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A Heterotic Standard Model
Volker Braun
November 3, 2005
A Heterotic Standard Model / University of North Carolina at Chapel Hill 1 / 54

Overview
Introduction
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
Introduction
The Calabi-Yau
The Vector Bundle
A First Heterotic Standard Model
Spectral Sequences
A New Heterotic Standard Model
Conclusion
hep-th/0410055: Elliptic Calabi-Yau Threefolds with Z 3 × Z 3 Wilson Lines
hep-th/0501070: A Heterotic Standard Model
hep-th/0502155: A Standard Model from the E 8 × E 8 Heterotic Superstring
hep-th/0505041: Vector Bundle Extensions, Sheaf Cohomology, and the Heterotic Standard Model
hep-th/0509051: Heterotic Standard Model Moduli
hep-th/0510142: Moduli Dependent µ-Terms in a Heterotic Standard Model
A Heterotic Standard Model / University of North Carolina at Chapel Hill 2 / 54

Introduction
❖ Introduction
❖ An Organizational
Principle
❖ Wilson Line
Breaking
❖ More Group
Theory
❖ Yet More Group
Theory
❖ Wish List
Introduction
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 3 / 54

Introduction
Introduction
❖ Introduction
❖ An Organizational
Principle
❖ Wilson Line
Breaking
❖ More Group
Theory
❖ Yet More Group
Theory
❖ Wish List
● Geometric compactification of the E 8 × E 8
heterotic string.
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 4 / 54

Introduction
Introduction
❖ Introduction
❖ An Organizational
Principle
❖ Wilson Line
Breaking
❖ More Group
Theory
❖ Yet More Group
Theory
❖ Wish List
● Geometric compactification of the E 8 × E 8
heterotic string.
● d = 4, N = 1 ⇒ stable background.
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 4 / 54

Introduction
Introduction
❖ Introduction
❖ An Organizational
Principle
❖ Wilson Line
Breaking
❖ More Group
Theory
❖ Yet More Group
Theory
❖ Wish List
● Geometric compactification of the E 8 × E 8
heterotic string.
● d = 4, N = 1 ⇒ stable background.
● SU(3) C × SU(2) L × U(1) Y .
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 4 / 54

Introduction
Introduction
❖ Introduction
❖ An Organizational
Principle
❖ Wilson Line
Breaking
❖ More Group
Theory
❖ Yet More Group
Theory
❖ Wish List
● Geometric compactification of the E 8 × E 8
heterotic string.
● d = 4, N = 1 ⇒ stable background.
● /////////////////////////////////////////
SU(3) C × SU(2) L × U(1) Y .
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 4 / 54

Introduction
Introduction
❖ Introduction
❖ An Organizational
Principle
❖ Wilson Line
Breaking
❖ More Group
Theory
❖ Yet More Group
Theory
❖ Wish List
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
● Geometric compactification of the E 8 × E 8
heterotic string.
● d = 4, N = 1 ⇒ stable background.
● /////////////////////////////////////////
SU(3) C × SU(2) L × U(1) Y .
●
SU(3) C × SU(2) L × U(1) Y × U(1) B−L
⇒ proton decay suppressed.
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 4 / 54

Introduction
Introduction
❖ Introduction
❖ An Organizational
Principle
❖ Wilson Line
Breaking
❖ More Group
Theory
❖ Yet More Group
Theory
❖ Wish List
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
● Geometric compactification of the E 8 × E 8
heterotic string.
● d = 4, N = 1 ⇒ stable background.
● /////////////////////////////////////////
SU(3) C × SU(2) L × U(1) Y .
●
●
SU(3) C × SU(2) L × U(1) Y × U(1) B−L
⇒ proton decay suppressed.
No exotic matter.
A New Heterotic
Standard Model
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 4 / 54

Introduction
Introduction
❖ Introduction
❖ An Organizational
Principle
❖ Wilson Line
Breaking
❖ More Group
Theory
❖ Yet More Group
Theory
❖ Wish List
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
● Geometric compactification of the E 8 × E 8
heterotic string.
● d = 4, N = 1 ⇒ stable background.
● /////////////////////////////////////////
SU(3) C × SU(2) L × U(1) Y .
●
●
●
SU(3) C × SU(2) L × U(1) Y × U(1) B−L
⇒ proton decay suppressed.
No exotic matter.
All of the ordinary matter fields
(including right-handed Neutrino).
A Heterotic Standard Model / University of North Carolina at Chapel Hill 4 / 54

An Organizational Principle
Introduction
❖ Introduction
❖ An Organizational
Principle
❖ Wilson Line
Breaking
❖ More Group
Theory
❖ Yet More Group
Theory
❖ Wish List
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Ancient Lore: Spin(10) GUT with Z 3 × Z 3
Wilson lines “works”:
16 of Spin(10): Breaks into one family of
quarks and leptons including a
right-handed Neutrino.
16 of Spin(10): Anti-family.
10 = 10 of Spin(10): Higgs and color triplets.
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 5 / 54

An Organizational Principle
Introduction
❖ Introduction
❖ An Organizational
Principle
❖ Wilson Line
Breaking
❖ More Group
Theory
❖ Yet More Group
Theory
❖ Wish List
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
Ancient Lore: Spin(10) GUT with Z 3 × Z 3
Wilson lines “works”:
16 of Spin(10): Breaks into one family of
quarks and leptons including a
right-handed Neutrino.
16 of Spin(10): Anti-family.
10 = 10 of Spin(10): Higgs and color triplets.
Compactification scale ∼ GUT scale
... but nice way to package representations.
A Heterotic Standard Model / University of North Carolina at Chapel Hill 5 / 54

Wilson Line Breaking
Introduction
❖ Introduction
❖ An Organizational
Principle
❖ Wilson Line
Breaking
❖ More Group
Theory
❖ Yet More Group
Theory
❖ Wish List
Spin(10) ⊃ SU(3)×SU(2)×U(1)×U(1)×Z 3 × Z 3
{
Standard Model
gauge group
}
×U(1) B−L ×{Wilson lines}
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Z 3 × Z 3 is smallest Wilson line possible.
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 6 / 54

Wilson Line Breaking
Introduction
❖ Introduction
❖ An Organizational
Principle
❖ Wilson Line
Breaking
❖ More Group
Theory
❖ Yet More Group
Theory
❖ Wish List
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
Spin(10) ⊃ SU(3)×SU(2)×U(1)×U(1)×Z 3 × Z 3
16 = χ 2 1χ 2
(
3, 2, 1, 1
)
⊕ χ
2
1
(
1, 1, 6, 3
)
⊕
⊕ χ 2 1 χ2 2(
3, 1, −4, −1
)
⊕ χ
2
2
(
3, 1, 2, −1
)
⊕
⊕ ( 1, 2, −3, −3 ) ⊕ χ 1
(
1, 1, 0, 3
)
( ) ( )
10 = χ 1 1, 2, 3, 0 ⊕ χ1 χ 2 3, 1, −2, −2 ⊕
) ( )
⊕ χ1( 2 1, 2, −3, 0 ⊕ χ
2
1 χ 2 2 3, 1, 2, 2
A Heterotic Standard Model / University of North Carolina at Chapel Hill 6 / 54

Wilson Line Breaking
Introduction
❖ Introduction
❖ An Organizational
Principle
❖ Wilson Line
Breaking
❖ More Group
Theory
❖ Yet More Group
Theory
❖ Wish List
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
Spin(10) ⊃ SU(3)×SU(2)×U(1)×U(1)×Z 3 × Z 3
16 = χ 2 1χ 2
(
3, 2, 1, 1
)
⊕ χ
2
1
(
1, 1, 6, 3
)
⊕
⊕ χ 2 1 χ2 2(
3, 1, −4, −1
)
⊕ χ
2
2
(
3, 1, 2, −1
)
⊕
⊕ ( 1, 2, −3, −3 ) ⊕ χ 1
(
1, 1, 0, 3
)
( ) ( )
10 = χ 1 1, 2, 3, 0 ⊕ χ1 χ 2 3, 1, −2, −2 ⊕
) ( )
⊕ χ1( 2 1, 2, −3, 0 ⊕ χ
2
1 χ 2 2 3, 1, 2, 2
Right-handed Neutrino
A Heterotic Standard Model / University of North Carolina at Chapel Hill 6 / 54

More Group Theory
Introduction
❖ Introduction
❖ An Organizational
Principle
❖ Wilson Line
Breaking
❖ More Group
Theory
❖ Yet More Group
Theory
❖ Wish List
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
G = Z 3 × Z 3 = G 1 × G 2
Fix generators g 1 and g 2 .
Characters (=1-d representations): Denote
generators by χ 1 and χ 2 , where (ω = e 2πi
3 )
χ 1 (g 1 ) = ω χ 1 (g 2 ) = 1
χ 2 (g 1 ) = 1 χ 2 (g 2 ) = ω .
All other characters are products of χ 1 and χ 2 .
A Heterotic Standard Model / University of North Carolina at Chapel Hill 7 / 54

Yet More Group Theory
Introduction
❖ Introduction
❖ An Organizational
Principle
❖ Wilson Line
Breaking
❖ More Group
Theory
❖ Yet More Group
Theory
❖ Wish List
The Calabi-Yau
Maximal regular subgroup
SU(4) × Spin(10) ⊂ E 8 :
SU(4)
Spin(10)
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
The adjoint of E 8 (fermions in the E 8 × E 8
heterotic string) decomposes as
248 = ( 1, 45 ) ⊕ ( 15, 1 ) ⊕ ( 4, 16 ) ⊕ ( 4, 16 ) ⊕ ( 6, 10 )
A Heterotic Standard Model / University of North Carolina at Chapel Hill 8 / 54

