daniel persson - chalmers tekniska högskola · 2018-01-29 · daniel persson mathematical sciences...

The dark side of the moon- new connections in geometry, number theory and physics -

January 29, 2018

Daniel Persson

Mathematical Sciences Colloquium

SymmetriesTransformations that leave a geometric object invariant

A circle has infinitely many rotational symmetries

Infinite rotational symmetry

Finite reflection symmetry

Symmetry groups

The set of symmetries of an object form a mathematical structure called a group.

Two consecutive transformations is also a symmetry

There exists an identity transformation

Each transformation has an inverse

Consecutive transformations are associative

Permutation symmetries

The set {1,2,3} has 6 permutations

(1,2,3) (1,3,2) (2,1,3) (2,3,1) (3,1,2) (3,2,1)

These belong to the permutation group of 3 elements.

“Ile de feu 2”(Island of fire)

Olivier Messiaen 1949

This piece of music has the symmetry group of a 45-dim algebraic structure

M12M12

“Ile de feu 2”(Island of fire)

Olivier Messiaen 1949

This piece of music has the symmetry group of a 45-dim algebraic structure

M12

P1 = (1, 7, 10, 2, 6, 4, 5, 9, 11, 12)(3, 8)

This corresponds to the following permutation:

P1 = (1, 7, 10, 2, 6, 4, 5, 9, 11, 12)(3, 8)

P2 = (1, 6, 9, 2, 7, 3, 5, 4, 8, 10, 11)(12)


Messiaen also uses the following permutation in another part:

P1 = (1, 7, 10, 2, 6, 4, 5, 9, 11, 12)(3, 8)

P2 = (1, 6, 9, 2, 7, 3, 5, 4, 8, 10, 11)(12)


Messiaen also uses the following permutation in another part:

Theorem (Diaconis, Graham, Kantor 1985): M12 = hP1, P2i

Of course, Messiaen did not use all 95040 permutations in the Mathieu group!

Finite simple groups

Finite groups that can not be divided into smaller pieces are called simple.

These are like building blocks of symmetries

Finite simple groups

Finite groups that can not be divided into smaller pieces are called simple.

These are like building blocks of symmetries

Elementary particles

2, 3, 5, 7, 11, 13,

17, 19, 23, 29, 31,

37, 41, 43, 47, 53,

59, 61, 67, 71, 73,

79, 83, 89, 97

Prime numbers

Classification of finite simple groups

1832: Galois discovers the first infinite family (alternating groups)


1861-1873: Mathieu discovers M11,M12,M22,M23,M24



1861-1873: Mathieu discovers M11,M12,M22,M23,M24

1972-1983: Gorenstein program to classify all finite simple groups


The monster

Conjecture (Fischer & Griess, 1973): There exists a huge finite simple group of order

The monster


The monster

Theorem (Griess 1982): The monster group exists. It is the symmetry group of a certain 196884-dimensional algebraic structure. (commutative, non-associative algebra)


The monster

1983: The classification program is completed! This was a heroic joint effort of 100’s of mathematicians.

Theorem (Griess 1982): The monster group exists. It is the symmetry group of a certain 196884-dimensional algebraic structure. (commutative, non-associative algebra)

[https://irandrus.files.wordpress.com/2012/06/periodic-table-of-groups.pdf]

https://irandrus.files.wordpress.com/2012/06/periodic-table-of-groups.pdf

ClassificationSeveral infinite families: cyclic, alternating, Lie type

26 sporadic cases

Several infinite families: cyclic, alternating, Lie type

26 sporadic casesMonster M

Classification


26 sporadic casesThe happy family

(all related to the monster)

Classification


26 sporadic cases

Pariah groups

(all related to the monster)The happy family

(not related to the monster)

Classification


26 sporadic cases

Pariah groups

(all related to the monster)The happy family

(not related to the monster)

Classification

Why do these groups exist?

The first hints of moonshine…


In 1978 John McKay was taking a break from the classification program of finite groups and was doing some recreational

reading in number theory.


He then stumbled upon the following series expansion:



J(q) =1

q+ 196884q + 21493760q2 + 864299970q3 + 20245856256q4 + · · ·


He then stumbled upon the following series expansion:

Being a finite group theorist he immediately opened up the Atlas…



J(q) =1

q+ 196884q + 21493760q2 + 864299970q3 + 20245856256q4 + · · ·

196884 = 1 + 196883 McKay’s equation

196884 = 1 + 196883 McKay’s equation

21493760 = 1 + 196883 + 21296876 Thompson’s equation

196884 = 1 + 196883 McKay’s equation

21493760 = 1 + 196883 + 21296876 Thompson’s equation

What does this really mean?

