Notes on the K3 Surface and the Mathieu Group M24 Tohru Eguchi, Hirosi Ooguri, and Yuji Tachikawa

Home , Mathieu group M24, Monster group

Experimental Mathematics, 20(1):91–96, 2011 Copyright C Taylor & Francis Group, LLC ISSN: 1058-6458 print DOI: 10.1080/10586458.2011.544585

CONTENTS We point out that the elliptic genus of the K3 surface has a natural 1. Introduction and Conclusions decomposition in terms of dimensions of irreducible represen- 2. Appendix: Data on M 24 tations of the largest Mathieu group M24. The reason remains a 3. Appendix: M24 and the classical geometry of K3 mystery. Acknowledgments References

1. INTRODUCTION AND CONCLUSIONS The elliptic genus of a complex D-dimensional hyper- K¨ahler manifold M is deﬁned as

3 FL +FR L 0 L¯ 0 4πizJ ,L Zell(τ; z)=TrR×R(−1) q q¯ e 0 in terms of the two-dimensional supersymmetric sigma model whose target space is M [Witten 87]. Since M is assumed to be hyper-K¨ahler, the two-dimensional theory has N = 4 superconformal algebra as its symmetry. Then L0 and L¯0 are zero modes of the left- and right- J 3 moving Virasoro operators; 0 is the zero mode of the third component of the aﬃne SU(2) algebra; FL and FR are the left- and right-moving fermion numbers. The trace is taken over the Ramond sector of the theory. This elliptic genus is a Jacobi form of weight zero and index D/2. The elliptic genus for the K3 surface was explicitly calculated in [Eguchi et al. 89] and is given by

Zell(K3)(τ; z) (1–1) θ (τ; z) 2 θ (τ; z) 2 θ (τ; z) 2 =8 2 + 3 + 4 . θ2 (τ;0) θ3 (τ;0) θ4 (τ;0)

Here θi(τ; z)(i =2, 3, 4) are Jacobi theta functions. Ac- tually, the space of Jacobi forms of weight zero and index one is known to be one-dimensional, and thus the above result could have been guessed without explicit computation. We ﬁnd that Zell(K3)(τ; z = 0) = 24 and Zell(K3)(τ; z =1/2) = 16 + O(q), and thus (1–1) repro- duces the Euler number and signature of K3. In [Eguchi et al. 89] and more recently in 2000 AMS Subject Classiﬁcation: 20D08, 14J28, 58J26 [Eguchi and Hikami 09], the expansion of the K3 elliptic Keywords: K3 surface, elliptic genus, Mathieu groups genus in terms of the irreducible representations of the

