Cumrun Vafa

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Strings and Geometry
Cumrun Vafa June 6, 2011 University
of Pennsylvania
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Geometry and Physics have a long joint history,
dating at least all the way back to the
Greek philosophers and geometers.
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Modern examples of this deep link include
Einsteins 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.
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Newton
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Apple Where it all starts!
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Classical Mechanics
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FLAT SPACE
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Add Time Space Not Flat!
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Geometry of Space and Time is Curved
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Particles move on geodesics
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curvature geodesic ? looks like force
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Matter ? Curvature
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Even light bends as it moves on geodesics
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The planetary orbits can also be explained
geometrically Sun curves spacetime and planets
follow geodesics
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Theory is relevant also for cosmological
questions
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Particles are fuzzy at smaller scales
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Quantum mechanics was born
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Bohr
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This was so radical, even Einstein was intrigued!
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Principle of quantum mechanics seems valid even
at nuclear and subnuclear scales
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These ideas were checked by scattering
experiments of high energy particles
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Feynman
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Interaction takes place by exchange of particles
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GravityQuantum Mechanics??
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Particlesquantum mechanicsgravity
Difficult to make sense of !
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Strings come to the rescue!
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At yet smaller scales elementary particles
look like strings
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String Interaction
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Joining of strings
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Joining and splitting of strings
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String interactions are described by the
beautiful geometry of surfaces
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Everything seems to be in place with strings at
a very tiny (at present unobservable) scale
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Smooth geometry of strings seems to explain all
known interactions (at least in principle)
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Strings connection between geometry and physics
seems to start on the wrong foot String theory
demands that spacetime not be four dimensional!
d10 (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?!
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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?!
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• But this answer is not totally satisfactory
• What are the extra dimensions
• good for?!

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U(3)
U(2)
U(1)
Charged matter
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SO(10)
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2
1/g
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Naïve continuation does not make
geometric sense
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Instead a new geometry opens up!
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Smooth Transition
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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 SA/4 A area of the horizon
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Black hole as wrapped branes in the internal
dimensions
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Wheeler Quantum mechanics? wild fluctuations of
space at Planck Scale
-33 (10 cm)
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Not only the space is fluctuating in size at
Planck scale but even its topology
should fluctuate. Very difficult to describe.
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Space and time should look like A quantum
foam at Planck scale
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The geometry of the corners is exactly the same
as for strings moving in a complex 3D space!
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Macroscopic crystal geometry ?? Stringy
geometry Atomic structure of crystal ??
?
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A simple model for melting crystal as removing
atoms starting from the atoms near the corners
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Typical configuration of molten
crystal
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We are witnessing an ongoing revolution
linking Geometry and Physics
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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!
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