TYpe II supergravity
295
B(2) is the 2-form associated with the Kalb-Ramond field and the (1
dimensional) Hodge dual '"C(p) is
(-0)1/2
*C -
L
C"I"2 ... "p dx'" A dx"2 A
A dx"lO-p
(p) = p!(1O _ p)!""I"2"'''IO-plllll2 ... Ilp
. . .
(10.10
It is important to bear in mind that the effective action (10.105) is a
approximation that is good in the low-energy limit a' ~ O. Higher-order term
in the curvature are negligible provided JR.(G)a' « 1. Roughly speaking, w
may say that the metric obtained by solving the lowest-order field equations i
only well defined on length scales I » I,. Note that all terms from the N
NS sector are mUltiplied by e-'2t/I , while terms from the R-R sector are n
coupled to the dilaton. This is a feature of the 'string frame' in which the actio
(10.105) is written. To remove the dilaton factor from the curvature term, as i
the conventional 'Einstein frame' in which we have worked hitherto, we perfor
the following field redefinition
G - ~/2
(10.10
Il" - e gll'"
Then the effective bosonic action in the Einstein frame is
SUA = --;- {j dlOx (_g)I/2JR.(g) - ~ j[dt/> A *dt/> + e-~ !H(3) A * H(3)
2"10
2
+ e3~/2F(2) A * F(2) + e~/2F(4) A * F(4) + B(2) A F(4) A F(4)')}
(10.11
lYpe 11 string theory has.N = 2 supersymmetry
aa,
which. in 10 dimension
is realized by two Majorana-Weyl spinors Qa and
each having have 16 re
components, so
a
there is a total of 32 supersymmetry charges; Q acts on the righ
movers, and
on the left-movers. In type HA theory, the two spinors ha
opposite chirality. while type liB both spinors have the same chirality. Thu
type liB is a chiral theory and type HA non-chiral. The R-R states in type liB ar
also represented by form fields but now with components transforming as eve
ranked tensors C(O). C(2) and C(4)-the only subtlety is that the [4)+ = 3S
SO (8) is self-dual. Analogous forms of the actions (10.105) and (10.1 10) ma
also be written in terms of the field strengths F(I), F(3) and F(5) derived fro
these R-R sector fields. Otherwise, the structure of the two forms of the actio
is very similar to the type IIA case and we shall not reproduce them here. T
self-duality constraint (of the 5-form field strength) must be applied as an ext
condition on the solution of the field equations. The important point is that
may consistently truncate the type HA and type lIB effective actions to inclu
only the graviton. dilaton plus one field-strength tensor F(n) (or H(3). This
non-trivial because it must be verified that the (local) supersymmetry variation
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295
B(2) is the 2-form associated with the Kalb-Ramond field and the (1
dimensional) Hodge dual '"C(p) is
(-0)1/2
*C -
L
C"I"2 ... "p dx'" A dx"2 A
A dx"lO-p
(p) = p!(1O _ p)!""I"2"'''IO-plllll2 ... Ilp
. . .
(10.10
It is important to bear in mind that the effective action (10.105) is a
approximation that is good in the low-energy limit a' ~ O. Higher-order term
in the curvature are negligible provided JR.(G)a' « 1. Roughly speaking, w
may say that the metric obtained by solving the lowest-order field equations i
only well defined on length scales I » I,. Note that all terms from the N
NS sector are mUltiplied by e-'2t/I , while terms from the R-R sector are n
coupled to the dilaton. This is a feature of the 'string frame' in which the actio
(10.105) is written. To remove the dilaton factor from the curvature term, as i
the conventional 'Einstein frame' in which we have worked hitherto, we perfor
the following field redefinition
G - ~/2
(10.10
Il" - e gll'"
Then the effective bosonic action in the Einstein frame is
SUA = --;- {j dlOx (_g)I/2JR.(g) - ~ j[dt/> A *dt/> + e-~ !H(3) A * H(3)
2"10
2
+ e3~/2F(2) A * F(2) + e~/2F(4) A * F(4) + B(2) A F(4) A F(4)')}
(10.11
lYpe 11 string theory has.N = 2 supersymmetry
aa,
which. in 10 dimension
is realized by two Majorana-Weyl spinors Qa and
each having have 16 re
components, so
a
there is a total of 32 supersymmetry charges; Q acts on the righ
movers, and
on the left-movers. In type HA theory, the two spinors ha
opposite chirality. while type liB both spinors have the same chirality. Thu
type liB is a chiral theory and type HA non-chiral. The R-R states in type liB ar
also represented by form fields but now with components transforming as eve
ranked tensors C(O). C(2) and C(4)-the only subtlety is that the [4)+ = 3S
SO (8) is self-dual. Analogous forms of the actions (10.105) and (10.1 10) ma
also be written in terms of the field strengths F(I), F(3) and F(5) derived fro
these R-R sector fields. Otherwise, the structure of the two forms of the actio
is very similar to the type IIA case and we shall not reproduce them here. T
self-duality constraint (of the 5-form field strength) must be applied as an ext
condition on the solution of the field equations. The important point is that
may consistently truncate the type HA and type lIB effective actions to inclu
only the graviton. dilaton plus one field-strength tensor F(n) (or H(3). This
non-trivial because it must be verified that the (local) supersymmetry variation
0.
8)
n
s
e
s
Sot
n
n
m
9)
0)
s,
al
tve
s,
e
nof
y
m
n
he
ra
we
de
is
of