Extended monstrous moonshine - Institute for Mathematical ... · Extended monstrous moonshine Scott Carnahan Department of Mathematics University of Tsukuba ... Classi cation of nite

Monstrous MoonshineAdditional Moonshine PhenomenaGeneralized Monstrous Moonshine

Modular MoonshineUnification?

Extended monstrous moonshine

Scott Carnahan

Department of MathematicsUniversity of Tsukuba

2018-6-27Pan-Asia Number Theory Conference

Scott Carnahan Extended monstrous moonshine



What is moonshine?Strange connections between finite groups andmodular forms

Symmetry

Finite groups

Complex analysis/

Number theory

Modular formsMoonshine




What is moonshine?Strange connections between finite groups andmodular forms

The connections should be “very special”

Infinitely many cases ⇒ not moonshine!




Monstrous Moonshine




Classification of finite simple groups

Any finite simple group is one of the following

A cyclic group of prime order

An alternating group An (n ≥ 5)

A group of Lie type (16 infinite families)

One of 26 sporadic simple groups

Largest sporadic: Monster M, about 8 · 1053

elements (Griess 1982).194 irred. repres. of dim 1, 196883, 21296876, . . .




Hauptmodul (or principal modulus)

A Hauptmodul for a discrete subgroup Γ < SL2(R)is a holomorphic function H→ C invariant under Γ,that generates the function field of Γ\H.

J-function as Hauptmodul

The quotient space SL2(Z)\H has genus zero.Function field generated by J . Fourier expansion:q−1 + 196884q + 21493760q2 + · · · (q = e2πiz)




Coefficients of J and Irreducible Monster reps

196884 = 1 + 196883 (McKay, 1978)21493760 = 1+196883+21296876 (Thompson, 1979)

864299970 = 2×1+2×196883+21296876+842609326

......

How to continue this sequence?

McKay-Thompson conjecture: Natural graded rep⊕∞n=0 Vn of M such that

∑dimVnq

n−1 = J .




Idea: Physics forms a bridge

Monster J functionConformal field theory

(Vertex operator algebras)

Solution: Frenkel, Lepowsky, Meurman 1988

Constructed a vertex operator algebraV \ =

⊕n≥0 V

\n (the Moonshine Module), such that∑

n≥0(dimV \n)qn−1 = J and AutV \ = M.




Refined correspondence

Thompson’s suggestion: replace graded dimensionwith graded trace of non-identity elements.

Monstrous Moonshine Conjecture (Conway, Norton1979)

There is a faithful graded representationV =

⊕n≥0 Vn of the monster M such that for all

g ∈M, the series Tg(τ) =∑

n≥0 Tr(g |Vn)qn−1 isthe q-expansion of a congruence Hauptmodul (=“generates function field of genus 0 H-quotient”).




First proof (Atkin, Fong, Smith 1980)

Theorem: A virtual representation of M existsyielding the desired functions.No construction.

Second proof (Borcherds 1992)

Theorem: The Conway-Norton conjecture holds forV \.




Outline of Borcherds’s proof

FLM construction: V \

Add torus and quantize

Lie algebra m

Automorph. ∞ prod.

gens. and rels.

Lie algebra LIsom.m ∼= L

Twisted Denominator Identities

Recursion relations

Hauptmoduln




Additional Moonshine Phenomena




Theorem (Ogg 1974)

The primes p such that X0(p)+ = X0(p)/〈wp〉 hasgenus zero are

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71

These are the primes p such that all supersingularelliptic curves over Fp have j invariant in Fp.

Ogg’s Jack Daniels problem

Explain why these are precisely the primes thatdivide the order of the Monster.




Borcherds’s half-solution

For each p|#M, there is a conjugacy class pA, suchthat Tg(τ) is a Hauptmodul of X0(p)+ for g in pA.

The other half (still open)

Explain why V \ has so many automorphisms.




Positivity phenomena

For g in pA, the coefficients of Tg(τ) arenon-negative integers.E.g. for 2A, Tg(τ) = q−1 + 4372q + 96256q2 + · · ·is a Hauptmodul for Γ0(2)+.

Extra phenomena (Conway-Norton, Queen)

The coefficients of Tg(τ) appear to be dimensionsof representations of centralizers.E.g., For g in 2A, CM(g) ∼= 2.B, with irreduciblerepresentations: 1, 4371, 96255, 96256, . . .




Two explanations conjectured!1 Generalized Moonshine (Norton 1987): Graded

representations V (g) of CM(g) in characteristiczero, traces of centralizing elements areHauptmoduln.

2 Modular moonshine (Ryba 1994): Gradedrepresentations V p of CM(g) in characteristicp, Brauer characters of p-regular centralizingelements are Hauptmoduln.




Complementary history

Generalized Modular1990s Objects V (g) Hauptmoduln

exist (Dong- assuming existenceLi-Mason 1997) (Borcherds-Ryba 1996)

2010s Hauptmoduln Objects V p exist

Key advance:

Theory of cyclic orbifolds of vertex operatoralgebras.




