Cumrun Vafa

June 6, 2011University of Pennsylvania

Strings and Geometry

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

Modern examples of this deep link include Einstein’s geometrization of gravity and more recently the unificationof 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.

Newton

Apple: Where it all starts!

Classical Mechanics

FLAT SPACE

Add Time + Space Not Flat!

Geometry of Space and Time is Curved

Particles move on geodesics

curvature + geodesic looks like force

Matter Curvature

Even light bends as it moves on geodesics

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

Theory is relevant also for cosmological questions

Particles are ``fuzzy’’ at smaller scales

Quantum mechanics was born

Bohr

This was so radical, even Einstein was intrigued!

Principle of quantum mechanics seems valideven at nuclear and subnuclear scales

These ideas were checked by scattering experiments of high energy particles

Feynman

Interaction takes place by exchange of particles

Gravity+Quantum Mechanics?

Particles+quantum mechanics+gravity

Difficult to make sense of !

Strings come to the rescue!

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

String Interaction

Joining of strings

Joining and splitting of strings

String interactions are described by the beautiful geometry of surfaces

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

Smooth geometry of stringsseems to explain all known interactions (at least in principle)

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 toanswer questions relevant to 4d physics?!

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?!

But this answer is not totally satisfactory:

What are the extra dimensions

good for?!

Charged matter

U(3)

U(1)

U(2)

SO(10)

1/g2

Naïve continuation does not make geometric sense

Instead a new geometry opens up!

Smooth Transition

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/4A= area of the horizon

Black holeas wrapped branes in the internal dimensions

Wheeler: Quantum mechanics wildfluctuations of space at Planck Scale -33 (10 cm)

Not only the space is fluctuating in sizeat Planck scale but even its topology shouldfluctuate. Very difficult to describe.

Space and time should look likeA ``quantum foam’’ at Planck scale

The geometryof the corners is exactly the same as for strings movingin a complex 3D space!

Macroscopic crystal geometry Stringy geometryAtomic structure of crystal ?

A simple modelfor melting crystal as removing atoms starting from the atoms near the corners

Typical configuration of molten crystal

We are witnessing anongoing revolution linkingGeometry and Physics

We are witnessing anongoing revolution linkingGeometry and Physics

We have learned a lot, buta lot remains to be learned.It is a `work in progress’!