SLIDE 1 Calabi-Yau Metrics and the Spectrum of the Laplace Operator
Volker Braun
Department of Physics University of Pennsylvania (soon: DIAS, Dublin)
16 July 2008
SLIDE 2 Introduction
1
Introduction String Theory Interesting Things to Calculate
2
Calabi-Yau Metrics
3
The Z5 × Z5 Quotient
4
The Laplace Operator
5
Pretty Pictures
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 2 / 61
SLIDE 3 Introduction String Theory
String Theory
Field theory (Supergravity) limit of string theory: MPl > MGUT
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 3 / 61
SLIDE 4 Introduction String Theory
String Theory
Field theory (Supergravity) limit of string theory: MPl > MGUT 10-dimensional space-time = R3,1 ×X
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 3 / 61
SLIDE 5 Introduction String Theory
String Theory
Field theory (Supergravity) limit of string theory: MPl > MGUT 10-dimensional space-time = R3,1 ×X Kaluza-Klein compactification on internal Calabi-Yau threefold X
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 3 / 61
SLIDE 6 Introduction String Theory
String Theory
Field theory (Supergravity) limit of string theory: MPl > MGUT 10-dimensional space-time = R3,1 ×X Kaluza-Klein compactification on internal Calabi-Yau threefold X Laplace equation on the threefold ∆ Φ(6)
i
= λiΦ(6)
i
determines KK modes.
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 3 / 61
SLIDE 7 Introduction Interesting Things to Calculate
Wish List
Zero modes λn = 0 determine the light 4-d particles. Success: Reduces to cohomology.
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 4 / 61
SLIDE 8 Introduction Interesting Things to Calculate
Wish List
Zero modes λn = 0 determine the light 4-d particles. Success: Reduces to cohomology. Normalization of Fields
Φ ∧ ∗¯ Φ = 1. Calabi-Yau Metric?
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 4 / 61
SLIDE 9 Introduction Interesting Things to Calculate
Wish List
Zero modes λn = 0 determine the light 4-d particles. Success: Reduces to cohomology. Normalization of Fields
Φ ∧ ∗¯ Φ = 1. Calabi-Yau Metric? Yukawa couplings. Product in cohomology. If we only knew
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 4 / 61
SLIDE 10 Introduction Interesting Things to Calculate
Wish List
Zero modes λn = 0 determine the light 4-d particles. Success: Reduces to cohomology. Normalization of Fields
Φ ∧ ∗¯ Φ = 1. Calabi-Yau Metric? Yukawa couplings. Product in cohomology. If we only knew
Massive modes λn > 0. Higher-order couplings. A∞ products.
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 4 / 61
SLIDE 11 Calabi-Yau Metrics
1
Introduction
2
Calabi-Yau Metrics Kähler Geometry Donaldson’s Algorithm Implementation Testing the Metric
3
The Z5 × Z5 Quotient
4
The Laplace Operator
5
Pretty Pictures
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 5 / 61
SLIDE 12 Calabi-Yau Metrics Kähler Geometry
Kähler Metrics on the Quintic
Let’s consider our favourite CY threefold: QF =
0 + z5 1 + z5 2 + z5 3 + z5 4 = 0
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SLIDE 13 Calabi-Yau Metrics Kähler Geometry
Kähler Metrics on the Quintic
Let’s consider our favourite CY threefold: QF =
0 + z5 1 + z5 2 + z5 3 + z5 4 = 0
The metric is completely determined by the Kähler potential K(z, ¯ z): gi¯
j(z, ¯
z) = ∂i ¯ ∂¯
jK(z, ¯
z) ω = gi¯
j(z, ¯
z) dzi d¯ z
¯ j = ∂ ¯
∂K(z, ¯ z).
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 6 / 61
SLIDE 14 Calabi-Yau Metrics Kähler Geometry
Kähler Metrics on the Quintic
Let’s consider our favourite CY threefold: QF =
0 + z5 1 + z5 2 + z5 3 + z5 4 = 0
The metric is completely determined by the Kähler potential K(z, ¯ z): gi¯
j(z, ¯
z) = ∂i ¯ ∂¯
jK(z, ¯
z) ω = gi¯
j(z, ¯
z) dzi d¯ z
¯ j = ∂ ¯
∂K(z, ¯ z). Locally, K is a real function. ω is a (1, 1)-form.
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 6 / 61
SLIDE 15 Calabi-Yau Metrics Kähler Geometry
Fubini-Study Metric
Unique SU(5) invariant Kähler metric on P4 KFS = ln
4
zi¯ z¯
i
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 7 / 61
SLIDE 16 Calabi-Yau Metrics Kähler Geometry
Fubini-Study Metric
Unique SU(5) invariant Kähler metric on P4 KFS = ln
4
zi¯ z¯
i
Generalize to KFS = ln
4
β=0
hα¯
βzα¯
z¯
β
with h a hermitian 5 × 5 matrix.