What is a vertex operator algebra?1 V =

⊕n∈Z Vn, a graded vector space

2 1 ∈ V0, an “identity element”3 ω ∈ V2 a “Virasoro element”4 Y : V ⊗ V → V ((z)), “multiplication”

satisfying1 Y (1, z)x = x , Y (x , z)1 ∈ x + zV [[z ]]2 coefficients of Y (ω, z) give Virasoro action3 “commutativity and associativity”.4 ∀n, dimVn <∞. For n� 0, Vn = 0.




Lattice vertex operator algebras

L an even positive definite lattice.VL = C[L]⊗ SymC x(L⊗ C)[x ].

Graded dimension is θL(τ)ηrank L

.E.g., For Leech lattice, get J + 24.




V-ModulesFor V - vertex operator algebra, a V -module is avector space M with an action mapYM : V ⊗M → M((z)) satisfying somecompatibility.V is holomorphic if all V -modules are direct sumsof V .

Theorem (Dong 1994)

All VL-modules are direct sums of VL+α forα ∈ L∨/L.In particular, VL holomorphic ⇔ L unimodular.




Theorem (van Ekeren, Moller, Scheithauer 2015)

V holomorphic VOA, g “anomaly-free”automorphism of order n. Then there exist:

a “generalized VOA” gV =⊕

i ,j∈Z/nZgV i ,j .

a pair of commuting automorphisms (g , g ∗)giving decomposition into V i ,j .

An isomorphism V =⊕

i∈Z/nZgV i ,0

a holomorphic VOA V /g =⊕

j∈Z/nZgV 0,j

Cyclic orbifold duality: (V , g)↔ (V /g , g ∗)




Special case: First construction of V \ (1988)

VΛ - Leech lattice VOAσ lifted from the −1-automorphism of Λ.Then V \ = (VΛ)/σ.

Now we have 51 constructionsWe can take any σ that is fixed-point free, with “nomassless states”. (51 algebraic conjugacy classes).




Generalized Monstrous Moonshine




Generalized Moonshine Conjecture (Norton 1987):

g ∈M⇒ V (g) graded proj. rep. of CM(g)

(g , h), gh = hg ⇒ Z (g , h; τ) holomorphic on H

1 q-expansion of Z (g , h; τ) is graded trace of (alift of) h on V (g).

2 Z is invariant under simultaneous conjugationof the pair (g , h) up to scalars.

3 Z (g , h; τ) constant or a Hauptmodul.4 Z (g , h; aτ+b

cτ+d ) proportional to Z (g ahc , g bhd ; τ).5 Z (g , h; τ) = J(τ) if and only if g = h = 1.




Brute force solution (like Atkin-Fong-Smith)?

This is a finite problem:

Finitely many conjugacy classes of commutingpairs, and possible levels are bounded.

Central extensions of centralizers “can becomputed”.

Not finite enough for 2018

We still haven’t classified the commuting pairs.

We still don’t know character tables of allcentralizers, let alone central extensions.




Physics Language (Dixon, Ginsparg, Harvey 1988)

V (g) - twisted sectors of a conformal field theorywith M-symmetry.Z (g , h; τ) - genus 1 partition functions (with(g , h)-twisted boundary conditions).All except Hauptmodul claim (3) “follow” fromconformal field theory considerations.

Algebraic Interpretation

V (g) = irreducible g -twisted V \-module V \(g)Z (g , h; τ) = Tr(hqL(0)−1|V (g)) for a lift h.




Geometric interpretation of Z

Physicists draw boundary conditions as colorings.

g

h

g

h

Commuting pair (g , h) gives hom π1(E , e)→M.SL2(Z) action changes generating pair.Ignoring scalar ambiguities, claims (2) and (4) saythat Z is a function on the moduli space of ellipticcurves with principal M-bundles.




First Breakthrough (Dong, Li, Mason 1997)

- Existence and uniqueness (up to isom.) of V \(g).- Convergence of power series defining Z .- Settles claims (1), (2), (5).- Reduces SL2(Z) claim (4) to “g -rationality”.

Theorem (C, Miyamoto 2016)

Category of g -twisted V \-modules is semisimple.This resolves the SL2(Z)-compatibility claim (4).




On to claim (3)

We now need to show that all Z (g , h; τ) areHauptmoduln or constant.

Second Breakthrough (Hohn 2003)

Generalized Moonshine for 2A (Baby monster case).- Gives outline for proving Hauptmodul claim (3).




Borcherds-Hohn program for Hauptmoduln

Ab. intertw. alg. gNV

\

Add torus and quantize

Lie algebra mg

Automorph. ∞ prod.

gens. and rels.

