Mock Modular Mathieu Moonshine

Shamit Kachru Strings 2014

Saturday, June 21, 14 Any new material in this talk is based on:SU/ITP-14/17

Mock Modular Mathieu Moonshine Modules

Miranda C. N. Cheng1, Xi Dong2, John F. R. Duncan3, Sarah Harrison2, Shamit Kachru2, and Timm Wrase2

1Institute of Physics and Korteweg-de Vries Institute for Mathematics, University of

which just appearedAmsterdam, on the Amsterdam, arXiv, the Netherlands and work⇤ in progress. 2Stanford Institute for Theoretical Physics, Department of Physics, and Theory Group, SLAC, Stanford University, Stanford, CA 94305, USA 3Department of Mathematics, Applied Mathematics and Statistics, Case Western Reserve University,Outline Cleveland, of OH talk: 44106, USA

1. IntroductionAbstract We construct vertexII. operatorGeometric super-algebras which motivation lead to modules for moonshine relations connecting the sporadic simple Mathieu groups M22 and M23, with distinguished mock modular

forms. Starting withIII. an orbifoldM22/M23 of a free fermion moonshine theory, any subgroup of Co0 that fixes a 3- dimensional subspace of its unique non-trivial 24-dimensional representation commutes with a certain = 4 superconformal algebra. Similarly, any subgroup of Co that fixes a 2-dimensional N 0 subspace of the 24-dimensional representation commutes with a certain = 2 superconformal N Saturday, June 21, 14 algebra. Through the decomposition of the corresponding twined partition functions into characters of the =4(resp. = 2) superconformal algebra, we arrive at mock modular forms N N which coincide with the graded characters of an infinite-dimensional Z-graded module for the corresponding group. The two Mathieu groups are singled out amongst various other possibili- ties by the moonshine property: requiring the corresponding mock modular forms to be regular at all cusps inequivalent to the low temperature cusp at i . Our constructions constitute the 1 first examples of explicitly realized modules underlying moonshine phenomena relating mock modular forms to sporadic simple groups. Modules for other groups, including the sporadic groups of McLaughlin and Higman–Sims, are also discussed.

⇤On leave from CNRS, Paris.

1 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:

⌧ a⌧+b A ⌧ ! c⌧+d ⌘ · Then a modular function is a meromorphic function which satisﬁes:

f(A ⌧)=f(⌧) ·

while a modular form of weight k satisﬁes instead:

f(A ⌧)=(c⌧ + d)kf(⌧) ·

Saturday, June 21, 14 Saturday, June 21, 14 Wednesday, October 2, 13 1

Physics 451 lectures: Spring 2014 Shamit Kachru, Stanford

I. LECTURE 12: MONSTROUS MOONSHINE

Today, we’re going to take a bit of a detour from geometric ideas and talk about a famous story from the mid 1980s called “Monstrous moonshine.” In part this is because there are new developments surrounding this class of ideas that we will revisit later on. In part it is because this will directly relate to other ‘counting problems’ in AdS/CFT which we will encounter starting in the next lecture; what we learn today will turn out fortuitously to be crucial background there. Monstrous moonshine is a set of interesting relations between three distinct classes of objects: * Algebraic structures (finite groups) * Modular forms *Solutionsofstringtheory Lets first introduce some baby facts we’ll need from the first two areas.

A. Algebraic structures

The simple finite groups have been classified. They fall into 18 infinite families, and in additionThe thereMonster are 26 oddballs is the - the largest “sporadic groups.”of the Of sporadic these, the largest groups. is the Fischer- Griess Monster M, with

JfunctionM =246 320 59 76 112 133 17 19 23 29 31 41 47 59 71 8 1053 . (1) || || · · · · · · · · · · · · · · ⇠ ⇥ All modular functions may be written as rational The Monster is hard to visualize. Its ﬁrst non-trivial irreps occur in high dimensions: functions of Monstrous moonshine originated in the observation It is not an easy beast to visualize. The smallest dim = 1; 196that:, 883; 21, 296, 876; (2) irreducible representations··· are: A natural question for people from other areas of mathematics, or from physics (where group theory has often played a role, and we have a psychological yearning for deep unique John McKay noticed that Dims of irreps of M structures to determine the world): are the sporadic groups deep and interesting, or are they d1 1 accidents to be forgotten as mathematics progresses? d2 196,883

d3 21,296,876

d4 842,609,326 Monstrous Moonshine!A basic result says that any modular function can be It was being constructed in ’78 when John McKay took written as a rationalwhile function of a break to study some number theory papers.

Saturday, June 21, 14 1 j(⌧)= + 744 + 196, 884 q + 21, 493, 760 q2 + Jfunctionq ··· All modularq functions= e2⇡i⌧ may be written as rational functions of Saturday,John June 21,What 14 McKay, John taking McKay a break noticed from (with hard help work from on sporadicJohn groups in the late 1970s,Thompson) came across was: this expansion in a

number John McKaytheory noticed paper. that He noticed: Dims of irreps of M

d1 1

196, 884 = 196, 883 + 1 ddims2 of irreps196,883 of Monster! d 21,296,876 21, 493, 760 = 21, 296, 876 + 196, 883 + 1 3 d4 842,609,326 ... Monstrous Moonshine! Saturday,Wednesday, June October 21, 14 2, 13 A basic result says that any modular function can be written as a rational function of