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 7 / 61
SLIDE 17 Calabi-Yau Metrics Kähler Geometry
Fubini-Study Metric
Unique SU(5) invariant Kähler metric on P4 KFS = ln
4
zi¯ z¯
i
Generalize to KFS = ln
4
β=0
hα¯
βzα¯
z¯
β
with h a hermitian 5 × 5 matrix. Restrict to Q ⊂ P4. Not Ricci flat.
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 7 / 61
SLIDE 18 Calabi-Yau Metrics Donaldson’s Algorithm
Donaldson’s Ansatz
Let’s try [Donaldson] K(z, ¯ z) = ln
P¯ jℓ=k
h(i1,...,ik),(¯
j1,...,¯ jk) zi1 1 · · · zik k
¯ z
¯ j1 1 · · · z ¯ jk k
for some hermitian N × N matrix h
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 8 / 61
SLIDE 19 Calabi-Yau Metrics Donaldson’s Algorithm
Donaldson’s Ansatz
Let’s try [Donaldson] K(z, ¯ z) = ln
P¯ jℓ=k
h(i1,...,ik),(¯
j1,...,¯ jk) zi1 1 · · · zik k
¯ z
¯ j1 1 · · · z ¯ jk k
for some hermitian N × N matrix h N = 5 + k − 1 k
- =
- # deg k monomials
- Volker Braun (UPenn)
Metrics and the Laplace Operator Universität Hamburg 8 / 61
SLIDE 20 Calabi-Yau Metrics Donaldson’s Algorithm
Technicalities
On the quintic z5
0 + z5 1 + z5 2 + z5 3 + z5 4 = 0. So not all
monomials are independent in degrees k ≥ 5.
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 9 / 61
SLIDE 21 Calabi-Yau Metrics Donaldson’s Algorithm
Technicalities
On the quintic z5
0 + z5 1 + z5 2 + z5 3 + z5 4 = 0. So not all
monomials are independent in degrees k ≥ 5. Let sα be a basis for C[z0, . . . , z4]
0 + z5 1 + z5 2 + z5 3 + z5 4 = 0
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 9 / 61
SLIDE 22 Calabi-Yau Metrics Donaldson’s Algorithm
Technicalities
On the quintic z5
0 + z5 1 + z5 2 + z5 3 + z5 4 = 0. So not all
monomials are independent in degrees k ≥ 5. Let sα be a basis for C[z0, . . . , z4]
0 + z5 1 + z5 2 + z5 3 + z5 4 = 0
Donaldson’s Ansatz K(z, ¯ z) = ln
β
hα¯
βsα¯
s¯
β
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 9 / 61
SLIDE 23 Calabi-Yau Metrics Donaldson’s Algorithm
More Technical
sα are sections of OQ(k)
0→H0
P4, O(k −5)
P4, O(k)
Q, OQ(k)
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 10 / 61
SLIDE 24 Calabi-Yau Metrics Donaldson’s Algorithm
More Technical
sα are sections of OQ(k)
0→H0
P4, O(k −5)
P4, O(k)
Q, OQ(k)
Metric on the line bundle OQ(k) (σ, τ) = σ(z)¯ τ(¯ z) hα¯
βsα(z)¯
s¯
β(¯
z)
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 10 / 61
SLIDE 25 Calabi-Yau Metrics Donaldson’s Algorithm
More Technical
sα are sections of OQ(k)
0→H0
P4, O(k −5)
P4, O(k)
Q, OQ(k)
Metric on the line bundle OQ(k) (σ, τ) = σ(z)¯ τ(¯ z) hα¯
βsα(z)¯
s¯
β(¯
z) Metric on the space of sections H0 Q, OQ(k)
z) dVol
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 10 / 61
SLIDE 26 Calabi-Yau Metrics Donaldson’s Algorithm
More Technical
sα are sections of OQ(k)
0→H0
P4, O(k −5)
P4, O(k)
Q, OQ(k)
Metric on the line bundle OQ(k) (σ, τ) = σ(z)¯ τ(¯ z) hα¯
βsα(z)¯
s¯
β(¯
z) ∈ C∞(Q, C) Metric on the space of sections H0 Q, OQ(k)
z) dVol ∈ C
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 10 / 61
SLIDE 27 Calabi-Yau Metrics Donaldson’s Algorithm
Balanced Metrics
h is “balanced” if the matrices representing the metrics coincide, that is:
β≤N = h−1
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 11 / 61
SLIDE 28 Calabi-Yau Metrics Donaldson’s Algorithm
Balanced Metrics
h is “balanced” if the matrices representing the metrics coincide, that is:
β≤N = h−1
Theorem Let h be the balanced metric for each k. Then the sequence of metrics ωk = ∂ ¯ ∂ ln
βsα¯
s¯
β
converges to the Calabi-Yau metric as k → ∞.
