Calabi-Yau Metrics and the Spectrum of the Laplace Operator Volker - PowerPoint PPT Presentation

Calabi-Yau Metrics Donaldson’s Algorithm Balanced Metrics h is “balanced” if the matrices representing the metrics coincide, that is: �� �� β ≤ N = h − 1 s α , s β 1 ≤ α, ¯ Theorem Let h be the balanced metric for each k. Then the sequence of metrics � h α ¯ ω k = ∂ ¯ β s α ¯ ∂ ln s ¯ β converges to the Calabi-Yau metric as k → ∞ . Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 11 / 61

Calabi-Yau Metrics Donaldson’s Algorithm T-Operator How to solve �� �� − 1 s α , s β = h ? Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 12 / 61

Calabi-Yau Metrics Donaldson’s Algorithm T-Operator How to solve �� �� − 1 s α , s β = h ? Donaldson’s T-operator: � � T ( h ) α ¯ β = s α , s β � s α ¯ s ¯ β = � h α ¯ z ) dVol β s α ( z )¯ β (¯ s ¯ Q Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 12 / 61

Calabi-Yau Metrics Donaldson’s Algorithm T-Operator How to solve �� �� − 1 s α , s β = h ? Donaldson’s T-operator: � � T ( h ) α ¯ β = s α , s β � s α ¯ s ¯ β = � h α ¯ z ) dVol β s α ( z )¯ β (¯ s ¯ Q One can show that iterating T ( h n ) − 1 = h n + 1 converges! Fixed point is balanced metric. Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 12 / 61

Calabi-Yau Metrics Implementation Donaldson’s Algorithm Pick a basis of sections s α Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 13 / 61

Calabi-Yau Metrics Implementation Donaldson’s Algorithm Pick a basis of sections s α Iterate h = T ( h ) − 1 where � s α ¯ s ¯ β T ( h ) α ¯ β = dVol s α h α ¯ β ¯ s ¯ Q β Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 13 / 61

Calabi-Yau Metrics Implementation Donaldson’s Algorithm Pick a basis of sections s α Iterate h = T ( h ) − 1 where � s α ¯ s ¯ β T ( h ) α ¯ β = dVol s α h α ¯ β ¯ s ¯ Q β The approximate Calabi-Yau metric is � s α h α ¯ j = ∂ i ¯ β ¯ g i ¯ ∂ ¯ j ln s ¯ β Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 13 / 61

Calabi-Yau Metrics Implementation Details Exact Calabi-Yau volume form � d 4 z dVol = Ω ∧ ¯ Ω , Ω = Q ( z ) Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 14 / 61

Calabi-Yau Metrics Implementation Details Exact Calabi-Yau volume form � d 4 z 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

Calabi-Yau Metrics Implementation Details Exact Calabi-Yau volume form � d 4 z 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

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 ∼ ω 3 Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 15 / 61

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 ∼ ω 3 So normalize both volume forms and define � � � � 1 − Ω ( z ) ∧ ¯ � � Ω (¯ z ) � � σ k = � dVol ω 3 ( z , ¯ z ) Q On a Calabi-Yau manifold σ k = O ( k − 2 ) Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 15 / 61

0 . 5 σ k Fit for k ≥ 3: σ k = 3 . 23 / k 2 − 4 . 55 / k 3 0 . 4 0 . 3 σ 0 . 2 0 . 1 0 k = k = k = k = k = k = k = k = k = 0 1 2 3 4 5 6 7 8

The Z 5 × Z 5 Quotient Outline Introduction 1 Calabi-Yau Metrics 2 The Z 5 × Z 5 Quotient 3 Symmetric Quintics Invariant Theory The Laplace Operator 4 Pretty Pictures 5 Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 17 / 61

The Z 5 × Z 5 Quotient Symmetric Quintics Symmetric Quintics The Fermat quintic is part of a 5-dimensional family of quintics with a free Z 5 × Z 5 group action. Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 18 / 61

The Z 5 × Z 5 Quotient Symmetric Quintics Symmetric Quintics The Fermat quintic is part of a 5-dimensional family of quintics with a free Z 5 × Z 5 group action. It is numerically much easier to work on the �� � four-generation quotient Q Z 5 × Z 5 . �� �� � � Q = � Q Z 5 × Z 5 , O Q ( k ) = O e Q ( k ) Z 5 × Z 5 . Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 18 / 61

