Ramond-Ramond potential and field strenght

I have a doubt about R-R potential in Superstring theory. The known facts are (according many books, for example "Basic Concepts in String Theory" by R. Blumenhagen, D. Lüst and S. Theisen):

1. in the R-R spectrum of Type IIA string I have a one form $C_1$ and a three form $C_3$;
2. in the R-R spectrum of Type IIB string I have a zero form $C_0$, a two form $C_2$ and a four (anti)self-dual form $C_4$. So we can say there exist $C_p$ with $p=0,1,2,3,4$ somewhere.
3. To these forms I can associate a field strength via the usual $F_{n+1}=\mathrm{d}C_n$. Of course I will have $F_5 = *F_5$ where the asterisk is the Hodge dual.

The problem is that going ahead in the study (in the same book but in general wherever there are branes) I found $C_p$ with any $p=0,1,2,3,4,5,6,7,8,9$.

1. So, how are these $C_p$ with $p>4$ defined? Simply by some duality with respect of the others?
2. Consequently and consistently how are their field strength defined? With some Hodge dual?
3. For instance $F_3=\mathrm{d}C_2$, then I can define $F_7=*F_3$, knowing that $\mathrm{d}F_3=0$ how can I say that $\mathrm{d}*F_3=\mathrm{d}F_7=0$?
4. Once I got $\mathrm{d}F_7=0$, how can I say $F_7=\mathrm{d}C_6$ globally? Why have I assumed the cohomology to be trivial? or equivalently: is this valid only in flat space?
• Maybe the solution is that $C_p$ and $C_{8-p}$ have the same group content, so I can decide which include in the spectrum. When I construct vertex operator, the only thing that I have are spin filed which I combine with gammas to give the R-R field I want.
– MaPo
Apr 15 '16 at 13:45

For Type II superstrings, the Ramond-Ramond fields comes from bi-spinors formed by the two supersymmetries indices. In the RNS formulation for example, they are described by the vertex operator $$(\Theta F\tilde\Theta)$$, where $$\Theta,\tilde\Theta$$ are the Spin Field for the left and right movers. It is a known fact that the spin fields are the supercharges of the target space, up to picture number changing.

In Type IIA we have two supersymmetries with opposite chirality $$Q^{\alpha}$$ and $$Q_{\hat{\beta}}$$, so the Ramond-Ramond fields are described by the bi-spinor

$$(F)_{\hat{\beta}}\,^{\alpha}=F\,\delta_{\hat{\beta}}\,^{\alpha}+ F_{mn}\,(\gamma^{mn})_{\hat{\beta}}\,^{\alpha}+F_{mnpq}(\gamma^{mnpq})_{\hat{\beta}}\,^{\alpha}$$

while for Type IIB we have supersymmetries of the same chirality $$Q_{\alpha}$$ and $$Q_{\hat{\beta}}$$, so the Ramond-Ramond fields are described by

$$(F)_{\alpha\hat{\beta}}=F_{m}\,(\gamma^{m})_{\alpha\hat{\beta}}+F_{mnp}(\gamma^{mnp})_{\alpha\hat{\beta}} + F_{mnpqr}(\gamma^{mnpqr})_{\alpha\hat{\beta}}$$

The linearized equations of motion that you get from requiring that the RR vertex operators are representatives of the BRST cohomology are Dirac equations for the spinorial indices of the bi-spinor $$F$$ that made up $$V_{RR}=(\Theta F\tilde\Theta)$$

$$(\gamma^{m})\,_{\alpha\beta}\,\partial_{m}(F)_{\hat{\gamma}}\,^{\beta}=0$$

$$(\gamma^{m})\,^{\hat\alpha\hat\beta}\,\partial_{m}(F)_{\hat{\beta}}\,^{\alpha}=0$$

$$(\gamma^{m})\,^{\alpha\beta}\,\partial_{m}(F)_{\beta\hat\gamma}=0$$

$$(\gamma^{m})\,^{\hat\alpha\hat\beta}\,\partial_{m}(F)_{\gamma\hat\beta}=0$$

These equations imply for all the $$p$$-forms the equations $$dF=d*F=0$$. Note that there is no gauge potential, only the field strengths (also, no gauge invariance coming from $$V\cong V + Q \chi$$). This is so because the strings do not source RR fields. When you wrote the $$C$$'s (the gauge potentials), you made a choice of viewing $$dF=0$$ as a constraint and $$d*F=0$$ as an equation of motion, but the theory is symmetrical about this choice - it is possible to instead view $$d*F=0$$ as a constraint and $$dF$$ as an equation of motion. The only way to break this symmetry is to introduce a source for the RR fields, and there are no sources for RR fields in absence of branes.

When you introduced the $$C$$'s you made a particular choice for describing this theory. This choice may or may not be convenient depending on the possible sources that you add to the system. What is really fundamental are the $$F$$'s, not the $$C$$'s. See here to learn more about the role of gauge potentials in gauge theories.