Lie algebra LgIsom.mg∼= Lg

Twisted Denominator Identities

Recursion relations

Hauptmoduln




Right side (C 2009)

Borcherds products of the form:

Tg(σ)−Tg(−1/τ) = p−1∏

m>0,n∈ 1NZ

(1− pmqn)cgm,n(mn)

- Exponent cgm,n(mn) is qmn-coefficient of a v.v.

modular function formed from {Tg i (τ)}N−1i=0 .

- Lg is a Z⊕ 1NZ-graded BKM Lie algebra.

- Simple roots of multiplicity cg1,n(n) in degree (1, n).




Add a torus and quantize

- Take a graded tensor product with a latticeabelian intertwining algebra VII1,1(−1/N)

- Get conformal VA, c = 26, graded by 2d lattice,has invariant form.- Apply a bosonic string quantization functor.- For Fricke g (i.e., Tg(τ) = Tg(−1/Nτ)), get aBKM Lie algebra mg with real simple root.- graded by II1,1(−1/N) ∼= Z⊕ 1

NZ.




Comparison

Borcherds-Kac-Moody Lie algebras:- mg has canonical projective action of CM(g).- Lg has “nice shape”: known simple roots, goodhomology.Isomorphism from matching root multiplicities:dim(Lg)m,n = (mg)m,n = cgm,n(mn).

Transport de structure ⇒ Lg gets CM(g) action.




End of proof (C 2016)

Virtual CM(g)-module isom H∗(Eg ,C) ∼=∧∗ Eg

implies equivariant Hecke operators nTn given bynTnZ (g , h, τ) =

∑ad=n,0≤b<d

Z (g d , g−bha, aτ+bd )

act by monic polynomials on Z (g , h, τ).

Hauptmodul condition follows (C 2008).

Constants come from (g , h) such that all g ahc

are non-Fricke when (a, c) = 1, using claim (4).

This resolves the final claim (3).




Modular Moonshine




Ryba’s conjecture

For each g in conjugacy class pA (p|#M), there isa vertex algebra V p =

⊕n≥0 V

pn over Fp with an

action of CM(g), such that for all p-regularelements h, the Brauer character∑

n≥0

Tr(h|V pn )qn−1

is the q-expansion of the Hauptmodul Tgh(τ).




Borcherds-Ryba interpretation 1996

V p = H0(g ,V \Z), where V \

Z is a self-dual integralform of V \ (i.e., a VOA over Z with M-symmetry).

Theorem (Borcherds-Ryba 1996, Borcherds 1998)

If V \Z exists, then V p := H0(g ,V \

Z) works.

Theorem (C 2017)

V \Z exists.




How to construct a self-dual integral form?

Existing constructions (e.g., by cyclic orbifold) havedenominators.For example, order n orbifold construction requires1n and eπi/n.

SolutionDo orbifold constructions of many different orders,and glue using faithfully flat descent.




Unification?




Recall

Conjugacy classes pA yield reps. of CM(g):V (g) (char. 0) and V g (char. p)Same p-regular characters!In fact, for any g ∈M, we get reps of CM(g):

V (g) (char. 0) and H∗(g ,V \Z) (char. |g |).

Question

Is there an integral structure that produces both?




Main obstruction

Sometimes H1(g ,V \Z) 6= 0.

Conjecture (Borcherds-Ryba 1996)

H1(g ,V \Z) = 0 if and only if

Tg(τ) =∑

Tr(g |V \n)qn−1 has a pole at 0.

Definition

We say g is Fricke if Tg(τ) has a pole at 0.Equivalently, Tg(τ) is invariant under the Frickeinvolution wN : τ 7→ − 1

Nτ .g is non-Fricke if Tg(τ) is regular at 0.




Fricke versus non-FrickeM has 141 Fricke classes, and 53 non-Frickeclasses

Tg(τ) non-negative coeffs. ⇔ g Fricke.

Z (g , h, τ) has a pole at ∞ if and only if g isFricke.

V \/g ∼=

{V \ g is Fricke

VΛ g non-Fricke




Conjecture (Borcherds 1998)

There is a rule that assigns to any g ∈M of ordern, a 1

nZ-graded Z[e2πi/n]-module Vg with an actionof a central extension Z/nZ.CM(g), such that

Vg ⊗Z[e2πi/n] C ∼= V (g) as Z/nZ.CM(g)-reps.

g Fricke ⇒ Vg ⊗Z[e2πi/n] Z/nZ ∼= H0(g ,V \Z) as

Z/nZ.CM(g)-reps.

H∗(h,Vg) = Vgh ⊗ Z/|h|Z when g , h commuteand have coprime order.

(additional compatibilities)




Current progress

Twisted modules V (g) can be defined overZ[1

n , e2πi/n], but removing 1

n is tricky.

Looks like H1(g ,V \Z) = 0 for Fricke g (in progress)

What we really need

Canonical lifts of H∗(g ,V \Z) to characteristic

zero.

Meaningful interpretation of these objects.


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Extended monstrous moonshine - Institute for Mathematical ... · Extended monstrous moonshine Scott Carnahan Department of Mathematics University of Tsukuba ... Classi cation of nite