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Geometric Models of Matter

Michael Atiyah (Edinburgh)

Joint work with J. Figueroa-O’Farrill (Edinburgh), N. S. Manton (Cambridge) and B. J. Schroers (Heriot-Watt)

University of Leeds 8 July, 2011 I Weyl, Kaluza-Klein

Electro-Magnetism = Curvature of 5th (circular) dimension

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I Einstein

Gravity = Curvature of Space-Time 2

I Einstein

Gravity = Curvature of Space-Time

I Weyl, Kaluza-Klein

Electro-Magnetism = Curvature of 5th (circular) dimension I Heisenberg

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Quantum Mechanics

I Bohr 3

Quantum Mechanics

I Bohr

I Heisenberg 4

Non-abelian Gauge Theories of Matter

I Yang-Mills Generalization of Maxwell’s Equations with U(1) replaced by non-abelian groups SU(2), SU(3), ... I degree f = baryon number

I Energy function E(f ) : Dynamics

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Skyrme Model

I Non-linear (soliton) model of proton/neutron

f : R3 → SU(2) f (x) → 1 as x → ∞ I Energy function E(f ) : Dynamics

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Skyrme Model

I Non-linear (soliton) model of proton/neutron

f : R3 → SU(2) f (x) → 1 as x → ∞

I degree f = baryon number 5

Skyrme Model

I Non-linear (soliton) model of proton/neutron

f : R3 → SU(2) f (x) → 1 as x → ∞

I degree f = baryon number

I Energy function E(f ) : Dynamics 2. Adopt Kaluza-Klein circular dimension (asymptotically) 3. Interchange roles of electric/magnetic

I ⇒ 4-dimensional Riemannian geometry

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New Speculative Idea

I 1. Static only (ignore time) 3. Interchange roles of electric/magnetic

I ⇒ 4-dimensional Riemannian geometry

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New Speculative Idea

I 1. Static only (ignore time) 2. Adopt Kaluza-Klein circular dimension (asymptotically) I ⇒ 4-dimensional Riemannian geometry

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New Speculative Idea

I 1. Static only (ignore time) 2. Adopt Kaluza-Klein circular dimension (asymptotically) 3. Interchange roles of electric/magnetic 6

New Speculative Idea

I 1. Static only (ignore time) 2. Adopt Kaluza-Klein circular dimension (asymptotically) 3. Interchange roles of electric/magnetic

I ⇒ 4-dimensional Riemannian geometry I leptons (electrons, neutrino)

I Improves on Skyrme model by incorporating electric charge and leptons

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First Objective

I Precise models for

I baryons (proton, neutron) I Improves on Skyrme model by incorporating electric charge and leptons

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First Objective

I Precise models for

I baryons (proton, neutron) I leptons (electrons, neutrino) 7

First Objective

I Precise models for

I baryons (proton, neutron) I leptons (electrons, neutrino)

I Improves on Skyrme model by incorporating electric charge and leptons I Einstein: Ricci = constant scalar

I Weyl: conformally invariant

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Conformal Geometry

L I Riemann = Weyl Ricci I Weyl: conformally invariant

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Conformal Geometry

L I Riemann = Weyl Ricci

I Einstein: Ricci = constant scalar 8

Conformal Geometry

L I Riemann = Weyl Ricci

I Einstein: Ricci = constant scalar

I Weyl: conformally invariant − I Manifold self-dual if W = 0

I Models of matter (anti-self-dual : anti-matter)

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Dimension 4

+ L − I W = W W self dual, anti-self-dual (depends on orientation) I Models of matter (anti-self-dual : anti-matter)

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Dimension 4

+ L − I W = W W self dual, anti-self-dual (depends on orientation) − I Manifold self-dual if W = 0 9

Dimension 4

+ L − I W = W W self dual, anti-self-dual (depends on orientation) − I Manifold self-dual if W = 0

