Introduction to String Theory and D–Branes - School of Natural ...

the massless states, with multiplicity 24 2 , which is appropriate, since there are 24 2 physical states in the
graviton multiplet (Gµν, Bµν, Φ). Introducing a common piece of terminology, a term q w1 ¯q w2 , represents
the appearance of a “weight” (w1, w2) field in the 1+1 dimensional conformal field theory, denoting its
left–moving and right–moving weights or “conformal dimensions”.
4 Strings on Circles and T–Duality
In this section we shall study the spectrum of strings propagating in a spacetime which has a compact
direction. The theory has all of the properties we might expect from the knowledge that at low energy we
are placing gravity and field theory on a compact space. Indeed, as the compact direction becomes small, the
parts of the spectrum resulting from momentum in that direction become heavy, and hence less important,
but there is much more. The spectrum has additional sectors coming from the fact that closed strings can
wind around the compact direction, contributing states whose mass is proportional to the radius. Thus,
they become light as the circle shrinks. This will lead us to T–duality, relating a string propagating on a
large circle to a string propagating on a small circle.[18] This is just the first of the remarkable symmetries
relating two string theories in different situations that we shall encounter here. It is a crucial consequence
of the fact that strings are extended objects. Studying its consequences for open strings will lead us to
D–branes, since T–duality will relate the Neumann boundary conditions we have already encountered to
Dirichlet ones[13, 15], corresponding to open strings ending on special hypersurfaces in spacetime.
4.1 Closed Strings on a Circle
The mode expansion (74) for the closed string theory can be written as:
X µ (z, ¯z) = xµ
˜xµ α ′
+ − i
2 2 2 (αµ
α ′
0 + ˜αµ 0 )τ +
2 (αµ 0 − ˜αµ 0 )σ + oscillators . (181)
We have already identified the spacetime momentum of the string:
p µ = 1
√ 2α ′ (αµ
0
+ ˜αµ 0 ) . (182)
If we run around the string, i.e., take σ → σ + 2π, the oscillator terms are periodic and we have
X µ (z, ¯z) → X µ
α ′
(z, ¯z) + 2π
2 (αµ 0 − ˜αµ 0 ) . (183)
So far, we have studied the situation of non–compact spatial directions for which the embedding function
X µ (z, ¯z) is single–valued, and therefore the above change must be zero, giving
α µ
0 = ˜αµ 0 =
α ′
2 pµ . (184)
Indeed, momentum p µ takes a continuum of values reflecting the fact that the direction X µ is non–compact.
Let us consider the case that we have a compact direction, say X 25 , of radius R. Our direction X 25
therefore has period 2πR. The momentum p 25 now takes the discrete values n/R, for n ∈ Z. Now, under
σ ∼ σ + 2π, X 25 (z, ¯z) is not single valued, and can change by 2πwR, for w ∈ Z. Solving the two resulting
equations gives:
α 25
0 + ˜α 25
0 = 2n
α 25
0
− ˜α25
0 =
48
R
α ′
2
2
wR (185)
α ′