Jeff Harvey University of Chicago

Cheng, Duncan & JH 1204.2779, 1307.5793Cheng, Duncan 1605.04480Rayhaun, JH 1504.08179Duncan, Rayhaun, JH to appear 18xx.yyyyy ?

Umbral Penumbral

Moonshine

and

Thanks to

Thanks to

Organizers for a very stimulating and well run meeting

Thanks to

Program Committee for inviting me to speak

Organizers for a very stimulating and well run meeting

Thanks to

Program Committee for inviting me to speak

??? for arranging the moonshine talks to be within hours of the full moon in Okinawa.

Organizers for a very stimulating and well run meeting

penumbra

Sun .

:

.

Umbra

.

Modular forms play an important role in number theory and physics because they count things:

1

⌘(⌧)24=

q�1

Qn(1� qn)24 States of open bosonic string2 M !

�12

�ell(K3; ⌧, z) 2 J0,1

⇥E8(⌧) =X

p2�8

e⇡i⌧p2 2 M4 Points in E8 root lattice

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 for G.”

Introduction

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

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

Since Moonshine involves many of the same forms and techniques as CFT and BH counting we hope to learn something new about string theory.

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

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).

Since Moonshine involves many of the same forms and techniques as CFT and BH counting we hope to learn something new about string theory.

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

n+ e, f, · · ·✓a bc d

◆2 SL2(Z)

c = 0 mod n

Atkin-Lehner

A class of genus zero subgroups of SL(2,R):

Fricke: n 2 e, f, · · · or non-Fricke otherwise⌧ ! �1/n⌧

�0(2)\H

Z(2B, 1; ⌧) =⇣

⌘(⌧)⌘(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.

⌧h

twist

twine

g

Given ( ) g, h 2 M gh = hg ⇡1(T2) ! M

= Z(g, h; ⌧) = TrVghqL0�c/24

Genus zero groups:Govern all twists and twines of Monstrous moonshine.

Classify cases of umbral and penumbral moonshine.

⌧h

twist

twine

g

Given ( ) g, h 2 M gh = hg ⇡1(T2) ! M

= Z(g, h; ⌧) = TrVghqL0�c/24

In (generalized) Monstrous Moonshine these are all genus zero functions (Conway, Norton, Queen, Borcherds, Carnahan)

Ogg: for p prime is genus zero precisely when p divides the order of the Monster.

p+ p

Umbral Moonshine

M24 K3

Umbral Moonshine

M24 K3

Umbral Moonshine

Niemeier Lattices(genus zero)

Umbral Moonshine

M24 K3

Umbral Moonshine

Niemeier Lattices(genus zero)

Umbral Moonshine

optimal mock Jacobi theta

39 non-Fricke genus zero groups

(Cheng&Duncan)

Penumbral Moonshine

Th

Penumbral Moonshine

Th2.F4(2)

Th 3.G2(3) L2(7)

Lifts, Gen. Moonshine

???

Penumbral Moonshine

Th2.F4(2)

Th 3.G2(3) L2(7)

Lifts, Gen. Moonshine

???

skew-holomorphic Jacobi forms

84 Fricke genus zero groups

Penumbral Moonshine

Two Parallel Worlds of Weight 1/2 Moonshine

Umbral PenumbralModular objects

Moonshine Groups

Explicit Constructions

Physics Connections

Genus zero groups

Optimal weight 1/2, mock Jacobi forms

non-Fricke

M24, Aut(Niemeier)

For several, not uniform

K3 elliptic genus

Optimal weight 1/2 modular skew-holo Jacobi formsFricke

Groups of generalized moonshine? Lattices?

None yet

BPS counting functions at attractor points and Lifts (S. Harrison)

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

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

What is a skew-holomorphic Jacobi form?

The rest of the talk is devoted to brief answers to some of 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 some of 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 some of 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 some of 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?

�(⌧, z) =X

r mod 2m

hr(⌧)✓m,r(⌧, z)

Eichler-Zagier Jacobi form: � 2 J1,m

(✓m,r(⌧, z) =X

n=r mod 2m

qn2/4myn)

Weight 1/2, these transform under a double cover of the modular group “Weil representation of the metaplectic group”

✓m,r ⇠ M1/2(⇢m) hr ⇠ M1/2(⇢m)

Skew-holo Jacobi: �1,m 2 Jsk1,m hr antiholomorphic in .⌧

Jacobi: hr ⇠ M1/2(⇢m)

Skew-holo Jacobi: hr ⇠ M1/2(⇢m)

' J1,m

' Jsk1,m

Examples at m=1 ( )

f0 = ✓(⌧) = 1 + 2q + 2q4 + 2q9 + 2q16 +O(q25)

f3 = q�3 � 248q + 26752q4 � 85995q5 + 1707264q8 +O(q9)

f4 = q�4 + 492q + 143376q4 + 565760q5 + 18473000q8 +O(q9)

f7 = q�7 � 4119q + 8288256q4 � 52756480q5 + 5734772736q8 +O(q9)

f(⌧) = h0(4⌧) + h1(4⌧)

(Borcherds, Zagier)

. . .

