Moonshine With A Twist Of Geometry

Sean Colin-Ellerin

November 14, 2016

Outline

I What is Moonshine?

I Groups and Representations

I Modular Forms

I Monstrous Moonshine

I K3 Surfaces and Topological Invariants

I Discriminant Property

Outline

I What is Moonshine?

I Groups and Representations

I Modular Forms

I Monstrous Moonshine

I K3 Surfaces and Topological Invariants

I Discriminant Property

Outline

I What is Moonshine?

I Groups and Representations

I Modular Forms

I Monstrous Moonshine

I K3 Surfaces and Topological Invariants

I Discriminant Property

Outline

I What is Moonshine?

I Groups and Representations

I Modular Forms

I Monstrous Moonshine

I K3 Surfaces and Topological Invariants

I Discriminant Property

Outline

I What is Moonshine?

I Groups and Representations

I Modular Forms

I Monstrous Moonshine

I K3 Surfaces and Topological Invariants

I Discriminant Property

Outline

I What is Moonshine?

I Groups and Representations

I Modular Forms

I Monstrous Moonshine

I K3 Surfaces and Topological Invariants

I Discriminant Property

What is Moonshine?

+ Geometry

Groups and Representations

I Group = Symmetries

Ex. D8 = symmetries of square

I Representations give action of group on objects in terms ofmatrices

Groups and Representations

I Group = Symmetries

Ex. D8 = symmetries of square

I Representations give action of group on objects in terms ofmatrices

Modular Forms

I A torus is given by two vectors (periods) in upper-half planeH, but one parameter τ

I Remains untwisted if we act withSL2(Z) = {

(a bc d

)| ad − bc 6= 0} by z 7→ az+b

cz+d

I Modular form of weight k is holomorphic map f : H→ C thatis the same on equivalent tori (up to prefactor):

f

(az + b

cz + d

)= (cz + d)k f (z)

(a bc d

)∈ SL2(Z)

I Important in string theory because tori are good candidatesfor string compactifications, modular forms can be expandedin Laurent series

f (z) =∞∑n=0

anqn q = e2πiz ,

which is partition function

Modular Forms

I A torus is given by two vectors (periods) in upper-half planeH, but one parameter τ

I Remains untwisted if we act withSL2(Z) = {

(a bc d

)| ad − bc 6= 0} by z 7→ az+b

cz+d

I Modular form of weight k is holomorphic map f : H→ C thatis the same on equivalent tori (up to prefactor):

f

(az + b

cz + d

)= (cz + d)k f (z)

(a bc d

)∈ SL2(Z)

I Important in string theory because tori are good candidatesfor string compactifications, modular forms can be expandedin Laurent series

f (z) =∞∑n=0

anqn q = e2πiz ,

which is partition function

Modular Forms

I A torus is given by two vectors (periods) in upper-half planeH, but one parameter τ

I Remains untwisted if we act withSL2(Z) = {

(a bc d

)| ad − bc 6= 0} by z 7→ az+b

cz+d

I Modular form of weight k is holomorphic map f : H→ C thatis the same on equivalent tori (up to prefactor):

f

(az + b

cz + d

)= (cz + d)k f (z)

(a bc d

)∈ SL2(Z)

I Important in string theory because tori are good candidatesfor string compactifications, modular forms can be expandedin Laurent series

f (z) =∞∑n=0

anqn q = e2πiz ,

which is partition function

Modular Forms

I A torus is given by two vectors (periods) in upper-half planeH, but one parameter τ

I Remains untwisted if we act withSL2(Z) = {

(a bc d

)| ad − bc 6= 0} by z 7→ az+b

cz+d

I Modular form of weight k is holomorphic map f : H→ C thatis the same on equivalent tori (up to prefactor):

f

(az + b

cz + d

)= (cz + d)k f (z)

(a bc d

)∈ SL2(Z)

I Important in string theory because tori are good candidatesfor string compactifications, modular forms can be expandedin Laurent series

f (z) =∞∑n=0

anqn q = e2πiz ,

which is partition function

Monstrous Moonshine

I Special modular function of weight 0 called j-function:

j(z) = q−1 + 744 + 196884q + 21493760q2 + . . .

gives change of coordinates on sphere

I Character table of Monster group M

1+196883 = 1968841+196883+21296876 = 21493760

I 1978: McKay noticed this and sent it to Thompson and hewas astounded, called it Moonshine

Monstrous Moonshine

I Special modular function of weight 0 called j-function:

j(z) = q−1 + 744 + 196884q + 21493760q2 + . . .

