Enriques moonshine

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Enriques moonshine

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IOP PUBLISHING

JOURNAL OF PHYSICS A: MATHEMATICAL AND THEORETICAL

doi:10.1088/1751-8113/46/31/312001

J. Phys. A: Math. Theor. 46 (2013) 312001 (11pp)

FAST TRACK COMMUNICATION

Enriques moonshine Tohru Eguchi 1 and Kazuhiro Hikami 2 1 Department of Physics and Research Center for Mathematical Physics, Rikkyo University, Tokyo 171-8501, Japan 2 Faculty of Mathematics, Kyushu University, Fukuoka 819-0395, Japan

E-mail: [email protected] and [email protected]

Received 7 May 2013, in final form 3 July 2013 Published 19 July 2013 Online at stacks.iop.org/JPhysA/46/312001 Abstract We propose a new moonshine phenomenon associated with the elliptic genus of the Enriques surface ( 12 of the elliptic genus of K3) with the symmetry group given by the Mathieu group M12 . PACS numbers: 11.25.Hf, 02.40.Tt

1. Mathieu moonshine Recently a new moonshine phenomenon associated with the elliptic genus of the K3 surface has been discovered and is receiving some attention. It was first observed in [8] that when one expands the elliptic genus of K3 in terms of irreducible characters of the N = 4 superconformal algebra (SCA) the expansion coefficients A(n) at lower values of n are decomposed into a sum of dimensions of irreducible representations (irreps.) of the Mathieu group M24 . Subsequently the twisted elliptic genera of the K3 surface for each conjugacy class g of M24 (analogues of the McKay–Thompson series of monstrous moonshine) have been constructed and used to determine systematically the decomposition of expansion coefficients up to very high values of n (∼1000) [1, 5, 9, 10]. Finally a mathematical proof has been given to show that expansion coefficients are in fact decomposed into a sum of dimensions of irreps. of M24 with positive and integral multiplicities for all values of n [11]. Thus the ‘Mathieu moonshine’ phenomenon has now been established although its physical or mathematical origin are not yet explained. We present the character table and list of conjugacy classes of M24 in tables 1 and 2. We also present the data of the decomposition of expansion coefficients A(n) of the elliptic genus of K3 ∞ A(n)chRh=n+ 1 ,= 1 (z; τ ) (1.1) Z K3 (z; τ ) = 24chRh= 1 ,=0 (z; τ ) + 4

n=0

4

2

into irreps. of M24 in table 3. Note that here Z K3 denotes the elliptic genus of K3 and chRh= 1 ,

and chRh=n+ 1 , are massless (BPS) and massive (non-BPS) characters (with h = n + 4

1751-8113/13/312001+11$33.00 © 2013 IOP Publishing Ltd

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4

1 4

and 1

χ1 χ2 χ3 χ4 χ5 χ6 χ7 χ8 χ9 χ10 χ11 χ12 χ13 χ14 χ15 χ16 χ17 χ18 χ19 χ20 χ21 χ22 χ23 χ24 χ25 χ26

1A

2A

2B

3A

1 23 45 45 231 231 252 253 483 770 770 990 990 1035 1035 1035 1265 1771 2024 2277 3312 3520 5313 5544 5796 10 395

1 7 −3 −3 7 7 28 13 35 −14 −14 −18 −18 27 −21 −21 49 −21 8 21 48 64 49 −56 −28 −21

