Umbral Moonshine, Niemeier Lattices, and String Theory

String-Math 2013, Stony Brook

Umbral Moonshine,
Niemeier Lattices,
and String Theory

Miranda Cheng

(CNRS & IMJ, Paris)

Wednesday, June 19, 13

Outline

Review:

• Moonshine
• The First Case of Umbral Moonshine: M24

New:

• Umbral Moonshine and Niemeier Lattices

math.RT:1306.xxxx w. J. Duncan, J. Harvey

• Umbral Moonshine and String Theory

het-th:1306.xxxx w. Xi Dong, J. Duncan, J. Harvey, S. Kachru,T.Wrase

Wednesday, June 19, 13

I. Moonshine

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Finite Moonshine Modular
Groups Objects

Wednesday, June 19, 13

Finite Moonshine Modular
Groups Objects

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Modular Forms

A modular form f(τ) transforms “covariantly” under a subgroup Γ of SL(2,R).
upper-half plane

f(τ) :

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Modular Forms

Example: the J-function

-1 -1/2 1/2 1

M CHENG

Wednesday, June 19, 13

String theory is good at producing
automorphic forms!

eg. The SL(2,Z) symmetry of the partition function.
time

Wednesday, June 19, 13

String theory is good at producing
automorphic forms!

eg. The SL(2,Z) symmetry of the partition function.
time
q-series

Wednesday, June 19, 13

String theory is good at producing
automorphic forms!

eg. The SL(2,Z) symmetry of the partition function.

time q-series SL(2,Z)

Wednesday, June 19, 13 modular under
SL(2,Z)

String theory is good at producing
automorphic forms!

More generally, there can be space-time symmetries
(such as E-M, S-,T-dualities) as well as world-sheet
symmetries (such as SL(2,Z)).

All symmetries have to be reflected in suitable
partition functions.

Wednesday, June 19, 13

Finite Moonshine Modular
Groups Objects

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Finite Groups

Discrete Symmetries of Objects

Example 1. Symmetry of a square = the dihedral group “Dih4”.

Wednesday, June 19, 13

Finite Groups

Discrete Symmetries of Objects

Example 2. Close Packing Lattices: considering the most efficient way to
stack up identical balls.

Face-Centered Cubic
(fcc) Lattices
e.g. Cu,Ag,Au

M2.CBaHckEgNrouGnd/Finite Groups

Wednesday, June 19, 13

Finite Groups

Discrete Symmetries of Objects

Example 2. Close Packing Lattices: considering the most efficient way to
stack up identical balls.

Face-Centered Cubic
(fcc) Lattices
e.g. Cu,Ag,Au, oranges, ...

M2.CBaHckEgNrouGnd/Finite Groups

Wednesday, June 19, 13

Finite Groups V
G
the Representations

A representation V of a finite group G is a space that G acts on.

e.g. G = Dih4 =Symmetry group of a square.

y

2-dimensional representation

0x

M2.CBaHckEgNrouGnd/Finite Groups

Wednesday, June 19, 13

Some Interesting Examples

the “Sporadic Groups”

The only 26 groups that do not fall in infinite families.
They are usually symmetries of very interesting objects.

Example 1. M24 = the oldest sporadic “Mathieu Groups”. [Mathieu 1860]

|M24| = the number of elements in M24 = 244,823,040.

Example 2. “The Monster” = the largest sporadic groups. [Fischer, Griess 1970-80]

|M| ~ 1054.
The smallest non-trivial representation has 196,883
dimensions.

M2.CBaHckEgNrouGnd/Finite Groups

Wednesday, June 19, 13

Finite Moonshine Modular
Groups Objects

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Moonshine

relates modular forms and finite groups.

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Moonshine

relates modular forms and finite groups.

[McKay late 70’s]

Wednesday, June 19, 13

Moonshine

relates modular forms and finite groups.

[McKay late 70’s]

dim irreps of Monster

??

Monstrous Moonshine
Conjecture

[Conway–Norton ’79]

The Most Natural The Largest
Modular Function Sporadic Group

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String Theory

explains Monstrous Moonshine.

Q:What’s the most efficient way to stack up 24-dimensional identical balls?

A: It’s given by the Leech lattice ΛLeech. [Leech 1967]

ΛLeech has very interesting sporadic symmetries.

Q: But it’s 24-dimensional! What can we do with it?
A: Just the right number of dimensions for string theory!

Wednesday, June 19, 13

String Theory

explains Monstrous Moonshine.

Strings in the Leech lattice background:

R24/ΛLeech

Wednesday, June 19, 13

String Theory

explains Monstrous Moonshine.

Strings in the Leech lattice background: Modular Symmetry

The Partition Function = J(τ)

Monstrous
Moonshine

R24/ΛLeech Sporadic Symmetry

Wednesday, June 19, 13 ΛLeech
Co1 Z/2 Monster

Monstrous Moonshine

A Meeting Place of Different Subjects

To prove the Monstrous Moonshine Conjecture

• Use string theory

(orbifold conformal field theory, no-ghost theorem, ...)

• Led to important developments in algebra and

representation theory

(vertex operator algebra, Borcherds–Kac–Moody algebra, ...)

