UmbralandPenumbral

Moonshine

Jeff Harvey
University of Chicago
Cheng, Duncan & JH 1204.2779, 1307.5793
Cheng, Duncan 1605.04480
Rayhaun, JH 1504.08179
Duncan, Rayhaun, JH to appear 18xx.yyyyy ?

Thanks toThanks to
Organizers for a very stimulating and well run meetingThanks to
Organizers for a very stimulating and well run meeting
Program Committee for inviting me to speakThanks to
Organizers for a very stimulating and well run meeting
Program Committee for inviting me to speak??? for arranging the moonshine talks to be within
hours of the full moon in Okinawa.

penumbra
Sun:.

.

Umbra.

Introduction
Modular forms play an important role in number theory andphysics because they count things:
1q 1!
2 M 12 States of open bosonic string=Q24n24⌘(⌧ )
q )
n (1X
⇡i⌧ p22 M4⇥E8 (⌧ ) =e
Points in E8 root latticep2ell (K3; ⌧, z)

82 J0,1

BPS states on K3xS1

When the things being counted live in vector spaces
which are representations of a finite group G we often say
informally that the modular form “exhibits moonshine forG.”

But Moonshine should be more: exceptional, special,sporadic, mysterious, finite in number. Ideas of whatmoonshine is and is not are evolving.

But Moonshine should be more: exceptional, special,sporadic, mysterious, finite in number. Ideas of whatmoonshine is and is not are evolving.
Since Moonshine involves many of the same forms andtechniques as CFT and BH counting we hope to learnsomething new about string theory.

But Moonshine should be more: exceptional, special,sporadic, mysterious, finite in number. Ideas of whatmoonshine is and is not are evolving.
Since Moonshine involves many of the same forms andtechniques as CFT and BH counting we hope to learnsomething new about string theory.
I will discuss two kinds of Moonshine with these properties:
1. Umbral: Started with EOT K3/M24 observation (2010).
Generalized to other Jacobi forms and groups classified
by Niemeier lattices (Cheng, Duncan, H).
2. Penumbral: Started with Thompson moonshine (H,Rayhaun), generalized to other skew-holo Jacobi forms(Duncan, H, Rayhaun, to appear).

Common element of Monstrous, Umbral and Penumbral
moonshine: Role of genus zero groups

A class of genus zero subgroups of SL(2,R):✓n + e, f, · · ·◆a b
2 SL2 (Z)c dc = 0 mod n

Atkin-Lehner

Fricke: n 2 e, f, · · ·⌧ ! 1/n⌧or non-Fricke otherwiseZ(2B, 1; ⌧ ) =0 (2)\H

⇣

⌘(⌧ )
⌘(2⌧ )⌘24+ 24

Genus zero groups:
Govern all twists and twines of Monstrous moonshine.
Classify cases of umbral and penumbral moonshine.

Genus zero groups:
Govern all twists and twines of Monstrous moonshine.
Classify cases of umbral and penumbral moonshine.
2⇡(T) ! M)
Given g, h 2 M gh = hg ( 1twine h⌧gtwist= Z(g, h; ⌧ ) = TrVg hq L0 c/24

Genus zero groups:
Govern all twists and twines of Monstrous moonshine.
Classify cases of umbral and penumbral moonshine.
2⇡(T) ! M)
Given g, h 2 M gh = hg ( 1twine h⌧g= Z(g, h; ⌧ ) = TrVg hq L0 c/24

twist

In (generalized) Monstrous Moonshine these are all genus
zero functions (Conway, Norton, Queen, Borcherds,Carnahan)
Ogg: for p prime p + p is genus zero precisely whenp divides the order of the Monster.

Umbral Moonshine

Umbral Moonshine

M24K3

Umbral Moonshine

UmbralMoonshine

M24
Niemeier
Lattices
K3(genus zero)Umbral Moonshine

optimal mock Jacobi theta
UmbralMoonshine

39 non-Fricke genus
M24 zero groups
Niemeier
Lattices
K3(genus zero)(Cheng&Duncan)

Penumbral MoonshinePenumbral MoonshineTh

Penumbral MoonshineTh
3.G2(3) L2(7)2.F 4(2)Th
Lifts, Gen.
Moonshine???

