i 32 2M 1 4( 2e4`4)a1a2;(a3a4) + 8 3M 1 MAB `AI `BJ (e2)a1a2;a3a4;IJ + 32 + 8 3M AB CD`AI1 `CI2 `BI3 `DI4 F M iNi I1I2I3I4;a1a2(a3a4) eM eeN f 1 (MEF `EI `F J )eM eeNf a3a4 HIMJ;iaN1 ai2 + 5 more + 32 4 AB CD`AI1 `CI2 `BI3 `DI4 EM i I1I2I3I4;a1a2(a3a4) + 8 3M 1 (MEF `EI `F J a3a4 GIMJ;ia1a2 + 5 more 4 MAB ` AK `BN e~Ki eM (eeNf ) exp 2 MAB `AI `BJ e~Iie~Ji ; (A.28) where we have used EIi I1I2I3I4;a1a2(a3a4)e~Ij = 0 and GIi I1I2;a1a2 e~Ij = 0. Let us now study the di erent powers of ` terms in the integrand. Note that since we can replace `AI `BJ e~Iie~Ji by 21 @M@AB , e~Ii should be treated as ` 1 in the power counting. Also note that the tensors G; H; E; F contain factors of e~Ii. First let us note that the `6 terms cancel as in the 4-derivative case after integration by parts. Moving on to the `4 terms, they can be organized to be X f~(6) a1a2;a3a4;efii `4 3 G H d X exp hi AB`A `Bi 10( )M 21 `1;`22 64 4 ef ( 2e4`4)a1a2;(a3a4) a3a4 ( 2e4`4)ef;(a1a2) + a1a2 ( 2e4`4)ef;(a3a4) 2 a2a3 ( 2e4`4)ef;(a1a4) 1 2 a2a4 ( 2e4`4)ef;(a1a3) 2 a1a3 ( 2e4`4)ef;(a2a4) +16 4 a1e( 2e4`4)fa2;(a3a4) + a2e( 2e4`4)a1f;(a3a4) + a3e( 2e4`4)a1a2;(fa4) 1 1 + a4e( 2e4`4)a1a2;(a3f) : u1 a1a2 a3a4 a1(a3 a4)a2 ef + 2u2 ef a1a2 a3a4 + 2u4 ea3 a4)f a1a2 + (a1 $ a3; a2 $ a4) = 0: e(a1 a2)f a3a4 e(a1 a2)(a3 a4)f Again, the symmetrization on the indices (ef ) is implicitly understood. ef (a3(a1 a2)a4) + e(a1 a2)(a3 a4)f This already xes ui's to be u1 = In the following we will show that the terms with `2 and `0 in the integrand also satis es the same di erential equation (A.20) with the same values of ui's. Let us start with the `0 term in the covariant Hessian (l.h.s. of (A.20)), X f~(6) a1a2;a3a4;efii `0 / 3 G H d 6 GH 1 X `1;`22 exp exp i AB`A Hence we need to show that the righthand side of (A.20) is also zero when replacing fa(61a)2;a3a4 by its `0 term in the integrand, namely, fa(61a)2;a3a4 ! ( a1a2 a3a4 a1(a3 a4)a2 ). Indeed, under this replacement the righthand side of (A.20) is zero with ui's given by (A.30) (A.29) (A.30) (A.31) (A.32) exp i AB`A B 2 MAB`AI `BJ e~Iie~Ji eJf nh 4 3h 8 3 ef (e2)a1a2;a3a4;IJ 16 3 a1(a3 a4)a2 eIeeJf 4 3 2 a1e a2f eIa3 eJa4 + 2 a3e a4f eIa1 eJa2 a1e a3f eIa2 eJa4 a1e a4f eIa2 eJa3 a2e a3f eIa1 eJa4 a2e a4f eIa1 eJa3 a3a4 ( a1eeIa2 + a2eeIa1 ) a1a2 ( a3eeIa4 + a4eeIa3 ) 1 2 a2a3 ( a1eeIa4 + a4eeIa1 ) + 3 moreio: Xf~(6) a1a2;a3a4;efii `2 3 G H d 6 GH 1 X i (A.33) (A.34) We need to match the second derivative of f (6) given above with the righthand side of (A.20) at the `2 order in the integrand. For example, the coe cient for ef (e2)a1a2;a3a4;IJ on the righthand side of (A.20) is 8 3(u1+u2) = 8 3, which agrees with the coe cient the second derivative f~(6). Similarly one can show that the `2 terms agree on both sides of (A.20). In conclusion, we have checked that fa(61a)2;a3a4 given in (4.31) satis es the following 2r(e rf)fa(61a)2;a3a4 = 2fa(61a)2;a3a4 ef + fe(f6;)a1a2 a3a4 + fe(f6;)a3a4 a1a2 + fe((6a)3;a1a2 fa4) + fe((6a)1;a3a4 fa2) + (e $ f ) ; modulo the (f (4))2 term that is determined in section 4 and appendix B. B Relation to 5d MSYM amplitudes In section 4, we discuss how the numerical coe cients v1; v2; v3 for the (f (4))2 term in (3.4) can be xed from the 6d (2; 0) SCFT limit, where a similar di erential equation holds [35]. The four-point 4- and 6-derivative couplings on the tensor branch of the 6d (2; 0) SCFT can be in turn computed