gauge theory and the geometry of elliptic curves
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Gauge Theory and the Geometry of Elliptic Curves Andreas Kapfer Particle Physics School Munich Colloquium: April 2015 Andreas Kapfer Gauge Theory and Elliptic Curves PPSMC April 2015 1 / 17 Outline Outline mathematical theory connection

  1. Gauge Theory and the Geometry of Elliptic Curves Andreas Kapfer Particle Physics School Munich Colloquium: April 2015 Andreas Kapfer Gauge Theory and Elliptic Curves PPSMC April 2015 1 / 17

  2. Outline Outline mathematical theory connection via F-theory gauge theories of elliptic curves Andreas Kapfer Gauge Theory and Elliptic Curves PPSMC April 2015 2 / 17

  3. Outline Outline mathematical theory connection via F-theory gauge theories of elliptic curves 1 Simple Introduction to Elliptic Fibrations and F-theory 2 Recap of Anomalies in Quantum Field Theory 3 Anomaly Cancelation in F-theory ⇔ Symmetries of Elliptic Fibrations Andreas Kapfer Gauge Theory and Elliptic Curves PPSMC April 2015 2 / 17

  4. Geometry of Elliptic Fibrations Fibrations Fibrations Fibration (roughly): base space fiber space total space (fibration) Fibration “To each point in a base space a fixed fiber space is attached!” Andreas Kapfer Gauge Theory and Elliptic Curves PPSMC April 2015 3 / 17

  5. Geometry of Elliptic Fibrations Fibrations Fibrations Fibration (roughly): base space fiber space total space (fibration) Fibration “To each point in a base space a fixed fiber space is attached!” total space looks locally like “base x fiber” Andreas Kapfer Gauge Theory and Elliptic Curves PPSMC April 2015 3 / 17

  6. Geometry of Elliptic Fibrations Fibrations Example Trivial Fibration: Line x Line Building blocks: Base Fiber Fibration: Andreas Kapfer Gauge Theory and Elliptic Curves PPSMC April 2015 4 / 17

  7. Geometry of Elliptic Fibrations Fibrations Example Trivial Fibration: Line x Line Building blocks: Base Fiber Fibration: − → → trivial fibration because simple product space: line x line Andreas Kapfer Gauge Theory and Elliptic Curves PPSMC April 2015 4 / 17

  8. Geometry of Elliptic Fibrations Fibrations Non-Trivial Fibration: M¨ obius Strip Building blocks: Base Fiber Fibration: Andreas Kapfer Gauge Theory and Elliptic Curves PPSMC April 2015 5 / 17

  9. Geometry of Elliptic Fibrations Fibrations Non-Trivial Fibration: M¨ obius Strip Building blocks: Base Fiber Fibration: − → → non-trivial fibration because the fiber gets twisted when going around the circle → still locally trivial: line x line Andreas Kapfer Gauge Theory and Elliptic Curves PPSMC April 2015 5 / 17

  10. Geometry of Elliptic Fibrations Fibrations Elliptic Fibration Building blocks: Fiber Base arbitrary space Fibration: elliptic curve: torus with special points (K-rational points) → later more shape of the torus varies over the base Andreas Kapfer Gauge Theory and Elliptic Curves PPSMC April 2015 6 / 17

  11. Geometry of Elliptic Fibrations Sections Sections Section: smooth map: base → total space point in the base �→ point in the fiber over it Andreas Kapfer Gauge Theory and Elliptic Curves PPSMC April 2015 7 / 17

  12. Geometry of Elliptic Fibrations Sections Sections Section: smooth map: base → total space point in the base �→ point in the fiber over it Example: line x line ↑ Andreas Kapfer Gauge Theory and Elliptic Curves PPSMC April 2015 7 / 17

  13. Geometry of Elliptic Fibrations Sections Sections Section: smooth map: base → total space point in the base �→ point in the fiber over it Example: line x line ↑ ↑ → smooth embedding of the base space into the total space Andreas Kapfer Gauge Theory and Elliptic Curves PPSMC April 2015 7 / 17

  14. Geometry of Elliptic Fibrations Sections Example: a simple section of the M¨ obius strip �  Andreas Kapfer Gauge Theory and Elliptic Curves PPSMC April 2015 8 / 17

  15. Geometry of Elliptic Fibrations Sections Sections of elliptic fibrations: elliptic curve: torus with special points (K-rational points) elliptic fibration: fiber elliptic curve over some base space a section marks a single point in each fiber ⇒ the K-rational points of elliptic curves define (rational) sections of the elliptic fibration Andreas Kapfer Gauge Theory and Elliptic Curves PPSMC April 2015 9 / 17

  16. Geometry of Elliptic Fibrations Sections Sections of elliptic fibrations: elliptic curve: torus with special points (K-rational points) elliptic fibration: fiber elliptic curve over some base space a section marks a single point in each fiber ⇒ the K-rational points of elliptic curves define (rational) sections of the elliptic fibration F-theory rational sections ⇔ U (1) gauge symmetries Andreas Kapfer Gauge Theory and Elliptic Curves PPSMC April 2015 9 / 17