I Models of matter (anti-self-dual : anti-matter) I M self-dual 4-manifold has twistor space Z 1. Z complex analytic 3-dimensional 2. Z real fibration over M with S 2 as fibre 3. Z has anti-linear involution, anti-podal map on each fibre 4. Z encodes conformal structure of M (and Einstein equations)

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Penrose Twistor Theory 1. Z complex analytic 3-dimensional 2. Z real fibration over M with S 2 as fibre 3. Z has anti-linear involution, anti-podal map on each fibre 4. Z encodes conformal structure of M (and Einstein equations)

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Penrose Twistor Theory

I M self-dual 4-manifold has twistor space Z 2. Z real fibration over M with S 2 as fibre 3. Z has anti-linear involution, anti-podal map on each fibre 4. Z encodes conformal structure of M (and Einstein equations)

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Penrose Twistor Theory

I M self-dual 4-manifold has twistor space Z 1. Z complex analytic 3-dimensional 3. Z has anti-linear involution, anti-podal map on each fibre 4. Z encodes conformal structure of M (and Einstein equations)

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Penrose Twistor Theory

I M self-dual 4-manifold has twistor space Z 1. Z complex analytic 3-dimensional 2. Z real fibration over M with S 2 as fibre 4. Z encodes conformal structure of M (and Einstein equations)

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Penrose Twistor Theory

I M self-dual 4-manifold has twistor space Z 1. Z complex analytic 3-dimensional 2. Z real fibration over M with S 2 as fibre 3. Z has anti-linear involution, anti-podal map on each fibre 10

Penrose Twistor Theory

I M self-dual 4-manifold has twistor space Z 1. Z complex analytic 3-dimensional 2. Z real fibration over M with S 2 as fibre 3. Z has anti-linear involution, anti-podal map on each fibre 4. Z encodes conformal structure of M (and Einstein equations) 2 4 1 I Use quaterions H = C = R X = HP 4 + − I Note: S conformally flat W = W = 0 (2 twistor spaces)

I Converts Riemannian Geometry into Holomorphic Geometry

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Basic Example

4 3 I X = S , Z = CP 4 + − I Note: S conformally flat W = W = 0 (2 twistor spaces)

I Converts Riemannian Geometry into Holomorphic Geometry

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Basic Example

4 3 I X = S , Z = CP 2 4 1 I Use quaterions H = C = R X = HP I Converts Riemannian Geometry into Holomorphic Geometry

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Basic Example

4 3 I X = S , Z = CP 2 4 1 I Use quaterions H = C = R X = HP 4 + − I Note: S conformally flat W = W = 0 (2 twistor spaces) 11

Basic Example

4 3 I X = S , Z = CP 2 4 1 I Use quaterions H = C = R X = HP 4 + − I Note: S conformally flat W = W = 0 (2 twistor spaces)

I Converts Riemannian Geometry into Holomorphic Geometry I 1. Complex moduli 2. H2(X ) plays role of H1 of Riemann Surfaces 3. Connected sums (but obstructions)

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Analogy with Riemann Surfaces 2. H2(X ) plays role of H1 of Riemann Surfaces 3. Connected sums (but obstructions)

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Analogy with Riemann Surfaces

I 1. Complex moduli 3. Connected sums (but obstructions)

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Analogy with Riemann Surfaces

I 1. Complex moduli 2. H2(X ) plays role of H1 of Riemann Surfaces 12

Analogy with Riemann Surfaces

I 1. Complex moduli 2. H2(X ) plays role of H1 of Riemann Surfaces 3. Connected sums (but obstructions) I WARNING: including anti-matter will present difficulties

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Long-Term Aim

I Use twistor-spaces to model interactions of matter 13

Long-Term Aim

I Use twistor-spaces to model interactions of matter

I WARNING: including anti-matter will present difficulties I Models of electrically neutral particles (neutrons, neutrino) will be compact

