Gravitational couplings in \( \mathcal{N}=2 \) string compactifications and Mathieu Moonshine

Journal of High Energy Physics, May 2018

Abstract We evaluate the low energy gravitational couplings, Fg in the heterotic E8 ×E8 string theory compactified on orbifolds of K3 × T 2 by g′ which acts as a ℤ N automorphism on K3 together with a 1/N shift along T 2. The orbifold g′ corresponds to the conjugacy classes of the Mathieu group M24. The holomorphic piece of F g is given in terms of a polylogarithm with index 3−2g and predicts the Gopakumar-Vafa invariants in the corresponding dual type II Calabi-Yau compactifications. We show that low lying Gopakumar-Vafa invariants for each of these compactifications including the twisted sectors are integers. We observe that the conifold singularity for all these compactifications occurs only when states in the twisted sectors become massless and the strength of the singularity is determined by the genus zero Gopakumar-Vafa invariant at this point in the moduli space.

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Gravitational couplings in \( \mathcal{N}=2 \) string compactifications and Mathieu Moonshine

HJE 2 string compactifications and Mathieu Moonshine Aradhita Chattopadhyaya 0 1 Justin R. David 0 1 0 C.V. Raman Avenue , Bangalore 560012 , India 1 Centre for High Energy Physics, Indian Institute of Science We evaluate the low energy gravitational couplings, Fg in the heterotic E8 ×E8 on K3 together with a 1/N shift along T 2. The orbifold g′ corresponds to the conjugacy classes of the Mathieu group M24. The holomorphic piece of Fg is given in terms of a polylogarithm with index 3−2g and predicts the Gopakumar-Vafa invariants in the corresponding dual type II Calabi-Yau compactifications. We show that low lying Gopakumar-Vafa invariants for each of these compactifications including the twisted sectors are integers. We observe that the conifold singularity for all these compactifications occurs only when states in the twisted sectors become massless and the strength of the singularity is determined by the genus zero Gopakumar-Vafa invariant at this point in the moduli space. Moonshine; Superstrings and Heterotic Strings; Discrete Symmetries; Topological Strings - 1 Introduction 2 Heterotic string on orbifolds of K3 × T 2 3 The integral for gravitational thresholds 4 Evaluating the integral for gravitational thresholds 4.1 5 Gopakumar-Vafa invariants and conifold singularities 5.1 5.2 5.3 Low lying coefficients of F¯ghol Integrality of the Gopakumar-Vafa invariants Conifold singularities 6 Conclusions A Modular transformations B Details for g = 1 threshold integral C Details for the g > 1, threshold integral D Genus zero GV invariants for CHL orbifolds E List of twisted elliptic genera features is the R2 term in N = 4 theories in d = 4 which can be evaluated by a one-loop computation in the type II frame, but yields predictions for an infinite sum of space time instanton effects due to five branes in the heterotic frame [1, 2]. Let us focus on the gravitational coupling Fg of the low energy effective action of N = 2 string theories in d = 4. These couplings appear as the following terms in the effective action S = Z Fg(y, y¯)F 2g−2R2, (1.1) where F, R are the self dual part of the graviphoton and the Riemann curvature. The coupling Fg depends on the vector multiplets y, y¯ of the theory. The canonical and well studied example of such a theory is the E8 × E8 heterotic string theory compactified on K3 × T 2 with the standard embedding of the spin connection in to a SU(2) of one of the E8. Building on the earlier works [3–8], a detailed study of these couplings for this theory has been carried out by [9] for g = 1 and in [10] for g > 1 by explicitly evaluating the one loop threshold integrals. This compactification is also the standard and well studied example of N = 2 string duality [11, 12]. The result from the heterotic side is one loop exact. Moduli dependence of the one loop threshold corrections in this model have also been studied earlier in [13–17]. On the type II A side, the theory is compactified on the Calabi-Yau manifold X with Euler number χ(X) = −480. In fact the Euler number of the dual Calabi-Yau is predicted from Fg. Though the full expression of Fg is intricate, the holomorphic1 part of this coupling, which can be obtained by sending y → ∞ is simple and predicts certain topological invariants of the dual Calabi-Yau three fold X. Lets call this holomorphic part F¯g(y), the genus g topological amplitude. The dual Calabi-Yau is generally a K3 fibration over a base T 2. The heterotic side corresponds to the semi-classical limit when the volume of the base is large. The topological amplitude F¯g(y) predicts the number of genus g holomorphic curves in the K3 fibers of the dual Calabi-Yau X in the semi-classical limit. The most convenient way to extract this information is using the observation of Gopakumar and Vafa who showed that a generating function for F¯g(y) for any Calabi-Yau can be written in terms of integer invariants which are called Gopakumar-Vafa invariants [18, 19]. Therefore the explicit calculation of F¯g(y) from the heterotic side yields a prediction for the Gopakumar-Vafa invariants of the dual Calabi-Yau X in the semi-classical limit. In this paper we generalize the evaluation of Fg to the heterotic E8 × E8 string theory compactified on orbifolds of K3 × T 2. The orbifold g′ acts as a Z K3 together with the 1/N shift on one of the circles of T 2. We consider the standard N automorphism on embedding in which the SU(2) spin connection of the K3 is embedded in one of the E8 of the theory. The orbifold g′ corresponds to the conjugacy classes of Mathieu group M24 listed in table 1 and they all preserve N = 2 supersymmetry. These compactifications were introduced in [20, 21] with the motivation of exploring the role of M24 in string compactifications which was first studied in the original K3 × T 2 model by [22]. We would like to emphasise that these compactifications are distinct from that studied by [23]. In this work the K3 was realized as a Z N orbifold of T 4, with N = 2, 3, 4, 5 and the 1/N shifts 1It is actually the anti-holomorphic part of Fg but we will take the complex conjugate and refer to it as the holomorphic part. – 2 – where restricted to lie along the internal E8 × E8 lattice and not on the external T 2. Note that the results of our paper does not hinge on the realization of K3 as an orbifold of T 4 and the shifts we consider are along the external T 2. One of our goals in evaluating the Fg for the orbifolds under consideration in this paper is to determine the properties of the dual Calabi-Yau geometry on the type II A side and to initiate a study of these geometries and understand if at all M24 plays a role in these geometries. Recently the role of sporadic symmetry groups