Some Comments on Equivariant Gromov-Witten Theory and 3D Gravity (A Talk in Two Parts) This Theory Has Appeared in Various Interesting Contexts

Home , Shamit Kachru

Some comments on equivariant Gromov-Witten theory and 3d gravity (a talk in two parts) This theory has appeared in various interesting contexts.

For instance, it is a candidate dual to (chiral?) supergravity with deep negative cosmologicalShamit constant. Kachru Witten (Stanford & SLAC)

Sunday, April 12, 15

Tuesday, August 11, 15 Much of the talk will be review of things well known to various parts of the audience.

To the extent anything very new appears, it is based on two recent collaborative papers:

* arXiv : 1507.00004 with Benjamin, Dyer, Fitzpatrick

* arXiv : 1508.02047 with Cheng, Duncan, Harrison

Some slightly older work with other (also wonderful) collaborators makes brief appearances.

Tuesday, August 11, 15 Today, I’d like to talk about two distinct topics. Each is related to the general subjects of this symposium, but neither has been central to any of the talks we’ve heard yet.

1. Extremal CFTs and quantum gravity ...where we see how special chiral CFTs with sparse spectrum may be important in gravity...

II. Equivariant Gromov-Witten & K3

...where we see how certain enumerative invariants of K3 may be related to subjects we’ve heard about...

Tuesday, August 11, 15 I. Extremal CFTs and quantum gravity

A fundamental role in our understanding of quantum gravity is played by the holographic correspondence between conformal ﬁeld theories and AdS gravity.

In the basic dictionary between these subjects

conformal symmetry AdS isometries $ primary field of dimension bulk quantum field of mass m() $ ......

Tuesday, August 11, 15 I For compact target manifold, discrete string spectrum ! discrete symmetry groups

I For one-loop partition function, worldsheet is torus any trace I For compact target manifold,! discretefunction string spectrumshould be invariant under ! discreteSL symmetry(2, Z) groups

I For one-loop partition function, worldsheet is torus any trace ! function should be invariant under SL(2, Z)

observables, which of course are also the ones we will be studying here, because of their phenomenological importance, and because, since the B model is soluble classically, the relation given byB. mirror The symmetryelliptic betweengenus observables and Mathieu of the A (moonshineX)modelandobservables observables, which of course are also the ones we will be studying here, because of their of the B(Y )modelisparticularlyuseful. phenomenological importance, and because, since the B model is soluble classically, the What is moonshine? The theory describingOutline strings propagating on a K3 relation given byB. mirror The symmetryelliptic betweengenusMonstrous moonshine observables and Mathieu of the A (moonshineX)modelandobservables There are famous examplesConway moonshine where the correspondence can surface is a 2d non-linearConclusions sigma model with target a of2. the PreliminariesbeB(Y made)modelisparticularlyuseful. very precise. One, relevant to this conference, Appearance in stringWhat is moonshine? theory The theoryK3 describing surface,Outline i.e.strings a theory propagating of maps on a K3 To begin with, we recall theMonstrous standard moonshine supersymmetric nonlinear sigma model in two relates the sigmaConway model moonshine with target Hilb(K3) to Conclusions 2.dimensions. Preliminariessurface2 It governsis a 2d maps non-linearΦ : Σ → X ,sigma with Σ modelbeing a Riemannwith target surface anda X a superstringsI In string theory wecompactified me consider 2d on AdS3xS3xK3. Riemannian manifoldAppearanceK3 of metricsurface, in stringg.Ifwepicklocalcoordinates theory i.e. a theory of mapsz, z on Σ and φI on X, To begin with, wesigma recall models,: ⌃ the maps standardK from3 supersymmetric nonlinear sigma model in two then Φ can be describedworldsheet locally to! target via manifold functions φI (z, z¯). Let K and K be the canonical dimensions.2 It governs maps Φ : Σ → X, with Σ being a Riemann surface and X a and anti-canonicalI In string line theory bundles we me of considerΣ (the 2d bundles of one formsoftypes(1, 0) and (0, 1), Riemannian manifold of metric g.Ifwepicklocalcoordinatesz, z on Σ and φI on X, sigma models,: ⌃ mapsK from31/2 respectively), and let K1/2 and K be square roots of these. Let TX be the complexified then Φ canThis be described2dworldsheet field locally totheory! target via manifold functions has someφI (z, z¯basic). Let Kfieldsand K be (along the canonical with I 1/2 ∗ tangent bundle of X. The fermi fields of the model are ψ+,asectionofK ⊗ Φ (TX), and anti-canonicalsuperpartners), line bundles1/2 of Σ (theand bundles an action of one formschematicallysoftypes(1, 0) like: and (0, 1), and ψI ,asectionofK ⊗ Φ∗1(/TX2 ). The Lagrangian is3 respectively),This− 2d and letfieldK1/ 2theoryand K hasbe square some roots basic of these. fields Let TX be(along the complexified with tangent bundle of X. The fermi fields1 of the model arei ψI ,asectionofK1/2 ⊗ Φ∗(TX), L =2t d2z g (Φ)∂ φI ∂ φJ + g + ψI D ψJ I superpartners),1/2 ∗ andIJ anz actionz¯ schematically3IJ − z − like: and ψ ,asectionofK !⊗ Φ ("TX2 ).Sarah The M. Harrison Lagrangianc =12Moonshine2 is − (2.1) i I J 1 I J K L 2 1+ gIJψ+IDzψJ+ + i RIJKLI ψ+ψJ+ψ− ψ− . L =2t d z gIJ2(Φ)∂zφ ∂z¯φ + 4gIJψ−Dzψ− # ! "2 Sarah M. Harrison c =12Moonshine2 (2.1) Monday, August 10, 15 ¯ Here t is a coupling constant, andi RIJKLI isJ the Riemann1 tensorI J K of XL . Dz¯ is the ∂ operator + gIJψ Dzψ + RIJKLψ ψ ψ ψ . on K1/2 ⊗ Φ∗(TX)constructedusingthepullbackoftheLevi-Civitaconnect2 + + 4 + + − −# ion on TX. InTuesday, formulas August 11, 15 (using a local holomorphic trivialization of K1/2), HereMonday,t is August a coupling 10, 15 constant, and RIJKL is the Riemann tensor of X. Dz¯ is the ∂¯ operator 1/2 ∗ J on K ⊗ Φ (TX)constructedusingthepullbackoftheLevi-CivitaconnectI ∂ I ∂φ I K ion on TX. Dz¯ψ = ψ + Γ ψ , (2.2) In formulas (using a local holomorphic+ trivialization∂z + ∂z of KJK1/2+),

I J 1/2 ∗ with ΓJK the affine connectionI of X.Similarly∂ I ∂φDzIis theK ∂ operator on K ⊗ Φ (TX). Dz¯ψ+ = ψ+ + ΓJKψ+ , (2.2) The supersymmetries of the model∂z are generated∂z by infinitesimal transformations 1/2 with ΓI the affine connection ofI X.SimilarlyI D Iis the ∂ operator on K ⊗ Φ∗(TX). JK δΦ = i%−ψ+ + i%+ψz − The supersymmetries of theI model are generatedI K byI infinitesiM mal transformations δψ+ = −%−∂zφ − i%+ψ− ΓKMψ+ (2.3) I I I δΦ =I i%−ψ + i%I+ψ K I M δψ− = −%++∂zφ −−i%−ψ+ ΓKMψ− , I I K I M δψ+ = −%−∂zφ − i%+ψ− ΓKMψ+ (2.3) 2 This discussion will be at the classical level, and we will notworryabouttheanomaliesthat I I K I M arise and spoil some assertionsδψ− if= the−% target+∂zφ space− i%− isψ+ notΓKM a Caψlabi-Yau− , manifold. 3 2 2 Here d z is the measure −idz ∧ dz¯.Thusifa and b are one forms, a ∧ b = This2 discussion will be at the classical level, and we will notworryabouttheanomaliesthat i d z (azbz¯ − az¯bz). The Hodge ! operator is defined by !dz = idz, !dz¯ = −idz¯. $ arise and spoil some assertions if the target space is not a Calabi-Yau manifold. 3$ 2 Here d z is the measure −idz ∧ dz¯.Thusif4 a and b are one forms, a ∧ b = 2 i d z (azbz¯ − az¯bz). The Hodge ! operator is defined by !dz = idz, !dz¯ = −idz¯. $ $ 4 Here is a natural question: given a desired spectrum of bulk AdS fields (“set of elementary particles”), does there exist a dual CFT which defines it non-perturbatively?

This question is far beyond our reach. But we can explore simple examples. Perhaps the simplest is pure 3d quantum gravity.

Do there exist conformal ﬁeld theory duals to pure 3d (super)gravity?

Tuesday, August 11, 15 You see that for eigenstates of the “Hamiltonian” it becomes obviousThe spectrum that thisof pure is just: 3d gravity would be just the graviton multipletThese transformations and the associated are multiparticleassociated with the TheseH transformations are associatedgroup SL(2,Z). with the Tr e , 2⇡R states. ⇠ group SL(2,Z). Now instead ofWe a canparticle use the moving Thisspectrum means in imaginaryto that read the off partition time,the partition function function we were This meanscomputing, ofthat the the conjectural partition that summarizes functiondual CFT: wethe were spectrum of string computing,consider that a string: summarizesstates, should the behave spectrum well of under string SL(2,Z): states, should behave well under SL(2,Z): n 2⇡i⌧ Z = n cnq q = e n 2⇡i⌧ Z = cnq q = e n P cn = # of states at mass level n P cn = # of states at mass level n more properly, There are now two parameters that ﬁx the shapea⌧+b of should behave as Z( c⌧+d )moreZ properly,(⌧) a⌧+b should⇠ behave as a modular function the torus, instead of justZ (ac ⌧radius.+d ) Z(⌧) ⇠ a modular function or form.... or form....

Monday, August 10, 15 Tuesday, August 11, 15 Thursday, May 15, 14 Monday, August 10, 15 Thursday, May 15, 14 Monday, August 10, 15 Thursday, May 15, 14 factorization, and mainly consider only the holomorphic sector of the theory. The full partition function is the product of the function we determine and a similar antiholomor- phic function. These partition functions have been studied before [31,32], with diﬀerent motivation.

3.1. The Bosonic Case

We begin with the bosonic case. What are the physical states ofpuregravityina spacetime asymptotic at inﬁnity to AdS3? Since there are no gravitational waves in the theory, the onlystatethatisobvious at ﬁrst sight is the vacuum, corresponding in the classical limit to Anti de Sitter space.