1 j(⌧)= + 744 + 196, 884 q + 21, 493, 760 q2 + Jfunctionq ··· All modularq functions= e2⇡i⌧ may be written as rational functions of JohnWhat McKay, John taking McKay a break noticed from (with hard help work from on sporadicJohn Thompson) was: groupsWhyWhy should in thewould therelate there 1970s, be a be relationshipcame a relation across betweenbetween this expansion thesethe j-function two in a One numbernotices theoryimmediately paper. the He “coincidence”: noticed: Johnand McKay irrepsclasses noticed of of the objects?? that Monster? Dims of irreps of M WhyWhy should would there there be a be relationship a relation betweenbetween thesethe dj-function1 two1 196, 884 = 196, 883 + 1 ddims2 of irreps196,883 of Monster! * A positive integerand irrepsclasses begs interpretationof of the objects?? Monster? as dim(V)d for21,296,876 some 21, 493, 760 = 21, 296, 876 + 196, 883 + 1 3 vector space V. d4 842,609,326 * A positive integer begs interpretation... as dim(V) for some The “McKay-Thompson Monstrous Moonshine! series” greatly strengthen Saturday,Wednesday, June* October 21,So 14 2, 13 suppose there vectorexists anspace inﬁnite V. dimensional graded evidence for some real relationship: representation: Suppose* So suppose there isthere a physical exists theoryan inﬁnite whose dimensional partition gradedfunction V = Vrepresentation:1 V1 V2 V3 ... is j, and which has Monster symmetry. Then:

V = V 1 V1whereV2 V3 ...

V 1 = ⇢0,V1 = ⇢1where⇢0,V2 = ⇢2 ⇢1 ⇢0,...

Saturday, June 21, 14 Wednesday, October 2, 13 Saturday, June 21, 14 V 1 = ⇢0,V1 = ⇢1 ⇢0,V2 = ⇢2 ⇢1 ⇢0,...

Saturday, June 21, 14 Wednesday, October 2, 13 1 i j(⌧) 744 = dim(V 1) q + 1 dim(Vi) q i=1 P But if the Monster M has a natural action on V, this also suggests that we can fruitfully study the McKay-Thompson This suggests to alsoseries study: the MT series:

ch (g)=Tr(⇢(g)),g M ⇢ 2

Now, if we imagine the Hilbert space V arises1 from the i 1 Tg(⌧)=chV 1 (g) q + i=1 chVi (g) q Now, statespaceif we imagine of a the2d CFT,Hilbert we spacewould V identify: arises from the statespace of a 2d CFT, we would identify:P For eachZ(⌧ )=Tr(conjugacyqL0 )= jclass(⌧) . of M, we get such a series. Now For each conjugacy class in M (there are 194), we get such while Zthe(⌧)=Tr( partitionqL0 )= jfunction(⌧) . is modular under SL(2,Z), a series. In a similar way, we incould general write the the MT MT series series as: are not: In a similar way, we could write the MT series as: Now the point is the following.L0 Let us imagine the modular ModularZ[g](⌧)=Tr( propertiesgq )=T[g](⌧) . functionSaturday,Wednesday, j is Junebeing October 21, 14 2, 13generated by a partition function of a 2d Now the point is the following.L0 Let us imagine the modular Z[g](⌧)=Tr(gq )=T[g](⌧) . Now,CFT onwhile aIndeed, torus. ZModular is standard SL(2,Z) If we orbifold propertiestwist invariant, arguments the boundary suggestthe MT that series conditions: aren’t:...but we should still get function j isunder being a modular generated transformation by a partition functionmodular of a 2d functions under CFT on a torus. If we twist the boundary conditions: Now, whileIndeed, Z is standard SL(2,Z) orbifoldab invariant, arguments suggestthe MT that series aren’t:a subgroup of SL(2,Z) under a modular transformation0 cd1 that preserves the B.C. h @ A hdgc ⇥ ggab ahb 0 cd1 For twining genus: h=1. Thus to get samed diagramc Saturday, June 21, 14 h @ A h g need: ⇥ ggahb we shouldSaturday, June still 21, 14 get‣ c=0a modular mod N=o(g) form for a subgroup of the For twining a=1 genus: mod N=o(g)h=1. Thus [actually to get gcd(a,N)=1 same sufficient]diagram Saturday, June 21, 14 ‣ modular groupneed: that preserves the form of the BC. we shouldThursday, July 19, still 12 get‣ c=0a modular mod N=o(g) form for a subgroup of the Thus,modular any conjecture group‣ a=1that for mod thepreserves N=o(g) decompositions [actually the gcd(a,N)=1 form sufficient] isof subjected the BC. to manyThursday, checks, July 19, 12 because the spaces of these modular Thus, any conjectureforms are for quite the constrained.decompositions is subjected to many checks, because the spaces of these modular Wednesday, October 2, 13 forms are quite constrained.

Wednesday, October 2, 13 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

Saturday, June 21, 14 II. Enter the Calabi-Yau manifold A. The elliptic genus of K3 II. Geometric Motivation The partition function is, in general, difﬁcult to computeThis is whereand highly matters dependent stood untilupon the the work precise of EOTpoint (2010) in moduli brought space K3 one into studies. the story: More robust objects arise by counting BPS states. *A For prototype any (2,2) (in SCFT, 2d (2,2) one theories)can compute is the the elliptic elliptic genus: genus:

¯ Z (q, )=Tr ( 1)FL qL0 eiJL ( 1)FR q¯L0 ell L RR

It is a generalizationLets unpack of this a modular for a moment. form, known as a

Saturday, June 21, 14 Jacobi form.

Saturday, June 21, 14 1Introduction.

