Note on Twisted Elliptic Genus of K3 Surface

We discuss the possibility of Mathieu group M24 acting as symmetry group on the K3 elliptic genus as proposed recently by Ooguri, Tachikawa and one of the present authors. One way of testing this proposal is to derive the twisted elliptic genera for all conjugacy classes of M24 so that we can determine the unique decomposition of expansion coefficients of K3 elliptic genus into irreducible representations of M24. In this paper we obtain all the hitherto unknown twisted elliptic genera and find a strong evidence of Mathieu moonshine.


Now consider a graded vector space
where the space V (n) has a dimension dim V (n) = A(n). Since the dimension of the representation space is given by the trace of the identity element, we may rewrite the sum (1.7) as Twisted elliptic genus is defined instead by considering an arbitrary group element g − q Since the trace of g depends only on its conjugacy class, there exists a twisted elliptic genus Z g (z; τ ) corresponding to each conjugacy class. As a generalization of (1.1), we define the twisted elliptic genus by the decomposition where χ g ∈ Z is the Witten index Z g (z = 0; τ ). The q-series Σ g (τ ) is thus the analogue of the McKay-Thompson series of the monstrous moonshine [4].
Twisted elliptic genera of the conjugacy classes of type I have already been obtained in the literature [2,13]. On the other hand, twisted elliptic genera of type II are yet largely unknown. In this paper we obtain all the twisted elliptic genera of type II and then use the character formula of the Mathieu group to derive the coefficients of the decomposition of K3 elliptic genus into a sum of irreducible representations of M 24 . We have checked that we always obtain the positive integral coefficients in the decomposition up to q 600 . We thus provide a very strong support for the Mathieu moonshine conjecture. 1 2. Twisted elliptic genus of the set type I Twisted elliptic genera of the type I has been obtained previously [2,5,13]. Type I genera for basic classes, pA (p = 2, 3, 5, 7) were discussed by A.Sen and his collaborators [5,16] (also [15]) in connection with the counting problem of 1 4 BPS monopoles and dyons.
We first introduce the standard notation in the theory of Jacobi forms [12] φ 0,1 (z; τ ) = 4 θ 10 (z; τ ) and Here the Jacobi theta functions are defined in Appendix A. φ M,N denotes a Jacobi form with weight M and index N. We also use the Eisenstein series The elliptic genus for K3 is given by [10,17] Note that the class 1A consists of the identity element and hence Z 1A is the original untwisted elliptic genus.
In the case of classes pA (p = 2, 3, 5, 7) there is a general formula for the twisted elliptic genera [2] Twisted elliptic genera for other classes are given by [2,13] (2. 16) In the class 23A we have used the newforms [23] f 23,1 (τ ) = 2 q − q 2 − q 4 − 2 q 5 − 5 q 6 + 2 q 7 + 4 q 9 + 6 q 10 − 6 q 11 + 5 q 12 + 6 q 13 3. Twisted elliptic genus of the set type II Our main task in this paper is to obtain all the twisted elliptic genera belonging to type II. Here, unfortunately there is no definite guiding principle. We have to make an educated guess for the candidate elliptic genera which reproduce the correct coefficients of lower order q-expansions (see Table 3) and have the correct weight and level as modular forms.
By trial and error we have obtained the following elliptic genera which are written in the form of η-product; For the class 21A one has a linear combination of η-products One sees that Σ g (τ ) is the η-product which is modular on congruence subgroup Γ 0 (ord(g)) with character.
We note the following relation among the genera of type I and type II

Mathieu moonshine
In Table 2 the character formula for the Mathieu group M 24 is presented. We denote its elements as χ g R where R runs over irreducible representations and g runs over conjugacy classes. It is well-known that the character formula obeys the orthogonality relation where n g is the number of elements in the conjugacy class g and |G| is the order of the group G. Let us denote the multiplicity of the representation R in the decomposition of the K3 elliptic genus at level n as c R (n). We then obtain the value of the twisted genus of the class g at level n as where A g (n) is defined in (1.10). Note that by choosing g = 1A in (4.2) we find In fact c R (n) is the multiplicity of representation R at level n.
If one uses the orthogonality relation (4.1), we can invert the relation (4.2) and find a formula for the multiplicities We have checked by computer that the multiplicities c R (n) are positive integers for all representations up to n = 600. See Table 4. This provides a very strong support of the Mathieu moonshine conjecture.

Entropy
In Table 3, we have tabulated the values of A g (n), the expansion coefficients of twisted genera Z g (z; τ ). In the untwisted case (g = 1A) we have applied the the method of Bringmann-Ono [1] and obatined the Poincaré series [6] where −4 • is the Legendre symbol, and I denotes the Bessel function, We have identified the expotential growth of {A(n)} at large n as the entropy of K3 surface See [7,8] for the discussion of entropy of higher-dimensional complex manifolds with reduced holonomy.
In case of g ∈ type I, twisted elliptic genus Z g (z; τ ) is modular on the congruence subgroup Γ 0 (ord(g)) of SL(2; Z). Correspondingly the q-series Σ g (τ ) is a mock theta function on Γ 0 (ord(g)), and by using the same method as above the Fourier coefficients A g (n) are given by See [6] where a case of Γ 0 (2) was studied. The above formula shows that the entropy S g of "twisted" K3 is given by .
Thus the entropy of twisted K3 is reduced by a factor 1/ ord(g). This coincides with the result of [20] that the entropy of the Z N twisted CHL model is 1/N times the entropy of the untwisted model.

Discussions
We have completed the analysis initiated in [2,13] on Mathieu moonshine phenomenon by providing all the twisted elliptic genera for K3 surface. Making use of them we are able to decompose uniquely the expansion coefficients {A g (n)} into a sum of irreducible representations of M 24 . We find that the multiplicities of all irreducible representations are positive integers up to the level n = 600.
For the sake of illustration we present the decomposition at the level n = 98; Thus the observation of [9] may well be proved to be true.
We are, however, still very far from satisfactory understanding of the origin of the symmetry of the Mathieu group M 24 . As is well-known, there are special classes of K3 surfaces which possess automorphism under subgroups of M 23 [18,19] (see [22] for a recent result). Thus it appears that M 24 emerges as an enhanced symmetry in string theory. Hopefully the twisted genera we have obtained offer some clue in our search for the action of M 24 on the string Hilbert space in K3 compactification.

Acknowledgments
We thank H.Ooguri and Y.Tachikawa for their collaboration at the early stage of this work. K.H. thanks D.Zagier for useful discussions at MFO, and participants in "Prospects in q-series and modular forms" at Dublin for discussions. Some computations are performed by use of SAGE [21].       Table 3. Values A g (n) from twisted elliptic genera for lower levels n.