1 −1 5 5 −9 −9 12 −11 3 10 10 −10 −10 35 −5 −5 −15 11 24 −19 16 0 9 24 36 −45

1 5 0 0 −3 −3 9 10 6 5 5 0 0 0 0 0 5 16 −1 0 0 10 −15 9 −9 0

3B 1 −1 3 3 0 0 0 1 0 −7 −7 3 3 6 −3 −3 8 7 8 6 −6 −8 0 0 0 0

4A 1 −1 −3 −3 −1 −1 4 −3 3 2 2 6 6 3 3 3 −7 3 8 −3 0 0 1 −8 −4 3

4B 1 3 1 1 −1 −1 4 1 3 −2 −2 2 2 −1 3 3 1 −5 0 1 0 0 −3 0 4 −1

4C 1 −1 1 1 3 3 0 1 3 −2 −2 −2 −2 3 −1 −1 −3 −1 0 −3 0 0 −3 0 0 3

5A 1 3 0 0 1 1 2 3 −2 0 0 0 0 0 0 0 0 1 −1 −3 −3 0 3 −1 1 0

6A 1 1 0 0 1 1 1 −2 2 1 1 0 0 0 0 0 1 0 −1 0 0 −2 1 1 −1 0

6B 1 −1 −1 −1 0 0 0 1 0 1 1 −1 −1 2 1 1 0 −1 0 2 −2 0 0 0 0 0

7A

7B

1 2

1 2

0 0 0 1 0 0 0

0 0 0 1 0 0 0

√ −1+i 7 2√ −1−i 7 2

√ −1+i 7 2√ −1−i 7 2

−1 √ −1 + i 7 √ −1 − i 7 −2 0 1 2 1 −1 0 0 0 0

√ −1−i 7 2√ −1+i 7 2

√ −1−i 7 2√ −1+i 7 2

−1 √ −1 − i 7 √ −1 + i 7 −2 0 1 2 1 −1 0 0 0 0

8A 1 1 −1 −1 −1 −1 0 −1 −1 0 0 0 0 1 −1 −1 1 −1 0 −1 0 0 −1 0 0 1

10A 1 −1 0 0 1 1 2 −1 −2 0 0 0 0 0 0 0 0 1 −1 1 1 0 −1 −1 1 0

11A 1 1 1 1 0 0 −1 0 −1 0 0 0 0 1 1 1 0 0 0 0 1 0 0 0 −1 0

12A 1 −1 0 0 −1 −1 1 0 0 −1 −1 0 0 0 0 0 −1 0 −1 0 0 0 1 1 −1 0

12B 1 −1 1 1 0 0 0 1 0 1 1 1 1 0 −1 −1 0 −1 0 0 0 0 0 0 0 0

14A

14B

1 0 √ 7 − −1+i 2√ 7 − −1−i 2 0 0 0 −1 0 0 0 √

1 0 √ 7 − −1−i 2√ 7 − −1+i 2 0 0 0 −1 0 0 0 √

−1 0 0 0 0 1 0 −1 1 0 0 0 0

−1 0 0 0 0 1 0 −1 1 0 0 0 0

−1+i 7 2√ −1−i 7 2

−1−i 7 2√ −1+i 7 2

15A

15B

1 0 0 0

1 0 0 0

√ −1+i 15 2√ −1−i 15 2

−1 0 1 0 0 0 0 0 0 0 0 1 −1 0 0 0 0 −1 1 0

√ −1−i 15 2√ −1+i 15 2

−1 0 1 0 0 0 0 0 0 0 0 1 −1 0 0 0 0 −1 1 0

21A

21B

1 −1

1 −1

0 0 0 1 0 0 0

0 0 0 1 0 0 0

√ −1+i 7 2√ −1−i 7 2

√ −1+i 7 2√ −1−i 7 2

−1 √ 7 − −1+i 2√ 7 − −1−i 2 1 0 1 −1 1 −1 0 0 0 0

√ −1−i 7 2√ −1+i 7 2

√ −1−i 7 2√ −1+i 7 2

−1 √ 7 − −1−i 2√ 7 − −1+i 2 1 0 1 −1 1 −1 0 0 0 0

23A

23B

1 0 −1 −1 1 1 −1 0 0

1 0 −1 −1 1 1 −1 0 0

1 1 0 0 0 0 0 0 0 0 1 0 1 0 −1

1 1 0 0 0 0 0 0 0 0 1 0 1 0 −1

√ −1+i 23 2√ −1−i 23 2

J. Phys. A: Math. Theor. 46 (2013) 312001

2 Table 1. Character table of M24 . |M24 | = 244 823 040.

√ −1−i 23 2√ −1+i 23 2

Fast Track Communication



Table 2. Cycle shapes of conjugacy classes of M24 .

g

Size

1A 2A 2B 3A 3B 4A 4B 4C 5A 6A 6B 7A 7B 8A 10A 11A 12A 12B 14A 14B 15A 15B 21A 21B 23A 23B

1 11 385 31 878 226 688 485 760 637 560 1912 680 2550 240 4080 384 10 200 960 10 200 960 5829 120 5829 120 15 301 440 12 241 152 22 256 640 20 401 920 20 401 920 17 487 360 17 487 360 16 321 536 16 321 536 11 658 240 11 658 240 10 644 480 10 644 480