2Ellipticgenus.Math object definition slide In this section, I recall some basic facts about elliptic genera for Calabi- Yau manifolds, mostly from [5], and I explicitly derive it for4-folds.Let CAbe Jacobi a complex form manifold of level of complex 0 and dimension indexd ,withm behavesSU(d)holonomy. as Then its elliptic genus is a function φ(τ,z)withthefollowingtransformation properties aτ + b z mcz2 ab φ , = e φ(τ,z), SL2(Z)(1) !cτ + d cτ + d" #cτ + d$ ! cd" ∈ φ(τ,z+ λτ + µ)=( 1)2m(λ+µ)e[ m(λ2τ +2λz)]φ(τ,z), λ,µ Z (2) − − ∈ d where m = 2 ,andithasanexpansionoftheform φ(τ,z)= c(n, r)qnyr (3) under modular transformations≥ ∈Z and elliptic n 0,r% +m Iuseherethenotationstransformations.e[x]= Thee2πix latter, q = e [τis], encodingy = e[z]. The the coefficients c(0, behaviorm + p)for0 underp c havethe the “spectral following geometrical flow” of meaning N=2 − ≤ ≤ SCFTs. c c(0, m + p)=χ = ( 1)p+qhp,q (4) − p − q%=0 Saturday, June 21, 14 p,q Saturday, Junewhere 21, 14 h are the Hodge-numbers of C.Furthermore φ(τ, 0) = χ (5)

1 B. The elliptic genus of K3 and EOT

K3 is the simplest non-trivial Calabi-Yau space - i.e., it admits a family of Ricci ﬂat Kahler metrics and preserves Becausespace-time it supersymmetry.is an index independent of (many) moduli, Herethe is elliptica picture: genus is highly computable.

If this thing was computed in 1989, why talk about it now? It was computed for K3 in 1989 It was bycomputed Eguchi, Ooguri, for Taormina, K3 in Yang: 1989 by EOTY; expanded In 2010,in N=4 the characters following wasin Ooguri’s noticed. thesis; It is naturaland interpreted (given the in supersymmetryterms of theof moonshine K3 CFT) to in expand2010: this in Wednesday, October 2, 13

characters4 of the2 N=4 superconformal algebra: If this thing was computed✓i(⌧, in) 1989, why talk about it now? (⌧, )=8 2 ✓i(⌧, 0) In 2010, the followingi=2 was noticed. It is natural (given the X short short long (⌧,z) = 20 chshort (⌧,z) 2 chshort (⌧,z)+ 1 A ch long (⌧,z) supersymmetry(K⌧3, ) = 20 ch of1/ 4the,(0⌧, K3) CFT)2 ch to1/ 4expand,1/2 (⌧, this)+ in n=11 An ch1/4+n,1/2 (⌧, ) The Jacobi theta functions1/ are4,0 famous objects1/4,1/2 you n=1 n 1/4+n,1/2 characters of the N=4 superconformal algebra:P can look up. P

short short long short short 1 long (K⌧3,(⌧),z =) =20 20chchThe1/4,(0⌧(,⌧,z) )values22 chch1/4,1/2( (⌧⌧of,z, )+)+ then=11 A AsAn chch1/ 4+aren,1/2( ⌧,zgiven(⌧), ) by: 1/4,0 1/4,1/2 n=1 n 1/4+n,1/2 Saturday, June 21, 14 P P From: Hirosi Ooguri Subject: My PhD thesis The valuesA1 = 90of the = 45 As+ are45 given by: Date: May 15, 2014 2:32:56 PM PDT To:dims Shamit of Kachruirreps Cc: Ooguri Hirosi A1 A=2 90= = 45 462+ 45 = 231 + 231 of M24! dims of irreps Dear Shamit, A2 = 462 = 231 + 231 of M24! A3 = 1540 = 770 + 770 Here is a copy of my PhD thesis. Please see A3 = 1540 = 770 + 770 (3.16). I did not know how to divide these by two.

Regards, Hirosi Eguchi, Ooguri, and Tachikawa noticed this in 2010, and Saturday,Eguchi, June 21, 14 Ooguri, and Tachikawa noticed this in 2010, and Saturday, Juneconjectured 21, 14 a “Mathieu Moonshine” relating K3 conjecturedcompactiﬁcation toa “Mathieuthe sporadic groupMoonshine” M24. relating K3 Saturday, Junecompactiﬁcation 21, 14 to the sporadic group M24.

Thursday, May 15, 14 Wednesday, October 2, 13

Wednesday, October 2, 13 1Introduction.

In thisTheThe section, Mathieu Mathieu I recall group group some M24M24 basic isis a facts sporadicsporadic about groupgroup elliptic of of order: gen order:era for Calabi- Yau manifolds,M24 mostly is froma sporadic [5], and group I explicitly of order derive it for4-folds.Let C be a complex manifold of complex dimension d,withSU(d)holonomy. A Jacobi M24form=2 of10 3level3 5 7 011 and23 =index 244, 823 m, 040 behaves as Then its ellipticM24| genus|=210 is a·3 function3 ·5 · 7 · φ11(τ·,z23)withthefollowingtransformation = 244, 823, 040 properties | | · · · · · “Automorphismaτ + b z group mczof the2 unique doubly evenab self-dual φ“Automorphism, group= e of the φunique(τ,z), doubly even self-dualSL2(Z )(1) !cτbinary+ d c τcode+ d" of length#cτ +24d $with no words! ofcd length" ∈ 4 binary code(extended of length binary 24 with Golay no wordscode).” of length 4 φ(τ,z+ λτ + µ)=((extended1)2m(λ binary+µ)e[ m Golay(λ2τ +2 code).”λz)]φ(τ,z), λ,µ Z (2) − − ∈ where m = d ,andithasanexpansionoftheform * Consider a sequence2 of 0s andOr 1s in plain English: φ(τOr,z)= in plain English:c(n, r)qnyr (3) under modular transformations andM22 elliptic and M23 are * Any lengthWednesday, October 24 2, word13 in G has even overlapn≥0,r with∈Z+m all % the subgroups of Wednesday, Octobercodewords 2, 13 in G iff it is in G Iuseherethenotationstransformations.e[x]= Thee2πix latter, q = e [τis], encodingy = e[permutationsz]. The the coe ﬃincients * cThe(0, numberbehaviorm + p of)for0 1s in eachunderp elementc havethe is divisible the “spectral following by 4 geometrical ﬂow”M24 of meaning that N=2 stabilize − but not≤ equal≤ to 4 one or two points SCFTs. c * The subgroup of S24 thatc(0 preserves, m + p G)= is M24χ = ( 1)p+qhp,q (4) − p − q%=0 Friday, May 9, 14 p,q Saturday, June 21, 14 Why does M24 appear in relation to K3? Saturday, June 21, where14 h are the Hodge-numbers of C.Furthermore

No one really knows! There are someφ(τ hints, 0) = χ (5) from geometry.