Wish List
Introduction
❖ Introduction
❖ An Organizational
Principle
❖ Wilson Line
Breaking
❖ More Group
Theory
❖ Yet More Group
Theory
❖ Wish List
To make use of this group theory, we would like
● A Calabi-Yau threefold X with Z 3 × Z 3
fundamental group.
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 9 / 54

Wish List
Introduction
❖ Introduction
❖ An Organizational
Principle
❖ Wilson Line
Breaking
❖ More Group
Theory
❖ Yet More Group
Theory
❖ Wish List
To make use of this group theory, we would like
● A Calabi-Yau threefold X with Z 3 × Z 3
fundamental group.
●
The Calabi-Yau should be torus fibered.
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 9 / 54

Wish List
Introduction
❖ Introduction
❖ An Organizational
Principle
❖ Wilson Line
Breaking
❖ More Group
Theory
❖ Yet More Group
Theory
❖ Wish List
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
To make use of this group theory, we would like
● A Calabi-Yau threefold X with Z 3 × Z 3
fundamental group.
●
●
The Calabi-Yau should be torus fibered.
A SU(4) ⊂ E 8 instanton leaves Spin(10)
unbroken, so we want a rank 4 stable
holomorphic vector bundle V on X.
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 9 / 54

Wish List
Introduction
❖ Introduction
❖ An Organizational
Principle
❖ Wilson Line
Breaking
❖ More Group
Theory
❖ Yet More Group
Theory
❖ Wish List
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
To make use of this group theory, we would like
● A Calabi-Yau threefold X with Z 3 × Z 3
fundamental group.
●
●
●
The Calabi-Yau should be torus fibered.
A SU(4) ⊂ E 8 instanton leaves Spin(10)
unbroken, so we want a rank 4 stable
holomorphic vector bundle V on X.
With the “right” cohomology groups (low
energy spectrum).
A Heterotic Standard Model / University of North Carolina at Chapel Hill 9 / 54

Introduction
The Calabi-Yau
❖ Calabi-Yau
Introduction
❖ Calabi-Yau
Construction
❖ Calabi-Yau
Properties
❖ Group Actions on
the Base I
❖ Group Actions on
the Base II
❖ Invariant
Cohomology
❖ Divisors on the
Base
❖ Divisors on the
Calabi-Yau
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 10 / 54

Calabi-Yau Introduction
Introduction
The Calabi-Yau
❖ Calabi-Yau
Introduction
❖ Calabi-Yau
Construction
❖ Calabi-Yau
Properties
❖ Group Actions on
the Base I
❖ Group Actions on
the Base II
❖ Invariant
Cohomology
❖ Divisors on the
Base
❖ Divisors on the
Calabi-Yau
Calabi-Yau
threefold X with
π 1 (X) = Z 3 × Z 3
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 11 / 54

Calabi-Yau Introduction
Introduction
The Calabi-Yau
❖ Calabi-Yau
Introduction
❖ Calabi-Yau
Construction
❖ Calabi-Yau
Properties
❖ Group Actions on
the Base I
❖ Group Actions on
the Base II
❖ Invariant
Cohomology
❖ Divisors on the
Base
❖ Divisors on the
Calabi-Yau
Work with
Simply connected
Calabi-Yau
threefold ˜X with
free Z 3 × Z 3 action
=
Have in mind
Calabi-Yau
threefold X with
π 1 (X) = Z 3 × Z 3
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 11 / 54

Calabi-Yau Introduction
Introduction
The Calabi-Yau
❖ Calabi-Yau
Introduction
❖ Calabi-Yau
Construction
❖ Calabi-Yau
Properties
❖ Group Actions on
the Base I
❖ Group Actions on
the Base II
❖ Invariant
Cohomology
❖ Divisors on the
Base
❖ Divisors on the
Calabi-Yau
Work with
Simply connected
Calabi-Yau
threefold ˜X with
free Z 3 × Z 3 action
=
Have in mind
Calabi-Yau
threefold X with
π 1 (X) = Z 3 × Z 3
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
elliptically fibered
torus fibered
(torus fibered (assuming Z 3 × Z 3
with section)
preserves fibration)
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 11 / 54

Calabi-Yau Construction
Introduction
The Calabi-Yau
❖ Calabi-Yau
Introduction
❖ Calabi-Yau
Construction
❖ Calabi-Yau
Properties
❖ Group Actions on
the Base I
❖ Group Actions on
the Base II
❖ Invariant
Cohomology
❖ Divisors on the
Base
❖ Divisors on the
Calabi-Yau
Start with two dP 9 surfaces B 1 and B 2 .
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 12 / 54

Calabi-Yau Construction
Introduction
The Calabi-Yau
❖ Calabi-Yau
Introduction
❖ Calabi-Yau
Construction
❖ Calabi-Yau
Properties
❖ Group Actions on
the Base I
❖ Group Actions on
the Base II
❖ Invariant
Cohomology
❖ Divisors on the
Base
❖ Divisors on the
Calabi-Yau
B 1
β 1
B 2
β 2
Start with two dP 9 surfaces and .
P 1
P 1
Note: dP 9 are elliptically fibered; Fibers over a
generic point x ∈ P 1 are
β −1
1 (x) ≃ T 2 ⊂ B 1 , β −1
2 (x) ≃ T 2 ⊂ B 2 .
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 12 / 54

Calabi-Yau Construction
Introduction
The Calabi-Yau
❖ Calabi-Yau
Introduction
❖ Calabi-Yau
Construction
❖ Calabi-Yau
Properties
❖ Group Actions on
the Base I
❖ Group Actions on
the Base II
❖ Invariant
Cohomology
❖ Divisors on the
Base
❖ Divisors on the
Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
B 1
β 1
B 2
β 2
Start with two dP 9 surfaces and .
P 1
P 1
Note: dP 9 are elliptically fibered; Fibers over a
generic point x ∈ P 1 are
β −1
1 (x) ≃ T 2 ⊂ B 1 , β −1
2 (x) ≃ T 2 ⊂ B 2 .
The fiber product B 1 × P
1 B 2 is the fibration
over P 1 with fiber
β −1 (x) = β1 −1 (x) × β−1 2 (x)
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 12 / 54

Calabi-Yau Construction
Introduction
The Calabi-Yau
❖ Calabi-Yau
Introduction
❖ Calabi-Yau
Construction
❖ Calabi-Yau
Properties
❖ Group Actions on
the Base I
❖ Group Actions on
the Base II
❖ Invariant
Cohomology
❖ Divisors on the
Base
❖ Divisors on the
Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
B 1
β 1
B 2
β 2
Start with two dP 9 surfaces and .
P 1
P 1
Note: dP 9 are elliptically fibered; Fibers over a
generic point x ∈ P 1 are
β −1
1 (x) ≃ T 2 ⊂ B 1 , β −1
2 (x) ≃ T 2 ⊂ B 2 .
The fiber product B 1 × P
1 B 2 is the fibration
over P 1 with fiber
β −1 (x) = β1 −1 (x) × β−1 2 (x)
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 12 / 54

Calabi-Yau Properties
Introduction
The Calabi-Yau
❖ Calabi-Yau
Introduction
❖ Calabi-Yau
Construction
❖ Calabi-Yau
Properties
❖ Group Actions on
the Base I
❖ Group Actions on
the Base II
❖ Invariant
Cohomology
❖ Divisors on the
Base
❖ Divisors on the
Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
●
˜X
def
= B 1 × P
1 B 2 is a simply connected
Calabi-Yau threefold, c 1 ( ˜X) = 0.
A Heterotic Standard Model / University of North Carolina at Chapel Hill 13 / 54

Calabi-Yau Properties
Introduction
The Calabi-Yau
❖ Calabi-Yau
Introduction
❖ Calabi-Yau
Construction
❖ Calabi-Yau
Properties
❖ Group Actions on
the Base I
❖ Group Actions on
the Base II
❖ Invariant
Cohomology
❖ Divisors on the
Base
❖ Divisors on the
Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
●
●
def
˜X = B 1 × P
1 B 2 is a simply connected
Calabi-Yau threefold, c 1 ( ˜X) = 0.
Every elliptically fibered Calabi-Yau over a
dP 9 is such a fiber product.
A Heterotic Standard Model / University of North Carolina at Chapel Hill 13 / 54

Calabi-Yau Properties
Introduction
The Calabi-Yau
❖ Calabi-Yau
Introduction
❖ Calabi-Yau
Construction
❖ Calabi-Yau
Properties
❖ Group Actions on
the Base I
❖ Group Actions on
the Base II
❖ Invariant
Cohomology
❖ Divisors on the
Base
❖ Divisors on the
Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
●
●
●
def
˜X = B 1 × P
1 B 2 is a simply connected
Calabi-Yau threefold, c 1 ( ˜X) = 0.
Every elliptically fibered Calabi-Yau over a
dP 9 is such a fiber product.
h 1,1( ˜X) = 19 = h
2,1 ( ˜X)
A Heterotic Standard Model / University of North Carolina at Chapel Hill 13 / 54

Calabi-Yau Properties
Introduction
The Calabi-Yau
❖ Calabi-Yau
Introduction
❖ Calabi-Yau
Construction
❖ Calabi-Yau
Properties
❖ Group Actions on
the Base I
❖ Group Actions on
the Base II
❖ Invariant
Cohomology
❖ Divisors on the
Base
❖ Divisors on the
Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
●
●
●
●
def
˜X = B 1 × P
1 B 2 is a simply connected
Calabi-Yau threefold, c 1 ( ˜X) = 0.
Every elliptically fibered Calabi-Yau over a
dP 9 is such a fiber product.
h 1,1( ˜X) = 19 = h
2,1 ( ˜X)
Group actions on B 1 , B 2 lift to ˜X if their
action on the common base P 1 is identical.
A Heterotic Standard Model / University of North Carolina at Chapel Hill 13 / 54