Cycle shape 124 18 28 212 6 6 13 38 4 4 24 14 22 44 46 4 4 15 12 22 32 62 64 13 73 13 73 12 21 41 82 22 102 12 112 21 41 61 121 122 11 21 71 141 11 21 71 141 11 31 51 151 11 31 51 151 31 211 31 211 11 231 11 231

spin-) of N = 4 SCA in the R-sector with (−1)F insertion. For later use we also record the data of expansion coefficients Ag (n) of twisted elliptic genera ZgK3 (z; τ ) of K3 for each conjugacy class g ∈ M24

ZgK3 (z; τ ) = χgchRh= 1 ,=0 (z; τ ) + 4

∞ n=0

Ag (n)chRh=n+ 1 ,= 1 (z; τ ), 4

(1.2)

2

in table 4. Note that A(n) ≡ A1A (n). Recently there has been an attempt at generalizing Mathieu moonshine [2] based on suitable Jacobi forms with higher values of indices >1 and again expanding them in terms of N = 4 superconformal characters using the data of [4]. This ‘umbral moonshine’ sequence has smaller symmetry groups than M24 . Unfortunately, its Jacobi forms do not correspond to the elliptic genera of any complex manifolds and the connection to geometry is not clear in umbral moonshine. In [6] we have discussed yet another example of moonshine based on N = 2 SCA instead of N = 4. 2. Enriques moonshine In this communication we want to propose a new example of the moonshine phenomenon which may be called ‘Enriques moonshine’. It is defined by the elliptic genus of the Enriques surface expanded in terms of N = 4 characters. Its symmetry group is M12 . Recall that the 3

χ1

χ2

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

−2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 2 0 8 6 12 16 26 34

0 0 0 0 0 0 0 0 0 0 0 0 2 0 0 0 2 2 6 4 14 20 32 40 80 108 174 252 398 560 876

χ3 = χ4

χ5 = χ6

0 1 0 0 0 0 0 0 0 0 0 0 0 2 0 2 2 8 6 18 20 40 55 98 132 234 322 514 742 1154 1642

0 0 1 0 0 0 0 0 0 0 2 0 2 2 8 8 18 25 50 68 126 182 314 460 744 1106 1742 2560 3922 5758 8642

χ7

χ8

χ9

0 0 0 0 0 0 0 0 0 0 0 0 2 4 4 12 18 30 50 80 128 214 328 512 798 1232 1860 2836 4238 6328 9368

0 0 0 0 0 0 0 0 0 0 2 0 4 2 6 8 22 26 58 72 138 200 346 496 824 1208 1904 2802 4310 6286 9486

0 0 0 0 0 0 0 0 0 2 2 0 4 6 14 18 36 54 100 150 254 396 640 972 1544 2336 3602 5394 8160 12 090 18 008

χ10 = χ11

χ12 = χ13

0 0 0 1 0 0 0 0 0 2 0 4 6 10 16 38 50 94 148 252 390 652 988 1590 2426 3764 5677 8688 12 912 19 380 28 580

0 0 0 0 0 0 0 0 1 0 2 4 8 14 24 40 72 116 194 318 516 814 1298 2020 3140 4814 7348 11 092 16 686 24 840 36 824

χ14 0 0 0 0 0 0 0 0 0 2 2 6 4 18 22 46 68 130 192 346 520 872 1336 2144 3236 5084 7626 11 666 17 356 26 078 38 368

χ15 = χ16 0 0 0 0 0 0 0 0 1 2 2 4 8 14 24 44 72 124 202 332 536 860 1348 2118 3278 5038 7670 11 618 17 418 25 994 38 480

χ17

χ18

χ19

χ20

χ21

χ22

χ23

χ24

χ25

χ26

0 0 0 0 0 0 0 0 2 0 4 2 12 16 34 46 100 140 256 394 676 1020 1686 2546 4050 6108 9444 14 100 21 414 31 636 47 172

0 0 0 0 0 0 0 2 0 2 4 8 12 26 38 78 122 212 342 582 904 1476 2302 3638 5584 8654 13 090 19 914 29 772 44 512 65 776

0 0 0 0 0 0 0 2 0 2 4 10 12 30 46 86 140 246 388 664 1036 1684 2630 4162 6376 9892 14 968 22 744 34 026 50 892 75 158

0 0 0 0 2 0 0 0 2 2 6 8 18 28 58 88 170 262 454 722 1196 1862 3000 4624 7248 11 042 16 940 25 462 38 434 57 068 84 776

0 0 0 0 0 0 0 0 2 4 6 14 26 44 80 138 232 392 654 1062 1716 2742 4324 6768 10 500 16 112 24 566 37 148 55 764 83 146 123 176

