1. Introduction and statement of results This paper concerns the deep properties of the modular forms and mock modular forms.

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1 MOONSHINE FOR M 4 AND DONALDSON INVARIANTS OF CP ANDREAS MALMENDIER AND KEN ONO Abstract. Eguchi, Ooguri, and Tachikawa recently conjectured 9] a new moonshine phenomenon. They conjecture that the coefficients of a certain mock modular form H(τ, which arises from the K3 surface elliptic genus, are sums of dimensions of irreducible representations of the Mathieu group M 4. We prove that H(τ surprisingly also plays a significant role in the theory of Donaldson invariants. We prove that the Moore-Witten 15] u-plane integrals for H(τ are the SO(3-Donaldson invariants of CP. This result then implies a moonshine phenomenon where these invariants conjecturally are expressions in the dimensions of the irreducible representations of M 4. Indeed, we obtain an explicit expression for the Donaldson invariant generating function Z(p, S in terms of the derivatives of H(τ. 1. Introduction and statement of results This paper concerns the deep properties of the modular forms and mock modular forms which arise from a study of the K3 surface elliptic genus. To define these objects, we require Dedekind s eta-function η(τ := q 1 4 n=1 (1 qn (note. τ H throughout and q := e πiτ, and the classical Jacobi theta function ϑ ab (v τ := n Z q (n+a 8 e πi (n+a(v+ b, where a, b {0, 1} and v C. We recall some standard identities. ϑ 1 (v τ = ϑ 11 (v τ ϑ 1 (0 τ = 0 ϑ 1(0 τ = πη 3 (τ ϑ (v τ = ϑ 10 (v τ ϑ 3 (v τ = ϑ 00 (v τ ϑ 4 (v τ = ϑ 01 (v τ ϑ (0 τ = n Z q (n+1 8 ϑ (0 τ = 0 ϑ 3 (0 τ = n Z q n ϑ 3(0 τ = 0 ϑ 4 (0 τ = n Z ( 1n q n ϑ 4(0 τ = 0 Moreover, for convenience we let ϑ j (τ := ϑ j (0 τ for j =, 3, 4. The K3 surface elliptic genus 7] is given by (ϑ ( (z τ ϑ3 (z τ Z(z τ = ϑ (τ ϑ 3 (τ ( ] ϑ4 (z τ. ϑ 4 (τ The second author thanks the NSF and the Asa Griggs Candler Fund for their generous support. 1

2 ANDREAS MALMENDIER AND KEN ONO This expression is obtained by an orbifold calculation on T 4 /Z in 6]. Its specializations at z = 0, z = 1/ and z = (τ + 1/ gives the classical topological invariants χ=4, σ=16 and  = respectively. Here we consider the following alternate representation obtained by Eguchi and Hikami 8] motivated by superconformal field theory: Z(z τ = ϑ 1(z τ ( 4 µ(z; τ + H(τ. η(τ 3 Here H(τ is defined by (1.1 H(τ := 8 w { 1, 1+τ, τ } where µ(z; τ is the famous function µ(z; τ = µ(w; τ = q 1 8 i eπiz ϑ 1 (z τ n Z ( 1 + A n q n, n=1 ( 1 n q 1 n(n+1 e πinz 1 q n e πiz defined by Zwegers 18] in his thesis on Ramanujan s mock theta functions. As explained in 8], H(τ is the holomorphic part of a weight 1/ harmonic Maass form, a so-called mock modular form. Its first few coefficients A n are: n A n Amazingly, Eguchi, Ooguri, and Tachikawa 9] recognized these numbers as sums of dimensions of the irreducible representations of the Mathieu group M 4. Indeed, the dimensions of the irreducible representations are (in increasing order: 1, 3, 45, 45, 31, 31, 5, 53, 483, 770, 770, 990, 990, 1035, 1035, 1035, 165, 1771, 04, 77, 331, 350, 5313, 5544, 5796, One sees that A 1, A, A 3, A 4 and A 5 are dimensions, while A 6 = and A 7 = We have their moonshine conjecture 1 also referred to as umbral moonshine 4]: Conjecture (Moonshine. The Fourier coefficients A n of H(τ are given as special sums of dimensions of irreducible representations of the simple sporadic group M 4. Here we prove that the coefficients of H(τ encode further deep information. We compute the numbers D m,n H(τ], the Moore-Witten 15] u-plane integrals for H(τ, and we prove that they are, up to a multiplicative factor of 1, the SO(3-Donaldson invariants for CP. 1 This is analogous to the Monstrous Moonshine conjecture by Conway and Norton which related the coefficients of Klein s j-function to the representations of the Monster 1]. By work of Frenkel, Lepowsky and Meurman 10], and Borcherds ] (among others, moonshine for j(τ is now understood. As in the case of the Montrous Moonshine Conjecture, there are many representations of the generic A n, and so the proper formulation of this conjecture requires a precise description of these sums 11].