by the one- and two-loop amplitudes in 5d maximal SYM on its Coulomb branch [27]. Therefore, to determine these coe cients, we will x the relative normalization between the F 4 and D2F 4 couplings in the Coulomb branch e ective action of 5d maximal SYM and the T 5 compacti ed heterotic string amplitudes in this appendix. B.1 In this subsection, we would like to x the relative normalization between the F 4 coupling from one-loop heterotic string amplitude and that from one-loop 5d maximal SYM on its Coulomb branch by looking at a point of enhanced ADE gauge symmetry in the heterotic moduli space and a degeneration limit of the genus one Riemann surface (see gure 3). A similar reduction of the genus one and two amplitudes in the type II string theory to supergravity amplitudes was considered in [34]. one-loop amplitude A1SYM in 5d maximal SYM. Recall that the heterotic one-loop amplitude is with the theta function de ned by A1jF 4 = Z d i `2R+2 i` y = e 2 2 y y X e i ` ` 2 2`2R+2 i` y: 1 (B.2) (B.4) Let us inspect the contributions to the integral in the large 2 regime, where ( ) can be approximated by q = e2 i . Then 2 L `2R = 2, and we have is dominated by the contribution from ` ` = A1jF 4 ! (2 ) `aL1 `aL2 `aL3 `aL4 e 2 2`2R : In the limit of the moduli space where `R ! 0 for some of the ` ` = 2 lattice vectors, the dominant contribution takes the form of the one-loop contribution from integrating out W -bosons labeled the root vectors ` in 5d maximal SYM. Here `2R is proportional to the W -boson mass squared, and `aL labels the charge of the W -boson with respect to the a-th Cartan generator. To compare the normalization with the 5d SYM one-loop amplitude, we use the Schwinger parametrization to write down the contribution from the diagrams involving light internal W -bosons, which are labeled by the root vectors `L, A1 SYM = = X (`L)2=2 1 26 25 Z 3 Z dt t 3 3! `aL1 `aL2 `aL3 `a4 e t(p2+m2) dt t 21 X (`L)2=2 `aL1 `aL2 `aL3 `aL4 e tm2 `2 `2 X ` `=2 Identifying m2 = 2 `2R, we x the relative normalization to be A1jF 4 ! 210 123 A1SYM: B.2 Six-derivative coupling f (6) In this subsection, we would like to x the relative normalization between the D2F 4 coupling from two-loop heterotic string amplitude and that from two-loop 5d maximal SYM on its Coulomb branch by looking at a point of enhanced ADE gauge symmetry in the heterotic moduli space and a degeneration limit of the genus two Riemann surface (see gure 4). Recall that the heterotic two-loop amplitude is t u 3 + (2 perms) # Z Q 10( ) = e2 i( + + ) e2 i(n +k +` ) : Y (n;k;`)>0 with the theta function given by X `1;`22 X `1;`22 e i AB`LA `LB i AB`RA `RB+2 i`A yA+ 2 ((Im ) 1)AByA yB e i AB`A `B 2 Im AB`RA `RB+2 i`A yA+ 2 ((Im ) 1)AByA yB : Each component of Re AB has periodicity 1. The imaginary part of the period matrix can be written as Im t1 + t3 ; with det Im = t1t2 + t1t3 + t2t3. In the limit of large positive t1; t2; t3, this corresponds to the genus two Riemann surface degenerating into three long tubes, of length t1; t2; t3 respectively. We can also write Im AB`A `B = t1(`1)2 + t2(`2)2 + t3(`1 + `2)2; ((Im ) 1)AByA yB = t1y22 + t2y12 + t3(y1 y2)2 : t1t2 + t1t3 + t2t3 In the limit of large positive t1; t2; t3, the theta function, apart from the term 1 which vanishes upon taking y-derivative, is dominated by the terms involving lattice vectors ` such that `2L + `2R is close to 2, when