Aspects of Umbral Moonshine Jeff Harvey Heterotic Strings and (0,2) QFT Texas A&M April 28-May 2, 2014 based on work with Miranda Cheng, John Duncan and Sameer Murthy and work in progress with Sameer and Caner Nazaroglu.
1 Thursday, May 8, 14 Amuse-bouche Ramanujan’s third order mock theta function:
2 1 qn f(q)= =1+q 2q2 +3q3 3q4 +3q5 5q6 + (1 + q)2 (1 + qn)2 ··· n=0 X ··· The first component of a two-component mock modular form appearing in works of Eguchi-Hikami and Dabholkar- Murthy-Zagier: (3,1) 1/12 2 3 4 H (q)=2q ( 1 + 16q + 55q + 144q + 330q + 704q5 + 1397q6 + ) ··· replace Tr 1A = 2 Tr 2B = 2 2 1 ! 2 1 Tr 1A =2 16 Tr 2B =0 4+ 5 ⇥ ! 4+ 5 Tr 1A =2 55 Tr 2B = 2 2 9 ⇥ ! 2 9 etc. using the character table of 2.M12 to find
(3,1) 1/12 2 H (q)= 2q f(q ) 2B
Thursday, May 8, 14 Modular objects M M O O N S H I N E
G S Groups (Finite) String theory A Algebras
Thursday, May 8, 14 The original example of this structure is Monstrous Moonshine G The Monster sporadic group M The modular function j ( ) and other genus zero hauptmoduln appearing as McKay- Thompson series. A Vertex Operator Algebra (OPE of chiral Vertex operators) S Bosonic or Heterotic String on an asymmetric 24 orbifold background (R / L)/(Z/2)
Thursday, May 8, 14 1 1 m J(⌧)=q + 196884q + = a(m)q ··· m= 1 X
196883+1 SL(2,Z) i \H [ { 1} dimensions of first two irreps of the Monster simple group In the asymmetric orbifold J(⌧)=TrqL0 c/24 CFT construction
Thursday, May 8, 14 One of the most remarkable, and still poorly understood parts of Monstrous Moonshine is that the McKay-Thompson series L c/24 T (⌧)=Trgq 0 ,gM g 2 are all hauptmoduls, that is analogs of the modular function J ( ⌧ ) in that they map genus zero quotients of the UHP by subgroups of SL (2 ,R ) into the Riemann sphere. The development of the Monster CFT by Frenkel- Lepowsky-Meurman and the proof of the above genus-zero property by Borcherds introduced a great deal of new technology into mathematics and string theory: orbifolds, vertex operator algebras...
Thursday, May 8, 14 OUTLINE
• A new kind of moonshine relating K3 and the Mathieu group M24 (Eguchi, Ooguri,Tachikawa 2010) • A mathematical extension called Umbral Moonshine (M.Cheng, J.Duncan, JH) • Trying to relate Mathieu and Umbral Moonshine to Black Holes and BPS state counting via Fivebranes (S.Murthy and JH)
2 Thursday, May 8, 14 K3 elliptic genus and M24 moonshine
A K3 surface X defines a (4,4) SCFT with c=6 and from that a modular object known as the elliptic genus:
X J0 J¯0 L0 1/4 L¯0 1/4 J0 Zell( ,z)=TrR R( 1) q q¯ y 2 i 2 iz q = e ,y = e ; H,z C X Zell( ,z) is a (weak) Jacobi form of weight 0, index 1 The elliptic genus is independent of moduli and can be computed directly, say in an orbifold limit. It can also be determined by an indirect argument using the fact that there is a unique such Jacobi form up to normalization.
3 Thursday, May 8, 14 K3 elliptic genus and M24 moonshine A weak Jacobi form of (weight,index)=(k,m) is a holomorphic map : H C C obeying 2 a + b z k 2 imcz modular: , =(c + d) e c +d ( ,z) c + d c + d
m 2 2m elliptic: ( ,z + + µ)=q y ( ,z)
ab SL(2, Z); , µ Z cd
n r weak: ( ,z)= c(n, r)q y ,c(n, r) = 0 unless n 0 n,r
4 Thursday, May 8, 14 K3 elliptic genus and M24 moonshine
It is known that the ring of weak Jacobi forms for k even has the form (Eichler-Zagier)
weak m j m j Jk,m = j=0Mk+2j 2,1 0,1 ·
2 Modular forms of 11( ,z) wt. k+2j, generated ( )6 by E4,E6