Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

Recent advances in moonshine
Scott Carnahan
Department of Mathematics
University of Tsukuba

2017-3-25
JMS Special Lecture

Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

What is moonshine?
Strange connections between finite groups andmodular forms

Symmetry

Moonshine

Finite groups

Complex analysis/
Number theory
Modular forms

Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

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

Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

Monstrous Moonshine (1978-1992)

Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

Classification of finite simple groups (1982-2004)
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, about 8 · 1053 elements(Griess 1982).
194 irred. repres. of dim 1, 196883, 21296876, . . .
Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

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

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

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

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

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

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

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

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

First proof (Atkin, Fong, Smith 1980)
Theorem: A virtual representation of M exists
yielding the desired functions.
No construction.
Second proof (Borcherds 1992)
Theorem: The Conway-Norton conjecture holds for
V \.

Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

Outline of Borcherds’s proof
FLM construction: V \Automorph. ∞ prod.

Add torus and quantize

gens. and rels.

Lie algebra m

Isom.
m∼
=L

Lie algebra L

Twisted Denominator Identities
Recursion relations
Hauptmoduln
Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

Add a torus
Functor: tensor with lattice VA V \ 7→ V \ ⊗ VII1,1 .
Central charge increases by 2 (from 24 to 26).
Quantize (need central charge 26 = critical dim)Old canonical quantization: Primary mod spurious.
1
Equivalent functor: HBRST.
Get Lie algebra m with monster action.
Oscillator cancellation (no-ghost theorem)\
mm,n ∼
when (m, n) 6= (0, 0).
=V
1+mn

Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

Infinite product identity (Koike-Norton-Zagier)Y−1J(σ) − J(τ ) = p(1 − p m q n )c(mn)m>0,n∈Z

where J(τ ) =n2πiσ, q = e 2πiτ .
n≥−1 c(n)q , p = ePRemarkable property
Left side is pure in p and q.

Vanishing of pq 2 term ⇒ c(4) = c(3) + c(1)
2 .
\ ∼
\
\
2
Get isom V5 = V4 ⊕ Λ (V2 ) of monster reps.
Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

End of Borcherds’s proof
All M-reps Vn\ are determined by (Vn\ )6n=0 .
Same for coefficientsof McKay-Thompson
P
series Tg (τ ) = n≥0 Tr(g |Vn )q n−1 .
Theorem (Koike): Conway-Norton’s candidate
functions satisfy the same recursion relations.
suffices to check first 7 terms.
Theorem (Cummins, Gannon 1997)
The recursion relations alone are sufficient to getΓ0 (N)-invariant Hauptmodul property.
Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

Generalized Monstrous Moonshine(1987-2016)

Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

Moonshine away from the monster?
Suggested by Conway-Norton 1979.
Computations by Queen 1980.
Example: Baby monster irreps 1, 4371, 96255, . . .
q −1 + 4372q + 96256q 2 + · · · is Hauptmodul forΓ0 (2)+ .
Strange observation (Norton)
Only groups “inside” the monster are interesting.
(central extensions of centralizers of elements)
Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

The 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

2

3
4
5

q-expansion of Z (g , h; τ ) is graded trace of (alift of) h on V (g ).
Z is invariant under simultaneous conjugationof the pair (g , h) up to scalars.
Z (g , h; τ ) constant or a Hauptmodul.
aτ +b
a c
b d
Z (g , h; cτ+d ) proportional to Z (g h , g h ; τ ).
Z (g , h; τ ) = J(τ ) if and only if g = h = 1.
Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

Brute force solution (like Atkin-Fong-Smith)?
This is a finite problem:
Finitely many conjugacy classes of commuting
pairs, and possible levels are bounded.
Central extensions of centralizers “can becomputed”.
Not finite enough for 2017
We still haven’t classified the commuting pairs.
We still don’t know character tables of all
centralizers, let alone central extensions.
Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

Physics Language (Dixon, Ginsparg, Harvey 1988)
V (g ) - twisted sectors of a monster CFT.
Z (g , h; τ ) - genus 1 partition functions (withtwisted boundary conditions).
All except Hauptmodul claim (3) “follow” from
conformal field theory considerations.
Algebraic Interpretation
V (g ) = irreducible g -twisted V \ -module V \ (g )Z (g , h; τ ) = Tr(h̃q L(0)−1 |V (g )).
Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

Geometric interpretation of Z
Physicists draw boundary conditions as colorings.
hgghCommuting pair (g , h) describes hom π1 (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 elliptic
curves with principal M-bundles.
Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

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

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

g -rationality is really a corollary
Main theorem of [C-Miyamoto] is: If V is strongly
regular, then so is the fixed-point subVOA V g .
Here, “strongly regular” means roughly “module
category is a modular tensor category”.
This gives modular functions for traces ofautomorphisms of VOAs in infinitely many cases(therefore not really moonshine).

Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

Main steps of proof
1
V a C2 -cofinite VOA, CFT type, σ finite orderaut, ⇒ V σ is C2 -cofinite (Miyamoto 2013)2
If V is also regular, then V σ is a projective
V σ -module (uses Huang-Lepowsky-Zhang
2007-2011).
3
Any irreducible V σ -module W is rigid, i.e., getisom. W  V σ → W  (W ∨  W ) →(W  W ∨ )  W → V σ  W (uses Huang’s
genus 1 fcns + Verlinde + Miyamoto’s
pseudo-trace).
Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

On to claim (3)We now need to show that all Z (g , h; τ ) are
Hauptmoduln or constant.
Second Breakthrough (Höhn 2003)
Generalized Moonshine for 2A (Baby monster case).
- Gives outline for proving Hauptmodul claim (3).

Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

Borcherds-Höhn program for Hauptmoduln
Ab. intertw. alg. gN V \Automorph. ∞ prod.

Add torus and quantize

gens. and rels.

Lie algebra mg

Isom.
mg ∼
= Lg

Lie algebra Lg

Twisted Denominator Identities
Recursion relations
Hauptmoduln
Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

Right side (C 2009)
Borcherds products of the form:Y−1
Tg (σ) − Tg (−1/τ ) = p

g(1 − p m q n )cm,n (mn)m>0,n∈ N1 Z
g
- Exponent cm,n(mn) is q mn -coefficient of a v.v.
N−1modular function formed from {Tg i (τ )}i=0.
1- Lg is a Z ⊕ N Z-graded BKM Lie algebra.
g- Simple roots of multiplicity c1,n(n) in degree (1, n).

Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

Third Breakthrough (van Ekeren, Möller,
Scheithauer 2015)- There exists an abelian intertwining algebra
structure on|g |−1
g

\

V :=

M

V \ (g i )i=0- Dimensions of eigenspaces match coefficientsgcm,n(k) of v.v. modular function.
Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

Add a torus and quantize- Take a graded tensor product with a lattice
abelian 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 a
BKM Lie algebra mg with real simple root.
- graded by II1,1 (−1/N) ∼= Z ⊕ N1 Z.

Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

Comparison
Borcherds-Kac-Moody Lie algebras:- mg has canonical projective action of CM (g ).
- Lg has “nice shape”: known simple roots, good
homology.
Isomorphism from matching root multiplicities:gdim(Lg )m,n = (mg )m,n = cm,n(mn).
Transport de structure ⇒ Lg gets C^
M (g ) action.

Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

End of proof (C 2016)∼ V∗ Eg
Virtual C^
M (g )-module isom H∗ (Eg , C) =implies equivariant HeckenT̂n given by
P operatorsd−b a aτ +bnT̂n Z (g , h, τ ) =Z (g , g h , d )ad=n,0≤b<dact by monic polynomials on Z (g , h, τ ).
Hauptmodul condition follows (C 2008).
Constants come from (g , h) such that all g a hc
are non-Fricke when (a, c) = 1, using claim (4).
This resolves the final claim (3).
Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

Stronger version of conjecture?
Folklore: constant ambiguities are precisely
controlled by a “Moonshine element”
γ \ ∈ H 3 (M, C× ).
H 3 (M, C× ) not known to be nontrivial.
G. Mason says |γ \ | ∈ 24Z if γ \ exists.
Existence of canonical element γ \ follows from
non-abelian twisted fusion (in progress).
M is enhanced to “categorical group” M̃.
Z naturally lives on space MM̃
1,1Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

Connections to elliptic cohomology and tmf?
Segal and Stolz-Teichner: interpretation of tmf
in terms of CFTs (hence VOAs).
Dependence on commuting pairs looks like
Hopkins-Kuhn-Ravenel ”higher character”
theory at height 2.
Claims (1), (2), (4) suggest V \ ∈ tmf (B M̃)Equivariant Hecke operators T̂n , used in
Hauptmodul proof, first appeared ascohomology operations for E``(BG ). Explicit
formula given in (Ganter 2007) .
Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

Rademacher sums and quantum
gravity (2009-)

Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

Rademacher’s sum, 1938
Try to make an SL2 (Z)-invariant function fromthe B(Z) = {± ( 10 n1 )}-invariant function q −1 .
P
Problem: the sum γ∈B(Z)\SL2 (Z) e(−γτ )diverges everywhere.
Regularize: subtract
at infinity. P constantsaτ +be(−τ ) + lim
e − cτ +d − e − ca
K →∞

0<c<K
−K 2 <d<K 2(c,d)=1

converges conditionally to J(τ ) + 12.
Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

Generalization by Duncan-Frenkel 2009
Allow large class of groups Γ in SL2 (R).
Allow arbitrary poles at distinguished cusp.
Arbitrary non-positive weight.
For weight 0, adjustments to constant terms.
Connection to Hauptmodul
Weight 0 sum is Γ-Hauptmodul ⇔ Γ is genus 0.
If not, modular function plus weight 2 cusp form.

Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

Rademacher sums are natural in quantum gravity
Cosets B(Z)\SL2 (Z) enumerate asymptotically
AdS3 spacetimes with torus boundary.
Non-trivial cosets correspond to BTZ blackhole solutions of Einstein’s equations.
This gives a semiclassical “sum over histories”
when computing quantum gravity partition
function.
(Dijkgraaf-Maldacena-Moore-Verlinde: ”A blackhole Farey tale”, Manschot-Moore: ”A modern
Farey tale”)
Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

Moonshine-gravity proposal (Duncan, Frenkel 2009)
Generalized moonshine is connected to“second-quantized twisted chiral gravity”
through AdS/CFT.
Denominator formulas for Monstrous Lie
algebras mg come from totalized Rademacher
sums, which also describe gravity Fock spaces.
Warning: Quantization of 2+1 dimensional gravity
is still far from rigorous.

Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

K3 Mathieu Moonshine (2010-)

Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

K3 Mathieu moonshine
An experimental mathematical observation
motivated by physics.
K3 surfaces
A K3 surface is a compact complex surface that is
simply connected and has trivial holomorphic
canonical class.
ExamplesFermat quartic: V (x 4 + y 4 + z 4 + w 4 ) ⊂ P3C
Kummer: Blow up orbifold points in (C2 /Λ)/{±1}.
Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

Theorem (Kodaira 1964)
The underlying smooth 4-manifolds of any two K3
surfaces are diffeomorphic.
Moduli space of complex structures
The moduli space of K3 surfaces is a connectedcomplex 20-manifold. Algebrizable part is
19-dimensional.

Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

Elliptic genus (Landweber-Stong, Ochanine 1980s)
Homomorphism ΩSO → M(Γ0 (2)).
{closed oriented mfds} → {modular forms}
Enhancements by Witten, Hirzebruch,
Krichever.
2-variable Elliptic genus
M a complex d-manifold. DefineEll(M)L∈ y d/2 Z[y , y −1 ][[q]] as holom. Euler char. ofy −d/2 (Λ−yqn−1 T̄M ⊗ Λ−y −1 qn TM ⊗ Sqn T̄M ⊗ Sqn TM )n≥1
Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

Theorem (Borisov, Libgober 1999)
If M is Calabi-Yau, then Ell(M) is a Jacobi form of
weight 0 and index d/2. In particular, Ell(K 3) has
index 1.
UniquenessThe space of Jacobi forms of weight 0 and index 1
is one-dimensional, spanned by Ell(K 3) = 2φ0,1 .
2y + 20 + 2y −1 + q(20y 2 − 128y + 216 − 128y −1 +20y −2 ) + O(q 2 )Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

Superconformal elliptic genus (Witten)
For a representation H of N = 2 superconformal
algebra, one defines the elliptic genus asEll(H) = TrHRR (q L0 −c/24 y J0 (−1)F q̄ L̄0 −c̄/24 (−1)F̄ )Physics conjecture (Witten)
Given a sigma model CFT with target Calabi-Yau X
and Hilbert space H, Ell(H) = Ell(X ).

Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

Enhanced supersymmetry for K3
K3 surfaces have hyperKähler structure, so their
CFTs have action of N = 4 superconformal algebra.
Natural question
Decompose Ell(K 3) into elliptic genera for
irreducible N = 4 representations?
- the genera are linearly independent

Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

Eguchi, Ooguri, Tachikawa 2010
Decomposition into N = 4 characters:X
Ell(K 3)(τ, z) = 20χ1/4,0 −2χ1/4,1/2 +An χ1/4+n,1/2
n≥1where A1 = 2 × 45, A2 = 2 × 231, A3 = 2 × 770.
Surprising observation
The numbers 45, 231, 770 are dimensions ofirreducible reps of the sporadic group M24
Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