in the elliptic genera of Calabi-Yau 5-folds has been investigated [ 24 ]. We now summarize our main result for the holomorphic gravitational coupling Fg for the orbifolds studied in this paper. Consider the twisted elliptic genus of K3 by an automorphism g′ of order N , which is defined as E2k are Eisenstein series of weight 2k and ζ refers to the zeta function. P2g is related to the Schur polynomial S of order g by P2g(x1, x2, · · · xg) = −S 1 1 x1, 2 x2, · · · g xg . – 3 – , are Jacobi forms which transform under SL(2, Z) with index 1 and weight 0 and −2 respec(r,s) are numerical constants and βg(r′,s) is a weight 2 modular form under Γ0(N ). After the discovery of the Mathieu moonshine symmetry in the elliptic genus of K3 [25], the twining character F (0,1) for all the M24 conjugacy classes was first found in [26–28]. For g′ given in table 1, the corresponding to the full twisted elliptic genus M24 have been explicitly determined in [29, 30]. We list them in appendix E for completeness. Now given the twisted elliptic genus, consider the following weight 2g quasi-modular form under Γ0(N ) f (r,s)(τ )P2g(G2, G4, G6, · · · , G2g) = where Then the topological amplitude F¯g is given by The sum over lattice points m > 0 refers to the following lattice points (n1, n2), n1 ∈ n1T + n2U . The functions Li3−2g are polylogarithm functions of order 3 − 2g. From comparing (1.5) and (1.7) we see that indeed it is the coefficients of the twisted elliptic genus of K3 which forms the basic input data for the topological amplitude F¯g(y). We wish to emphasise that this observation is the key result of this paper. This is a generalization of the observation by [22] in which the elliptic genus of K3 which determines the factor E4E6/η24, is the crucial input data for the topological amplitude for the unorbifolded model. Our result is also a generalisation of the work of [10] which evaluated the Fg for the un-orbifolded heterotic compactification on K3 × T 2. Now comparing the instanton contributions in F¯g(y) with the form of the topological amplitude written in terms of the Gopakumar-Vafa invariants we can extract out these invariants. It is apriori not clear that the invariants will be integers since the coefficients c(gr−,s1) in (1.5), themselves are not integers. Gopakumar-Vafa invariants are all integral. This forms a simple consistency check of our result. In fact we will see that once the genus zero Gopakumar-Vafa invariants are integers, the higher genus invariants are assured to be integers. This is shown for genus g = 1, 2, 3. The constant term in (1.7) contains the information of the Euler character of the dual Calabi-Yau X. The Euler character of the Calabi-Yau dual to all the orbifolds considered However to the level we have verified the in this paper is listed in table 4. We then study the conifold singularities of the F¯hol which correspond to points in g moduli space of enhanced gauge symmetry. We observe that there are no conifold singularities from the untwisted sector and all conifold singularities are due to twisted sector states becoming massless. The strength of this singularity is proportional to the genus zero Gopakumar-Vafa invariant corresponding to this state. The list of low lying genus zero Gopakumar-Vafa invariants for g′ which we refer to as CHL orbifolds2 are provided in appendix D. For the rest of the g′ in table 1. we provide an ancillary Mathematica code from which the genus zero Gopakumar-Vafa invariants can be evaluated and seen to be integral. We briefly mention the method we adopt to evaluate Fg. In the heterotic frame it is given in terms of a one loop integral over the fundamental domain. We basically follow the 2These are geometric actions, which keep the Hodge number of K3; h1,1 ≥ 1. – 4 – method of orbits adopted in many works in this subject starting from [31]. See [32] for a recent discussion of these methods. Therefore the methods used in this paper have been developed and established by several previous works. However we do not directly apply the lattice reduction theorem of Borcherds [33] as done by earlier works [10, 34, 35] for evaluating Fg, g > 1. This is because the application of the reduction theorem when the integrand has modular forms of Γ0(N ) is rather intricate and we find it is easier to proceed directly and carry out each of the steps involved in the integrations. We would like to emphasise that using the direct method we pursue, there are several manipulations which are straightforward, but nevertheless important to arrive the standard form of Fg.3 For this reason we find it instructive to provide the details of the way we implement the method of orbits in the appendices B, C. Since this area of research has a rich history with several contributions, it is important to mention that similar integrals involving modular forms of Γ0(N ) have occurred earlier for the specific case of N = 4 in [36] and for N = 2 in [37] and [34]. Modular forms transforming under Γ0(N ) for arbitrary N occurring in the theta lift of the twisted elliptic genus of K3 were done in [38] and [39]. As far as we are aware the present work is the first evaluation of gravitational couplings Fg for the orbifold compactifictions introduced in [20] which generalizes the result of [10] to integrands involving Γ0(N ) forms. The organization of the paper is as follows. In section 2, we review aspects of the Z N we will present the new supersymmetric index which forms the basic ingredient for the gravitational threshold integral. Next in section 3 we write down the expressions for the first evaluate the gravitational coupling F1. We do this because the integrand as well as performing the integral is a straight forward generalization of the one done for the K3 × T 2 theory by [9] and provides checks for our calculation. Next we present the result for the gravitational coupling Fg(y, y¯) for g > 1 and extract the holomorphic coupling F¯hol(y). In g section 5 we study the properties of the genus g topological amplitude and observe that the Gopakumar-Vafa invariants of the dual Calabi-Yau predicted by our calculation. We then study the conifold singularities of F¯hol(y). In section 6 we present our conclusions. g The appendix A contains the modular properties of the integrand involved in evaluating Fg(y, y¯). Appendices B, C contain the details of the integrations to obtain the gravitational thresholds. Appendix D lists the Gopakumar-Vafa invariants of for all g′ corresponding to CHL orbifolds. Finally appendix E lists the elliptic genera of all the orbifolds considered in the paper for completeness. 