The Z 5 × Z 5 Quotient Symmetric Quintics Symmetric Quintics The Fermat quintic is part of a 5-dimensional family of quintics with a free Z 5 × Z 5 group action. It is numerically much easier to work on the �� � four-generation quotient Q Z 5 × Z 5 . �� �� � � Q = � Q Z 5 × Z 5 , O Q ( k ) = O e Q ( k ) Z 5 × Z 5 . To do this, we only have to replace the sections s α of Q ( k ) by invariant sections! O e = H 0 � � Z 5 × Z 5 H 0 � � � Q , O Q ( k ) Q , O e Q ( k ) Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 18 / 61

The Z 5 × Z 5 Quotient Symmetric Quintics Symmetry Group � z 0 � � 0 0 0 0 1 � � z 0 � z 1 z 1 1 0 0 0 0 g 1 = z 2 z 2 0 1 0 0 0 z 3 z 3 0 0 1 0 0 z 4 z 4 0 0 0 1 0   1 0 0 0 0 � z 0 � � z 0 � 2 π i  0 e 0 0 0  5 z 1 z 1   e 2 2 π i g 2 = z 2 z 2 0 0 5 0 0   z 3 z 3 e 3 2 π i 0 0 0 0 z 4 5 z 4 e 4 2 π i 0 0 0 0 5 2 π i Note that g 1 g 2 g − 1 1 g − 1 5 , so they generate the = e 2 Heisenberg group 0 → Z 5 → G → Z 5 × Z 5 → 0 Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 19 / 61

The Z 5 × Z 5 Quotient Invariant Theory Invariant Theory The invariant sections are C [ z 0 , z 1 , z 2 , z 3 , z 4 ] G Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 20 / 61

The Z 5 × Z 5 Quotient Invariant Theory Invariant Theory The invariant sections are 100 � C [ z 0 , z 1 , z 2 , z 3 , z 4 ] G = η i C [ θ 1 , θ 2 , θ 3 , θ 4 , θ 5 ] i = 1 (“Hironaka decomposition”) Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 20 / 61

The Z 5 × Z 5 Quotient Invariant Theory Invariant Theory The invariant sections are 100 � C [ z 0 , z 1 , z 2 , z 3 , z 4 ] G = η i C [ θ 1 , θ 2 , θ 3 , θ 4 , θ 5 ] i = 1 (“Hironaka decomposition”) where def z 5 0 + z 5 1 + z 5 2 + z 5 3 + z 5 θ 1 = 4 def θ 2 z 0 z 1 z 2 z 3 z 4 = def z 3 0 z 1 z 4 + z 0 z 3 1 z 2 + z 0 z 3 z 3 4 + z 1 z 3 2 z 3 + z 2 z 3 θ 3 3 z 4 = def z 10 0 + z 10 1 + z 10 2 + z 10 3 + z 10 θ 4 = 4 def z 8 0 z 2 z 3 + z 0 z 1 z 8 3 + z 0 z 8 2 z 4 + z 8 1 z 3 z 4 + z 1 z 2 z 8 θ 5 = 4 Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 20 / 61

The Z 5 × Z 5 Quotient Invariant Theory Secondary Invariants ... and the “secondary invariants” η i are polynomials in degrees 0, 5, 10, 15, 20, 25, 30: def η 1 = 1 def = z 2 0 z 1 z 2 2 + z 2 1 z 2 z 2 3 + z 2 2 z 3 z 2 4 + z 2 3 z 4 z 2 0 + z 2 4 z 0 z 2 η 2 1 . . . def = z 30 0 + z 30 1 + z 30 2 + z 30 3 + z 30 η 100 4 Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 21 / 61

The Z 5 × Z 5 Quotient Invariant Theory Secondary Invariants ... and the “secondary invariants” η i are polynomials in degrees 0, 5, 10, 15, 20, 25, 30: def η 1 = 1 def = z 2 0 z 1 z 2 2 + z 2 1 z 2 z 2 3 + z 2 2 z 3 z 2 4 + z 2 3 z 4 z 2 0 + z 2 4 z 0 z 2 η 2 1 . . . def = z 30 0 + z 30 1 + z 30 2 + z 30 3 + z 30 η 100 4 All invariants are in degrees divisible by 5 No invariant sections in O e Q ( k ) unless 5 | k ? Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 21 / 61

