geometric models of matter - university of leeds2.adopt kaluza-klein circular dimension...
TRANSCRIPT
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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
20
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