1 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 2 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 3 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 5 Skyrme Model I Non-linear (soliton) model of proton/neutron f : R3 ! SU(2) f (x) ! 1 as x ! 1 I Energy function E(f ) : Dynamics 5 Skyrme Model I Non-linear (soliton) model of proton/neutron f : R3 ! SU(2) f (x) ! 1 as x ! 1 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 ! 1 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 6 New Speculative Idea I 1. Static only (ignore time) 3. Interchange roles of electric/magnetic I ) 4-dimensional Riemannian geometry 6 New Speculative Idea I 1. Static only (ignore time) 2. Adopt Kaluza-Klein circular dimension (asymptotically) I ) 4-dimensional Riemannian geometry 6 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 7 First Objective I Precise models for I baryons (proton, neutron) I Improves on Skyrme model by incorporating electric charge and leptons 7 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 8 Conformal Geometry L I Riemann = Weyl Ricci I Weyl: conformally invariant 8 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) 9 Dimension 4 + L − I W = W W self dual, anti-self-dual (depends on orientation) I Models of matter (anti-self-dual : anti-matter) 9 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) 10 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) 10 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) 10 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) 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 4. Z encodes conformal structure of M (and Einstein equations) 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 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 11 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 11 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 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) 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) 12 Analogy with Riemann Surfaces 2. H2(X ) plays role of H1 of Riemann Surfaces 3. Connected sums (but obstructions) 12 Analogy with Riemann Surfaces I 1. Complex moduli 3. Connected sums (but obstructions) 12 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 13 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 14 Compactness I Models of electrically charged particles (proton, electron) will be non-compact but complete 14 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) 15 Symmetries I self-dual, Einstein I Zero scalar curvature (non-compact) I Positive scalar curvature (compact) 15 Symmetries I All 4 basic particles will have SO(3)-symmetry I Zero scalar curvature (non-compact) I Positive scalar curvature (compact) 15 Symmetries I All 4 basic particles will have SO(3)-symmetry I self-dual, Einstein I Positive scalar curvature (compact) 15 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). 16 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). 16 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 17 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 17 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 17 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 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 17 Non-Compact Models (Electron) I Taub-NUT manifold, mass