The Z 5 × Z 5 Quotient Invariant Theory Secondary Invariants ... and the “secondary invariants” η i are polynomials in degrees 0, 5, 10, 15, 20, 25, 30: def η 1 = 1 def = z 2 0 z 1 z 2 2 + z 2 1 z 2 z 2 3 + z 2 2 z 3 z 2 4 + z 2 3 z 4 z 2 0 + z 2 4 z 0 z 2 η 2 1 . . . def = z 30 0 + z 30 1 + z 30 2 + z 30 3 + z 30 η 100 4 All invariants are in degrees divisible by 5 No invariant sections in O e Q ( k ) unless 5 | k ? Q ( k ) only equivariant if 5 | k . O e Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 21 / 61

The Laplace Operator Outline Introduction 1 Calabi-Yau Metrics 2 The Z 5 × Z 5 Quotient 3 The Laplace Operator 4 Solving the Laplace Equation Example: P 3 Pretty Pictures 5 Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 22 / 61

The Laplace Operator Solving the Laplace Equation The Laplace-Beltrami Operator The scalar Laplace operator � � � � � φ i � φ i ∆ = λ i Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 23 / 61

The Laplace Operator Solving the Laplace Equation The Laplace-Beltrami Operator The scalar Laplace operator � � � � � φ i � φ i ∆ = λ i In terms of some (non-orthogonal) basis of functions { f s } , we can write � � � � �� � � � ˜ � φ i � f t = f t φ i t Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 23 / 61

The Laplace Operator Solving the Laplace Equation The Laplace-Beltrami Operator The scalar Laplace operator � � � � � φ i � φ i ∆ = λ i In terms of some (non-orthogonal) basis of functions { f s } , we can write � � � � �� � � � ˜ � φ i � f t = f t φ i t (Generalized) eigenvalue equation � � � � � φ i � φ i ∆ = λ i � � � � � � � � � � � � � � ∆ � f t � ˜ � f t � ˜ ⇒ f s f t φ i = λ i f s f t φ i � �� � � �� � � � v v Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 23 / 61

The Laplace Operator Solving the Laplace Equation Spherical Harmonics Using an approximate finite basis { f s } , we only have to solve the generalized eigenvalue problem � � � � � � � � � � ∆ � f t � f t f s v = λ i f s v Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 24 / 61

The Laplace Operator Solving the Laplace Equation Spherical Harmonics Using an approximate finite basis { f s } , we only have to solve the generalized eigenvalue problem � � � � � � � � � � ∆ � f t � f t f s v = λ i f s v Nice basis: Recall that P 4 = S 9 � U ( 1 ) So take the U ( 1 ) -invariant spherical harmonics on S 9 . Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 24 / 61

The Laplace Operator Solving the Laplace Equation Homogeneous Coordinates In homogeneous coordinates, the spherical harmonics are � �� � degree k monomial degree k monomial � � k | z 0 | 2 + | z 1 | 2 + | z 2 | 2 + | z 3 | 2 + | z 4 | 2 So, for example k = 1 on P 1 : z 0 ¯ z 1 ¯ z 0 ¯ z 1 ¯ z 0 z 0 z 1 z 1 Homog. | z 0 | 2 + | z 1 | 2 | z 0 | 2 + | z 1 | 2 | z 0 | 2 + | z 1 | 2 | z 0 | 2 + | z 1 | 2 ¯ x ¯ 1 x x x Inhomog. 1 + | x | 2 1 + | x | 2 1 + | x | 2 1 + | x | 2 Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 25 / 61

Example: P 3 The Laplace Operator Example: P 3 Analytic result: Multiplicities of eigenvalues � n + 3 � 2 � n + 2 � 2 µ n = − , n = 0 , 1 , . . . n n − 1 Eigenvalues (normalize Vol P 3 = 1) λ n , 0 = · · · = λ n ,µ n − 1 = 4 π √ n ( n + 3 ) 3 6 Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 26 / 61