Group Actions on the Base I
Introduction
The Calabi-Yau
❖ Calabi-Yau
Introduction
❖ Calabi-Yau
Construction
❖ Calabi-Yau
Properties
❖ Group Actions on
the Base I
❖ Group Actions on
the Base II
❖ Invariant
Cohomology
❖ Divisors on the
Base
❖ Divisors on the
Calabi-Yau
The Vector Bundle
We classified all Z 3 × Z 3 actions on dP 9
surfaces.
The moduli space looks like this:
A one parameter family
3 special limits
3 isolated cases
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 14 / 54

Group Actions on the Base II
Introduction
The Calabi-Yau
❖ Calabi-Yau
Introduction
❖ Calabi-Yau
Construction
❖ Calabi-Yau
Properties
❖ Group Actions on
the Base I
❖ Group Actions on
the Base II
❖ Invariant
Cohomology
❖ Divisors on the
Base
❖ Divisors on the
Calabi-Yau
All such dP 9 surfaces with G = Z 3 × Z 3 action
give rise to a G action on ˜X.
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 15 / 54

Group Actions on the Base II
Introduction
The Calabi-Yau
❖ Calabi-Yau
Introduction
❖ Calabi-Yau
Construction
❖ Calabi-Yau
Properties
❖ Group Actions on
the Base I
❖ Group Actions on
the Base II
❖ Invariant
Cohomology
❖ Divisors on the
Base
❖ Divisors on the
Calabi-Yau
All such dP 9 surfaces with G = Z 3 × Z 3 action
give rise to a G action on ˜X.
●
The 3 isolated cases never yield a free
Z 3 × Z 3 action.
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 15 / 54

Group Actions on the Base II
Introduction
The Calabi-Yau
❖ Calabi-Yau
Introduction
❖ Calabi-Yau
Construction
❖ Calabi-Yau
Properties
❖ Group Actions on
the Base I
❖ Group Actions on
the Base II
❖ Invariant
Cohomology
❖ Divisors on the
Base
❖ Divisors on the
Calabi-Yau
The Vector Bundle
All such dP 9 surfaces with G = Z 3 × Z 3 action
give rise to a G action on ˜X.
●
●
The 3 isolated cases never yield a free
Z 3 × Z 3 action.
The one-parameter family and its limits can
give a free Z 3 × Z 3 action on ˜X.
We only consider this one-parameter family in
the following.
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 15 / 54

Invariant Cohomology
Introduction
The Calabi-Yau
❖ Calabi-Yau
Introduction
❖ Calabi-Yau
Construction
❖ Calabi-Yau
Properties
❖ Group Actions on
the Base I
❖ Group Actions on
the Base II
❖ Invariant
Cohomology
❖ Divisors on the
Base
❖ Divisors on the
Calabi-Yau
G = Z 3 × Z 3 action free
⇒
H p,q( X ) = H p,q( ˜X)
G
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 16 / 54

Invariant Cohomology
Introduction
The Calabi-Yau
❖ Calabi-Yau
Introduction
❖ Calabi-Yau
Construction
❖ Calabi-Yau
Properties
❖ Group Actions on
the Base I
❖ Group Actions on
the Base II
❖ Invariant
Cohomology
❖ Divisors on the
Base
❖ Divisors on the
Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
G = Z 3 × Z 3 action free
⇒ H p,q( X ) = H p,q( G ˜X)
Hodge diamond h p,q (X) = 1 0 0 1
0 3 3 0
0 3 3 0
1 0 0 1
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 16 / 54

Invariant Cohomology
Introduction
The Calabi-Yau
❖ Calabi-Yau
Introduction
❖ Calabi-Yau
Construction
❖ Calabi-Yau
Properties
❖ Group Actions on
the Base I
❖ Group Actions on
the Base II
❖ Invariant
Cohomology
❖ Divisors on the
Base
❖ Divisors on the
Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
G = Z 3 × Z 3 action free
⇒
H p,q( X ) = H p,q( ˜X)
G
Hodge diamond h p,q (X) = 1 0 0 1
0 3 3 0
0 3 3 0
1 0 0 1
h 1,1( X ) = 3 dimensional space
of divisor classes.
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 16 / 54

Divisors on the Base
Introduction
The Calabi-Yau
❖ Calabi-Yau
Introduction
❖ Calabi-Yau
Construction
❖ Calabi-Yau
Properties
❖ Group Actions on
the Base I
❖ Group Actions on
the Base II
❖ Invariant
Cohomology
❖ Divisors on the
Base
❖ Divisors on the
Calabi-Yau
dim C = 3 : ˜X
π 1
π2
dim C = 2 : B 1
β 1
dim C = 1 : P 1
B 2
β 2
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 17 / 54

Divisors on the Base
Introduction
The Calabi-Yau
❖ Calabi-Yau
Introduction
❖ Calabi-Yau
Construction
❖ Calabi-Yau
Properties
❖ Group Actions on
the Base I
❖ Group Actions on
the Base II
❖ Invariant
Cohomology
❖ Divisors on the
Base
❖ Divisors on the
Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
dim C = 3 : ˜X
π 1
π2
dim C = 2 : B 1
β 1
dim C = 1 : P 1
B 2
β 2
Invariant divisors on the base B 1 , B 2 :
H 1,1( B 1 ) G = Cf 1 ⊕ Ct 1
H 1,1( B 2 ) G = Cf 2 ⊕ Ct 2
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 17 / 54

Divisors on the Calabi-Yau
Introduction
The Calabi-Yau
❖ Calabi-Yau
Introduction
❖ Calabi-Yau
Construction
❖ Calabi-Yau
Properties
❖ Group Actions on
the Base I
❖ Group Actions on
the Base II
❖ Invariant
Cohomology
❖ Divisors on the
Base
❖ Divisors on the
Calabi-Yau
Pull-back of divisors from the base
⎧
⎪⎨
˜X
π1 −1 (f 1) =
⎪⎩ T 4 fiber of
π −1
1 (t 1) def
⎫
⎪⎬
= π −1
⎪ ⎭
P 1
= τ 1 π2 −1 (t 2) = def
τ 2
2 (f 2) = def
φ
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 18 / 54

Divisors on the Calabi-Yau
Introduction
The Calabi-Yau
❖ Calabi-Yau
Introduction
❖ Calabi-Yau
Construction
❖ Calabi-Yau
Properties
❖ Group Actions on
the Base I
❖ Group Actions on
the Base II
❖ Invariant
Cohomology
❖ Divisors on the
Base
❖ Divisors on the
Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Pull-back of divisors from the base
⎧
⎪⎨
˜X
π1 −1 (f 1) =
⎪⎩ T 4 fiber of
π −1
1 (t 1) def
⎫
⎪⎬
= π −1
⎪ ⎭
P 1
= τ 1 π2 −1 (t 2) = def
τ 2
H 1,1( G ˜X) = Cπ
−1
1 (f 1) + Cπ1 −1 (t 1)+
+ Cπ2 −1 (f 2) + Cπ2 −1 (t 2) =
= Cφ ⊕ Cτ 1 ⊕ Cτ 2
2 (f 2) = def
φ
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 18 / 54

Introduction
The Calabi-Yau
The Vector Bundle
❖ Line Bundles
❖ Equivariant Line
Bundles I
❖ Equivariant Line
Bundles II
❖ Notation
❖ The Serre
Construction
❖ Equivariant Vector
Bundles
❖ Equivariant
Example
❖ Constructing
Vector Bundles
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 19 / 54

Line Bundles
Introduction
The Calabi-Yau
The Vector Bundle
❖ Line Bundles
❖ Equivariant Line
Bundles I
❖ Equivariant Line
Bundles II
❖ Notation
❖ The Serre
Construction
❖ Equivariant Vector
Bundles
❖ Equivariant
Example
❖ Constructing
Vector Bundles
A First Heterotic
Standard Model
On any variety Y , we have
{ }/ {
}
Divisors D ∼ = Line bundles O Y (D)
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 20 / 54

Line Bundles
Introduction
The Calabi-Yau
The Vector Bundle
❖ Line Bundles
❖ Equivariant Line
Bundles I
❖ Equivariant Line
Bundles II
❖ Notation
❖ The Serre
Construction
❖ Equivariant Vector
Bundles
❖ Equivariant
Example
❖ Constructing
Vector Bundles
A First Heterotic
Standard Model
On any variety Y , we have
{ }/ {
}
Divisors D ∼ = Line bundles O Y (D)
Linear equivalence
For ˜X, B 1 , B 2 , P 1 that is just cohomology class
of the divisor in H 1,1 .
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 20 / 54

Line Bundles
Introduction
The Calabi-Yau
The Vector Bundle
❖ Line Bundles
❖ Equivariant Line
Bundles I
❖ Equivariant Line
Bundles II
❖ Notation
❖ The Serre
Construction
❖ Equivariant Vector
Bundles
❖ Equivariant
Example
❖ Constructing
Vector Bundles
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
On any variety Y , we have
{ }/ {
}
Divisors D ∼ = Line bundles O Y (D)
Linear equivalence
For ˜X, B 1 , B 2 , P 1 that is just cohomology class
of the divisor in H 1,1 .
Every line bundle is of the form
● O ˜X(x 1 τ 1 + x 2 τ 2 + x 3 φ) , x 1 , x 2 , x 3 ∈ Z.
● O Bi (y 1 t i + y 2 f i ) , y 1 , y 2 ∈ Z.
● O P
1(n) , n ∈ Z.
A Heterotic Standard Model / University of North Carolina at Chapel Hill 20 / 54