I Models of electrically charged particles (proton, electron) will be non-compact but complete

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Compactness I Models of electrically charged particles (proton, electron) will be non-compact but complete

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Compactness

I Models of electrically neutral particles (neutrons, neutrino) will be compact 14

Compactness

I Models of electrically neutral particles (neutrons, neutrino) will be compact

I Models of electrically charged particles (proton, electron) will be non-compact but complete I All 4 basic particles will have SO(3)-symmetry

I self-dual, Einstein

I Zero scalar curvature (non-compact)

I Positive scalar curvature (compact)

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Symmetries I self-dual, Einstein

I Zero scalar curvature (non-compact)

I Positive scalar curvature (compact)

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Symmetries

I All 4 basic particles will have SO(3)-symmetry I Zero scalar curvature (non-compact)

I Positive scalar curvature (compact)

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Symmetries

I All 4 basic particles will have SO(3)-symmetry

I self-dual, Einstein I Positive scalar curvature (compact)

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Symmetries

I All 4 basic particles will have SO(3)-symmetry

I self-dual, Einstein

I Zero scalar curvature (non-compact) 15

Symmetries

I All 4 basic particles will have SO(3)-symmetry

I self-dual, Einstein

I Zero scalar curvature (non-compact)

I Positive scalar curvature (compact) Neutron complex projective plane CP2 Neutrino 4-sphere S4 with standard metrics (symmetries SU(3), SO(5)).

STOP PRESS May want to replace CP2 by one of the Hitchin “manifolds” H(N) which are self-dual Einstein orbifolds (on space CP2) with conical angle 4π/(N + 2) along RP2. (Note: N = 2 gives CP2).

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Compact Models STOP PRESS May want to replace CP2 by one of the Hitchin “manifolds” H(N) which are self-dual Einstein orbifolds (on space CP2) with conical angle 4π/(N + 2) along RP2. (Note: N = 2 gives CP2).

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Compact Models

Neutron complex projective plane CP2 Neutrino 4-sphere S4 with standard metrics (symmetries SU(3), SO(5)). 16

Compact Models

Neutron complex projective plane CP2 Neutrino 4-sphere S4 with standard metrics (symmetries SU(3), SO(5)).

STOP PRESS May want to replace CP2 by one of the Hitchin “manifolds” H(N) which are self-dual Einstein orbifolds (on space CP2) with conical angle 4π/(N + 2) along RP2. (Note: N = 2 gives CP2). I Taub-NUT manifold, mass parameter m > 0

I ”dual of Dirac monopole” 2 I Topology of C , U(2) symmetry 3 2 1 I Asymptotically fibration S → S fibre S

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Non-Compact Models (Electron) I ”dual of Dirac monopole” 2 I Topology of C , U(2) symmetry 3 2 1 I Asymptotically fibration S → S fibre S

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Non-Compact Models (Electron)

I Taub-NUT manifold, mass parameter m > 0 2 I Topology of C , U(2) symmetry 3 2 1 I Asymptotically fibration S → S fibre S

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Non-Compact Models (Electron)

I Taub-NUT manifold, mass parameter m > 0

I ”dual of Dirac monopole” 3 2 1 I Asymptotically fibration S → S fibre S

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Non-Compact Models (Electron)

I Taub-NUT manifold, mass parameter m > 0

I ”dual of Dirac monopole” 2 I Topology of C , U(2) symmetry 17

Non-Compact Models (Electron)

I Taub-NUT manifold, mass parameter m > 0

I ”dual of Dirac monopole” 2 I Topology of C , U(2) symmetry 3 2 1 I Asymptotically fibration S → S fibre S I Atiyah-Hitchin manifold AH (moduli space of centred SU(2)-monopoles of charge 2) 2 2 I Topology of CP − RP , SO(3)-symmetry 2 I Asymptotically unoriented circle-bundle over RP

I asymptotic metric Taub-NUT with m < 0 Note: Hitchin manifolds H(N) → AH as N → ∞

