aa r X i v : . [ h e p - t h ] J un Enriques Moonshine
TOHRU
EGUCHI
Department of Physics and Research Center for Mathematical Physics,Rikkyo University, Tokyo 171-8501, Japan.
KAZUHIRO
HIKAMI
Faculty of Mathematics, Kyushu University, Fukuoka 819-0395, Japan.
Abstract.
We propose a new moonshine phenomenon associated with the ellipticgenus of the Enriques surface ( of the elliptic genus of K
3) with the symmetry groupgiven by the Mathieu group M .
1. Mathieu moonshine
Recently a new moonshine phenomenon associated with the elliptic genus of K K 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.) ofthe Mathieu group M . Subsequently the twisted elliptic genera of K g of M (analogues of McKay–Thompson series of monstrous moon-shine) have been constructed and used to determine systematically the decomposition ofexpansion coefficients up to very high values of n ( ∼ M with positive and integral multiplicities for allvalues of n [11]. Thus the “Mathieu moonshine” phenomenon has now been establishedalthough its physical or mathematical origin is not yet explained.We present the character table and list of conjugacy classes of M in Tables 1 and 2.We also present the data of the decomposition of expansion coefficients A ( n ) of ellipticgenus of K Z K ( z ; τ ) = 24 ch e Rh = ,ℓ =0 ( z ; τ ) + ∞ X n =0 A ( n ) ch e Rh = n + ,ℓ = ( z ; τ ) (1.1) E-mail addresses : [email protected], [email protected] . Date : May 6, 2013. into irreps. of M in Table 3. Note that here Z K denotes the elliptic genus of K e Rh = ,ℓ and ch e Rh = n + ,ℓ are massless (BPS) and massive (non-BPS) characters (with h = n + and spin- ℓ ) of N = 4 SCA in R-sector with ( − F insertion. For later use wealso record the data of expansion coefficients A g ( n ) of twisted elliptic genera Z K g ( z ; τ )of K g ∈ M Z K g ( z ; τ ) = χ g ch e Rh = ,ℓ =0 ( z ; τ ) + ∞ X n =0 A g ( n ) ch e Rh = n + ,ℓ = ( z ; τ ) , (1.2)in Table 4. Note that A ( n ) ≡ A ( n ).Recently there has been an attempt at generalizing Mathieu moonshine [2] basedon suitable Jacobi forms with higher values of indices > N = 4 superconformal characters using the data of [4]. This “umbralmoonshine” sequence has smaller symmetry groups than M . Unfortunately, its Jacobiforms do not correspond to elliptic genera of any complex manifolds and the connectionto geometry is not clear in umbral moonshine. In [6] we have discussed a still anotherexample of moonshine based on N = 2 SCA instead of N = 4.
2. Enriques moonshine
In this paper we want to propose a new example of moonshine phenomenon whichmay be called as “Enriques moonshine”. It is defined by the elliptic genus of Enriquessurface expanded in terms of N = 4 characters. Its symmetry group is M . Recall thatEnriques surface is closely related to K
3: it is obtained by quotienting K K Z Enriques ( z ; τ ) = 12 Z K ( z ; τ ) = 4 "(cid:18) θ ( z ; τ ) θ (0; τ ) (cid:19) + (cid:18) θ ( z ; τ ) θ (0; τ ) (cid:19) + (cid:18) θ ( z ; τ ) θ (0; τ ) (cid:19) . (2.1)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 decompositionof A ( n ) contains a complex representation of M , it also contains its complexconjugate representation, and (ii) when A ( n ) contains a real representation itsmultiplicity is always even [11].2. Thus in order to keep integrality of the decomposition when we divide by 2the K G of M where all thecomplex representations of M become real representations of G . It turns outthat this is the case of M .3. Geometrical considerations on Enriques surface suggests the relevance of thesymmetry group M [12].Let us first derive the decomposition of M representations (reps.) into those of M in order to examine the reality of representations. For this purpose we want tomake a correspondence between the conjugacy classes of the two groups. In Table 5 we NRIQUES MOONSHINE 3 list the conjugacy classes of M and their permutation representations. We recall thatMathieu group M is the symmetry group of Golay code and permutes dodecads intoeach other. M is the subgroup of M which fixes a dodecad [3]. Conjugacy class of2A of M , for instance, has a cycle shape 2 and it is natural that this correspondsto the conjugacy class 2B of M with a cycle shape 2 . Thus in general a class g of M should correspond to a class g ′ of M whose cycle shape is the square of that of g . There are exceptions to this rule when there exists a non-trivial outer automorphismbetween conjugacy classes of M . From the Table 5 we note that the sizes of conjugacyclasses are equal for the pair 4A ,