Equivariant Line Bundles I
Introduction
The Calabi-Yau
The Vector Bundle
❖ Line Bundles
❖ Equivariant Line
Bundles I
❖ Equivariant Line
Bundles II
❖ Notation
❖ The Serre
Construction
❖ Equivariant Vector
Bundles
❖ Equivariant
Example
❖ Constructing
Vector Bundles
Line bundles on
X = ˜X/G
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 21 / 54

Equivariant Line Bundles I
Introduction
The Calabi-Yau
Work with
Have in mind
The Vector Bundle
❖ Line Bundles
❖ Equivariant Line
Bundles I
❖ Equivariant Line
Bundles II
G-equivariant line
bundles on ˜X
=
Line bundles on
X = ˜X/G
❖ Notation
❖ The Serre
Construction
❖ Equivariant Vector
Bundles
❖ Equivariant
Example
❖ Constructing
Vector Bundles
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 21 / 54

Equivariant Line Bundles I
Introduction
The Calabi-Yau
Work with
Have in mind
The Vector Bundle
❖ Line Bundles
❖ Equivariant Line
Bundles I
❖ Equivariant Line
Bundles II
❖ Notation
❖ The Serre
Construction
❖ Equivariant Vector
Bundles
❖ Equivariant
Example
❖ Constructing
Vector Bundles
A First Heterotic
Standard Model
G-equivariant line
bundles on ˜X
=
An equivariant line bundle is a
line bundle L together with a
group action γ : G × L → L:
Line bundles on
X = ˜X/G
L
˜X
γ g
g
L
˜X
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 21 / 54

Equivariant Line Bundles II
Introduction
The Calabi-Yau
The Vector Bundle
❖ Line Bundles
❖ Equivariant Line
Bundles I
❖ Equivariant Line
Bundles II
❖ Notation
❖ The Serre
Construction
❖ Equivariant Vector
Bundles
❖ Equivariant
Example
❖ Constructing
Vector Bundles
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
●
Most line bundles on ˜X cannot be made
equivariant.
A Heterotic Standard Model / University of North Carolina at Chapel Hill 22 / 54

Equivariant Line Bundles II
Introduction
The Calabi-Yau
The Vector Bundle
❖ Line Bundles
❖ Equivariant Line
Bundles I
❖ Equivariant Line
Bundles II
❖ Notation
❖ The Serre
Construction
❖ Equivariant Vector
Bundles
❖ Equivariant
Example
❖ Constructing
Vector Bundles
●
Most line bundles on ˜X cannot be made
equivariant.
● Only the line bundles O ˜X(x 1 τ 1 + x 2 τ 2 + x 3 φ)
, x 1 , x 2 , x 3 ∈ Z with x 1 + x 2 ≡ 0 mod 3
allow for a G = Z 3 × Z 3 action.
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 22 / 54

Equivariant Line Bundles II
Introduction
The Calabi-Yau
The Vector Bundle
❖ Line Bundles
❖ Equivariant Line
Bundles I
❖ Equivariant Line
Bundles II
❖ Notation
❖ The Serre
Construction
❖ Equivariant Vector
Bundles
❖ Equivariant
Example
❖ Constructing
Vector Bundles
A First Heterotic
Standard Model
●
Most line bundles on ˜X cannot be made
equivariant.
● Only the line bundles O ˜X(x 1 τ 1 + x 2 τ 2 + x 3 φ)
, x 1 , x 2 , x 3 ∈ Z with x 1 + x 2 ≡ 0 mod 3
allow for a G = Z 3 × Z 3 action.
●
In these cases, there is always more than
one G action
⇒ Different equivariant line bundles!
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 22 / 54

Notation
Introduction
The Calabi-Yau
The Vector Bundle
❖ Line Bundles
❖ Equivariant Line
Bundles I
❖ Equivariant Line
Bundles II
❖ Notation
❖ The Serre
Construction
❖ Equivariant Vector
Bundles
❖ Equivariant
Example
❖ Constructing
Vector Bundles
Consider the trivial line bundle O ˜X
= ˜X × C.
●
Obvious equivariant action
γ g : ˜X × C → ˜X × C, (p, v) ↦→ ( g(p), v )
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 23 / 54

Notation
Introduction
The Calabi-Yau
The Vector Bundle
❖ Line Bundles
❖ Equivariant Line
Bundles I
❖ Equivariant Line
Bundles II
❖ Notation
❖ The Serre
Construction
❖ Equivariant Vector
Bundles
❖ Equivariant
Example
❖ Constructing
Vector Bundles
Consider the trivial line bundle O ˜X
= ˜X × C.
●
●
Obvious equivariant action
γ g : ˜X × C → ˜X × C, (p, v) ↦→ ( g(p), v )
Different equivariant action by multiplying
with a character
χγ g : ˜X × C → ˜X × C, (p, v) ↦→ ( g(p), χ(g)v )
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 23 / 54

Notation
Introduction
The Calabi-Yau
The Vector Bundle
❖ Line Bundles
❖ Equivariant Line
Bundles I
❖ Equivariant Line
Bundles II
❖ Notation
❖ The Serre
Construction
❖ Equivariant Vector
Bundles
❖ Equivariant
Example
❖ Constructing
Vector Bundles
A First Heterotic
Standard Model
Spectral Sequences
Consider the trivial line bundle O ˜X
= ˜X × C.
●
●
●
Obvious equivariant action
γ g : ˜X × C → ˜X × C, (p, v) ↦→ ( g(p), v )
Different equivariant action by multiplying
with a character
χγ g : ˜X × C → ˜X × C, (p, v) ↦→ ( g(p), χ(g)v )
We write χO ˜X
for this different equivariant
line bundle.
A New Heterotic
Standard Model
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 23 / 54

The Serre Construction
Introduction
The Calabi-Yau
The Vector Bundle
❖ Line Bundles
❖ Equivariant Line
Bundles I
❖ Equivariant Line
Bundles II
❖ Notation
❖ The Serre
Construction
❖ Equivariant Vector
Bundles
❖ Equivariant
Example
❖ Constructing
Vector Bundles
A First Heterotic
Standard Model
A way to construct may stable rank 2 vector
bundles on a surface (here: B 1 and B 2 ).
● Take two line bundles L 1 , L 2 .
●
●
An ideal sheaf I (sheaf of functions
vanishing at some fixed points).
Define S as an extension
0 −→ L 1 −→ S −→ L 2 ⊗ I −→ 0
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
●
Cayley-Bacharach property ⇒ generic
extension is a rank 2 vector bundle.
A Heterotic Standard Model / University of North Carolina at Chapel Hill 24 / 54

Equivariant Vector Bundles
Introduction
The Calabi-Yau
The Vector Bundle
❖ Line Bundles
❖ Equivariant Line
Bundles I
❖ Equivariant Line
Bundles II
❖ Notation
❖ The Serre
Construction
❖ Equivariant Vector
Bundles
❖ Equivariant
Example
❖ Constructing
Vector Bundles
Vector bundles on
X = ˜X/G
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 25 / 54

Equivariant Vector Bundles
Introduction
The Calabi-Yau
Work with
Have in mind
The Vector Bundle
❖ Line Bundles
❖ Equivariant Line
Bundles I
❖ Equivariant Line
Bundles II
❖ Notation
❖ The Serre
Construction
❖ Equivariant Vector
Bundles
❖ Equivariant
Example
G-equivariant
vector bundles on
˜X
=
Vector bundles on
X = ˜X/G
❖ Constructing
Vector Bundles
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 25 / 54

Equivariant Vector Bundles
Introduction
The Calabi-Yau
Work with
Have in mind
The Vector Bundle
❖ Line Bundles
❖ Equivariant Line
Bundles I
❖ Equivariant Line
Bundles II
❖ Notation
❖ The Serre
Construction
❖ Equivariant Vector
Bundles
❖ Equivariant
Example
❖ Constructing
Vector Bundles
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
G-equivariant
vector bundles on
˜X
Problem: Even if E, F are equivariant,
=
0 −→ E −→ V −→ F −→ 0
Vector bundles on
X = ˜X/G
Extension is not necessarily equivariant!
Only extensions in Ext 1 ( F, E ) G
are
equivariant.
A Heterotic Standard Model / University of North Carolina at Chapel Hill 25 / 54

Equivariant Example
0 −→ O B2 (−2f 2 ) −→ W −→ χ 2 O B2 (2f 2 ) ⊗ I 9 −→ 0
A Heterotic Standard Model / University of North Carolina at Chapel Hill 26 / 54

Equivariant Example
0 −→ O B2 (−2f 2 ) −→ W −→ χ 2 O B2 (2f 2 ) ⊗ I 9 −→ 0
●
O B2 (−2f 2 ), χ 2 O B2 (2f 2 ) are equivariant.
A Heterotic Standard Model / University of North Carolina at Chapel Hill 26 / 54

Equivariant Example
0 −→ O B2 (−2f 2 ) −→ W −→ χ 2 O B2 (2f 2 ) ⊗ I 9 −→ 0
●
●
O B2 (−2f 2 ), χ 2 O B2 (2f 2 ) are equivariant.
I 9 is the ideal sheaf of one G orbit.
A Heterotic Standard Model / University of North Carolina at Chapel Hill 26 / 54

Equivariant Example
0 −→ O B2 (−2f 2 ) −→ W −→ χ 2 O B2 (2f 2 ) ⊗ I 9 −→ 0
●
●
●
O B2 (−2f 2 ), χ 2 O B2 (2f 2 ) are equivariant.
I 9 is the ideal sheaf of one G orbit.
Has the Cayley-Bacharach property.
A Heterotic Standard Model / University of North Carolina at Chapel Hill 26 / 54

