Superstring Theory

Compactification

Since it is via compactification, which yields local non-Abelian gauge symmetries, and other symmetry groups (to describe all the known particles and forces, plus some unknow ones), the mathematical treatment will be presented in the followings for a better understanding of the subject. Compactification of one dimension will be described in more details. The result is then generalized to compactify the extra 16 dimensions and finally to curl up six of the remaining ten dimensions.

		Considering the simple example of a bosonic string with one dimension (for both right and left movers) compactified on a circle, say X²⁵ in a form similar to Eq.(4) with = 25. Therefore, the value of a point in X²⁵ must satisfy: x²⁵ = x²⁵ + 2nR ---------- (40)
Figure 04a One Dimension Compactification [view large image]	Figure 04b Modes of Motion for Strings [view large image]	where R is the radius of the circle, and n is any integer known as winding number for the string configuration (Figure 04a). The momentum p²⁵ is then constrained by the requirement that exp(ip²⁵x²⁵) should be single valued (see also 5-D Space-time).

Consequently,

p²⁵ = m / R ---------- (41)

where m is any integer. Thus, while the rest of the degrees of freedom take on the form of Eq.(4) with

= 0, 1, ..., 24; the 25th one can be expressed as:

Figure 04b shows serveral kinds of motion within the compactified space - a cylinder in this case. Diagram (a) illustrates some point particles moving on this cylinder. A loop of string can undergo similar motion, with one difference being that it oscillates as it moves around on the surface as shown in Diagram (b). The oscillations of the string imbue it with characteristics such as mass and charge. The string can also wrap around the cylinder as shown in Diagram (c). The string will continue to slide around and oscillate, but it will do so in this extended configuration. In fact, the string can wrap around any number of times. This kind of motion is known as motion in winding mode. A wrapped string has a minimum mass, determined by the size of the circular dimension and the number of times it wraps around as shown in Eqs.(45) and (46). The string's oscillatory motion adds more mass in excess of this minimum. In the light cone gauge: X⁺ = ( X⁰ + X²⁴), X^- = ( X⁰ - X²⁴), the 25-dimensional mass-squared operator including all the contributions from different kind of motion has the expression:

where the number operator for the right- and left-movers is given by:

Two massless vectors with N = 1, = 1, and m = n = 0. They are constructed by taking one index to be associated with the 25-dimensional space-time and the other with the compactified dimension:

This is similar to the Kaluza-Klein compactification when a dimension was compactified on a circle with components of the 26-dimensional metric tensor. The one index associated with the compactified dimension becomes the components of a U(1) gauge field for 25 dimensions. There are two vectors in this case corresponding to the left and right mover respectively. They can be considered as the components of the U(1) X U(1) gauge group.

Another scheme to make massless states is to set N = 1, = 0, m = n = 1, and R = 1 / (2)^1/2. This is purely a stringy phenomenon depending on the existence of winding number. If the state with a particular pair of m and n is denoted by |m,n>, then four extra massless vectors can be constructed according to the different combination of m and n:

Thus, for the particular choice of the radius R of the compactified dimension, there are present in the theory the six massless vector fields required for the adjoint representation of the gauge group SU(2) X SU(2).

In the heterotic string, the left mover is a bosonic string in 26 dimensions; while the right mover is a superstring in 10 dimensions:

---------- (53a)

---------- (53b)

---------- (53c)

---------- (53d)

---------- (53e)

The extra 16 dimensions can be considered as 16 "internal" degrees of freedom. Gauge group can be constructed by compactifying these degrees of freedom on a 16-dimensional torus, which is defined by introducing a lattice with basis vector e_a^I, a = 1, ..., 16, chosen to have length 2^1/2, then the x and p terms in X_L^I can be written as:

where the R_a are radii, the n_a, and m_a are arbitrary integers.

In the light cone gauge: X⁺ = ( X⁰ + X⁹), X^- = ( X⁰ - X⁹), the ten-dimensional mass-squared operator for the physical states is given by:

M² = M_R² + M_L² ---------- (55)
with M_R² = M_L² ---------- (56)
For the superstring right movers,

For the bosonic string left movers with 16 dimensions compactified on a torus,

where a sum over i from 1 to 8, and over I from 1 to 16 is understood.