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 11 / 61
SLIDE 29 Calabi-Yau Metrics Donaldson’s Algorithm
T-Operator
How to solve
−1 = h?
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SLIDE 30 Calabi-Yau Metrics Donaldson’s Algorithm
T-Operator
How to solve
−1 = h? Donaldson’s T-operator: T(h)α¯
β =
sα¯ s¯
β
hα¯
βsα(z)¯
s¯
β(¯
z) dVol
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 12 / 61
SLIDE 31 Calabi-Yau Metrics Donaldson’s Algorithm
T-Operator
How to solve
−1 = h? Donaldson’s T-operator: T(h)α¯
β =
sα¯ s¯
β
hα¯
βsα(z)¯
s¯
β(¯
z) dVol One can show that iterating T(hn)−1 = hn+1 converges! Fixed point is balanced metric.
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 12 / 61
SLIDE 32 Calabi-Yau Metrics Implementation
Donaldson’s Algorithm
Pick a basis of sections sα
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SLIDE 33 Calabi-Yau Metrics Implementation
Donaldson’s Algorithm
Pick a basis of sections sα Iterate h = T(h)−1 where T(h)α¯
β =
sα¯ s¯
β
sαhα¯
β¯
s¯
β
dVol
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 13 / 61
SLIDE 34 Calabi-Yau Metrics Implementation
Donaldson’s Algorithm
Pick a basis of sections sα Iterate h = T(h)−1 where T(h)α¯
β =
sα¯ s¯
β
sαhα¯
β¯
s¯
β
dVol The approximate Calabi-Yau metric is gi¯
j = ∂i ¯
∂¯
j ln
β¯
s¯
β
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 13 / 61
SLIDE 35 Calabi-Yau Metrics Implementation
Details
Exact Calabi-Yau volume form dVol = Ω ∧ ¯ Ω, Ω =
Q(z)
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 14 / 61
SLIDE 36 Calabi-Yau Metrics Implementation
Details
Exact Calabi-Yau volume form dVol = Ω ∧ ¯ Ω, Ω =
Q(z) Integrate by summing over random points. [Douglas,Karp,Lukic,Reinbacher]
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 14 / 61
SLIDE 37 Calabi-Yau Metrics Implementation
Details
Exact Calabi-Yau volume form dVol = Ω ∧ ¯ Ω, Ω =
Q(z) Integrate by summing over random points. [Douglas,Karp,Lukic,Reinbacher] Implemented in C++ Parallelizable (MPI) Use 10 node dual-core Opteron cluster (Evelyn Thomson, ATLAS).
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 14 / 61
SLIDE 38 Calabi-Yau Metrics Testing the Metric
Testing the Result
How do we test whether the metric is the Calabi-Yau metric? We could compute the Ricci tensor, but its easier to test that Ω ∧ ¯ Ω ∼ ω3
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 15 / 61
SLIDE 39 Calabi-Yau Metrics Testing the Metric
Testing the Result
How do we test whether the metric is the Calabi-Yau metric? We could compute the Ricci tensor, but its easier to test that Ω ∧ ¯ Ω ∼ ω3 So normalize both volume forms and define σk =
Ω(¯ z) ω3(z, ¯ z)
On a Calabi-Yau manifold σk = O(k −2)
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 15 / 61
SLIDE 40 0.1 0.2 0.3 0.4 0.5
k= k= 1 k= 2 k= 3 k= 4 k= 5 k= 6 k= 7 k= 8
σ σk Fit for k ≥ 3: σk = 3.23/k 2 − 4.55/k 3
SLIDE 41 The Z5 × Z5 Quotient
Outline
1
Introduction
2
Calabi-Yau Metrics
3
The Z5 × Z5 Quotient Symmetric Quintics Invariant Theory
4
The Laplace Operator
5
Pretty Pictures
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SLIDE 42 The Z5 × Z5 Quotient Symmetric Quintics
Symmetric Quintics
The Fermat quintic is part of a 5-dimensional family of quintics with a free Z5 × Z5 group action.
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SLIDE 43 The Z5 × Z5 Quotient Symmetric Quintics
Symmetric Quintics
The Fermat quintic is part of a 5-dimensional family of quintics with a free Z5 × Z5 group action. It is numerically much easier to work on the four-generation quotient Q
Q = Q
OQ(k) = Oe
Q(k)
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 18 / 61
SLIDE 44 The Z5 × Z5 Quotient Symmetric Quintics
Symmetric Quintics
The Fermat quintic is part of a 5-dimensional family of quintics with a free Z5 × Z5 group action. It is numerically much easier to work on the four-generation quotient Q
Q = Q
OQ(k) = Oe
Q(k)
To do this, we only have to replace the sections sα of Oe
Q(k) by invariant sections!
