Monstrous Moonshine (1978-1992)
More monstrous moonshines
Cyclic orbifolds over small rings
Monster symmetry
Gluing forms over small rings
Further questions

Monstrous Moonshine over the
Integers
Scott Carnahan
Department of Mathematics
University of Tsukuba

2019-9-2
The 64th Algebra Symposium
Scott Carnahan

Monstrous Moonshine over the Integers

Monstrous Moonshine (1978-1992)
More monstrous moonshines
Cyclic orbifolds over small rings
Monster symmetry
Gluing forms over small rings
Further questions

What is moonshine?
Strange connections between finite groups andmodular forms

Symmetry

Moonshine

Finite groups

Complex analysis/
Number theory
Modular forms

Scott Carnahan

Monstrous Moonshine over the Integers

Monstrous Moonshine (1978-1992)
More monstrous moonshines
Cyclic orbifolds over small rings
Monster symmetry
Gluing forms over small rings
Further questions

What is moonshine?
Strange connections between finite groups andmodular forms
The connections should be “very special”
Infinitely many cases ⇒ not moonshine!

Scott Carnahan

Monstrous Moonshine over the Integers

Monstrous Moonshine (1978-1992)
More monstrous moonshines
Cyclic orbifolds over small rings
Monster symmetry
Gluing forms over small rings
Further questions

Monstrous Moonshine (1978-1992)

Scott Carnahan

Monstrous Moonshine over the Integers

Monstrous Moonshine (1978-1992)
More monstrous moonshines
Cyclic orbifolds over small rings
Monster symmetry
Gluing forms over small rings
Further questions

Classification of finite simple groups (1982 or 2004)
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

Monstrous Moonshine over the Integers

Monstrous Moonshine (1978-1992)
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Cyclic orbifolds over small rings
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SL2 (Z) action on complex upper half-plane H
Generators: ( 10 11 ) : z 7→ z + 1, ( 01 −1
0 ) : z 7→ −1/z

(Wikipedia)

J-function as Hauptmodul
The quotient space SL2 (Z)\H has genus zero. J
generates the function field. Fourier expansion:q −1 + 196884q + 21493760q 2 + · · · (q = e 2πiz )Scott Carnahan

Monstrous Moonshine over the Integers

Monstrous Moonshine (1978-1992)
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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:graded rep
L∞
P Naturaln−1
dim Vn q= J.
n=0 Vn of M such that
Scott Carnahan

Monstrous Moonshine over the Integers

Monstrous Moonshine (1978-1992)
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Cyclic orbifolds over small rings
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Further questions

Idea: Physics forms a bridge

formal field theoryConerator algeb
Monstertex opras)re(V

J function

Solution: Frenkel, Lepowsky, Meurman 1988
Constructeda vertex operator algebraL\
V = n≥0 Vn\ (the Moonshine Module), such thatP
\ n−1
= J and Aut V \ = M.
n≥0 (dim Vn )q
Scott Carnahan

Monstrous Moonshine over the Integers

Monstrous Moonshine (1978-1992)
More monstrous moonshines
Cyclic orbifolds over small rings
Monster symmetry
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Further questions

Refined correspondence
Thompson’s suggestion: replace graded dimension
with graded trace of non-identity elements.
Monstrous Moonshine Conjecture (Conway, Norton
1979)
ThereLis a faithful graded representation
V = n≥0 Vn of the monster M such that for allPg ∈ M, the series Tg (τ ) = n≥0 Tr(g |Vn )q n−1 isthe q-expansion of a congruence Hauptmodul (=“generates function field of genus 0 H-quotient”).
Scott Carnahan

Monstrous Moonshine over the Integers

Monstrous Moonshine (1978-1992)
More monstrous moonshines
Cyclic orbifolds over small rings
Monster symmetry
Gluing forms over small rings
Further questionsFirst proof (Atkin, Fong, Smith 1980)
Theorem: A virtual representation of M exists
yielding the trace functions Tg .
No construction.
Second proof (Borcherds 1992)
Theorem: The Conway-Norton conjecture holds for
V \.