In a conformal field theory with central charge c =24k,thegroundstateenergyisL0 = c/24 = k.ThecontributionofthegroundstateΩ to the partition function Z(q)= − − | " L0 k Tr q is therefore q− . Of course, there is more to the theory than just the ground state. According to Brown and Henneaux [21], a proper treatment of the behavior at infinity leads to the construction of a Virasoro algebra that acts on the physical Hilbert space. The Virasoro generators You would quickly see that the spectrum of a theory Ln,n 1annihilate Ω ,butbyactingwithL 2,L 3,...,wecanmakenewstatesof ≥− | " − − withs njust graviton and its descendants is not the general form n∞=2 L n Ω ,withenergy k+ n nsn.(Weassumethatallbutfinitely consistent.− | " This would− give a partition function: many of the sn vanish.)! If these are the only states" to consider, then the partition function would be k ∞ 1 Z (q)=q− . (3.1) 0 1 qn n=2 # − This cannot be the complete answer, because the function Z0(q)isnotmodular- invariant. Therewhere must bec = additional 24k is the states central such thatchargeZ0( qof)iscompletedtoamodular- the theory, invariant function. and determines the AdS3 cosmological term. Additional states are expected, because the theory also has BTZ black holes. The main reason for writing the present paper, after all, is to understand the role of the BTZ black This alone is not modular. Can we fix it up? holes in the quantum theory. We will assume that black holes account for the difference between the naive partition function Z0(q)andtheexactoneZ(q). To use this assumption

to determineTuesday,Z August(q 11,), 15 we need to know something about the black holes. The classical BTZ black hole is characterized by its mass M and angular momentum J.IntermsoftheVirasorogenerators, 1 M = (L0 + L0) ! (3.2) J =(L L ), 0 − 0 29 Yes! We forgot that in gravity, we expect also black holes. The spectrum should contain:

Happily, the “anything goes” portion of the spectrum is then determined uniquely by modularity!

Tuesday, August 11, 15 In the present case, as the pole in Z(q)atq =0isoforderk, Z must be a polynomial in J of degree k.Thus k In the present case, as the pole in Z(q)atq =0isoforderk, Z must be a polynomial in Z(q)= f J r, (3.7) J of degree k.Thus r k r=0 r Z(q)= frJ!, (3.7) with some coefficients f .Thesek +1coer=0 fficients can be adjusted in a unique fashion to r ! ensure thatwithThe someZ (k+1q)takestheform(3.4),orinotherwordstoensurethattheter coefficients conditionsfr.Thesek +1coe onffi cientsthe can polar be adjusted terms in a unique determine fashion to thems in Z(q) ensure that Z(q)takestheform(3.4),orinotherwordstoensurethattheterms in Z(q) n of order q− , n =0n ,...,k,coincidewiththenaivefunctionZ0(q). of order q− , n =0,...,k,coincidewiththenaivefunctionfunction uniquely.Z0(q). WhenWhen we do we this, do this, we we get get a a function function that we that will we call Z willk(q), callk =1Z, 2k,(3q,...),.Thisk =1, 2, 3,....This functionfunction is our is candidate our candidate for for the the generating generating function function that counts that the quantum counts states the of quantum states of three-dimensionalThe first gravitycase, in k=1, a spacetime yields asymptotic the to AdSJ function.3.Forexample,for Ak =1wefew more: three-dimensionalhave simply Z1 gravity(q)=J(q), in and a the spacetime next few examples asymptotic are to AdS3.Forexample,fork =1we have simplyZ (Zq)=1(qJ)=(q)2 J393767(q), and the next few examples are 2 − 2 2 =2q− +1+42987520q +40491909396q + ... Z2(q)=J(q) 3393767 no CFTs Z3(q)=J(−q) 590651J(q) 64481279 − − (3.with8) such Z 2 3 1 2 2 =q− =+1+42987520q− + q− +1+2593096794q +40491909396q +12756091394048q +q ...+ ... are known to 4 2 Z4(q)=3J(q) 787535J(q) 8597555039J(q) 644481279 exist Z3(q)=J(q) 590651− J(q) − 64481279 − 4 2 1 2 =q−−+ q− + q− +2+81026609428− q +1604671292452452276q + .... (3.8) 3 1 2 =q− + q− +1+2593096794q +12756091394048q + ... Following [31], we refer to a holomorphic CFT with c =24k and partition function 4 2 Z4(qZ)=k(q)asanextremalCFT.Accordingtoourproposals,thedualoftJ(q) 787535J(q) 8597555039J(q) 644481279hree-dimensional Tuesday, August 11, 15 − − − gravity should4 be2 an extremal1 CFT. 2 =Asq− was+ alreadyq− + notedq− in+2+81026609428 the introduction, Frenkel,q +1604671292452452276 Lepowsky,andMeurmancon-q + .... structed [24,25] an extremal CFT with k =1,thatis,aholomorphicCFTwithc =24and Following [31], we refer to a holomorphic CFT with c =24k and partition function partition function J(q)=Z1(q). They also conjectured its uniqueness. If that conjecture Zk(q)asanextremalCFT.Accordingtoourproposals,thedualoftas well as the ideas in the present paper are correct, then the FLM theory must be the hree-dimensional dual to quantum gravity for k =1.Unfortunately,asalsonotedintheintroduction,for gravity should be an extremal CFT. k>1, extremal CFT’s are not known, though their possible existence has been discussed Asin was the literature already [31-33] noted for reasons in the not introduction, related to three-dimensional Frenkel, gravity. Lepowsky Our reasoning,andMeurmancon- structedin [24,25] this paper an suggests extremal that such CFT theories with shouldk =1,thatis,aholomorphicCFTwith exist and be unique for each k. c =24and The main point of the FLM construction was that their theory has as a group of partitionsymmetries function theJ Fischer-Griess(q)=Z1(q monster). They group alsoM,thelargestofthesporadicfinitegroups. conjectured its uniqueness. If that conjecture as well asArguably, the ideas the FLM in theory the present is the most paper natural are known correct, structure thenwith M thesymmetry.FLM The theory must be the dual tocoe quantumfficients in gravity the q-expansion for k of=1.Unfortunately,asalsonotedintheintroduction,for the J-function are integers for number-theoretic reasons, k>1, extremal CFT’s are not known, though31 their possible existence has been discussed in the literature [31-33] for reasons not related to three-dimensional gravity. Our reasoning in this paper suggests that such theories should exist and be unique for each k. The main point of the FLM construction was that their theory has as a group of symmetries the Fischer-Griess monster group M,thelargestofthesporadicfinitegroups. Arguably, the FLM theory is the most natural known structure with M symmetry. The coefficients in the q-expansion of the J-function are integers for number-theoretic reasons,

31 Interestingly, the k=1 case yields as a candidate dual to pure Thisgravity gave manywith furtherdeep non-trivialnegative checks.cosmological Eventually, constant, a beautifulprecisely and complete the (?) FLM story theory! 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 Tuesday, August 11, 15 Monday, August 10, 15 Caveats:

* k=1 is, at most, barely geometry -- but note

log(196, 883) 4⇡ Bekenstein-Hawking entropy of lightest ⇠ black holes is close to working...

* There is a subtlety with holomorphic factorization. Perhaps this is best thought of as a duality between “chiral CFT” and “chiral gravity.” I will not discuss this further here.

Tuesday, August 11, 15 In mathematical terms, this means the following: One can run the same story with supersymmetry.

* The Hilbert spaceNow, admits c=12k a is Z/2Z accessible. grading. * Even states are called “bosons,” odd states “fermions.” *With The symmetryminimal SUSY, generator at k=1, is again an odd a candidate supercharge CFT Q. exists!

Tuesday, August 11, 15

Monday, August 10, 15 simple groups. In 3, we describe how the groups G = M ,Co and Co arise as the global § ∼ 24 2 3 symmetry groups when considering the present SCFT as a representation of the Spin(7) algebra. In 4, we brieﬂy review the representation theory of the Spin(7) superalgebra. In 5, we ﬁrst § § discuss the partition functions c L0− Zg =Trgq 24 (1.1)

simple groups. In 3, we describe how the groups G = M24,Co2 and Co3 arise as the global twined by all elements§ g G of the global symmetry groups∼ mentioned above. Subsequently, we symmetry groups when∈ considering the present SCFT as a representation of the Spin(7) algebra. discuss the mock modular properties of the 2-vector-valued functions that arise when decompos- In 4, we briefly review the representation theory of the Spin(7) superalgebra. In 5, we first ing Z§g into Spin(7) characters. In 6 we close the paper with some discussions. The appendices§ discuss the partitionB. functionsMoonshines§ beyond the Monster include our conventions for standard modular functionsc (Appendix A); character tables for the L0− Zg =Trgq 24 (1.1) groups which appear (Appendix B); and the coefficient tables (Appendix C) together with the correspondingtwinedThere by all are elementsrepresentations eveng self-dualG of for the the global twined unimodular symmetry partition groups function mentionedlatticess (Appendix above. in D).lower Subsequently, we ∈ dimensionsdiscuss the mock than modular the properties Leech of the lattice. 2-vector-valued In dimension functions that arise 8, whenthere decomposing Z into Spin(7) characters. In 6 we close the paper with some discussions. The appendices 2TheFreeFieldTheoryg is a unique§ structure. include our conventions for standard modular functions (Appendix A); character tables for the Thegroups two-dimensional which appearc (Appendix= 12 chiral conformalB); and the field coe theoryfficient that tables we (Appe will considerndix C) was together first des withcribed the incorresponding [2], and an alternative representations construction for the was twined given partition in [14]. In function [15], it wass (Appendix shown that D). the NS sector of theory is described by a distinguished super vertex operator algebra, V s!,andcanberegarded as2TheFreeFieldTheory the natural analogue of the Frenkel-Lepowsky-MeurmanThe supersymmetrized moonshine module theory for Conway’s largest group. In this section we describe the two constructionsof bosons propagating of this 2d chiral on conformal the field theory,The two-dimensional and review howc its= 12 NS-sector chiral conformal twined partition field theoryE8 functionslattice, that werealizes ar wille given consider moonshine by certain was first normalized described principalin [2], and moduli an alternative with vanishing construction constant was terms. given infor [14]. Conway’s In [15], it was largest shown sporadic that the NS sector of theoryThe is construction described by presented a distinguished in [2] contains super vertex 8 free operator bosons algebXgroup!i ra,compactifiedV s!,andcanberegarded on the eight- 8 i dimensionalas the natural torus analogueR /ΛE8 ofgiven the Frenkel-Lepowsky-Meurman by the E8 lattice plus their moon 8 fermionicshine module superpartners for Conway’sψ , subjectlargest to group. a Z/2 In orbifold this section we describe the two constructions of this 2d chiral conformal field i i i i Monday,theory, August 10, and 15 review how its NS-sector(X twined, ψ ) partition( X , ψ functions) . are given by certain normalized(2.1) → − − principal moduli with vanishing constant terms. Clearly, this theory exhibits a manifest = 1 supersymmetry. In addition, the orbifolding The construction presented in [2] containsN 8 free bosons Xi compactified on the eight- ensures that there are no NS primary fields of dimension 1 1. The partition function of the It has an equivalentR8 description as 2the orbifold of the i dimensional torusThe/Λ ENS8 given sector by the Epartition8 lattice plus function their 8 fermionic is: superpartners ψ , theory in the NS sector is given by subject to a Z/2 orbifoldtheory of 24 free chiral fermions: i i i 4 i 4 4 (LX0−,c/ψ24) 1( XE4,θ3ψ ) . θ4 θ2 (2.1) ZNS,E8(τ)=trNS q →= − − +16 +16 (τ) (2.2) 2 η12 θ4 θ4 FLM; i(z), ! 2 4 " Duncan Clearly, this theory exhibits= aq− manifest1/2 +0+276=q!1 1/2 supersymmetry.+2048q +11202 Inq3/ addition,2 + , the orbifolding(2.3) N ··· ensures that there are no NS primary fields of dimension 1 1. The partition function of the The NS sector partition function2 is: theory in the NS sector isThe given bypartition function is: 2πix whereTuesday, August we 11, write 15 q = e(τ) and use the shorthand notation e(x) := e .IntheaboveE4(τ) ∞ is the weight 4 Eisenstein series defined in (A.5), η(τ)=q1/24 (1 qn) is the Dedekind 4 4 n=1 4− L0−c/24 1 E4θ3 θ4 θ2 1 The coefficientsZNS,E8(τ)=tr NSinq the q-series= +16decompose+16 (τ )nicely (2.2) Here and in the rest of the paper we use the common terminology2 η12 andθ# refer4 toθ the4 conformal weight of a state, ! 2 4 " or the L0-eigenvalue of an eigenstate, as “dimension”.− into representations= q 1/2 +0+276 of theq1/2 +2048sporadicq +11202 groupq3/2 + Co , 0 . (2.3) ···