91 92 Experimental Mathematics, Vol. 20 (2011), No. 1

N = 4 superconformal algebra has been discussed in detail. We first provide the data of representation theory n 123456 7 8 9··· [Eguchi and Taormina 87, Eguchi and Taormina 88]. An 45 231 770 2277 5796 13915 30843 65550 132825 ··· For a rigorous mathematical exposition, see, for example, [Kac 98, Kac and Wakimoto 04]. TABLE 1. The initial coefficients An of the series (1–4) Let us introduce the character formula of the BPS (short) representation of spin = 0 in the Ramond sector On the other hand, the asymptotic behavior of An F with (−1) insertion at large n has recently been derived using an ana- θ τ z 2 logue of the Rademacher expansion of modular forms Rˆ τ z 1 ( ; ) µ τ z , chh= 1 ,=0( ; )= ( ; ) (1–2) 4 η(τ)3 [Eguchi and Hikami 09]: ∞ πiz 1 n(n+1) 2πinz √ −ie n q 2 e 2 2π 1 (n− 1 ) µ(τ; z)= (−1) . An ≈ √ e 2 8 . (1–5) θ (τ; z) 1 − qn e2πiz 8n − 1 1 n=−∞ A The BPS representation has nonvanishing index It turns out that (1–5) gives a good estimate of n even at smaller values of n, and this confirms the Rˆ τ z . chh 1 , ( ; =0)=1 An = 4 =0 positivity of the coefficients . Note that the series µ τ z We also introduce the character of a non-BPS (long) rep- ( ; ) (1–2) has the form of a Lerch sum (or mock resentation with conformal dimension h: theta function) and thus has a complex modular 2 transformation law that involves Mordell’s integral. h− 3 θ1 (τ; z) q 8 . In such a situation we can use the method recently η τ 3 ( ) developed in [Zwegers 02, Bringmann and Ono 06, Then the elliptic genus is expanded as Bringmann and Ono 08, Zagier 07] and construct the θ τ z 2 Poincaré–Maass series to derive the above asymptotic Z K τ z Rˆ τ z τ 1 ( ; ) , ell( 3)( ; )=24chh= 1 ,=0( ; )+Σ( ) 4 η(τ)3 formula. (1–3) Table 1 contains a surprise: the first five coefficients, A ,...,A , are equal to the dimensions of the irreducible where the expansion function Σ(τ) is given by 1 5 representations of M24, the largest Mathieu group; see 1 1+τ A A Σ(τ)=−8 µ τ; z = + µ τ; z = (1–4) Section 2. The coefficients 6 and 7 can also be nicely 2 2 decomposed as sums of dimensions:1 τ + µ τ; z = 2 A6 = 3520 + 10395, ∞ −1/8 n A = 10395 + 5796 + 5544 + 5313 + 2024 + 1771. = −2 q 1 − An q . 7 n =1 For n ≥ 8, it is still possible to decompose An into a If one uses the relation that the non-BPS representation sum of dimensions of irreducible representations of M24, splits into a sum of BPS representations at the unitarity but the decompositions are not as unique.2 bound h =1/4, then This observation can be compared to the famous ob- 2 − 1 θ1 (τ; z) Rˆ Rˆ servation in [Thompson 79], where the first few terms of q 8 τ z τ z , =2chh= 1 ,=0( ; )+chh= 1 ,= 1 ( ; ) J q η(τ)3 4 4 2 the expansion coefficients of ( ), the polar term in Σ may be eliminated, and the decom- J q 1 q2 ··· , ( )=q + 196884 + 21493760 + (1–6) position (1–3) can also be written as Z K τ z Rˆ τ z − Rˆ τ z could be naturally decomposed into the sum of the di- ell( 3)( ; )=20chh= 1 ,=0( ; ) 2chh= 1 ,= 1 ( ; ) 4 4 2 mension of the irreducible representation of the monster ∞ 2 n− 1 θ1 (τ; z) +2 An q 8 . η τ 3 n ( ) =1 1 The tentative decomposition of A7 shown in a previous version The coefficients An have been computed explicitly for of this paper was incorrect in view of a later study of twisted elliptic genus in [Cheng 10, Gaberdiel et al. 10]. Here it is corrected lower orders by expanding the series (1–4) (see Table 1), according to those papers. and it has been conjectured that they are all positive 2 It may also be interesting to point out that 2, 3, 5, 7, 11, 23 appear integers [Ooguri 89, Taormina and Wendlend 89, private in the prime factorization of An more frequently than any other prime numbers and with certain periodicities in n.Thesearealso communication]. prime factors of the order of M24. Eguchi et al.: Notes on the K3 Surface and the Mathieu Group M24 93 simple group. In [Conway and Norton 79] it is formulated ant separately in left and right sectors, and the discussion in terms of an infinite-dimensional graded representation is effectively the same as considering the elliptic genus. of the monster group i Vi such that dim Vi is the coef- It will be extremely interesting to see whether the i ficient of q of J(q), and Conway and Norton called this Mathieu group M24 in fact acts on the elliptic genus observation monstrous moonshine. of K3. It was then found [Frenkel et al. 88] that this representation is naturally associated with the two-dimensional 2. APPENDIX: DATA ON M24 26 string propagating on R /Λ/Z2 , where Λ is the Leech The largest Mathieu group, M24,has lattice. See, for example, [Gannon 04] for a recent review. In our case, the existence of a natural vector 210 · 33 · 5 · 7 · 11 · 23 = 244823040 q space whose graded dimension gives Σ( ) is guaran- elements. There are 26 conjugacy classes and 26 teed by the construction: it is the Hilbert space of the irreducible representations. The character table is two-dimensional supersymmetric conformal field theory given in Table 2, whose data are taken from whose target space is K3. The problem is to identify the [Math. Soc. Japan 07, Conway et al. 85]. The conju- M 3 action of 24 on it. gacy class is labeled according to the convention of The nonabelian symplectic symmetry of K3 was stud- ± [Conway et al. 85]. In the character table, ep stands for ied mathematically in [Mukai 88, Kondo 98]. Mukai enu- ± √ merated eleven K3 surfaces that possess finite nonabelian ep =(± −p − 1)/2. automorphism groups. It turns out that all these groups The dimensions of the irreducible representations are, M are various subgroups of the Mathieu group 24; see Sec- in increasing order, tion 3 for more details. Is it possible that these automorphism groups at isolated points in the moduli space of the 1, 23, 45, 45, 231, 231, 252, 253, 483, 770, 770, K3 surface are enhanced to M24 over the whole moduli 990, 990, 1035, 1035, 1035, 1265, 1771, 2024, 2277, space when we consider the elliptic genus? This question 3312, 3520, 5313, 5796, 5544, 10395. is currently under study using Gepner models and matrix factorization. Here the irreducible representations of dimensions As discussed in [Eguchi and Hikami 10], expansion co- 45, 231, 770, 990, 1035 efficients of elliptic genera of hyper-Kähler manifolds in general have an exponential growth and are closely re- come in complex conjugate pairs. There is in addition an lated to the black hole entropy. In particular, in the case extra real 1035-dimensional irreducible representation. of the kth symmetric product of the K3 surfaces we ob- 3. APPENDIX: M AND THE CLASSICAL GEOMETRY tain the leading behavior 24 OF K3 2 2 2π k n− k A ≈ e k +1 ( 2(k +1) ) , n Here we briefly summarize the relation between the classical geometry of the K3 surface and M ,firstfoundin which gives√ the entropy of the standard D1-D5 black hole 24 [Mukai 88] and elaborated in [Kondo 98]. S ≈ 2π kn at large k (k = Q1 Q5 , where Q1 and Q5 are the numbers of D1 and D5 branes). Thus the elliptic Before proceeding, we need to recall the definition of M genus of the K3 surface may be considered as describ- 24. Of many equivalent ways to define it, one that is ing the multiplicity of microstates of a small black hole most understandable to string theorists is to use an even self-dual lattice of dimension 24. Consider the root lattice with Q1 = Q5 =1. A Here the situation is somewhat similar to a model of of 1 whose generator has squared length 2. Let us denote A∗ black hole described in [Witten 07], where microstates of its weight lattice by 1 , whose generator has squared / A 24 a small black hole span the representation space of the length 1 2. Take the 24-dimensional lattice 1 . This is A∗ 24 monster group. Although the partition function of the even but not self-dual, because the dual lattice is 1 . N A 24 theory is discussed, the relevant CFT is modular invari- An even self-dual lattice containing 1 will have the structure A 24 ⊂ N ⊂ A∗ 24. 3 Dong and Mason pursued the analogue of monstrous moonshine 1 1 M in the case of 24; see, for example, [Dong and Mason 94] and G N/A 24 references therein. So far, there is no direct connection between Let = 1 , which is a vector subspace of A∗ 24/A 24 Z 24 G their work and the geometry of K3. 1 1 2 . Let us represent an element of by a 94 Experimental Mathematics, Vol. 20 (2011), No. 1

