Mock Modular Mathieu Moonshine Shamit Kachru Strings 2014 Saturday, June 21, 14
Any new material in this talk is based on: Mock Modular Mathieu Moonshine Modules Miranda C. N. Cheng 1 , Xi Dong 2 , John F. R. Duncan 3 , Sarah Harrison 2 , Shamit Kachru 2 , and Timm Wrase 2 which just appeared on the arXiv, and work in progress. Outline of talk: 1. Introduction II. Geometric motivation III. M22/M23 moonshine Saturday, June 21, 14
1. Introduction A talk about moonshine needs to begin with some recap of the story to date, and the objects involved. At the heart of the story are two classes of beautiful and enigmatic objects in mathematics: First off, we have the sporadic finite groups -- the 26 simple finite groups that do not come in infinite families. Saturday, June 21, 14
And secondly, we have the modular functions and forms: objects which play well with the modular group (c.f. worldsheet partition functions, S-duality invt. space-time actions, ...). A common example: consider SL(2,Z) acting on the UHP via fractional linear transformations: τ → a τ + b c τ + d ≡ A · τ Then a modular function is a meromorphic function which satisfies: f ( A · τ ) = f ( τ ) while a modular form of weight k satisfies instead: f ( A · τ ) = ( c τ + d ) k f ( τ ) Saturday, June 21, 14
� Monstrous moonshine originated in the observation that: � Dims of irreps of M d 1 1 d 2 196,883 d 3 21,296,876 d 4 842,609,326 � while j ( τ ) = 1 q + 744 + 196 , 884 q + 21 , 493 , 760 q 2 + · · · � q = e 2 π i τ Saturday, June 21, 14 � �
� One notices immediately the “coincidence”: � The “McKay-Thompson series” greatly strengthen � evidence for some real relationship: Suppose there is a physical theory whose partition function is j, and which has Monster symmetry. Then: V = V − 1 ⊕ V 1 ⊕ V 2 ⊕ V 3 ⊕ . . . V − 1 = ρ 0 , V 1 = ρ 1 ⊕ ρ 0 , V 2 = ρ 2 ⊕ ρ 1 ⊕ ρ 0 , . . . Saturday, June 21, 14
This suggests to also study the MT series: ch ρ ( g ) = Tr ( ρ ( g )) , g ∈ M T g ( τ ) = ch V − 1 ( g ) q − 1 + P ∞ i =1 ch V i ( g ) q i For each conjugacy class of M, we get such a series. Now while the partition function is modular under SL(2,Z), in general the MT series are not: Z [ g ] ( τ ) = Tr( gq L 0 ) = T [ g ] ( τ ) ...but we should still get modular functions under a subgroup of SL(2,Z) 0 1 @ a b that preserves the B.C. A c d h d g c h − − − − − − − − − − → g a h b g Saturday, June 21, 14
This gave many further non-trivial checks. Eventually, a beautiful and complete (?) story was worked out, by Frenkel-Lepowsky-Meurman, Borcherds, .... Summary on Monster Unique 24-dim’l even self-dual lattice with no points of length-squared 2 Bosonic strings on Leech lattice orbifold Frenkel, Lepowsky, Meurman; Dixon, Ginsparg, Harvey; Borcherds Modular invariant Monster symmetry partition function ���� Saturday, June 21, 14
II. Geometric Motivation This is where matters stood until the work of EOT (2010) brought K3 into the story: * For any (2,2) SCFT, one can compute the elliptic genus: q ¯ Z ell ( q, γ L ) = Tr RR ( − 1) F L q L 0 e i γ J L ( − 1) F R ¯ L 0 It is a generalization of a modular form, known as a Jacobi form. Saturday, June 21, 14
Math object definition slide A Jacobi form of level 0 and index m behaves as � mcz 2 � a τ + b � � � � z a b = e φ ( τ , z ) , ∈ SL 2 ( Z ) ( φ c τ + d, c d c τ + d c τ + d φ ( τ , z + λτ + µ ) = ( − 1) 2 m ( λ + µ ) e [ − m ( λ 2 τ + 2 λ z )] φ ( τ , z ) , λ , µ ∈ Z under modular transformations and elliptic transformations. The latter is encoding the behavior under the “spectral flow” of N=2 SCFTs. Saturday, June 21, 14
It was computed for K3 in 1989 by EOTY; expanded in N=4 characters in Ooguri’s thesis; and interpreted in terms of moonshine in 2010: 4 θ i ( τ , γ ) 2 X φ ( τ , γ ) = 8 θ i ( τ , 0) 2 i =2 n =1 A n ch long γ ) = 20 ch short 1 / 4 , 0 ( τ , γ ) − 2 ch short 1 / 4 , 1 / 2 ( τ , γ ) + P ∞ 1 / 4+ n, 1 / 2 ( τ , γ ) From: Hirosi Ooguri <h.ooguri@gmail.com> The values of the As are given by: Subject: My PhD thesis Date: May 15, 2014 2:32:56 PM PDT To: Shamit Kachru <shamit.kachru@gmail.com> Cc: Ooguri Hirosi <h.ooguri@gmail.com> A 1 = 90 = 45 + 45 dims of irreps Dear Shamit, of M24! A 2 = 462 = 231 + 231 Here is a copy of my PhD thesis. Please see A 3 = 1540 = 770 + 770 (3.16). I did not know how to divide these by two. Regards, Hirosi Saturday, June 21, 14