H0 Q, OQ(k)
Q(k)
Z5×Z5
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SLIDE 45 The Z5 × Z5 Quotient Symmetric Quintics
Symmetry Group
g1 z0
z1 z2 z3 z4
0 0 0 0 1
1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0
z0
z1 z2 z3 z4
z0
z1 z2 z3 z4
1 0 e
2πi 5
e2 2πi
5
e3 2πi
5
e4 2πi
5
z0
z1 z2 z3 z4
1 g−1 2
= e
2πi 5 , so they generate the
Heisenberg group 0 → Z5 → G → Z5 × Z5 → 0
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 19 / 61
SLIDE 46 The Z5 × Z5 Quotient Invariant Theory
Invariant Theory
The invariant sections are C[z0, z1, z2, z3, z4]G
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SLIDE 47 The Z5 × Z5 Quotient Invariant Theory
Invariant Theory
The invariant sections are C[z0, z1, z2, z3, z4]G =
100
ηiC[θ1, θ2, θ3, θ4, θ5] (“Hironaka decomposition”)
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 20 / 61
SLIDE 48 The Z5 × Z5 Quotient Invariant Theory
Invariant Theory
The invariant sections are C[z0, z1, z2, z3, z4]G =
100
ηiC[θ1, θ2, θ3, θ4, θ5] (“Hironaka decomposition”) where θ1
def
= z5
0 + z5 1 + z5 2 + z5 3 + z5 4
θ2
def
= z0z1z2z3z4 θ3
def
= z3
0z1z4 + z0z3 1z2 + z0z3z3 4 + z1z3 2z3 + z2z3 3z4
θ4
def
= z10
0 + z10 1 + z10 2 + z10 3 + z10 4
θ5
def
= z8
0z2z3 + z0z1z8 3 + z0z8 2z4 + z8 1z3z4 + z1z2z8 4
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SLIDE 49 The Z5 × Z5 Quotient Invariant Theory
Secondary Invariants
... and the “secondary invariants” ηi are polynomials in degrees 0, 5, 10, 15, 20, 25, 30: η1
def
= 1 η2
def
= z2
0z1z2 2 + z2 1z2z2 3 + z2 2z3z2 4 + z2 3z4z2 0 + z2 4z0z2 1
. . . η100
def
= z30
0 + z30 1 + z30 2 + z30 3 + z30 4
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 21 / 61
SLIDE 50 The Z5 × Z5 Quotient Invariant Theory
Secondary Invariants
... and the “secondary invariants” ηi are polynomials in degrees 0, 5, 10, 15, 20, 25, 30: η1
def
= 1 η2
def
= z2
0z1z2 2 + z2 1z2z2 3 + z2 2z3z2 4 + z2 3z4z2 0 + z2 4z0z2 1
. . . η100
def
= z30
0 + z30 1 + z30 2 + z30 3 + z30 4
All invariants are in degrees divisible by 5 No invariant sections in Oe
Q(k) unless 5|k?
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 21 / 61
SLIDE 51 The Z5 × Z5 Quotient Invariant Theory
Secondary Invariants
... and the “secondary invariants” ηi are polynomials in degrees 0, 5, 10, 15, 20, 25, 30: η1
def
= 1 η2
def
= z2
0z1z2 2 + z2 1z2z2 3 + z2 2z3z2 4 + z2 3z4z2 0 + z2 4z0z2 1
. . . η100
def
= z30
0 + z30 1 + z30 2 + z30 3 + z30 4
All invariants are in degrees divisible by 5 No invariant sections in Oe
Q(k) unless 5|k?
Oe
Q(k) only equivariant if 5|k.
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 21 / 61
SLIDE 52 The Laplace Operator
Outline
1
Introduction
2
Calabi-Yau Metrics
3
The Z5 × Z5 Quotient
4
The Laplace Operator Solving the Laplace Equation Example: P3
5
Pretty Pictures
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SLIDE 53 The Laplace Operator Solving the Laplace Equation
The Laplace-Beltrami Operator
The scalar Laplace operator ∆
- φi
- = λi
- φi
- Volker Braun (UPenn)
Metrics and the Laplace Operator Universität Hamburg 23 / 61
SLIDE 54 The Laplace Operator Solving the Laplace Equation
The Laplace-Beltrami Operator
The scalar Laplace operator ∆
- φi
- = λi
- φi
- In terms of some (non-orthogonal) basis of functions {fs},
we can write
φi
Metrics and the Laplace Operator Universität Hamburg 23 / 61
SLIDE 55 The Laplace Operator Solving the Laplace Equation
The Laplace-Beltrami Operator
The scalar Laplace operator ∆
- φi
- = λi
- φi
- In terms of some (non-orthogonal) basis of functions {fs},
we can write
φi
- (Generalized) eigenvalue equation
∆
ft
φi
= λi
ft
φi
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 23 / 61
SLIDE 56 The Laplace Operator Solving the Laplace Equation
Spherical Harmonics
Using an approximate finite basis {fs}, we only have to solve the generalized eigenvalue problem
v = λi
v
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SLIDE 57 The Laplace Operator Solving the Laplace Equation
Spherical Harmonics
Using an approximate finite basis {fs}, we only have to solve the generalized eigenvalue problem
v = λi
v Nice basis: Recall that P4 = S9 U(1) So take the U(1)-invariant spherical harmonics on S9.
