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Aspects of Umbral Moonshine – Jeff Harvey 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 − 2χ1 − ! 2χ1 − Tr 1A =2 16 Tr 2B =0 χ4+χ5 ⇥ ! χ4+χ5 Tr 1A =2 55 Tr 2B = 2 2χ9 ⇥ ! 2χ9 − 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 ( τ ) 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 /ΛL)/(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(τ,z)=TrR R( 1) − q − q¯ − y × − 2πiτ 2πiz q = e ,y = e ; τ H,z C ∈ ∈ X Zell(τ,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 φ : H C C obeying × → 2 aτ + b z k 2πimcz modular: φ , =(cτ + d) e cτ+d φ(τ,z) cτ + d cτ + d mλ2 2mλ elliptic: φ(τ,z + λτ + µ)=q− y− φ(τ,z) ab SL(2, Z); λ, µ Z cd∈ ∈ n r weak: φ(τ,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 φ 2,1φ0,1− ⊕ · − 2 Modular forms of θ11(τ,z) wt. k+2j, generated η(τ)6 by E4,E6 θ (τ,z) 2 θ (τ,z) 2 θ (τ,z) 2 4 01 + 10 + 00 θ (τ,0) θ (τ,0) θ (τ,0) 01 10 00 5 Thursday, May 8, 14 K3 elliptic genus and M24 moonshine So, for example weak J0,1 = aφ0,1 weak { 2 } 2 J0,2 = aφ0,1,bE4φ 2,1 { − } weak 3 2 3 J0,3 = aφ0,1,bE4φ0,1φ 2,1, cE6φ 2,1 { − − } weak Since J 0 , 1 is one-dimensional and X Zell(τ,0) = Euler(X) = 24 we have X Zell(τ,z)=2φ0,1(τ,z) 6 Thursday, May 8, 14 K3 elliptic genus and M24 moonshine Since we have an N=4 SCA we can decompose into characters of unitary reps: X ˜ Zell = mh,chh, (h,) The unitary reps fall into two classes. For c=6 we have “massless” h =1/4,=0, 1/2 ch˜ = complicated Lerch sum “massive” 2 h 3/8 θ11(τ,z) h>1/4,=1/2 ch˜ = q − h,1/2 η(τ)3 2 1/8 θ11 At h=1/4: ch˜ +2ch˜ = q− 1/4,1/2 1/4,0 η3 7 Thursday, May 8, 14 K3 elliptic genus and M24 moonshine One finds the decomposition ZX = 20ch˜ 2ch˜ + 2c(n 1/8)ch˜ ell 1/4,0 − 1/4,1/2 − n+1/4,1/2 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 positive integer irreps of multiplicity M24 8 Thursday, May 8, 14 K3 elliptic genus and M24 moonshine It is useful to repackage this using an identity between massless and massive reps as θ2 ZX = 24ch˜ + H(2)(τ) 11 ell 1/4,0 η3 with (2) 1/8 n 1/8 H (τ)= 2q− + 2c(n 1/8)q − − − n 1 ≥ This gives us a q series with coefficients related to dimensions of M24 representations. This is reminiscent of monstrous moonshine for the modular 1 function J(τ)=q− + const + 196884q + ··· 196883 + 1 9 Thursday, May 8, 14 K3 elliptic genus and M24 moonshine Monstrous Moonshine K3/M24 n (2) n 1/8 J(τ)= a(n)q H (τ)=2 c(n 1/8)q − − n 1 n 0 ≥− ≥ a(n)=dim of Monster reps, c(n-1/8)=dim of M24 reps, (2) J=modular function for SL(2, Z) H =mock modular form of weight 1/2 for SL(2, Z) FLM: ?? V = m 1Vm,a(m) = dimVm W = m 0Wm 1/8 ⊕ ≥− ⊕ ≥ − Constructed as c=24 asymmetric ?? orbifold CFT In either case we can define McKay-Thompson (2) series J g ,H g by dimV = tr1 trg m |Vm → |Vm 10 Thursday, May 8, 14 Mock modular forms Recall a modular form of weight k is a holomorphic function f ( τ ) obeying aτ + b k ab f( )=(cτ + d) f(τ) SL(2, Z) cτ + d cd∈ h ( τ ) is a mock modular form of weight k if there is a pair ( h ( τ ) ,g ( τ )) where g ( τ ) is a holomorphic modular form of weight 2-k, known as the shadow of h, such that the non-holomorphic function ∞ k hˆ(τ)=h(τ) + const g( z¯)(z + τ)− dz τ¯ − − is modular of weight k. (2) 3 ( H ( τ ),η(τ) ) is such a pair with k=1/2 11 Thursday, May 8, 14 Mock modular forms 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 had found them or how they were defined. 12 Thursday, May 8, 14 Mock modular forms The modern understanding of mock modular forms is a recent development (Zwegers 2002) and is starting to appear in a number of different contexts including BH counting and topological N=4 SYM on four manifolds. It seems that the right mathematical framework for the connection between the K3 elliptic genus and M24 is that of Jacobi forms and mock modular forms but the origin of M24 and a concrete description of a module for M24 associated to the elliptic genus remain obscure. Rather than solving these problems we present a generalization of this structure which suggests looking for the solution in new directions. 14 Thursday, May 8, 14 Umbral Moonshine In Mathieu moonshine we have a triple H(2)(⌧), 24⌘3(⌧),M24 of a mock modular form, its shadow, and a finite group which acts on the mock modular form and its shadow. Are there generalizations of this structure? After much work, both theoretical and “experimental” the answer is “yes” and some remarkable new structure is revealed, but at the moment I cannot tell you the origin of this structure. Thursday, May 8, 14 Umbral Moonshine Let X be a root system with A,D,E components, rank 24 and with all components having equal Coxeter number. For each of the 23 X above we (M. Cheng, J. Duncan and JH) claim that there exist 6 mathematical objects determined by X HX ,SX ,GX , ΓX ,TX ,LX Thursday, May 8, 14 Umbral Moonshine The Niemeier lattice L X Recall that even, unimodular (self-dual) lattices occur only in dimensions that are a multiple of 8: 8 : E8 16: E8xE8, Spin(32)/Z2 24: 23 Niemeier lattices Γ X and the Leech lattice with no roots.
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