Wednesday, October 2, 13 1 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 Then, in addition to trying to more deeply understand the EOT result, it is logical to explore wider classes of geometries to look for similar connections.

We move to the setting of 4d N=2 string vacua. Our ﬁrstFor thought Calabi-Yau was to threefolds, explore heterotic there is a strings story involvingon the

Such vacua arise in two simple avatars, related byCheng, duality: Dong, Duncan, K3, where additionalheterotic/type data comes IIinto duality: play. SK, Harvey, Wrase; Harrison,SK, Vafa; SK, Paquette Ferrara, Harvey, Strominger, Vafa

Heterotic strings on Type IIA on Calabi-Yau K3xT2 The Hirzebruch surfacethreefolds is a P 1 bundle over P 1 . In fact, string dualities also relate these to elliptic ﬁbration over c2(V1)+c2(V2) type II strings on K3 Z Fn,n1 = 12 + n, n2 = 12 n Calabi-Yau threefolds Size of “base” of ﬁbration n + n = 24 maps to heterotic dilaton ⌘ 1 2

Can we ﬁnd evidence for similar moonshine in the spectrum of BPS states of this much wider class 1of dilaton S sizetheories? of base P

Thursday, June 20, 13

Morrison, A related M24They can structure be lifted to appears 6d dualities in in theF-theory. heteroticVafa string Saturday, June 21, 14 Wednesday, October 2, 13 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 3 Elliptic genera of more general fourfolds

For a generic Calabi-Yau fourfold, the elliptic genus is a weak Jacobi form of index 2 and weight 0. That is, it satisﬁes

a⌧ + b z 4⇡icz2 ab ell , =exp ell(⌧,z), for all SL(2, Z) . (3.1) Z c⌧ + d c⌧ + d c⌧ + d Z cd 2 ✓ ◆ ✓ ◆ ! The relevant ring of weak Jacobi forms is generated by the basis elements

10,1(q, y) 1 2 8 2 3 Elliptic2,1(q, y genera)= of more general= fourfolds2+y +12 8y +2y q + ... , (3.2) ⌘(q)24 y y2 y ✓ ◆ ✓ ◆ (q, y) 1 10 64 For a generic Calabi-Yau fourfold,12,1 the elliptic genus is a weak Jacobi form of index 2 and 2 0,1(q, y)= 24 = +10+y + 2 +108 64y +10y q + ...(3.3) , weight 0. That is, it satisﬁes ⌘(q) y y y ✓ ◆ ✓ ◆ a⌧ + b z 4⇡icz2 ab whereell ,i,k are the=exp unique cuspell(⌧,z forms), for all of index i andSL(2, weightZ) . (3.1) k and we use the conventions Z c⌧ + d c⌧ + d c⌧ + d Z cd 2 ✓ ◆ ✓ ◆ ! The relevant ring of weak Jacobi forms is generated byq thee basis2⇡i⌧ elements,y e2⇡iz . (3.4) ⌘ ⌘ 10,1(q, y) 1 2 8 2 2,1(q, y)= = 2+y +12 8y +2y q + ... , (3.2) ⌘(q)24 y y2 y ✓ ◆ ✓ ◆ (q, y) 1 10 64 (q, y)= 12,1 = +10+y + +108 64y +10y2 q + ...(3.3) , Expanding0,1 ⌘ the(q)24 ellipticy genus oney2 hasy ✓ ◆ ✓ ◆ where i,k are the unique cusp forms of index i and weight k and we use the conventions n r ell(q, y)= c(n, r)q y . (3.5) q e2⇡iZ⌧ ,y e2⇡iz . (3.4) We were then⌘ led⌘ to thinkn 0 ,raboutZ+m Calabi-Yau fourfolds. X2

Expanding the elliptic genus one has For a given fourfold X, the coecients of 2,1 and 0,1 are determined by the topology of X. For aThe given elliptic Euler(q, y character)= genusc ((Xofn, r)and) qnayr .Calabi-Yau(3.5) fourfold has an Zell n 0,r Z+m X2 interesting4 structure. For a given fourfold X, the coecients of 2,1 and 0,1 are determined by the topology of k 0,k X. For a given Euler character (X)and (X)= ( 1) h (X) , (3.6) 0 k=0 It is a linear 4combinationX of two Jacobi forms of (X)= ( 1)kh0,k(X) , (3.6) 3 Elliptic genera0 of more general fourfolds C.D.D. Neumann, the elliptic genus is givenX byk=0 weight 0 and index 2: 1996 the elliptic genus is given by For a generic Calabi-Yau fourfold, the elliptic genus is( aX weak) Jacobi form of index 2 and weight 0. That is, it satisﬁes 2 2 2 (⌧,z)= (X) E2 (q)(X) (q, y)2 + 2 (q, y) E (q) (q, y) . (3.7) ell(⌧,zell)=0(X) E4(q0) 2,1(q, y)4 + 2,10,1(q, y) E4(q) 2,1(q, y) 0,1. (3.7) 4 2,1 Z Z 144 144 a⌧ + b z 4⇡icz 2 ab Here E4 is theell Eisenstein, series of weight=exp four: ell(⌧,z), for all SL(2, Z) . (3.1) Z c⌧ + d c⌧ + d c⌧ + d Z cd 2 Here E4 is✓ the Eisenstein◆ series✓ of◆ weight four: ! 1 2k 2 4 6 TheE4( relevantq)=1+240 ring of weak3(k) Jacobiq =1+240 formsq is+2160 generatedq +6720 by theq basis+ elements(3.8) ··· k=1 1 X 2k 2 4 6 E4(q)=1+24010,1(q, y) 1 3(k)q2 =1+2408 q +21602 q +6720Standardq + generators for (3.8) 2,1(q, y)= = 2+y +12 8y +2y q + ... , (3.2) ⌘(q)24 y y2 y ring of weak··· Jacobi forms ✓ k=1 ◆ ✓ ◆ (q, y) 1 X 10 64 (q, y)= 12,1 = +10+y + +108 64y +10y2 q + ...(3.3) , 0,1 ⌘(q)24 y 7 y2 y ✓ ◆ ✓ ◆

where i,k are the unique cusp forms of index i and weight k and we use the conventions