Equivariant Example
0 −→ O B2 (−2f 2 ) −→ W −→ χ 2 O B2 (2f 2 ) ⊗ I 9 −→ 0
●
●
●
O B2 (−2f 2 ), χ 2 O B2 (2f 2 ) are equivariant.
I 9 is the ideal sheaf of one G orbit.
Has the ( Cayley-Bacharach property.
)
● Ext 1 χ 2 O B2 (2f 2 ) ⊗ I 9 , O B2 (−2f 2 ) = C 9
A Heterotic Standard Model / University of North Carolina at Chapel Hill 26 / 54

Equivariant Example
0 −→ O B2 (−2f 2 ) −→ W −→ χ 2 O B2 (2f 2 ) ⊗ I 9 −→ 0
●
●
●
O B2 (−2f 2 ), χ 2 O B2 (2f 2 ) are equivariant.
I 9 is the ideal sheaf of one G orbit.
Has the ( Cayley-Bacharach property.
)
● Ext 1 χ 2 O B2 (2f 2 ) ⊗ I 9 , O B2 (−2f 2 ) = Reg(G) =
1 ⊕ χ 1 ⊕ χ 2 1 ⊕ χ 2 ⊕ χ 1 χ 2 ⊕ χ 2 1 χ 2 ⊕ χ 2 2 ⊕ χ 1χ 2 2 ⊕ χ2 1 χ2 2
A Heterotic Standard Model / University of North Carolina at Chapel Hill 26 / 54

Equivariant Example
0 −→ O B2 (−2f 2 ) −→ W −→ χ 2 O B2 (2f 2 ) ⊗ I 9 −→ 0
●
●
●
O B2 (−2f 2 ), χ 2 O B2 (2f 2 ) are equivariant.
I 9 is the ideal sheaf of one G orbit.
Has the ( Cayley-Bacharach property.
)
● Ext 1 χ 2 O B2 (2f 2 ) ⊗ I 9 , O B2 (−2f 2 ) = Reg(G) =
1 ⊕ χ 1 ⊕ χ 2 1 ⊕ χ 2 ⊕ χ 1 χ 2 ⊕ χ 2 1 χ 2 ⊕ χ 2 2 ⊕ χ 1χ 2 2 ⊕ χ2 1 χ2 2
so there exist extensions.
A Heterotic Standard Model / University of North Carolina at Chapel Hill 26 / 54

Constructing Vector Bundles
Introduction
The Calabi-Yau
The Vector Bundle
❖ Line Bundles
❖ Equivariant Line
Bundles I
❖ Equivariant Line
Bundles II
❖ Notation
❖ The Serre
Construction
❖ Equivariant Vector
Bundles
❖ Equivariant
Example
❖ Constructing
Vector Bundles
Building blocks:
●
Line bundles on ˜X.
● Rank 2 bundles pulled back from B 1 , B 2 .
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 27 / 54

Constructing Vector Bundles
Introduction
The Calabi-Yau
The Vector Bundle
❖ Line Bundles
❖ Equivariant Line
Bundles I
❖ Equivariant Line
Bundles II
❖ Notation
❖ The Serre
Construction
❖ Equivariant Vector
Bundles
❖ Equivariant
Example
❖ Constructing
Vector Bundles
Building blocks:
●
Line bundles on ˜X.
● Rank 2 bundles pulled back from B 1 , B 2 .
Operations:
●
Tensor product of bundles.
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 27 / 54

Constructing Vector Bundles
Introduction
The Calabi-Yau
The Vector Bundle
❖ Line Bundles
❖ Equivariant Line
Bundles I
❖ Equivariant Line
Bundles II
❖ Notation
❖ The Serre
Construction
❖ Equivariant Vector
Bundles
❖ Equivariant
Example
❖ Constructing
Vector Bundles
Building blocks:
●
Line bundles on ˜X.
● Rank 2 bundles pulled back from B 1 , B 2 .
Operations:
●
●
Tensor product of bundles.
Sums of bundles.
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 27 / 54

Constructing Vector Bundles
Introduction
The Calabi-Yau
The Vector Bundle
❖ Line Bundles
❖ Equivariant Line
Bundles I
❖ Equivariant Line
Bundles II
❖ Notation
❖ The Serre
Construction
❖ Equivariant Vector
Bundles
❖ Equivariant
Example
❖ Constructing
Vector Bundles
A First Heterotic
Standard Model
Building blocks:
●
Line bundles on ˜X.
● Rank 2 bundles pulled back from B 1 , B 2 .
Operations:
●
Tensor product of bundles.
● ///////// Sums///// of/////////////// bundles. Never (slope-) stable!
●
Extensions of bundles.
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 27 / 54

Introduction
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
❖ The Gauge Bundle
❖ Particle Spectrum
❖ The Lagrangian
❖ The String Miracle
Spectral Sequences
A New Heterotic
Standard Model
A First Heterotic Standard
Model
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 28 / 54

The Gauge Bundle
Introduction
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
❖ The Gauge Bundle
❖ Particle Spectrum
❖ The Lagrangian
❖ The String Miracle
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
Define these two rank 2 vector bundles
V 1
V 2
def
= χ 2 O ˜X(−τ 1 + τ 2 ) ⊕ χ 2 O ˜X(−τ 1 + τ 2 ) =
= 2χ 2 O ˜X(−τ 1 + τ 2 )
def
= O ˜X(τ 1 − τ 2 ) ⊗ π2(W)
∗
We define the rank 4 bundle V finally as a
generic extension
0 −→ V 2 −→ V −→ V 1 −→ 0
hep-th/0501070: A Heterotic Standard Model
hep-th/0502155: A Standard Model from the E 8 × E 8 Heterotic Superstring
hep-th/0505041: Vector Bundle Extensions, Sheaf Cohomology, and the Heterotic Standard Model
A Heterotic Standard Model / University of North Carolina at Chapel Hill 29 / 54

Particle Spectrum
Introduction
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
❖ The Gauge Bundle
❖ Particle Spectrum
❖ The Lagrangian
❖ The String Miracle
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
●
●
●
●
●
●
3 families of quarks and leptons.
Zero anti-families.
4 Higgs (twice MSSM).
Doublets and triplets are completely split,
all triplets are projected out.
Hidden pure E 7 or Spin(12) with 2 matter
fields.
6 geometric moduli, 19 vector bundle
moduli, some hidden E 8 bundle moduli.
hep-th/0509051: Heterotic Standard Model Moduli
A Heterotic Standard Model / University of North Carolina at Chapel Hill 30 / 54

The Lagrangian
Introduction
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
❖ The Gauge Bundle
❖ Particle Spectrum
❖ The Lagrangian
❖ The String Miracle
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
Of course, we do not know the Kähler
potential. What can we learn from the
superpotential W ?
●
●
Field
φ
H
¯H
Q i
Higgs µ-terms φH ¯H
Yukawa couplings Q i H ¯Q i + Q i ¯H ¯Qi
¯Q i
Name
Vector bundle moduli
Higgs
Higgs-conjugate
Quarks & leptons of the i-th family
Anti-Q i
A Heterotic Standard Model / University of North Carolina at Chapel Hill 31 / 54

The String Miracle
Introduction
The Calabi-Yau
Compactifying on ( ˜X, V)/G, we found
The Vector Bundle
A First Heterotic
Standard Model
❖ The Gauge Bundle
❖ Particle Spectrum
❖ The Lagrangian
❖ The String Miracle
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 32 / 54

The String Miracle
Introduction
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
❖ The Gauge Bundle
❖ Particle Spectrum
❖ The Lagrangian
❖ The String Miracle
Compactifying on ( ˜X, V)/G, we found
● Higgs µ-terms φH ¯H with 4 out of the 19
vector bundle moduli.
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
hep-th/0510142: Moduli Dependent µ-Terms in a Heterotic Standard Model
A Heterotic Standard Model / University of North Carolina at Chapel Hill 32 / 54

The String Miracle
Introduction
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
❖ The Gauge Bundle
❖ Particle Spectrum
❖ The Lagrangian
❖ The String Miracle
Compactifying on ( ˜X, V)/G, we found
● Higgs µ-terms φH ¯H with 4 out of the 19
vector bundle moduli.
●
No Yukawa couplings.
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
hep-th/0510142: Moduli Dependent µ-Terms in a Heterotic Standard Model
A Heterotic Standard Model / University of North Carolina at Chapel Hill 32 / 54

The String Miracle
Introduction
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
❖ The Gauge Bundle
❖ Particle Spectrum
❖ The Lagrangian
❖ The String Miracle
Spectral Sequences
A New Heterotic
Standard Model
Compactifying on ( ˜X, V)/G, we found
● Higgs µ-terms φH ¯H with 4 out of the 19
vector bundle moduli.
●
No Yukawa couplings.
Yukawa textures
without symmetries!?!
Conclusion
hep-th/0510142: Moduli Dependent µ-Terms in a Heterotic Standard Model
A Heterotic Standard Model / University of North Carolina at Chapel Hill 32 / 54

Introduction
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
❖ Leray Spectral
Sequence
❖ An Example
❖ Leray Degrees
❖ Leray Degree Table
❖ The
Superpotential
❖ More on Leray
Degrees
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 33 / 54

Leray Spectral Sequence
Introduction
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
❖ Leray Spectral
Sequence
❖ An Example
❖ Leray Degrees
❖ Leray Degree Table
❖ The
Superpotential
❖ More on Leray
Degrees
How did we compute all these cohomology
groups?
Leray spectral sequence for any sheaf F on
˜X → B 2 :
)
( )
E p,q
2 = H
(B p 2 , R q π 2∗ F ⇒ H p+q ˜X, F
A New Heterotic
Standard Model
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 34 / 54