2πix where weFor write instance,q = e(τCo) and 276=276 use the shorthand while:3 notation e(x) := e .IntheaboveE4(τ) with simple 0 decompositions, e.g. 276∞ = 276 and: is the weight 4 Eisenstein series deﬁned in (A.5), η(τ)=q1/24 (1 qn) is the Dedekind The coefﬁcients in the q-series decomposen=1 − nicely 1Here and in the rest of the paper we use2048 the common = 24 terminology + 2024and# refer to the conformal weight of a state, or the L0-eigenvalueinto of an representations eigenstate, as “dimension”. of the sporadic group Co 0 . 11202 = 1 + 276 + 299 + 1771 + 8855 For instance, 276=276 while:3

Monday,Sunday, AugustApril 12, 10, 15 15 2048 = 24 + 2024 11202 = 1 + 276 + 299 + 1771 + 8855

Sunday, April 12, 15 simple groups. In 3, we describe how the groups G = M ,Co and Co arise as the global § ∼ 24 2 3 symmetry groups when considering the present SCFT as a representation of the Spin(7) algebra. In 4, we brieﬂy review the representation theory of the Spin(7) superalgebra. In 5, we ﬁrst § § discuss the partition functions c L0− Zg =Trgq 24 (1.1)

simple groups. In 3, we describe how the groups G = M24,Co2 and Co3 arise as the global twined by all elements§ g G of the global symmetry groups∼ mentioned above. Subsequently, we symmetry groups when∈ considering the present SCFT as a representation of the Spin(7) algebra. discuss the mock modular properties of the 2-vector-valued functions that arise when decompos- In 4, we briefly review the representation theory of the Spin(7) superalgebra. In 5, we first ing Z§g into Spin(7) characters. In 6 we close the paper with some discussions. The appendices§ discuss the partition functions § include our conventions for standard modular functionsc (Appendix A); character tables for the L0− Zg =Trgq 24 (1.1) groups which appear (Appendix B); and the coefficient tables (Appendix C) together with the correspondingtwined by all elementsrepresentationsg G of for the the global twined symmetry partition groups function mentioneds (Appendix above. D). Subsequently, we ∈ discuss the mock modular properties of the 2-vector-valued functions that arise when decomposing Z into Spin(7) characters. In 6 we close the paper with some discussions. The appendices 2TheFreeFieldTheoryg § include our conventions for standard modular functions (Appendix A); character tables for the Thegroups two-dimensional which appearc (Appendix= 12 chiral conformalB); and the field coe theoryfficient that tables we (Appe will considerndix C) was together first des withcribed the incorresponding [2], and an alternative representations construction for the was twined given partition in [14]. In function [15], it wass (Appendix shown that D). the NS sector of theory is described by a distinguished super vertex operator algebra, V s!,andcanberegarded as2TheFreeFieldTheory the natural analogue of the Frenkel-Lepowsky-Meurman moonshine module for Conway’s largest group. In this section we describe the two constructions of this 2d chiral conformal field theory,The two-dimensional and review howc its= 12 NS-sector chiral conformal twined partition field theory functions that we ar wille given consider by certain was first normalized described principalin [2], and moduli an alternative with vanishing construction constant was terms. given in [14]. In [15], it was shown that the NS sector of theoryThe is construction described by presented a distinguished in [2] contains super vertex 8 free operator bosons algebXi ra,compactifiedV s!,andcanberegarded on the eight- 8 i dimensionalas the natural torus analogueR /ΛE8 ofgiven the Frenkel-Lepowsky-Meurman by the E8 lattice plus their moon 8 fermionicshine module superpartners for Conway’sψ , subjectlargest to group. a Z/2 In orbifold this section we describe the two constructions of this 2d chiral conformal field theory, and review how its NS-sector(Xi twined, ψi) partition( Xi, ψ functionsi) . are given by certain normalized(2.1) → − − principal moduli with vanishing constant terms. Clearly, this theory exhibits a manifest = 1 supersymmetry. In addition, the orbifolding The construction presented in [2] containsN 8 free bosons Xi compactified on the eight- ensures that there are no NS primary fields of dimension 1 1. The partition function of the It has an equivalentR8 description as 2the orbifold of the i dimensional torusThe/Λ ENS8 given sector by the Epartition8 lattice plus function their 8 fermionic is: superpartners ψ , theory in the NS sector is given by subject to a Z/2 orbifoldtheory of 24 free chiral fermions: i i i 4 i 4 4 (LX0−,c/ψ24) 1( XE4,θ3ψ ) . θ4 θ2 (2.1) ZNS,E8(τ)=trNS q →= − − +16 +16 (τ) (2.2) 2 η12 θ4 θ4 i(z), ! 2 4 " Clearly, this theory exhibits a− manifest1/2 =!1 1/2 supersymmetry. In3/ addition,2 the orbifolding = q +0+276N q +2048q +11202q + , (2.3) 1 1 ··· ensures that there are no NS primary fields of dimension 2 . The partition function of the The NS sector partition function2πix is: wheretheory we in write the NSq = sectore(τ) is andThe given use bypartition the shorthand function notation e(x )is: := e .IntheaboveE4(τ) ∞ is the weight 4 Eisenstein series defined in (A.5), η(τ)=q1/24 (1 qn) is the Dedekind 4 4 n=1 4− L0−c/24 1 E4θ3 θ4 θ2 1 The coefficientsZNS,E8(τ)=tr NSinq the q-series= +16decompose+16 (τ )nicely (2.2) Here and in the rest of the paper we use the common terminology2 η12 andθ# refer4 toθ the4 conformal weight of a state, ! 2 4 " or the L0-eigenvalue of an eigenstate, as “dimension”.− into representations= q 1/2 +0+276 of theq1/2 +2048sporadicq +11202 groupq3/2 + Co , 0 . (2.3) ···

Tuesday,Sunday, August April 11, 12,15 15 in passing that this is a strong assumption at large c; a much weaker criterion, fixing only a coefficients up to the power q− , with a growing less quickly than c as c , would satisfy the →∞ conditions we can justifiably place on a theory of pure gravity as well. In any case, at k = 1, a conformal theory satisfying this constraint is available. Remarkably, it is the theory constructed by Frenkel, Lepowsky and Meurman in connection with Monstrous moonshine some thirty years ago [2]! At higher values of k, which correspond to less negative values of the cosmological term, precise candidates do not yet exist. Various explorations resulting in constraintsIncreasing of thethe possibilitiessupersymmetry at these has values always appear been in [a3–6]. strategy1 A similar discussionthat string in [1] theorists of pure find= 1useful. supergravity Let’s do in AdSthat3 hereyields also. a potential family of N partition functions at c =12k, and again, at k = 1 a known superconformal theory fits the bill Gaberdiel, Gukov, Keller, Moore and Ooguri asked a Z –a 2 orbifold of thenatural supersymmetrized question inE 8thelattice context theory of [2, 2d 12]. N=(2,2) In [13], a criterion was similarly proposed for =(2, 2) superconformal theories dual to pure supersymmetry.N extended supergravity in AdS3. These authors argued that the elliptic genus of the conformal field theory, definedTo as an N=2 theory, we can associate a Jacobi form:

c ¯ c FL 2πizJ0 L0 24 FR L0 24 ZEG(τ,z)=Tr RR ( 1) e q − ( 1) q¯ − , (1.2) H − − is strongly constrained by the requirements of pure = 2 supergravity in the bulk. On general the elliptic genus (we could useN the partition function grounds, ZEG is a weak Jacobi formif the of weight theory 0 andis chiral). index m ,wherem = c/6. These authors were able to argue that for sufficiently large central charge, such extremal elliptic genera do not exist; at smallTuesday,m August, however, 11, 15 there are candidate Jacobi forms waiting to be matchedtoactual = 2 superconformal field theories. N Here, we propose that the case with m = 4 – of interest also because of the representation theory visible in the q-expansion of the extremal elliptic genus, as we describe below – can actually be realized. In fact the realization uses in a crucial way a known conformal field theory, 24 that based on the A1 Niemeier lattice. Extending the work of [14], we note that an enlarged, Z double cover of the A24/Z orbifold admits an = 2 superconformal symmetry, and in fact 2 1 2 N its chiral partition function matches the desired extremal elliptic genus. As our construction is chiral, this theory likely corresponds to a chiral gravity [15] analogue of the pure (super)gravity constructions envisioned in earlier works. (For discussion of the supersymmetric extension of chiral gravity, see e.g. [16].) As holomorphic factorization of these gravity partition functions in any case engenders many confusions, it is plausible that the earlier two examples of extremal theories should also be viewed as duals of chiral gravity theories in 3d. The organization of this note is as follows. In 2, we describe the m = 4 extremal elliptic § genus. In 3, we prove that an enlarged version of the A24 Niemeier theory both admits an =2 § 1 N 1Related research on 3d gravity partition functions [7], general constraints on the gap in 2d CFTs [8, 9], and the relationship of sparse CFT spectra with the emergence of spacetime [10, 11] has also appeared in recent years.