0 0 0 0 0 0 2 0 2 4 8 12 30 44 86 144 252 410 704 1116 1836 2902 4616 7166 11 192 17 084 26 148 39 436 59 330 88 280 131 020

0 0 0 0 0 0 0 2 4 4 12 22 40 70 128 218 378 630 1044 1702 2764 4384 6950 10 856 16 834 25 840 39 428 59 564 89 490 133 356 197 596

0 0 0 0 0 0 0 2 2 8 10 24 38 80 126 238 382 670 1074 1800 2846 4622 7204 11 376 17 504 27 056 41 022 62 294 93 218 139 342 205 970

0 0 0 0 0 2 0 2 2 8 10 26 40 84 132 246 400 704 1120 1880 2980 4828 7532 11 898 18 294 28 288 42 894 65 114 97 456 145 690 215 318

0 0 0 0 0 0 2 2 6 10 24 40 80 136 254 424 742 1222 2058 3320 5408 8572 13 620 21 204 32 976 50 524 77 176 116 494 175 146 260 828 386 724


n


4 Table 3. Multiplicities of the decomposition of A(n) into irreducible representations of M24 in Mathieu moonshine.

1A

2A

2B

3A

3B

4A

4B

4C

5A

6A

6B

7AB

8A

10A

11A

12A

12B

14AB

15AB

21AB

23AB

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

−2 90 462 1540 4554 11 592 27 830 61 686 131 100 265 650 521 136 988 770 1830 248 3303 630 5844 762 10 139 734 17 301 060 29 051 484 48 106 430 78 599 556 126 894 174 202 537 080 319 927 608 500 376 870 775 492 564 1191 453 912 1815 754 710 2745 870 180 4122 417 420 6146 311 620 9104 078 592

−2 −6 14 −28 42 −56 86 −138 188 −238 336 −478 616 −786 1050 −1386 1764 −2212 2814 −3612 4510 −5544 6936 −8666 10 612 −12 936 15 862 −19 420 23 532 −28 348 34 272

−2 10 −18 20 −38 72 −90 118 −180 258 −352 450 −600 830 −1062 1334 −1740 2268 −2850 3540 −4482 5640 −6968 8550 −10 556 13 064 −15 930 19 268 −23 460 28 548 −34 352

−2 0 −6 10 0 −18 20 0 −30 42 0 −60 62 0 −90 118 0 −156 170 0 −228 270 0 −360 400 0 −510 600 0 −762 828

−2 6 0 −14 12 0 −16 30 0 −42 42 0 −70 84 0 −110 126 0 −166 210 0 −282 300 0 −392 462 0 −600 660 0 −840

−2 −6 −2 4 −6 −8 6 6 −4 −14 0 18 −8 −18 10 22 −12 −36 14 36 −18 −40 24 54 −28 −72 22 84 −36 −92 48

−2 2 −2 −4 2 8 −2 −10 4 10 −8 −14 8 22 −6 −26 12 28 −18 −36 14 48 −16 −58 28 64 −34 −76 36 100 −40

−2 2 6 −4 −6 0 6 −2 −12 10 16 −6 −16 6 18 −10 −28 12 38 −20 −42 16 48 −18 −60 32 78 −36 −84 36 96