1A 2A 3A 5A 4B 7A 7B 8A 6A 11A 15A 15B 14A 14B 23A 23B 12B 6B 4C 3B 2B 10A 21A 21B 4A 12A 11111111111111111111111 111 23753322111000000−1 −1 −1 −1 −1 −1 −1 −1 −1 −1 252 28 9 2 4 0 0 0 1 −1 −1 −100−1 −1 0000122 0 04 1 253 13 10 3 1 1 1 −1 −2000−1 −1001111−11 −111−30 1771 −21 16 1 −500−100110000−1 −1 −17111 0 03 0 3520 64 10 0 0 −1 −10−2000 1 111000−800−1 −10 0 − e+ e− − −e+ −e− − − − e− e+ − 45 30017 7 10 1 0 0 7 7 1 11113 5 0 7 7 30 − e− e+ − −e− −e+ − − − e+ e− − 45 30017 7 10 1 0 0 7 7 1 11113 5 0 7 7 30 − e+ e− e+ e− − − − e− e+ 990 18 0 0 2 7 7 00 0 0 0 7 7 1111 2310 0 7 7 60 − e− e+ e− e+ − − − e+ e− 990 18 0 0 2 7 7 00 0 0 0 7 7 1111 2310 0 7 7 60 − e+ e− − − − − − −e− −e+ 1035 21 0 0 3 2 7 2 7 101000000111 3 507 7 30 − e− e+ − − − − − −e+ −e− 1035 21 0 0 3 2 7 2 7 101000000111 3 507 7 30 1035 27 0 0 −1 −1 −110 1 0 0−1 −1 0 0 0236350−1 −13 0 − − − e+ e− − − − 231 7 3110011 0 15 15 0 011003091 0 01 1 − − − e− e+ − − − 231 7 3110011 0 15 15 0 011003091 0 01 1 − − e+ e− − − − 770 14 5 0 20001000 0 023 23 112 7100 0 02 1 − − e− e+ − − − 770 14 5 0 20001000 0 023 23 112 7100 0 02 1 483 35 6 −2300−12−111000000303−20030 1265 49 5 0 1 −2 −211000 0 00000−38−15 0 1 1 −7 −1 2024 8 −1 −10110−10−1 −1 1 1 0 0 000824−1118−1 2277 21 0 −3122−10000000002−36−19 1 −1 −1 −30 3312 48 0 −301100100−1 −1000−20−61611 100 5313 49 −15 3 −300−11000000000−30 9−10011 5796 −28 −914000−1 −1 1 1 0 0 0 0 0000361 0 0−4 −1 5544 −56 9 −1000010−1 −1 0 0 1 1 000024−100−81 10395 −21 0 0 −10010000 0 0−1 −10030−45 0 0 0 3 0