gives change of coordinates on sphere

I Character table of Monster group M

1+196883 = 1968841+196883+21296876 = 21493760

I 1978: McKay noticed this and sent it to Thompson and hewas astounded, called it Moonshine

Monstrous Moonshine

I Special modular function of weight 0 called j-function:

j(z) = q−1 + 744 + 196884q + 21493760q2 + . . .

gives change of coordinates on sphere

I Character table of Monster group M

1+196883 = 1968841+196883+21296876 = 21493760

I 1978: McKay noticed this and sent it to Thompson and hewas astounded, called it Moonshine

Explanation

I McKay and Thompson hypothesized a function for eachconjugacy class Tg (z)

I 1988: Harvey et. al in “Beauty and the Beast” showed that a24-dimensional bosonic torus model has partition functionj(z)− 720 and symmetry group M, where each Tg

corresponds to different boundary conditions

I 1992: Borcherds, using results from string theory, shows eachTg is a change of coordinates on sphere (with finite numberof points removed) and modular for Gg < SL2(R)

I Prize question: Does there exist Monster space (manifold)?Professor Harvey and I are looking at candidate M8

0 fromphysics perspective

I Now there are many types of Moonshine: Umbral, Thompson,Conway, Rudvalis, Baby Monster, Mathieu,...all different!

Explanation

I McKay and Thompson hypothesized a function for eachconjugacy class Tg (z)

I 1988: Harvey et. al in “Beauty and the Beast” showed that a24-dimensional bosonic torus model has partition functionj(z)− 720 and symmetry group M, where each Tg

corresponds to different boundary conditions

I 1992: Borcherds, using results from string theory, shows eachTg is a change of coordinates on sphere (with finite numberof points removed) and modular for Gg < SL2(R)

I Prize question: Does there exist Monster space (manifold)?Professor Harvey and I are looking at candidate M8

0 fromphysics perspective

I Now there are many types of Moonshine: Umbral, Thompson,Conway, Rudvalis, Baby Monster, Mathieu,...all different!

Explanation

I McKay and Thompson hypothesized a function for eachconjugacy class Tg (z)

I 1988: Harvey et. al in “Beauty and the Beast” showed that a24-dimensional bosonic torus model has partition functionj(z)− 720 and symmetry group M, where each Tg

corresponds to different boundary conditions

I 1992: Borcherds, using results from string theory, shows eachTg is a change of coordinates on sphere (with finite numberof points removed) and modular for Gg < SL2(R)

I Prize question: Does there exist Monster space (manifold)?Professor Harvey and I are looking at candidate M8

0 fromphysics perspective

I Now there are many types of Moonshine: Umbral, Thompson,Conway, Rudvalis, Baby Monster, Mathieu,...all different!

Explanation

I McKay and Thompson hypothesized a function for eachconjugacy class Tg (z)

I 1988: Harvey et. al in “Beauty and the Beast” showed that a24-dimensional bosonic torus model has partition functionj(z)− 720 and symmetry group M, where each Tg

corresponds to different boundary conditions

I 1992: Borcherds, using results from string theory, shows eachTg is a change of coordinates on sphere (with finite numberof points removed) and modular for Gg < SL2(R)

I Prize question: Does there exist Monster space (manifold)?Professor Harvey and I are looking at candidate M8

0 fromphysics perspective

I Now there are many types of Moonshine: Umbral, Thompson,Conway, Rudvalis, Baby Monster, Mathieu,...all different!

Explanation

I McKay and Thompson hypothesized a function for eachconjugacy class Tg (z)

I 1988: Harvey et. al in “Beauty and the Beast” showed that a24-dimensional bosonic torus model has partition functionj(z)− 720 and symmetry group M, where each Tg

corresponds to different boundary conditions

I 1992: Borcherds, using results from string theory, shows eachTg is a change of coordinates on sphere (with finite numberof points removed) and modular for Gg < SL2(R)

I Prize question: Does there exist Monster space (manifold)?Professor Harvey and I are looking at candidate M8

0 fromphysics perspective

I Now there are many types of Moonshine: Umbral, Thompson,Conway, Rudvalis, Baby Monster, Mathieu,...all different!