3 MOONSHINE FOR M 4 AND DONALDSON INVARIANTS OF CP 3 These invariants are a sequence of rational numbers which together form a diffeomorphism class invariant for CP (for background see 5, 1, 13, 14]. Theorem 1.1. For all m, n N 0, the SO(3-Donaldson invariants Φ m,n for CP satisfy 1 Φ m,n = D m,n H(τ]. Remark. The u-plane integrals D m,n H(τ] are given explicitly in terms of the coefficients of H(τ (see 3.1. Therefore, the Eguchi-Ooguri-Tachikawa Moonshine Conjecture implies that these Donaldson invariants are given explicitly in terms of the dimensions of the irreducible representations of M 4. We will discuss the numerical identities implied by Theorem 1.1 in Section 3.. We also describe the Donaldson invariant generating function in terms of derivatives of H(τ. This paper builds upon earlier work by the authors 14] on the Moore-Witten Conjecture for CP. We shall make substantial use of the results in that paper, and we will recall the main facts that we need to prove Theorem 1.1. In Section we recall basic facts about those weight 1/ harmonic Maass forms whose shadow is the cube of Dedekind s eta-function. In Section 3 we recall and apply the main results from 14]. In particular, we recall the relationship between the u-plane integrals for such forms and the SO(3-Donaldson invariants for CP. We then conclude with the proof of Theorem Certain harmonic Maass forms We let M(τ be a weight 1/ harmonic Maass form 3 (for definitions see 3, 16, 17] for Γ( Γ 0 (4 whose shadow 4 is η(τ 3. Namely, we have that d (.1 i d τ M (τ = 1 η 3 (τ. Imτ For such M(τ, we write M(τ = M + (τ + M (τ, where the holomorphic part, a mock modular form, is M + (τ = q 1/8 n 0 H n q n/. The non-holomorphic part M (τ is M (τ = i ( 1 ( 1 l (l + 1 Γ, π Imτ q (l+1 8, π l 0 where Γ(1/, t is the incomplete Gamma function. This follows from Jacobi s identity η(τ 3 = q 1 8 ( 1 n (n + 1q n +n. n=0 Remark. Note that the non-holomorphic part M (τ is the same for every weight 1/ harmonic Maass form with shadow η 3 (τ since this part is obtained as the Eichler-Zagier integral of the shadow. However, the holomorphic part is not uniquely determined. It is unique up to the addition of a weakly holomorphic modular form, a form whose poles (if any are supported at cusps. 3 These forms were first defined by Bruinier and Funke 3] in their work on geometric theta lifts. 4 The term shadow was coined by Zagier in 17].