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 24 / 61
SLIDE 58 The Laplace Operator Solving the Laplace Equation
Homogeneous Coordinates
In homogeneous coordinates, the spherical harmonics are
- degree k monomial
- degree k monomial
- |z0|2 + |z1|2 + |z2|2 + |z3|2 + |z4|2
k So, for example k = 1 on P1: Homog.
z0¯ z0 |z0|2+|z1|2 z1¯ z0 |z0|2+|z1|2 z0¯ z1 |z0|2+|z1|2 z1¯ z1 |z0|2+|z1|2
Inhomog.
1 1+|x|2 x 1+|x|2 ¯ x 1+|x|2 x¯ x 1+|x|2
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SLIDE 59 The Laplace Operator Example: P3
Example: P3
Analytic result: Multiplicities of eigenvalues µn = n + 3 n 2 − n + 2 n − 1 2 , n = 0, 1, . . . Eigenvalues (normalize Vol P3 = 1) λn,0 = · · · = λn,µn−1 = 4π
3
√ 6 n(n + 3)
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SLIDE 60 The Laplace Operator Example: P3
Example: P3
Analytic result: Multiplicities of eigenvalues µn = n + 3 n 2 − n + 2 n − 1 2 , n = 0, 1, . . . Eigenvalues (normalize Vol P3 = 1) λn,0 = · · · = λn,µn−1 = 4π
3
√ 6 n(n + 3) Numeric result: k = 3, Np = 100,000.
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 26 / 61
SLIDE 61
Spectrum on P3: k = 3, Np = 100,000
20 40 60 80 100 120 140 50 100 150 200 250 300 350 400 λn n Eigenvalues k = 3, Np = 100,000
SLIDE 62 Spectrum on P3: k = 3
50 100 150 200 10000 100000 1e+06
16π
3
√ 6 40π
3
√ 6 72π
3
√ 6
λ Np µ0 = 1 µ1 = 15 µ2 = 84 µ3 = 300 Eigenvalues, k = 3
SLIDE 63
Spectrum on P3: Np = 100,000
50 100 150 200 250 300 1 2 3 4 5 27.6622 69.1554 124.48 193.635 276.622 λ k µ0 = 1 µ1 = 15 µ2 = 84 µ3 = 300 µ4 = 825 µ5 = 1911
SLIDE 64 Pretty Pictures
1
Introduction
2
Calabi-Yau Metrics
3
The Z5 × Z5 Quotient
4
The Laplace Operator
5
Pretty Pictures Random Quintic Fermat Quintic Quintic Quotient Families Differential Forms
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SLIDE 65 Pretty Pictures Random Quintic
Random Quintic
Now, take some quintic Q(z) = (−0.3192 + 0.7096i)z5
0 + (−0.3279 + 0.8119i)z4 0z1
+ (0.2422 + 0.2198i)z4
0z2 + · · · + (−0.2654 + 0.1222i)z5 4
with 126 random (nonzero) coefficients.
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 31 / 61
SLIDE 66 Pretty Pictures Random Quintic
Random Quintic
Now, take some quintic Q(z) = (−0.3192 + 0.7096i)z5
0 + (−0.3279 + 0.8119i)z4 0z1
+ (0.2422 + 0.2198i)z4
0z2 + · · · + (−0.2654 + 0.1222i)z5 4
with 126 random (nonzero) coefficients. No symmetry expect all eigenvalues to have multiplicity 1.
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 31 / 61
SLIDE 67 Pretty Pictures Random Quintic
Random Quintic
Now, take some quintic Q(z) = (−0.3192 + 0.7096i)z5
0 + (−0.3279 + 0.8119i)z4 0z1
+ (0.2422 + 0.2198i)z4
0z2 + · · · + (−0.2654 + 0.1222i)z5 4
with 126 random (nonzero) coefficients. No symmetry expect all eigenvalues to have multiplicity 1. Metric: kh = 8. Integrate T-operator using 3,000,000 points. Normalize Vol(Q) = 1. Laplacian: kφ = 3. Integrate using Np = 200,000 points.
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SLIDE 68
Random Quintic: kφ = 3, Np = 200,000
50 100 150 200 100 200 300 400 500 λn n Eigenvalues on random quintic
SLIDE 69
Random Quintic: Np = 200,000
50 100 150 200 1 2 3 λ kφ Eigenvalues
SLIDE 70 Pretty Pictures Random Quintic
Weyl’s Formula
Theorem (Weyl) lim
n→∞
λd/2
n
n = (4π)
d 2 Γ
d
2 + 1
, where d [= 6] is the dimension of the manifold.
