In the Green-Schwarz formalism, for F1 strings, we have the action

$$S=S_1+S_2$$

Where

$${S_1} = - T\int_{}^{} {\sqrt { - \det \left( {{\Pi _{\alpha \mu }}\Pi _\beta ^\mu } \right)} {{\text{d}}^2}\sigma } $$

and $S_2$ is the additional action term that arises through Kappa Symmetry.

The transformations of kappa symmetry for F1 strings are intuitively related to that for D0 branes, etc. They are given by:

$$\begin{gathered} \delta {X^\mu } = {{\bar \Theta }^A}{\gamma ^\mu } \\ \delta {\Theta ^A} = - \delta {{\bar \Theta }^A}{\gamma ^\mu }{\Theta ^A} \\ \end{gathered} $$

So that:

$$\delta \Pi _\alpha ^\mu = - 2\delta {\bar \Theta ^A}{\gamma ^\mu }{\partial _\alpha }{\Theta ^A}$$

From this transformation, the variation in $S_1$ by kappa symmetry, is almost trivially (not that trivial, thiough...):

$$\delta {S_1} = \frac{2}{\pi }\int_{}^{} {\sqrt { - \lambda } {\lambda ^{\alpha \beta }}\Pi _\alpha ^\mu \delta {{\bar \Theta }^A}{\gamma _\mu }{\partial _\beta }{\Theta ^A}{{\text{d}}^2}\sigma } $$

Here,

$$\begin{gathered} \lambda = \det {\lambda ^{\alpha \beta }} \\ {\lambda ^{\alpha \beta }} = {\Pi _{\alpha \mu }}\Pi _\beta ^\mu \\ \end{gathered} $$

To determine $S_2$, howefver, is not all that trivial since the simple procedure I've mentioned here, for example, is just not practical by any means.

We there fore use a 2-form, form $\Omega_2$, such that:

$${S_2} = \int_{}^{} {{\Omega _2}} = \int {{\epsilon ^{\alpha \beta }}{\Omega _{\alpha \beta }}{{\text{d}}^2}\sigma } $$

We may also (formally) define a 3-form $\Omega_3=\mbox{d}\Omega_2,$ so that by Stokes' theorem, we have it that:

$$ \int_M {{\Omega _2}} = \int_D^{} {{\Omega _3}} $$

$M$ would be the worldsheet whereas $D$ would be it's interior, ; $M=\partial D$.

There are 3 supersymmetric 1-forms prominent to us namely, ${\text{d}}{\Theta ^1},{\text{d}}{\Theta ^2},{\Pi ^\mu }$. If ${\Omega _3}$ is to be supersymmetryic, which it better be, then, we shoul'd have it that ${\Omega _3}$ involves just these 3 1-forms.

A very sensible choice of 3-form for $\Omega_3$ is:

$${\Omega _3} = A\left( {{\text{d}}{{\bar \Theta }^1}{\gamma _\mu }{\text{d}}{\Theta ^1} -{\kern 1pt} {\kern 1pt} {\kern 1pt} {\text{d}}{{\bar \Theta }^2}{\gamma _\mu }{\text{d}}{\Theta ^2}} \right){\Pi ^\mu }$$

Now, when I first learnt this, I was pretty confused at the "$-$" si1gn, (probably because of the fact that I was too confused to bother reading the next line of bbs : ) ...).

As I guess many others may also be confused at this, let me ask and answer the question:

Is this minus sign, by any chance, related to the fact that Type IIA String Theory is chiral? Does this mean that the above expression only holds for the Type IIA, and not for the Type IIB ? .