4 4 ANDREAS MALMENDIER AND KEN ONO The next result gives families of modular forms from such an M(τ using Cohen brackets. To make this precise, we recall the two Eisenstein series E (τ := 1 4 dq n and Ê (τ := E (τ 3 πimτ. n=1 The authors proved the following lemma in 14]. d n Lemma.1. Lemma 4.10 of 14]] Assuming the hypotheses above, we have that k ( ( k Γ 1 ( E k 1 M(τ] := ( 1 j j Γ ( 1 + j j 3 j E k j (τ q d j M (τ dq j=0 is modular of weight k + 1/ for Γ( Γ 0 (4, and it satisfies d i d τ E k 1 M(τ] = 1 Ê k (τ η 3 (τ. Imτ This lemma implies the following corollary: Corollary.. If M(τ and M(τ are weight 1 harmonic Maass forms on Γ( Γ 0(4 whose shadow is η(τ 3, then ] E k 1 M(τ] E k 1 M(τ = E k 1 M(τ M(τ ] = E k 1 M + (τ M ] + (τ is a weakly holomorphic modular form of weight k + 1/..1. The Q(q series. Here we recall one explicit example of a harmonic Maass form which plays the role of M(τ in the previous subsection. To this end, we define modular forms A(τ and B(τ by A(τ := A(8τ = a(nq n := η(4τ8 η(8τ = 7 q 1 8q 3 + 7q 7, B(τ := B(8τ = n= 1 n= 1 b(nq n := η(8τ5 η(16τ 4 = q 1 5q 7 + 9q 15. We sieve on the Fourier expansion of A(τ to define the modular forms A 3,8 (τ := A 3,8 (8τ = a(nq n = 8q 3 56q 11 +, A 7,8 (τ := A 7,8 (8τ = n 3 (mod 8 n 7 (mod 8 a(nq n = q 1 + 7q q We also recall the definition of the following mock theta function M(q := q 1 n=0 ( 1 n+1 q 8(n+1 n k=1 (1 q16k 8 n+1 k=1 (1 + q16k 8 = q 7 + q 15 3q 3 +,

5 MOONSHINE FOR M 4 AND DONALDSON INVARIANTS OF CP 5 We define and so we have that Q + (q = Q + (τ := 7 A 3,8(τ + 3 A 7,8(τ 1 B(τ + 4M(q, (. Q + (τ/8 = 1 q 1 8 ( q q q q +... In terms of this q-series, the authors proved the following theorem in 14]. Theorem.3. Theorem 7. of 14]] The function Q + (τ/8 is the holomorphic part of a weight 1/ harmonic Maass form on Γ( Γ 0 (4 whose shadow is η(τ u-plane integrals, Donaldson invariants and the proof of Theorem 1.1 Suppose again that M(τ is a weight 1/ harmonic Maass form on Γ( Γ 0 (4 whose shadow is η(τ 3. For m, n N 0, the authors proved that the quantities (3.1 D m,n M + (τ] := n k=0 ( 1 k+1 n 1 3 n (n! (n k! k! ϑ 9 4(τ ϑ 4 (τ + ϑ 4 3(τ] m+n k ϑ (τ ϑ 3 (τ] m+n+3 E k 1 M + (τ ]] where.] q 0 denotes the constant coefficient term, are the Moore-Witten u-plane integrals for M(τ (cf. 14]. Notice that if M(τ is another such form, then (3. D m,n M + (τ] D m,n M + (τ] = D m,n M + (τ M + (τ]. In their seminal paper 15], Moore and Witten essentially conjectured that the u-plane integrals in (3.1 for a suitable M + (τ should give the SO(3-Donaldson invariants of CP. These invariants are an infinite sequence of rational numbers Φ m,n labeled by integers m, n N that can be assembled in a generating function in the two formal variables p, S: Z(p, S = m,n 0 Φ m,n p m S n m! (n!. This power series is a diffeomorphism invariant for CP. The main theorem in 14] proved this conjecture for Q + (τ/8. Theorem 3.1. Theorem 1.1 of 14]] For m, n N 0 we have that Φ m,n = D m,n Q + (τ/8]. Using the work in 14], we prove the following important theorem. Theorem 3.. Let M(τ be as above, then for all m, n N 0 we have: D m,n Q + (τ/8] D m,n M + (τ] = D m,n Q + (τ/8 M + (τ] = 0. q 0,

