Physics Notes: String Theory

Lecture 10: Nov 29, 2010 Back to PHY32

Topics: T-duality, D-branes, modeling QFT, QCD and EM

Q: How do we classify manifolds that are good for string theory?

The characterization is a local characterization. The manifold must be a solution of the Einstein field equation. .

We also need to satisfy some supersymmetry related properties. This is not strictly required, but we only have the mathematical structure to deal with super symmetric theories. The property is known as Kahler. [Kahler manifolds have the property that the metric changes slowly from place to place. The metric’s rate of change is limited to quadratic with separation. Another important property is that rotation commutes with parallel transport. This means that you can transport and then rotate or rotate and then transport a vector and get the same result. It is easy to see that commutation of rotation and transport adds a lot of structure.]

[See http://en.wikipedia.org/wiki/Moduli_(physics) about restrictions on supersymmetic manifolds.]

Given a 10 dimensional space, there would be 6 compact dimensions. The set of manifolds satisfying the requirements are known as Calabi-Yau manifolds. It is believed that there are a finite but large number of these manifolds. Fluxes and D-branes can be added to the theories, increasing the number of possibilities. Finding our world among the possibilities is a big problem.

String theory math is consistent, contains gravity, quantum mechanics, fermions and bosons. It provides at least one way to make QM and gravity consistent.

Lecture

T-duality review

Strings can move around compact dimensions. They can also be wound around compact dimensions. For closed strings, even if you start with no wound strings, you will get them. A non-wound closed string can become excited, grow in size, and meet itself around the compact dimension. When it does so, it has a probability of interacting and leaving two wound strings behind. Since strings are oriented, it is easy to see that one string will be positively wound and the other negatively wound. A consequence in this model of closed strings is that the total winding number is conserved. This winding number will turn out to behave like electric charge.

Two wound strings can collide and join into one string that is not wound if the original strings had one unit of winding in opposite directions.

The energy spectrums of wound and non-wound strings are both quantized. The energy levels of a non-wound string are based on quantization of angular momentum. The energy levels of a wound string are based on the energy per unit length of a string, which we set to 1 for now.

Suppose that all you could measure about a system was the energy spectrum. Then you can’t tell what r is. You can find a value that is either r, or 1/r.

So, there is no point to thinking of compact radii < 1.

Now let’s develop some expressions for the winding number and the angular momentum.

[The traditional way to compute a winding number about a point in the plane is to integrate the angle change along the closed curve. The same thing works here if we think of the position in the compact dimension as an angle.]

In the equation below, W proportional to the number of times a string is wound, with one direction being positive and the other negative. Likewise, N is proportional to the number of units of angular momentum carried by the string.

The quantized angular momentum has a similar expression. In this case we have to add up the momentum of all the mass points on the string. The mass per unit length is a constant, so we can divide it out and just integrate the velocity.

T-duality says that the equations governing the motion of the string are unchanged if you exchange 3 things

Winding and momentum quantum numbers are like electric charge. Two strings with the same winding number repel each other. Two strings with different winding numbers attract.

This happens through the extension of gravity to the compact dimensions.

The metric tensor becomes

A symmetric tensor

is a 4 vector corresponding with the vector potential .

is a scalar field, usually denoted . Called the diliton field.

is the size of the 5^th dimension.

You can have waves in the size of the dimension.

is be derived from the flow of momentum. Quantized momentum behaves like an electric charge.

The winding number W also behaves like charge, but not the same charge. For the electric charge, photons are the gauge boson. What is the gauge boson for wound strings? We start with a closed string and excite it.

n is the mode and i is the direction in space.

But, we have to remember the level matching rule, which requires that the left moving energy equals the right moving energy.

This gives us a lowest legal excitation of

The special cases involving the compact dimension y are:

The sum is a polarized state that looks like a photon.

We can also subtract, giving a different field which we associate with winding.

We now add one more element to T-duality that will be important for open strings.

Open Strings

We need to add a new element to the theory.

Dn-branes

Brane is a generalization of a membrane. A brane is an extended n dimensional object. ‘n’ in the above notation stands for the dimensionality. N=0 would correspond to a point in space, n=1 a line, and n=2 a membrane, n can range up to the dimension of space.

D stands for Dirichlet boundary conditions, which seems counter intuitive for open strings, which have Neumann boundary conditions (derivative of coordinates go to 0 at the end of the strings).

Dn-Branes are essential to consistency of string theory. They also lead to equivalences between manifolds, which was a surprise to mathematicians.

Open strings don’t have winding numbers, they have endpoints.

At the ends we have Neumann boundary conditions.

The ends become straight, and there is no net tension at the ends of strings.

Now suppose that r <<< 1 and we apply T-duality.

But this means that the coordinates of the endpoints are stuck. The implication is that the theory must have objects that mail down the ends of strings, but only in the compact direction.

The T-duality interchange exposes a new object.

The new object is a line object, a D-brane cutting across the compact dimension. Because of relativity, these branes must be bendable and moveable. We can think of strings as being attached to them.

What kind of D-brane is this example? If space has 9 dimensions, then this brane must be a D8-brane. Each constraint removes one dimension from the D-brane. In this case the y coordinate of the brane is a function by the other 8 dimensions + time.

Endpoints of strings are confined to branes. Both endpoints can be on the same brane, or on different branes.

A D1-brane can even be a loop.

What are the string interaction processes?

Strings can also break, leaving new endpoints on a brane, which could be a different one. One can use the string events to build up QFT processes. At low energies this model reduces to QFT and is a commonly used tool for studying QFT.

These strings also have orientation, so the right endpoints have to come together to join.

Suppose you have 3 branes. We will label them the red, green and blue branes.

How do we make quarks? The strings have only one end on these branes and the other end goes off somewhere else.

We can also model electromagnetism with a single brane.

A D-string on a D3-brane works and makes a magnetic monopole.

D-branes are now a primary tool for studying QCD.