Similar to the case of one dimension compactification, the massless states (M_R² = M_L² = 0) with vanishing winding numbers generate 16 massless vectors in a U¹⁶(1) gauge group (because only the left mover is involved). The p_L^I in the compactified dimensions produce more massless states. However with the requirements of absence of anomalies and finiteness of loop contributions to the scattering matrix, the radii of the torus is restricted to R_a = 1/2^1/2 and only two forms of lattices in 16 dimensions are compatible. One of them leads to a SO(32) gauge group, and the other can be identified to the gauge fields of an E₈ X E₈ gauge group. The heterotic string contains no tachyons because the only negative mass-squared right-mover state has M_R² = -2 (see Eqs.(57) and (59)); while the only negative mass-squared left-mover state has M_L² = -4 (see Eqs.(60)). These negative mass states are invalid because they fail to satisfy Eq.(56). Thus, unlike the case of the superstring, the absence of tachyons in the Heterotic theory is not enforced by a GSO projection.

Any string theory that is to be a candidate theory of the world we live in will have to possess just four observable space-time dimensions, or, if there are extra spatial dimensions, they will have to be compactified on a sufficiently small scale as to be unobservable with the energies that are currently available to us. To complete the construction of a four-dimensional theory it is necessary next to compactify six of the remaining ten dimensions in some way for both right and left movers. The simplest possibility is to repeat the previous procedure employed for the compactification of the 16 extra dimensions in the left-mover. Unfortunately, it yields massless states not compatible with the world we live in because they are always non-chiral. However, a simple modification of the toroidal compactification to orbifold compactification can overcome this difficulty. An orbifold is a 6-dimensional space obtained by identifying points on the torus that are mapped into one another (by rotating 2

/3) referred to as the point group as shown in Figure 05. The diagram actually shows a two-dimensional orbifold surface. By

		combining three such orbifolds together, it is possible to generate a six-dimensional space with 3 X 3 X 3 = 27 singular points, which can be identified to the 27 fermionic fields in the E₆ group. In this way, one of the E₈ in the 16-dimansional compactification breaks correctly into SU(3) and E₆. The E₆ itself has yet to be broken into an even finer structure. It turns out that the orbifold predicts 36
Figure 05 Orbifold [view large image]	Figure 06 Calabi-Yau Manifolds [view large image]	generations of elementary particles. This is clearly far too many (for the observed 3 generations), but at least the theory is on the right track.

Orbifold compactification is the simplest and it preserves the equation of string in its simple form. However, the formulation is not on a truly manifold because it involves singular points. If one is prepared to pay the price of much more difficult equations of string, the Calabi-Yau manifolds (see Figure 06) may be constructed by cutting off the singular points, patching them up smoothly, and shrinking the patch-ups to zero. A typical Calabi-Yau shape contains holes as shown in Figure 07. There is

a family of lowest-energy string vibrations associated with each hole in the Calabi-Yau portion of space. Because the familiar elementary particles should correspond to the lowest-energy oscillatory patterns, the existence of multiple holes means that the patterns of string vibrations will fall into multiple families. If the curled-up Calabi-Yau has three holes, then we will find three families of elementary particles as observed. Unfortunately, the number of holes in each of the tens of thousands of known Calabi-Yau shapes spans a wide range from 3, 4, 5, 25, ... 480.

Figure 07 Calabi-Yau Holes [view large image]

The problem is that at present no one knows how to deduce from the equations of string which of the Calabi-Yau shapes constitutes the extra spatial dimensions. The properties of the force and matter particles can be extracted from the boundaries of the various multidimensional holes,

which intersect and overlap with one another. The idea is that as strings vibrate through the extra curled-up dimensions, the precise arrangement of the various holes and the way in which the Calabi-Yau shape folds around them has a direct impact on the possible resonant patterns of vibration. It seems then the string theory can provide us with a framework for answering questions - such as why the electron and other particles have the masses they do. Once again, though, carrying through with such calculations requires that we know which Calabi-Yau space to take for the extra dimensions. Since the Calabi-Yau shape can be deformed in many ways (see Figure 07), there are literally an infinite variety of them. They are now referred to as vacua, and each of them could be a different universe with a different set of physical parameters.

See more in "Calabi-Yau Manifold".