  17. Geometry of Elliptic Fibrations Rational Points Rational points on elliptic curves Describe spaces by polynomial equations: Sphere (in R 3 ) x 2 + y 2 + z 2 = 1 Elliptic Curve/Torus (in P 2 2 , 3 , 1 ) y 2 = x 3 + fxz 4 + gz 6 , with f , g ∈ C Andreas Kapfer Gauge Theory and Elliptic Curves PPSMC April 2015 10 / 17

  18. Geometry of Elliptic Fibrations Rational Points Rational points on elliptic curves Describe spaces by polynomial equations: Sphere (in R 3 ) x 2 + y 2 + z 2 = 1 Elliptic Curve/Torus (in P 2 2 , 3 , 1 ) y 2 = x 3 + fxz 4 + gz 6 , with f , g ∈ C Rational points: Solutions to these equations with ( x , y , z ) ∈ Q 3 ( → cf. Fermat’s last theorem: integer solutions to x n + y n = z n for n ≥ 3) Andreas Kapfer Gauge Theory and Elliptic Curves PPSMC April 2015 10 / 17

  19. Geometry of Elliptic Fibrations Rational Points Suppose you have found n linear independent rational solutions ( x , y , z ) ∈ Q 3 : rational solutions can be “added” in a tricky way to get further rational solutions → group structure for rational points ∼ = Z n − 1 Example: Group of rational points for n = 3 : Z 2 zero-point/origin generators of the group of rational points Andreas Kapfer Gauge Theory and Elliptic Curves PPSMC April 2015 11 / 17

  20. Geometry of Elliptic Fibrations Rational Points Suppose you have found n linear independent rational solutions ( x , y , z ) ∈ Q 3 : rational solutions can be “added” in a tricky way to get further rational solutions → group structure for rational points ∼ = Z n − 1 Example: Group of rational points for n = 3 : Z 2 zero-point/origin generators of the group of rational points Andreas Kapfer Gauge Theory and Elliptic Curves PPSMC April 2015 11 / 17

  21. Geometry of Elliptic Fibrations Rational Points Suppose you have found n linear independent rational solutions ( x , y , z ) ∈ Q 3 : rational solutions can be “added” in a tricky way to get further rational solutions → group structure for rational points ∼ = Z n − 1 Example: Group of rational points for n = 3 : Z 2 zero-point/origin generators of the group of rational points Andreas Kapfer Gauge Theory and Elliptic Curves PPSMC April 2015 11 / 17

  22. Geometry of Elliptic Fibrations Rational Points Suppose you have found n linear independent rational solutions ( x , y , z ) ∈ Q 3 : rational solutions can be “added” in a tricky way to get further rational solutions → group structure for rational points ∼ = Z n − 1 Example: Group of rational points for n = 3 : Z 2 zero-point/origin generators of the group of rational points Andreas Kapfer Gauge Theory and Elliptic Curves PPSMC April 2015 11 / 17

  23. Geometry of Elliptic Fibrations Rational Points Suppose you have found n linear independent rational solutions ( x , y , z ) ∈ Q 3 : rational solutions can be “added” in a tricky way to get further rational solutions → group structure for rational points ∼ = Z n − 1 Example: Group of rational points for n = 3 : Z 2 zero-point/origin generators of the group of rational points Andreas Kapfer Gauge Theory and Elliptic Curves PPSMC April 2015 11 / 17

  24. Geometry of Elliptic Fibrations Rational Points Suppose you have found n linear independent rational solutions ( x , y , z ) ∈ Q 3 : rational solutions can be “added” in a tricky way to get further rational solutions → group structure for rational points ∼ = Z n − 1 Example: Group of rational points for n = 3 : Z 2 zero-point/origin generators of the group of rational points Andreas Kapfer Gauge Theory and Elliptic Curves PPSMC April 2015 11 / 17

  25. Geometry of Elliptic Fibrations Rational Points Suppose you have found n linear independent rational solutions ( x , y , z ) ∈ Q 3 : rational solutions can be “added” in a tricky way to get further rational solutions → group structure for rational points ∼ = Z n − 1 Example: Group of rational points for n = 3 : Z 2 zero-point/origin generators of the group of rational points Andreas Kapfer Gauge Theory and Elliptic Curves PPSMC April 2015 11 / 17

  26. Geometry of Elliptic Fibrations Rational Points Suppose you have found n linear independent rational solutions ( x , y , z ) ∈ Q 3 : rational solutions can be “added” in a tricky way to get further rational solutions → group structure for rational points ∼ = Z n − 1 Example: Group of rational points for n = 3 : Z 2 zero-point/origin generators of the group of rational points Andreas Kapfer Gauge Theory and Elliptic Curves PPSMC April 2015 11 / 17

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