Example: P 3 The Laplace Operator Example: P 3 Analytic result: Multiplicities of eigenvalues � n + 3 � 2 � n + 2 � 2 µ n = − , n = 0 , 1 , . . . n n − 1 Eigenvalues (normalize Vol P 3 = 1) λ n , 0 = · · · = λ n ,µ n − 1 = 4 π √ n ( n + 3 ) 3 6 Numeric result: k = 3, N p = 100 , 000. Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 26 / 61

Spectrum on P 3 : k = 3 , N p = 100 , 000 140 120 λ n 100 80 60 40 20 Eigenvalues k = 3, N p = 100 , 000 0 0 50 100 150 200 250 300 350 400 n

Spectrum on P 3 : k = 3 200 Eigenvalues, k = 3 150 µ 3 = 300 λ 72 π √ 3 6 100 µ 2 = 84 40 π √ 3 6 50 µ 1 = 15 16 π √ 3 6 µ 0 = 1 0 10000 100000 1e+06 N p

Spectrum on P 3 : N p = 100 , 000 300 µ 5 = 1911 276.622 λ 250 µ 4 = 825 200 193.635 150 µ 3 = 300 124.48 100 µ 2 = 84 69.1554 50 µ 1 = 15 27.6622 µ 0 = 1 0 0 0 1 2 3 4 5 k

Pretty Pictures Introduction 1 Calabi-Yau Metrics 2 The Z 5 × Z 5 Quotient 3 The Laplace Operator 4 Pretty Pictures 5 Random Quintic Fermat Quintic Quintic Quotient Families Differential Forms Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 30 / 61

Pretty Pictures Random Quintic Random Quintic Now, take some quintic Q ( z ) = ( − 0 . 3192 + 0 . 7096 i ) z 5 0 + ( − 0 . 3279 + 0 . 8119 i ) z 4 0 z 1 + ( 0 . 2422 + 0 . 2198 i ) z 4 0 z 2 + · · · + ( − 0 . 2654 + 0 . 1222 i ) z 5 4 with 126 random (nonzero) coefficients. Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 31 / 61

Pretty Pictures Random Quintic Random Quintic Now, take some quintic Q ( z ) = ( − 0 . 3192 + 0 . 7096 i ) z 5 0 + ( − 0 . 3279 + 0 . 8119 i ) z 4 0 z 1 + ( 0 . 2422 + 0 . 2198 i ) z 4 0 z 2 + · · · + ( − 0 . 2654 + 0 . 1222 i ) z 5 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

Pretty Pictures Random Quintic Random Quintic Now, take some quintic Q ( z ) = ( − 0 . 3192 + 0 . 7096 i ) z 5 0 + ( − 0 . 3279 + 0 . 8119 i ) z 4 0 z 1 + ( 0 . 2422 + 0 . 2198 i ) z 4 0 z 2 + · · · + ( − 0 . 2654 + 0 . 1222 i ) z 5 4 with 126 random (nonzero) coefficients. No symmetry expect all eigenvalues to have multiplicity 1. Metric: k h = 8. Integrate T-operator using 3 , 000 , 000 points. Normalize Vol ( Q ) = 1. Laplacian: k φ = 3. Integrate using N p = 200 , 000 points. Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 31 / 61

Random Quintic: k φ = 3 , N p = 200 , 000 200 λ n 150 100 50 Eigenvalues on random quintic 0 0 100 200 300 400 500 n

Random Quintic: N p = 200 , 000 200 Eigenvalues λ 150 100 50 0 0 1 2 3 k φ

Pretty Pictures Random Quintic Weyl’s Formula Theorem (Weyl) � d � � = 384 π 3 � d λ d / 2 2 Γ = ( 4 π ) 2 + 1 n lim , n Vol n →∞ where d [= 6 ] is the dimension of the manifold. Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 34 / 61

Pretty Pictures Random Quintic Weyl’s Formula Theorem (Weyl) � d � � = 384 π 3 � d λ d / 2 2 Γ = ( 4 π ) 2 + 1 n lim , n Vol n →∞ where d [= 6 ] is the dimension of the manifold. Independent check on the volume normalization. Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 34 / 61

Weyl’s Limit 25000 k φ = 1 k φ = 2 20000 k φ = 3 λ 3 n n 15000 384 π 3 10000 5000 0 0 100 200 300 400 500 n