Scott Carnahan

Monstrous Moonshine over the Integers

Monstrous Moonshine (1978-1992)
More monstrous moonshines
Cyclic orbifolds over small rings
Monster symmetry
Gluing forms over small rings
Further questions

More monstrous moonshines

Scott Carnahan

Monstrous Moonshine over the Integers

Monstrous Moonshine (1978-1992)
More monstrous moonshines
Cyclic orbifolds over small rings
Monster symmetry
Gluing forms over small rings
Further questions

Moonshine for other groups?
Conway-Norton 1979, computations by Queen 1980.
Example: Baby monster B has irreps of dim
1, 4371, 96255, . . ., and the Hauptmodul for Γ0 (2)+is q −1 + 4372q + 96256q 2 + · · · .
Main observation
If g has prime order p and g is in conjugacy classpA ⊂ M, then Tg has positive integer coefficientsthat “look like” representations of CM (g ).
For p = 2, CM (g ) ∼= 2.B, a central extension ofBaby monster.
Scott Carnahan

Monstrous Moonshine over the Integers

Monstrous Moonshine (1978-1992)
More monstrous moonshines
Cyclic orbifolds over small rings
Monster symmetry
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Further questions

Conj 1: Generalized Moonshine (Norton 1987)
For each g ∈ M exists V (g ) graded proj. rep. ofCM (g ). Trace functions Z (g , h; τ ) for commuting
pairs satisfy modularity properties.
Conj 2: Modular Moonshine (Ryba 1994)
For each g in class pA, there is a vertex algebra Vg
over Fp with CM (g ) action. For each p-regularh ∈ CM (g ), the graded Brauer character of h on Vg
is equal to Tgh .
Scott Carnahan

Monstrous Moonshine over the Integers

Monstrous Moonshine (1978-1992)
More monstrous moonshines
Cyclic orbifolds over small rings
Monster symmetry
Gluing forms over small rings
Further questions

Interpretation of Generalized Moonshine
V (g ) - twisted sectors of a monster CFT.
Z (g , h; τ ) - genus 1 partition functions (withtwisted boundary conditions).
(Dixon, Ginsparg, Harvey 1988)
Interpretation of Modular Moonshine
Vg = Ĥ 0 (g , VZ\ ) - Tate cohomology of VZ\ , aself-dual integral form of V \ with M symmetry.
(Borcherds, Ryba 1996)
Scott Carnahan

Monstrous Moonshine over the Integers

Monstrous Moonshine (1978-1992)
More monstrous moonshines
Cyclic orbifolds over small rings
Monster symmetry
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Further questionsFirst advances (1990s)
Generalized moonshine: Existence and uniqueness(up to isom.) of V \ (g ) (Dong, Li, Mason 1997).
Modular moonshine: Good properties, assuming
existence of VZ\ (Borcherds, Ryba 1996, 1998).
Later advances (2010s)
Generalized: Good properties of Z (g , h; τ ).
Modular: Existence of VZ\ .
Scott Carnahan

Monstrous Moonshine over the Integers

Monstrous Moonshine (1978-1992)
More monstrous moonshines
Cyclic orbifolds over small rings
Monster symmetry
Gluing forms over small rings
Further questions

Main breakthrough for both moonshines- If V is strongly regular and holomorphic, andg ∈ Aut(V ) is finite order, then there exists anabelian intertwining algebra structure on the direct
sum of irreducible twisted modules|g |−1g

V :=M

V (g i )

i=0(van Ekeren, Möller, Scheithauer 2015)
Scott Carnahan

Monstrous Moonshine over the Integers

Monstrous Moonshine (1978-1992)
More monstrous moonshines
Cyclic orbifolds over small rings
Monster symmetry
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Further questions

Corollary (Cyclic orbifold construction)
Let V be a strongly regular and holomorphic vertex
operator algebra, and g ∈ Aut(V ) finite order.
Assume g is “anomaly-free” (i.e., eigenvalues ofL(0) on V (g ) are in |g1 | Z).
L|g |−1iDecompose g V:=
L i,j i=0 V (g ) under
L 0,jcanonical gaction toL
get
V , where V =V . Theni,0
V /g :=
V is a strongly regular holomorphic
vertex operator algebra, and there is a canonical√automorphism g ∗ such that V i,0 is the e 2π −1i/|g |eigenspace.
Scott Carnahan

Monstrous Moonshine over the Integers

Monstrous Moonshine (1978-1992)
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Cyclic orbifolds over small rings
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Cyclic orbifold constructions of V \ from Leechlattice vertex operator algebra VΛ
1Order 2 orbifold (Frenkel, Lepowsky, Meurman
1988)
2
Order 3 orbifold (Chen, Lam, Shimakura 2016)3
Orders 5, 7, 13 (Abe, Lam, Yamada 2017)4
46 classes of composite order (C 2017)- confirms Tuite’s orbifold correspondence (1992):
Massless classes in Co0 ↔ non-Fricke classes in M.
For p ∈ {2, 3, 5, 7, 13}, (VΛ , pa) ↔ (V \ , pB).
Scott Carnahan