2 Heterotic string on orbifolds of K3 × T 2 In this section we briefly describe the general class of N = 2 compactifications we will study. Consider the E8 × E8 heterotic string theory compactified on K3 × T 2 in which the SU(2) spin connection of K3 is embedded in one of the E8’s which is called the standard embedding. We then orbifold by a freely acting Z N which acts as a g′ automorphism on 3For example we obtain the form given in (C.7), which requires the next steps to cast in the standard form (C.11). – 5 – 1A 2A 3A 5A 7A 11A 23A/B 4B 6A 8A 14A/B 15A/B 2B 3B 1 2 3 5 7 11 23 4 6 8 14 15 2 3 are such that the twining genera matches with the F (0,1) listed in [25]. These orbifolds are of order 4 and 9 respectively in our analysis. We refer to the classes 2A, 3A, 5A, 7A, 4B, 6A, 8A as the CHL orbifolds. K3 together with a 1/N shift along one of the circles of K3. The g′ action corresponds to any of the 26 conjugacy classes of Mathieu group listed in the table. In the standard embedding one of the E8 lattice breaks to a D6 ⊗ D2 lattice. The SU(2) spin connection of K3 couples to the fermions of D2 lattice. These left moving fermions with the left moving bosons of the K3 combines with the right moving supersymmetric K3 CFT to form a (6, 6) conformal field theory. The internal conformal field theory is given by H (6,6) (6,0) internal = HD2K3 ⊗ HD6 ⊗ HE8 (8,0) ⊗ HT 2 (2,3) (2.1) The g′ orbifold acts as a Z N automorphism on the (6, 6) CFT which preserves its SU(2) R-symmetry together with a 1/N shift on one of the circles of the T 2 CFT. Thus these compactifications preserves N theories as compactifications of the E8 × E8 theory on the orbifold K3 × T 2/ZN . = 2 supersymmetry in 4 dimensions. We refer to these – 6 – The starting point in evaluating both gauge and gravitational threshold corrections is to obtain the new supersymmetric index of the internal CFT. This is defined by Znew = 1 moving fermion number of the T 2 CFT together with the right moving fermion number of β(0,s)(τ ) = − p + 1 Ep(τ ), for 1 ≤ s ≤ p − 1 β(r,rk)(τ ) = π(p − 1) ∂τ [ln η(τ ) − ln η(N τ )] – 7 – The trace in (2.2) is taken over the Ramond sector of the right moving supersymmetric HJEP05(218) internal conformal field theory. Note that (c, c˜) = (22, 9). In [21], the new supersymmetric index for compactifications on the orbifolds K3 × T 2/ZN was evaluated and it was shown that it can be written in terms of the twisted elliptic genus of K3. Let us briefly review the result. First we define the twisted elliptic genus of K3 which is given by 1 N genus can be written in the general form F (0,0)(τ, z) = αg(0′,0)A(τ, z), are Jacobi forms which transform under SL(2, Z) with index 1 and weight 0 and −2 respec(r,s) are numerical constants and βg(r′,s) is a weight 2 modular form under Γ0(N ). For example for g′ ∈ pA with p = 2, 3, 5, 7 we have where Here 2 2 12i (2.2) (2.3) (2.4) (2.5) (2.6) (2.7) (2.8) is a modular from which transforms with weight 2 under Γ0(N ). The twisted elliptic genus transforms under SL(2, Z) as F (r,s) 2T2U2 | − m1U + m2 + n1T + n2T U |2, 12 p2R + m1n1 + m2n2 for a, b, c, d ∈ Z, ad − bc = 1 The indices in (2.9) cs + ar and ds + br are taken to be mod N . Appendix E lists the twisted elliptic genus F (r,s)(τ, z) for corresponding to the conjugacy classes g′ in table. Given the twisted elliptic genus, we an read out αg′ the general form given in (2.5). Then the new supersymmetric index for the standard embedding compactifications of the E8 × E8 heterotic string on the orbifold K3 × T 2/ZN (r,s), βg(r′,s) using is given by Znew(q, q¯) = −2 η24(τ ) Γ(2r,2,s)E4 4 αg′ the lattice sum on T 2 which is defined as Here it is understood that the indices r, s are summed from 0 to N − 1. In (2.11), Γ(2r,2,s) is 1 1 = X c(r,s)(l)ql. l∈ NZ – 8 – and T, U are the K¨ahler and complex structure of the torus T 2. It is also understood in (2.11), that there is a sum over the lattice momenta m1, m2 as well as the winding numbers n1, n2. Note that in (2.11) its only the lattice sum that depends on both q and q¯, the Eisenstein series E4, E6 as well as the Γ0(N ) weight 2 form βg(r′,s) depends only on the holomorphic co-ordinate q. It is instructive to examine the situation in which there is no orbifold performed. Then p = 1 and α1A = 8 and there are no twisted sectors. Thus the new supersymmetric index reduces to Znew(q, q¯) = −4 E4(τ )E6(τ ) Here Γ2,2 is the usual lattice sum without phases and twists. For later purpose we define the following Γ0(N ) form, which occurs in the new supersymmetric index given in (2.11). has the following transformation property under the SL(2, Z) generators Using the fact that βg(r′,s)(τ ) is a Γ0(N ) form and (2.9) it is easy to see that the f (r,s)(τ ) f (r,s)(τ + 1) = f (r,s+r)(τ ), f (r,s) 1 − τ = (−iτ )−2f (N−r,s)(τ ) = (−iτ )−2f (r,N−s)(τ ). Given the new supersymmetric index we can read out the difference between the number of vectors and hypers in the spectrum [9]. This is given by Nh − Nv = − N−1 X c(0,s)(0). s=0 (2.15) (2.17) (2.16) Here, we have also included the four U(1)’s resulting from the metric and the NeveuSchwarz B-field with one index along the T 2 in the counting of the vectors. This data also gives the Euler number of the corresponding type II dual Calabi-Yau compactification which is given by χ(X) = −2(Nh − Nv) = 2 X c(r,s)(0). N−1 s=0 For the compactification on K3 × T 2 the value of Nh − Nv = 240 and therefore the corresponding Euler number of the dual K3-fibered Calabi-Yau manifold is χ(X) = −480. These data for all the orbifolds considered in this paper is provided in table 4. In the subsequent sections we will use the new supersymmetric index in (2.11) as a starting point and evaluate the gravitational one-loop corrections for heterotic compactifications on orbifolds K3 × T 2/ZN . 3 The integral for gravitational thresholds In this section we obtain the integral for gravitational thresholds involving N = 2 heterotic compactifications on orbifolds considered in this paper. Though these integrals were obtained for the unorbifolded situation earlier in [8, 9] we find it instructive to go through the analysis. This is because we wish to arrive at the key result that the integrals which evaluate the thresholds for the orbifolds involve the twisted elliptic genus of K3. It is first easy to discuss the one loop integral which captures the gravitational correction with no graviphoton insertions. The result of this integral provides the moduli dependence of the following higher derivative term in the N = 2 effective action for these compactifications I1(T, U ) = R+2F1(T, U ). (3.1) Here R+ refers to the anti-self-dual Riemann tensor. From the earlier works of [3, 4], a nice compact expression for the F1 has been obtained in [9]. This is given by4 1 F τ2 compactification for N = 1. 