Pretty Pictures Random Quintic Massive Gravitons Consider KK modes of the graviton that are spin-2 in 4 dimensions: � h 10 d h 4 d n ,µν ( x 0 , x 1 , x 2 , x 3 ) · φ 6 d µν = n ( y 1 , . . . , y 6 ) n Mass m n = √ λ n . Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 36 / 61

Pretty Pictures Random Quintic Massive Gravitons Consider KK modes of the graviton that are spin-2 in 4 dimensions: � h 10 d h 4 d n ,µν ( x 0 , x 1 , x 2 , x 3 ) · φ 6 d µν = n ( y 1 , . . . , y 6 ) n Mass m n = √ λ n . Gravitational potential between two test masses M 1 and M 2 : ∞ � M 1 M 2 e −√ λ n r V ( r ) = − G 4 r n = 0 Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 36 / 61

1 0 ( G 4 M 1 M 2 ) -1 -2 M 1 M 2 � V ( r ≫ 1 ) = − G 4 V ( r ) r -3 V ( r ≪ 1 ) = − 15 G 4 M 1 M 2 -4 8 π 3 r 7 -5 0 . 6 0 . 8 1 . 0 1 . 2 1 . 4 r

Pretty Pictures Fermat Quintic Introduction 1 Calabi-Yau Metrics 2 The Z 5 × Z 5 Quotient 3 The Laplace Operator 4 Pretty Pictures 5 Random Quintic Fermat Quintic Quintic Quotient Families Differential Forms Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 38 / 61

Fermat Quintic: N p = 500 , 000 200 λ 150 µ 5 = 30 100 µ 4 = 60 µ 3 = 4 µ 2 = 20 50 µ 1 = 20 0 µ 0 = 1 0 1 2 3 k φ

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

Pretty Pictures Fermat Quintic Symmetries of the Fermat Quintic The first massive eigenmode has degeneracy 20. The automorphisms group of the Fermat quintic Q F is �� � � � � � 4 Aut Q F = S 5 × Z 2 ⋉ Z 5 Lemma (Representation theory of Aut ( � Q F ) ) The 80 irreps of Aut ( � Q F ) 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

Pretty Pictures Fermat Quintic Symmetries of the Fermat Quintic The first massive eigenmode has degeneracy 20. The automorphisms group of the Fermat quintic Q F is �� � � � � � 4 Aut Q F = S 5 × Z 2 ⋉ Z 5 Lemma (Representation theory of Aut ( � Q F ) ) The 80 irreps of Aut ( � Q F ) 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

Pretty Pictures Fermat Quintic Donaldson’s Operator A (conjectural) alternative calculation of the spectrum of the scalar Laplacian. Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 41 / 61

Pretty Pictures Fermat Quintic Donaldson’s Operator A (conjectural) alternative calculation of the spectrum of the scalar Laplacian. Specific to the scalar Laplacian only. Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 41 / 61

Pretty Pictures Fermat Quintic Donaldson’s Operator A (conjectural) alternative calculation of the spectrum of the scalar Laplacian. Specific to the scalar Laplacian only. “Compares” balanced metrics at k and 2 k . � e ∆ ∼ Q α ¯ γδ = ( s α , s β )( s γ , s δ ) dVol β, ¯ � � � Recall: T ( h ) α ¯ β = ( s α , s β ) dVol Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 41 / 61

Fermat Quintic: Donaldson’s Operator 140 120 100 80 λ n 60 40 Donaldson’s operator, k h = 1 20 Donaldson’s operator, k h = 2 Donaldson’s operator, k h = 3 0 0 20 40 60 80 100 120 140 n

Fermat Quintic: scalar Laplacian 140 120 100 80 λ n 60 40 20 k φ = 3, k h = 3 (low precision metric) k φ = 3, k h = 8 (high precision metric) 0 0 20 40 60 80 100 120 140 n

Donaldson vs. scalar Laplacian 140 120 100 80 λ n 60 Donaldson’s operator, k h = 1 40 Donaldson’s operator, k h = 2 Donaldson’s operator, k h = 3 20 k φ = 3, k h = 3 (low precision metric) k φ = 3, k h = 8 (high precision metric) 0 0 20 40 60 80 100 120 140 n