Penumbral Moonshine

skewThholomorphic
3.G2(3)L2(7)Jacobiforms
2.F 4(2)Th
Lifts, Gen.
84 Fricke
genus
Moonshine
zero???
groupsTwo Parallel Worlds of Weight 1/2 Moonshine
UmbralPenumbral

Modularobjects

Optimal weight 1/2,
mock Jacobi forms

Genus zerogroupsnon-Fricke

Optimal weight 1/2
modular skew-holo
Jacobi forms
FrickeMoonshine
GroupsM24, Aut(Niemeier)Groups of generalized
moonshine? Lattices?

Explicit
Constructions

For several, notuniform

None yetPhysics
Connections

K3 elliptic genus

BPS counting functionsat attractor points andLifts (S. Harrison)

The rest of the talk is devoted to brief answers to someof the following questions:

The rest of the talk is devoted to brief answers to someof the following questions:
What is a skew-holomorphic Jacobi form?

The rest of the talk is devoted to brief answers to someof the following questions:
What is a skew-holomorphic Jacobi form?
What are the Fricke and non-Fricke genus zero groups?

The rest of the talk is devoted to brief answers to someof the following questions:
What is a skew-holomorphic Jacobi form?
What are the Fricke and non-Fricke genus zero groups?
What role does genus zero play?

The rest of the talk is devoted to brief answers to someof the following questions:
What is a skew-holomorphic Jacobi form?
What are the Fricke and non-Fricke genus zero groups?
What role does genus zero play?
What are some examples of penumbral moonshine?

The rest of the talk is devoted to brief answers to someof the following questions:
What is a skew-holomorphic Jacobi form?
What are the Fricke and non-Fricke genus zero groups?
What role does genus zero play?
What are some examples of penumbral moonshine?
What does it all mean?

What is a skew-holomorphic Jacobi form?
Eichler-Zagier Jacobi form:(⌧, z) =X

hr (⌧ )✓m,r (⌧, z)

2 J1,m(✓m,r (⌧, z) =X

qn2 /4m ny )n=r mod 2mr mod 2m

Weight 1/2, these transform under a double
cover of the modular group “Weil representationof the metaplectic group”✓m,r ⇠ M1/2 (⇢m )Skew-holo Jacobi:
Jacobi:
Skew-holo Jacobi:sk
1,m 2 J1,mhr ⇠ M1/2 (⇢m )hr antiholomorphic in ⌧ .

hr ⇠ M1/2 (⇢m ) ' J1,msk
hr ⇠ M1/2 (⇢m ) ' J1,mExamples at m=1 ( f (⌧ ) = h0 (4⌧ ) + h1 (4⌧ ) )49f0 = ✓(⌧ ) = 1 + 2q + 2q + 2q + 2q
f3 = q 3248q + 26752q 4

1625+ O(q )85995q 5 + 1707264q 8 + O(q 9 )f4 = q 4 + 492q + 143376q 4 + 565760q 5 + 18473000q 8 + O(q 9 )f7 = q74119q + 8288256q

458952756480q + 5734772736q + O(q )...

(Borcherds, Zagier)Skew-holo Jacobi forms in strings/BPS counting will bediscussed by S. Harrison

What are the non-Fricke and Fricke genus 0 groups?
39 non-Fricke genus zero groups234
5
66+26+378
9

10
10+210+5
12
12+312+4
13
14+715+5
16

18
18+2
18+9
20+4
21+3
22+11
24+8
25
28+730+15

30+3,5,15
30+6,10,15
33+11
36+4
42+6,14,21
46+23
60+12,15,20
70+10,14,35
78+6,26,39

UmbralWhat are the non-Fricke and Fricke genus 0 groups?
39 non-Fricke genus zero groups
2345
66+26+3789

10
10+210+5
12
12+312+4
13
14+715+516

18
18+2
18+9
20+4
21+3
22+11
24+82528+7
30+15

Umbral30+3,5,15
30+6,10,15
33+11
36+4
42+6,14,21
46+23
60+12,15,20
70+10,14,35
78+6,26,39