Monstrous Moonshine over the Integers

Monstrous Moonshine (1978-1992)
More monstrous moonshines
Cyclic orbifolds over small rings
Monster symmetry
Gluing forms over small rings
Further questions

Cyclic orbifolds over small rings

Scott Carnahan

Monstrous Moonshine over the Integers

Monstrous Moonshine (1978-1992)
More monstrous moonshines
Cyclic orbifolds over small rings
Monster symmetry
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Vertex algebras
A vertex algebra over a commutative ring R is an
R-module V , with an element 1 ∈ V and amultiplication map V P⊗R V → V ((z)), writtenu ⊗ v 7→ Y (u, z)v = un vz −n−1 , satisfying:1
Y (1, z) = idV z 0 and Y (a, z)1 ∈ a + zV [[z]].
2For
P anyr r , s, t ∈ Z, and any u, v , w ∈ V ,(u v )r +s−i w =Pi≥0 i i t+iti≥0 (−1) i (ur +t−i (vs+i w ) −(−1)t vs+t−i (ur +i w ))Scott Carnahan

Monstrous Moonshine over the Integers

Monstrous Moonshine (1978-1992)
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Example: Lattice vertex algebras over Z
For any positive definite even unimodular lattice L
there is a self-dual vertex algebra (VL )Z over Z(Borcherds 1986). It is a Z-form ofSym(t −1 (C ⊗ L)[t −1 ]) ⊗ C[L] spanned by
sα1 ,n1 · · · sαk ,nk e α , where e α is a basis element of
C[L], αi are chosen from a basis of L, and theoperator sα,k is the z k -coefficient of
P
n
−1
−1
exp( n>0 α(−n)
n z ). Here, Sym(t (C ⊗ L)[t ]) isa representation of the Heisenberg algebra, withgenerators α(n) = αt −n ∈ L[t, t −1 ] ⊕ CK .
Scott Carnahan

Monstrous Moonshine over the Integers

Monstrous Moonshine (1978-1992)
More monstrous moonshines
Cyclic orbifolds over small rings
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Vertex operator algebras
A vertex operator algebra over R with half central
charge c is a vertex algebra V over R equipped witha “conformal
element” ω and a Z-grading
LV =Vn , such that1
If u ∈ Vm , v ∈ Vn , then uk v ∈ Vm+n−k−1 .
P
2
The coefficients of Y (ω, z) = Ln z −n−2satisfy Virasoro relations:[Lm , Ln ] = (m − n)Lm+n + c m+1
3 δm+n,0 id.
3Each Vn is a finite rank projective R-module,and L0 acts on Vn by n · id.
Scott Carnahan

Monstrous Moonshine over the Integers

Monstrous Moonshine (1978-1992)
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Abelian intertwining algebras over subrings of C
An abelian intertwining algebra is a “braided
commutative” generalization of vertex operator
algebra, graded by an abelian group A with
Eilenberg-MacLane abelian 3-cocycle (F , Ω). Thesecan be defined over any subring R of C that×
contains not only all values of F : A×3√ → C and
Ω : A×2 → C× , but also 1/N and e π −1/N , whereΩ(a, a)N = 1 for all a ∈ A.

Scott Carnahan

Monstrous Moonshine over the Integers

Monstrous Moonshine (1978-1992)
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Cyclic orbifolds over small rings
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Key lemmaL
Let V = i,j∈Z/NZ V i,j be a self-dual abelianintertwining algebra over C, where each
V i,j is anLirreducible
V 0,0 -module, and let U =V 0,j andL i,0W =V . If R is a suitable subring of C, andwe are given self-dual R-forms UR and WR suchthat UR ∩ V 0,0 = WR ∩ V 0,0 , then they generate aself-dual R-form of V .