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SLIDE 71 Pretty Pictures Random Quintic
Weyl’s Formula
Theorem (Weyl) lim
n→∞
λd/2
n
n = (4π)
d 2 Γ
d
2 + 1
, where d [= 6] is the dimension of the manifold. Independent check on the volume normalization.
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SLIDE 72 Weyl’s Limit
384π3 5000 10000 15000 20000 25000 100 200 300 400 500 λ3
n
n n kφ = 1 kφ = 2 kφ = 3
SLIDE 73 Pretty Pictures Random Quintic
Massive Gravitons
Consider KK modes of the graviton that are spin-2 in 4 dimensions: h10d
µν =
h4d
n,µν(x0, x1, x2, x3) · φ6d n (y1, . . . , y6)
Mass mn = √λn.
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 36 / 61
SLIDE 74 Pretty Pictures Random Quintic
Massive Gravitons
Consider KK modes of the graviton that are spin-2 in 4 dimensions: h10d
µν =
h4d
n,µν(x0, x1, x2, x3) · φ6d n (y1, . . . , y6)
Mass mn = √λn. Gravitational potential between two test masses M1 and M2: V(r) = −G4 M1M2 r
∞
e−√λnr
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 36 / 61
SLIDE 75
1 0.6 0.8 1.0 1.2 1.4 V(r)
r V(r ≫ 1) = −G4 M1M2 r V(r ≪ 1) = −15G4 8π3 M1M2 r 7
SLIDE 76 Pretty Pictures Fermat Quintic
1
Introduction
2
Calabi-Yau Metrics
3
The Z5 × Z5 Quotient
4
The Laplace Operator
5
Pretty Pictures Random Quintic Fermat Quintic Quintic Quotient Families Differential Forms
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 38 / 61
SLIDE 77
Fermat Quintic: Np = 500,000
50 100 150 200 1 2 3 λ kφ µ0 = 1 µ1 = 20 µ2 = 20 µ3 = 4 µ4 = 60 µ5 = 30
SLIDE 78 Pretty Pictures Fermat Quintic
Symmetries of the Fermat Quintic
The first massive eigenmode has degeneracy 20.
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 40 / 61
SLIDE 79 Pretty Pictures Fermat Quintic
Symmetries of the Fermat Quintic
The first massive eigenmode has degeneracy 20. The automorphisms group of the Fermat quintic QF is Aut
4 Lemma (Representation theory of Aut( QF)) The 80 irreps of Aut( QF) are in dimension Dimension d 1 2 4 5 6 8 10 Irreps in dim d 4 4 4 4 2 4 4 · · · · · · 12 20 30 40 60 80 120 2 8 8 12 18 4 2
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 40 / 61
SLIDE 80 Pretty Pictures Fermat Quintic
Symmetries of the Fermat Quintic
The first massive eigenmode has degeneracy 20. The automorphisms group of the Fermat quintic QF is Aut
4 Lemma (Representation theory of Aut( QF)) The 80 irreps of Aut( QF) are in dimension Dimension d 1 2 4 5 6 8 10 Irreps in dim d 4 4 4 4 2 4 4 · · · · · · 12 20 30 40 60 80 120 2 8 8 12 18 4 2
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 40 / 61
SLIDE 81 Pretty Pictures Fermat Quintic
Donaldson’s Operator
A (conjectural) alternative calculation of the spectrum
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 41 / 61
SLIDE 82 Pretty Pictures Fermat Quintic
Donaldson’s Operator
A (conjectural) alternative calculation of the spectrum
Specific to the scalar Laplacian only.
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 41 / 61
SLIDE 83 Pretty Pictures Fermat Quintic
Donaldson’s Operator
A (conjectural) alternative calculation of the spectrum
Specific to the scalar Laplacian only. “Compares” balanced metrics at k and 2k. e∆ ∼ Qα¯
β,¯ γδ =
- (sα, sβ)(sγ, sδ) dVol
- Recall:
T(h)α¯
β =
- (sα, sβ) dVol
- Volker Braun (UPenn)
Metrics and the Laplace Operator Universität Hamburg 41 / 61
SLIDE 84
Fermat Quintic: Donaldson’s Operator
20 40 60 80 100 120 140 20 40 60 80 100 120 140 λn n Donaldson’s operator, kh = 1 Donaldson’s operator, kh = 2 Donaldson’s operator, kh = 3
SLIDE 85
Fermat Quintic: scalar Laplacian
20 40 60 80 100 120 140 20 40 60 80 100 120 140 λn n kφ = 3, kh = 3 (low precision metric) kφ = 3, kh = 8 (high precision metric)
SLIDE 86
Donaldson vs. scalar Laplacian
20 40 60 80 100 120 140 20 40 60 80 100 120 140 λn n Donaldson’s operator, kh = 1 Donaldson’s operator, kh = 2 Donaldson’s operator, kh = 3 kφ = 3, kh = 3 (low precision metric) kφ = 3, kh = 8 (high precision metric)
SLIDE 87 Pretty Pictures Quintic Quotient
1
Introduction
2
Calabi-Yau Metrics
3
The Z5 × Z5 Quotient
4
The Laplace Operator
5
Pretty Pictures Random Quintic Fermat Quintic Quintic Quotient Families Differential Forms
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 45 / 61
SLIDE 88 Pretty Pictures Quintic Quotient
Fermat Quintic
QF = QF
- Z5 × Z5
- Volker Braun (UPenn)
Metrics and the Laplace Operator Universität Hamburg 46 / 61
SLIDE 89 Pretty Pictures Quintic Quotient
Fermat Quintic
QF = QF
- Z5 × Z5
- Simply use only invariant functions
- ft
- .