[Frenkel–Lepowsky–Meurman, 80’s]
[Borcherds, 80-90’s]

Wednesday, June 19, 13

II. The First Case of
Umbral Moonshine:

M24

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K3 Sigma-Model

In a 2d N>=(2,2) SCFT, BPS states are counted by the elliptic
genus, which transforms nicely (Jacobi form) under SL(2,Z).

Wednesday, June 19, 13

K3 Sigma-Model

In a 2d N>=(2,2) SCFT, BPS states are counted by the elliptic
genus, which transforms nicely (Jacobi form) under SL(2,Z).

Recall: 2d sigma model on K3 is a N=(4,4) SCFT.

[Eguchi–Ooguri–Taormina–Yang ’88]

Wednesday, June 19, 13

K3 Sigma-Model

In a 2d N>=(2,2) SCFT, BPS states are counted by the elliptic
genus, which transforms nicely (Jacobi form) under SL(2,Z).

Recall: 2d sigma model on K3 is a N=(4,4) SCFT.

[Eguchi–Ooguri–Taormina–Yang ’88]

number of long N=4 multiplets
also dimensions of irreps of M24!!!

[Eguchi–Ooguri–Tachikawa ’10]

Wednesday, June 19, 13

K3 Sigma-Model

In a 2d N>=(2,2) SCFT, BPS states are counted by the elliptic
genus, which transforms nicely (Jacobi form) under SL(2,Z).

Recall: 2d sigma model on K3 is a N=(4,4) SCFT.

[Eguchi–Ooguri–Taormina–Yang ’88]

number of long N=4 multiplets
also dimensions of irreps of M24!!!

[Eguchi–Ooguri–Tachikawa ’10]

Counting the number of long multiplets:

Mock Modular Form

Wednesday, June 19, 13

Mock Modular Form

: a variant of modular form that transforms with a
holomorphic anomaly, given by the shadow (or umbra) of the mmf.

modular mmf shadow [Ramanujan ’20]
.........
[Zwegers ’02]

Wednesday, June 19, 13

String Theory on K3 and M24:
A Conjecture (Theorem)

We can also consider the twisted (or equivariant) K3 elliptic
genus EGg(τ,z;K3) and extract the interesting part Hg(τ).

Conjecture/Theorem:
There exists a M24–module K=⊕nKn such that for all [g]⊂M24, Hg(τ) coincides
with the graded character:

Status:
1) The candidate mock modular forms Hg(τ) have been proposed for all
[g]⊂M24.
[M.C. /Gaberdiel-Hohenneger-Volpato/Eguchi-Hikami ’10]

2) With these given Hg(τ), the conjecture is recently proven. [T. Gannon ’12]
3) However, an explanation of this M24 moonshine is yet to be found.

Wednesday, June 19, 13

III. Umbral Moonshine

based on work with J. Duncan and J. Harvey
1204.2779, 1306.xxxx

Wednesday, June 19, 13

Umbral Moonshine

A New Type of Moonshine

“Niemeier” Umbral Moonshine “Mock”
Finite Modular
Groups
Forms

Wednesday, June 19, 13

Niemeier Groups

More Symmetries From Packing Balls

M CHENG

Wednesday, June 19, 13

Niemeier Groups

More Symmetries From Packing Balls

:“deep holes” of the sphere packing
= the points where the distance to any
a ball is maximum

M CHENG

Wednesday, June 19, 13

Niemeier Groups

More Symmetries From Packing Balls

:“deep holes” of the sphere packing
= the points where the distance to any
a ball is maximum

M CHENG

Wednesday, June 19, 13

Niemeier Groups

More Symmetries From Packing Balls

:“deep holes” of the sphere packing
= the points where the distance to any
centre of a ball is maximum

The deep holes define a lattice
themselves.

In the 24-dimensional sphere packing given
by the Leech lattice ΛLeech, there are 23
such deep hole lattices, N(1), N(2), ..., N(23),
called the Niemeier Lattices.

Theorem (Niemeier 1973)
In 24 dimensions, there are exactly 24 interesting (even, self-dual, positive-
definite) lattices.
They are ΛLeech and the 23 Niemeier lattices N(1), N(2), ..., N(23).

M CHENG

Wednesday, June 19, 13

Niemeier Groups

More Symmetries From Packing Balls

Consider the symmetries of these 23 Niemeier lattices, we obtain the 23 finite
groups, which we call the Niemeier Groups.

Def: G X := Aut(NX)/ WX
X = the roots (<x,x>=2) of the Niemeier lattice
WX= Weyl reflection w.r.t. X

Example:
Consider the “simplest” Niemeier lattice with X=24 A1.
Its symmetry gives GX=M24!

Remark:
X is the root system of a Niemeier lattice

X is one of the 23 unions of simply-laced root systems such that 1. all irred.
components have the same Coxeter number. 2. the total rank is 24

Wednesday, June 19, 13

Niemeier Mock Forms

For each of the 23 Niemeier lattice NX, we associate a mock modular
form HX via a (unique) procedure given by an A-D-E classification.

