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
Monstrous MoonshineAdditional Moonshine PhenomenaGeneralized Monstrous Moonshine
Modular MoonshineUnification?
What is moonshine?Strange connections between finite groups andmodular forms
Symmetry
Finite groups
Complex analysis/
Number theory
Modular formsMoonshine
Scott Carnahan Extended monstrous moonshine
Monstrous MoonshineAdditional Moonshine PhenomenaGeneralized Monstrous Moonshine
Modular MoonshineUnification?
What is moonshine?Strange connections between finite groups andmodular forms
The connections should be “very special”
Infinitely many cases ⇒ not moonshine!
Scott Carnahan Extended monstrous moonshine
Monstrous MoonshineAdditional Moonshine PhenomenaGeneralized Monstrous Moonshine
Modular MoonshineUnification?
Monstrous Moonshine
Scott Carnahan Extended monstrous moonshine
Monstrous MoonshineAdditional Moonshine PhenomenaGeneralized Monstrous Moonshine
Modular MoonshineUnification?
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, . . .
Scott Carnahan Extended monstrous moonshine
Monstrous MoonshineAdditional Moonshine PhenomenaGeneralized Monstrous Moonshine
Modular MoonshineUnification?
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)
Scott Carnahan Extended monstrous moonshine
Monstrous MoonshineAdditional Moonshine PhenomenaGeneralized Monstrous Moonshine
Modular MoonshineUnification?
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 .
Scott Carnahan Extended monstrous moonshine
Monstrous MoonshineAdditional Moonshine PhenomenaGeneralized Monstrous Moonshine
Modular MoonshineUnification?
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.
Scott Carnahan Extended monstrous moonshine
Monstrous MoonshineAdditional Moonshine PhenomenaGeneralized Monstrous Moonshine
Modular MoonshineUnification?
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”).
Scott Carnahan Extended monstrous moonshine
Monstrous MoonshineAdditional Moonshine PhenomenaGeneralized Monstrous Moonshine
Modular MoonshineUnification?
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 \.
Scott Carnahan Extended monstrous moonshine
Monstrous MoonshineAdditional Moonshine PhenomenaGeneralized Monstrous Moonshine
Modular MoonshineUnification?
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
Scott Carnahan Extended monstrous moonshine
Monstrous MoonshineAdditional Moonshine PhenomenaGeneralized Monstrous Moonshine
Modular MoonshineUnification?
Additional Moonshine Phenomena
Scott Carnahan Extended monstrous moonshine
Monstrous MoonshineAdditional Moonshine PhenomenaGeneralized Monstrous Moonshine
Modular MoonshineUnification?
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.
Scott Carnahan Extended monstrous moonshine
Monstrous MoonshineAdditional Moonshine PhenomenaGeneralized Monstrous Moonshine
Modular MoonshineUnification?
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.
Scott Carnahan Extended monstrous moonshine
Monstrous MoonshineAdditional Moonshine PhenomenaGeneralized Monstrous Moonshine
Modular MoonshineUnification?
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, . . .
Scott Carnahan Extended monstrous moonshine
Monstrous MoonshineAdditional Moonshine PhenomenaGeneralized Monstrous Moonshine
Modular MoonshineUnification?
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.
Scott Carnahan Extended monstrous moonshine
Monstrous MoonshineAdditional Moonshine PhenomenaGeneralized Monstrous Moonshine
Modular MoonshineUnification?
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.
Scott Carnahan Extended monstrous moonshine
Monstrous MoonshineAdditional Moonshine PhenomenaGeneralized Monstrous Moonshine
Modular MoonshineUnification?
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.
Scott Carnahan Extended monstrous moonshine
Monstrous MoonshineAdditional Moonshine PhenomenaGeneralized Monstrous Moonshine
Modular MoonshineUnification?
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.
Scott Carnahan Extended monstrous moonshine
Monstrous MoonshineAdditional Moonshine PhenomenaGeneralized Monstrous Moonshine
Modular MoonshineUnification?
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.
Scott Carnahan Extended monstrous moonshine
Monstrous MoonshineAdditional Moonshine PhenomenaGeneralized Monstrous Moonshine
Modular MoonshineUnification?
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 ∗)
Scott Carnahan Extended monstrous moonshine
Monstrous MoonshineAdditional Moonshine PhenomenaGeneralized Monstrous Moonshine
Modular MoonshineUnification?