Scott Carnahan

Monstrous Moonshine over the Integers

Monstrous Moonshine (1978-1992)
More monstrous moonshines
Cyclic orbifolds over small rings
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Intermediate orbifolds (after Abe, Lam, Yamada)
Let P0 = {2, 3, 5, 7, 13}. If p, q are distinct in P0 ,and pq 6∈ {65, 91}, then there is an automorphismḡ of the Leech lattice of order pq, such that no
non-identity power of ḡ has fixed points, and anorder pq lift g ∈ Aut(VΛ ). Then:
1
VΛ /g p ∼= V\
= VΛ /g q ∼2VΛ /g ∼
= VΛ .
\
In particular, there are 2 copies
L of V i inside theabelian intertwining algebra i VΛ (g ), which is
generated by 2 copies of VΛ .
Scott Carnahan

Monstrous Moonshine over the Integers

Monstrous Moonshine (1978-1992)
More monstrous moonshines
Cyclic orbifolds over small rings
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Corollary
Let p, q be distinct elements of P0 = {2, 3, 5, 7, 13},such that pq 6∈ {65,91}, and let√
π −1/pq]. Then, there is a self-dual
Rpq = Z[1/pq, e
Rpq -form of the abelian intertwining algebraLii VΛ (g ), and it contains 2 isomorphic self-dual
Rpq -forms of V \ .

Scott Carnahan

Monstrous Moonshine over the Integers

Monstrous Moonshine (1978-1992)
More monstrous moonshines
Cyclic orbifolds over small rings
Monster symmetry
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Monster symmetry

Scott Carnahan

Monstrous Moonshine over the Integers

Monstrous Moonshine (1978-1992)
More monstrous moonshines
Cyclic orbifolds over small rings
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Further questions

Symmetries of VΛ
The Leech lattice Λ has Co0 = 2.Co1 symmetry.
Aut VΛ ∼= (C× )24 .Co0 (non-split extension).
Let p ∈ P0 , ḡ ∈ Co0 fixed-point free, order p. Then
any order p lift g ∈ Aut VΛ has centralizer(Z/pZ)24/(p−1) .CCo0 (ḡ ). The same is true forsuitably chosen automorphisms√of the R-form, aslong as R contains 1/p and e π −1/p .

Scott Carnahan

Monstrous Moonshine over the Integers

Monstrous Moonshine (1978-1992)
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Cyclic orbifolds over small rings
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Symmetries of VR\
The self-dual Rpq -forms of V \ naturally inherit anaction of Gp = p 1+24/(p−1) .(CCo0 (g¯q )/g¯q ) from anabelian intertwining algebra containing VR\pq and(VΛ )Rpq (and similarly for Gq ).
Maximal subgroups (from Wilson 2015)
When p ∈ P0 , Gp contains the Sylow p-subgroup ofM, and when p ∈ {2, 3, 5}, Gp is contained in aunique maximal subgroup of M.
Scott Carnahan

Monstrous Moonshine over the Integers

Monstrous Moonshine (1978-1992)
More monstrous moonshines
Cyclic orbifolds over small rings
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Monster symmetry
If p, q are distinct elements of P0 such thatpq 6∈ {65, 91}, then Gp and Gq generate M. In
particular, the self-dual Rpq -forms of V \ have M
symmetry,

Scott Carnahan

Monstrous Moonshine over the Integers

Monstrous Moonshine (1978-1992)
More monstrous moonshines
Cyclic orbifolds over small rings
Monster symmetry
Gluing forms over small rings
Further questions

Gluing forms over small rings

Scott Carnahan

Monstrous Moonshine over the Integers

Monstrous Moonshine (1978-1992)
More monstrous moonshines
Cyclic orbifolds over small rings
Monster symmetry
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Further questions

Gluing data
Given a diagram R1 → R3 ← R2 of commutative
rings, a gluing datum for vertex operator algebras isa triple (V 1 , V 2 , f ), where1
V 1 is a vertex operator algebra over R1 ,2
V 2 is a vertex operator algebra over R2 , and3f : V 1 ⊗R1 R3 → V 2 ⊗R2 R3 is an isomorphismof vertex operator algebras over R3 .
These form a category, where morphisms are pairsof maps satisfying a commutative square condition.
Scott Carnahan

Monstrous Moonshine over the Integers

Monstrous Moonshine (1978-1992)
More monstrous moonshines
Cyclic orbifolds over small rings
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Effective gluing lemma
Let i1 : R → R1 and i2 : R → R2 be maps ofcommutative rings, such that either
1
i1 and i2 form a Zariski open cover, or2
i1 and i2 are faithfully flat.
Then, the category of gluing data for
R1 → R1 ⊗R R2 ← R2 is equivalent to the categoryof vertex operator algebras over R.

