Gravity and Strings

14.2 Quantum theories of strings 419
The vacuum is defined to be annihilated by all the oscillators αi n and ˜αi n with n > 0 and
states are created by acting on it with creation operators αi −n and ˜αi −n , with n > 0, on the
momentum eigenstates |0, k〉
p i |0, k〉=k i |0, k〉, p + |0, k〉=k + |0, k〉. (14.43)
Relative to this vacuum, the mass operator M 2 =−2p + H − p i p i takes the form, for
open strings with only N boundary conditions,
M 2 = 1
(N + A). (14.44)
α ′
For open strings with DD boundary conditions in one coordinate
and for closed strings
In all these cases
M 2
x2 − x1
=
2πα ′
N =
2
+ 1
(N + A), (14.45)
α ′
M 2 = 2
(N + Ñ + A + Ã). (14.46)
α ′
n>0
nα i −n αi n
, Ñ =
n>0
n ˜α i −n ˜αi n , (14.47)
are the level operators that take only positive integer values and A and à are constants that
arise in the normal ordering of the Hamiltonian and take the value A = Ã = (2 − d)/24.
In the closed-string case there is still one constraint that has not been eliminated, which
is associated with the ξ 1 shift symmetry:
N = Ñ. (14.48)
Let us now consider the lightest states of these three theories. The lightest states of
the open string with N boundary conditions are the |0, k〉, whose mass is, according to
Eq. (14.44), M2 = (2 − d)/(24α ′ ),which is negative for d > 2, corresponding to a spacetime
scalar tachyon and indicating the instability of the open-bosonic-string vacuum. The
next lightest states are obtained by acting with the α i −1
operators on |0, k〉 and have
masses M2 = (26 − d)/(24α ′ ). They fill a vector representation of SO(d − 2), just like a
d-dimensional massless spacetime vector particle. Poincaré invariance then requires the
mass of these states to be zero and d = 26 and the spectrum contains a scalar tachyon, a
massless vector, and massive and higher-spin states.
The lightest state of the DD open string is again |0, k〉, with (d = 26) M2 =
[(x2 − x1)/(2πα ′ )] 2 − 1/α ′ ,whose value and sign depend on the distance between the hypersurfaces
x = x1, x2 to which the string endpoints are attached. When x2 = x1 there are
massless states αi −1 |0, k〉, i = x and αx −1 |0, k〉 (x being the DD direction), a massless vector,
and a scalar in d − 1 dimensions. When x2 = x1 we have the d − 2 states of a massive
(d − 1)-dimensional vector. The scalar plays the role of a Higgs scalar.