−2 0 2 0 −6 2 0 6 0 −10 6 0 8 0 −18 4 0 14 0 −24 14 0 18 0 −36 12 0 30 0 −50 22

−2 0 2 2 0 −2 −4 0 2 2 0 −4 −2 0 6 6 0 −4 −6 0 4 6 0 −8 −8 0 10 8 0 −10 −12

−2 −2 0 2 4 0 0 −2 0 6 2 0 −6 −4 0 2 6 0 −6 −6 0 6 4 0 −8 −10 0 8 12 0 −8

−2 −1 0 0 4 0 −2 2 −3 0 0 6 0 −6 0 −4 0 0 8 0 −6 4 −7 0 0 12 0 −10 2 −6 0

−2 −2 −2 0 −2 0 2 −2 0 −2 −4 2 0 2 2 −2 0 −4 −2 0 −2 4 4 −2 0 −4 −6 4 0 4 4

−2 0 2 0 2 2 0 −2 0 −2 −2 0 0 0 −2 4 0 −2 0 0 −2 0 2 0 4 4 0 −2 0 −2 −2

−2 2 0 0 0 −2 0 −2 2 0 0 2 2 0 0 0 −4 0 −2 2 0 −2 0 4 0 0 0 −2 0 −2 4

−2 0 −2 −2 0 −2 0 0 2 −2 0 0 −2 0 −2 −2 0 0 2 0 0 2 0 0 −4 0 −2 0 0 −2 0

−2 2 0 2 0 0 0 −2 0 −2 −2 0 2 0 0 2 2 0 2 −2 0 −2 0 0 0 2 0 0 0 0 0

−2 1 0 0 0 0 2 2 −1 0 0 −2 0 −2 0 0 0 0 0 0 2 0 −1 0 0 0 0 −2 −2 2 0

−2 0 −1 0 0 2 0 0 0 2 0 0 2 0 0 −2 0 −1 0 0 2 0 0 0 0 0 0 0 0 −2 −2

−2 −1 0 0 −2 0 −2 2 0 0 0 0 0 0 0 2 0 0 2 0 0 −2 −1 0 0 0 0 2 2 0 0

−2 −2 2 −1 0 0 0 0 0 0 2 0 0 2 2 0 0 0 −2 0 0 0 0 2 0 0 −1 0 0 0 0

5


n


Table 4. Expansion coefficients of Ag (n) in Mathieu moonshine.



Table 5. Cycle shapes of conjugacy classes of M12 .

g

Size

Cycleshape

1A 2A 2B 3A 3B 4A 4B 5A 6A 6B 8A 8B 10A 11A 11B

1 396 495 1760 2640 2970 2970 9504 7920 15 840 11 880 11 880 9504 8640 8640

112 26 14 24 13 33 34 22 42 14 42 12 52 62 11 21 31 61 41 81 12 21 81 21 101 11 111 11 111

Enriques surface is closely related to K3: it is obtained by quotienting K3 by a fixed-point free involution and has an Euler number 12. Its elliptic genus is one half of that of K3 2 2 2 θ (z; τ ) (z; τ ) (z; τ ) 1 θ θ 10 00 01 Z Enriques (z; τ ) = Z K3 (z; τ ) = 4 + + . (2.1) 2 θ10 (0; τ ) θ00 (0; τ ) θ01 (0; τ ) Enriques moonshine is motivated by the following simple considerations. (1) It is known that in the case of Mathieu moonshine the expansion coefficients A(n) are always even for any n 1: this is because (i) when the decomposition of A(n) contains a complex representation of M24 , it also contains its complex conjugate representation, and (ii) when A(n) contains a real representation its multiplicity is always even [11]. (2) Thus in order to keep integrality of the decomposition when we divide by 2 the K3 elliptic genus we just need to find a subgroup G of M24 where all the complex representations of M24 become real representations of G. It turns out that this is the case of M12 . (3) Geometrical considerations on the Enriques surface suggest the relevance of the symmetry group M12 [12]. Let us first derive the decomposition of M24 representations (reps.) into those of M12 in order to examine the reality of the representations. For this purpose we want to make a correspondence between the conjugacy classes of the two groups. In table 5 we list the conjugacy classes of M12 and their permutation representations. We recall that the Mathieu group M24 is the symmetry group of the Golay code and permutes dodecads into each other. M12 is the subgroup of M24 which fixes a dodecad [3]. The conjugacy class of 2A of M12 , for instance, has a cycle shape 26 and it is natural that this corresponds to the conjugacy class 2B of M24 with a cycle shape 212 . Thus in general a class g of M12 should correspond to a class g of M24 whose cycle shape is the square of that of g. There are exceptions to this rule when there exists a non-trivial outer automorphism between conjugacy classes of M12 . From table 5 we note that the sizes of the conjugacy classes are equal for the pair 4A, 4B and 8A, 8B and 11A, 11B. It is known [3] that these pairs are tied by a non-trivial outer automorphism σ . If one takes a class g of M12 the corresponding class of M24 should become g ∪ σ (g). In the case of g = 4A, σ (4A) = 4B, for instance, the cycle shape of g ∪ σ (g) equals 42 22 ∪ 42 14 and that 6



Table 6. Character table of M12 . |M12 | = 95 040.