Thm (Cappelli–Itzykson–Zuber ’87):

The (m 1) ⇥ (m 1) matrices N X such that

1. Pm 1 i(⌧ )NiX,j j(⌧ ) is modular invariant.
i,j=1

2. NiX,j 2 Z 0, N1X,1 = 1

are in 1-1 to ADE root systems X, where m = Coxeter
number of X.

i : spin i 1 characters of A1 a ne Kac–Moody alge-
bra at level m 2.

M CHENG 8

Wednesday, June 19, 13

Niemeier Mock Forms

For each of the 23 Niemeier lattice NX, we associate a mock modular
form HX via a (unique) procedure given by an A-D-E classification.

Thm :

Consider vector-valued modular form h such that Pm 1 hi✓ˆm,i
i=1
is a Jacobi form of weight 1 and index m.

The (m 1) ⇥ (m 1) matrices N X such that

1. Pm 1 hi(⌧ )NiX,j ✓m,j is a Jacobi form of weight 1
i,j=1
and index m.

2. NiX,j 2 Z 0, N1X,1 = 1

are in 1-1 to ADE root systems X, where m = Coxeter
number of X.

✓ˆm,i = ✓m,i ✓m, i, ✓m,i = X qk2/4myk

. k=i km2oZd 2m

M CHENG 9

Wednesday, June 19, 13

Niemeier Mock Forms

For each of the 23 Niemeier lattice NX, we associate a mock modular
form HX via a (unique) procedure given by an A-D-E classification.

Thm (Dabholkar–Murthy–Zagier ’12):

For a meromorphic Jacobi form of weight 1 and index

m with simple poles at z 2 Q + Q⌧ , there is a canonical

way of separating into the ”polar part” and ”finite

part” = P + F such that the theta-decomposition
Pm 1
of the finite part i=1 hi✓ˆm,i leads to a vector-valued

mock modular form h = (hi) with shadow given by the

theta function

Sm,i = @ .
@y ✓m,i
y=1

M CHENG 10

Wednesday, June 19, 13

Niemeier Mock Forms

For each of the 23 Niemeier lattice NX, we associate a mock modular
form HX via a (unique) procedure given by an A-D-E classification.

Thm (Cheng–Duncan–Harvey ’13):
For every ADE root system, consider a weight 1/2
vector-valued mock modular form HX = (HiX) with
shadow SX = (SiX), SiX = NiX,jSm,j and satisfies the
growth condition

HiX(⌧ ) = 2q 1/4m i,1 + O(1) , ⌧ ! i1.
Such a mock modular form is unique if it exists.

M CHENG 11

Wednesday, June 19, 13

Niemeier

Lattices NX

GX := Aut(NX)/ WX
Niemeier Finite

Groups GX

M CHENG

Wednesday, June 19, 13

Niemeier Mock Forms

For each of the 23 Niemeier lattice NX, we associate a mock modular
form HXg for every element g of GX via a (unique) procedure related to
the A-D-E classification of modular invariants.

[Inspired by Cappelli–Itzykson–Zuber ’87, Dabholkar–Murthy–Zagier ’12]

Niemeier

Lattices NX

X ↦ SX (the shadow) ↦ HX GX := Aut(NX)/ WX

Mock Modular Niemeier Finite

Forms HX Groups GX

M CHENG

Wednesday, June 19, 13

Niemeier Mock Forms

For each of the 23 Niemeier lattice NX, we associate a mock modular
form HXg for every element g of GX via a (unique) procedure related to
the A-D-E classification of modular invariants.

[Inspired by Cappelli–Itzykson–Zuber ’87, Dabholkar–Murthy–Zagier ’12]

Niemeier

Lattices NX

X ↦ SX (the shadow) ↦ HX GX := Aut(NX)/ WX

Mock Modular Niemeier Finite

Forms HX Groups GX

M CHEHNXgGfor every [g]⊂ GX

Wednesday, June 19, 13

Umbral Moonshine

Relating Mock Modular Forms and Niemeier Symmetries

Mock Modular Umbral Moonshine Niemeier Finite

Forms HX MC–Duncan–Harvey Groups GX
’12-’13

M CHENG

Wednesday, June 19, 13

Umbral Moonshine

Relating Mock Modular Forms and Niemeier Symmetries

Mock Modular Umbral Moonshine Niemeier Finite

Forms HX MC–Duncan–Harvey Groups GX
’12-’13

Umbral Moonshine Conjecture (1306.xxxx)
For all of the 23 Niemeier Lattices NX, there exists an infinite-
dimensional GX–module KX= ⊕α KXα (the “umbral mondule”) such

that the mmf HXg coincides with its (graded) character:

M CHENG

Wednesday, June 19, 13

Umbral Moonshine

An M24 Example

Corresponding to the “simplest” Niemeier lattice NX, X=24 A1

They are dimensions of irreps of the
largest Mathieu sporadic group GX=M24 !

UM

Wednesday, June 19, 13

Umbral Moonshine

A Second Example

Corresponding to the “next simplest” Niemeier lattice NX, X=12 A2

They are dimensions of irreps of the
Mathieu sporadic group GX=2.M12 !

UM

Wednesday, June 19, 13