TABLE 2. Character table of M24.

sequence of 24 zeros and ones, and define the weight of a Now let us recall that the cohomology lattice of K3, vector to be the number of ones in it. Λ=H∗(K3, Z), The self-duality of N translates to the fact that G is 12-dimensional. The evenness translates to the fact that is also an even self-dual lattice of dimension 24, but with only the weight of every element of G is a multiple of 4. signature (4, 20). The close connection between M24 and Let us further demand that only the vectors of N whose the geometry of the K3 surface stems from this fact. 24 length squared is two be the roots of A1 and not more. TakeaK3surfaceS, and let G be its symmetry pre- Then G does not have an element with weight 4. serving the holomorphic 2-form. Let ΛG be the part of These conditions fix the form of G uniquely, and G is Λ preserved by G, and ΛG its orthogonal complement. 1,1 known as the extended binary Golay code. The group Then ΛG is inside the primitive part of H , and thus M24 is defined as the subgroup of the permutation group it is negative definite. Using Nikulin’s result, it can be 24 S24 of the coordinates of Z2 that preserves G. shown that ΛG is a sublattice of N. Therefore G is a The lattice N thus constructed defines a chiral CFT subgroup of M24. 24 with c = 24 whose current algebra is A1 . Therefore M24 The subgroup G cannot be M24 itself, however. The is the discrete symmetry of this chiral CFT. action of G on S preserves at least H0 , H4 , H2,0 , Eguchi et al.: Notes on the K3 Surface and the Mathieu Group M24 95

H0,2 , and the Kähler form. Hence ΛG is at least five- [Conway et al. 85] J. Conway, R. Curtis, S. Norton, R. Parker, dimensional, and ΛG is at most 19-dimensional. This and R. Wilson. Atlas of Finite Groups. Maximal Sub- implies that the action of G on N as real linear maps groups and Ordinary Characters for Simple Groups. Ox- ford: Clarendon Press, 1985. should at least have a five-dimensional fixed subspace. This translates to the fact that the action of G on 24 [Dong and Mason 94] C. Y. Dong and G. Mason. “An Orb- points as a subgroup of M24 splits them into at least five ifold Theory of Genus Zero Associated to the Spo- M orbits. radic Group 24.” Commun. Math. Phys. 164 (1994) 87–104. Similarly, starting from a subgroup G of M24 that acts on 24 points with at least five orbits, one can construct [Eguchi and Hikami 09] T. Eguchi and K. Hikami. “Su- G H1,1 the action of on . Using the global Torelli theorem, perconformal Algebras and Mock Theta Functions 2: this translates to the existence of a K3 surface S whose Rademacher Expansion for K3 Surface.” Commun. Num- symmetry is G. ber Theor. and Phys. 3 (2009), 531–554. One example is the Fermat quartic X4 + Y 4 + Z4 + 4 3 2 [Eguchi and Hikami 10] T. Eguchi and K. Hikami, “N = 4 Su- W =0inCP . The symmetry is (Z4 ) S4 , with 384 M perconformal Algebra and the Entropy of Hyper-Kähler elements. This is indeed a subgroup of 24 that decom- Manifolds.” JHEP 1002 (2010), 019. poses 24 points into five orbits, of lengths 1, 1, 2, 4, and 16. [Eguchi and Taormina 87] T. Eguchi and A. Taormina. “Uni- More examples and details of the analysis can be found tary Representations of N = 4 Superconformal Algebra.” in [Mukai 88, Kondo 98]. Phys. Lett. B 196 (1987), 75–81. [Eguchi and Taormina 88] T. Eguchi and A. Taormina. N ACKNOWLEDGMENTS “Character Formulas for the = 4 Superconformal Al- gebra.” Phys. Lett. B 200 (1988), 315–322. The authors thank the hospitality of the Aspen Center of Physics during the workshop “Unity of String Theory,” [Eguchi et al. 89] T. Eguchi, H. Ooguri, A. Taormina, and when the observation reported in this paper was made. S. K. Yang. “Superconformal Algebras and String Com- pactification on Manifolds with SU(N) Holonomy.” Nucl. We would like to thank S. Mukai for discussions. Phys. B 315 (1989), 193–221. T. E. is supported in part by a Grant in Aid from the Japan Ministry of Education, Culture, Sports, Science, [Frenkel et al. 88] I. B. Frenkel, J. Lepowsky and A. Meur- and Technology (MEXT). H. O. is supported in part by man. Vertex Operator Algebras and the Monster,Pure U.S. Department of Energy grant DE-FG03-92-ER40701, and Applied Math. 134. New York: Academic Press, the World Premier International Research Center Initia- 1988. tive, Grant-in-Aid for Scientific Research (C) 20540256 [Gaberdiel et al. 10] M. R. Gaberdiel, S. Hohenegger, and of MEXT, and the Humboldt Research Award. Y. T. is R. Volpato. “Mathieu Twining Characters for K3.” supported in part by NSF grant PHY-0503584, and by arXiv:1006.0221 [hep-th], 2010. a Marvin L. Goldberger membership at the Institute for Advanced Study. [Gannon 04] T. Gannon. “Monstrous Moonshine: The First Twenty-Five Years.” arXiv:math/0402345, 2004.