3 We assume that the Hilbert space of our theory is a direct sum ofunitaryhighest weight representations of the =2algebra.ThisallowsustodeﬁnetheRR-sector N partition function

L0 c/24 2πizJ0 L˜0 c/24 2πiz¯J˜0 iπ(J0 J˜0) ZRR(τ,z;¯τ, z¯):=Tr q − e q¯ − e e − (2.1) HRR which has good modular properties under the SL(2, Z)action(τ,z) ( aτ+b , z ). Here, → cτ+d cτ+d as usual, q = e2πiτ and y = e2πiz,andsimilarlyfor¯q andy ¯. In these conventions, the elliptic genus of an =(2, 2) superconformal field theory N C is defined to be χ(τ,z; ):=Z (τ,z;¯τ, 0) . (2.2) C RR It is holomorphic in (τ,z)bythestandardpropertiesoftheWittenindex.Forreferences on the elliptic genus see [4, 5, 19, 29, 30, 31, 40, 41, 42, 43, 44, 46, 47]. =2algebrashavethecrucialspectralflowisomorphism[45],which allows us to N relate the NS and R-sector partition functions. Recall that spectral flow SF for θ 1 Z is θ ∈ 2 an isomorphism of =2superconformalalgebraswhichmapseigenvalues N L L + θJ + θ2m (2.3) 0 → 0 0 J J +2θm. (2.4) 0 → 0

The spectral ﬂow operators act on Z = ZRR as: 1 1 (SF SF Z)=e mθ2τ +2mθ(z + ) e mθ˜2τ¯ +2mθ˜(¯z ) Z(τ,z+ τθ;¯τ, z¯ + θ˜τ¯), θ θ˜ 2 · − 2 " # " # (2.5) ! where e(x):=e2πix.Forsimplicitywerestrictourattentiontotheorieswithintegral spectrum of left- and right-moving U(1) charges J0, J˜0.Again,itshouldbepossible,and would be interesting, to relax this assumption. Spectral-ﬂow invariant theories with integral U(1) charges satisfy

Z =(SF SF )Z θ, θ˜ Z (2.6) RR θ θ˜ RR ∈ 1 Z =(SF SF )Z θ, θ˜ Z + . (2.7) NSNS θ!θ˜ RR ∈ 2

As is well-known [29], the modularity! properties of ZRR together with spectral flow invariance and unitarity imply that the elliptic genus is a weak Jacobi form of index m and weight zero [22]. A weak Jacobi form φ(τ,z)ofweightw and index m Z,with w ∈ (τwith,z) c(Hn, !We)=(C,satisfiesthetransformationlaws can1) declarec(n, !). that It follows a theory from the is spectralextremal flow if identity the elliptic that c( n, !)=0 ∈ × 2 − 2 − ˜ for 4mn genus! < ism “as.Following[22],wedenoteby close as possible to thatJw,m determinedthe vector space by of the weak Jacobi − − 2 forms of weightaτ + bw andz index m.AJacobiformisthenaweakJacobiformwhosepolarw 2πim cz ab φ( , vacuum)=( ccharacterτ + d) e ofcτ+ dtheφ(τ,z N=2) algebra.”SL(2, Z) , (2.8) part vanishescτ + (seed below).cτ + d $cd% ∈ Suppose we are given an integer m Z .If(!,n) Z2 is a lattice point we refer 2πim($2τ+2∈ $z)+ ∈ Z to its polarityφGiven(τas,zp+=4 %τa +Fouriermn%")=!e2−.If expansionφ J˜ φ( letτfor,z us) a defineweak the Jacobipolar%, %" partform, of φ,denoted(2.9) − ∈ 0,m ∈ andφ−,tobethesumofthetermsintheFourierexpansioncorrespon has a Fourier expansion of index m and weight 0: ding to lattice points of negative polarity. By spectral flow one can always relate the degeneracies to those in the φ(τ,z)= c(n, %)qny$ (2.10) fundamental domain with ! m.Ifweimposethemodulartransformation(2.8)with | | ≤ n 0,$ Z 1 SL(2, Z), which implements charge≥ conjugation,&∈ then c(n, !)=c(n, !)andtherefore − ∈ − the polar coefficients which cannot be related to each other by spectral flow and charge conjugation are c(n,it! )where(can be! ,nuseful)isvaluedinthe to definepolar the regionpolar part:(of index m), defined to P be –4– (m) := (!,n):1 ! m, 0 n, p =4mn !2 < 0 . (2.11) P { ≤ ≤ ≤ − } For an example, see figure 1.

n Tuesday, August 11, 15

Figure 1: Acartoonshowingpolarstates(representedby“ ”) in the region (m).Spectralﬂow • P by θ = 1 relates these states to particle states in the NS sector of an =2superconformalﬁeld 2 N theory which are holographically dual to particle states in AdS3.

Given any Fourier expansion

ψ(τ,z)= ψˆ(n, !)qny! (2.12) !,n Z !∈ we deﬁne its polar polynomial (of index m)tobethesumrestrictedtothepolarregion (m): P Pol(ψ):= ψˆ(n, !)qny! . (2.13) (!,n) (m) !∈P

Let us moreover denote by Vm the space of polar polynomials, i.e. the vector space generated by the monomials qny! with (!,n) (m). ∈ P

–5– with c(n, !)=( 1)wc(n, !). It follows from the spectral flow identity that c(n, !)=0 − − for 4mn !2 < m2.Following[22],wedenotebyJ˜ the vector space of weak Jacobi − − w,m forms of weight w and index m.AJacobiformisthenaweakJacobiformwhosepolar part vanishes (see below). Suppose we are given an integer m Z .If(!,n) Z2 is a lattice point we refer ∈ + ∈ to its polarity as p =4mn !2.Ifφ J˜ let us define the polar part of φ,denoted − ∈ 0,m φ−,tobethesumofthetermsintheFourierexpansioncorresponding to lattice points of negative polarity. By spectral flow one can always relate the degeneracies to those in the fundamental domain with ! m.Ifweimposethemodulartransformation(2.8)with | | ≤ 1 SL(2, Z), which implements charge conjugation, then c(n, !)=c(n, !)andtherefore − ∈ − the polar coefficients which cannot be related to each other by spectral flow and charge conjugation are c(n, !)where(!,n)isvaluedinthepolar region (of index m), defined to P be (m) := (!,n):1 ! m, 0 n, p =4mn !2 < 0 . (2.11) P { ≤ ≤ ≤ − } For an example, see figure 1.

Given any Fourier expansion

ψ(τ,z)= ψˆ(n, !)qny! (2.12) !,n Z Using facts from Eichler !and∈ Zagier, GGKMO were we deﬁne its polar polynomial (of index m)tobethesumrestrictedtothepolarregion able to(m ):prove that for large index m, no extremal SCFT P (by thisPol( deﬁnition)ψ):= ψˆ(n, ! )canqny! . exist. (2.13) (!,n) (m) !∈P

Let us moreover denote by Vm the space of polar polynomials, i.e. the vector space generated by the monomials qny! with (!,n) (m). ∈ P

Tuesday, August 11, 15

–5– However, there could be theories at

c =6, 12, 18, 24, 30, 42, 48, 66, 78

We can provide explicit constructions for chiral CFTs realizing the preferred Jacobi forms at c=12,24. This doubles the list of known “extremal CFTs”.

The construction at c=12 arose “by accident” in a project attempting to do something else.

Tuesday, August 11, 15 In [?] it was shown that one can extend this construction by building larger superalgebras out of the fermions. A choice of two (three) fermions is required to generate the U(1) (SU(2)) R-symmetry currents of an =2( = 4) superalgebra. This breaks the symmetry group of N N the theory to a subgroup of Co0 which stabilizes a two-(three-)plane in the Leech lattice and by construction commutes with the corresponding superalgebra. Consequently the chiral CFT, when viewed as =2( = 4) superconformal field theory, results in modules for subgroups N N G0 of Co0 that stabilize two- (three-)planes in the Leech lattice. In other words, decomposing the (U(1)-graded) partition function twined by g G0 into characters of the corresponding 2 superalgebra yields a two-component vector-valued mock modular form whose coecients are given by certain g-characters. It was also established in [?] that the groups M23 and M22 are distinguished among the groups which preserve an = 2 or = 4 algebra, respectively. This N N is because all of their twining functions satisfy a certain “optimal pole” condition, and are thus candidates for the moonshine phenomenon. In the rest of this paper, we explore a similar story for an extended = 1 superalgebra, the N algebra which, as discussed in [?], is associated with c = 12 conformal field theories with target manifolds of exceptional holonomy Spin(7). This algebra, which can be defined as an =1 N superalgebraThe extended global by symmetry a single copy of of the the Ising N=4 model, theory breaks the globalwill be symmetry the group of the Conway module from Co0 to a subgroup which stabilizes a line in the Leech lattice. The subgroup of Co 0 which fixes the given 3-plane.1 choice of line corresponds to the choice of single fermion used to generate the c = 2 Ising sector of the extended = 1. We compute twining functions for a number of groups which preserve N suchThe a “1-plane” book in by the LeechConway lattice and& showSloane they arecontains given in terms a nice of two-component discussion vector- valued mockofSubgroups the modular various forms. of See subgroups the Table Conway 1 for a summaryone group can of the realizepreserving extended by superalgebras fixing the in the Conway module and examples of the groups they preserve. The Mathieu groups M22, M23 and 3-planes,k-planeOne can2-planes,... act view on thethe in super-E8corresponding the Leech theory lattice. superconformalas a theoryFor the with rest M24 are singled out here as the 3-, 2-, 1-plane preserving groups in the Leech lattice that admit a representationofa as variety permutationthe talk of groupsIdifferent focusfield on 24on supersymmetry objects.theory. one See set [? ]of for morechoices: algebras. details on these groups and their relation to the Leech lattice. Superalgebra Geometrical Representation Global symmetry group =0 R24 Spin(24) N =1 ⇤ Co N Leech 0 Spin(7) ⇤Leech, fixed 1-plane M24 =2 ⇤ , fixed 2-plane M N Leech 23 =4 ⇤ , fixed 3-plane M N Leech 22 Table 1: The extended superconformal algebras compatible with the chiral CFT, the resulting unbroken globalThe symmetry fixed groups,planes andgive the one, relevant two, geometric or three objects fermions these symmetry out of groups act on. Sunday, Aprilwhich 12, 15 one can write currents to enhance the N=1 algebra 3. Mock modular formsto other play choices. an important role in the Cheng, Dong, Duncan, Harrison, SK, Wrase; moonshine. Benjamin, Harrison, Paquette, SK, Whalen; Cheng, Harrison, SK, Whalen