the lattice embedding is near an ADE point in the moduli space. The Igusa cusp form 10( ), on the other hand, behaves as 10( ) ! e2 iRe( 11+ 22 12)e 2 (t1+t2+t3); where we have used the product expression for 10( ), (B.5) (B.6) (B.8) (B.10) (B.11) giving the factor HJEP12(05)4 exp 2 (t1(`1R)2 + t2(`2R)2 + t3(`1R + `2R)2) + 2 i`A yA : We are interested in the limit where (`1R)2, (`2R)2, and (`1R + `2R)2 are small, and correspond to W -boson masses of three propagators in the two-loop diagram. We have (in the rest of this section we will not distinguish `Ia with (`L)Ia since in the limit of interest (`R)Ia ! 0) A2jD2F 4 ! 4 X (`1)2=(`2)2=(`1+`2)2=2 IJ KL`Ia1 `aJ2 `aK3 `aL4 3 u t Z dt1dt2dt3 (t1t2 + t1t3 + t2t3) 2 1 e 2 (t1(`1R)2+t2(`2R)2+t3(`1R+`2R)2) +(cyclic perms in 2; 3; 4): Here (n; k; `) > 0 means that n; k 0, ` 2 Z, and in the case when n = k = 0, the product is only over ` < 0. In the above expression we parametrize as The integration over Re AB then picks out the terms in the theta function with 1 `1 = `2 `2 = (`1 + `2)2 = 2; (B.12) (B.13) (B.14) (B.15) Here `Ia is the eigenvalue of the Cartan generator Ta on the W -boson labeled by the root vector `I , on the propagator of length tI , I = 1; 2. On the third propagator of length t3, the W -boson has charge `1a + `2a with respect to Ta. Let us compare this with the two-loop amplitude at 6-derivative order in 5d SYM, whose contribution from the diagrams involving two light internal W -bosons takes the form A2 SYM = 2 X (`1L)2=(`2L)2=(`1L+`2L)2=2 t21t22`1a1 `1a2 `2a3 `2a4 + 5 more Z dt1dt2dt3 t12t2t3`1a1 `1a2 ( ` 1 `2a3 )`2a4 t12t2t3( ` 1 a1 `2a1 )`2a2 `1a3 `1a4 + 10 more Z d5p1d5p2 (2 )10 e Pi3=1 ti(pi2+mi2) + (cyclic perms in 2; 3; 4); where the rst and the second lines come from the rst and the second two-loop diagrams in gure 4, respectively. The term proportional to t21t22, for instance, comes from the twoloop diagram with two external lines (with Cartan label a1; a2) attached to the propagator of length t1 and two external lines (with Cartan label a3; a4) attached to the propagator of length t2. The `1; `2; ` 1 `2 to each internal propagator. stand for all the other possible assignments of the W -boson root vectors 4 2 two-loop amplitudes A2SYM in 5d maximal SYM. We can identify m21 = 2 (`1R)2, m22 = 2 (`2R)2, m23 = 2 (`1R + `2R)2. The factor in the bracket, after multiplication by s and summation over permutations, can be organized into the form (taking into account s + t + u = 0) s(t1t2 + t1t3 + t2t3)2 `1a1 `1a2 `2a3 `2a4 + `2a1 `2a2 `1a3 `1a4 + (cyclic perms in 2; 3; 4) 2 3 s(t1t2 + t1t3 + t2t3)2 `1a1 `1a2 `2a3 `2a4 + `2a1 `2a2 `1a3 `1a4 2`(1a1 `2a2)`(a3 `a4) 1 2 + (cyclic perms in 2; 3; 4) 3 (t1t2 + t1t3 + t2t3)2( IK JL + IL JK )`Ia1 `aJ2 `aK3 `aL4 + (cyclic perms in 2; 3; 4): Notice that only terms with two `1 and two `2 will survive after summing over the s; t; u channels. Hence the SYM two-loop amplitude can be put into the form (B.18) (B.19) A2 SYM = 2 11 s 3 Z ( IK JL + IL JK )`Ia1 `aJ2 `aK3 `aL4 + (cyclic perms in 2; 3; 4) dt1dt2dt3 (t1t2 + t1t3 + t2t3) 2 1 e Pi timi2 : This is indeed proportional to (B.15), A2jD2F 4 ! 215 9 SYM: A2 Open Access. 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Ying-Hsuan Lin, Shu-Heng Shao, Yifan Wang, Xi Yin. Supersymmetry Constraints and String Theory on K3, Journal of High Energy Physics, 2015, 1-42, DOI: 10.1007/JHEP12(2015)142