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).
Scott Carnahan Extended monstrous moonshine
Monstrous MoonshineAdditional Moonshine PhenomenaGeneralized Monstrous Moonshine
Modular MoonshineUnification?
Generalized Monstrous Moonshine
Scott Carnahan Extended monstrous moonshine
Monstrous MoonshineAdditional Moonshine PhenomenaGeneralized Monstrous Moonshine
Modular MoonshineUnification?
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.
Scott Carnahan Extended monstrous moonshine
Monstrous MoonshineAdditional Moonshine PhenomenaGeneralized Monstrous Moonshine
Modular MoonshineUnification?
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.
Scott Carnahan Extended monstrous moonshine
Monstrous MoonshineAdditional Moonshine PhenomenaGeneralized Monstrous Moonshine
Modular MoonshineUnification?
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.
Scott Carnahan Extended monstrous moonshine
Monstrous MoonshineAdditional Moonshine PhenomenaGeneralized Monstrous Moonshine
Modular MoonshineUnification?
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.
Scott Carnahan Extended monstrous moonshine
Monstrous MoonshineAdditional Moonshine PhenomenaGeneralized Monstrous Moonshine
Modular MoonshineUnification?
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).
Scott Carnahan Extended monstrous moonshine
Monstrous MoonshineAdditional Moonshine PhenomenaGeneralized Monstrous Moonshine
Modular MoonshineUnification?
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).
Scott Carnahan Extended monstrous moonshine
Monstrous MoonshineAdditional Moonshine PhenomenaGeneralized Monstrous Moonshine
Modular MoonshineUnification?
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
Scott Carnahan Extended monstrous moonshine
Monstrous MoonshineAdditional Moonshine PhenomenaGeneralized Monstrous Moonshine
Modular MoonshineUnification?
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).
Scott Carnahan Extended monstrous moonshine
Monstrous MoonshineAdditional Moonshine PhenomenaGeneralized Monstrous Moonshine
Modular MoonshineUnification?
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.
Scott Carnahan Extended monstrous moonshine
Monstrous MoonshineAdditional Moonshine PhenomenaGeneralized Monstrous Moonshine
Modular MoonshineUnification?
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.
Scott Carnahan Extended monstrous moonshine
Monstrous MoonshineAdditional Moonshine PhenomenaGeneralized Monstrous Moonshine
Modular MoonshineUnification?
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).
Scott Carnahan Extended monstrous moonshine
Monstrous MoonshineAdditional Moonshine PhenomenaGeneralized Monstrous Moonshine
Modular MoonshineUnification?
Modular Moonshine
Scott Carnahan Extended monstrous moonshine
Monstrous MoonshineAdditional Moonshine PhenomenaGeneralized Monstrous Moonshine
Modular MoonshineUnification?
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).
Scott Carnahan Extended monstrous moonshine
Monstrous MoonshineAdditional Moonshine PhenomenaGeneralized Monstrous Moonshine
Modular MoonshineUnification?
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.
Scott Carnahan Extended monstrous moonshine
Monstrous MoonshineAdditional Moonshine PhenomenaGeneralized Monstrous Moonshine
Modular MoonshineUnification?
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.
Scott Carnahan Extended monstrous moonshine
Monstrous MoonshineAdditional Moonshine PhenomenaGeneralized Monstrous Moonshine
Modular MoonshineUnification?
Unification?
Scott Carnahan Extended monstrous moonshine
Monstrous MoonshineAdditional Moonshine PhenomenaGeneralized Monstrous Moonshine
Modular MoonshineUnification?
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?
Scott Carnahan Extended monstrous moonshine
Monstrous MoonshineAdditional Moonshine PhenomenaGeneralized Monstrous Moonshine
Modular MoonshineUnification?
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.
Scott Carnahan Extended monstrous moonshine
Monstrous MoonshineAdditional Moonshine PhenomenaGeneralized Monstrous Moonshine
Modular MoonshineUnification?
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
Scott Carnahan Extended monstrous moonshine
Monstrous MoonshineAdditional Moonshine PhenomenaGeneralized Monstrous Moonshine
Modular MoonshineUnification?
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)
Scott Carnahan Extended monstrous moonshine
Monstrous MoonshineAdditional Moonshine PhenomenaGeneralized Monstrous Moonshine
Modular MoonshineUnification?
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.
Scott Carnahan Extended monstrous moonshine