1A χ1 χ2 χ3 χ4 χ5 χ6 χ7 χ8 χ9 χ10 χ11 χ12 χ13 χ14 χ15

1 11 11 16 16 45 54 55 55 55 66 99 120 144 176

2A 1 −1 −1 4 4 5 6 −5 −5 −5 6 −1 0 4 −4

2B 1 3 3 0 0 −3 6 7 −1 −1 2 3 −8 0 0

3A 1 2 2 −2 −2 0 0 1 1 1 3 0 3 0 −4

3B 1 −1 −1 1 1 3 0 1 1 1 0 3 0 −3 −1

4A 1 −1 3 0 0 1 2 −1 3 −1 −2 −1 0 0 0

4B 1 3 −1 0 0 1 2 −1 −1 3 −2 −1 0 0 0

5A 1 1 1 1 1 0 −1 0 0 0 1 −1 0 −1 1

6A 1 −1 −1 1 1 −1 0 1 1 1 0 −1 0 1 −1

6B 1 0 0 0 0 0 0 1 −1 −1 −1 0 1 0 0

8A 1 −1 1 0 0 −1 0 −1 −1 1 0 1 0 0 0

8B 1 1 −1 0 0 −1 0 −1 1 −1 0 1 0 0 0

10A 1 −1 −1 −1 −1 0 1 0 0 0 1 −1 0 −1 1

11A

11B

1 0 0

1 0 0

1 −1 0 0 0 0 0 −1 1 0

1 −1 0 0 0 0 0 −1 1 0

√ −1+i 11 2√ −1−i 11 2

√ −1−i 11 2√ −1+i 11 2

of g becomes 44 22 14 which is class 4B of M24 . Thus 4A, 4B of M12 should both correspond to 4B of M24 . In this way we can construct the following table of correspondences. g ∈ M12 1A 2A 2B 3A 3B 4A 4B 5A 6A 6B 8A 8B 10A 11A 11B g ∈ M24 1A 2B 2A 3A 3B 4B 4B 5A 6B 6A 8A 8A 10A 11A 11A (2.2) Let us now determine the branching rule of the irreps. of M24 into those of M12 . We consider the following ‘inner product’ of character tables of M24 and M12 to derive the multiplicity of a representation r of M12 contained in the representation R of M24 r χ (M24 )Rt(g) χ (M12 )−1 g = multiplicity of irrep. r in irrep. R. (2.3) g

Here t(g) = g of (2.2), and χ (M12 )−1 is the inverse of the character table of M12 in the sense of a matrix. Using the character tables of M24 , M12 in tables 1, 6, we find the above multiplicities as given by table 7. Note that as we mentioned already, the decomposition of complex representations of M24 contains only real representations of M12 or the sum of pairs of complex conjugate representations of M12 . Therefore if we substitute M24 reps. by their M12 decompositions in the Mathieu moonshine of table 3, and divide by an overall factor 2, we maintain the integrality of the multiplicities of M12 representations. One obtains the decomposition of the elliptic genus of the Enriques surface given in terms of M12 reps. See table 8. There is in fact a more elegant way to derive the decomposition of the Enriques elliptic genus. This is to use the method of the twisted elliptic genus. We have at hand the twisted genera for all conjugacy classes in Mathieu moonshine (tabulated in [5]) and we can use these results. We introduce an ansatz that the twisted elliptic genera for Enriques moonshine are one half of those of Mathieu moonshine of the corresponding conjugacy classes K3 (z; τ ) for all conjugacy classes g of M12 . ZgEnriques (z; τ ) = 12 Zt(g)

(2.4)

Then by introducing the expansion coefficients AEnriques (n) for all classes g ∈ M12 g ∞ ZgEnriques (z; τ ) = χgEnriques chh= 1 ,=0 (z; τ ) + AEnriques (n)chh=n+ 1 ,= 1 (z; τ ) (2.5) g 4

4

2

n=0

7



Table 7. Branching of M24 representations into those of M12 . Only non-zero multiplicities are written.

M24 \M12 χ1 χ2 χ3 χ4 χ5 χ6 χ7 χ8 χ9 χ10 χ11 χ12 χ13 χ14 χ15 χ16 χ17 χ18 χ19 χ20 χ21 χ22 χ23 χ24 χ25 χ26

1 23 45 45 231 231 252 253 483 770 770 990 990 1035 1035 1035 1265 1771 2024 2277 3312 3520 5313 5544 5796 10 395