4Note that we have normalized F1 so that the result for the integral agrees with [9] for the K3 × T 2 – 9 – The subtraction by bgrav is to ensure that the integral is well defined for τ2 → ∞. Note that the trace over the internal conformal field theory is precisely what occurs in the new supersymmetric index given in (2.2). Therefore to obtain the moduli dependence of the gravitational correction for the orbifold compactifications in this paper, we substitute the result for the new supersymmetric index (2.11) in (3.2) This results in , where we have ignored the constant necessary to regulate the integral. From the properties (A.6) and (2.15) we see that the integrand is invariant under SL(2, Z) transformations. Note that the key input that goes into the integral are the coefficients of the twisted elliptic genus of K3 represented by the functions αg′ integral over the fundamental domain which is done in section 4.1. (r,s), βg(r′,s). The next step is to perform the Let us now discuss the integral which results in the moduli dependence of the following term in the one loop effective action Ig(T, U ) = R+2F+2g−2Fg(T, U ), where F+ is the anti-self dual field strength of the graviphoton and g > 1. From the world sheet analysis of [8] we see that the one loop integral is given by (3.3) (3.4) (3.5) (3.6) The super script (r, s) for pR indicates the sector over which the right moving momentum belongs to. When r 6= 0 the winding number n1 on one of the circles of the T 2 is quantized in units of Z + Nr since the right moving momentum belongs to the rth twisted sector of the Fg = 1 2π2(g!)2 Z d2τ ( τ2 1 τ22η2(τ ) TrR h(i∂¯X)(2g−2)(−1)F F qL0− 2c4 q¯L˜0− 2c˜4 i ×h Yg Z i=1 the transverse non-compact space time bosons. The above result is obtained by carrying out the Wick contractions of the vertex operators for the graviton and the graviphoton insertions as mentioned in [8]. Note that the correlators of the space time bosons and the internal conformal field theory factorize. The 2g − 2 insertions of ∂¯X arise from the (2g − 3) graviphoton vertices in the (0)-ghost picture and one ∂¯X that appears in the picture changing operator. Since the correlator is evaluated on the torus, the insertions of ∂¯X can contribute only through the zero modes. Therefore this can be replaced as follows (i∂¯X)(2g−2) → . 1/N shift action. Then trace over the internal conformal field theory can be written as5 This trace is evaluated using the input in (3.6) and repeating the steps which led to the result in (2.11). The only difference is the insertions of the lattice momenta p(r,s). Note that the sectors (r, s) are summed over, and it is always understood that the lattice momenta and winding are summed. Now we need to simplify the correlators over the free non-compact bosons. To this end we follow [8] and use the generating function for these correlators G(λ, τ, τ¯) = X ∞ 1 g=1 (g!)2 λ τ2 2g Yg Z h i=1 The correlation functions are normalized free field correlators of the space-time bosons. This generating function is given by To use this generating function for the correlators, we consider G(λ, τ, τ¯) = 2πiλη3 2 θ1(λ, τ ) e− πτλ22 . F (λ, T, U ) = X λ2gFg(T, U ). ∞ g=1 Then using the result (3.7) and (3.9) in (3.5), the one loop amplitude simplifies to function admits the expansion p(r,s)λ λ˜ = √2T2U2 R . k=0 Here again note that the key input that goes into the integrand (3.11) for the gravitational couplings for the orbifold compactifications considered in this paper is the twisted elliptic To perform this integral we follow the approach in [10]. The reciprocal of the theta 2πλ˜η3 ! θ1(λ˜, τ ) 2 ∞ e− πτλ˜22 = X λ˜2kP2k(Gˆ2, . . . , G2k), 5In the analysis of [8], the trace over the internal conformal field theory was referred to as Cǫ(τ¯). Further more, what we call as left movers is right movers in [8] and vice-versa. We follow the notations of [10]. where P2k is a polynomial related to the Schur polynomial by P2k(Gˆ2, . . . , , G2k) = −Sk Gˆ2, 2 G4, 3 G6, . . . , k G2k , 1 1 1 where the Schur polynomials are in turn defined by the expansion exp " ∞ X xkzk k=1 # = X Sk(x1, · · · xk)zk, ∞ k=0 x 2 2 x 3 6 P2(Gˆ2) = −Gˆ2, P4(Gˆ2, G4) = − 21 (Gˆ22 + G4), P3(Gˆ2, G4, G6) = − 16 (Gˆ23 + Gˆ2Gˆ4) − 3 G6. 1 S1(x1) = x1, S2(x1, x2) = where the G’s are normalized Eisenstein series given by Note that P is holomorphic, except for the occurrence of Gˆ2 and it transforms as a modular form of weight 2k. Substituting the expansion (3.13) in (3.11), we obtain F (λ, T, U ) = X ∞ k=0 λ2(k+1) π2(2T2U2)k 1 (r,s)E6 − βg(r′,s)(τ )E4 (p(Rr,s))2kP2k+2. Note that the integrand is an invariant under SL(2, Z). This can be seen using the properties in (A.9) and (2.15) 4 4.1 g = 1 Evaluating the integral for gravitational thresholds We will first perform the integral in (3.3). As it will be clear subsequently, it is simpler to treat the case of g = 1 separately from the integral for g > 1. We will follow the unfolding method developed in [9] for performing this integral. However we generalize the discussion to integrands which contain modular forms transforming under Γ0(N ) rather than SL(2, Z). Let us first define the Fourier expansions of the expressions in the integrand. 1 2η24 E4 4 αg′ Note that as expected, the twisted sectors admit fractional q expansions. Substituting this expansion and the values of p2L and p2R from (2.12) into (3.3) we obtain The first step to do the integral is to perform the Poisson re-summation over the momenta m1, m2 using the formula X f (m)e2πism/N = X m∈Z Z ∞ k∈Z+ Ns −∞ duf (u) exp(2πiku). (4.3) Using this identity and performing the integral over the corresponding variables u1, u2, we Z d2τ Therefore using (4.4) in (4.2) we can write the integral as πT2 n1 k1 ! n2 k2 , G(~n, ~k) = − U2τ2 |A|2 − 2πiT (det)A, J (A, τ ) = T2 exp − U2 |A|2 − 2πiT detA f (r,s)(τ )Eˆ2(τ ), Using (4.6), we can think of J as a function of the matrix A. Then the sum over r, s and ~n, ~k in the right hand side of (4.7) can be thought of as the sum over matrices of the I1(T, U ) = F τ22 A 2 Z d τ X J (A, τ ). Now using the modular transformation given in (2.15) and the definition of J in (4.8), it This symmetry allows us to extend the integration over the fundamental domain to its images under SL(2, Z) together with the restriction of the summation over A to its inequivalent SL(2, Z) orbits. Lets denote the sum over inequivalent SL(2, Z) orbits as P′A, then F1 becomes I = X′ Z A d2τ Now the label r, s is to interpreted as N n1 mod N and N k1 mod N respectively. The region of integration FA depends on the orbit represented by A. From the analysis of [9] we see that there are three inequivalent orbits. These are as follows: the zero orbit the non-degenerate orbit and the degenerate orbit ζ(3) T2U2 3 1 2π (4.10) (4.12) (4.13) (4.14) (4.15) (4.16)             (4.17) The contribution from each of these orbits has been evaluated in the appendix B. Taking the sum of these contributions the result for F1 is given by F1 = T2 π6 E22(q)f (0,0)(q) q0 N−1 + X s=0       +4Re X   k≥0,l≥−1 (k,l)6=(0,0) k∈Z+r/N where P˜ is given by c˜(0,s)(0) − log(T2U2) + U2 − κ − c(0,s)(0) π3 T2 6ζ(4, s/N ) U22 + 6 c(0,s)(0) c(0,0)(0) c(0,1)(0) 2A 3A 5A 7A 11A 23A −120 −80 −48 −240/7 −240/11 −240/23 136 109 77 829/14 442/11 473/23 From the appendix it will be clear that the above result holds for T2 > U2 as in the analysis of [9]. Let us examine the result for N = 1, the unorbifolded K3 × T 2 compactification. We obtain For unorbifolded K3 this answer reduces to F1 = −48πT2 − 264(− log(T2U2) − κ) − 88πU2 (4.18) Let us recall that for each of the orbifolds we can read out the coefficients c˜(r,s) and c(r,s) using their definition given in (4.1) and the explicit expressions for the twisted elliptic genera given in the appendix E. Let us make a few observations from the result for F1. Note that the coefficients which determine the moduli dependence of F1 in the first two lines of (4.15) depends on low lying topological data of the new supersymmetric index. This dependence does not involve exponentials in moduli T, U . The low lying coefficients of the new supersymmetric index for the various orbifold compactifications corresponding to the Mathieu moonshine conjugacy classes are listed in tables 2 and 3. One also observes that for an order N orbifold c(0,s)(−1) = 1/N and c˜(0,s)(0) can be written as c˜(0,s)(0) = c(0,s)(0) − N . 24 (4.19) Using this topological data we can evaluate the linear and quadratic dependences of the moduli T and U which result from the contribution of the integral along the zero orbit and the degenerate orbit. These coefficients for the various orbifolds are listed in table 2 and 3. Note that the coefficient of Tζ2(U3)2 is determined by the difference Nh − Nv or the Euler character. An interesting observation is the occurrence of the low lying toplogical data weighted with the Hurwitz-zeta function ζ(2, s/N ) as well as ζ(4, s/N ) in the orbifolds. This can be seen only for the orbifold compactifications. In fact if one is just given the HJEP05(218) c(0,0)(0) c(0,1)(0) s1 c(0,s1)(0) c(0,s2)(0) X πT2 three coefficients of U2, and U22/T2 and 1/T2U2 in the gravitational threshold F1 and the properties of the low lying coefficients in, we can determine c(0,s) for all the orbifold compactifications corresponding to the conjugacy classes of Mathieu Moonshine considered here. A summary of these coefficients, including the Euler number for all the orbifolds considered in this paper is given in table 4. The higher level coefficients of the new supersymmetric index control the exponentially suppressed terms in the last line of (4.15). Note that the exponential dependence of T moduli carries the information of the twisted sectors. 4.2 g > 1 The gravitational threshold integral for g > 1 is given by Fg = 1 where k = g − 1 > 0. Note that here the summation over (r, s) is implied. The steps to perform the integral are similar to the case when g = 1, however one needs to keep track of the extra insertions of the momentum p(r,s). First we perform the Poisson re-summation over the variables (m1, m2) to obtain Fg(T, U ) = N−1 X r,s=0 n2,k2∈Z,n1,k1∈Z+ Nr T2 2U2 k 2 ZF dτ 2τ J˜(A, τ ), 2 J˜(A, τ ) = T2A − U2 |A|2 − 2πiT detA f (r,s)(τ )P2k+2(τ ), 6The Calabi Yau manifolds with χ values −480, 32, 276, 400, 520, 524, 644, 768 are known to exist and they are listed in [40]. Now similar to the g = 1 case we can think of the sum over r, s and ~n, ~k as sum over the matrices of the form in (4.6) with n2, k2 ∈ . Thus (4.21) can be written as Ig(T, U ) = Now using the modular transformation given in (2.15) and the definition of J˜ in (4.22), it This symmetry allows us to extend the integration over the fundamental domain to its images under SL(2, Z) together with the restriction of the summation over A to its inequivalent SL(2, Z) orbits. Let us again denote the sum over inequivalent SL(2, Z) orbits as P′A, then Fg becomes Fg = X′ Z A FA T2 2U2 k d2τ 2 τ 2 T2A Now the label r, s is to interpreted as N n1 mod N and N k1 mod N respectively. The region of integration FA depends on the orbit represented by A. We can now look at the contribution of the three inequivalent orbits. First note that the contribution of the zero orbit vanishes due to the presence of A Thus we are left with the non-degenerate orbit which is characterized by the set of matrices 2k in the integrand. A = and the degenerate orbit n1 j ! 0 p , A = p ∈ Z, n1, j ∈ N 1 Z, n1 > j 0 j ! Note that here we have included the p = 0 case also in the non-degenerate orbit for convenience. This is because the p = 0 can be treated uniformly together with the p 6= 0 situation in the non-degenerate orbit. The detailed evaluation of the integral in the two orbits is carried out in the appendix C. This integral was done in [10] for the case of K3 × T 2 compactification and in [34] for the FHSV compactification. The latter situation involves only modular forms under Γ0(2) in the integrand. Here we have generalized the evaluation of the integral for integrands containing modular forms transforming under Γ0(N ). Further more as it will be clear in the appendix C, we do not directly apply the reduction theorem of Borcherds [33] as done in the earlier works to perform the integral. The application of the reduction theorem is rather intricate and it easier to proceed straightforwardly from (4.25) and carry out the steps involved in the integration. To write the result in a convenient form, let us define the following two dimensional vectors and the inner products. m = (n1, n2), y = (T, U ), m · y = n1T1 + n2U2 + i(n1T1 + n2T2), We also define the Fourier coefficients involved in the expansion of the modular forms as Then, from the evaluation in appendix C, the result for the contribution from the nondegenerate orbit is given by Fgn>on1deg = 22(g−1)π2 r,s=0 m6=0 t=0 h=0 and m 6= 0 refers to any of the following cases, sˆ + 1/2 = |ν|, |ν| = 2k − h − j − t − 1/2, k = g − 1 n1 > 0, n2 > 0; n1 < 0, n2 < 0; r r N n1 = 0, n2 > 0 or n2 < 0; n2 = 0, n1 > 0 or n1 < 0; n1 = , n2 < 0 or n1 = − N , n2 > 0 with r|n2| ≤ N, r > 0. In (4.30), the index r in c(r,s) is related to n1 by r = N n1 result for the non-degenerate orbit is valid for T2, U2 > 0.7 Performing the integral for the mod N and n2 ∈ Z. The 7Note that the result for the integral also contains the complex conjugate F¯g, which we have suppressed for simplicity. (4.28) (4.29) (4.30) (4.31) (4.32) In the (0,0) sector one has and j < n1, j = 0, 1, . . . n1 −1 and p ∈ Z. So we can proceed for the τ1 integral as done in [9] for the un-orbifolded K3. We write τ1′ = j + pU1 + n1τ1, then the relevant part for the τ1 integration in (B) becomes 2 Iˆ(τ1) = T2 Z d τ Here we have focused on the τ1 dependent terms, and kept a generic term using the Fourier expansion of f (0,0(q)Eˆ2(τ ). Replacing τ1 by τ1′ we see that the only j dependence comes from Now performing the sum over j forces n1n2 = n where n2 is an integer. Rest of the τ1 integration in the untwisted sector is Gaussian and the result is given by, If we are in the untwisted sectors (0, s), we have j = j′ + s/N with j′ being an integer the above result holds but with an additional factor of e−2πin2s/N and replacing c(0,0)(n1n2) and c˜(0,0)(n1n2) by c(0,s)(n1n2) and c˜(0,s)(n1n2). Let us now look at the τ2 integration. The τ2 integrand is given by, I2 = pT2U2e−2πin2s/N Z ∞ dτ2 where F is given by, F = −2πτ2n2n1 − U2τ2 πT2 (n1τ2 + pU2)2 − 2πiT n1p − 2πpn2U1 − πn22U2τ2 . T2 The τ2 integration is of the Bessel form and we use the following formulae to evaluate it, Z dx x3/2 e−ax+b/x = pπ/be−2√ab, Z dx x5/2 e−ax+b/x = √πe−2√ab 1 + 2√ab ! 