Leray Spectral Sequence
Introduction
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
❖ Leray Spectral
Sequence
❖ An Example
❖ Leray Degrees
❖ Leray Degree Table
❖ The
Superpotential
❖ More on Leray
Degrees
A New Heterotic
Standard Model
Conclusion
How did we compute all these cohomology
groups?
Leray spectral sequence for any sheaf F on
˜X → B 2 :
)
( )
E p,q
2 = H
(B p 2 , R q π 2∗ F ⇒ H p+q ˜X, F
R q π 2∗ is just the degree q cohomology along
the fiber.
A Heterotic Standard Model / University of North Carolina at Chapel Hill 34 / 54

Leray Spectral Sequence
Introduction
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
❖ Leray Spectral
Sequence
❖ An Example
❖ Leray Degrees
❖ Leray Degree Table
❖ The
Superpotential
❖ More on Leray
Degrees
A New Heterotic
Standard Model
Conclusion
How did we compute all these cohomology
groups?
Leray spectral sequence for any sheaf F on
˜X → B 2 :
)
( )
E p,q
2 = H
(B p 2 , R q π 2∗ F ⇒ H p+q ˜X, F
R q π 2∗ is just the degree q cohomology along
the fiber.
Think of E p,q
2 as the “forms with p legs along
the base and q legs along the fiber”.
A Heterotic Standard Model / University of North Carolina at Chapel Hill 34 / 54

An Example
Introduction
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
❖ Leray Spectral
Sequence
❖ An Example
❖ Leray Degrees
❖ Leray Degree Table
❖ The
Superpotential
❖ More on Leray
Degrees
A New Heterotic
Standard Model
Conclusion
( ) ( )
Example: H 1 ˜X, ∧ 2 V = H 1 ˜X, 2χ2 π2 ∗(W) )
π 2∗
(2χ 2 π2(W)
∗ )
R 1 π 2∗
(2χ 2 π2(W)
∗
= 2χ 2 W
= 2χ 1 χ 2 W ⊗ O B2 (−f 2 )
Compute H p (B 1 , · · · ) by two more Leray SS...
⇒ E p,q
2 =
q=1 0 2⊕2χ 1 ⊕2χ 2 ⊕2χ 2 1⊕2χ 2 2⊕2χ 1 χ 2 2⊕2χ 2 1χ 2 0
q=0 0 2⊕2χ 1 ⊕2χ 2 ⊕2χ 2 1 ⊕2χ2 2 ⊕2χ 1χ 2 2 ⊕2χ2 1 χ 2 0
p=0 p=1 p=2
A Heterotic Standard Model / University of North Carolina at Chapel Hill 35 / 54

Leray Degrees
Introduction
The Calabi-Yau
The two fibrations
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
❖ Leray Spectral
Sequence
❖ An Example
❖ Leray Degrees
❖ Leray Degree Table
❖ The
Superpotential
❖ More on Leray
Degrees
A New Heterotic
Standard Model
Conclusion
dim C = 3 : ˜X
π 1
π2
dim C = 2 : B 1
β 1
dim C = 1 : P 1
B 2
β 2
allow us to refine the cohomology degree
according to # of legs in the π 1 fiber, the base,
and the π 2 fiber direction.
A Heterotic Standard Model / University of North Carolina at Chapel Hill 36 / 54

Leray Degree Table
Field Cohomology Fiber 1 Base Fiber 2
Q i , ¯Q i H 1( ) ˜X, V 0 0 1
H 1 , H 2 H 1( ˜X, ∧ 2 V ) 0 1 0
¯H 1 , ¯H 2 H 1( ˜X, ∧ 2 V ) 0 0 1
φ 1 , . . . , φ 4 H 1( ˜X, V ⊗ V
∨ ) 1 0 0
φ 5 , . . . , φ 19 H 1( ˜X, V ⊗ V
∨ ) 0 0 1
A Heterotic Standard Model / University of North Carolina at Chapel Hill 37 / 54

Leray Degree Table
Field Cohomology Fiber 1 Base Fiber 2
Q i , ¯Q i H 1( ) ˜X, V 0 0 1
H 1 , H 2 H 1( ˜X, ∧ 2 V ) 0 1 0
¯H 1 , ¯H 2 H 1( ˜X, ∧ 2 V ) 0 0 1
φ 1 , . . . , φ 4 H 1( ˜X, V ⊗ V
∨ ) 1 0 0
φ 5 , . . . , φ 19 H 1( ˜X, ) V ⊗ V
∨
0 0 1
¯Ω H 3( ) ˜X, O ˜X
1 1 1
A Heterotic Standard Model / University of North Carolina at Chapel Hill 37 / 54

The Superpotential
The cubic terms in the superpotential are
A Heterotic Standard Model / University of North Carolina at Chapel Hill 38 / 54

The Superpotential
The cubic terms in the superpotential are
● Higgs µ-terms (note: ∧ 2 V = ∧ 2 V ∨ )
( ) ( ) ( )
H 1 ˜X, V ⊗ V
∨
⊗ H 1 ˜X, ∧ 2 V ⊗ H 1 ˜X, ∧ 2 V ∨
( )
−→ H 3 ˜X, O ˜X
= C
A Heterotic Standard Model / University of North Carolina at Chapel Hill 38 / 54

The Superpotential
The cubic terms in the superpotential are
● Higgs µ-terms (note: ∧ 2 V = ∧ 2 V ∨ )
( ) ( ) ( )
H 1 ˜X, V ⊗ V
∨
⊗ H 1 ˜X, ∧ 2 V ⊗ H 1 ˜X, ∧ 2 V ∨
( )
−→ H 3 ˜X, O ˜X
= C
●
Yukawa couplings
H 1( ˜X, V
)
⊗ H 1( ˜X, V
)
⊗ H 1( ˜X, ∧ 2 V ∨)
−→ H 3 (
˜X, O ˜X
)
= C
A Heterotic Standard Model / University of North Carolina at Chapel Hill 38 / 54

More on Leray Degrees
Introduction
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
❖ Leray Spectral
Sequence
❖ An Example
❖ Leray Degrees
❖ Leray Degree Table
❖ The
Superpotential
❖ More on Leray
Degrees
The products respect the additional Leray
degrees!
Field Fiber 1 Base Fiber 2
H 1 , H 2 0 1 0
¯H 1 , ¯H 2 0 0 1
φ 1 , . . . , φ 4 1 0 0
A New Heterotic
Standard Model
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 39 / 54

More on Leray Degrees
Introduction
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
❖ Leray Spectral
Sequence
❖ An Example
❖ Leray Degrees
❖ Leray Degree Table
❖ The
Superpotential
❖ More on Leray
Degrees
A New Heterotic
Standard Model
Conclusion
The products respect the additional Leray
degrees!
Field Fiber 1 Base Fiber 2
H 1 , H 2 0 1 0
¯H 1 , ¯H 2 0 0 1
φ 1 , . . . , φ 4 1 0 0
The only allowed cubic coupling is
W =
∑
λ iab φ i H a ¯Hb
i=1..4
a,b=1,2
A Heterotic Standard Model / University of North Carolina at Chapel Hill 39 / 54

Introduction
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
❖ Ideal Sheaves
❖ Serre Construction
❖ The Gauge Bundle
❖ Low Energy
Spectrum
❖ Gauge Group
Breaking
❖ Vector Bundle
Breaking
❖ The Higgs Sector
❖ Cohomology
❖ Doublet-Triplet
Splitting
❖ Leray Degrees
A New Heterotic Standard
Model
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 40 / 54

Ideal Sheaves
Introduction
The Calabi-Yau
The Vector Bundle
I thought your solution was unique!
Whats new?
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
❖ Ideal Sheaves
❖ Serre Construction
❖ The Gauge Bundle
❖ Low Energy
Spectrum
❖ Gauge Group
Breaking
❖ Vector Bundle
Breaking
❖ The Higgs Sector
❖ Cohomology
❖ Doublet-Triplet
Splitting
❖ Leray Degrees
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 41 / 54

Ideal Sheaves
Introduction
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
I thought your solution was unique!
Whats new?
●
On ˜X the G = Z 3 × Z 3 action is free.
A New Heterotic
Standard Model
❖ Ideal Sheaves
❖ Serre Construction
❖ The Gauge Bundle
❖ Low Energy
Spectrum
❖ Gauge Group
Breaking
❖ Vector Bundle
Breaking
❖ The Higgs Sector
❖ Cohomology
❖ Doublet-Triplet
Splitting
❖ Leray Degrees
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 41 / 54

Ideal Sheaves
Introduction
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
❖ Ideal Sheaves
❖ Serre Construction
❖ The Gauge Bundle
❖ Low Energy
Spectrum
❖ Gauge Group
Breaking
❖ Vector Bundle
Breaking
❖ The Higgs Sector
❖ Cohomology
❖ Doublet-Triplet
Splitting
❖ Leray Degrees
I thought your solution was unique!
Whats new?
●
On ˜X the G = Z 3 × Z 3 action is free.
● But on B 1 , B 2 there are orbits of length 3
and 9.
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 41 / 54

Ideal Sheaves
Introduction
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
❖ Ideal Sheaves
❖ Serre Construction
❖ The Gauge Bundle
❖ Low Energy
Spectrum
❖ Gauge Group
Breaking
❖ Vector Bundle
Breaking
❖ The Higgs Sector
❖ Cohomology
❖ Doublet-Triplet
Splitting
❖ Leray Degrees
I thought your solution was unique!
Whats new?
●
On ˜X the G = Z 3 × Z 3 action is free.
● But on B 1 , B 2 there are orbits of length 3
and 9.
Observation: We can split up the ideal sheaf of
9 points in 3 + 6 points!
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 41 / 54

