1. Cumrun Vafa June 6, 2011 University of Pennsylvania Strings and Geometry

2. Geometry and Physics have a long joint history, dating at least all the way back to the Greek philosophers and geometers.

3. Modern examples of this deep link include Einstein’s geometrization of gravity and more recently the unification of forces and gravity in the context of string theory. My aim in this talk is to review some deep links we have witnessed between geometry and physics in the context of string theory.

4. Newton

5. Apple: Where it all starts!

6. Classical Mechanics

10. FLAT SPACE

11. Add Time + Space Not Flat!

12. Geometry of Space and Time is Curved

14. Particles move on geodesics

15. curvature + geodesic  looks like force

16. Matter  Curvature

17. Even light bends as it moves on geodesics

18. The planetary orbits can also be explained geometrically: Sun curves spacetime and planets follow geodesics

19. Theory is relevant also for cosmological questions

20. Particles are ``fuzzy’’ at smaller scales

21. Quantum mechanics was born

22. Bohr

23. This was so radical, even Einstein was intrigued!

24. Principle of quantum mechanics seems valid even at nuclear and subnuclear scales

25. These ideas were checked by scattering experiments of high energy particles

27. Feynman

28. Interaction takes place by exchange of particles

31. Gravity+Quantum Mechanics  ?

32. Particles+quantum mechanics+gravity Difficult to make sense of !

33. Strings come to the rescue!

34. At yet smaller scales ``elementary particles’’ look like strings

36. String Interaction

39. Joining of strings

40. Joining and splitting of strings

42. String interactions are described by the beautiful geometry of surfaces

43. Everything seems to be in place with strings at a very tiny (at present unobservable) scale

44. Smooth geometry of strings seems to explain all known interactions (at least in principle)

45. String’s connection between geometry and physics seems to start on the wrong foot: String theory demands that spacetime not be four dimensional! d=10 (or 11 in M-theory) This seems to raise a dilemma: How can we hope to use string theory to answer questions relevant to 4d physics?!

46. First attempt at the answer: We can demand that the extra dimensions be curled up into a tiny compact space, thus rendering it unobservable and avoiding a direct clash with reality! Not totally satisfactory: What are the extra dimensions good for?!

47. But this answer is not totally satisfactory: What are the extra dimensions good for?!

50. Charged matter U(3) U(1) U(2)

51. SO(10)

52. 1/g 2

53. Naïve continuation does not make geometric sense

54. Instead a new geometry opens up!

55. Smooth Transition

56. On the other hand there are a number of ons for 4d physics which require atter answer: Black Hole Entropy: Where are the microstates of the black hole hidden? Bekenstein-Hawking formula: S=A/4 A= area of the horizon

57. Black hole as wrapped branes in the internal dimensions

58. Wheeler: Quantum mechanics  wild fluctuations of space at Planck Scale -33 ( 10 cm)

59. Not only the space is fluctuating in size at Planck scale but even its topology should fluctuate. Very difficult to describe.

61. Space and time should look like A ``quantum foam’’ at Planck scale

62. The geometry of the corners is exactly the same as for strings moving in a complex 3D space!

63. Macroscopic crystal geometry  Stringy geometry Atomic structure of crystal  ?

64. A simple model for melting crystal as removing atoms starting from the atoms near the corners

65. Typical configuration of molten crystal

66. We are witnessing an ongoing revolution linking Geometry and Physics

67. We are witnessing an ongoing revolution linking Geometry and Physics We have learned a lot, but a lot remains to be learned. It is a `work in progress’!