196884 = 1 + 196883

We can think of 196883 as the dimension of a vector space on which the monster acts

This is the smallest non-trivial irreducible representation of MThe factor 1 then corresponds to the trivial reps.

21493760 = 1 + 196883 + 21296876

Fun with the monster

196884 = 1 + 196883



1 �! V1 dimR V1 = 1

196883 �! V2 dimR V2 = 196883

21493760 = 1 + 196883 + 21296876

21296876 �! V3 dimR V3 = 21296876


196884 = 1 + 196883



1 �! V1 dimR V1 = 1

196883 �! V2 dimR V2 = 196883

21493760 = 1 + 196883 + 21296876

21296876 �! V3 dimR V3 = 21296876

= Tr�e��V1

�

= Tr�e��V2

�

= Tr�e��V3

�


196884 = 1 + 196883



1 �! V1 dimR V1 = 1

196883 �! V2 dimR V2 = 196883

21493760 = 1 + 196883 + 21296876

21296876 �! V3 dimR V3 = 21296876

= Tr�e��V1

�

= Tr�e��V2

�

= Tr�e��V3

�

characters of the identity element e 2 M


McKay and Thompson conjectured that there exists an infinite-dimensional,graded monster module, called , such that V \

V \ =1M

n=�1

V (n)

V (�1) = V1

V (1) = V1 � V2

V (2) = V1 � V2 � V3


V (0) = 0 and the J-function is the graded dimension

J(q) =1X

n=�1

Tr(e|V (n))qn

V \ =1M

n=�1

V (n)

V (�1) = V1

V (1) = V1 � V2

V (2) = V1 � V2 � V3


V (0) = 0 and the J-function is the graded dimension

J(q) =1X

n=�1

Tr(e|V (n))qn

This suggests to also consider for each the McKay-Thompson series

Tg(q) =1X

n=�1

Tr(g|V (n))qn

g 2 M

V \ =1M

n=�1

V (n)

Modular forms

J(q) =1

q+ 196884q + 21493760q2 + 864299970q3 + 20245856256q4 + · · ·

Modular forms

The J-function has a very special invariance property. If we set q = e2⇡i⌧

⌧ 2 H = {z 2 C |=(z) > 0}

then we have

f

✓a⌧ + b

c⌧ + d

◆= f(⌧)

✓a bc d

◆2 SL(2,Z)

J(q) =1

q+ 196884q + 21493760q2 + 864299970q3 + 20245856256q4 + · · ·

Modular forms

The J-function has a very special invariance property. If we set q = e2⇡i⌧

⌧ 2 H = {z 2 C |=(z) > 0}

then we have

✓a bc d

◆2 SL(2,Z)

This is a modular form of weight 0 (or a modular function).

Up to a constant there is a unique such function which is holomorphic away from a simple pole at the cusp =(⌧) ! 1

J(q) =1

q+ 196884q + 21493760q2 + 864299970q3 + 20245856256q4 + · · ·

f

✓a⌧ + b

c⌧ + d

◆= f(⌧)

It generates the field of rational functions on the sphere (Hauptmodul)

SL(2, Z)\H ⇠

any modular function =polynomial in J(⌧)

polynomial in J(⌧)

It generates the field of rational functions on the sphere (Hauptmodul)

SL(2, Z)\H ⇠

any modular function =polynomial in J(⌧)

polynomial in J(⌧)

Monstrous moonshine conjecture (Conway-Norton 1979): For all elements The McKay-Thompson seriesare hauptmoduls with respect to some genus zero �g ⇢ SL(2,R)

Tg(⌧)g 2 M

genus zero �g\H ⇠�g

“The stuff we were getting was not supported by logical argument. It had the feeling of mysterious moonbeams lighting up dancing Irish leprechauns. Moonshine can also refer to

illicitly distilled spirits, and it seemed almost illicit to be working on this stuff.”

- John Conway

Monster group

M

Modular function

J(⌧)

????

2d conformal field theory (vertex operator algebra)

Monster group

M

Modular function

J(⌧)

Enter physics!