Saturday, June 21, 14 q e2⇡i⌧ ,y e2⇡iz . (3.4) ⌘ ⌘ 7

Expanding the elliptic genus one has

(q, y)= c(n, r)qnyr . (3.5) Zell n 0,r Z+m X2

For a given fourfold X, the coecients of 2,1 and 0,1 are determined by the topology of X. For a given Euler character (X)and

4 (X)= ( 1)kh0,k(X) , (3.6) 0 Xk=0 the elliptic genus is given by

2 (X) 2 2 ell(⌧,z)=0(X) E4(q) 2,1(q, y) + 0,1(q, y) E4(q) 2,1(q, y) . (3.7) Z 144 Here E4 is the Eisenstein series of weight four:

1 E (q)=1+240 (k)q2k =1+240q2 +2160q4 +6720q6 + (3.8) 4 3 ··· Xk=1

7 The piece that is present universally has a nice expression:

Z 1 4 ✓ (⌧, 2z)✓ (⌧, 0)11 univ ⇠ ⌘(⌧)12 i=1 i i P 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. TheMock Platonic modular realm moonshine of M22/M23 for M22 moonshine and M23

We begin with the chiral SCFT on the E8 root lattice.

It can be formulated in terms of 8 bosons and their Fermi superpartners. Chiral orbifold theory on E8 lattice

Next, wePerform orbifold: orbifold: Constant 8 disappears from partition function:

Saturday,Saturday, June June 21, 21, 14 14

All coefficients are dimensions of Conway reps.

for TheChiral partition orbifold functiontheory on of E8 this lattice theory can be Chiral computedorbifold Perform theory by orbifold: elementary on E8 lattice means. It is given by: Constant 8 disappears from partition function: Perform orbifold: Constant 8 disappears from partition function:

All coefficients are dimensions of Conway reps.

The coefﬁcients are interesting:for All coefficients are dimensions of Conway reps.

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 hasChiral a (not E8 so orbifold manifest) possesses Conway Conwaysymmetry, symmetry which commutes with the N=1 supersymmetry.Duncan ’05 FLM; Duncan Conway symmetry commutes with N=1 SUSY. ’05 Expand partition function in N=1 characters:

Coefficients of N=1 characters have “nicer” Thisdecompositions CFT played a in signiﬁcant terms of dimensions role in attempts of irreps of to ﬁnd. a holographic dual of pure supergravity in AdS3. Witten ’07

Saturday,Saturday, June June 21, 21, 14 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 Saturday, June 21, 14 One can construct a full string theory simply related to this module by considering the asymmetric orbifold generated by this Z 2 acting separately on left/right.

Elliptic genus: Elliptic genus:Its elliptic genus is:

of asymmetric orbifold on E8 factorizes: of asymmetric orbifold on E8 factorizes:

where is left partition function with grading: where is left partition function with grading:

Weak Jacobi form of weight and index Weak Jacobi form of weight and index Also partition function of chiral E8 orbifold Also partition function of chiral E8 orbifold Now, expand this thing in N=4 characters:

Saturday, June 21, 14 Saturday, June 21, 14 BPS BPS Z(q, y) = 21 chh=1/2,l=0(q, y)+chh=1/2,l=1(q, y) Expand + 560 ch (q, y) + 8470 ch (q, y)+ h=3/2,l=1/2 h=5/2,l=1/2 ··· + 210in N=4 chh=3 characters:/2,l=1(q, y) + 4444 chh=5/2,l=1(q, yCheng,)+ XD, Duncan, Harrison,··· Kachru, these two lines are governed by mock modularWrase, informs! preparation

The coefﬁcients are dimensions of M22 representations: Coefficients are dimensions of M22 representations!

Analogous to M24/K3 moonshine NoAnd, virtual no representationsvirtual representations appear.

Saturday, June 21, 14

Saturday, June 21, 14 1Introduction.

2Ellipticgenus.Math object definition slide In this section, I recall some basic facts about elliptic genera for Calabi- Yau manifolds, mostly from [5], and I explicitly derive it for4-folds.Let A C “mockAbe Jacobi a complex modular form manifold of form” level of complex 0arises and dimension indexwhend ,with ma theorybehavesSU(d)holonomy. has as Then its elliptic genus is a function φ(τ,z)withthefollowingtransformation to makeproperties a choice between modularity and holomorphy. aτ + b z mcz2 ab φ , = e φ(τ,z), SL2(Z)(1) Typical example:!cτ + d cτ +supersymmetricd" #cτ + d$ index! incd a "theory∈ with φ(τ,z+ λτ + µ)=( 1)2m(λ+µ)e[ m(λ2τ +2λz)]φ(τ,z), λ,µ Z (2) Right-moving primaries − − ∈ d where m = 2 ,andithasanexpansionoftheform ¯ Z (q, )=Tr ( 1)FL qL0 eiJL ( 1)FR q¯L0 ell L RR φ(τ,z)= c(n, r)qnyr (3) Differenceunder in Spectral modular Densities transformations and elliptic n≥0,r∈Z+m Non-Hol may% not quite manage to localize Iuseherethenotationstransformations.e[x]= Theeon2πix latter,right-movingq = e [τis], encodingy =grounde[z]. Thestates, the coe duefficients c(0, behaviorm + p)for0 underp c havethe the to “spectral mismatch following geometrical in flow” densities of meaning of N=2 fermions − ≤ ≤ Mock Unpaired Ground States and cbosons at finite energy. modular SCFTs. c(0, m + p)=χ = ( 1)p+qhp,q (4) − p − q%=0 Saturday,Thursday, June 21,May 14 8, 14 Saturday, June 21, 14 where hp,q are the Hodge-numbers of C.Furthermore φ(τ, 0) = χ (5)

1 There is an equivalent description of the E8 orbifold theory as a theory of 24 free fermions orbifolded by the action of ( 1) F .