6 6 ANDREAS MALMENDIER AND KEN ONO Proof. We prove that the constant terms vanish in expressions of the form n ( 1 k+1 (n! ϑ 9 4(τ ϑ 4 (τ + ϑ 4 3(τ] m+n k n 1 3 n (n k! k! ϑ (τ ϑ 3 (τ] m+n+3 E k 1 Q + (τ/8 M(τ ]. k=0 It is sufficient to show that this is the case for each summand. Therefore, after rescaling τ 8τ and q q 8 it is enough to show that the constant vanishes in (3.3 Θ 9 4(τ 16 Θ 4 (τ + Θ 4 3(τ] m+n k Θ (τ Θ 3 (τ] m+n+3 E k 1 Θ 9 = 4(τ Θ (τθ 3 (τη(8τ 3 Q + (τ M(8τ ] 16 Θ 4 (τ + Θ 4 3(τ] m+n k η(8τ 3 Θ (τ Θ 3 (τ] m+n k (Θ (τθ 3 (τ E k k+ 1 Q + (τ M(8τ ]. Here the classical theta functions are defined by Θ (τ := η(16τ = q (n+1 = q + q 9 + q 5 +, η(8τ Θ 3 (τ := n=0 η(8τ 5 η(4τ η(16τ = 1 + q 4n = 1 + q 4 + q 16 + q 36 +, n=1 Θ 4 (τ := η(4τ η(8τ = 1 + ( 1 n q 4n = 1 q 4 + q 16 q n=1 These are related to the theta functions ϑ (τ, ϑ 3 (τ and ϑ 4 (τ by ( τ ( τ ( τ ϑ (τ = Θ, ϑ 3 (τ = Θ 3, ϑ 4 (τ = Θ We define a weakly holomorphic modular function by (3.4 Ẑ 0 (q = Ẑ0(τ := where E (4τ is the weight Eisenstein series with E (4τ Θ (τ Θ 3 (τ E (4τ = 16Θ (τ 4 + Θ 3 (τ 4 = 1 + 4q 4 + 4q +, and Ẑ0(τ/8 is a Hauptmodul for Γ 0 (4. A calculation shows that Equation (3.3 becomes q d dq Ẑ0(q = Θ 4 (τ 9 Θ (τθ 3 (τη(8τ 3. (3.5 q d dq Ẑ0(q Ẑ0(q m+n k H k (q, where (3.6 H k (q := η(8τ 3 (Θ (τθ 3 (τ k+ E k 1 Q + (τ M(8τ ].

7 MOONSHINE FOR M 4 AND DONALDSON INVARIANTS OF CP 7 To prove the theorem, it suffices to show that the constant term in (3.5 vanishes. Hence, it is enough to show that H k (q is a polynomial in Ẑ0(q. To this end, we define M0 (Γ 0 (8 to be the space of modular function on Γ 0 (8 which are holomorphic away from infinity, and is a subspace of C((q. One can easily verify that M0 (Γ 0 (8 is precisely the set of polynomials in Ẑ0(q. From Corollary. we can observe that H k (q is modular with weight 0. A calculation shows that (Θ (τθ 3 (τ = q f(q 4 is holomorphic away from infinity, and f(q Zq]]. We also have η(8τ 3 = q g(q 8 and E k 1 Q + (τ M(8τ] = q 1 h(q 4, where g(q, h(q Zq]]. Hence, H k (q C((q is modular of weight 0 on M0 (Γ 0 (8, and so is a polynomial in Ẑ0(τ Proof of Theorem 1.1. Since H(τ is the mock modular part of a weight 1/ harmonic Maass form on Γ( Γ 0 (4 whose shadow is the 8 3 η(τ 3 / = 1 η(τ 3, it follows from Theorem 3.1 and 3. that for m, n N 0 we have: ] Φ m,n = D m,n Q + (τ/8] = D m,n H(τ/1] + D m,n Q + (τ/8 H(τ/1 = D m,n H(τ/1] = 1 1 D m,n H(τ]. 3.. Discussion of the identities implied by Theorem 1.1. In the table below we list the first non-vanishing SO(3-Donaldson invariants Φ m,n of CP as well as the coefficients D m,n M + (τ] when the mock modular form is given as M + (τ = q 1/8 k 0 H k q k/. In general, D m,n M + (τ] is nonvanishing for m + n 0 (mod and a rational linear combination of the first (m + n/ + 1 coefficients of M + (τ. (m, n Φ m,n D m,n M + (τ] (0, H 1 + 6H 0 (0, H H H 0 (1, H H H 0 (, H 1 4 H H 0 (0, H H H H 0 (1, H H H H 0 (, H H H H 0 (3, H H H H 0 (4, H H H H 0 Theorem 3.1 states that choosing M + (τ = Q + (τ/8 from (. we find equality of the Donaldson invariants Φ m,n and the u-plane integral D m,n M + (τ]. In fact, setting H 0 = 1, H 1 = 8, H = 39, H 3 = 196 in the third column of the table above gives the Donaldson invariants of the second column. On the other hand, the choice M + (τ = H(τ/1 from (1.1 implies that H 0 = 1/6, H k = A k /6, H k+1 = 0 for k N. Theorem 1.1 states that choosing M + (τ = H(τ/1 we still find equality of the Donaldson invariants Φ m,n and the u-plane integral D m,n M + (τ].