Pretty Pictures Quintic Quotient Introduction 1 Calabi-Yau Metrics 2 The Z 5 × Z 5 Quotient 3 The Laplace Operator 4 Pretty Pictures 5 Random Quintic Fermat Quintic Quintic Quotient Families Differential Forms Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 45 / 61

Pretty Pictures � Quintic Quotient ( Z 5 × Z 5 ) Fermat Quintic �� � Q F = � Q F Z 5 × Z 5 Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 46 / 61

Pretty Pictures � Quintic Quotient ( Z 5 × Z 5 ) Fermat Quintic �� � Q F = � Q F Z 5 × Z 5 � � � f t Simply use only invariant functions . Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 46 / 61

Pretty Pictures � Quintic Quotient ( Z 5 × Z 5 ) Fermat Quintic �� � Q F = � Q F Z 5 × Z 5 � � � f t Simply use only invariant functions . Yields eigenvalues λ Z 5 × Z 5 . n Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 46 / 61

Pretty Pictures � Quintic Quotient ( Z 5 × Z 5 ) Fermat Quintic �� � Q F = � Q F Z 5 × Z 5 � � � f t Simply use only invariant functions . Yields eigenvalues λ Z 5 × Z 5 . n Rescale volume to one: � 1 25 = Vol Q F ) − → 1 λ Z 5 × Z 5 → 25 − 1 / 3 λ Z 5 × Z 5 − = λ n n n Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 46 / 61

� Fermat Quintic ( Z 5 × Z 5 ) : N p = 100 , 000 400 125 λ Z 5 × Z 5 300 100 λ n n 75 200 50 k φ = 0 k φ = 1 100 25 k φ = 2 0 0 0 20 40 60 80 100 n

Pretty Pictures Families Introduction 1 Calabi-Yau Metrics 2 The Z 5 × Z 5 Quotient 3 The Laplace Operator 4 Pretty Pictures 5 Random Quintic Fermat Quintic Quintic Quotient Families Differential Forms Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 48 / 61

� Pretty Pictures Families ( Z 5 × Z 5 ) Family #1 Quintic A family of Quintics � � � � � z 5 Q ψ = i − 5 ψ z i = 0 �� � Q ψ = � Q ψ Z 5 × Z 5 Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 49 / 61

� Pretty Pictures Families ( Z 5 × Z 5 ) Family #1 Quintic A family of Quintics � � � � � z 5 Q ψ = i − 5 ψ z i = 0 �� � Q ψ = � Q ψ Z 5 × Z 5 � Q 0 is the Fermat quintic. Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 49 / 61

� Pretty Pictures Families ( Z 5 × Z 5 ) Family #1 Quintic A family of Quintics � � � � � z 5 Q ψ = i − 5 ψ z i = 0 �� � Q ψ = � Q ψ Z 5 × Z 5 � Q 0 is the Fermat quintic. ψ = 1 is the conifold point. Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 49 / 61

� Pretty Pictures Families ( Z 5 × Z 5 ) Family #1 Quintic A family of Quintics � � � � � z 5 Q ψ = i − 5 ψ z i = 0 �� � Q ψ = � Q ψ Z 5 × Z 5 � Q 0 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

Pretty Pictures Families Conifold Point of the Quintic Conifold Point z 5 0 + z 5 1 + z 5 2 + z 5 3 + z 5 4 − 5 z 0 z 1 z 2 z 3 z 4 = 0 Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 50 / 61

Pretty Pictures Families Conifold Point of the Quintic Conifold Point z 5 0 + z 5 1 + z 5 2 + z 5 3 + z 5 4 − 5 z 0 z 1 z 2 z 3 z 4 = 0 is singular at z C = [ 1 : 1 : 1 : 1 : 1 ] : Q 1 ( z C ) = 0 = ∂ Q 1 ( z C ) ∂ z i Volker Braun (UPenn) Metrics and the Laplace Operator Universität Hamburg 50 / 61

� Quintic ( Z 5 × Z 5 ) Family #1 100 80 λ 60 40 � � ��� � 20 � z 5 Q ψ = i − 5 ψ z i = 0 Z 5 × Z 5 0 -1.5 -1 -0.5 0 0.5 1 1.5 ψ