ADE FIVEBRANES, MOCK MODULAR FORMS AND UMBRAL MOONSHINE Sameer Murthy, JH; SM, Caner Nazaroglu and JH
arXiv:1307.77717 and to appear
Recent Developments in String Theory 21-25 July 2014 Ascona Switzerland
Tuesday, July 22, 14 OUTLINE
Review of Matheiu and Umbral Moonshine
Physical Origin and Properties of the Fivebrane SCFT
BPS State Counting and Connections to Umbral Moonshine
Some Fun Math
Open Problems and Conclusions
Tuesday, July 22, 14 Review of Mathieu and Umbral Moonshine
The elliptic genus of a K3 surface X has a decomposition into characters of the N=4 SCA:
X R˜ R˜ 1 R˜ Zell = 20ch1, 1 ,0 +2c( 1/8)ch1, 1 , 1 + c(n 1/8)ch1,n+ 1 , 1 4 4 2 4 2 n=1 X (2) 1/8 n 1/8 H ( )= 2q + 2c(n 1/8)q n 1 c(7/8) = 45 c(15/8) = 231 dimensions of M24 irreps c(23/8) = 770 c(31/8) = 2277 (Eguchi, Ooguri, Tachikawa) 26 c(n 1/8) = ri dimR n i i
Tuesday, July 22, 14 This gives us a q series with coefficients related to representations of M24 with positive multiplicity and is reminiscent of the monstrous moonshine connection between representations of the Monster group and the modular function
1 J( )=q + const + 196884q + ···
However H (2) ( ⌧ ) is not a modular 196883 + 1 function, rather it is a mock modular function or mock theta function. Mock theta functions first appeared in 1920 in the last letter Ramanujan wrote to Hardy. He wrote them down as q expansions but did not explain how he found them or fully define their properties.
Tuesday, July 22, 14 Recall a modular form of weight k is a holomorphic function f
a + b k ab f( )=(c + d) f( ) SL(2, Z) c + d cd h ( ) is a mock modular form of weight k if there is a pair ( h ( ) ,g ( )). where g ( ) is a holomorphic modular form of weight 2-k, known as the shadow of h ( ⌧ ) , such that the non-holomorphic function