4B and 8A ,
8B and 11A , σ . If one takes a class g of M thecorresponding class of M should become g ∪ σ ( g ). In the case of g = 4A, σ (4A) = 4B,for instance, the cycle shape of g ∪ σ ( g ) equals 4 ∪ and that of g ′ becomes 4 which is class 4B of M . Thus 4A ,
4B of M both should correspond to 4B of M . Inthis way we can construct the following table of correspondences. g ∈ M
1A 2A 2B 3A 3B 4A 4B 5A 6A 6B 8A 8B 10A 11A 11B g ′ ∈ M
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 M into those of M . Weconsider the following “inner product” of character tables of M and M to derive themultiplicity of a representation r of M contained in the representation R of M X g χ ( M ) t ( g ) R χ ( M ) − rg = multiplicity of irrep. r in irrep. R (2.3)Here t ( g ) = g ′ of (2.2), and χ ( M ) − is the inverse of the character table of M in thesense of a matrix. Using the character tables of M , M in Tables 1, 6, we find the abovemultiplicities as given by Table 7. Note that as we mentioned already, decomposition ofcomplex representations of M contain only real representations of M or the sum ofpairs of complex conjugate representations of M .Therefore if we substitute M reps. by their M decompositions in the Mathieumoonshine of Table 3, and divide by an overall factor 2, we maintain the integrality ofmultiplicities of M representations. One obtains the decomposition of the elliptic genusof Enriques surface given in terms of M reps. See Table 8.There is in fact a more elegant way to derive the decomposition of Enriques ellipticgenus. This is to use the method of twisted elliptic genus. We have at hand the twistedgenera for all conjugacy classes in Mathieu moonshine (tabulated in [5]) and we can usethese results. We introduce an ansatz that twisted elliptic genera for Enriques moonshineare one half of those of Mathieu moonshine of the corresponding conjugacy classes Z Enriques g ( z ; τ ) = 12 Z K t ( g ) ( z ; τ ) for all conjugacy classes g of M (2.4)Then by introducing the expansion coefficients A Enriques g ( n ) for all classes g ∈ M Z Enriques g ( z ; τ ) = χ Enriques g ch h = ,ℓ =0 ( z ; τ ) + ∞ X n =0 A Enriques g ( n ) ch h = n + ,ℓ = ( z ; τ ) (2.5) T. EGUCHI AND K. HIKAMI where χ Enriques g is the Euler number χ Enriques g = Z Enriques g (0; τ ), g ∈ M
1A 2A 2B 3A 3B 4A 4B 5A 6A 6B 8A 8B 10A 11A 11B χ Enriques g
12 0 4 3 0 2 2 2 0 1 1 1 0 1 1we obtain the multiplicity for the M representation r at level n X g n g | G | χ ( M ) gr A Enriques g ( n ) = c Enriques r ( n ) . (2.6)Here | G | denotes the order of M (=95040) and n g is the size of M conjugacy class g .By using the orthogonality relation of the character table it is possible to provethat the above formula in fact reproduces the data of Table 8. First we recall that themultiplicity of representation R in Mathieu moonshine is given by X g ′ n g ′ | G ′ | χ ( M ) g ′ R A g ′ ( n ) = c K R ( n ) (2.7)Here g ′ runs over conjugacy classes of M , and | G ′ | is the order of M . We covert M representations into M representations and divide by 2 to obtain multiplicities inEnriques moonshine c Enriques r ( n ) = 12 X R "X g ′ n g ′ | G ′ | χ ( M ) g ′ R A g ′ ( n ) × "X g χ ( M ) t ( g ) R ( χ ( M ) − ) rg = 12 X g,g ′ δ ( g ′ , t ( g )) ( χ ( M ) − ) rg A g ′ ( n ) = 12 X g n g | G | χ ( M ) gr A t ( g ) ( n )= X g n g | G | χ ( M ) gr A Enriques g ( n ) . (2.8)
3. Discussions
In this article we have taken one half of the elliptic genus of K Z Enriques .We have shown that M in fact acts on Z Enriques . We should note, however, that asymmetry group still larger than M may possibly act on the elliptic genus. We haveevidence that a maximal subgroup M : 2 of M (binary extension of M ) acts on Z Enriques .It was crucial for the existence of Enriques moonshine that all the multiplicities ofreal representations of M are even integers in Mathieu moonshine. We have recentlynoticed that similar phenomena take place in the Umbral moonshine and thus it is quitelikely that we can take one half of Jacobi forms of Umbral moonshine and construct a NRIQUES MOONSHINE 5 new moonshine series with reduced symmetry groups. This issue will be discussed in aforthcoming publication [7].
