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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 Leeds8 July, 2011

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I Einstein

Gravity = Curvature of Space-Time

I Weyl, Kaluza-Klein

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

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I Einstein

Gravity = Curvature of Space-Time

I Weyl, Kaluza-Klein

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

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Quantum Mechanics

I Bohr

I Heisenberg

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Quantum Mechanics

I Bohr

I Heisenberg

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Non-abelian Gauge Theories of Matter

I Yang-MillsGeneralization of Maxwell’s Equations with U(1) replaced bynon-abelian groups SU(2), SU(3), ...

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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

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

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

I Energy function E (f ) : Dynamics

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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

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

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

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

I ⇒ 4-dimensional Riemannian geometry

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First Objective

I Precise models forI baryons (proton, neutron)

I leptons (electrons, neutrino)

I Improves on Skyrme model by incorporating electric chargeand leptons

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First Objective

I Precise models forI baryons (proton, neutron)I leptons (electrons, neutrino)

I Improves on Skyrme model by incorporating electric chargeand leptons

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First Objective

I Precise models forI baryons (proton, neutron)I leptons (electrons, neutrino)

I Improves on Skyrme model by incorporating electric chargeand leptons

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Conformal Geometry

I Riemann = Weyl⊕

Ricci

I Einstein: Ricci = constant scalar

I Weyl: conformally invariant

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Conformal Geometry

I Riemann = Weyl⊕

Ricci

I Einstein: Ricci = constant scalar

I Weyl: conformally invariant

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Conformal Geometry

I Riemann = Weyl⊕

Ricci

I Einstein: Ricci = constant scalar

I Weyl: conformally invariant

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Dimension 4

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)

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Dimension 4

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)

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Dimension 4

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)

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Penrose Twistor Theory

I M self-dual 4-manifold has twistor space Z

1. Z complex analytic 3-dimensional2. Z real fibration over M with S2 as fibre3. Z has anti-linear involution, anti-podal map on each fibre4. 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-dimensional2. Z real fibration over M with S2 as fibre3. Z has anti-linear involution, anti-podal map on each fibre4. 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 S2 as fibre3. Z has anti-linear involution, anti-podal map on each fibre4. 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-dimensional2. Z real fibration over M with S2 as fibre

3. Z has anti-linear involution, anti-podal map on each fibre4. 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-dimensional2. Z real fibration over M with S2 as fibre3. 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-dimensional2. Z real fibration over M with S2 as fibre3. Z has anti-linear involution, anti-podal map on each fibre4. Z encodes conformal structure of M (and Einstein equations)

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Basic Example

I X = S4,Z = CP3

I Use quaterions H = C 2 = R4 X = HP1

I Note: S4 conformally flat W+ = W− = 0 (2 twistor spaces)

I Converts Riemannian Geometry into Holomorphic Geometry

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Basic Example

I X = S4,Z = CP3

I Use quaterions H = C 2 = R4 X = HP1

I Note: S4 conformally flat W+ = W− = 0 (2 twistor spaces)

I Converts Riemannian Geometry into Holomorphic Geometry

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Basic Example

I X = S4,Z = CP3

I Use quaterions H = C 2 = R4 X = HP1

I Note: S4 conformally flat W+ = W− = 0 (2 twistor spaces)

I Converts Riemannian Geometry into Holomorphic Geometry

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Basic Example

I X = S4,Z = CP3

I Use quaterions H = C 2 = R4 X = HP1

I Note: S4 conformally flat W+ = W− = 0 (2 twistor spaces)

I Converts Riemannian Geometry into Holomorphic Geometry

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Analogy with Riemann Surfaces

I 1. Complex moduli2. H2(X ) plays role of H1 of Riemann Surfaces3. 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 Surfaces3. Connected sums (but obstructions)

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Analogy with Riemann Surfaces

I 1. Complex moduli2. 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 moduli2. H2(X ) plays role of H1 of Riemann Surfaces3. Connected sums (but obstructions)

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Long-Term Aim

I Use twistor-spaces to model interactions of matter

I WARNING: including anti-matter will present difficulties

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Long-Term Aim

I Use twistor-spaces to model interactions of matter

I WARNING: including anti-matter will present difficulties

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Compactness

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

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

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Compactness

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

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

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Compactness

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

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

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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)

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)

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)

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)

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)

I Positive scalar curvature (compact)

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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 spaceCP2) 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)).

STOP PRESS May want to replace CP2 by one of the Hitchin“manifolds” H(N) which are self-dual Einstein orbifolds (on spaceCP2) 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)).