Skew-holo Jacobi forms in strings/BPS counting will be discussed by S. Harrison

39 non-Fricke genus zero groups234566+26+3789

1010+210+51212+312+41314+715+516

1818+218+920+421+322+1124+82528+730+15

30+3,5,1530+6,10,1533+1136+442+6,14,2146+2360+12,15,2070+10,14,3578+6,26,39

What are the non-Fricke and Fricke genus 0 groups?Umbral

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+29 30+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

84 Fricke genus zero groups

39 non-Fricke genus zero groups234566+26+3789

1010+210+51212+312+41314+715+516

1818+218+920+421+322+1124+82528+730+15

30+3,5,1530+6,10,1533+1136+442+6,14,2146+2360+12,15,2070+10,14,3578+6,26,39

What are the non-Fricke and Fricke genus 0 groups?Umbral

SL2(Z)q�1j(⌧)Rademacher: can be obtained by averaging

over modulo its stabilizer, but one must regularize. This can be generalized to other genus zero hauptmoduls (Knopp, Duncan&Frenkel)

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

f0(⌧) = Reg

0

@X

�2�inv/�0

fpolar0 |�

1

Aholomorphic, not obviously modular

What role does genus zero play?

j(⌧) = q�1 + 196884q + 21493760q2 + 864299970q3 + · · ·

Farey Tales (Dijkgraaf, Maldacena, Moore,Verlinde, Manschot,…):An interpretation as a sum over asymptotic geometries with boundary in context of BH counting

T 2AdS3

Obstruction to modularity of Rademacher sums:

0 ! M !k ! Mk ! S2�k ! 0

Mock Cusp (shadow)Modularon � < 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.

s(⌧)d⌧ = s(⌧ 0)d⌧ 0Weight 2:

genus ( ) >0

holo 1-form on

�\H

�\H

When Rademacher gives mock modular forms, they have rational coefficients only when the shadows are theta functions in one variable

0 ! M0(m) ! M0(m) ! S2(m) ! 0

0 ! M1/2(⇢m) ! M1/2(⇢m) ! S3/2(⇢̄m) ! 0

Shimura, Shintani, Brunier, Ono, Skoruppa, Zagier

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?

SL2(R)

D0For a given such forms exist for finitely many m (lambency) indexed by genus zero (Fricke) subgroups of .

Optimal skew-holo Jacobi forms of penumbral moonshine

�(m,D0) 2 Jsk1,m hs = 2qD0/4m +O(1)

hr = O(1), r 6= s

D0 = s2 mod 4mwith

What are some examples of penumbral moonshine?

SL2(R)

D0For a given such forms exist for finitely many m (lambency) indexed by genus zero (Fricke) subgroups of .

Optimal skew-holo Jacobi forms of penumbral moonshine

�(m,D0) 2 Jsk1,m hs = 2qD0/4m +O(1)

hr = O(1), r 6= s

D0 = s2 mod 4mwith

What are some examples of penumbral moonshine?

F3(⌧) = 2f3(⌧) + 248✓(⌧) =X

m

c(m)qmThompson:

The first few allowed m values are 1,3,7,13,19,21,31.

m=1 is the Thompson moonshine example.

m=13,19,31 are the other prime values m dividing the order of Th. For 19,31 the centralizers are Abelian cyclic groups of these orders.

m=3 leads to moonshine for which is related to the centralizer of an element of order 3 in Th.

3.G2(3)

m=7 leads to moonshine for which is related to the centralizers of an element of order 7 in Th.

L2(7)

D0 = �3

(m,D0) = (1,�4)F0 = q�1 � 492 + 2⇥ 142884q + 2⇥ 18473000q2 + · · ·F1 = 2⇥ 565760q5/4 + 2⇥ 51179520q9/4 + · · ·

Moonshine for 2.F4(2)

(m,D0) = (1,�4)F0 = q�1 � 492 + 2⇥ 142884q + 2⇥ 18473000q2 + · · ·F1 = 2⇥ 565760q5/4 + 2⇥ 51179520q9/4 + · · ·

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�1 � 492 + 2⇥ 142884q + 2⇥ 18473000q2 + · · ·F1 = 2⇥ 565760q5/4 + 2⇥ 51179520q9/4 + · · ·

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) = (6,�23)⌘J(⌧) = q�23/24 � q1/24 + 196883q25/24 + 21296876q49/24 + · · ·

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

What does it all mean?

There are many connections to generalized Monstrous Moonshine via Groups and Lifts.

Mathematically umbral and penumbral moonshine look like two sides of a single coin:

M1/2(⇢m)

non-FrickeM1/2(⇢m)

Fricke

For each prime p dividing there is a weakly holomorphic weight 1/2 form

|M|f (p) =

X

n

c(p)(n)qn

Z(g, 1, ⌧) = q�c1Y

n=1

(1� qn)c(n2)

There is also an intricate relation between lifts of the twined weight 1/2 forms of Thompson moonshine and the twines of the 3C twisted generalize moonshine weight 0 forms of L. Queen

There is also an intricate relation between lifts of the twined weight 1/2 forms of Thompson moonshine and the 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 the twined weight 1/2 forms of Thompson moonshine and the twines of the 3C twisted generalize moonshine weight 0 forms of L. Queen

On the physics side we would like to understand

Is 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 the twined weight 1/2 forms of Thompson moonshine and the twines of the 3C twisted generalize moonshine weight 0 forms of L. Queen

On the physics side we would like to understand

Is 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 the mathematical similarity between holo and skew-holo Jacobi forms?

There is also an intricate relation between lifts of the twined weight 1/2 forms of Thompson moonshine and the twines of the 3C twisted generalize moonshine weight 0 forms of L. Queen

On the physics side we would like to understand

Is 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 the mathematical similarity between holo and skew-holo Jacobi forms?

Does string theory provide an understanding of the vector spaces and representations being counted in these new examples of moonshine?

Thank You(and please enjoy the moonshine tonight)

盛者必衰