Theorem (Gannon 2012) - like Atkin-Fong-Smith
There is a N = 4-representation with faithfulcommuting action of M24 , whose elliptic genus isEll(K 3), such that taking traces of elements of M24
yields Jacobi forms of small level.
Additional suggestive evidenceP
Setting An = dim Hn , the seriesP An q n is amock modular form, and so isTr(g |Hn )q n .
Analogue of Hauptmodul property (Cheng,
Duncan 2012): The trace forms are weight 1/2
Rademacher sums.
Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

Big mystery: where do we get M24 symmetry?
No M24 -symmetry on K3 surfaces (Mukai 1988,
Kondo 1998). Only get subgroups of M23 with
≥ 5 orbits.
No M24 -symmetry of K3 CFTs (Gaberdiel,
Hohenegger, Volpato 2011). Moduli space isAut(II4,20 )\O4,20 (R)/(O4 (R) × O20 (R)).
Stabilizers fix 4-dim subspace - naturally live in
Co1 , but too small.

Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

How much structure do we need?
N = 4 rep.

???

N = 4 superCFT

less structure

more structure

more symmetry

less symmetry

∞-dim

M24

Scott Carnahan

small groups

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

Holomorphic vertex operator superalgebras?
Generalized Mathieu Moonshine (Gaberdiel,
Persson, Ronellenfitsch, Volpato 2012)suggests good orbifold behavior.
Chiral de Rham constructions proposed, but
few computations.
Conway moonshine module V s\ (Duncan,
Mack-Crane 2015) may be manipulated to
produce some K3-like characters.

Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

Symmetry surfing (Taormina, Wendland 2013)
Moduli space of K3 CFTs is 80-dimensional, with
scattered symmetries. Thus, try gluing symmetries
from different points.
Works well for Kummer surfaces - get max
subgroup (Z/2Z)4 o A8 ⊂ M24
Recent progress on connections with V s\

Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

Newer Moonshines (2012-, 2014-,
2017-)

Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

Umbral moonshine (Cheng, Duncan, Harvey 2012)For each Niemeier (even unimod. pos. def. rank24) lattice N, get:the umbral group G N = Aut N/Weyl(N)graded representations K N of G N , such that
graded traces are vector valued mock modular
forms (vector rep. tied to Coxeter # of N)
shadows are specific theta functions.
K3 Mathieu moonshine is the case N = A24
1 .
Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

Theorem (Duncan, Griffin, Ono 2015)
Umbral moonshine modules exist.
Only N = E83 case has a known construction.
Many umbral functions come from V s\ .
Connections to physics and geometry are still
speculative, and the subject of active research.
Example (Cheng, Harrison 2014): Niemeier
lattices ↔ duVal degenerations of marked K3.
Ell(K 3) = sum of umbral genus and singular
local genus.
Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

Thompson moonshine observation (Piezas 2014)
Coefficients of the weight 1/2 modular form
f3 = q −3 − 248q + 26752q 4 − · · · “come from”
sporadic group Th.
Partial (Generalized Monstrous) explanationp
For g in class 3C, Z (g , 1; τ ) = 3 j(τ /3)= q −1/9 + 248q 2/9 + 4124q 5/9 + · · · , andCM (g ) = Z/3Z × Th. Coeffs
give reps of Th, andp
3chars are Hauptmoduln. j(τ /3) ∼ theta lift of f3 .
Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

Problem:This only explains Th representations for2
coefficients of q n in f3 .
Refined observation (Harvey, Rayhoun 2015)
There is a 21 Z-graded Th-module whose gradedsuper-dimension is the weight 1/2 form 2f3 + 248θ.
Graded traces are also “nice” weight 1/2 forms.
Theorem (Griffin, Mertens 2016)
A Thompson moonshine module exists.
No construction or natural explanation.
Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

Skew-holomorphic moonshine (Duncan, Harvey,
Rayhoun ≥ 2017)
Thompson moonshine appears to be the level 1case of a more general phenomenon involving
weight 1/2 forms that lift to Hauptmoduln for
Fricke-containing genus zero groups.
Calculations are still underway.
Physics is still quite unclear.

Scott Carnahan

Recent advances in moonshine

Monstrous Moonshine (1978-1992)
Generalized Monstrous Moonshine (1987-2016)
Rademacher sums and quantum gravity (2009-)
K3 Mathieu Moonshine (2010-)
Newer Moonshines (2012-, 2014-, 2017-)

Summary
Generalized Monstrous Moonshine: Controlled
by the vertex operator algebra V \ . Hauptmodul
property comes from string quantization andpossibly 3d quantum gravity.
Mathieu and umbral moonshine: possibly
controlled by V s\ and K3 surfaces.
Thompson and skew-holomorphic moonshine:
unknown.
Thank you.
Scott Carnahan

Recent advances in moonshine