Gravity and Strings

426 String theory
14.3 Compactification on S 1 :Tduality and D-branes
We can obtain four-dimensional string theories by compactification. The simplest compactification
would be on a circle. Already in this case we are going to start seeing stringy
effects (T duality, first discovered in [622, 819], and enhancement of gauge symmetry) that
we did not see in the field-theory (KK) case, which are a manifestation of the extendedobject
nature of strings and a suggestion that there is a minimal length in string theory.
General references on T duality are [35, 456].
We are going to study first the compactification of closed bosonic strings on a circle.
14.3.1 Closed bosonic strings on S 1
If Z ≡ X ˆd−1 is the compact coordinate, it is convenient to identify Z ∼ Z + 2π Rz, where
Rz is the compactification radius, and keep using the Minkowski metric. Now, in the mode
expansion Eq. (14.40) of Z the following applies.
1. There is another zero mode compatible with the periodicities of ξ 1 and Z:
Rzw
ℓ ξ 1 , w ∈ Z. (14.58)
When we go around the closed string once, ξ 1 → ξ 1 + 2πℓ,wegow times around
the compact dimension: Z → Z + 2π Rzw. This is a winding mode, apurely stringy
animal that corresponds to the capacity of closed strings to be wrapped w times
around compact dimensions.
2. There are also string KK modes as in Chapter 11,
Quantization leads to the mass formula and constraint
M 2 = n2
R 2 z
+ R2 z w2
α
′ 2 + 2
n
Rz p + ξ 0 , n ∈ Z. (14.59)
(N + Ñ − 2), N = Ñ + nw. (14.60)
α ′
Observe that the mass of the w = 1 mode agrees with the definition of the string tension
on page 409: it is the product of the length of the compact dimension and the string tension.
The spectrum is now that of the uncompactified theory (the n = w = 0 sector) plus new
sectors with non-vanishing KK momentum or winding number. The spectrum of the uncompactified
theory has to be interpreted now in ˆd − 1 dimensions: the ˆd-dimensional
graviton gives rise to a graviton, a (KK) vector and a (KK) scalar in ˆd − 1 dimensions,
while the KR 2-form gives rise to another 2-form and another (winding) vector and the
dilaton gives another dilaton. For generic values of the compactification radius Rz there are
no more massless states and the vector gauge symmetry group is U(1) 2 .Asdiscussed in
Chapter 11, KK modes are charged with respect to the KK U(1) vector field. We will see
that winding modes are charged with respect to the winding U(1) vector field.