Tuesday, August 11, 15 Monday, August 10, 15 5 of m as: The ﬁrst few “extremal Jacobi forms” are:

m=1 ZEG = ϕ0,1 (2.6) m=2 1 2 5 2 ZEG = ϕ0,1 + ϕ 2,1E4 (2.7) 6 6 − m=3 1 3 7 2 13 3 ZEG = ϕ0,1 + ϕ0,1ϕ 2,1E4 + ϕ 2,1E6 (2.8) 48 16 − 24 − m=4 67 4 2 11 3 1 2 2 1 4 ZEG = ϕ 2,1E4 + ϕ 2,1ϕ0,1E6 + ϕ 2,1ϕ0,1E4 + ϕ0,1 . (2.9) 144 − 27 − 8 − 432

We note in passing that the m = 2 extremal elliptic genus arises as the chiral partition Thefunction m=2 of the case= 2precisely theory discussed coincides in 7 of [20]. the That with theory chiral enjoys partition an M symmetry. N § 23 Here,function our focus of will the be on N=2 constructing formulation an example ofof an this extremal c=12m =4theory. theory.2 It We now give a few more detailshas M23 about thesymmetry. would-be theory at m =4. We1. The have character also expansion been able of the partitionto engineer, function iswith Benjamin, Dyer

m=4 N=2 N=2 and Fitzpatrick,Z an=ch N=27 +47ch theory7 with c=24 matching EG 2 ;1,4 2 ;1,0 N=2 the+(23+61984 m=4 caseq + above.)ch 7 ;2 4 ··· 2 , Tuesday, August 11, 15 N=2 N=2 +(2024+485001q + )(ch 7 ;2,3 +ch7 ;2, 3) ··· 2 2 − N=2 N=2 +(14168+1659174q + )(ch 7 ;2,2 +ch7 ;2, 2) ··· 2 2 − N=2 N=2 +(32890+2969208q + )(ch 7 ;2,1 +ch7 ;2, 1) . (2.10) ··· 2 2 −

Our conventions for characters are as in [20]; chN=2 denotes the =2superconformal l;h,Q N character with l m 1 = c 1 .ForBPScharactersh = m = c , while for non-BPS ≡ − 2 6 − 2 4 24 characters h = m + n with n Z. But because the non-BPS characters at various values 4 ∈ N=2 of n diﬀer only by an overall power of q, we write all of them as ch m and multiply l; 4 +1,Q by the appropriate power of q in front.

2. The degeneracies in (2.10) are suggestive of an interesting (sporadic) discrete symmetry

group. The Mathieu group M24 has representations of dimension 23 and 2024, for instance. In fact we will see below that this is no coincidence; our construction of an =2SCFT N 24 with this chiral partition function will be based on the A1 Niemeier lattice, which enjoys M symmetry. The choice of = 2 algebra breaks the symmetry group to M , which 24 N 23 is the symmetry group of the resulting = 2 superconformal ﬁeld theory. N 3. Because of the properties of the ring of weak Jacobi forms of weight 0 and index 4, such a form is determined by four coeﬃcients in its q, y expansion (where y = e2πiz). For instance

2We note that it is plausible that more than one such theory exists, i.e. there could be additional constructions which also match the extremal m =4ellipticgenus.

5 of m as: m=1 ZEG = ϕ0,1 (2.6) m=2 1 2 5 2 ZEG = ϕ0,1 + ϕ 2,1E4 (2.7) 6 6 − m=3 1 3 7 2 13 3 ZEG = ϕ0,1 + ϕ0,1ϕ 2,1E4 + ϕ 2,1E6 (2.8) 48 16 − 24 − m=4 67 4 2 11 3 1 2 2 1 4 ZEG = ϕ 2,1E4 + ϕ 2,1ϕ0,1E6 + ϕ 2,1ϕ0,1E4 + ϕ0,1 . (2.9) 144 − 27 − 8 − 432

We note in passing that the m = 2 extremal elliptic genus arises as the chiral partition function of the = 2 theory discussed in 7 of [20]. That theory enjoys an M symmetry. N § 23 Here, our focus will be on constructing an example of an extremal m =4theory.2 We now give a few more details about the would-be theory at m =4. The m=4 case has character expansion: 1. The character expansion of the partition function is

m=4 N=2 N=2 Z =ch7 +47ch7 EG 2 ;1,4 2 ;1,0 N=2 +(23+61984q + )ch 7 ;2,4 suggestive of ··· 2 N=2 N=2 Mathieu +(2024+485001q + )(ch 7 ;2,3 +ch7 ;2, 3) ··· 2 2 − group N=2 N=2 +(14168+1659174q + )(ch 7 ;2,2 +ch7 ;2, 2) ··· 2 2 − N=2 N=2 +(32890+2969208q + )(ch 7 ;2,1 +ch7 ;2, 1) . (2.10) ··· 2 2 −

Our conventions for characters are as in [20]; chN=2 denotes the =2superconformal l;h,Q N character withBecausel m 1 = ofc the1 .ForBPScharacters small size of theh = spacem = c ,of while Jacobi for non-BPS forms ≡ − 2 6 − 2 4 24 characters h = m + n with n Z. But because the non-BPS characters at various values in the caseof4 atindex hand, 4 the ∈and Ramond weight sector 0, it elliptic is fixed genus just has by a q, a y fewexpansion coeffs: N=2 of n differ only by an overall power of q, we write all of them as ch m and multiply l; 4 +1,Q by the appropriate power of q in front. 1 Zm=4,RR = +46+y4 + (q) . (2.11) EG y4 O 2. The degeneracies in (2.10) are suggestive of an interesting (sporadic) discrete symmetry 0 group.Matching TheTuesday, Mathieu August the 11, 15 group coeffiMcientshas of representations(q ) in (2.11) of dimension suffices 23 to and completely 2024, for instance. determine this weak 24 O In factJacobi we will form. see below that this is no coincidence; our construction of an =2SCFT N 24 with this chiral partition function will be based on the A1 Niemeier lattice, which enjoys M symmetry. The choice of = 2 algebra breaks the symmetry group to M , which 24 24 Z N 23 3Theis the symmetryA1 group/ 2 of theorbifold resulting = 2 superconformal field theory. N 3.The Because best of known the properties chiral conformal of the ring fieldof weak theories Jacobi formsare associated of weight 0 to and theories index 4, of such chiral a bosons prop- agatingform is determined on even self-dual by four coe unimodularfficients in its latticesq, y expansion of dimension (where y 24= ke2,π asiz). well For instance as their orbifolds. At 2We notedimension that it is plausible 24, there that are more precisely than one 24 such such theory lattices exists, i.e. – the there Leech could lattice be additionaland constructionsthe 23 Niemeier lat- which also match the extremal m =4ellipticgenus. tices [21]. The chiral conformal field theories at c = 24 are conjecturally classified as well, starting with the work of Schellekens [22], as extended in [23]. For a recent review of progress 5 in this classification, see e.g. [24]. 24 We focus here on one of the theories associated to Niemeier lattices,theA1 theory. The 24 24 A1 Niemeier lattice contains the lattice vectors in the A1 root lattice, as well as additional vectors generated by the “gluing vectors.” We discuss aspects of our construction in some detail below. But first, we pause to give a general description of the theory, just by using simple facts 24 about the A1 lattice. These facts are as follows:

1. The SU(2) current algebra (associated to each of the A1 factors) has three currents, and 24 so the A1 theory is expected to have 72 states at conformal dimension ∆ =1. (This is correct in the full theory – the additional gluing vectors do not add states at this low conformal dimension.)

2. This lattice theory, like all such theories, admits a canonical Z2 symmetry – the one inverting the lattice vectors. (In the language of 24 chiral bosons, it acts as Xi Xi). →− The dimension of the “twist ﬁelds” σa creating the twisted sector ground states from 3 the untwisted sector is ∆twist = 2 . This is the right dimension for a supercharge; and following the general construction of [14], in fact this theory can be promoted to an =1 N superconformal theory. The total number of such twist ﬁelds is 212 = 4096 [25], and we will carefully choose two linear combinations of them to be the supercharges G . ±

3. Furthermore, in the A24 theory, one can actually promote to =2superconformal 1 N invariance. For an = 2 superconformal algebra, we require an additional U(1) current N and a pair of supercharges with charges under the U(1). For the U(1) we select the Z ± 2 invariant current, i√2X1 i√2X1 J =2(e + e− ) , (3.1)

6 In fact, a familiar friend from recent talks allows us to construct this theory.

The c=24 theories are conjecturally classiﬁed. The simplest examples are associated with Niemeier lattices.

From these a simple construction follows:

24 * The theory associated to the A 1 Niemeier lattice can be orbifolded by the canonical Z 2 symmetry.

* The resulting theory has a supersymmetry, by analogy with the Dixon-Ginsparg-Harvey construction.

Tuesday, August 11, 15 * In fact, by choosing a U(1) generator in one of the 24 copies of A 1 , one can promote this to N=2 supersymmetry.

* The 72 currents associated with the 24 copies of SU(2) current algebra split into 48 which have half-integral of m as: singularities with the supercharge, and 24 which have integral singularities. m=1 ZEG = ϕ0,1 (2.6) * Thesem=2 facts1 2 immediately5 2 allow one to prove that ZEG = ϕ0,1 + ϕ 2,1E4 (2.7) this theory6 has6 chiral− partition function given by m=3 1 3 7 2 13 3 ZEG = ϕ0,1 + ϕ0,1ϕ 2,1E4 + ϕ 2,1E6 (2.8) 48 16 − 24 − m=4 67 4 2 11 3 1 2 2 1 4 ZEG = ϕ 2,1E4 + ϕ 2,1ϕ0,1E6 + ϕ 2,1ϕ0,1E4 + ϕ0,1 . (2.9) 144 − 27 − 8 − 432

We note in passing that the m = 2 extremal elliptic genus arises as the chiral partition Tuesday, August 11, 15 function of the = 2 theory discussed in 7 of [20]. That theory enjoys an M symmetry. N § 23 Here, our focus will be on constructing an example of an extremal m =4theory.2 We now give a few more details about the would-be theory at m =4.

1. The character expansion of the partition function is

m=4 N=2 N=2 Z =ch7 +47ch7 EG 2 ;1,4 2 ;1,0 N=2 +(23+61984q + )ch 7 ;2 4 ··· 2 , N=2 N=2 +(2024+485001q + )(ch 7 ;2,3 +ch7 ;2, 3) ··· 2 2 − N=2 N=2 +(14168+1659174q + )(ch 7 ;2,2 +ch7 ;2, 2) ··· 2 2 − N=2 N=2 +(32890+2969208q + )(ch 7 ;2,1 +ch7 ;2, 1) . (2.10) ··· 2 2 −

2. The degeneracies in (2.10) are suggestive of an interesting (sporadic) discrete symmetry

2We note that it is plausible that more than one such theory exists, i.e. there could be additional constructions which also match the extremal m =4ellipticgenus.

