298
Black holes in string theory
or (10.110) also involves the field strengths F(n) (with n even) that derive from
the R-R fields C(II-l)' However, unlike the NS-NS field B(2), these fields are
not coupled electrically to the string world sheet or its excitations. In fact, the
vertex operator for the emission of an R-R state involves the corresponding field
strength rather than the gauge field. It therefore vanishes at zero momentum and,
in consequence, the pertubative string states are electrically neutral with respect
to the charge associated with the R-R gauge fields. The foregoing discussion
suggests that the R-R field strengths F(II) are naturally associated with branes
baving p = n - 2 dimensions. Thus, type HA superstring theory is also associated
with O-branes (i.e. point particles), 2-branes (membranes) and 4-branes baving
non-zero values of the associated R-R charges. Similarly, type lIB superstring
theory is associated with I-branes (R-R strings), 3-branes and 5-branes. Since
the perturbative string states are neutral with respect to the charges associated
with the R-R fields, type 11 (A or B) string theory has to be augmented with
non-perturbative dynamical objects (p-branes) that do have R-R charges, i.e.
electric charges associated with the R-R gauge form fields. The branes are called
'Dirichlet p-branes' or 'Dp-branes', because besides the closed-string sector,
there has to be an open-string sector in which the open string ends on these
p-dimensional hyperplanes [17, 18]. Hence, the open-string world sheet has
Dirichlet boundary conditions in the directions perpendicular to the branes. It
turns out that these Dp-branes do couple electrically to the associated (closedstring) R-R fields, just as the fundamental string is coupled electrically to the
NS-NS Kalb-Ramond field B,.,.I)' (More precisely, D-branes act as a source for
the associated R-R gauge fields.) It is this enlargement of (the theory formerly
known as) string theory that has led to the understanding of black -hole entropy in
terms of the associated microstates. We now turn to the construction of the explicit
black-hole solutions of the type 11 supergravity field equations whose entropy we
shall eventually be able to explain in terms of Dp-branes.
10.8 Black holes in string theory
As we have just noted, besides fluctuations of the string, string theory also
has various non-perturbative soli tons. These are static, finite-energy solutions
of the classical field equations, just as RN black holes are static finite-energy
solutions of the classical Einstein-MaxweU field equations. It follows from the
previous discussion that a ('black brane') solution of the field equations deriving
from the action (l0. I 11) for g,.,.I), with non-zero F(n), will give the gravitational
field associated with an (n - 2)-brane having the associated NS-NS or R-R
charge. The field equations may be simplified by looking for solutions that have
Poincare invariance in the p + I = n - I dimensions associated with the p-brane
world volume and rotational invariance in the remaining transverse directions.
The coordinates x"" are, therefore, split into longitudinal ones, denoted x G with
a = 0, 1, ... , p, and transverse ones, denoted yi with; = (p + I), ... ,9. The
Black holes in string theory
or (10.110) also involves the field strengths F(n) (with n even) that derive from
the R-R fields C(II-l)' However, unlike the NS-NS field B(2), these fields are
not coupled electrically to the string world sheet or its excitations. In fact, the
vertex operator for the emission of an R-R state involves the corresponding field
strength rather than the gauge field. It therefore vanishes at zero momentum and,
in consequence, the pertubative string states are electrically neutral with respect
to the charge associated with the R-R gauge fields. The foregoing discussion
suggests that the R-R field strengths F(II) are naturally associated with branes
baving p = n - 2 dimensions. Thus, type HA superstring theory is also associated
with O-branes (i.e. point particles), 2-branes (membranes) and 4-branes baving
non-zero values of the associated R-R charges. Similarly, type lIB superstring
theory is associated with I-branes (R-R strings), 3-branes and 5-branes. Since
the perturbative string states are neutral with respect to the charges associated
with the R-R fields, type 11 (A or B) string theory has to be augmented with
non-perturbative dynamical objects (p-branes) that do have R-R charges, i.e.
electric charges associated with the R-R gauge form fields. The branes are called
'Dirichlet p-branes' or 'Dp-branes', because besides the closed-string sector,
there has to be an open-string sector in which the open string ends on these
p-dimensional hyperplanes [17, 18]. Hence, the open-string world sheet has
Dirichlet boundary conditions in the directions perpendicular to the branes. It
turns out that these Dp-branes do couple electrically to the associated (closedstring) R-R fields, just as the fundamental string is coupled electrically to the
NS-NS Kalb-Ramond field B,.,.I)' (More precisely, D-branes act as a source for
the associated R-R gauge fields.) It is this enlargement of (the theory formerly
known as) string theory that has led to the understanding of black -hole entropy in
terms of the associated microstates. We now turn to the construction of the explicit
black-hole solutions of the type 11 supergravity field equations whose entropy we
shall eventually be able to explain in terms of Dp-branes.
10.8 Black holes in string theory
As we have just noted, besides fluctuations of the string, string theory also
has various non-perturbative soli tons. These are static, finite-energy solutions
of the classical field equations, just as RN black holes are static finite-energy
solutions of the classical Einstein-MaxweU field equations. It follows from the
previous discussion that a ('black brane') solution of the field equations deriving
from the action (l0. I 11) for g,.,.I), with non-zero F(n), will give the gravitational
field associated with an (n - 2)-brane having the associated NS-NS or R-R
charge. The field equations may be simplified by looking for solutions that have
Poincare invariance in the p + I = n - I dimensions associated with the p-brane
world volume and rotational invariance in the remaining transverse directions.
The coordinates x"" are, therefore, split into longitudinal ones, denoted x G with
a = 0, 1, ... , p, and transverse ones, denoted yi with; = (p + I), ... ,9. The