84 Fricke genus zero groups
1
2+2
3+3
4+4
5+5
6+2,3,6
6+6
7+7
8+8
9+9
10+2,5,10
10+10
11+11
12+4,3,12
12+12
13+13
14+2,3,7
14+14
15+3,5,15
15+15
16+16

17+17
18+2,9,18
18+18
19+19
20+4,5,20
20+20
21+3,7,21
21+21
22+2,11,22
23+23
24+3,8,24
24+24
25+25
26+2,13,26
26+26
27+27
28+4,7,28
29+2930+2,3,5,6,10,15,30
30+2,15,30
30+5,6,30

31+31
32+32
33+33
34+34
35+5,7,35
35+35
36+4,9,36
36+36
38+38
39+3,13,39
39+39
41+41
42+2,3,7,6,14,21,42
42+3,14,42
44+4,11,44
45+5,9,45
46+2,23,46
47+47
49+49
50+2,25,50
50+50

51+3,17,51
54+2,27,54
55+5,11,55
56+7,8,58
59+59
60+3,4,5,12,15,20,60
60+4,15,60
62+2,31,62
66+2,3,11,6,22,33,66
66+6,11,66
69+3,23,69
70+2,5,7,10,14,35,70
71+71
78+2,3,13,6,26,39,78
87+3,29,87
92+4,23,92
94+2,47,94
95+5,19,95
105+3,5,7,15,21,35,105
110+2,5,11,10,22,55,110
119+7,17,119

What role does genus zero play?
j(⌧ ) = q123+ 196884q + 21493760q + 864299970q + · · ·1j(⌧)
q
Rademacher:can be obtained by averaging
over SL2 (Z) modulo its stabilizer, but one must
regularize. This can be generalized to other genus
zero hauptmoduls (Knopp, Duncan&Frenkel)

0f0 (⌧ ) = Reg @

2X

inv /

polarf0|01A holomorphic, notobviously modular

Extended to other weights and multiplier systems byKnopp, Niebur, Bringmann-Ono, Cheng-Duncan.

Farey Tales (Dijkgraaf, Maldacena, Moore,Verlinde,Manschot,…):An interpretation as a sum over asymptotic AdS32geometries with T boundary in context of BH
counting
Obstruction to modularity of Rademacher sums:
Modular MockCusp (shadow)

0 ! Mk! ! Mk ! S2 k ! 0 on

< SL(2, R)

Main Point: At weight zero, Rademacher gives modular
functions when there are no weight two cusp forms
and mock modular forms otherwise.
00s(⌧)d⌧=s(⌧)d⌧Weight 2:
holo 1-form ongenus ( \H) >0\H0 ! M0 (m) ! M0 (m) ! S2 (m) ! 0

Shimura, Shintani, Brunier,Ono, Skoruppa, Zagier
0 ! M1/2 (⇢m ) ! M1/2 (⇢m ) ! S3/2 (¯⇢m ) ! 0

When Rademacher gives mock modular forms, they
have rational coefficients only when the shadows aretheta functions in one variable
Cheng-Duncan: Genus zero classification of optimal
mock Jacobi theta functions (including Umbral forms)
Duncan, H, Rayhaun: Genus zero classification of
possible skew-holo Jacobi forms of penumbral
moonshineWhat are some examples of penumbral moonshine?

What are some examples of penumbral moonshine?
Optimal skew-holo Jacobi forms of penumbral moonshine(m,D0 )

sk2 J1,mhs = 2q

D0 /4m+ O(1)2D=s
mod 4mwith 0hr = O(1), r 6= s

For a given D0 such forms exist for finitely many m(lambency) indexed by genus zero (Fricke)
subgroups of SL2 (R) .

