String Theory Demystified

CHAPTER 8 Compactifi cation and T-Duality 165
endpoints that satisfy Dirichlet boundary conditions. In English this means that the
endpoints of an open string are attached to a D-brane.
A D-brane can be classifi ed by the number of spatial dimensions it contains. A point
is a zero-dimensional object and therefore is a D0-brane. A line, which is a onedimensional
object is a D1-brane (so strings can be thought of as D1-branes). Later we
will see that the physical world of three spatial dimensions and one time dimension that
we can perceive directly is a D3-brane contained in the larger world of 11-dimensional
hyperspace. In the example studied in this chapter, we considered a D24-brane, with one
spatial dimension compactifi ed that leaves 24 dimensions for the hyperplane surface.
Using the procedure outlined here, other dimensions can be compactifi ed. If we
choose to compactify n dimensions then that leaves behind a D(25-n)-brane. The
procedure outlined here is essentially the same in superstring theory, but in that
case compactifying n dimensions gives us a D(9-n)-brane. Note that:
• The ends of an open string are free to move in the noncompactifi ed
directions—including time. So in bosonic theory, if we have compactifi ed
n directions, the endpoints of the string are free to move in the other
1 + (25-n) directions. In superstring theory, the endpoints will be free to
move in the other 1 + (9-n) directions. In the example considered in this
chapter where we compactifi ed 1 dimension in bosonic string theory, the
end points of the string are free to move in the other 1 + 24 dimensions.
We can consider the existence of D-branes to be a consequence of the symmetry
of T-duality. The number, types, and arrangements of D-branes restrict the open
string states that can exist. We will have more to say about D-branes and discuss
T-duality in the context of superstrings in future chapters.
In this chapter we described compactifi cation which involves taking a spatial dimension
and compactifying it to a small circle of radius R. Going through this procedure, it was
discovered that a symmetry emerges called T-duality, which relates theories with small
R to equivalent theories with large R. An important consequence of T-duality was
discovered when it was learned that open strings with Neumann boundary conditions
are transformed into open strings with Dirichlet boundary conditions in the dual theory.
The result is the endpoints of the string are fi xed to a hyperplane called a D-brane.
Quiz
1. Translational invariance along σ leads to the condition ( L − L ) ψ = for
0 0 0
25 25
physical states ψ . Use this to fi nd a relation between pL , NL, and pR , N R .
Summary