Scott Carnahan

Monstrous Moonshine over the Integers

Monstrous Moonshine (1978-1992)
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Cyclic orbifolds over small rings
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Comparison of fixed points√

Let Rn = Z[1/n, e π −1/n ] and let g ∈ pB. Recall(V \ , pB) is orbifold dual to (VΛ , pa), andg ∼Vpq= (VΛ )σRpq . Theng
Vpq
⊗Rpq Rpqr ∼= (VΛ )σRpqr ∼= Vprg ⊗Rpr Rpqr .
pB-pure elementary subgroups (Wilson 1988)
For each p ∈ P0 , there is an elementary subgroupHp ⊂ M of order p 2 , whose non-identity elements
lie in conjugacy class pB.
Scott Carnahan

Monstrous Moonshine over the Integers

Monstrous Moonshine (1978-1992)
More monstrous moonshines
Cyclic orbifolds over small rings
Monster symmetry
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Construction of gluing datum
Vpq and Vpr are generated by g -fixed point
subalgebras for g ranging over Hp , so
Vpq ⊗Rpq Rpqr ∼= Vpr ⊗Rpr Rpqr by uniqueness ofgenerated self-dual forms.
Sufficiency of gluing data
From our isomorphisms
Vpq ⊗Rpq Rpqr ∼= Vpr ⊗Rpr Rpqr , we may produce aself-dual Z-form with M-symmetry by repeatedgluing. Uniqueness comes from the fact that
M\M/M is a singleton.
Scott Carnahan

Monstrous Moonshine over the Integers

Monstrous Moonshine (1978-1992)
More monstrous moonshines
Cyclic orbifolds over small rings
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Main result
There is a unique self-dual Z-form VZ\ of V \ suchthat VZ\ ⊗ Rpq ∼= Vpq . This form has M-symmetry,
and the natural inner product is positive definite.
Corollary
Modular moonshine conjecture.
Corollary
There exists a positive definite unimodular lattice ofrank 196884 with a faithful monster action.
Scott Carnahan

Monstrous Moonshine over the Integers

Monstrous Moonshine (1978-1992)
More monstrous moonshines
Cyclic orbifolds over small rings
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Further questions

Further questions

Scott Carnahan

Monstrous Moonshine over the Integers

Monstrous Moonshine (1978-1992)
More monstrous moonshines
Cyclic orbifolds over small rings
Monster symmetry
Gluing forms over small rings
Further questions

Borcherds-Ryba proof of modular moonshineP\
1
Easy part: n T̃r(h|Ĥ ∗ (g , VZ,n))q n−1 =P\ n−1, where Ĥ ∗ is the virtualn Tr(gh|Vn )q
module Ĥ 0 Ĥ 1
2
Hard part: Ĥ 1 (g , VZ\ ) = 0 for g ∈ pA.
Can we extend this to composite order g ?
Conjecture (Borcherds-Ryba 1996)
Ĥ 1 (g , VZ\ ) = 0 for all Fricke classes g .
Scott Carnahan

Monstrous Moonshine over the Integers

Monstrous Moonshine (1978-1992)
More monstrous moonshines
Cyclic orbifolds over small rings
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Unifying Conjecture (Borcherds 1998)
For each g ∈ M, let Rg = Z[e 2πi/|g | ]. Then there isa free |g1 | Z-graded Rg -supermodule V̂g with anaction of Z/|g |Z.CM (g ), such that:
1V̂1 = VZ\ .
2
If h ∈ CM (g ) satisfies (|g |, |h|) = 1, thenV̂gh ⊗Rgh Z/|h|Z ∼= Ĥ ∗ (h̃, V̂g ) for a lift h̃ of h.
3
If g is Fricke, then V̂g ⊗R C ∼= V \ (g ).
g

4

If g is non-Fricke, then V̂g is a self-dual
conformal vertex superalgebra.
Scott Carnahan

Monstrous Moonshine over the Integers

Monstrous Moonshine (1978-1992)
More monstrous moonshines
Cyclic orbifolds over small rings
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Connections to other moonshines
Numerical evidence relating
non-Fricke classes in M ↔ Umbral moonshine
Fricke classes ↔ Skew-holomorphic moonshine
No concrete conjectures yet (as far as I know).

Scott Carnahan

Monstrous Moonshine over the Integers

Monstrous Moonshine (1978-1992)
More monstrous moonshines
Cyclic orbifolds over small rings
Monster symmetry
Gluing forms over small rings
Further questions

Thank you

Scott Carnahan

Monstrous Moonshine over the Integers