Ideal Sheaves
Introduction
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
❖ Ideal Sheaves
❖ Serre Construction
❖ The Gauge Bundle
❖ Low Energy
Spectrum
❖ Gauge Group
Breaking
❖ Vector Bundle
Breaking
❖ The Higgs Sector
❖ Cohomology
❖ Doublet-Triplet
Splitting
❖ Leray Degrees
Conclusion
I thought your solution was unique!
Whats new?
●
On ˜X the G = Z 3 × Z 3 action is free.
● But on B 1 , B 2 there are orbits of length 3
and 9.
Observation: We can split up the ideal sheaf of
9 points in 3 + 6 points! Define
I 3
Ideal sheaf on B 1 , 3 points in 3 fibers.
I 6 Ideal sheaf on B 2 ,
Singular point in 3I 1 with multiplicity 2.
(i.e. function and a first derivative = 0)
A Heterotic Standard Model / University of North Carolina at Chapel Hill 41 / 54

Serre Construction
Introduction
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
❖ Ideal Sheaves
❖ Serre Construction
❖ The Gauge Bundle
❖ Low Energy
Spectrum
❖ Gauge Group
Breaking
❖ Vector Bundle
Breaking
❖ The Higgs Sector
❖ Cohomology
❖ Doublet-Triplet
Splitting
❖ Leray Degrees
Define rank 2 bundles W i on B i
0 → χ 1 O B1 (−f 1 ) → W 1 → χ 2 1O B1 (f 1 ) ⊗ I 3 → 0
0 → χ 2 2O B2 (−2f 2 ) → W 2 → χ 2 O B2 (2f 2 )⊗I 6 → 0
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 42 / 54

The Gauge Bundle
Introduction
The Calabi-Yau
Define these two rank 2 vector bundles
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
V 1
V 2
def
= O ˜X(−τ 1 + τ 2 ) ⊗ π1(W ∗ 1 )
def
= O ˜X(τ 1 − τ 2 ) ⊗ π2(W ∗ 2 )
A New Heterotic
Standard Model
❖ Ideal Sheaves
❖ Serre Construction
❖ The Gauge Bundle
❖ Low Energy
Spectrum
❖ Gauge Group
Breaking
❖ Vector Bundle
Breaking
❖ The Higgs Sector
❖ Cohomology
❖ Doublet-Triplet
Splitting
❖ Leray Degrees
We define the rank 4 bundle V finally as a
generic extension
0 −→ V 1 −→ V −→ V 2 −→ 0
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 43 / 54

Low Energy Spectrum
Introduction
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
The massless spectrum
= zero modes of /D E8
= H 1 cohomology of the adjoint bundle E V/G
8 .
A New Heterotic
Standard Model
❖ Ideal Sheaves
❖ Serre Construction
❖ The Gauge Bundle
❖ Low Energy
Spectrum
❖ Gauge Group
Breaking
❖ Vector Bundle
Breaking
❖ The Higgs Sector
❖ Cohomology
❖ Doublet-Triplet
Splitting
❖ Leray Degrees
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 44 / 54

Low Energy Spectrum
Introduction
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
❖ Ideal Sheaves
❖ Serre Construction
❖ The Gauge Bundle
❖ Low Energy
Spectrum
❖ Gauge Group
Breaking
❖ Vector Bundle
Breaking
❖ The Higgs Sector
❖ Cohomology
❖ Doublet-Triplet
Splitting
❖ Leray Degrees
The massless spectrum
= zero modes of /D E8
= H 1 cohomology of the adjoint bundle E V/G
8 .
( )
H 1 X, E V/G
8 =
)
= H
(X, 1 E V 8 /G
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 44 / 54

Low Energy Spectrum
Introduction
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
The massless spectrum
= zero modes of /D E8
= H 1 cohomology of the adjoint bundle E V/G
8 .
A New Heterotic
Standard Model
❖ Ideal Sheaves
❖ Serre Construction
❖ The Gauge Bundle
❖ Low Energy
Spectrum
❖ Gauge Group
Breaking
❖ Vector Bundle
Breaking
❖ The Higgs Sector
❖ Cohomology
❖ Doublet-Triplet
Splitting
❖ Leray Degrees
Conclusion
Work with
H 1 (
˜X, E
V
8
) G
=
Have in mind
( )
H 1 X, E V/G
8 =
)
= H
(X, 1 E V 8 /G
A Heterotic Standard Model / University of North Carolina at Chapel Hill 44 / 54

Gauge Group Breaking
Introduction
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
❖ Ideal Sheaves
❖ Serre Construction
❖ The Gauge Bundle
❖ Low Energy
Spectrum
❖ Gauge Group
Breaking
❖ Vector Bundle
Breaking
❖ The Higgs Sector
❖ Cohomology
❖ Doublet-Triplet
Splitting
❖ Leray Degrees
248 = ( 1, 45 ) ⊕ ( 15, 1 ) ⊕
⊕ ( 4, 16 ) ⊕ ( 4, 16 ) ⊕ ( 6, 10 )
( ) ( )
10 = χ 2 1, 2, 3, 0 ⊕ χ
2
1 χ 2 3, 1, −2, −2 ⊕
) ( )
⊕ χ2( 2 1, 2, −3, 0 ⊕ χ1 χ 2 2 3, 1, 2, 2
Correspondingly, the fermions split as...
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 45 / 54

Vector Bundle Breaking
Introduction
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
❖ Ideal Sheaves
❖ Serre Construction
❖ The Gauge Bundle
❖ Low Energy
Spectrum
❖ Gauge Group
Breaking
❖ Vector Bundle
Breaking
❖ The Higgs Sector
❖ Cohomology
❖ Doublet-Triplet
Splitting
❖ Leray Degrees
( ) (
)
E V 8 = O ˜X
⊗ θ(45) ⊕ ad(V) ⊗ θ(1) ⊕
( ) ( ) ( )
⊕ V⊗θ(16) ⊕ V ∨ ⊗θ(16) ⊕ ∧ 2 V⊗θ(10)
where θ(· · · ) is the trivial bundle.
[
θ(10) = χ 2 θ ( 1, 2, 3, 0 )] [
⊕ χ 2 1χ 2 θ ( 3, 1, −2, −2 )] ⊕
[
⊕ χ 2 2θ ( 1, 2, −3, 0 )] ⊕
[χ 1 χ 2 2θ ( 3, 1, 2, 2 )]
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 46 / 54

The Higgs Sector
Introduction
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
❖ Ideal Sheaves
❖ Serre Construction
❖ The Gauge Bundle
❖ Low Energy
Spectrum
❖ Gauge Group
Breaking
❖ Vector Bundle
Breaking
❖ The Higgs Sector
❖ Cohomology
❖ Doublet-Triplet
Splitting
❖ Leray Degrees
For example, focus on the fields in the 10:
H 1( ) G
˜X, E
V
8 = (lots of other fields) ⊕
[
⊕ χ 2 ⊗ H 1( G ( ) ˜X, ∧ V)] 2 ⊗ 1, 2, 3, 0 ⊕
[ ( G ( )
⊕ χ 2 1χ 2 ⊗ H 1 ˜X, ∧ V)] 2 ⊗ 3, 1, −2, −2 ⊕
[ ( G ( )
⊕ χ 2 2 ⊗ H 1 ˜X, ∧ V)] 2 ⊗ 1, 2, −3, 0 ⊕
[ ( G ( )
⊕ χ 1 χ 2 2 ⊗ H 1 ˜X, ∧ V)] 2 ⊗ 3, 1, 2, 2 .
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 47 / 54

Cohomology
Introduction
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
❖ Ideal Sheaves
❖ Serre Construction
❖ The Gauge Bundle
❖ Low Energy
Spectrum
❖ Gauge Group
Breaking
❖ Vector Bundle
Breaking
❖ The Higgs Sector
❖ Cohomology
❖ Doublet-Triplet
Splitting
❖ Leray Degrees
The necessary cohomology groups for V are
( )
H 1 ˜X, V = 3 Reg(G)
( )
H 1 ˜X, V
∨
= 0
( ) ( )
H 1 ˜X, ∧ 2 V = H 1 ˜X, V1 ⊗ V 2 =
= χ 1 χ 2 ⊕ χ 2 1χ 2 2 ⊕ χ 2 ⊕ χ 2 2
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 48 / 54

Doublet-Triplet Splitting
Introduction
The Calabi-Yau
The Vector Bundle
( )
H 1 ˜X, ∧ 2 V
= χ 1 χ 2 ⊕ χ 2 1χ 2 2 ⊕ χ 2 ⊕ χ 2 2
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
❖ Ideal Sheaves
❖ Serre Construction
❖ The Gauge Bundle
❖ Low Energy
Spectrum
❖ Gauge Group
Breaking
❖ Vector Bundle
Breaking
❖ The Higgs Sector
❖ Cohomology
❖ Doublet-Triplet
Splitting
❖ Leray Degrees
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 49 / 54

Doublet-Triplet Splitting
Introduction
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
( )
H 1 ˜X, ∧ 2 V
= χ 1 χ 2 ⊕ χ 2 1χ 2 2 ⊕ χ 2 ⊕ χ 2 2
Spectral Sequences
A New Heterotic
Standard Model
❖ Ideal Sheaves
❖ Serre Construction
❖ The Gauge Bundle
❖ Low Energy
Spectrum
❖ Gauge Group
Breaking
❖ Vector Bundle
Breaking
❖ The Higgs Sector
❖ Cohomology
❖ Doublet-Triplet
Splitting
❖ Leray Degrees
Conclusion
1 =
[
χ 2 ⊗ H 1 (
˜X, ∧ 2 V)] G
up Higgs
A Heterotic Standard Model / University of North Carolina at Chapel Hill 49 / 54