Math object definition slide M24 is a sporadic group of order | M24 | = 2 10 · 3 3 · 5 · 7 · 11 · 23 = 244 , 823 , 040 “Automorphism group of the unique doubly even self-dual binary code of length 24 with no words of length 4 (extended binary Golay code).” * Consider a sequence of 0s and 1s M22 and M23 are * Any length 24 word in G has even overlap with all the subgroups of codewords in G iff it is in G permutations in * The number of 1s in each element is divisible by 4 M24 that stabilize but not equal to 4 one or two points * The subgroup of S24 that preserves G is M24 Saturday, June 21, 14
This M24 moonshine, and a list of generalizations associated with each of the Niemeier lattices called “umbral moonshine,” have been investigated intensely in the past few years. The full analogue of the story of Monstrous moonshine is not yet clear. Cheng, Duncan, Harvey In particular, no known K3 conformal field theory (or auxiliary object associated with it) gives an M24 module with the desired properties. Gaberdiel, Hohenegger, Volpato The starting point for the work I’ll report was the desire to extend these kinds of results to Calabi-Yau manifolds of higher dimension. Saturday, June 21, 14
For Calabi-Yau threefolds, there is a story involving the heterotic/type II duality: Heterotic strings on Type IIA on Calabi-Yau K3xT2 threefolds elliptic fibration over Z c 2 ( V 1 ) + c 2 ( V 2 ) K 3 F n , n 1 = 12 + n, n 2 = 12 − n ≡ n 1 + n 2 = 24 dilaton S size of base P 1 A related M24 structure appears in the heterotic string on K3 as a (dualizable) quantity governing space-time threshold corrections. So it appears in GW invariants of the dual Calabi-Yau spaces. Cheng, Dong, Duncan, Harvey, Kachru, Wrase Saturday, June 21, 14
We were then led to think about Calabi-Yau fourfolds. The elliptic genus of a Calabi-Yau fourfold has an interesting structure. It is a linear combination of two Jacobi forms of weight 0 and index 2: C.D.D. Neumann, 1996 Z ell ( τ , z ) = χ 0 ( X ) E 4 ( q ) φ − 2 , 1 ( q, y ) 2 + χ ( X ) φ 0 , 1 ( q, y ) 2 − E 4 ( q ) φ − 2 , 1 ( q, y ) 2 � � . 144 ∞ σ 3 ( k ) q 2 k = 1 + 240 q 2 + 2160 q 4 + 6720 q 6 + · · · X E 4 ( q ) = 1 + 240 k =1 ✓ 2 Standard generators for φ 10 , 1 ( q, y ) ✓ 1 ◆ y 2 − 8 ◆ y + 12 − 8 y + 2 y 2 φ − 2 , 1 ( q, y ) = = y − 2 + y q + . . . , − ring of weak Jacobi forms η ( q ) 24 ✓ 1 ◆ ✓ 10 ◆ φ 12 , 1 ( q, y ) y 2 − 64 y + 108 − 64 y + 10 y 2 φ 0 , 1 ( q, y ) = = y + 10 + y + q + . . . η ( q ) 24 Saturday, June 21, 14
The piece that is present universally has a nice expression: P 4 1 i =1 θ i ( τ , 2 z ) θ i ( τ , 0) 11 Z univ ∼ η ( τ ) 12 As we’ll see momentarily, this has a very suggestive q- expansion and hints at many interesting things. But first, we switch to a setting where all statements can be made precise, without randomly selecting a fourfold. We can consider this a move to Platonic ideals instead of real-world grubby fourfolds... Saturday, June 21, 14
III. The Platonic realm of M22/M23 moonshine We begin with the chiral SCFT on the E8 root lattice. It can be formulated in terms of 8 bosons and their Fermi superpartners. Next, we orbifold: � � orbifold: � Saturday, June 21, 14 � �
The partition function of this theory can be computed by elementary means. It is given by: � � � � � � � The coefficients are interesting: � � for Natural decomposition � into Co1 reps! Saturday, June 21, 14 Saturday, June 21, 14
In fact, it was realized some years ago that this model has a (not so manifest) Conway symmetry, which commutes with the N=1 supersymmetry. FLM; � Duncan ’05 � This CFT played a significant role in attempts to find � a holographic dual of pure supergravity in AdS3. Witten ’07 Saturday, June 21, 14 Saturday, June 21, 14
We have realized several new things about this model; explaining them will occupy the rest of this talk. 1. It admits an N=4 description with more or less manifest M22 moonshine. 2. The elliptic genus and twining functions that arise match expectations for every conjugacy class. They have a beautiful interpretation as Rademacher sums. 3. The natural objects appearing in the genera are mock modular forms. This gives a completely explicit example of mock moonshine with a full construction of the module. 4. An analogous story holds for an N=2 description with M23 mock moonshine. Saturday, June 21, 14
One can construct a full string theory simply related to this module by considering the asymmetric orbifold generated by this acting separately on left/right. Z 2 Its elliptic genus is: � � � � Now, expand this thing in N=4 characters: Saturday, June 21, 14
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