8 8 ANDREAS MALMENDIER AND KEN ONO In fact, setting H 0 = 1/6, H 1 = 0, H = 45/6, H 3 = 0 in the third column of the table gives the Donaldson invariants of the second column as well. The proof of Theorem 1.1 implies the following form for the generating function Z(p, S of the SO(3-Donaldson invariants of CP in terms of the mock modular form H(τ: (3.7 Z(p, s = p m S n m+3n+4 3 n+1 m! n! m,n 0 q d n ( ] n dq Ẑ0(q ( 1 k Ẑ 0 (q m+n k Ê k k H(8τ], k=0 q 0 where Ẑ0(q was defined in (3.4 and we have set Ê k H(8τ] = η(8τ 3 (Θ (τθ 3 (τ k+ E k 1 H(8τ]. References 1] J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. London Math. Soc. 11 (1979, ] R. E. Borcherds, Monstrous Moonshine and Monstrous Lie superalgebras, Invent. Math. 109 (199, ] J. H. Bruinier and J. Funke, On two geometric theta lifts, Duke Math. J. 15 (004, ] M. C. N. Cheng, J. F. R. Duncan, J. A. Harvey, Umbral Moonshine, arxiv: v math.rt]. 5] S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds, Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, ] T. Eguchi, H. Ooguri, A. Taormina and S. K. Yang, Superconformal Algebras And String Compactification On Manifolds With SU(N Holonomy, Nuclear Phys. B 315 (1989, no. 1, ] T. Eguchi, Y. Sugawara, A. Taormina, Modular forms and elliptic genera for ALE spaces. Adv. Stud. Pure Math., 61, Math. Soc. Japan, Tokyo, ] T. Eguchi, K. Hikami, Superconformal algebras and mock theta functions. II. Rademacher expansion for K3 surface. Commun. Number Theory Phys. 3 (009, no. 3, ] T. Eguchi, H. Ooguri, Y. Tachikawa, Notes on the K3 surface and the Mathieu group M4. Exp. Math. 0 (011, no. 1, ] I. B. Frenkel, J. Lepowsky, and A. Meurman, A natural representation of the Fischer-Griess Monster with the modular J function as character, Proc. Natl. Acad. Sci. USA 81 (1984, ] M. R. Gaberdiel, R. Volpato, Mathieu Moonshine and Orbifold K3s, arxiv: v1 hep-th]. 1] L. Göttsche, Modular forms and Donaldson invariants for 4-manifolds with b + = 1, J. Amer. Math. Soc. 9 (1996, no. 3, ] L. Göttsche and D. Zagier, Jacobi forms and the structure of Donaldson invariants for 4-manifolds with b + = 1, Selecta Math. 4 (1998, no. 1, ] A. Malmendier, K. Ono, SO(3-Donaldson invariants of CP and mock theta functions. Geometry and Topology. accepted for publication. arxiv: math.dg]. 15] G. Moore and E. Witten, Integration over the u-plane in Donaldson theory, Adv. Theor. Math. Phys. 1 (1997, no., ] K. Ono, Unearthing the visions of a master: Harmonic Maass forms and number theory, Proceedings of the 008 Harvard-MIT Current Developments in Mathematics Conference, Int. Press, Somerville, Ma., 009,

9 MOONSHINE FOR M 4 AND DONALDSON INVARIANTS OF CP 9 17] D. Zagier, Ramanujan s mock theta functions and their applications d après Zwegers and Bringmann-Ono], Sém. Bourbaki (007/008, Astérisque, No. 36, Exp No. 986, vii-viii, ( ] S. P. Zwegers, mock theta functions, Ph.D. Thesis, Universiteit Utrecht, 00. Department of Mathematics, Colby College, Waterville, Maine address: andreas.malmendier@colby.edu Department of Mathematics and Computer Science, Emory University, Atlanta, Georgia 303 address: ono@mathcs.emory.edu