STOP PRESS May want to replace CP2 by one of the Hitchin“manifolds” H(N) which are self-dual Einstein orbifolds (on spaceCP2) with conical angle 4π/(N + 2) along RP2.(Note: N = 2 gives CP2).

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Non-Compact Models (Electron)

I Taub-NUT manifold, mass parameter m > 0

I ”dual of Dirac monopole”

I Topology of C 2, U(2) symmetry

I Asymptotically fibration S3 → S2 fibre S1

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Non-Compact Models (Electron)

I Taub-NUT manifold, mass parameter m > 0

I ”dual of Dirac monopole”

I Topology of C 2, U(2) symmetry

I Asymptotically fibration S3 → S2 fibre S1

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Non-Compact Models (Electron)

I Taub-NUT manifold, mass parameter m > 0

I ”dual of Dirac monopole”

I Topology of C 2, U(2) symmetry

I Asymptotically fibration S3 → S2 fibre S1

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Non-Compact Models (Electron)

I Taub-NUT manifold, mass parameter m > 0

I ”dual of Dirac monopole”

I Topology of C 2, U(2) symmetry

I Asymptotically fibration S3 → S2 fibre S1

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Non-Compact Models (Electron)

I Taub-NUT manifold, mass parameter m > 0

I ”dual of Dirac monopole”

I Topology of C 2, U(2) symmetry

I Asymptotically fibration S3 → S2 fibre S1

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Non-compact Models (Proton)

I Atiyah-Hitchin manifold AH(moduli space of centred SU(2)-monopoles of charge 2)

I Topology of CP2 − RP2,SO(3)-symmetry

I Asymptotically unoriented circle-bundle over RP2

I asymptotic metric Taub-NUT with m < 0Note: 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)

I Topology of CP2 − RP2,SO(3)-symmetry

I Asymptotically unoriented circle-bundle over RP2

I asymptotic metric Taub-NUT with m < 0Note: 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)

I Topology of CP2 − RP2, SO(3)-symmetry

I Asymptotically unoriented circle-bundle over RP2

I asymptotic metric Taub-NUT with m < 0Note: 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)

I Topology of CP2 − RP2, SO(3)-symmetry

I Asymptotically unoriented circle-bundle over RP2

I asymptotic metric Taub-NUT with m < 0Note: 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)

I Topology of CP2 − RP2, SO(3)-symmetry

I Asymptotically unoriented circle-bundle over RP2

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

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Self-intersection numbers

I ”Electron” compactifies to CP2

I CP1 (at∞) has self-intersection number +1

I ”Proton” compactifies to CP2

I RP2 (at∞) has self-intersection number -1(opposite signs for electron/proton since mass parameter hasdifferent sign)

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Self-intersection numbers

I ”Electron” compactifies to CP2

I CP1 (at∞) has self-intersection number +1

I ”Proton” compactifies to CP2

I RP2 (at∞) has self-intersection number -1(opposite signs for electron/proton since mass parameter hasdifferent sign)

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Self-intersection numbers

I ”Electron” compactifies to CP2

I CP1 (at∞) has self-intersection number +1

I ”Proton” compactifies to CP2

I RP2 (at∞) has self-intersection number -1(opposite signs for electron/proton since mass parameter hasdifferent sign)

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Self-intersection numbers

I ”Electron” compactifies to CP2

I CP1 (at∞) has self-intersection number +1

I ”Proton” compactifies to CP2

I RP2 (at∞) has self-intersection number -1(opposite signs for electron/proton since mass parameter hasdifferent sign)

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Self-intersection numbers

I ”Electron” compactifies to CP2

I CP1 (at∞) has self-intersection number +1

I ”Proton” compactifies to CP2

I RP2 (at∞) has self-intersection number -1

(opposite signs for electron/proton since mass parameter hasdifferent sign)

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Self-intersection numbers

I ”Electron” compactifies to CP2

I CP1 (at∞) has self-intersection number +1

I ”Proton” compactifies to CP2

I RP2 (at∞) has self-intersection number -1(opposite signs for electron/proton since mass parameter hasdifferent sign)

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Remaining Problems

1. Study baryon number > 1

2. Study moduli → evolution (dynamics)

3. Study spectral properties of the Dirac operator

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Remaining Problems

1. Study baryon number > 1

2. Study moduli → evolution (dynamics)

3. Study spectral properties of the Dirac operator

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Remaining Problems

1. Study baryon number > 1

2. Study moduli → evolution (dynamics)

3. Study spectral properties of the Dirac operator

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Remaining Problems

1. Study baryon number > 1

2. Study moduli → evolution (dynamics)

3. Study spectral properties of the Dirac operator