K3 Surfaces & Topological Invariants

I String theory: 26D → 11D → 10D, but we only see4-dimensions so we have R3,1 ×M, where M is Calabi-Yaumanifold

I Two types of 4-dimensional Calabi-Yau, one is K3 surface

Consider a 4-dimensional torus T4 and identify x ∼ −x to getM = T4/Z2

I Invariant of all models on K3 surfaces called elliptic genusZK3, which counts certain types of states of any model on aK3 surface

I Decompose into massless and massive partsZK3(τ, z) = 20 chR̃

h= 14,l=0

(τ, z)− 2 chR̃h= 1

4,l= 1

2

(τ, z) +∑∞

n=1 A(n) chR̃h=n+ 1

4,l= 1

2

(τ, z)

I 2010: Eguchi, Ooguri, Tachikawa notice A(n) is dimension ofnth irreducible representation of M24 <M

I Gannon showed true for all n and for Ag (n), where we twiststates by action of g ∈ M24, but it has been shown no K3theory has M24 symmetry!

K3 Surfaces & Topological Invariants

I String theory: 26D → 11D → 10D, but we only see4-dimensions so we have R3,1 ×M, where M is Calabi-Yaumanifold

I Two types of 4-dimensional Calabi-Yau, one is K3 surface

Consider a 4-dimensional torus T4 and identify x ∼ −x to getM = T4/Z2

I Invariant of all models on K3 surfaces called elliptic genusZK3, which counts certain types of states of any model on aK3 surface

I Decompose into massless and massive partsZK3(τ, z) = 20 chR̃

h= 14,l=0

(τ, z)− 2 chR̃h= 1

4,l= 1

2

(τ, z) +∑∞

n=1 A(n) chR̃h=n+ 1

4,l= 1

2

(τ, z)

I 2010: Eguchi, Ooguri, Tachikawa notice A(n) is dimension ofnth irreducible representation of M24 <M

I Gannon showed true for all n and for Ag (n), where we twiststates by action of g ∈ M24, but it has been shown no K3theory has M24 symmetry!

K3 Surfaces & Topological Invariants

I String theory: 26D → 11D → 10D, but we only see4-dimensions so we have R3,1 ×M, where M is Calabi-Yaumanifold

I Two types of 4-dimensional Calabi-Yau, one is K3 surface

Consider a 4-dimensional torus T4 and identify x ∼ −x to getM = T4/Z2

I Invariant of all models on K3 surfaces called elliptic genusZK3, which counts certain types of states of any model on aK3 surface

I Decompose into massless and massive partsZK3(τ, z) = 20 chR̃

h= 14,l=0

(τ, z)− 2 chR̃h= 1

4,l= 1

2

(τ, z) +∑∞

n=1 A(n) chR̃h=n+ 1

4,l= 1

2

(τ, z)

I 2010: Eguchi, Ooguri, Tachikawa notice A(n) is dimension ofnth irreducible representation of M24 <M

I Gannon showed true for all n and for Ag (n), where we twiststates by action of g ∈ M24, but it has been shown no K3theory has M24 symmetry!

K3 Surfaces & Topological Invariants

I String theory: 26D → 11D → 10D, but we only see4-dimensions so we have R3,1 ×M, where M is Calabi-Yaumanifold

I Two types of 4-dimensional Calabi-Yau, one is K3 surface

Consider a 4-dimensional torus T4 and identify x ∼ −x to getM = T4/Z2

I Invariant of all models on K3 surfaces called elliptic genusZK3, which counts certain types of states of any model on aK3 surface

I Decompose into massless and massive partsZK3(τ, z) = 20 chR̃

h= 14,l=0

(τ, z)− 2 chR̃h= 1

4,l= 1

2

(τ, z) +∑∞

n=1 A(n) chR̃h=n+ 1

4,l= 1

2

(τ, z)

I 2010: Eguchi, Ooguri, Tachikawa notice A(n) is dimension ofnth irreducible representation of M24 <M

I Gannon showed true for all n and for Ag (n), where we twiststates by action of g ∈ M24, but it has been shown no K3theory has M24 symmetry!

K3 Surfaces & Topological Invariants

I String theory: 26D → 11D → 10D, but we only see4-dimensions so we have R3,1 ×M, where M is Calabi-Yaumanifold

I Two types of 4-dimensional Calabi-Yau, one is K3 surface

Consider a 4-dimensional torus T4 and identify x ∼ −x to getM = T4/Z2

I Invariant of all models on K3 surfaces called elliptic genusZK3, which counts certain types of states of any model on aK3 surface

I Decompose into massless and massive partsZK3(τ, z) = 20 chR̃

h= 14,l=0

(τ, z)− 2 chR̃h= 1

4,l= 1

2

(τ, z) +∑∞

n=1 A(n) chR̃h=n+ 1

4,l= 1

2

(τ, z)

I 2010: Eguchi, Ooguri, Tachikawa notice A(n) is dimension ofnth irreducible representation of M24 <M

I Gannon showed true for all n and for Ag (n), where we twiststates by action of g ∈ M24, but it has been shown no K3theory has M24 symmetry!