5 24 It also has an M23 symmetry: the M24 of the A 1 Niemeier lattice is broken to M23 by the choice of N=2 R-current.

We are currently pursuing various ideas to ﬁnd analogous N=2 constructions at higher values of c, perhaps (by slightly relaxing the criteria) even values “not allowed” by GGKMO.

More generally, I think connections between the new objects playing a role in moonshine and natural objects in string theory or quantum gravity, are worth pursuing.

Tuesday, August 11, 15 II. Enumerative invariants of K3 surfaces

...and now for something completely different.

Happily, we have built up a stock of objects and facts which I can now use to simply explain observations in our paper which appeared yesterday on the arXiv.

Motivation: In summer 2014, three prominent mathematicians (Katz, Klemm, Pandharipande) wrote a paper “On the motivic stable pairs invariants of K3 surfaces,” whose abstract states: “Numerical data suggest the motivic invariants are linked to the Mathieu M24 phenomena.”

Tuesday, August 11, 15 Let me try to explain the relevant invariants.

There is a very easy way to count BPS states in type IIA string theory on K3. This theory enjoys a string duality, relating it to heterotic strings on T4.

In the heterotic theory, the string has right-moving supersymmetry. Any strings in the right-moving ground state but with arbitrary left excitations, are BPS.

1 BPS partition function = ⌘(q)24

Tuesday, August 11, 15 One might, however, want an intrinsically type II accounting. This was ﬁrst provided by Yau-Zaslow.

Let H denote the moduli space of holomorphic curves Mn of genus n equipped with a ﬂat U(1)-bundle, living inside K3.

Then if we deﬁne

d = ( H ) n Mn

n 1 1 1 2 3 n 0 dnq = (⌧) = q (1 + 24q + 324q + 3200q + ) ··· P

Tuesday, August 11, 15 A more intuitive version of this formula comes in yet another duality frame. Instead of imagining D2-branes 2wrapping Reﬁned K3curves, Invariants we can imagine n D0-branes inside one

To see how BPS indices of a K3 compactificationD4-brane. can be related to certain statements about the enumerative geometric properties of K3 surfaces, first recall the following equality. The generating functionThen of the the Eulersame characteristic counting of Kfunction3[n], the Hilbert arises scheme from of n points on an arbitrary K3 surface, is given [23] by Göttsche’s formula

[n] n 1 1 (K3 )q = , Gottsche (2.1) (⌧) n 0 X where q = e2⇡i⌧ and (⌧) denotes the unique (up to scale) cuspidal modular form of weight 12 for the modularand group,the construction with the Hilbert scheme of n points on a(⌧ K3)=⌘ 24surface(⌧)=q becomes(1 qk)24. manifest. (2.2) k>Y0 To see how the above quantity relates to BPS indices, consider the bound states of one D4-brane and n D0-branes on a K3 surface in type IIA string theory. The corresponding BPS index is given Tuesday, August 11, 15 by the Euler characteristic of the relevant moduli space, which in this case is of the form K3[n]. [n] (n) n Recall that K3 serves as a desingularisation of the n-th symmetric product K3 := K3 /Sn. As shown in [24], the value (K3[n]) coincides with the orbifold Euler characteristic of K3(n) (and directly similar statements hold for Hodge numbers). Using the string duality relating type IIA string theory compactified on K3 to heterotic string theory compactified on a torus, T 4,the right-hand side of (2.1) can be understood as the chiral partition function of a bosonic string. This is counting the Dabholkar–Harvey states—half BPS states of the heterotic theory—with the right-moving oscillators in their ground state and arbitrary left-moving excitations. The relation of (2.1) to K3 curve counting, discussed by Yau–Zaslow [16], building on the physical results of [25–27], is best understood when we go to another duality frame. Namely, the BPS system of one D4-brane and n D0-branes on K3 is dual to the system of one D2-brane wrapping a 2-cycle of K3 with self-intersection number 2n 2, and this can be realised as a holomorphic curve of genus n whenever n 0. Hence, in the D2-brane duality frame, the BPS index is argued to be equal to the Euler characteristic of the (compactification of the) moduli space of holomorphic curves of genus n with a choice of flat U(1)-bundle [25]. Let us denote this moduli space by H . In [16] it was shown that the contribution to ( H ) is localised on Mn Mn curves with genus 0 and n double (nodal) points. From the above one arrives at the Yau–Zaslow formula n 1 1 1 2 3 d q = = q (1 + 24q + 324q + 3200q + ), (2.3) n (⌧) ··· n 0 X 1 where dn counts the number of P ’s with n double points in a K3 surface. See [28] for a detailed proof of this identity.

4 The generating function formula (2.1) of G¨ottsche has been extended in [29, 30] from the Euler characteristic, to the elliptic genus [31–35]. Writing the Fourier expansion of the K3 elliptic genus as

2 n ` 1 Z (⌧,z; K3) = c(4n ` ) q y =2y +2y + 20 + O(q) (2.4) EG n,` Z nX20

2⇡iz (cf.Now, (3.7)), we where couldy = e hope, the second-quantised to gain more K3 elliptic information genus is given by than the DMVV formula: [n] n 1 1 s t r c(4rs t2) that providedZ (⌧ ,zby; K 3this)p function.= p (1 Forq y pinstance,) . the (2.5) EG n 0 r,s,t Z X r>Y0,s2 0 space-time physics of type II strings on K3 (times Recall that thea ellipticcircle) genus has is a generalisation SU(2) x of SU(2) Hirzebruch’s symmetry.y genus: for a compact complex manifold M with complex dimension d,wehave

d/2 lim ZEG(⌧,z; M)=y y(M), (2.6) ⌧ i Can’t we write a more! 1 reﬁned generating function which

p q p,q wherekeepsy(M track):= ofy ( the1) h SU(2)(M). As a Cartan result, taking quantum⌧ i in (2.5) numbers? (and then replacing p,q ! 1 p with q), we arriveP at

[n] n n 1 1 k 2 k 20 1 k 2 y(K3 )y q = q (1 yq ) (1 q ) (1 y q ) n 0 k>0 X Y (2.7) 6 1 ⌘(⌧) 1 =( y +2 y ) 2 . ✓1(⌧,z) (⌧)

Given the geometric interpretation of the generating function (2.1) just discussed, it is natural Movingto ask whether to this the one-variable y reﬁnementgenus gives (2.7) also the admits “KKV a curve-counting invariants.” interpretation. r Following [1, 2], Katz–Klemm–Vafa proposed in [17] that the numbers nn, satisfying

Tuesday, August 11, 15 r r 1/2 1/2 2r n 1 1 k 2 k 20 1 k 2 ( 1) n (y y ) q = q (1 yq ) (1 q ) (1 y q ) , (2.8) n r 0 n 0 k>0 X X Y encode the (reduced) Gromov–Witten invariants of a K3 surface in the following way. Given a (non-singular, projective) K3 surface X, and a primitive class ↵ Pic(X), the Gromov–Witten 2 potential reads r 2r 2 x k↵ F↵(gs,x)= Rk↵ gs e · , (2.9) r 0 k 1 X X r where R is the (reduced) Gromov–Witten invariant of genus r and curve class . It can be

5 re-expressed as

2r 2 r 2r 2 1 sin(dgs/2) x d↵ F↵(gs,x)= n˜k↵ gs e · , (2.10) d gs/2 r 0 k 1 d>0 ✓ ◆ X X X wheren ˜r = nr for all curve classes with self-intersection number =2n 2. Note that n · r=0 by specialising (2.8) to z = 0 we recover the Yau–Zaslow formula (2.3) with dn = nn .The ButKKV we conjecture expect relating more (2.8) torefinement Gromov–Witten invariantsfrom both has recently sides been -- proven two in [36]. Cartan(See alsoU(1)s [37].) exist, and K3 cohmology admits two gradings. After refiningThis fromis the the Eulerrole characteristic of the “KKP (2.1) to theinvariants.”y genus (2.7), the next obvious refinement is a generating function for the Hodge polynomials of K3[n],

[n] n 1 Hodge(K3 )q n 0 X 1 k 1 1 k 1 k 20 1 k 1 1 1 k 1 = q (1 uyq ) (1 u yq ) (1 q ) (1 uy q ) (1 u y q ) (2.11) k>Y0 6 1 1 ⌘(⌧) 1 =(u y y + u ) , ✓ (⌧,z+ w)✓ (⌧,z w) (⌧) 1 1 where u = e2⇡iw. Here, following [18], we deﬁne

d/2 d/2 q p p,q From the q-expansions (M ):=of uthese y functions,( u) ( y) hvia(M formulae) a (2.12)bit Hodge p,q too elaborate for me to reproduceX here, one can obtain variousfor M a Kählerfamous manifold enumerative of complex dimension invariantsd. As before, K(“curve3[n] denotes thecounting Hilbert scheme of n points on an arbitraryformulae”) K3 surface. associated to K3. To discuss the enumerative geometric interpretation of (2.11) it will be helpful to first recall the physical origin of the (refined) Gopakumar–Vafa invariants. Given a Calabi–Yau threefold Tuesday, August 11, 15 ˜ jL,jR M, and a homology class H2(M,Z), the refined Gopakumar–Vafa invariant N is defined 2 as the BPS index counting the BPS states of M2-branes wrapping a 2-cycle of class ,withspin quantum numbers (j ,j ) under the little group SO(4) SU(2) SU(2) of a massive particle L R ' L ⇥ R in 5-dimensional Lorentzian space-time. More directly, consider M-theory on M S1 TN ⇥ ⇥~✏ where TN denotes the Taub-NUT space, and the subscript ~✏ =(✏1, ✏2) indicates a twist by the element (ei✏1 ,ei✏2 ) of the U(1) U(1) subgroup of the Taub-NUT isometry group, U(1) SO(3). ⇥ ⇥ Then, in particular, the geometry is locally given by C2 near the tip of the Taub-NUT factor, 1 and writing (z1,z2) for local coordinates near the tip, the twist along the S is given by

z ei✏1 z ,z ei✏2 z . 1 ! 1 2 ! 2

The BPS partition function of this M-theory compactiﬁcation, in the presence of suitable three-

6 Trace functions in the Moonshine module V s\ B. Moonshines beyond the Monster The interesting (but, perhaps, unsurprising) thing, is that all of these canThere be obtainedare even self-dualfrom trace unimodular functions latticesin an in lower a prioridimensions unconnected than theMoonshine Leech lattice. module. In dimension 8, there is a unique structure. Let us return to our friend:

The supersymmetrized theory of bosons propagating on the E8 lattice, realizes moonshine for Conway’s largest sporadic group!