1988: Frenkel, Lepowsky, Meurman constructed the moonshine module V \

M

A quantum field theory with conformal symmetry defined on a Riemann surface

It is a 2-dimensional orbifold conformal field theory (string theory),a. k. a. vertex operator algebra, whose automorphism group is

To explain moonshine we need a special class of CFT’s:

holomorphic CFT = self-dual Vertex Operator Algebra (VOA)

In general a VOA describes only part of a CFT


It is a 2-dimensional orbifold conformal field theory (string theory),a. k. a. vertex operator algebra, whose automorphism group is M


To explain moonshine we need a special class of CFT’s:

holomorphic CFT = self-dual Vertex Operator Algebra (VOA)

In general a VOA describes only part of a CFT

1992: Borcherds proved the full moonshine conjecture, earning him the Fields medal


It is a 2-dimensional orbifold conformal field theory (string theory),a. k. a. vertex operator algebra, whose automorphism group is M


Do we understand everything now?

Why genus zero? Proven by Borcherds but no understanding of its origin.

2016: Explanation proposed by Paquette-D.P.-Volpato using T-duality in string theory.(part of which is a mathematical proof using VOAs)




Long-standing conjecture by Frenkel-Lepowsky-Meurman:

V \

Partial results by Dong-Li-Mason, Tuite, Paquette-D.P.-Volpato, Carnahan-Komuro-Urano,…

The moonshine module is unique.




Long-standing conjecture by Frenkel-Lepowsky-Meurman:

V \

Partial results by Dong-Li-Mason, Tuite, Paquette-D.P.-Volpato, Carnahan-Komuro-Urano,…

The moonshine module is unique.

Generalized moonshine conjecture by Norton in 1987:

Moonshine for orbifolds of V \

Given as a PhD project by Borcherds to his student Carnahan around 2005.

Complete proof announced by Carnahan in 2017.

Generalized moonshine includes several more of the sporadic groups occurring as centralizers of elements in the monster.

Frenkel-Lepowsky-Meurman: The sporadic groups exist because of moonshine.

New moonshine!What does the future have in store?

New moonshine!

Moonshine revolution

2010: New moonshine for conjectured by Eguchi, Ooguri, Tachikawa

M24

The role of the J-function is now played by the elliptic genus of string theory on a K3-surface

(weak Jacobi form)


2010: New moonshine for conjectured by Eguchi-Ooguri-Tachikawa

M24

2012: Proven by Gannon

But no analogue of the monster module is constructed!

2013: Generalized Mathieu moonshineestablished by Gaberdiel-D.P.-Volpato The role of the J-function is now played by the

elliptic genus of string theory on a K3-surface

(weak Jacobi form)


2010: New moonshine for conjectured by Eguchi-Ooguri-Tachikawa

M24

2012: Proven by Gannon

But no analogue of the monster module is constructed!

2013: Generalized Mathieu moonshineestablished by Gaberdiel-D.P.-Volpato

But why??

There is no string theory on K3 with Mathieu symmetry(Gaberdiel-Hohenegger-Volpato)

Also a mathematical formulation of no-go theorem by Huybrechts using autoequivalences of derived categories of coherent sheaves on K3-surfaces.

The role of the J-function is now played by the elliptic genus of string theory on a K3-surface

(weak Jacobi form)

2012: Umbral moonshine proposed by Cheng-Duncan-Harvey

A class of 23 moonshines, including Mathieu moonshine. Classified by Niemeier lattices. Involves also non-sporadic groups!

But it doesn’t stop there!



2015: Umbral moonshine conjecture proven by Duncan-Griffin-Ono

2016: Generalized Umbral moonshine established by Cheng-de Lange-Whalen




2015: Umbral moonshine conjecture proven by Duncan-Griffin-Ono

2016: Generalized Umbral moonshine established by Cheng-de Lange-Whalen

Naturally formulated using mock modular forms

This goes back to Ramanujan’s last letter to Hardy


The Last Letter...

Towards the end of his stay in England, Ramanujan fell very ill and he returned to India in 1919

3 months before his death, Ramanujan wrote a final letter to Hardy, including a list of 17 mysterious functions:

It took nearly 100 years, and the efforts of generations of mathematicians before Ramanujan’s theory was finally understood...

“Mock Theta Functions”

(Zwegers)

The Last Letter...

Towards the end of his stay in England, Ramanujan fell very ill and he returned to India in 1919

3 months before his death, Ramanujan wrote a final letter to Hardy, including a list of 17 mysterious functions:

The mock theta functions arise as McKay-Thompson series in Umbral moonshine!