Note added:
After the original version of this paper has been submitted to arXiv wecame to know the paper [13] by S. Govindarajan where the group M is used as thesymmetry group of Mathieu moonshine. In this paper the relation (2.2) between theconjugacy classes of M and M has been obtained. Also the multiplicities of irreps. of M in the decomposition of expansion coefficients A ( n ) at smaller values of n have beenobtained in agreement with our results of Enriques moonshine upto an overall factor 2.We thank S. Govindarajan for informing us of this paper. Acknowledgments
T.E. would like to thank Y.Tachikawa for discussions on the relation between M and M . He also thanks discussions with S. Mukai on Enriques surface. Research of T.E.is supported in part by JSPS KAKENHI Grant Number 22224001, 23340115. Researchof K.H. is supported in part by JSPS KAKENHI Grant Number 23340115, 24654041. T . E G U C H I AN D K . H I K A M I
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 χ χ
23 7 − − − − − − − − − − χ − − − − √ − − i √ − − − √ − − − i √ − √ − − i √ − − χ − − − − − i √ − √ − − − − i √ − − √ − − i √ − √ − − χ
231 7 − − − − − − − √ − − i √ χ
231 7 − − − − − − − − i √ − √ χ
252 28 12 9 0 4 4 0 2 1 0 0 0 0 2 − − − − − χ
253 13 −
11 10 1 − − − − − − χ
483 35 3 6 0 3 3 3 − − − − χ −
14 10 5 − − − − − √ − − i √ χ −
14 10 5 − − − − − − i √ − √ χ − −
10 0 3 6 2 − − − √ − − i √ − √ − − i √ − √ − − i √ χ − −
10 0 3 6 2 − − − − i √ − √ − − i √ − √ − − i √ − √ χ − − − − − − − χ − − − − − √ − − i √ − − − − √ − − − i √ χ − − − − − − i √ − √ − − − − − i √ − − √ χ −
15 5 8 − − − − − χ −
21 11 16 7 3 − − − − − χ − − − − − − − χ −
19 0 6 − − − − − − χ − − − − − χ − − − − − − χ −
15 0 1 − − − − χ −
56 24 9 0 − − − − − χ −
28 36 − − − − − χ − −
45 0 0 3 − − − T a b l e . c h a r a c t e r t a b l e o f M . | M | = NRIQUES MOONSHINE 7 g size cycle shape1A 1 1
2A 11385 1
2B 31878 2
3A 226688 1
3B 485760 3
4A 637560 2
4B 1912680 1
4C 2550240 4
5A 4080384 1
6A 10200960 1
6B 10200960 6
7A 5829120 1
7B 5829120 1
8A 15301440 1
10A 12241152 2
11A 22256640 1
12A 20401920 2
12B 20401920 12
14A 17487360 1
14B 17487360 1
15A 16321536 1
15B 16321536 1
21A 11658240 3
21B 11658240 3
23A 10644480 1
23B 10644480 1 Table 2.