* The orbifold has a manifest Spin(24) symmetry.

* Choosing an N=1 superalgebra actually reduces this symmetry to Co 0 .

* Now, to construct an N=4 superalgebra, choose three of the fermions.

Saturday, June 21, 14 Saturday, June 21, 14 11 Introduction Introduction

22 A A Free Free Field Field Module Module

ReviewReview John’s John’s module module and and the the N=1 N=1 structure. structure. Talk Talk about about Conway Conway characters characters at at torsion torsion points? points? (MC(MC probablyprobably not?!) not?!)

33 The TheNN=4=4ModuleModule

LetLet us us construct construct an an == 4 4 SCA SCA in in the the free free fermion fermion orbifold orbifold theory. theory. Our Our strategy strategy is is to to first NN constructconstruct the theSUSU(2)(2) R-currents, R-currents, and and act act them them on on an an == 1 1 supercurrent supercurrent to to generate generate the the full NN == 4 4 SCA. SCA. In In this this process, process, we we break break the theCoCo0symmetrysymmetry group group ( (MCMC wewe can’t can’t twine twine the the module module NN 0 with the full Co so we might need to change this.) down to M . with the full Co so we might need to change this.) down to M2222. WeWe start start with with 24 24 real real free free fermions fermions↵, ,↵↵=1=1,,22,, ,,2424..PickingPicking out out the the first first three three fermions, fermions, The currents of the SU(2)↵ R-symmetry······ of N=4 are then: we obtain the currents J : we obtain the currents Ji:i

JJi== i✏i✏ijkjk,i,j,k,i,j,k11,,22,,33 .. (3.1)(3.1) i ijk j k 22{{ }}

TheyThey form formOne an an a a canneneSU SUcheck(2)(2) current current quickly algebra algebra that with with level level 2 2 as as may may be be seen seen from from their their OPE: OPE: 1 i Ji(z)Jj(0) 1 ij +i ✏ijkJk(0) . (3.2) Ji(z)Jj(0) z2ij + z✏ijkJk(0) . (3.2) ⇠⇠z2 z

The next step is to pick an = 1 supercurrent and act Ji on it. Recall that an =1 The next stepBy bosonizing is to pick an theN= 1fermions, supercurrent writing and act theJi on currents it. Recall thatin an N =1 supercurrent exists in this modelN and may be written as a linear combination of spin fieldsN in supercurrentbosonic exists in thislanguage, model and and may writing be written the as N=1 a linear supercurrent combination of spinin fields in this model [1]. To do this explicitly we first group the 24 real fermions into 12 complex ones andthis extract modelthe the[1]. same singular To do thisway, terms explicitly one of theircan we OPEs:show first group that the one 24 real obtains fermions from into 12OPEs complex ones and bosonize them: and bosonize them: i 1/2 iH Ji(z)W (0) 1/2 Wi(0) , iH (3.14) 2 ( + i ) = e a , ¯ 2 ( i ) = e a ,a=1, 2, , 12 . a 1/2 2a 1 2a ⇠ iHa ¯ a 1⇠/2 22za 1 2a ⇠ iHa a ⌘2 (2a 1 + i2a) ⇠= e , a ⌘2 (2a 1 i2a) ⇠= e ,a=1, 2, ···, 12 . ⌘ ⌘ ··· (3.3) (3.3) where Wi are slightly modified combinations of spin fields: InSaturday, termsSaturday, June of 21, June 14 the 21, 14 bosonic fields Ha, an =1supercurrentW may be written as (MC rewrite In terms of the bosonic fields Ha, an N=1supercurrentW may be written as (MC rewrite this sentence?) N this sentence?) is H W1 = 2s2wiRssHe · cs(p) , (3.15) W= wse · cs(p) , (3.4) W = s w eis Hc (p) , (3.4) Xs s · s Xs is H W2 = i X4s1s2wRse · cs(p) , (3.16) where each component of s =(s1,s2, ,s12) takes the values 1/2, and the coecients ws where each component of s =(s ,s , ···s,s ) takes the values ±1/2, and the coecients w 1 2 ···X 12 ± s are C-numbers. We have introduced cocycles cs(pis)H to ensure that the operators with integer are C-numbers. We have introducedW3 = cocyclesi 2s1cwss(ep)· tocs ensure(p) , that the operators with integer(3.17) spins commute with all other operators,s and the operators with half integral spins anticommute spins commute with all other operators,X and the operators with half integral spins anticommute among themselves. The cocycles depend on the zero mode operators p which are characterized whereamongR themselves.s ( s , Thes ,s cocycles, ,s depend). (MC on: the should zero mode we call operatorsRss0 porwhich something are characterized to make the ⌘ 1 2 3 ··· 12 formulas look a bit nicer? )

We claim that all three Wi deﬁned above are valid = 1 supercurrents. This is because we 3 N may obtain, for instance, W3 from W by rotating3 the 1-2 plane by ⇡, and the conditions (3.9), (3.10) for being an = 1 supercurrent are invariant under SO(24) rotations. We may obtain N W and W similarly. This shows that each of W is an =1supercurrent. 2 3 i N Furthermore, we may check the OPEs of Wi:

8 8 2 1 W (z)W (0) + T (0) +2i✏ J (0) + @J (0) , (3.18) i j ⇠ ij z3 z ijk z2 k z k   2 @ W (z)W (0) 2i + J (0) , (3.19) i ⇠ z2 z i ✓ ◆ i J (z)W (0) ( W + ✏ W ) . (3.20) i j ⇠ 2z ij ijk k

This shows that W , Wi, Ji, and the stress tensor T deﬁned as

1 1 T = @ = @H @H (3.21) 2 ↵ ↵ 2 a a form an = 4 SCA with central charge c = 12. We may recombine the four =1supercurrents N N W , W into the more conventional =4supercurrents i N

1/2 1/2 W ± 2 (W iW ) ,W± 2 i(W iW ) , (3.22) 1 ⌘ ± 3 2 ⌘ ± 1 ± 2 which transform as 2 + 2¯ of the SU(2) R-symmetry. In terms of these supercurrents we obtain

5 and extract the singular terms of their OPEs: and extract the singular terms of their OPEs: i Ji(z)W (0) Wi(0) , (3.14) ⇠2z i J (z)W (0) W (0) , (3.14) i ⇠2z i where Wi are slightly modified combinations of spin fields: where Wi are slightly modified combinations of spin fields:

is H W1 = 2s2wRse · cs(p) , (3.15) is H W= 2s w e · c (p) , (3.15) 1 s 2 Rs s X s X is H W2 = i 4s1s2wRse ·is cHs(p) , (3.16) W2 = i 4s1s2wRse · cs(p) , (3.16) s X s X is H W3 = i 2s1wse · iscHs(p) , (3.17) W3 = i 2s1wse · cs(p) , (3.17) s s XX

where Rwheres R(s s1, ( ss2,s, 3s, ,s ,s, 12).,s ().MC (MC: should: should we we call callRRssss0 0 oror something something to to make make the the ⌘ ⌘ 1 2···3 ··· 12 formulasformulas look a look bit nicer? a bit nicer? ) )

We claimWe that claim all that three all threeW definedWi defined above above are are valid valid == 1 1 supercurrents. supercurrents. This is is because because we we i NN may obtain, for instance, W3 from W by rotating the 1-2 plane by ⇡, and the conditions (3.9), may obtain, for instance, W3 from W by rotating the 1-2 plane by ⇡, and the conditions (3.9), (3.10) for being an = 1 supercurrent are invariant under SO(24) rotations. We may obtain (3.10) for being an =N 1 supercurrent are invariant under SO(24) rotations. We may obtain N W2 and W3 similarly. This shows that each of Wi is an =1supercurrent. TheW2 and algebraW3 similarly. of This the shows supercurrents that each of Wi is an isN =1supercurrent.then calculated to be: Furthermore, we may check the OPEs of Wi: N Furthermore, we may check the OPEs of Wi: 8 8 2 1 W (z)W (0) + T (0) +2i✏ J (0) + @J (0) , (3.18) i j ⇠ 8ij z38 z ijk 2z2 k 1z k Wi(z)Wj(0) ij + T (0) +2 i✏ijk  Jk(0) + @Jk(0) , (3.18) ⇠ z3 z2 @ z2 z W (z)W (0)  2i + J (0) ,  (3.19) i ⇠2 z2@ z i W (z)W (0) 2i ✓+ J◆(0) , (3.19) i ⇠ iz2 z i Ji(z)Wj(0) ✓ (ijW +◆✏ijkWk) . (3.20) i⇠ 2z J (z)W (0) ( W + ✏ W ) . (3.20) i j ⇠ 2z ij ijk k This shows that W , W , J , and the stress tensor T defined as and together withi i the stress-energy tensor This shows that W , Wi, Ji, and the stress1 tensor T defined1 as T = @ = @H @H (3.21) 2 ↵ ↵ 2 a a 1 1 T = ↵@↵ = @Ha@Ha (3.21) form an = 4 SCA with central charge2 c = 12. We2 may recombine the four =1supercurrents N N W , Wi into the more conventional =4supercurrents form an = 4 SCA with central chargeNc = 12. We may recombine the four =1supercurrents N N W , W into the more conventional1/2 =4supercurrents 1/2 thesei fill out anW ± N=4,2 (W c=12iW3) ,W superconformal± 2 i(W1 iW2 )algebra., (3.22) 1 ⌘ N ± 2 ⌘ ± ±

1/2 1/2 which transformW ± as 22+ 2¯ of(W the SUiW(2)) ,W R-symmetry.± In2 terms i( ofW theseiW supercurrents) , we obtain(3.22) The subgroup1 ⌘of Conway± 3 that 2commutes⌘ ± 1 ± with2 the choice which transform as 2 + 2¯ ofof the theSU (2)three-plane R-symmetry. In termsis M22. of these supercurrents we obtain

Saturday, June 21, 14

5 In this language, the elliptic genus is more naturally Equivalent formulationrewritten of as:E8 orbifold theory In terms of 24 free fermions orbifolded by :

Can be shown isomorphic to E8 orbifold theory. Duncan ’05

This isPossess equivalent symmetry to our earlier commuting expression with N=1. (from the “E8 viewpoint”) Possess M22 by symmetry nontrivial commuting identities withon Jacobi N=4! forms. Choosing N=4 SCA breaks Conway down to M22. The 21 “non N=4” fermions give the 21 of M22. All This free fermion orbifold makes M22 symmetry higher states in the module have transformation laws manifest (although it is hidden in E8 orbifold theory). that can then be derived from ﬁrst principles.Cheng, XD, Duncan, Harrison, Kachru, Wrase, in preparation

Saturday, June 21, 14

Saturday, June 21, 14 One can now compute twining genera by an explicit prescription. But also, excitingly:

* One can use the magic of Rademacher sums!

In the original Monstrous Moonshine, all twining functions were given as Hauptmoduln of genus zero subgroups of the modular group.