“Mock Theta Functions”

A mock modular form of weight is the first member of a pair k (h, g)

So what are mock modular forms?

is holomorphic with at most exponential growth at cusps h : H ! C

is a holomorphic cusp form with weight (the shadow) g : H ! C 2� k

Def (Zwegers, Zagier):





The completion with h+ g̃

g̃ = (4i)k�1

Z 1

�⌧̄(z + ⌧)�kg(�z̄)dz

is a non-holomorphic modular form (harmonic Maass form) of weight k






The completion with h+ g̃

g̃ = (4i)k�1

Z 1

�⌧̄(z + ⌧)�kg(�z̄)dz

is a non-holomorphic modular form (harmonic Maass form) of weight k

Note: modular forms are mock modular forms with zero shadow g = 0


Umbral moonshine arises for weight k = 1/2

Pariah moonshine

2017: Duncan, Mertens, Ono proposes moonshine for two of the Pariahs


F (⌧) = �q�4 + 2 + 26752q3 + 143376q4 + 8288256q7 + · · ·

Mock modular form of weight k = 3/2

Pariah moonshine


F (⌧) = �q�4 + 2 + 26752q3 + 143376q4 + 8288256q7 + · · ·

Mock modular form of weight k = 3/2

F (⌧) = �q�4 + 2 +X

D<0

a(D)q|D|

The coefficients encode information about elliptic curves of discriminant

a(D)

�D

Implications for the Birch-Swinnerton-Dyer conjecture on the asymptotics of L-functions?

New connections to number theory

Pariah moonshine

So what is moonshine, really?

“Moonshine is not a well defined term, but everyone in the area recognizes it when they see it.”

Richard E. Borcherds


Roles of vertex operator algebras/CFT?

monster Lie algebra

modular function

M

holomorphic VOA V \

m

J(⌧) = q�1 + 196884q + · · ·

Monstrous Moonshine

(Borcherds algebra)

Mathieu Moonshine

M24

Borcherds Lie algebra…???

mock modular forms string theory on K3?VOA?


chiral de Rham complex?(Song)


The fact that generalized moonshine holds for Umbral moonshine strongly suggest some kind of

underlying VOA structure

(Gaberdiel-D.P.-Volpato, Cheng-de Lange-Whalen)

Consider the case of the J-function: w = 0 modular form for SL(2,Z)

If it had a shadow it would have weight 2. But there are no weight 2 holomorphic modular forms for . This is due to the genus zero property. SL(2,Z)

“Rigidity” (Harvey)



“Rigidity”

Umbral moonshine Thompson moonshine

weight 1/2 mock modular forms

Pariah moonshine weight 3/2 mock modular forms

no genus zero property

(Harvey)



“Rigidity”

Umbral moonshine Thompson moonshine

weight 1/2 mock modular forms

Pariah moonshine weight 3/2 mock modular forms

no genus zero property

What is the analogue of genus zero?

A consequence of T-duality in string theory (Paquette-D.P.-Volpato)

McKay-Thompson series should be Rademacher summable(Duncan-Frenkel, Cheng-Duncan)

(Harvey)

Quantum black holes?

Witten proposed that the monster VOA describes the quantum properties of black holes in 3d gravity

(via the AdS/CFT correspondence)

McKay-Thompson series are partition functions of black hole microstates

Quantum black holes?

Witten proposed that the monster VOA describes the quantum properties of black holes in 3d gravity

(via the AdS/CFT correspondence)

A similar statement holds for Mathieu moonshine (Dabholkar-Murthy-Zagier)

Black hole microstates organized into conjugacy classes of the sporadic group

McKay-Thompson series are partition functions of black hole microstates

(Cheng, Raum, D.P.-Volpato)

“The mock theta-functions give us tantalizing hints of a grand synthesis still to be discovered. Somehow it should be possible to build them into a coherent group-theoretical structure, analogous to the structure of modular forms.... This remains a challenge for the future. My dream is that I will live to see the day when our young physicists, struggling to bring the predictions of superstring theory into correspondence with the facts of nature, will be led to enlarge their analytic machinery to include mock theta-functions...”

Freeman Dyson in 1987:

What else will we discover on the dark side of the moon?

daniel persson - chalmers tekniska högskola · 2018-01-29 · daniel persson mathematical sciences...

Documents