Tuesday, August 11, 15

Monday, August 10, 15 to the conjugacy classes of M24 (although these forms are di↵erent from those arising from the Mathieu moonshine observation of Eguchi–Ooguri–Tachikawa). In [14] stability conditions on K3 surfaces are used to identify orthogonal pairs of (complex)

Ramond sector ground states, a± ,a± 24. More specifically, given a (projective) K3 surface X Z 2 2,0 + 0,2 X, choose vectors a H (X) and a H (X) such that a± ,a⌥ = 1, where , X 2 X 2 h X X i h· ·i denotes the Mukai pairing on the K3 cohomology lattice H˜ (X, Z):= Hn(X, Z) (cf., e.g. [14] or [20]), extended linearly to H˜ (X, Z) C = Hp,q(X). (Note that H2,0(X) and H0,2(X) are ⌦Z L isotropic with respect to , .) Next, choose a stability condition in Bridgeland’s distinguished h· ·i L space [59] and write Z for the corresponding central charge, which we may regard as an element 0,0 1,1 2,2 of H (X) H (X) H (X). Then choose isotropic a± CZ CZ¯ such that a±,a⌥ = 1. Z 2 h Z Z i [JD: this paragraph is to be fixed...] Such pairs correspond to supersymmetric K3 sigma models, according to the description [60,61] in terms of positive-definite 4-planes in the real K3 Its Ramondcohomology space. ground (See [19] states for a succinct span summary a 24 of this.) the We Conway can also use group. such a pair to equip V s\ with an action of the = 4 superalgebra at central charge c = 6, as we will now N As Duncandemonstrate. explained in his talk, there are states which are Write , for the bilinear form on 24 determined by the unique Co -invariant copy of the h· ·i 0 Leechrelated lattice inby24 .the Assume operator-state as in [14] that the pairs correspondencea± and a± are orthogonal, to that a± { X } { Z } X + + and a± are isotropic, and that a ,a = a ,a = 1. Using the notation of [13, 14], define Z generatorsh ofX Xai c=6h Z N=4Z i algebra: c.f. Duncan, vectors in V s\ by setting Mack-Crane

3 1 1 + 1 1 + 1 | := (a ( )a ( )v + a( )a ( )v), 4 X 2 X 2 Z 2 Z 2 i 1 1 |± := a± ( )a±( )v, 2 X 2 Z 2 (3.3) + + ⌧ ± := p2 a (0) a (0) a⌥ (0)a(0)a (0) ⌧ , 1 X ± X ± X Z Z tw + + ⌧ ± := ip2 a(0) a (0) a⌥(0)a (0)a (0) ⌧ , 2 ⌥ Z ± Z ± Z X X tw s\ 3 where v is the vacuum of V .Then| and the |± generate an SU(2) current algebra, and + + the pairs ⌧ , ⌧ and ⌧ , ⌧ each span a copy of the natural 2-dimensional representation With these{ 1 2choices,} { 1 2the} U(1)-graded partition function of the correpsonding SU(2). Write Gj±(z) for the spin-3/2 fields corresponding to the ⌧j±, and of this3 chiral theory turns out3 to be: define J ±(z) and J (z) similarly in terms of the |± and | . Then a routine calculation verifies Tuesday, Augustthe 11, OPEs15

3 3 4 4J (w) 2T (w) 2@wJ (z) G±(z)G⌥(w) + , j j ⇠ (z w)3 ± (z w)2 (z w) ± (z w) 4J ±(w) 2@wJ ±(w) (3.4) G±(z)G±(w) , 1 2 ⇠ ±(z w)2 ± (z w) G±(z)G±(w) G±(z)G⌥(w) 0, j j ⇠ 1 2 ⇠

for j 1, 2 ,whereT (z) is a spin-2 current generating the Virasoro algebra at c = 6. That 2 { }

10 is to say, the Gj±(z) generate an action of the (small) = 4 superconformal algebra at c =6 is to say, the Gj±(z) generate an action of the (small) N= 4 superconformal algebra at c =6 (cf. [62]) on V s\. Note that T (z) is not the stress-energyN tensor of V s\, which we denote L(z), (cf. [62]) on V s\. Note that T (z) is not the stress-energy tensor of V s\, which we denote L(z), and which has c = 12. More details on (3.3) and (3.4) will appear in [63]. and which has c = 12. More details on (3.3) and (3.4) will appear in [63]. The Ramond sector index of this supersymmetric model is simply a constant, The Ramond sector index of this supersymmetric model is simply a constant,

s\ F L(0) c s\ Z (⌧):=Tr(( 1) q 24 V ) s\ F L(0) c s\tw Z (⌧):=Tr(( 1) q 24 V| ) 4 | tw (3.5) 1 1 4 i+1 12 =1 1 ( 1) ✓i (⌧, 0) = 24, (3.5) = 2 ⌘12(⌧) ( 1)i+1✓12(⌧, 0) = 24, 2 ⌘12(⌧) i=2 i i=2X X but once equipped with a U(1) current we may consider the corresponding U(1)-graded Ramond but once equipped with a U(1) current we may consider the corresponding U(1)-graded Ramond sector index. Taking J(z):=2J 3(z), the U(1)-graded index works out [14] to be sector index. Taking J(z):=2J 3(z), the U(1)-graded index works out [14] to be c s\ s\ F L(0) 24 J(0) Z (⌧,z):=Tr(( 1) q y Vtw ) c s\ s\ F L(0) 24 J(0) | Z (⌧,z):=Tr(( 1) q y Vtw ) 1 1 4 | (3.6) 4 i+1 2 10 = ( 1) ✓i (⌧,z)✓i (⌧, 0), (3.6) 1 2 ⌘112(⌧) i+1 2 10 = i=2( 1) ✓ (⌧,z)✓ (⌧, 0), 2 ⌘12(⌧) X i i i=2 X which is a weak Jacobi form of weight 0 and index 1 satisfying Zs\(⌧, 0) = Zs\(⌧) = 24. That s\ s\ whichis, we is have a weak recovered Jacobi the form K3 of elliptic weightAnd genus, 0 and happily, index 1 satisfying Z (⌧, 0) = Z (⌧) = 24. That is, we have recovered the K3 elliptic genus, s\ Z (⌧,z)=ZEG(⌧,z; K3). (3.7) s\ Z (⌧,z)=ZEG(⌧,z; K3). (3.7) Evidence is presented in [14] that the coincidence (3.7) is not an accident, but rather reflects Evidence is presented in [14] thats\ the coincidence (3.7) is not an accident, but rather reflects aNow, deep relationship the symmetries between V and of K3 V surfaces \ are geometry. under To excellent explain this, control. recall that Co 0 is a deep relationship between V s\ ands\ K3 surface geometry. To explain this, recall that Co is precisely the subgroup of Aut(V ) that fixes ⌧tw.DefineG⇧ to be the subgroup of this0Co0 s\ preciselyconsisting the of subgroup elements of that Aut( point-wiseV ) that fix fixes the⌧tw 4-space.Define⇧ :=G⇧ Spanto bea± the,a± subgroup< 24.Then of thisGCo0is Our choices equipping it with an N=4 structure{ X Z } here ⇧ consisting of elements that point-wise fix the 4-space ⇧ := Span a± ,a± < 24.ThenG is precisely the supersymmetry preserving symmetry group of the K3{ X sigmaZ } model corresponding⇧ leave symmetries preserving 4-planes in the 24 of Co 0 . preciselyto ⇧, according the supersymmetry to the main preserving result of symmetry [19]. Now group for g of theG⇧ K3we sigma may model consider corresponding the g-twined 2 c.f. Gaberdiel, Hohenegger, Volpato to U⇧(1)-graded, according Ramond to the sector main result index, of [19]. Now for g G⇧ we may consider the g-twined 2 UTuesday,(1)-graded August 11, 15 Ramond sector index, s\ F L(0) c J(0) s\ Z (⌧,z):=Tr(g( 1) q 24 y V ), (3.8) g | tw s\ F L(0) c J(0) s\ Z (⌧,z):=Tr(g( 1) q 24 y V ), (3.8) g | tw and we may compare it to the corresponding equivariant K3 sigma model elliptic genus, and we may compare it to the corresponding equivariant K3 sigma model elliptic genus, F +F¯ L(0) c L¯(0) c¯ J(0) Z (⌧,z; ⇧,g):=Tr (g( 1) q 24 q¯ 24 y ) (3.9) EG RR F +F¯ L(0) c L¯(0) c¯ J(0) Z (⌧,z; ⇧,g):=Tr (g( 1) q 24 q¯ 24 y ) (3.9) EG RR (cf. [19]). One can check [14] that (cf. [19]). One can check [14] that s\ Zg (⌧,z)=ZEG(⌧,z; ⇧,g), (3.10) s\ Zg (⌧,z)=ZEG(⌧,z; ⇧,g), (3.10)