Cycle shapes of conjugacy classes of M . T . E G U C H I AN D K . H I K A M I n χ χ χ = χ χ = χ χ χ χ χ = χ χ = χ χ χ = χ χ χ χ χ χ χ χ χ χ χ − T a b l e . m u l t i p li c i t i e s o f t h e d ec o m p o s i t i o n o f A ( n ) i n t o i rr e du c i b l e r e p r e - s e n t a t i o n s o f M i n M a t h i e u m oo n s h i n e N R I Q U E S M OO N S H I N E n
1A 2A 2B 3A 3B 4A 4B 4C 5A 6A 6B 7AB 8A 10A 11A 12A 12B 14AB 15AB 21AB 23AB0 − − − − − − − − − − − − − − − − − − − − −
21 90 − − − − − − −
22 462 14 − − − − − − − −
28 20 10 −
14 4 − − − −
14 4554 42 −
38 0 12 − − − − − −
56 72 −
18 0 − − − − −
90 20 −
16 6 − − − − −
138 118 0 30 6 − − − − − − − − −
30 0 − −
12 0 2 0 − − −
238 258 42 − −
14 10 10 −
10 2 6 0 − − − − −
352 0 42 0 − − − − −
478 450 −
60 0 18 − − − − −
600 62 − − −
16 8 − − − −
786 830 0 84 −
18 22 6 0 0 − − − − −
90 0 10 − −
18 6 0 0 2 − − − −
110 22 − −
10 4 6 2 − − − − − −
12 12 −
28 0 0 6 0 0 0 − − −
156 0 −
36 28 12 14 − − − − − −
166 14 −
18 38 0 − − − − −
219 78599556 − − − −
24 0 − − − −
228 0 −
18 14 −
42 14 4 0 − − − − − −
40 48 16 0 6 6 4 4 0 − − − − −
16 48 18 0 4 − − − − −
360 0 54 − −
18 0 − − − − −
28 28 − − − − − − −
72 64 32 12 0 −
10 12 − − −
510 0 22 −
34 78 0 10 0 0 − − −
127 2745870180 − −
600 84 − −
36 30 8 8 −
10 4 − − − − −
36 36 −
84 0 0 12 2 0 0 0 0 0 − − −
762 0 −
92 100 36 − −
10 0 − − − − − − −
840 48 −
40 96 22 − − − − T a b l e . E x p a n s i o n c o e ffi c i e n t s o f A g ( n ) i n M a t h i e u m oo n s h i n e . g size cycle shape1A 1 1
2A 396 2
2B 495 1
3A 1760 1
3B 2640 3
4A 2970 2
4B 2970 1
5A 9504 1
6A 7920 6
6B 15840 1
8A 11880 4
8B 11880 1
10A 9504 2
11A 8640 1
11B 8640 1 Table 5.
Cycle shapes of conjugacy classes of M .
1A 2A 2B 3A 3B 4A 4B 5A 6A 6B 8A 8B 10A 11A 11B χ χ − − − − − − χ − − − − − − χ
16 4 0 − − − √ − − i √ χ
16 4 0 − − − − i √ − √ χ
45 5 − − − − χ
54 6 6 0 0 2 2 − − − χ − − − − − χ − − − − − χ − − − − − χ
66 6 2 3 0 − − − χ − − − − − − χ
120 0 − − − χ
144 4 0 0 − − − χ − − − − Table 6. character table of M . | M | = 95040 NRIQUES MOONSHINE 11 M \ M χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ χ
23 1 1 1 χ
45 1 χ
45 1 χ
231 1 1 χ
231 1 1 χ
252 1 1 1 2 1 1 χ
253 1 1 1 1 1 1 χ
483 1 1 2 2 1 1 χ
770 1 2 2 1 χ
770 1 2 2 1 χ
990 1 1 1 1 2 1 2 χ
990 1 1 1 1 2 1 2 χ χ χ χ χ χ χ χ χ χ χ χ χ Table 7.
Branching of M representations into those of M . Only non-zeromultiplicities are written. n χ χ = χ χ = χ χ χ χ χ = χ χ χ χ χ χ − Table 8. multiplicities of irreducible representations of M in Enriquesmoonshine EFERENCES 13
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