What are some examples of penumbral moonshine?
Optimal skew-holo Jacobi forms of penumbral moonshine(m,D0 )

sk2 J1,mhs = 2q

D0 /4m+ O(1)2D=s
mod 4mwith 0

hr = O(1), r 6= s

For a given D0 such forms exist for finitely many m(lambency) indexed by genus zero (Fricke)
subgroups of SL2 (R) .
XmThompson: F3 (⌧ ) = 2f3 (⌧ ) + 248✓(⌧ ) =c(m)qm

D0 =3

The first few allowed m values are 1,3,7,13,19,21,31.
m=1 is the Thompson moonshine example.
m=3 leads to moonshine for 3.G2 (3) which is related
to the centralizer of an element of order 3 in Th.
m=7 leads to moonshine for L2 (7) which is related tothe centralizers of an element of order 7 in Th.
m=13,19,31 are the other prime values m dividing theorder of Th. For 19,31 the centralizers are Abelian
cyclic groups of these orders.

(m, D0 ) = (1, 4)

F0 = q 1

492 + 2 ⇥ 142884q + 2 ⇥ 18473000q 2 + ·F1 = 2 ⇥ 565760q 5/4 + 2 ⇥ 51179520q 9/4 + · · ·Moonshine for 2.F4(2)(m, D0 ) = (1, 4)

F0 = q 1

492 + 2 ⇥ 142884q + 2 ⇥ 18473000q 2 + ·F1 = 2 ⇥ 565760q 5/4 + 2 ⇥ 51179520q 9/4 + · · ·Moonshine for 2.F4(2)(m, D0 ) = (4, 15)

Moonshine for Baby Monster as
realized in Hohn’s c=23 1/2 CFT(m, D0 ) = (1, 4)

F0 = q 1492 + 2 ⇥ 142884q + 2 ⇥ 18473000q 2 + ·F1 = 2 ⇥ 565760q 5/4 + 2 ⇥ 51179520q 9/4 + · · ·Moonshine for 2.F4(2)(m, D0 ) = (4, 15)(m, D0 ) = (6, 23)

Moonshine for Baby Monster as
realized in Hohn’s c=23 1/2 CFT⌘J(⌧ ) = q 23/24

q 1/24 + 196883q 25/24 + 21296876q 49/24

Moonshine for Monster at weight 1/2(decomposition into Virasoro characters)

What does it all mean?
Mathematically umbral and penumbral moonshine look
like two sides of a single coin:

M1/2 (⇢m )M1/2 (⇢m )non-Fricke

FrickeThere are many connections to generalized Monstrous
Moonshine via Groups and Lifts.
For each prime p dividing |M| there X
is a weakly(p)(p)
nf=c(n)q
holomorphic weight 1/2 form
1nYcn c(n2 )Z(g, 1, ⌧ ) = q(1 q )
n=1

There is also an intricate relation between lifts of thetwined weight 1/2 forms of Thompson moonshine andthe twines of the 3C twisted generalize moonshine
weight 0 forms of L. QueenThere is also an intricate relation between lifts of thetwined weight 1/2 forms of Thompson moonshine andthe twines of the 3C twisted generalize moonshine
weight 0 forms of L. Queen
On the physics side we would like to understandThere is also an intricate relation between lifts of thetwined weight 1/2 forms of Thompson moonshine andthe twines of the 3C twisted generalize moonshine
weight 0 forms of L. Queen
On the physics side we would like to understandIs there a physical interpretation of weight 1/2 forms
that lift to twined partition functions of CFT?

There is also an intricate relation between lifts of thetwined weight 1/2 forms of Thompson moonshine andthe twines of the 3C twisted generalize moonshine
weight 0 forms of L. Queen
On the physics side we would like to understandIs there a physical interpretation of weight 1/2 forms
that lift to twined partition functions of CFT?
Where do skew-holo Jacobi forms arise in BPS
state counting and is there a physics version of themathematical similarity between holo and skewholo Jacobi forms?

There is also an intricate relation between lifts of thetwined weight 1/2 forms of Thompson moonshine andthe twines of the 3C twisted generalize moonshine
weight 0 forms of L. Queen
On the physics side we would like to understandIs there a physical interpretation of weight 1/2 forms
that lift to twined partition functions of CFT?
Where do skew-holo Jacobi forms arise in BPS
state counting and is there a physics version of themathematical similarity between holo and skewholo Jacobi forms?
Does string theory provide an understanding of thevector spaces and representations being counted
in these new examples of moonshine?

Thank You(and please enjoy the moonshine tonight)

盛者必衰