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Quantization

Supersymmetry and Superstring

Compactification

Types of Superstring Theory

M-theory

Problems and Future Development

The string is an one-dimensional object, which can move in various ways. Its movement sweeps out a two-dimensional sheet, called the world-sheet (see Figure 01a) similar to the 4D space-time world-line. In the diagram, X represents a vector defined in D-dimensional space-time that begins at the origin of a coordinate system and ends at some point along the two-dimensional world-sheet with components X

degrees of freedom X_{}(,) trace out a curve as varies at fixed . The curve may be open or closed and varies within a range between 0 and as the string is traced out from one end to the other for an open string, or 0 to 2 round the circle for a closed string. The string sweeps out a worldsheet as varies from one instant to another. However, these two parameters have nothing whatsoever to do with the real space and real time. Since nobody has ever seen or detected this artificial space-time, they have to be hidden from the observables in a theory, and thus the requirement of reparametrization invariance. It turns out that such theory can be formulated only with the one-dimensional string within the framework of artificial space-time.
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## Figure 01a String, World Sheet [view large image] |

Construction of the superstring theory starts with the "Action Principle", which confines the movement of the string such that the world sheet sweeps out a minimum area (similar to the shorest distance for the case of point particle). If we demand the action S to be Lorentz covariant in form with conformal and reparametrization invariant, then it can be written as (with = 1, c = 1):

The variations associated with the Poincare (generalized Lorentz transformation), reparametrization (change of coordinates), and conformal invariances (change of artificial space-time) are:

where , = (0, 1) = (, ), are coordinates on the world-sheet. The requirement of the invariances largely determines the form of the action. Reparametrization invariance is to ensure that physics doesn't change with relabelling of these coordinates.

The Lorentz transformation in special relativity has been generalized to D-dimensions. The requirement of Lorentz covariant is to make sure that the superstring theory behaves the same in all inertial frames. Just as in gauge field theory or general relativity, there are fewer independent dynamical degrees of freedom than appear explicitly in the action. The degrees of freedom in the formulation can be reduced via the requirement of conformal invariance under the conformal transformation (Figure 01b), which rescale the world sheet metric. Such change in topology makes it feasible to evaluate string diagrams. Among other things this makes it possible to compactify the world sheet, closing off the holes corresponding to incoming and outgoing strings. For example, a world sheet with one incoming and one outgoing string (as in (a) of Figure 01b) can be conformally mapped to the plane of (a') with the incoming string appearing at the origin and the outgoing string at infinity (not shown) or to the sphere of (a") with the incoming and outgoing strings appearing at the south and north poles^{¶}. The external string states in (b) of Figure 01b with four awkward legs are projected to points as indicated in (b'). By a suitable choice of gauge (known as covariant or conformal gauge with_{} in the following case - a gauge is a mathematical device to fix the redundant degrees of freedom), the string equations of motion can be derived by varying X^{} to minimize the action S, thus we obtain:
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## Figure 01b Conformal Transformations |

---------- (2) |

Then by varying the world sheet metric

---------- (3) |

For a closed string, the general solution of Eq.(2) consistent with the boundary conditions X

---------- (4) |

The solution for an open string with the boundary condition

is: |

---------- (5) |

The two examples in the followings serve to visualize the geometry of the closed and open strings. Let us first consider a closed string momentarily at rest in the (X

X^{0} = (l^{2}p^{0}) X ^{1} = R cos(2)X ^{2} = R sin(2)where p ^{0} = iE, and R ~ Rcos(2) near ~ 0 is the radius of the cylinder. The energy-momentum tensor defined in Eq.(3) yields the relation E = 2RT confirming that T is indeed the energy per unit length of the string. Figure 02a shows the world sheet swept out by this closed string for a small interval of in the form of cylindrical surface. In general, the closed string will vibrate in the three dimensional space (X^{1}, X^{2}, X^{3}) in various harmonic modes (see Figure 02b, by Steuard Jensen, Alma College). | ||

## Figure 02a Closed String |
## Figure 02b Vibrating String [view large image] |

The solution for open string is given by Eq.(5). The rotating open string in the (X

X^{0} = (l^{2}p^{0}) X ^{1} = (l) cos()cos()X ^{2} = (l) cos()sin()The energy-momentum tensor defined in Eq.(3) yields the relation E ^{2} = (/l)^{2} for this case.
As the total angular momentum is given by the formula: , it follows that = J^{2}/2 for this case of spinning string. Thus / EJ^{2} = 1/2T, which can be
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## Figure 02c Open String |
identified as the Regge slope in Figure 03b. Furthermore, since the velocity of the string V can be computed from the formula: V = {|X ^{1}/X^{0}|^{2} + |X^{2}/X^{0}|^{2}}^{1/2} = |cos()| , |

This is the original string theory developed in the late 1960s. It is called the

- < z < , and 0 < 2 the metric is in the form:

ds

where R is the radius of the cylinder. If the coordinate z is changed to:

z = 2R

then ds

The conformal invariance allows the rescaling of the metric by a factor:

= sin

The metric is now in the form:

ds

which is the metric for a 2-sphere of radius R. The initial and final string states at z = -, + correspond to = 0, , i.e. the north and south poles of the 2-sphere, as shown in Figure 01b diagram a".

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