1. BERLIN ICRANet Pescara Gravity and Defects Gravity and Defects Hagen Kleinert, FU BERLIN & ICRANet Pescara

2. Imagine: Single-Valued World Multivalued World

3. Warm-up: Ginzburg-Landau Theory FALSE! Chain Rule: set But this is Ising –like!

5. Jumps! In 1D, can be removed by going to covering group U(1) In >1D impossible Correct Chain Rule:

6. Vortex Gauge Field Axial Gauge Invariant Field Strength:

7. Simplest MULTIVALUED FIELD in 2D Solve:

8. NOTE: Mother of Two Important Green Functions

9. Correct Treatment: Ginzburg-Landau Theory Chain Rule: set XY–like!

10. Application of Multivalued Fields to Magnetostatics Application of Multivalued Fields to Magnetostatics Recall Thomson and Maxwell: W. Thomson (1824-1907) J.C.Maxwell (1831-1879)

11. Generate Magnetic Field by Multivalued Gauge Transformations Thin Flux Tube:

12. Also: Magnetic Monopoles Also: Magnetic Monopoles Paul Dirac (1902-1984)

13. Derive: Minimal Coupling From Non-holonomic Gauge Transformations Then action changes by surface terms only: For nonholonomic Nontrivial J.L.Lagrange (1736-1819)

14. Schrödinger Theory Momentum Use nonholonomic then Solved by with nonzero magnetic field E.Schrödinger (1887-1961)

15. Multivalued Description of Magnetism Magnetic Field

16. Action Gauge Invariance

17. Defect Current Conserv.: Integration by parts Integration of Omega Enforced as Bianchi Identity: Double Gauge Theory:

18. Recall: Action arises also Recall: Action arises also from GL Theory of superfluid He from GL Theory of superfluid He In London (hydrodynamic) Limit Thus same formalism holds for superfluid He!

19. GC Sum Over Lines can be transformed into Disorder QFT Result: Ginzburg-Landau Theory of Superfluid Helium

20. Double-Gauge QFT of Monopoles

22. Now : Nontrivial Geometry from Nonholonomic Coordinate Transformations Burgers vector b

23. Frank Vector  DISCLINATIONS

26. FUNDAMENTALS: Universality of FREE PARTICLE motion:

27. Nonholonomic image of is Autoparallel Instead of Geodesic E.Schrödinger (1887-1961) A.Einstein (1879-1955)

28. QUANTUM THEORY: Trajectory is fat fluctuation sausage!  Tidal forces on wave packet ?

29. Lattice Defect Theory vs Abelian QED on Lattice Lattice formulation Define

30. CURIOSITY: Induced Gravity in `World Crystal´ Elastic Gauge Tfs: Canonical Form Momentum Conservation Enforced as Bianchi Idty: Double Gauge Theory

31. Dual Representation

32. BUT NEED

33. Modify Elastic Action to and further to FLOPPY CRYSTAL

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