2b3/2 . The result of this exercise yields for the untwisted sectors I |untwisted = 4Re X n1>0,n2≥−1 n1n26=0 n1∈ Z c˜(0,s)(m2/2)e−2πisn2/N Li1(e2πi(n1T +n2U)) (B.7) (B.8) (B.9) (B.10) (B.11) (B.12) 3 c(0,s)(m2/2)e−2πisn2/N P˜(n1T + n2U ) , Let us go over to the twisted sectors. First consider an order N orbifold when N is prime. In the twisted sectors, the n in the Fourier coefficients c(r,s)(n) and c˜(r,s)(n) is such that n ∈ Z/N . While j = j′ + s/N, j′ ∈ Z and n1 ∈ r/N + Z. The modular forms f (r,s) has the property that the transformations of τ → τ + 1 can be used to relate it to f (r,0). After doing this, one effectively works in the (r, 0) twisted sector but now j runs from 0 to n1 − 1/N in steps of 1/N .13 Thus from (B.8) we see that the sum over all the values of j for a prime N would be N n1 iff n/(n1N ) = n2 is an integer, else it is zero. Therefore n = n1n2N , only integral values of n are picked up. However one observes that the coefficients of qn in f (r,s) are related to the coefficients of qn in f (r,0) under transformation τ → τ + 1. If n is fractional the relation is through a multiplicative phase. Using this property one can write the result in terms of the Fourier coefficients in the f (r,s) sector. The final result for a prime N for the non-degenerate orbit can be given by I The above argument also holds when N is composite and the twisted sector r is such that r, N are co-prime. However if r divides N , then there are various sub-orbits for the sectors (r, s) under transformations τ → τ + 1. For example if N = 4, then the sectors (2, 0), (2, 2) and the sectors (2, 1), (2, 3) form distinct sub-sectors not related by transformations of the kind τ → τ + 1. One then uses the argument developed for N prime and carries it out for each of the sub-orbits. The end result works out to be the same as that given in (B.14). Degenerate orbit. Here the determinant of A is 0. So we choose the matrix A as A = 0 j! 0 p , where j ∈ Z/N, p ∈ Z and (j, p) 6= (0, 0). Since n1 = 0 twisted sectors don’t contribute to the degenerate orbit. There are logarithmic divergences due to the constant term in the expansion of f (0,s)E2(τ ) which needs to be renormalized. We have Iˆ = Z d2τ X A′,n,s FA τ22 T2 exp πT2 − U2τ2 |A| 2 c˜(0,s)(n) − πτ2 3 c(0,s)(n) qn, (B.15) multiply the integrand for the c˜(0,s) with a factor of (1 − e−Λ/τ2 ). where |A|2 = |j + pU |2, p ∈ Z, j ∈ Z + s/N in (0, s) sector. To regularize the result we 13See [38] for a discussion of this step for integrands which involve a Γ0(2) from. HJEP05(218) The result of the integral from the second term is given by Now for coefficient of c(0,s) one gets after integrating τ2 the result − c˜(0,s)(0) log(Λ) + γE + 1 + log(2/3√3) . − 6 U 2 X j,p π3 T2 Z and use the following results [9] Note that here j = j′ + s/N where j′ ∈ Z. This is the difference as compared to the calculations of [9]. To evaluate these sums on j, p we need to divide the sum in j′ 6= 0, p = 0 X j′∈Z (j′ + B)2 + C2 = π " C 1 + e2πi(B+iC) 1 − e2πi(B+iC) + e−2πi(B−iC) # 1 − e−2πi(B−iC) Note that in applying these formula, the information of s is present in B as now B = U1p + s/N . When p = 0, the terms in the sum that contribute in the Λ → ∞ limit from (B.17) and (B.19) result in N−1 X s=0 c˜(0,s)(0) . Now we carry out the sum over j′ for p 6= 0 in (B.19) using (B.21). When one does this, there is a term that results from the action of the derivative on the first term of (B.20), that is the term independent of B. This results in The integration domain FA in this orbit is given by The only dependence of τ1 comes from the qn and since it is integrated from −1/2 ≤ τ1 < 1/2 it picks up only the q0 term from f (0,s)Eˆ2. Performing the τ2 integration for the term containing the coefficient c˜(0,s) we obtain  c˜(0,s)(0)  π  X j,p |j + pU |2 − |j + pU |2 + ΛU2/(πT2)  − 1  Now, lets carry out the sum over j′ for p 6= 0 in the first two terms of (B.17). Applying (B.20) from the B independent part we get ∞ X p=1 2 p − pp2 + Λ/πT2U2 2 = − ln πT2U2 Λ − ln 4 + 2γE . ! s=0 π2 N−1 6 c(0,s)(0) X ζ(3) T2U2 Z FA τ2 dτ2 (1 − e−Λ/τ2 ) . (B.16)   (B.17) (B.18) (B.19) (B.20) (B.21) (B.22) (B.23) (B.24) This again comes in all the sectors of (0, s) and taking care of the moduli dependence and combining (B.18) we tobain N−1 s=1 X c˜(0,s)(− log T2U2 − κ), (B.25) 4Re X l6=0,l≥−1 l Z ∈ Combining all these results where κ = 1 − γE + log( 38√π3 ). Finally keeping track of all the second and third terms in the r.h.s. of B.20 and its derivative B.21, that is the B dependent part both in the sums (B.17) and (B.19) we obtain c˜(0,s)(0)e2πisl/N Li1(e2πi(lU)) − πT2U2 3 c(0,s)(0)e2πisl/N P˜(lU ) . (B.26) HJEP05(218) I degen = (B.22) + (B.23) + (B.25) + (B.26). (B.27) C Details for the g > 1, threshold integral The integral we have to do is the one in (4.21). As discussed there is no zero orbit. We have to perform the integral over the non-degenerate orbit and the degenerate orbit. Non-degenerate orbit. This orbit are characterized by matrices of the form For the non-degenerate orbit we apply the unfolding technique and choose a representative of the matrix A and integrate over two copies of the upper half plane. In this case A becomes A = n1 j! 0 p , p ∈ Z, n1, j ∈ N Z , n1 ≥ j ≥ 0. This gives det A = n1p and A = n1τ + j + pU . The domain for the integration now is two copies of the upper half plane. In the (0, s) sectors and (r, s) sectors one can verify that the limits on j, n1 are as in the last section for g = 1. The j dependence is in the phase given in (B.8). Again after shifting τ1 to τ1′ = j + pU1 + n1τ1 which forces the n of the Fourier coefficient to be integer as before. The relevant part for the τ1 integration in (4.21) becomes I(τ1) = T2 Z d2τ τ 2+t 2 T2 2U2 k (τ1′ + i(pU2 + n1τ2))2k Note that the coefficients here would be written as c(r,s)(m2/2, t) in different sectors and a factor of e−2πisn2/N will come as an extra phase to the integral in the corresponding sectors along with e−2πiT n1p. Also a factor of 2 would be present due to the two copies of the upper half plane. We shall not write these in the integral to avoid cumbersome notation and will only reinstate them back in the final result. Let us denote the vector m~ with the components m~ = (n1, n2), m2/2 = n1n2, (C.3) where n2 ∈ Z and n1 ∈ Z + r/N . This integral in (C.2) can be done by using binomial expansion for (τ1′ +i(pU2+n1τ2))2k and shifting the variable τ1′ to τ˜1 = τ1′ − in2U2τ2/T2. Following through these replacements in C.2, we obtain 2 I(τ1) = T2 Z d τ T2 (τ˜1 + in2U2τ2/T2 + i(pU2 + n1τ2))2k (C.4) The τ˜1 can be integrated and we are left with I(τ1) = T2 τ 2+t 2U2 2 1 T2 h ! 