χ1 1

χ2 11

χ3 11

1 1

1

1

χ4 16

χ5 16

χ6 45

χ7 54

χ8 55

2

1 1 1 1 2

χ9 55

χ10 55

χ11 66

χ12 99

χ13 120

χ14 144

χ15 176

1 1 1

1 1 1

1 1 1

2

1 1 1

1

1 1 1 1 1

1 1

1 1

1

1 1 1 1 1

1

1 2 1

1 2 1

1

1

2 2

1

1

2 1 2 1

2 1 2 1

2 2 1 1 2 4 4 4

2 2 4 4 4 2 4 4

1 1

1 1

1 1 1 1 1 2 1 2 2 3 3 7

1 1 1 1 1 2 1 2 2 3 3 7

1 3 1 1 3 3 4 5 1 1 6

1 1 2

2

1 2 2 1 1 2 2 2 1 2 3 3 6 6 8 9 9 15

2 2 2 2 2 2

1 3 2 1 3 4 4 4 4 6

3 2 3 3 4 4 6 5 5 11

4 2 2 2 2 4 10 8 14

1 1 2 2 1 2 2 2 2 3 4 6 6 11 9 11 20

where χgEnriques is the Euler number χgEnriques = ZgEnriques (0; τ ), g ∈ M12 1A 2A 2B 3A 3B 4A 4B 5A 6A 6B 8A 8B 10A 11A 11B χgEnriques 12 0 4 3 0 2 2 2 0 1 1 we obtain the multiplicity for the M12 representation r at level n ng g χ (M12 )r AEnriques (n) = cEnriques (n). g r |G| g

1

0

1

1 (2.6)

Here |G| denotes the order of M12 (= 95 040) and ng is the size of M12 conjugacy class g. (Note that the Euler numbers χgEnriques listed above cannot be written as an integral linear combination of M12 characters (see table 6) unlike the case of Mathieu moonshine. This is a point worth studying if it possibly raises a question of consistency of Enriques moonshine. We are grateful for the referee for raising this point.) By using the orthogonality relation of the character table it is possible to prove that the above formula in fact reproduces the data of table 8. First we recall that the multiplicity of representation R in Mathieu moonshine is given by ng g χ (M24 )R Ag (n) = cK3 (2.7) R (n). |G | g Here g runs over conjugacy classes of M24 , and |G | is the order of M24 . We convert M24 representations into M12 representations and divide by 2 to obtain multiplicities in Enriques moonshine 8

n

χ 2 = χ3

χ 4 = χ5

χ6

χ7

χ8

χ9 = χ10

χ11

χ12

χ13

χ14

−1 0 0 0 0 0 0 0 1 1 3 4 10 19 30 52 94 151 252 412 669 1064 1692 2622 4082 6270 9555 14 433 21 711 32 314 47 909

0 0 0 0 0 0 3 2 9 14 33 51 115 183 346 576 1017 1658 2817 4508 7385 11 676 18 579 28 863 44 995 68 818 105 225 158 731 238 790 355 395 527 223

0 0 0 0 0 2 1 7 10 23 42 88 147 286 484 861 1444 2468 4020 6647 10 649 17 087 26 877 42 197 65 174 100 406 152 718 231 277 346 819 517 616 766 024

0 1 0 0 1 4 4 18 28 66 119 242 420 801 1364 2420 4069 6920 11 330 18 681 29 960 48 040 75 625 118 616 183 384 282 327 429 576 650 388 975 551 1455 614 2154 660

0 0 0 0 2 4 8 16 38 76 148 278 522 938 1664 2874 4922 8248 13 674 22 316 36 064 57 526 90 908 142 120 220 348 338 446 515 886 780 008 1171 218 1746 034 2586 488

0 0 1 0 3 1 10 15 43 70 162 272 546 933 1721 2896 5058 8340 14 000 22 644 36 844 58 442 92 775 144 536 224 690 344 382 525 845 793 968 1193 511 1777 621 2635 260

0 0 0 0 2 3 9 17 39 75 154 282 534 951 1698 2922 5022 8388 13 941 22 717 36 750 58 560 92 630 144 714 224 472 344 655 525 510 794 367 1193 023 1778 220 2634 546

0 0 0 1 1 4 10 23 43 94 179 346 633 1152 2018 3535 5994 10 099 16 689 27 318 44 021 70 371 111 037 173 798 269 200 413 792 630 341 953 589 1431 222 2134 316 3160 915

0 0 0 0 3 5 15 32 70 134 276 511 956 1716 3056 5263 9033 15 107 25 077 40 913 66 134 105 420 166 710 260 529 403 992 620 437 945 863 1429 925 2147 351 3200 923 4742 013

