String Theory and M-Theory

316 M-theory and string duality
transform as a doublet under the ¡ SL(2, ) symmetry group. Therefore, to
rewrite the action in a way that the symmetry is manifest, let us rename
the two-form potentials B2 = B (1)
2 and C2 = B (2)
2 and introduce a twocomponent
vector notation
B2 =
B (1)
2
B (2)
2
. (8.61)
¡
Similarly, H3 = dB2 is also a two-component column vector. Under a transformation
by
d
Λ =
b
c
∈ SL(2, ),
a
(8.62)
the B fields transform linearly by the rule
B2 → ΛB2. (8.63)
Since the parameters in Λ are constants, H3 transforms in the same way.
The complex scalar field τ, defined by
τ = C0 + ie −Φ , (8.64)
is useful because it transforms nonlinearly by the familiar rule
τ →
aτ + b
. (8.65)
cτ + d
The field C0 is sometimes referred to as an axion, because of the shift symmetry
C0 → C0+constant of the theory (in the supergravity approximation),
and then the complex field τ is referred to as an axion–dilaton field.
The action can be conveniently written in terms of the symmetric SL(2, ¡ )
matrix
M = e Φ
|τ| 2 −C0
−C0 1
which transforms by the simple rule
, (8.66)
M → (Λ −1 ) T MΛ −1 . (8.67)
The canonical Einstein-frame metric g E µν and the four-form C4 are SL(2, ¡ )
invariant. Note that since the dilaton transforms, the type IIB string-frame
metric gµν in the action (8.53), which is related to the canonical Einstein
metric by
gµν = e Φ/2 g E µν, (8.68)