In Mathieu moonshine, this was not true. However, the genus zero property is equivalent to arising as a Rademacher sum in the Monster case, and this property holds for the twining functions of Mathieu moonshine. Duncan, Frenkel; Cheng, Duncan

Here, the twinings also all arise as Rademacher sums.

Saturday, June 21, 14 Saturday, June 21, 14 1Introduction.

In this paper, I extend the work of Kawai [4], calculating N=2 heterotic string one-loop threshold corrections with a Wilson line turned on,toCalabi-Yau three- and fourfolds. (See also [10] for an alternative interpretation of Kawai’s result.) In full generality, this calculation provides a mapfromacertain class of Jacobi functions (including elliptic genera) to modular functions of certain subgroups of Sp4(Q), in a product form. In a number of cases, these products turn out to be equal to the denominator formula of a generalized RademacherKac-Moody Sums algebra. and It Rademacher seems natural to Series assume that this algebra is present3 in the corresponding string theory, and indeed in [9] it is argued that this algebra is formed by the vertex operators of vector multiplets and hypermultiplets. For instance, to obtain a modular form of even integral weight w =2k we may, following Poincaré(cf. (2.10)), take f(2Ellipticgenus.τ)=e(mτ)whereMathm is object an integer, definitionτ is a parameter slide on the upper- half plane H, and here and everywhere else in the article we employ the notation In this section, I recall some basic facts about elliptic genera for Calabi- Yau manifolds, mostly from [5], and I explicitly derive it for4-folds.Let i A “RademacherCAbe Jacobi a complexe( xform)= manifold sum”e2π ofx. level ofis complexa close0 and dimension relative indexd ,withm of behaves aSU Poincare(1.1)(d)holonomy. as series.Then You its ellipticcan obtain genus is aa function modularφ(τ,z )withthefollowingtransformationform by starting with an SLproperties Then the subgroup ofobjectΓ = 2 (Zinvariant) leaving f invariantunder isthe just stabilizer the subgroup of upper-triangular infinity, and then aτ + b z mcz2 ab matrices, which we denote Γ sinceφ its elements, are= preciselye thoseφ(τ that,z), fix the infinite cuspSL of (Z)(1) ∞summingcτ + d overcτ + d representativescτ + d of rightcd cosets:∈ 2 Γ (cf. 2.1). Thus we are led to consider! the sum " # $ ! " § φ(τ,z+ λτ + µ)=( 1)2m(λ+µ)e[ m(λ2τ +2λz)]φ(τ,z), λ,µ Z (2) aτ + −b 1 − Poincare series for ∈ where m = de,andithasanexpansionoftheformm , (1.2) 2 cτ + d (cτ + d)w modular form of ab Γ Γ " # ! cd!"∈ ∞\ φ(τ,z)= c(n, r)qnweightyr w=2k (3) under modular transformations≥ ∈Z and elliptic n 0,r% +m for w =2k, taken over a set of representativestransformations. for the right The cosets latter of Γ isin encodingΓ.Whenk> the1 RademacherIuseherethenotations sums aree[x]= modifiede2πix, q = ∞eversions[τ], y = e[z ].of The this coe thatfficients this sum is absolutely convergent,c(0, behaviorm locally+ p)for0 uniformly underp forc haveτtheH the , “spectral and following thus defines geometrical flow” a holomorphic of meaning N=2 − ≤ ≤ ∈ function on H which is invariant forimprove the weight w the=2 kconvergenceaction of Γ byc construction. properties. If m 0 SCFTs. p+q p,q ≥ c(0, m + p)=χp = ( 1) h SL (4) then it remains bounded as (τ) and is thus a modular− form of weight− 2k for Γ = 2(Z). # →∞ q%=0 Thursday, May 8, 14 This result was obtained by Poincaréinp,q [Poi11]. (See [Kow10] for a historical discussion.) Saturday, June 21, 14where h are the Hodge-numbers of C.Furthermore For many choices of w and m, however (e.g. for w 2), the infinite sum in (1.2) is not ≤ Saturday, June 21, 14 φ(τ, 0) = χ (5) absolutely convergent (and not even conditionally convergent if w<1). Nontheless, we may ask if there is some way to regularise (1.2) in the case that w 2. One solution to this 1≤ problem, for the case that w = 0, was established by Rademacher in [Rad39]. Let J(τ)denote the elliptic modular invariant normalised to have vanishing constant term, so that J(τ) is the unique holomorphic function on H satisfying J aτ+b = J(τ)whenever ab SL (Z)and cτ+d cd ∈ 2 1 also J(τ)=q− + O(q)as (τ) for q =e(τ$). % # →∞ & '

1 2 3 J(τ)=q− +196884q +21493760q +864299970q + (1.3) ···

In [Rad39] Rademacher established the validity of the expression

aτ + b a J(τ)+12=e( τ) + lim e e (1.4) K cτ + d c − →∞ − − − ab Γ Γ " # ! cd!"∈ ∞\ $ % 0

* The commutant of a choice of U(1) R-current (and consequent N=2 algebra) is M23.

* Again there is a manifest symmetry in the fermionic construction, twinings are computable by a simple prescription, and everything holds together nicely.

This gives examples of mock modular moonshine for M22 and M23 at c=12. The Mathieu case of M24 at c=6 remains mysterious.

Saturday, June 21, 14

Saturday, June 21, 14 * ConnectionsIn the to big geometry picture, appearedthis means: through EOT. They extend to some extent to the heterotic theory * We now have several examples of mock modular and via duality to Calabi-Yau threefolds. moonshine with explicit modules.

* They arise naturally in the interplayquantum/stringy of duality with moonshine geometry is a new thing; can it deepen or of certain asymmetricbroaden orbifolds. the lessons? * Connections to geometric models (CY fourfolds, * Somespin(7) fourfolds manifolds) give require modular more forms work, closely but relatedthere to those in the last arebit promisingof my talk. avenues... Does this mean anything?

Saturday,Saturday, June June 21, 21, 14 14