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 46 / 61
SLIDE 90 Pretty Pictures Quintic Quotient
Fermat Quintic
QF = QF
- Z5 × Z5
- Simply use only invariant functions
- ft
- .
Yields eigenvalues λZ5×Z5
n
.
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 46 / 61
SLIDE 91 Pretty Pictures Quintic Quotient
Fermat Quintic
QF = QF
- Z5 × Z5
- Simply use only invariant functions
- ft
- .
Yields eigenvalues λZ5×Z5
n
. Rescale volume to one: 1 25 = Vol
→ 1 λZ5×Z5
n
− → 25−1/3λZ5×Z5
n
= λn
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 46 / 61
SLIDE 92 Fermat Quintic
100 200 300 400 20 40 60 80 100 25 50 75 100 125 λZ5×Z5
n
λn n kφ = 0 kφ = 1 kφ = 2
SLIDE 93 Pretty Pictures Families
1
Introduction
2
Calabi-Yau Metrics
3
The Z5 × Z5 Quotient
4
The Laplace Operator
5
Pretty Pictures Random Quintic Fermat Quintic Quintic Quotient Families Differential Forms
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 48 / 61
SLIDE 94 Pretty Pictures Families
Quintic
A family of Quintics
z5
i − 5ψ
Qψ
- Z5 × Z5
- Volker Braun (UPenn)
Metrics and the Laplace Operator Universität Hamburg 49 / 61
SLIDE 95 Pretty Pictures Families
Quintic
A family of Quintics
z5
i − 5ψ
Qψ
- Z5 × Z5
- Q0 is the Fermat quintic.
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 49 / 61
SLIDE 96 Pretty Pictures Families
Quintic
A family of Quintics
z5
i − 5ψ
Qψ
- Z5 × Z5
- Q0 is the Fermat quintic.
ψ = 1 is the conifold point.
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 49 / 61
SLIDE 97 Pretty Pictures Families
Quintic
A family of Quintics
z5
i − 5ψ
Qψ
- Z5 × Z5
- Q0 is the Fermat quintic.
ψ = 1 is the conifold point. ψ → ∞ is the “large complex structure limit”.
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 49 / 61
SLIDE 98 Pretty Pictures Families
Conifold Point of the Quintic
Conifold Point z5
0 + z5 1 + z5 2 + z5 3 + z5 4 − 5z0z1z2z3z4 = 0
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 50 / 61
SLIDE 99 Pretty Pictures Families
Conifold Point of the Quintic
Conifold Point z5
0 + z5 1 + z5 2 + z5 3 + z5 4 − 5z0z1z2z3z4 = 0
is singular at zC = [1 : 1 : 1 : 1 : 1]: Q1(zC) = 0 = ∂Q1 ∂zi (zC)
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 50 / 61
SLIDE 100 Quintic
20 40 60 80 100
0.5 1 1.5 λ ψ Qψ = z5
i − 5ψ
SLIDE 101 Pretty Pictures Families
Large Complex Structure Limit
Qψ = z5
i − 5ψ
- zi = 0
- Z5 × Z5
- ψ → ∞ is called “large complex structure limit”.
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 52 / 61
SLIDE 102 Pretty Pictures Families
Large Complex Structure Limit
Qψ = z5
i − 5ψ
- zi = 0
- Z5 × Z5
- ψ → ∞ is called “large complex structure limit”.
Strominger-Yau-Zaslow conjecture Any Calabi-Yau threefold is fibered by special Lagrangian 3-tori over a (rational homotopy) 3-sphere. In the large complex structure limit the 3-torus fiber shrinks to zero size.
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 52 / 61
SLIDE 103 Pretty Pictures Families
Large Complex Structure Limit
Qψ = z5
i − 5ψ
- zi = 0
- Z5 × Z5
- ψ → ∞ is called “large complex structure limit”.
Strominger-Yau-Zaslow conjecture Any Calabi-Yau threefold is fibered by special Lagrangian 3-tori over a (rational homotopy) 3-sphere. In the large complex structure limit the 3-torus fiber shrinks to zero size. Hence, the spectrum of the Laplacian degenerates into the spectrum of the base space.
