Gravity and Strings

15.3 Example: the fundamental string (F1) 445
of g, such as the fundamental string whose action is the string σ -model action and whose
tension is T = 1/(2πℓ 2 s ).
15.3 Example: the fundamental string (F1)
Many string solutions (i.e. solutions of the string effective action Eq. (15.1)) are known.
For instance, all the vacuum Einstein solutions are string solutions with constant dilaton
and pure gauge KR 2-form and, for each of them that admits an isometry, we can find
aT-dual string solution (possibly with non-trivial dilaton and KR 2-form). To investigate
T duality, however, we must choose a convenient solution to which we can give a physical
and stringy interpretation, in the same spirit as that in which we chose a gravitational wave
in Section 11.2.3 to illustrate KK reduction, four-dimensional electric–magnetic duality
and KK oxidation. The so-called fundamental string solution [281, 282] (see also [337])
denoted by F1 represents a string at rest and can play this role.
We can understand the F1 solution as a solution of the (“bulk-plus-brane”) action that
results from the addition of the string effective action Eq. (15.1) to the string σ -model
action Eq. (15.31) which, denoted from now on by SF1, acts as a singular one-dimensional
source for the former. The equations of motion of the spacetime fields 10 are Eqs. (15.11)
plus the source terms
16πG ( ˆd)
N e2( ˆφ− ˆφ0) δ ˆSF1
|ˆg| δ ˆg ˆµˆν =+8πG( ˆd)
|ˆg|
16πG ( ˆd)
N e2( ˆφ− ˆφ0) δ ˆSF1
|ˆg| δ ˆφ
16πG ( ˆd)
N e2( ˆφ− ˆφ0) δ ˆSF1
|ˆg| δ ˆB ˆµˆν
= 0,
N e2( ˆφ− ˆφ0) T
=+ 8πG( ˆd)
N e2( ˆφ− ˆφ0)
T
|ˆg|
d 2 ξ √ |γ |γ ij ˆgi ˆµ ˆg j ˆνδ ( ˆd) ( ˆx − ˆX),
d 2 ξɛ ij ∂i ˆX ˆµ ∂ j ˆX ˆν δ ( ˆd) ( ˆx − ˆX).
We also have to solve the equations of motion of the worldvolume fields,
(15.68)
− 2
T √ δ ˆSF1
|γ | δγ ij =ˆgij − 1
2γij ˆg k k = 0,
1
T √ δ ˆSF1
|γ | δ ˆX ˆµ =ˆg
2
ˆµˆν ∇ ˆX ˆν + γ ij ˆƔij ˆν + ɛij
2 √ |γ | ˆH
(15.69)
ˆµij = 0.
We work in the static gauge, identifying the worldvolume coordinates with the first
spacetime coordinates ˆX i = ξ i ≡ (T, Y ), i = 0, 1. The remaining spacetime coordinates
are transverse to the string worldvolume and we make for them the Ansatz X m (ξ) = 0,
m = 1,..., dˆ − 2. If the solution is to describe a fundamental string at rest, it is natural
to make an Ansatz for the metric with Poincaré symmetry in the worldvolume directions.
On the other hand, our experience with ERN BHs tells us that a full solution of the equations
with sources can be expected only when there is supersymmetry/extremality and the
solution depends solely on a reduced number of functions that are harmonic in transverse
10 We add hats to all ˆ
d-dimensional fields.