Gravity and Strings

16.5 Toroidal compactification of the heterotic string 481
Now we want to eliminate H completely from the action by using its equation of motion:
where
H ν1ν2ν3 =
1 1
e2φ √ ɛ
(7 − n)! |g| µ1···µ7−nν1ν2ν3 ˜Hµ1···µ7−n , (16.153)
˜H = (7 − n)∂ ˜B (16.154)
is the (7 − n)-form dual to H.
Since B appears in the action only through H, B is completely eliminated from this
action and replaced by ˜B,and we obtain the dual action
S =
g2 h
16πG d)
N
d d x √ |g| e −2φ
Now we study two special cases.
R − 4(∂φ) 2 + (−1)d
2 · (7 − n)! e4φ ˜H 2 − 1
Tr (∂Mη∂Mη)
8
− 1
4 (ηMη) F F
+ (−1)6−p
8 · (6 − p)! η
1
√ ɛ ˜BF
|g| F
.
(16.155)
The n = 4, d = 6 case. In this case ˜H is just another 3-form field strength. The field content
of the effective action coincides with the massless sector of the type-IIA string compactified
on K3 and, in fact, there is a field redefinition that takes us from the massless fields of the
heterotic string compactified on T 4 to the massless fields of the type-IIA string compactified
on K3, which supports the duality between the two theories (including the massive modes)
[332, 336, 583, 847, 962]:
˜H = HIIA, φ =−φIIA, gµν = e −2φIIA gIIA µν. (16.156)
On performing this change of variables, we obtain
S =
g2 h
16πG (6)
N
d 6 x √ |gIIA|
e−2φIIA
RIIA − 4(∂φIIA) 2 + 1 2 HIIA 2 · 3!
− 1
8 Tr(∂Mη∂Mη) − 1
4 (ηMη) F F + 1
16 η
1
√
|gIIA| ɛ BIIAFF
.
(16.157)
Observe that, in these variables, the vector fields do not carry the factor e −2φIIA . This is
due to the fact that all of them are of RR origin.