11 11 is to say, the G±(z) generate an action of the (small) = 4 superconformal algebra at c =6 j N (cf. [62]) on V s\. Note that T (z) is not the stress-energy tensor of V s\, which we denote L(z), and which has c = 12. More details on (3.3) and (3.4) will appear in [63]. is to say, the G±(z) generate an action of the (small) = 4 superconformal algebra at c =6 j N The(cf. Ramond[62]) on V sectors\. Note index that ofT ( thisz) is supersymmetric not the stress-energy model tensor is simply of V s\, a which constant, we denote L(z), and which has c = 12. More details on (3.3) and (3.4) will appear in [63]. s\ F L(0) c s\ Z (⌧):=Tr(( 1) q 24 V ) The Ramond sector index of this supersymmetric | modeltw is simply a constant, 4 (3.5) 1 1 i+1 12 s\ = 12 F L((0)1)c ✓si\ (⌧, 0) = 24, Z (⌧):=Tr((2 ⌘ (⌧1)) q 24 V ) i=2 tw X | 1 1 4 (3.5) = ( 1)i+1✓12(⌧, 0) = 24, but once equipped with a U(1) current2 we⌘12 may(⌧) consider thei corresponding U(1)-graded Ramond i=2 sector index. Taking J(z):=2J 3(z), the U(1)-gradedX index works out [14] to be but once equipped with a U(1) current we may consider the corresponding U(1)-graded Ramond s\ 3 F L(0) c J(0) s\ sector index. TakingZJ(z():=2⌧,z):=Tr((J (z), the1)U(1)-gradedq 24 y indexV works) out [14] to be | tw 4 (3.6) s\ 1 1 F L(0) ci+1J(0)2 s\ 10 Z (⌧,z):=Tr((= 1) q ( 1) 24 y ✓ (⌧V,z))✓ (⌧, 0), 2 ⌘12(⌧) i | tw i i=2 1 1 X4 (3.6) = ( 1)i+1✓2(⌧,z)✓10(⌧, 0), 2 ⌘12(⌧) i i i=2 s\ s\ which is a weak Jacobi form of weight 0 andX index 1 satisfying Z (⌧, 0) = Z (⌧) = 24. That is, wewhich have is recovered a weak Jacobi the K3 form elliptic of weight genus, 0 and index 1 satisfying Zs\(⌧, 0) = Zs\(⌧) = 24. That is, we have recovered the K3 elliptic genus, s\ Z (⌧,z)=ZEG(⌧,z; K3). (3.7) s\ Z (⌧,z)=ZEG(⌧,z; K3). (3.7) Evidence is presented in [14] that the coincidence (3.7) is not an accident, but rather reflects s\ a deep relationshipEvidence is presented between inV [14]and that K3 the surface coincidence geometry. (3.7) is not To an explain accident, this, but recall rather that reflectsCo0 is ss\\ preciselya deep the relationship subgroup between of Aut(V )and that K3 fixes surface⌧tw.Define geometry.G To⇧ to explain be the this, subgroup recall that ofCo this0 isCo0 Let G ⇧ be thes\ subgroup preserving 4-plane ⇧ . consistingprecisely of the elements subgroup that of point-wise Aut(V ) that fix the fixes 4-space⌧tw.Define⇧ :=G Span⇧ to bea± the,a± subgroup< 24.Then of this CoG 0 is { X Z } ⇧ consisting of elements that point-wise fix the 4-space ⇧ := Span a± ,a± < 24.ThenG is precisely the supersymmetry preserving symmetry group of the K3{ X sigmaZ } model corresponding⇧ precisely theThen supersymmetry for any g preserving G ⇧ , we symmetry can define group ofthe the twined K3 sigma partition model corresponding to ⇧, according to the main result2 of [19]. Now for g G⇧ we may consider the g-twined to ⇧, according to the main result of [19].function Now for g 2 G we may consider the g-twined U(1)-graded Ramond sector index, 2 ⇧ U(1)-graded Ramond sector index,

c s\ s\ F L(0) 24 J(0) Zg (s⌧\,z):=Tr(g( 1) Fq L(0) c yJ(0) Vstw\ ), (3.8) Z (⌧,z):=Tr(g( 1) q 24 y |V ), (3.8) g | tw andand we may we may compare compareand it to it compare to the the corresponding corresponding with twinings equivariant equivariant of K3 K3 sigma sigma sigma model modelmodels: elliptic elliptic genus, genus,

c c¯ ¯ c ¯ c¯ FF++FF¯ LL(0)(0) 24 L¯L(0)(0) 24 JJ(0)(0) ZEGZ(⌧,z(⌧;,z⇧;,g⇧,g):=Tr):=TrRR(g(g(( 1)1) qq 24 q¯q¯ 24 yy )) (3.9) (3.9) EG RR

(cf. [19]).(cf. [19]). One One can can check check [14] [14] that that

Tuesday, August 11, 15 sZ\s\(⌧,z)=Z (⌧,z; ⇧,g), (3.10) Zg g(⌧,z)=ZEGEG(⌧,z; ⇧,g), (3.10)

s\ F L(0) c s\ Z (⌧):=Tr(( 1) q 24 V ) | tw 1 1 4 (3.5) = ( 1)i+1✓12(⌧, 0) = 24, 2 ⌘12(⌧) i i=2 X but once equipped with a U(1) current we may consider the corresponding U(1)-graded Ramond sector index. Taking J(z):=2J 3(z), the U(1)-graded index works out [14] to be

s\ F L(0) c J(0) s\ Z (⌧,z):=Tr(( 1) q 24 y V ) | tw 1 1 4 (3.6) = ( 1)i+1✓2(⌧,z)✓10(⌧, 0), 2 ⌘12(⌧) i i i=2 X which is a weak Jacobi form of weight 0 and index 1 satisfying Zs\(⌧, 0) = Zs\(⌧) = 24. That is, we have recovered the K3 elliptic genus,

s\ Z (⌧,z)=ZEG(⌧,z; K3). (3.7)

Evidence is presented in [14] that the coincidence (3.7) is not an accident, but rather reflects s\ a deep relationship between V and K3 surface geometry. To explain this, recall that Co0 is s\ precisely the subgroup of Aut(V ) that fixes ⌧tw.DefineG⇧ to be the subgroup of this Co0 consisting of elements that point-wise fix the 4-space ⇧ := Span a± ,a± < 24.ThenG is { X Z } ⇧ precisely the supersymmetry preserving symmetry group of the K3 sigma model corresponding to ⇧, according to the main result of [19]. Now for g G we may consider the g-twined 2 ⇧ U(1)-graded Ramond sector index,

s\ F L(0) c J(0) s\ Z (⌧,z):=Tr(g( 1) q 24 y V ), (3.8) g | tw and we may compare it to the corresponding equivariant K3 sigma model elliptic genus,

F +F¯ L(0) c L¯(0) c¯ J(0) Z (⌧,z; ⇧,g):=Tr (g( 1) q 24 q¯ 24 y ) (3.9) EG RR

(cf. [19]). One can check [14] that Natural question:

?? s\ Zg (⌧,z)=ZEG(⌧,z; ⇧,g), (3.10)

In many cases yes. Universally? There may be a role for other Umbral modules. 11 Now, this allows us to deﬁne twined enumerative invariants as well. Because although we have only compared the elliptic genus of K3 to the partition function of this Moonshine module, all of the previous generating functions are determined by (special limits of) the elliptic genus!

Tuesday, August 11, 15 This may not be obvious, as they involve Euler characters The generating function formula (2.1) of G¨ottsche has been extended in [29, 30] from the (andThe generalizations) generating function formula of Hilbert (2.1) of G¨ottsche schemes. has been extendedBut recall in [29, that 30] from via the Euler characteristic, to the elliptic genus [31–35]. Writing the Fourier expansion of the K3 Euler characteristic,the formula to the of elliptic Dijkgraaf-Moore-Verlinde-Verlinde, genus [31–35]. Writing the Fourier expansion of the K3 elliptic genus as elliptic genus as

2 n ` 1 ZEG(⌧,z; K3) = c(4n ` )2q yn `=2y +2y 1+ 20 + O(q) (2.4) ZEG(⌧,z; K3) = c(4n ` ) q y =2y +2y + 20 + O(q) (2.4) n,` Z nXn,20` Z nX20 (cf. (3.7)), where y = e2⇡iz, the second-quantised K3 elliptic genus is given by the DMVV (cf. (3.7)), where y = e2⇡iz, the second-quantiseddetermines K3 elliptic genus is given by the DMVV formula: formula: [n] n 1 1 s t r c(4rs t2) ZEG(⌧,z; K3 [n)p] n 1= p 1 (1 q ys pt )r c(4rs t2.) (2.5) ZEG(⌧,z; K3 )p = p (1 q y p ) . (2.5) n 0 r,s,t Z Xn 0 r>Yr,s,t0,s2 0Z X r>Y0,s2 0 Recall that the elliptic genus is a generalisation of Hirzebruch’s y genus: for a compact complex Recall that the elliptic genus is a generalisation of Hirzebruch’s y genus: for a compact complex manifold M with complex dimension d,wehave manifoldTheM formulawith complex of dimension KKV isd,wehave just the ⌧ i limit of DMVV. ! 1 d/2 lim ZEG(⌧,z; M)=y d/2 y(M), (2.6) ⌧ limi ZEG(⌧,z; M)=y y(M), (2.6) !⌧ 1i ! 1 p q p,q where y(M):= y (p 1) hq p,q(M). As a result, taking ⌧ i in (2.5) (and then replacing Tuesday,where August 11,( 15M):= p,q y ( 1) h (M). As a result, taking ⌧ i in (2.5) (and then replacing y p,q !!11 p with q), we arrive at p with q), we arrivePP at

[n] n n 1 1 k 2 k 20 1 k 2 y(K3 [n)y] qn n 1= q 1 (1 yq )k (12 q )k 20(1 y q1 )k 2 y(K3 )y q = q (1 yq ) (1 q ) (1 y q ) n 0 k>0 Xn 0 Yk>0 (2.7) X Y 6 (2.7) 1 ⌘(⌘⌧()⌧)6 11 =( y +2 y )1 . =( y +2 y ✓)2(2⌧,z) (⌧) . 1✓1(⌧,z) (⌧)

GivenGiven the the geometric geometric interpretation interpretation of of the the generating generating function function (2.1) (2.1) just just discussed, discussed, it it is is natural natural toto ask ask whether whether this this one-variable one-variable reﬁnement reﬁnement (2.7) (2.7) also also admits admits a a curve-counting curve-counting interpretation. interpretation. Following [1, 2], Katz–Klemm–Vafa proposed in [17] that the numbers nr r, satisfying Following [1, 2], Katz–Klemm–Vafa proposed in [17] that the numbers nnn, satisfying

r rr r 1/12/2 1/12/22r2rn n1 1 1 1 k k 2 2 k k 2020 1 1kk 2 2 ( (1)1)nnn(y(y yy ) ) q q==qq (1(1 yqyq))(1(1 qq)) (1(1 yyqq)), , (2.8)(2.8) n r r0 n0 n0 0 k>k>0 0 XXXX YY encodeencode the the (reduced) (reduced) Gromov–Witten Gromov–Witten invariants invariants of of a a K3 K3 surface surface in in the the following following way. way. Given Given a a (non-singular,(non-singular, projective) projective) K3 K3 surface surfaceXX, and, and a a primitive primitive class class↵↵ Pic(Pic(XX),), the the Gromov–Witten Gromov–Witten 22 potentialpotential reads reads r r 2r2r2 2x xk↵k↵ F↵F(↵g(sg,xs,x)=)= RRk↵k↵gsgsee· · , , (2.9)(2.9) r r0 k0 k1 1 XXXX r r wherewhereRRisis the the (reduced) (reduced) Gromov–Witten Gromov–Witten invariant invariant of of genus genusrrandand curve curve class class.. It It can can be be

55 Then, via the relationship between K3 sigma models and V s\ we can produce equivariant curve counts.

To get to the KKP invariants, which have an additional grading, one simply needs to grade the partition function in V s\ by an additional U(1).

This succeeds in relating the KKP (and more primitive) invariants to sporadic groups and moonshine, though perhaps not by the route they expected.

Tuesday, August 11, 15 Last but not least:

THANKS TO THE ORGANISERS FOR AN EXCELLENT PROGRAM!!

Tuesday, August 11, 15