296
Black holes in string theory
the (zero) fermion fields is constantly zero. In the Einstein frame. this gives an
action of the form
S,. = 2:~o { f dlOx (-g)1/2R(g) - ~ f [d4>" *d4> + e- a , F(II) " * F(II)] }.
(10.111)
The value of a determines the coupling to the dilaton and may be read off from
the type IIA or liB action. If the chosen field strength is H(3) deriving from
the NS-NS form B(2). then a = 1, whereas if the chosen field strength derives
from an R-R field, then a = (n - 5)/2. In the latter case. n = 2,4 •...
corresponds to type HA. and n = I. 3,5, ... corresponds to type lIB. The fact that
the type 11 string theory effective action (10.110) involves various field strengths
F(II) suggests that there should be some objects in the underlying string theory
that couple directly to the associated gauge form fields, just as an electron is
coupled by its charge to the Maxwell gauge potential A". We shall see in the
next section that these objects are extended objects, p-branes. generally having p
spatial dimensions.
10.7 Form fields and D-branes
There is a geometric aspect of the antisymmetric forms which gives important
in sights. A gauge field A" is coupled naturally to the world line X"(t') of a
charged particle by a term in the action of the form
dX"
5 '" f A" d;"' dt'.
(10.112)
Under a (U ( I» gauge transformation. the vector potential I-form
A" dx" == A(l) -. A(I) + dA(o)
(10.113)
where A(o) is a O-form, i.e. a function. The field strength 2-form F(2) = dA(I) is
gauge invariant and satisfies
dF(2) = 0
(10.114)
which in four dimensions are two of Maxwell's equations. The other two are
d * F(2) = * J(I)
(10.115)
where J(I) == J" dx" is the current I-form. * F(2) and * J(l) are the Hodge duals.
They are defined in ten dimensions in (10.108) but have an obvious generalization
to any dimensionality. In four dimensions. *J(l) is a 3-form. and we may use
Gauss' theorem to find the electric charge Q in some spatial volume V3 enclosed
by a surface 52:
Q = ( * J(l) = ( * F.
(10.116)
)V3
)S2
Black holes in string theory
the (zero) fermion fields is constantly zero. In the Einstein frame. this gives an
action of the form
S,. = 2:~o { f dlOx (-g)1/2R(g) - ~ f [d4>" *d4> + e- a , F(II) " * F(II)] }.
(10.111)
The value of a determines the coupling to the dilaton and may be read off from
the type IIA or liB action. If the chosen field strength is H(3) deriving from
the NS-NS form B(2). then a = 1, whereas if the chosen field strength derives
from an R-R field, then a = (n - 5)/2. In the latter case. n = 2,4 •...
corresponds to type HA. and n = I. 3,5, ... corresponds to type lIB. The fact that
the type 11 string theory effective action (10.110) involves various field strengths
F(II) suggests that there should be some objects in the underlying string theory
that couple directly to the associated gauge form fields, just as an electron is
coupled by its charge to the Maxwell gauge potential A". We shall see in the
next section that these objects are extended objects, p-branes. generally having p
spatial dimensions.
10.7 Form fields and D-branes
There is a geometric aspect of the antisymmetric forms which gives important
in sights. A gauge field A" is coupled naturally to the world line X"(t') of a
charged particle by a term in the action of the form
dX"
5 '" f A" d;"' dt'.
(10.112)
Under a (U ( I» gauge transformation. the vector potential I-form
A" dx" == A(l) -. A(I) + dA(o)
(10.113)
where A(o) is a O-form, i.e. a function. The field strength 2-form F(2) = dA(I) is
gauge invariant and satisfies
dF(2) = 0
(10.114)
which in four dimensions are two of Maxwell's equations. The other two are
d * F(2) = * J(I)
(10.115)
where J(I) == J" dx" is the current I-form. * F(2) and * J(l) are the Hodge duals.
They are defined in ten dimensions in (10.108) but have an obvious generalization
to any dimensionality. In four dimensions. *J(l) is a 3-form. and we may use
Gauss' theorem to find the electric charge Q in some spatial volume V3 enclosed
by a surface 52:
Q = ( * J(l) = ( * F.
(10.116)
)V3
)S2