2k! 2j h ×Γ(j + 1/2) (πT2/U2τ2)j+1/2 e−πU2τ2n22/T2. ×exp − U2τ2 Here we have written down only the terms that involve τ1 dependence. Now putting back all the terms which have τ2 dpendence we expand the complete integral as a polynomial in τ2 which gives × Z ∞ 0 dτ2 (τ2)l+h−j−3/2−t exp[−πT2U2p2/τ2 − πτ2/(T2U2)(n1T2 + n2U2)2] Integrating term by term and replacing h by 2k − h, which still keeps the limits as 0 to 2k, we obtain I˜ = X 2k 2k−h [k−h/2] 2k − h! 2k! X X T2 T2 2U2 h=0 l=0 j=0 ×2 |p|T2U2 2j h l+2k−h−j−t−1/2 ×Kl+2k−h−j−t−1/2(2πi|p|(n1T2 + n2U2)). 2c(r,s)(r2/2, t)e−2πisn2/N The result of this integral (C.7) seems manifestly different from that obtained in [10] at this step. To bring into in the from we perform changes in the order of summations and limits. We outline these now. First we change the order of summation on l and h which gives 2k 2k−h [k−h/2] X X X h=0 l=0 j=0 → X X X . k e−2πiT1n1pΓ(j + 1/2)(−1)k−j (in2U2τ2/T2)h−2j(ipU2 + in1τ2)2k−h (C.5) (C.6) (C.7) (C.8) Now we call h − l as h+ and change the orders of the sums on h+ and l. Now one needs to change the order of the sums j and l which can be done as l=0 Ks+1/2(x) = l Lim(x) = X∞ x . r π 2x l=1 lm s e−x X (s + k)! 1 k=0 k!(s − k)! (2x)k , (C.9) (C.10) (C.11) (C.12) (C.13) (C.14) (C.15) HJEP05(218) One can shift to the variable j′ = k − j which also runs from 0 to k. We will re-label this variable j. The resultant integral is a gamma function and since there is a sum over p it The final sum over l is doable and this gives us I˜ = 21−k X X e−2πisn2/N c(r,s)(m2/2, t) X X k+1 t=0 n1n26=0,p6=0 ×ph+(T2U2)j+h++1/2−k(n2U2 + n1T2)2k−h+−2j(−1)k−j ×K2k−h+−j−t−1/2(2π|p|(n1T2 + n2U2))e−2πip(n1T2+n2U2). (2k)! j!h+!(2k − h+ − 2j)!(4π)j |p|T2U2 2k−h+−j−t−1/2 The result (C.11 agrees with equation (4.29) of [10] when K3 × T 2 is not orbifolded. In (C.11), the summation over the orbifold sectors is implied. To rewrite the result in terms of poly-logarithms we use the relations This leads to k+1 I˜= 21−k X X X X X 2k [k−h+/2] s t=0 n1,n26=0 h+=0 j=0 a=0 j!h+!(2k − h+ − 2j)! a!(s − a)! (4π)j+a (2k)! (s + a)! (−1)k−j+h+ ×(T2U2)k−t(sgn Im(m · y))h+c(r,s)(m2/2, t)e−2πisn2/N Im(mˆ·y)t−j−aLi1+a+j+t−2k(e2πimˆ·y). The case m = 0 needs to be treated separately. From (C.2) and (C.6) we see that we can perform the integral also for the limit n1 = n2 = 0. After the τ1 integration, for the m = 0 limit we obtain I˜|m=0 = X 2k! j=0 2j × Z ∞ 0 T2 T2 2U2 k Γ(j + 1/2)(−1)k−j dτ2 (τ2)j−3/2−t exp[−πT2U2p2/τ2](pU2)2k−2j(U2/(πT2))j+1/2. forms a zeta function. A factor of 2 arises on taking two copies of the upper half plane. However for values t = k the zeta function blows up so one can use a regularization by adding −ǫ to the power of τ2. We have So from C.15 the contribution from m = 0 to the non-degenerate part of the integral is given by, I nondegen|m=0 = 2 X X(−1)j22j−4k+t k k+1 j=0 t=0 (2k)! (2j)!(k − j)! e−2πisn2/N c(0,s)(0, t) ! πt+1/2 ×(πT2U2)−ǫΓ(1/2 + j + t − k + ǫ)ζ(1 + 2ǫ + 2t − 2k). In the equation (C.17) we have used the fact Γ(n + 1/2) = (2n)!√π n!4n . We now use the equation (C.17) and consider ǫ → 0 limit to extract out the finite contribution as follows, (C.17) (C.18) (C.19) (C.20) HJEP05(218) (πT2U2)−ǫ = 1 − ǫ log (πT2U2) , Γ(1/2 + s + ǫ) = Γ(1/2 + s)(1 + ǫ ψ(1/2 + s)), ζ(1 + 2ǫ) = + γE, 1 2ǫ where ψ(x) is the logarithmic derivative of the gamma function. The resulting sums in C.17 can be explicitly computed for some cases. We use the results Γ(1/2+n) = π1/22−n(2n+1)!! and Γ(1/2 − n) = (−1)nπ1/22n/(2n − 1)!! with n ≥ 0 to arrive at X(−1)j2j j=0 Γ(1/2 + j) = π1/2δk,0. Thus results for k = 0 that is genus one and k 6= 0 are different. For k > 0 we can write the full non-degenerate orbit result as, Ikn>on0d,meg=en0 = 2 X N−1 Xk−1 c(0,s)(m2/2, t) πt+1/2 ζ(1 + 2(t − k))(2T2U2)(k−t) (C.21) s˜=0 × X(−1)s˜22(s˜−2k)+t N−1 c(0,s) k + X s=0 23k πk s˜=0 X(−1)s (2k)! (2s˜)!(k − s˜)! (2k)! (2s˜)!(k − s˜)! Γ(1/2 + s˜ + t − k) ψ(1/2 + s˜). Putting all the contributions together the contribution for Fgn>on1degen is given by Fgn>on1degen = X X X X r,s m6=0 t=0 h=0 X j=0 a=0 X e−2πisn2/N c(r,s)(m2/2, t) (C.22) (sgn(Im(m · y)))h (Im(mˆ·y))t−j−a (T2U2)t X(−1)s s˜=0 Here m 6= 0 is defined in (4.32) and sˆ is defined in (4.31). Degenerate orbit. The degenerate orbit is characterised by matrices of the form A = 0 j ! 0 0 , Z j ∈ N , j 6= 0. The evaluation of the integral in the degenerate orbit proceeds as follows. Let us first assume T2 > U2 for this part of the calculation we have Ideg = T k+1 Z j2kτ2−t−2dτ2e−πj2T2/U2τ2−2πn′τ2 Z 1/2 2 dτ1e2πin′τ1 . −1/2 Note that since we are in the un-twisted sector for this orbit n′ ∈ Z. Furthermore to obtain a non-zero result due to the integration we require n′ = 0. Then the result for Idegen is given by obtain [9, 33]14 (2U2)k N−1 X s=0 t N−1 X s=0 t Ideg = X π−t−1U2(T2/U2)k−tc(0,s)(0, t)t!ζ(2(t + 1 − k, s/N )). If U2 > T2, we need to in fact we need start by performing at T-duality and then we Idegen = X π−t−1T2(U2/T2)k−tc(0,s)(0, t)t!ζ(2(t + 1 − k), s/N ). Now reinstating all factors we obtain Fgd>eg1en = 1 T22g−3 N−1 g X X c(0,s)(0, t) 22g−2πt+3 t! T2 U2 t ζ(2(2 + t − g), s/N ). 14The reason this has to be done has to do with the convergence of the integral for Fourier coefficient when n′ = −1. 0 n(n1,n2) 0 n(n1,n2) 8606976768 115311621680 1242058447872 11292809553810 89550084115200 HJEP05(218) (3/2,1) (5/2,1) (7/2,1) 16 8128 1212576 47890048 1055720304 Genus zero GV invariants for CHL orbifolds In this appendix we list the genus zero Gopakumar-Vafa invariants n0m corresponding to the CHL orbifolds. We observe that they are integers. For the remaining orbifolds in table 1, we provide an ancillary Mathematica code from which these invariants can be evaluated and seen to be integers. One point worth mentioning, is that all the remaining orbifolds especially the case of 2B and 3B do not have any geometric interpretation as actions on the K3 surface. Therefore the fact that the Gopakumar-Vafa invariants turn out to be integers is interesting. E List of twisted elliptic genera In this appendix we provide the list of the twisted elliptic genera used to determine the f (r,s) defined in (2.14) in this paper. We call the classes 2A, 3A, 5A, 7A, 4B, 6A, 7A, CHL (r/3, 0) (1/3,3) (1/3,9) (1/3.12) 378 110646 6063822 164258118 2918573208 HJEP05(218) orbifolds. When g′ corresponding to these classes act on K3, it can be seen that the values of the Hodge number h1,1 ≥ 1. The twisted elliptic genus of classes 2B, 3B is such that their twining character coincides with that constructed in [26–28]. But they do not correspond to M24 symmetry. In fact in M24, the classes have order 2 and 3, however the orbifolds we construct have order 4 and 9. The 2B twisted elliptic genus has been constructed in [52] by considering K3 as 6, SU(2) WZW models at level 1. While 3B twisted elliptic genus we give here was constructed in [29]. Conjugacy class pA, p ={1,2,3,5,7}. F (0,0) = = A(τ, z), F (0,s) = F (r,rs) = 8 p 8 8 p + 1 Ep(τ ) τ + s p (E.1) 1490560 103674750 1820321750 (r/5, 0) (1/5,5) (2/5,5) (3/5,5) (4/5,5) 250 72750 3892000 103674750 1820321750 HJEP05(218) 8 6 2 (n1, n2) (1/5, −1) (1/5,1) (6/5,1) (11/5,1) (16/5,1) 14 1032 159648 7172574 175994068 (n1, n2) (2/5, −1) (2/5,1) (7/5,1) (12/5,1) (17/5,1) (n1, n2) (3/5, −1) (3/5,1) (8/5,1) (13/5,1) (18/5,1) 3040 450704 17378724 375804020 4062 605260 11893715 260500896 (n1, n2) (4/5, −1) (4/5,1) (9/5,1) (14/5,1) (19/5,1) 7082 1045848 39672656 840212212 Conjugacy class 11A. Going through these steps we obtain the following formula for the twisted elliptic genus for 11A. 2 33 2 33 8 11 F (0,0) = = A(τ, z), F (0,s) = F (r,rs) = A(τ, z) − B(τ, z) A(τ, z) + B(τ, z) 1 6 E11(τ ) − 52 η2(τ )η2(11τ ) , 1 66 E11 τ + s 11 − 55 2 η2(τ + s)η2 τ + s (E.2) −642 187 54309 -2881360 76296885 1333671600 (n1, n2) (1/7, −1) (1/7,1) (8/7,1) (15/7,1) (22/7,1) 12 519 81348 3813632 96969840 (n1, n2) (2/7, −1) (2/7,1) (9/7,1) (16/7,1) (23/7,1) (17/7,1) (24/7) (18/7,1) (25/7,1) 1525 227875 9059700 201601562 2034 304146 12117044 268891664 3540 522435 19770135 418523073 0 n(n1,n2) −400 256 16 8 4 Conjugacy class 23A/B. F (0,k)(τ, z) = F (r,rk)(τ, z) = where 1 1 23 3 1 1 23 3 A − B A + B 23 1 12 E23 (n1, n2) (r/4, 0), r = 1, 3 (1/4,12) (1/4,16) (1/4, −2) (1/4,−3) (1/2, −1) (1/2,0) (1/2,1) (1/2,2) (5/4,1) (3/2,1) (E.3) f23,1(τ ) = 2 η3(τ )η3(23τ ) η(2τ )η(46τ ) + 8η(τ )η(2τ )η(23τ )η(46τ ) + 8η2(2τ )η2(46τ ) + 5η2(τ )η2(23τ ) τ + k 1 − 22 f23,1 τ + k 0 n(n1,n2) −524 132 246 12 8 8 (n1, n2) (r/6, 0), r = 1, 5 (1/6,24) (1/6,30) (n1, n2) (r/3, 0), r = 1, 2 (1/2,0) 17990607384 39912 2466900 71374608 1318893168 17990607384 HJEP05(218) (1/6, −1) (5/6, −1) (1/6,1) (5/6,1) (1/6, −2) (1/6,−4) (1/2, −1) (1/3, −1) (2/3,−1) (1/3,1) (2/3,1) 2032 303152 Conjugacy class 4B. The twisted elliptic genus for this class is given by 1024 6072 4 6 4 (7/6,1) 155620 (1/2,1) 3054 (11/6,1) (3/2,1) (5/3,1) F (0,1)(τ, z) = F (0,3)(τ, z) = F (1,s)(τ, z) = F (3,3s) = F (2,1)(τ, z) = F (2,3) = F (0,2)(τ, z) = F (2,2s)(τ, z) = 1 4 1 4 8A 8A 1 4A 3 + B 4A 3 − 3 4 1 4 4B 2B 3 − 3 E2(τ ) , 3 + 3 E2 τ + s 2 − 3 E2(τ ) + 2E4(τ ) 1 − 6 E2 τ + s 2 1 2 E4 B (5E2(τ ) − 6E4(τ ) , τ + s 4 0 n(n1,n2) −644 6 8 8 16 60002304 (n1, n2) (r/8, 0), r = odd (1/8,16) (r/4, 0), r = odd (1/4,4) Conjugacy class 6A. The twisted elliptic genus for 6A are given by F (0,1) = F (0,5); F (0,2) = F (0,4); 1 1 5 3 − B − 6 E2(τ ) − 2 E3(τ ) + 2 E6(τ ) (1/8, −2) (1/8,−3) (1/8,−4) (1/8, −5) (1/8,−6) (1/8,−7) F (1,k) = F (5,5k) = 1 2A 3 + B 1 − 12 E2 τ + k 2 1 − 6 E3 τ + k 3 τ + k 6 128 37888 2167808 (1/4, −2) (1/4,−3) (1/2,−1) (1/8, −1) (1/4, −1) 16 2A − 2 BE3(τ ) , 3 1 8A 4 3 − 3 BE2(τ ) . 517 2028 2 6 2 4 4 (1/2, 0) (5,8,1) 4 (3/4,1) 8096 22976 (1/2,2) 22976 (7/8,1) 4056 2 (1/2,4) (9/8,1) 1 (7/4,1) (E.6) , (E.7) HJEP05(218) F(2,2k+1) = A9 + 3B6 E3 F(4,4k+1) = A9 + 3B6 E3 τ +k+1 , 3 3 F(3,1) = F(3,5) = A B 9 − 12E3(τ)− 72E2 B τ +1 + 8 E2 B 2 3τ +1 , 2 F(3,2) = F(3,4) = A F(2r,2rk) = F(3,3k) = 16 2A+ 12BE3 τ +k , 3 16 83A + 32BE2 2 τ +k . Conjugacy class 8A. where r = 1,3,5,7. F(0,0)(τ,z) = A(τ,z), F(0,1) = F(0,3) = F(0,5) = F(0,7), 81 23A − B −21E4(τ) + 73E8(τ) . F(r,rk)(τ,z) = 81 23A + 8 −E4 4 B τ + k + 3E8 8 7 τ + k (E.11) F(2,1) = F(6,3) = F(2,5) = F(6,7), F(2,3) = F(6,5) = F(2,7) = F(6,1), 1 23A + 8 18 23A + B 3 B −E2(2τ) + 2E4 4 3 3 2τ + 1 2τ + 3 4 ; F(4,4s) = 81 83A + 23BE2 τ + s , 2 F(4,2) = F(4,6) = 1 4A 8 3 − B3 (3E2(τ) − 4E2(2τ) , F(4,2k+1) = 81 23A + B 43E2(4τ) − 23E2(2τ) − 21E4(τ) . (E.10) (E.12) (E.13) F (0,1)(τ, z) = F (0,3) = F (0,5) = F (0,9) = F (0,11) = F (0,13); A 14 3 − B − 36 E2(τ ) − 12 E7(τ ) + 7 91 36 E14(τ ) 1 1 − 72 E2 − 3 η(τ + k)η η(τ )η(2τ )η(7τ )η(14τ ) τ + k 2 1 − 12 E7 τ + k 2 τ + k 7 τ + k 7 η τ + k τ + k ; F (7,2k+1) = F (7,2k) = F (0,2k) = F (2r,2rk) = F (0,7) = F (7,7k) = A A + B + B 1 1 14 3 14 3 14 3 1 8 14 3 1 2 7 3 7 7 3 − 3 η(τ + k)η(2τ + 2k)η − 12 E7(τ ) + + 7 eiπ11/12η(τ )η(7τ )η 7τ + 1 2 − 12 E7(τ ) + + η(τ )η(7τ )η τ 2 7τ 2 τ + k 7 2τ + 2k 7 2τ + 2k 7 1 . τ − 72 E2 τ + 1 2 1 − 72 E2 2 7τ 2 τ + 1 2 7τ + 1 2 7 1 4 2 A − 4 BE7(τ ) A + τ + k 7 A − 3 BE2(τ ) , A + τ + k 2 k runs from 1 to 6, ; k runs from 0 to 6. k runs from 0 to 1. where r=1,3,5,9,11,13 and rk is Mod 14. F (2r,2rk+7) = 1 A 14 3 + B − 6 E2(τ ) − 12 E7 1 τ + k 7 where k runs from 0 to 6 and except 3 and r from 1 to 6. (E.14) (E.15) (E.16) (E.18) (E.19) (E.20) F (0,1)(τ, z) = F (0,2) = F (0,4) = F (0,7) = F (0,8) = F (0,11) = F (0,13) = F (0,14); 1 1 15 3 A A + B − 48 E3 1 1 1 − 4 η(τ + k)η τ + k 3 τ + k 3 τ + k τ + k + 7 48 E15 τ + k τ + k ; 15 3 − B − 16 E3(τ ) − 24 E5(τ ) + 16 E15(τ ) − 4 η(τ )η(3τ )η(5τ )η(15τ ) where r=1,2,4,7,8,11,13,14 and rk is mod 15. The sectors belonging to the 5A and 3A sub-orbits are given by F (0,0) = F (0,3k) = F (3r,3rk) = F (0,5k) = F (5r,5rk) = 4 4 15 3 15 3 A − 3 BE5(τ ) A + 1 τ + k 2A − 2 BE3(τ ) ; 2A + τ + k 3 k runs from 1 to 4; ; k runs from 0 to 4. k runs from 0 to 2. Finally the remaining sectors are given by F (3r,5+3rk) = F (3r,10+3rk) = 1 A 3 A 3 + B + B 1 1 1 3 1 − 4 E3(τ ) − 24 E5 − 4 η(τ + k)η(3τ + 3k)η − 4 E3(τ ) − 24 E5 τ + k τ + k 3 + 8 E5 τ + k 3 + 8 E5 where k runs from 0 to 4 and s=1 to 4. F (5r,3s+5rk) = 1 + B τ + k 3 where k runs from 0 to 2 and s=1 to 2. 5 4 + η(τ + k)η(5τ + 5k)η τ + k 3τ + 3k τ + k 3 τ + k 3 5τ + 5k 3 (E.21) (E.22) (E.23) (E.24) (E.25) Conjugacy class 2B. Conjugacy class 3B. F (0,0)(τ, z) = 2A; F (0,1)(τ, z) = F (0,3)(τ, z), (E.28) F (0,1)(τ, z) = F (0,2)(τ, z) = − B(τ, z) F (1,s)(τ, z) = F (3,3s) = − F (2,2s)(τ, z) = − (E2(τ ) − E4(τ )), 3 B(τ, z) 4 B(τ, z) B(τ, z) 3 E2 1 E2 3 3 E2(τ ), τ + s τ + s − E4 τ + s 4 , − 6 E2(τ ) + 3 E2(2τ ) , F (0,0)(τ, z) = F (0,1)(τ, z) = F (0,2) = F (0,4) = F (0,5) = F (0,7) = F (0,8); (E.29) F (0,1)(τ, z) = − F (0,3)(τ, z) = − F (r,rs)(τ, z) = F (3,1)(τ, z) = − F (3,2)(τ, z) = − 3 A(τ, z) , η2(3τ ) B(τ, z) 4 η2( τ +3s ) , 2B(τ, z) η6(τ + s) 2B(τ, z) e2πi/3 η6(τ ) 2B(τ, z) e4πi/3 η6(τ ) η2(3τ ) η2(3τ ) , , E3(τ ), = F (3,4) = F (3,7) = F (6,2) = F (6,8) = F (6,5); r = 1, 2, 4, 5, 7, 8 = F (3,5) = F (3,8) = F (6,1) = F (6,7) = F (6,4); F (3r,3rk)(τ, z) = − A(τ, z) B(τ, z) τ + k 3 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. [1] J.A. 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Aradhita Chattopadhyaya, Justin R. David. Gravitational couplings in \( \mathcal{N}=2 \) string compactifications and Mathieu Moonshine, Journal of High Energy Physics, 2018, 211, DOI: 10.1007/JHEP05(2018)211