K3 Surfaces & Topological Invariants

I String theory: 26D → 11D → 10D, but we only see4-dimensions so we have R3,1 ×M, where M is Calabi-Yaumanifold

I Two types of 4-dimensional Calabi-Yau, one is K3 surface

Consider a 4-dimensional torus T4 and identify x ∼ −x to getM = T4/Z2

I Invariant of all models on K3 surfaces called elliptic genusZK3, which counts certain types of states of any model on aK3 surface

I Decompose into massless and massive partsZK3(τ, z) = 20 chR̃

h= 14,l=0

(τ, z)− 2 chR̃h= 1

4,l= 1

2

(τ, z) +∑∞

n=1 A(n) chR̃h=n+ 1

4,l= 1

2

(τ, z)

I 2010: Eguchi, Ooguri, Tachikawa notice A(n) is dimension ofnth irreducible representation of M24 <M

I Gannon showed true for all n and for Ag (n), where we twiststates by action of g ∈ M24, but it has been shown no K3theory has M24 symmetry!

Discriminant Property

I Expand massive part of ZK3∑∞n=1 A(n) chR̃

h=n+ 14,l= 1

2

(τ, z) = 2 θ1(τ,z)η3(τ)

(−q−1/8 + 45q7/8 + 231q15/8 + . . .)

I Look at exponent D/8; if D = nλ2 with n and λ coprime,then there exists representation ρ and g ∈ M24 such that

ρn(g), ρ∗n(g) ∈ Q[√−n] = {a + b

√−n | a, b ∈ Q}

I Conjecture: The corresponding representation given whosedimension is coefficient of qD/8 contains at least one pair ofsuch ρn, ρ∗n

I Physically, this says that the energy of states tells us in whichrepresentations they transform

I Checked to very high order and for 6 different types ofMoonshine—want to find some underlying physicalexplanation of why this is true

Discriminant Property

I Expand massive part of ZK3∑∞n=1 A(n) chR̃

h=n+ 14,l= 1

2

(τ, z) = 2 θ1(τ,z)η3(τ)

(−q−1/8 + 45q7/8 + 231q15/8 + . . .)

I Look at exponent D/8; if D = nλ2 with n and λ coprime,then there exists representation ρ and g ∈ M24 such that

ρn(g), ρ∗n(g) ∈ Q[√−n] = {a + b

√−n | a, b ∈ Q}

I Conjecture: The corresponding representation given whosedimension is coefficient of qD/8 contains at least one pair ofsuch ρn, ρ∗n

I Physically, this says that the energy of states tells us in whichrepresentations they transform

I Checked to very high order and for 6 different types ofMoonshine—want to find some underlying physicalexplanation of why this is true

Discriminant Property

I Expand massive part of ZK3∑∞n=1 A(n) chR̃

h=n+ 14,l= 1

2

(τ, z) = 2 θ1(τ,z)η3(τ)

(−q−1/8 + 45q7/8 + 231q15/8 + . . .)

I Look at exponent D/8; if D = nλ2 with n and λ coprime,then there exists representation ρ and g ∈ M24 such that

ρn(g), ρ∗n(g) ∈ Q[√−n] = {a + b

√−n | a, b ∈ Q}

I Conjecture: The corresponding representation given whosedimension is coefficient of qD/8 contains at least one pair ofsuch ρn, ρ∗n

I Physically, this says that the energy of states tells us in whichrepresentations they transform

I Checked to very high order and for 6 different types ofMoonshine—want to find some underlying physicalexplanation of why this is true

Discriminant Property

I Expand massive part of ZK3∑∞n=1 A(n) chR̃

h=n+ 14,l= 1

2

(τ, z) = 2 θ1(τ,z)η3(τ)

(−q−1/8 + 45q7/8 + 231q15/8 + . . .)

I Look at exponent D/8; if D = nλ2 with n and λ coprime,then there exists representation ρ and g ∈ M24 such that

ρn(g), ρ∗n(g) ∈ Q[√−n] = {a + b

√−n | a, b ∈ Q}

I Conjecture: The corresponding representation given whosedimension is coefficient of qD/8 contains at least one pair ofsuch ρn, ρ∗n

I Physically, this says that the energy of states tells us in whichrepresentations they transform

I Checked to very high order and for 6 different types ofMoonshine—want to find some underlying physicalexplanation of why this is true