REFERENCES [Kac 98] V. G. Kac. Vertex Algebras for Beginners,2nd ed., University Lecture Series 10. Providence: American [Bringmann and Ono 06] K. Bringmann and K. Ono. “The Mathematical Society, 1998. f(q) Mock Theta Function Conjecture and Partition Ranks.” Invent. Math. 165 (2006), 243–266. [Kac and Wakimoto 04] V. G. Kac and M. Wakimoto. “Quan- tum Reduction and Representation Theory of Supercon- [Bringmann and Ono 08] K. Bringmann and K. Ono. “Coef- formal Algebras.” Advances in Math. 185 (2004), 400– ﬁcients of Harmonic Maas Forms.” Preprint, 2008. 458.

[Cheng 10] M. C. N. Cheng. “K3 Surfaces, N =4 Dyons, [Kondo 98]S. Kondo. “Niemeier Lattices, Mathieu Groups and the Mathieu Group M24.” arXiv:1005.5415 [hep-th], and Finite Groups of Symplectic Automorphisms of K3 2010. Surfaces.” Duke Math. Journal 92 (1998), 593–603.

[Conway and Norton 79] J. H. Conway and S. P. Norton. [Mukai 88] S. M. “Finite Groups of Automorphisms of K3 Sur- “Monstrous Moonshine.” Bull. London Math. Soc. 11 faces and the Mathieu Group.” Invent. Math. 94 (1988), (1979), 308–339. 183–221. 96 Experimental Mathematics, Vol. 20 (2011), No. 1

[Ooguri 89] H. Ooguri. “Superconformal Symmetry and Ge- [Math. Soc. Japan 07] Mathematical Society of Japan. ometry of Ricci Flat Kähler Manifolds.” Int. J. Mod. Iwanami Suugaku Jiten, 4th Japanese ed. Tokyo: Phys. A 4 (1989), 4303–4324. Iwanami Shoten, 2007. (English translation of the 3rd Japanese edition is available as the 2nd English edition [Thompson 79] J. G. Thompson. “Some Numerology between of Encyclopedic Dictionary of Mathematics. Cambridge: the Fischer–Griess Monster and the Elliptic Modular MIT press, 1987.) Function.” Bull. London Math. Soc. 11 (1979), 352–353. [Zagier 07] D. Zagier, “Ramanujan’s Mock Theta Functions [Witten 87] E. Witten. “Elliptic Genera and Quantum Field and Their Applications.” Séminaire Bourbaki 60ème Theory.” Commun. Math. Phys. 109 (1987), 525–536. année (2006–2007), no. 986.

[Witten 07] E. Witten. “Three-Dimensional Gravity Revis- [Zwegers 02] S. P. Zwegers. “Mock Theta Functions.” Ph.D. ited.” arXiv:0706.3359, 2007. thesis, Universiteit Utrecht, 2002.

Tohru Eguchi, Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502 Japan ([email protected])

Hirosi Ooguri, California Institute of Technology, Pasadena, CA 91125, USA ([email protected])

Yuji Tachikawa, School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA ([email protected])

Received April 21, 2010; accepted May 5, 2010.