0 0 0 2 2 8 16 42 78 174 322 632 1144 2102 3666 6434 10 886 18 382 30 316 49 696 80 010 127 988 201 830 316 064 489 368 752 450 1145 966 1733 926 2602 046 3880 816 5746 832

0 0 0 2 3 9 21 46 98 206 390 753 1384 2506 4420 7697 13 087 22 027 36 427 59 567 96 094 153 496 242 298 379 145 587 424 902 705 1375 439 2080 389 3122 821 4656 537 6896 777

χ15 0 0 1 1 4 11 26 56 124 242 485 914 1699 3051 5423 9375 16 032 26 887 44 563 72 744 117 541 187 481 296 284 463 254 718 126 1103 084 1681 406 2542 299 3817 239 5690 817 8429 971

9


0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30

χ1


Table 8. Multiplicities of irreducible representations of M12 in Enriques moonshine.



⎤ ⎡ ng 1 g t(g) Enriques −1 r ⎣ cr (n) = χ (M24 )R Ag (n)⎦ × χ (M24 )R (χ (M12 ) )g 2 R |G | g g 1 1 ng g δ(g , t(g)) (χ (M12 )−1 )gr Ag (n) = χ (M12 )r At(g) (n) 2 g,g 2 g |G| ng g = χ (M12 )r AEnriques (n). g |G| g =

(2.8)

3. Discussion In this communication we have taken one half of the elliptic genus of K3 and obtained the Enriques moonshine. Consistency of the Enriques surface as a string theory background is a delicate issue since its canonical class does not quite vanish while it carries a Ricci flat Kähler metric. We do not consider such questions in this paper and are primarily concerned with the possibility of the action of the symmetry group on the elliptic genus Z Enriques . We have shown that M12 in fact acts on Z Enriques . We should note, however, that a symmetry group still larger than M12 may possibly act on the elliptic genus. We have evidence that a maximal subgroup M12 :2 of M24 (binary extension of M12 ) acts on Z Enriques . It was crucial for the existence of Enriques moonshine that all the multiplicities of real representations of M24 are even integers in Mathieu moonshine. We have recently noticed that similar phenomena take place in umbral moonshine and thus it is quite likely that we can take one half of the Jacobi forms of umbral moonshine and construct a new moonshine series with reduced symmetry groups. This issue will be discussed in a forthcoming publication [7]. Acknowledgments TE would like to thank Y Tachikawa for discussions on the relation between M24 and M12 . He also thanks S Mukai for discussions on Enriques surface. The research of TE is supported in part by JSPS KAKENHI grant nos 22224001, 23340115. The research of KH is supported in part by JSPS KAKENHI grant nos 23340115, 24654041. Note added. After the original version of this paper was submitted to arXiv we came across the paper [13] by S Govindarajan where the group M12 is used as the symmetry group of Mathieu moonshine. In this paper, the relation (2.2) between the conjugacy classes of M24 and M12 has been obtained. Also the multiplicities of irreps. of M12 in the decomposition of the expansion coefficients A(n) at smaller values of n have been obtained in agreement with our results of Enriques moonshine up to an overall factor 2. We thank S Govindarajan for informing us of this paper.

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[6] Eguchi T and Hikami K 2012 N = 2 moonshine Phys. Lett. B 717 266–73 (arXiv:1209.0610 [hep-th]) [7] Eguchi T and Hikami K 2013 in preparation [8] Eguchi T, Ooguri H and Tachikawa Y 2011 Notes on the K3 surface and the Mathieu group M24 Exp. Math. 20 91–96 (arXiv:1004.0956 [hep-th]) [9] Gaberdiel M R, Hohenegger S and Volpato R 2010 Mathieu twining characters for K3 J. High Energy Phys. JHEP09(2010)058 (arXiv:1006.0221 [hep-th]) [10] Gaberdiel M R, Hohenegger S and Volpato R 2010 Mathieu moonshine in the elliptic genus of K3 J. High Energy Phys. JHEP10(2010)062 (arXiv:1008.3778 [math.AG]) [11] Gannon T 2012 Much ado about Mathieu arXiv:1211.5531 [math.RT] [12] Mukai S 2012 Lecture notes on K3 and Enriques surfaces Contributions to Algebraic Geometry: Impanga Lecture Notes (EMS Series of Congress Reports) ed P Pragacz (Zürich: European Mathematical Society) pp 389–405 [13] Govindarajan S 2010 Brewing moonshine for Mathieu arXiv:1012.5732 [hep-th]

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