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