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Non-compact Models (Proton) 2 2 I Topology of CP − RP , SO(3)-symmetry 2 I Asymptotically unoriented circle-bundle over RP

I asymptotic metric Taub-NUT with m < 0 Note: Hitchin manifolds H(N) → AH as N → ∞

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Non-compact Models (Proton)

I Atiyah-Hitchin manifold AH (moduli space of centred SU(2)-monopoles of charge 2) 2 I Asymptotically unoriented circle-bundle over RP

I asymptotic metric Taub-NUT with m < 0 Note: Hitchin manifolds H(N) → AH as N → ∞

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Non-compact Models (Proton)

I Atiyah-Hitchin manifold AH (moduli space of centred SU(2)-monopoles of charge 2) 2 2 I Topology of CP − RP , SO(3)-symmetry I asymptotic metric Taub-NUT with m < 0 Note: Hitchin manifolds H(N) → AH as N → ∞

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Non-compact Models (Proton)

I Atiyah-Hitchin manifold AH (moduli space of centred SU(2)-monopoles of charge 2) 2 2 I Topology of CP − RP , SO(3)-symmetry 2 I Asymptotically unoriented circle-bundle over RP 18

Non-compact Models (Proton)

I Atiyah-Hitchin manifold AH (moduli space of centred SU(2)-monopoles of charge 2) 2 2 I Topology of CP − RP , SO(3)-symmetry 2 I Asymptotically unoriented circle-bundle over RP

I asymptotic metric Taub-NUT with m < 0 Note: Hitchin manifolds H(N) → AH as N → ∞ 2 I ”Electron” compactifies to CP 1 I CP (at ∞) has self-intersection number +1 2 I ”Proton” compactifies to CP 2 I RP (at ∞) has self-intersection number -1 (opposite signs for electron/proton since mass parameter has different sign)

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Self-intersection numbers 1 I CP (at ∞) has self-intersection number +1 2 I ”Proton” compactifies to CP 2 I RP (at ∞) has self-intersection number -1 (opposite signs for electron/proton since mass parameter has different sign)

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Self-intersection numbers

2 I ”Electron” compactifies to CP 2 I ”Proton” compactifies to CP 2 I RP (at ∞) has self-intersection number -1 (opposite signs for electron/proton since mass parameter has different sign)

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Self-intersection numbers

2 I ”Electron” compactifies to CP 1 I CP (at ∞) has self-intersection number +1 2 I RP (at ∞) has self-intersection number -1 (opposite signs for electron/proton since mass parameter has different sign)

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Self-intersection numbers

2 I ”Electron” compactifies to CP 1 I CP (at ∞) has self-intersection number +1 2 I ”Proton” compactifies to CP (opposite signs for electron/proton since mass parameter has different sign)

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Self-intersection numbers

2 I ”Electron” compactifies to CP 1 I CP (at ∞) has self-intersection number +1 2 I ”Proton” compactifies to CP 2 I RP (at ∞) has self-intersection number -1 19

Self-intersection numbers

2 I ”Electron” compactifies to CP 1 I CP (at ∞) has self-intersection number +1 2 I ”Proton” compactifies to CP 2 I RP (at ∞) has self-intersection number -1 (opposite signs for electron/proton since mass parameter has different sign) 1. Study baryon number > 1 2. Study moduli → evolution (dynamics) 3. Study spectral properties of the Dirac operator

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Remaining Problems 2. Study moduli → evolution (dynamics) 3. Study spectral properties of the Dirac operator

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Remaining Problems

1. Study baryon number > 1 3. Study spectral properties of the Dirac operator

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Remaining Problems

1. Study baryon number > 1 2. Study moduli → evolution (dynamics) 20

Remaining Problems

1. Study baryon number > 1 2. Study moduli → evolution (dynamics) 3. Study spectral properties of the Dirac operator