Doublet-Triplet Splitting
Introduction
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
( )
H 1 ˜X, ∧ 2 V
= χ 1 χ 2 ⊕ χ 2 1χ 2 2 ⊕ χ 2 ⊕ χ 2 2 ⊕ 0χ 1 χ 2 2
Spectral Sequences
A New Heterotic
Standard Model
❖ Ideal Sheaves
❖ Serre Construction
❖ The Gauge Bundle
❖ Low Energy
Spectrum
❖ Gauge Group
Breaking
❖ Vector Bundle
Breaking
❖ The Higgs Sector
❖ Cohomology
❖ Doublet-Triplet
Splitting
❖ Leray Degrees
Conclusion
1 =
0 =
[ ( G
χ 2 ⊗ H 1 ˜X, ∧ V)] 2 up Higgs
[ ( G
χ 2 1 χ 2 ⊗ H 1 ˜X, ∧ V)] 2 3
A Heterotic Standard Model / University of North Carolina at Chapel Hill 49 / 54

Doublet-Triplet Splitting
Introduction
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
( )
H 1 ˜X, ∧ 2 V
= χ 1 χ 2 ⊕ χ 2 1χ 2 2 ⊕ χ 2 ⊕ χ 2 2
Spectral Sequences
A New Heterotic
Standard Model
❖ Ideal Sheaves
❖ Serre Construction
❖ The Gauge Bundle
❖ Low Energy
Spectrum
❖ Gauge Group
Breaking
❖ Vector Bundle
Breaking
❖ The Higgs Sector
❖ Cohomology
❖ Doublet-Triplet
Splitting
❖ Leray Degrees
Conclusion
1 =
0 =
1 =
[ ( G
χ 2 ⊗ H 1 ˜X, ∧ V)] 2 up Higgs
[ ( G
χ 2 1 χ 2 ⊗ H 1 ˜X, ∧ V)] 2 3
[ ( G
χ 2 2 ⊗ H 1 ˜X, ∧ V)] 2 down Higgs
A Heterotic Standard Model / University of North Carolina at Chapel Hill 49 / 54

Doublet-Triplet Splitting
Introduction
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
( )
H 1 ˜X, ∧ 2 V
= χ 1 χ 2 ⊕ χ 2 1χ 2 2 ⊕ χ 2 ⊕ χ 2 2 ⊕ 0χ 2 1χ 2
Spectral Sequences
A New Heterotic
Standard Model
❖ Ideal Sheaves
❖ Serre Construction
❖ The Gauge Bundle
❖ Low Energy
Spectrum
❖ Gauge Group
Breaking
❖ Vector Bundle
Breaking
❖ The Higgs Sector
❖ Cohomology
❖ Doublet-Triplet
Splitting
❖ Leray Degrees
Conclusion
1 =
0 =
1 =
0 =
[ ( G
χ 2 ⊗ H 1 ˜X, ∧ V)] 2 up Higgs
[ ( G
χ 2 1 χ 2 ⊗ H 1 ˜X, ∧ V)] 2 3
[ ( G
χ 2 2 ⊗ H 1 ˜X, ∧ V)] 2 down Higgs
[ ( G
χ 1 χ 2 2 ⊗ H 1 ˜X, ∧ V)] 2 3
A Heterotic Standard Model / University of North Carolina at Chapel Hill 49 / 54

Doublet-Triplet Splitting
Introduction
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
( )
H 1 ˜X, ∧ 2 V
= χ 1 χ 2 ⊕ χ 2 1χ 2 2 ⊕ χ 2 ⊕ χ 2 2
Spectral Sequences
A New Heterotic
Standard Model
❖ Ideal Sheaves
❖ Serre Construction
❖ The Gauge Bundle
❖ Low Energy
Spectrum
❖ Gauge Group
Breaking
❖ Vector Bundle
Breaking
❖ The Higgs Sector
❖ Cohomology
❖ Doublet-Triplet
Splitting
❖ Leray Degrees
Conclusion
1 =
0 =
1 =
0 =
[ ( G
χ 2 ⊗ H 1 ˜X, ∧ V)] 2 up Higgs
[ ( G
χ 2 1 χ 2 ⊗ H 1 ˜X, ∧ V)] 2 3
[ ( G
χ 2 2 ⊗ H 1 ˜X, ∧ V)] 2 down Higgs
[ ( G
χ 1 χ 2 2 ⊗ H 1 ˜X, ∧ V)] 2 3
A Heterotic Standard Model / University of North Carolina at Chapel Hill 49 / 54

Leray Degrees
Field Cohomology Fiber 1 Base Fiber 2
Q 1 , ¯Q 1 H 1( ) ˜X, V 1 0 0
Q 2 , Q 3 , ¯Q 2 , ¯Q 3 H 1( ) ˜X, V 0 0 1
H 1 , ¯H 1 H 1( ˜X, ∧ 2 V ) 0 1 0
φ 1 , . . .? H 1( ˜X, V ⊗ V
∨ ) ? ? ?
A Heterotic Standard Model / University of North Carolina at Chapel Hill 50 / 54

Leray Degrees
Field Cohomology Fiber 1 Base Fiber 2
Q 1 , ¯Q 1 H 1( ) ˜X, V 1 0 0
Q 2 , Q 3 , ¯Q 2 , ¯Q 3 H 1( ) ˜X, V 0 0 1
H 1 , ¯H 1 H 1( ˜X, ∧ 2 V ) 0 1 0
φ 1 , . . .? H 1( ˜X, V ⊗ V
∨ ) ? ? ?
● No µ-terms, H 1 ∧ ¯H 1 = 0.
A Heterotic Standard Model / University of North Carolina at Chapel Hill 50 / 54

Leray Degrees
Field Cohomology Fiber 1 Base Fiber 2
Q 1 , ¯Q 1 H 1( ) ˜X, V 1 0 0
Q 2 , Q 3 , ¯Q 2 , ¯Q 3 H 1( ) ˜X, V 0 0 1
H 1 , ¯H 1 H 1( ˜X, ∧ 2 V ) 0 1 0
φ 1 , . . .? H 1( ˜X, V ⊗ V
∨ ) ? ? ?
● No µ-terms, H 1 ∧ ¯H 1 = 0.
●
Yukawa couplings.
A Heterotic Standard Model / University of North Carolina at Chapel Hill 50 / 54

Introduction
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
❖ Summary
❖ Important Lessons
❖ Future Directions
Conclusion
A Heterotic Standard Model / University of North Carolina at Chapel Hill 51 / 54

Summary
Introduction
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
❖ Summary
❖ Important Lessons
❖ Future Directions
The “new” Heterotic Standard Model has
●
●
●
●
●
●
3 families of quarks and leptons.
Zero anti-families.
1 Higgs–Higgs conjugate pair
(exact MSSM).
Doublets and triplets are completely split,
all triplets are projected out.
Yukawa couplings.
No Higgs µ-terms, but can get those from
D-terms.
A Heterotic Standard Model / University of North Carolina at Chapel Hill 52 / 54

Important Lessons
Introduction
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
❖ Summary
❖ Important Lessons
❖ Future Directions
●
Discrete symmetries are important
A Heterotic Standard Model / University of North Carolina at Chapel Hill 53 / 54

Important Lessons
Introduction
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
❖ Summary
❖ Important Lessons
❖ Future Directions
●
Discrete symmetries are important
✦
Doublet-triplet splitting.
A Heterotic Standard Model / University of North Carolina at Chapel Hill 53 / 54

Important Lessons
Introduction
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
❖ Summary
❖ Important Lessons
❖ Future Directions
●
Discrete symmetries are important
✦
✦
Doublet-triplet splitting.
Moduli reduction, e.g.
h 1,1( ˜X) = 19 −→ 3 = h 1,1 (X)
A Heterotic Standard Model / University of North Carolina at Chapel Hill 53 / 54

Important Lessons
Introduction
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
❖ Summary
❖ Important Lessons
❖ Future Directions
●
●
Discrete symmetries are important
✦
✦
Doublet-triplet splitting.
Moduli reduction, e.g.
h 1,1( ˜X) = 19 −→ 3 = h 1,1 (X)
Not at a special point in moduli space
⇒ no enhanced spectrum.
A Heterotic Standard Model / University of North Carolina at Chapel Hill 53 / 54

Important Lessons
Introduction
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
❖ Summary
❖ Important Lessons
❖ Future Directions
●
●
●
Discrete symmetries are important
✦
✦
Doublet-triplet splitting.
Moduli reduction, e.g.
h 1,1( ˜X) = 19 −→ 3 = h 1,1 (X)
Not at a special point in moduli space
⇒ no enhanced spectrum.
Unique solution?
A Heterotic Standard Model / University of North Carolina at Chapel Hill 53 / 54

Important Lessons
Introduction
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
❖ Summary
❖ Important Lessons
❖ Future Directions
●
●
●
●
Discrete symmetries are important
✦
✦
Doublet-triplet splitting.
Moduli reduction, e.g.
h 1,1( ˜X) = 19 −→ 3 = h 1,1 (X)
Not at a special point in moduli space
⇒ no enhanced spectrum.
Unique solution?
Equivariant actions are the key.
A Heterotic Standard Model / University of North Carolina at Chapel Hill 53 / 54

Future Directions
Introduction
The Calabi-Yau
The Vector Bundle
A First Heterotic
Standard Model
Spectral Sequences
A New Heterotic
Standard Model
Conclusion
❖ Summary
❖ Important Lessons
❖ Future Directions
●
●
●
●
●
Supersymmetry breaking.
U(1) B−L breaking.
Instanton corrections to Yukawa couplings.
Moduli stabilization.
Revisit SU(5) with Z 2 Wilson line: no
U(1) B−L .
A Heterotic Standard Model / University of North Carolina at Chapel Hill 54 / 54