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
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)
This corresponds to the following permutation:
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)
This corresponds to the following permutation:
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)
1832: Galois discovers the first infinite family (alternating groups)
1861-1873: Mathieu discovers M11,M12,M22,M23,M24
Classification of finite simple groups
1832: Galois discovers the first infinite family (alternating groups)
1861-1873: Mathieu discovers M11,M12,M22,M23,M24
Classification of finite simple groups
1832: Galois discovers the first infinite family (alternating groups)
1861-1873: Mathieu discovers M11,M12,M22,M23,M24
1972-1983: Gorenstein program to classify all finite simple groups
Classification of finite simple groups
The monster
Conjecture (Fischer & Griess, 1973): There exists a huge finite simple group of order
The monster
Conjecture (Fischer & Griess, 1973): There exists a huge finite simple group of order
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)
Conjecture (Fischer & Griess, 1973): There exists a huge finite simple group of order
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]
ClassificationSeveral infinite families: cyclic, alternating, Lie type
26 sporadic cases
Several infinite families: cyclic, alternating, Lie type
26 sporadic casesMonster M
Classification
Several infinite families: cyclic, alternating, Lie type
26 sporadic casesThe happy family
(all related to the monster)
Classification
Several infinite families: cyclic, alternating, Lie type
26 sporadic cases
Pariah groups
(all related to the monster)The happy family
(not related to the monster)
Classification
Several infinite families: cyclic, alternating, Lie type
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…
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.
The first hints of moonshine…
He then stumbled upon the following series expansion:
In 1978 John McKay was taking a break from the classification program of finite groups and was doing some recreational
reading in number theory.
J(q) =1
q+ 196884q + 21493760q2 + 864299970q3 + 20245856256q4 + · · ·
The first hints of moonshine…
He then stumbled upon the following series expansion:
Being a finite group theorist he immediately opened up the Atlas…
In 1978 John McKay was taking a break from the classification program of finite groups and was doing some recreational
reading in number theory.
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
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.
1 �! V1 dimR V1 = 1
196883 �! V2 dimR V2 = 196883
21493760 = 1 + 196883 + 21296876
21296876 �! V3 dimR V3 = 21296876
Fun with the monster
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.
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
�
Fun with the monster
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.
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
Fun with the monster
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
McKay and Thompson conjectured that there exists an infinite-dimensional,graded monster module, called , such that V \
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
McKay and Thompson conjectured that there exists an infinite-dimensional,graded monster module, called , such that V \
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
1988: Frenkel, Lepowsky, Meurman constructed the moonshine module V \
It is a 2-dimensional orbifold conformal field theory (string theory),a. k. a. vertex operator algebra, whose automorphism group is M
A quantum field theory with conformal symmetry defined on a Riemann surface
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
1988: Frenkel, Lepowsky, Meurman constructed the moonshine module V \
It is a 2-dimensional orbifold conformal field theory (string theory),a. k. a. vertex operator algebra, whose automorphism group is M
A quantum field theory with conformal symmetry defined on a Riemann surface
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)
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.
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.
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)
Moonshine revolution
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)
Moonshine revolution
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!
But it doesn’t stop there!
2012: Umbral moonshine proposed by Cheng-Duncan-Harvey
2015: Umbral moonshine conjecture proven by Duncan-Griffin-Ono
2016: Generalized Umbral moonshine established by Cheng-de Lange-Whalen
A class of 23 moonshines, including Mathieu moonshine. Classified by Niemeier lattices. Involves also non-sporadic groups!
But it doesn’t stop there!
2012: Umbral moonshine proposed by Cheng-Duncan-Harvey
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
A class of 23 moonshines, including Mathieu moonshine. Classified by Niemeier lattices. Involves also non-sporadic groups!
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):
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
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
Def (Zwegers, Zagier):
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
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
Def (Zwegers, Zagier):
Umbral moonshine arises for weight k = 1/2
Pariah moonshine
2017: Duncan, Mertens, Ono proposes moonshine for two of the Pariahs
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
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
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
So what is moonshine, really?
So what is moonshine, really?
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?
Roles of vertex operator algebras/CFT?
chiral de Rham complex?(Song)
Roles of vertex operator algebras/CFT?
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)
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”
Umbral moonshine Thompson moonshine
weight 1/2 mock modular forms
Pariah moonshine weight 3/2 mock modular forms
no genus zero property
(Harvey)
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”
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?