Gravity and Strings

420 String theory
Observe that the αns would represent oscillations of the strings in directions perpendicular
to the x = x1, x2 hypersurfaces, which is not possible. Hence they must represent
oscillations of the x = x1, x2 hypersurfaces which, thus, are the 25-dimensional worldvolumes
of dynamical 24-dimensional objects: D24-branes. These objects can be understood
as new physical string states that must be non-perturbative since they do not appear in the
perturbative spectrum. Indeed, as we will see, their tension is ∼ g−1 , where g is the string
coupling constant that we will define later.
In the presence of two D24-branes parallel at x = x1, x2, open strings can have both ends
attached to either one of them or have each end attached to each of them. 12 Since we are
considering oriented strings, we must distinguish between the strings that connect 1 to 2
and those that connect 2 to 1. There are, then, four sectors and, according to the preceding
paragraphs, two of them include two massless vectors and scalars and two massive vectors,
and scalars, all of them living in the directions parallel to the D-branes. When the two
24-branes coincide, we have two extra massless vector and scalar fields. The four massless
vector fields turn out to be U(2) gauge fields. In general, when n D-branes coincide, there
are massless U(n) vector fields labeled by two indices I, J = 1, 2 that indicate to which
D-brane each string endpoint is attached and the gauge symmetry is spontaneously broken
to smaller groups when some of the D-branes become separated: the Higgs scalars give
mass to the gauge fields.
It is possible to introduce labels (Chan–Paton factors) for open-string endpoints even if
all coordinates have N boundary conditions, and the theory will have U(n) gauge vector
fields. In this case we can think that the spacetime is filled with n D25-branes.
Symmetry enhancements at particular values of moduli are some of the most interesting
features of string theories.
Let us consider now closed strings. The lightest states obeying the constraint (14.48)
after the tachyon |0, 0; k〉 are of the form αi j
−1 ˜α −1 |0, 0; k〉 and, just in d = 26, they fit into
Poincaré representations: the part symmetric and traceless in ij corresponds to a massless
graviton, the trace part to a massless scalar, the dilaton, and the antisymmetric part to a
massless 2-form field: the Kalb–Ramond (KR) field.
After introducing interactions and following the reasoning of Chapter 3, closed-string
theories must contain gravity, which in a certain limit must coincide with Einstein’s GR.
Finally, let us consider unoriented strings. They can be obtained from oriented strings
by taking the quotient by the worldsheet parity operator : ξ 1 → 2πℓ− ξ 1 , which interchanges
right- and left-moving sectors,
X µ
± = X µ
∓, (14.49)
and is an involution, 2 = 1, and generates a Z2 symmetry group. Taking the quotient
means keeping only -invariant states. On level-N states
|N; k〉=(−1)N|N; k〉, |N, Ñ; k〉=|Ñ, N; k〉. (14.50)
Thus, the KR tensor is removed from the closed-string spectrum and the massless vector
is removed from the open-string spectrum unless the endpoints have Chan–Paton factors
12 This system is not stable in bosonic-string theory, as the presence of tachyons indicates, but it will be in
type-II superstring theories.