2.3
Fields and target space
When looking at string theory, it’s important to be aware of the similarities and differences between string theory and ordinary quantum field theory for point particles. For one, a classical point particle carves out a worldline, which can be parameterised by a parameter τ , and can be set equal to the particle’s proper time (see section 1.2). We can write down an action for a point particle, from which we can derive it’s equations of motion. A particle in quantum field theory is described by quantum fields φ(xµ), µ = 0, . . . , 3, living in D = 4 dimensions. A theory of these particles can be described by an action, involving the fields and derivatives of these fields. The four dimensions quantum fields live in, are equal to our spacetime. When we want to describe particles interacting with each other, we write down an action, involving the fields and derivatives, plus higher order terms and corresponding couplings.
This is not the case in string theory. First of all, strings are one- dimensional objects embedded in D spacetime dimensions. These strings are not generated by fields, as in quantum field theory. It is, however, pos- sible to define a field theory for strings. Such a theory is called String field theory.4 You could say that string theory relates to string field theory as a point particle does to quantum field theory.
In contrast to the point particle, a string has infinitely many more in- ternal degrees of freedom, the oscillator modes, or string modes. These string modes can be seen as waves propagating on the world-sheet at the speed of light. When we quantize the string, the string modes actually be- come quantum modes of a two dimensional quantum field theory. Since a string propagates in D spacetime dimensions, there are D such fields Xµ(τ, σ), µ = 0, . . . , D − 1. From the world-sheet’s point of view, the fields Xµcan be seen as the coordinates of a manifold, called target space. In the case of string theory this is just equal to spacetime.
So particles in quantum field theory are described by quantum fields, living in D = 4 dimensions. Interactions can be described by putting higher order terms in the Lagrangian. Particles in string theory are described by different modes of a string propagating in spacetime. From the world-sheet’s point of view, a string mode is some configuration of D two dimensional quantum fields, which are embedded in spacetime.
In the next chapter we will study conformal field theory. Since the two- dimensional string world-sheet is conformal invariant, it is not a bad idea to
4
In string field theory there are fields Φ[X(σ)] which create and annihilate strings in a certain configuration. However, such fields do not even live in ordinary spacetime anymore, but in some sort of ‘stringy’ target space. It is possible to write down an action for open string field theory, but it turns out to be very complicated, if not impossible, to write down an action for closed string fields. This subject is, however, far beyond the scope of this thesis.
investigate this property to full extent. Conformal field theory provides us with a good way of achieving this. After having studied the mathematical framework of conformal field theory, we will incorparate it into a theory of interacting strings.
Chapter 3
Conformal field theory
3.1
Complex coordinates
3.1.1 Wick rotation
Conformal field theory is a very important tool for string theory. Interac- tions between strings and other strings, or backgrounds can be very hard to describe if one would try to parameterize the theory, and apply correct boundary conditions. An important idea that has been put forward is that interactions (for example the emission or absorption of strings) should be transformed (rescaled) in such a way that they become ‘pointlike’ opera- tors on the world-sheet. Since the world-sheet is conformal invariant, such transformations leave the theory unchanged. After having applied these transformations, it becomes very interesting to see how these operators be- have in the vicinity of each other. When two operators approach each other on the world-sheet, the quantum effects become apparent. A very conve- nient setting in which to study these sorts of interactions is called conformal field theory (CFT). CFT is a very extensive subject, so we will just cover some bacis properties and give a global overview on the subject.
It turns out to be convenient to study interacting strings in a Euclidean framework, hab = δab. A Euclidean metric can easily be obtained by per-
forming a Wick rotation on one of the world-sheet coordinates (usually the propertime coordinate).1 So we let
w = σ0+ σ1 (3.1a)
= σ1+ iσ2, (3.1b)
and w = σ¯ 1− iσ2. (3.1c)
It is easily checked that with these coordinates, the world-sheet metric in- deed has a Euclidean signature.
1
One has to be careful that when performing a Wick rotation, no poles are crossed. In most examples, however, this is not the case.
σ0 σ1 σ0 σ0 0 σ 0 σ00 σ0
Figure 3.1: Left: The world-sheet of a closed string is a cylinder in space- time. Right: The conformal mapping to complex coordinates. In the complex plane, the origin corresponds to the string’s proper time at minus infinity, σ−∞0 . Points at |z| = ∞ correspond to the string’s proper time at plus
infinity, σ0∞
3.1.2 Conformal transformation
The next convenient choice is to perform a conformal transformation on these coordinates, such that
z = e−iσ1+σ2 (= e−iw) = z1+ iz2, (3.2a) ¯
z = eiσ1+σ2 (= ei ¯w) = z1− iz2. (3.2b) Both open and closed strings can be studied in these coordinates. However, since closed strings are most easily described in this setting, we will focus on them for the remainder of this chapter.
As we discussed before, the world-sheet of a closed string is a cylinder, where the string’s proper time σ0 runs from −∞ to +∞, and the σ1 coor- dinate runs from 0 to 2π. In our new z coordinates, however, the string’s proper time runs radially outwards, with z = 0 corresponding to σ0 → −∞,
and |z| → +∞ to σ0 → +∞. The σ1 coordinates at a fixed time σ2 are
represented by circles around the origin. See figure 3.1. Derivatives on the complex plane are defined by
∂ ≡ ∂z= 1 2(∂z1 − i∂z2), (3.3a) ¯ ∂ ≡ ∂¯z= 1 2(∂z1 + i∂z2), (3.3b)