2012
DOI: 10.1007/s11005-012-0569-2
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Twisted Elliptic Genus for K3 and Borcherds Product

Abstract: Abstract. We further discuss the relation between the elliptic genus of K3 surface and the Mathieu group M 24 . We find that some of the twisted elliptic genera for K3 surface, defined for conjugacy classes of the Mathieu group M 24 , can be represented in a very simple manner in terms of the η product of the corresponding conjugacy classes. It is shown that our formula is a consequence of the identity between the Borcherds product and additive lift of some Siegel modular forms.

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Cited by 13 publications
(15 citation statements)
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“…In recent years a new moonshine phenomenon has been discovered, dubbed Mathieu moonshine, which relates the representation theory of the Mathieu group M 24 with weak Jacobi forms and superstring theory on K3-surfaces [18,22,[35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53]. The role of the McKay-Thompson series is here played by the so called twining genera φ g (τ, z), which are weak Jacobi forms with respect to subgroups of SL(2, Z).…”
Section: Connection With Moonshinementioning
confidence: 99%
“…In recent years a new moonshine phenomenon has been discovered, dubbed Mathieu moonshine, which relates the representation theory of the Mathieu group M 24 with weak Jacobi forms and superstring theory on K3-surfaces [18,22,[35][36][37][38][39][40][41][42][43][44][45][46][47][48][49][50][51][52][53]. The role of the McKay-Thompson series is here played by the so called twining genera φ g (τ, z), which are weak Jacobi forms with respect to subgroups of SL(2, Z).…”
Section: Connection With Moonshinementioning
confidence: 99%
“…This conjecture, originated from an observation of Eguchi, Ooguri and Tachikawa (EOT) in [5], proposes a connection between the elliptic genus of K3 and a finite sporadic simple group, the Mathieu group M 24 . After the original EOT proposal, a considerable amount of evidence in favour of this conjecture has been compiled [6][7][8][9][10] and several different incarnations of the relationship between M 24 and various string compactifications on K3 have been uncovered [11][12][13][14][15][16][17][18][19]. Despite the amount of work on the subject, however, no satisfactory explanation of this phenomenon has been provided so far.…”
Section: Jhep08(2014)094mentioning
confidence: 99%
“…They are, she argued, related to the 1/4-BPS spectrum of the K3 × T 2 -compactified type II string theory [12]. In contrast to the expansions given in [15], they are, however, not modular by construction.…”
Section: Introductionmentioning
confidence: 96%