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 52 / 61
SLIDE 104
Large Complex Structure Limit
20 40 60 80 100 0 1 10 100 ∞ λ ψ
SLIDE 105 Pretty Pictures Families
Quintic
- (Z5 × Z5) Family #2
- Qψ = z5
i − 5ψ zi has residual permutation symmetry.
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 54 / 61
SLIDE 106 Pretty Pictures Families
Quintic
- (Z5 × Z5) Family #2
- Qψ = z5
i − 5ψ zi has residual permutation symmetry.
Another family of Quintics
i + ϕ
i + iϕ
0z1z4 + cyc
0z1z2 2 + cyc
0z2 1z3 + cyc
0z2z3 + cyc
Qϕ
- Z5 × Z5
- Volker Braun (UPenn)
Metrics and the Laplace Operator Universität Hamburg 54 / 61
SLIDE 107 Pretty Pictures Families
Quintic
- (Z5 × Z5) Family #2
- Qψ = z5
i − 5ψ zi has residual permutation symmetry.
Another family of Quintics
i + ϕ
i + iϕ
0z1z4 + cyc
0z1z2 2 + cyc
0z2 1z3 + cyc
0z2z3 + cyc
Qϕ
- Z5 × Z5
- Q0 is again the Fermat quintic.
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 54 / 61
SLIDE 108 Quintic
20 40 60 80 100 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 λ ϕ Qϕ
SLIDE 109 Pretty Pictures Families
Spectral Gap
Definition The spectral gap is the gap between the unique zero mode λ0 = 0 and λ1.
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 56 / 61
SLIDE 110 Pretty Pictures Families
Spectral Gap
Definition The spectral gap is the gap between the unique zero mode λ0 = 0 and λ1. Theorem On a non-negatively curved manifold (e.g. Calabi-Yau) π2 D2 ≤ λ1 ≤ 2d(d + 4) D2 , where d is the real dimension and D the diameter.
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 56 / 61
SLIDE 111 Pretty Pictures Families
Spectral Gap
Definition The spectral gap is the gap between the unique zero mode λ0 = 0 and λ1. Theorem On a non-negatively curved manifold (e.g. Calabi-Yau) π2 D2 ≤ λ1 ≤ 2d(d + 4) D2 , where d is the real dimension and D the diameter. Essentially determined by diameter!
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 56 / 61
SLIDE 112 Pretty Pictures Families
The Diameter
Turn inequality around and estimate diameter from the spectral gap.
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 57 / 61
SLIDE 113 Pretty Pictures Families
The Diameter
Turn inequality around and estimate diameter from the spectral gap. Fermat quintic has λ1 ≈ 41.1 ⇒ 0.490 ≈ π √λ1 ≤ D ≤
√λ1 ≈ 1.71
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 57 / 61
SLIDE 114 Pretty Pictures Differential Forms
1
Introduction
2
Calabi-Yau Metrics
3
The Z5 × Z5 Quotient
4
The Laplace Operator
5
Pretty Pictures Random Quintic Fermat Quintic Quintic Quotient Families Differential Forms
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 58 / 61
SLIDE 115 Pretty Pictures Differential Forms
Differential Forms
The Laplace-Dolbeault Operator ∆φ(p,q)
n
= λnφ(p,q)
n
, φ(p,q)
n
∈ Ω(p,q)
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 59 / 61
SLIDE 116 Pretty Pictures Differential Forms
Differential Forms
The Laplace-Dolbeault Operator ∆φ(p,q)
n
= λnφ(p,q)
n
, φ(p,q)
n
∈ Ω(p,q) Complex conjugation & Hodge star: λ(p,q)
n
= λ(q,p)
n
= λ(3−p,3−q)
n
= λ(3−q,3−p)
n
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 59 / 61
SLIDE 117 Differential Forms on P3
(0,0) (3,3) (1,1) (2,2) (1,2) (2,1) (0,3) (3,0) (1,0) (0,1) (2,3) (3,2) (2,0) (0,2) (1,3) (3,1)
20 40 60 80 100 λ(p,q)
SLIDE 118 Summary
Summary
Calabi-Yau metrics are easy.
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 61 / 61
SLIDE 119 Summary
Summary
Calabi-Yau metrics are easy. Scalar Laplace-operator solved numerically.
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 61 / 61
SLIDE 120 Summary
Summary
Calabi-Yau metrics are easy. Scalar Laplace-operator solved numerically. Outlook
Laplacian on differential forms (Done on P3)
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 61 / 61
SLIDE 121 Summary
Summary
Calabi-Yau metrics are easy. Scalar Laplace-operator solved numerically. Outlook
Laplacian on differential forms (Done on P3) Vector bundles.
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 61 / 61
SLIDE 122 Summary
Summary
Calabi-Yau metrics are easy. Scalar Laplace-operator solved numerically. Outlook
Laplacian on differential forms (Done on P3) Vector bundles. Special Lagrangians.
Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 61 / 61
