String Theory and M-Theory

314 M-theory and string duality
There are several different ways of dealing with the problem of the selfdual
field. The original approach is to not construct an action, but only
the field equations and the supersymmetry transformations. This is entirely
adequate for most purposes, since the supergravity theory is only an effective
theory, and not a quantum theory that one inserts in a path integral.
The basic idea is that the supersymmetric variation of an equation of motion
should give another equation of motion (or combination of equations
of motion). By pursuing this systematically, it turns out to be possible to
determine the supersymmetry transformations and the field equations simultaneously.
In fact, the equations are highly overconstrained, so one obtains
many consistency checks.
It is possible to formulate a manifestly covariant action with the correct
degrees of freedom if, following Pasti, Sorokin, and Tonin (PST), one introduces
an auxiliary scalar field and a compensating gauge symmetry in a
suitable manner. The extra gauge symmetry can be used to set the auxiliary
scalar field equal to one of the space-time coordinates as a gauge choice, but
then the resulting gauge-fixed theory does not have manifest general coordinate
invariance in one of the directions. Nonetheless, it is a correct theory,
at least for space-time topologies for which this gauge choice is globally well
defined.
An action
We do not follow the PST approach here, but instead present an action
that gives the correct equations of motion when one imposes the self-duality
condition as an extra constraint. Such an action is not supersymmetric, however,
because (without the constraint) it has more bosonic than fermionic
degrees of freedom. Moreover, the constraint cannot be incorporated into
the action for the reasons discussed above.
The way to discover this action is to first construct the supersymmetric
equations of motion, and then to write down an action that reproduces those
equations when the self-duality condition is imposed by hand. The bosonic
part of the type IIB supergravity action obtained in this way takes the form
S = SNS + SR + SCS. (8.53)
Here SNS is the same expression as for the type IIA supergravity theory in
Eq. (8.40), while the parts of the action describing the massless R–R sector
fields are given by
SR = − 1
4κ 2
d 10 x √
−g |F1| 2 + | F3| 2 + 1
2 | F5| 2
, (8.54)