= Solution
A $(p+1)$-form <gauge potential> $A_{p+1}$ couples electrically to a spatial $p$-brane through the <Wess-Zumino brane coupling>
$$
S_{\rm charge}=e\int_{\Sigma_{p+1}}X^*A_{p+1}.
$$
The <worldvolume> dimension supplies exactly the degree required for integration. Under $A\mapsto A+d\Lambda$, the change is $e\int_{\partial\Sigma}X^*\Lambda$, so a closed worldvolume is gauge invariant; an open one needs suitable endpoint couplings. If $F_{p+2}=dA_{p+1}$ has an ordinary dual potential, $*F$ has degree $d-p-2$ and the dual potential degree $d-p-3$. Thus the <electric and magnetic dimensions of a form-coupled brane> are
$$
\boxed{p_{\rm electric}=p,\qquad p_{\rm magnetic}=d-p-4.}
$$
Electric charge is measured by $\int_{S^{d-p-2}}*F$, magnetic charge by $\int_{S^{p+2}}F$, on spheres linking the corresponding objects. Flux normalization, and possible Chern-Simons modifications of charge definitions, must be fixed in interacting backgrounds; this is the basic linear form-field dictionary.
In ten dimensions the common NS-NS fields are the <graviton> $g_{\mu\nu}$, <Kalb–Ramond field> $B_2$, and <dilaton> $\Phi$. Their on-shell bosonic polarizations are $35,28,1$, respectively. The graviton count is $10(10-3)/2=35$, and a massless $k$-form has $\binom8k$ transverse polarizations. The RR fields distinguish the two theories:
|| Theory
|| RR potentials and physical polarizations
|| Total bosons
|| Fermions and chiralities
| <Type IIA superstring theory>
| $C_1:8$, $C_3:56$
| $35+28+1+8+56=128$
| Two opposite-chirality Majorana-Weyl <gravitini>; two opposite-chirality Majorana-Weyl <dilatini>
| <Type IIB superstring theory>
| $C_0:1$, $C_2:28$, $C_4:35$ with self-dual $F_5$
| $35+28+1+1+28+35=128$
| Two same-chirality Majorana-Weyl <gravitini>; two same-chirality Majorana-Weyl <dilatini> of the opposite chirality to the gravitini
Each gravitino has $56$ physical polarizations, and each dilatino $8$, giving $128$ fermions in either theory. In IIA the two supersymmetry sectors have opposite ten-dimensional chiralities, and the dilatino in each sector has chirality opposite to its gravitino. In IIB both gravitini can be designated positive-chirality and both dilatini negative-chirality. The self-duality of the IIB five-form halves the unconstrained four-form count $\binom84=70$ to $35$. These are the <massless type IIA spectrum> and <massless type IIB spectrum>, not raw component counts.
The <Neveu–Schwarz two-form> couples electrically to the <fundamental string> and magnetically to the <NS5-brane> in both theories. IIA $C_1$ couples electrically to a <D0-brane> and magnetically to a <D6-brane>; $C_3$ couples electrically to a <D2-brane> and magnetically to a <D4-brane>. The magnetic descriptions use dual seven-form and five-form potentials, not extra independent polarizations. In IIB, $C_2$ gives <D1-brane> and <D5-brane> charges, while self-dual $C_4$ gives the self-dual <D3-brane>. The scalar $C_0$ couples to a Euclidean <D-instanton>, whose magnetic partner is the <D7-brane>. Momentum and, after circle compactification, <Kaluza-Klein monopole> charge come from the metric sector.
The full brane inventory also involves nonpropagating top-form potentials. A <D8-brane> is an IIA domain-wall source for <Romans mass> $F_0$, described dually by a nine-form potential; it is not generated by the propagating $C_1,C_3$ massless spectrum alone. A space-filling IIB <D9-brane> has a ten-form coupling and requires appropriate global RR-charge consistency. Such fields add no ordinary massless polarizations. This distinction avoids confusing the list of propagating particles with all possible charged extended objects.
For <T-duality>, compactify a spatial coordinate with $X^9\sim X^9+2\pi R$. Closed strings have integer momentum $n$ and winding $w$, with
$$
p_L=\frac nR+\frac{wR}{\alpha'},\qquad p_R=\frac nR-\frac{wR}{\alpha'}.
$$
The map
$$
\boxed{R'=\frac{\alpha'}R,\qquad n'=w,\qquad w'=n}
$$
leaves $p_L$ unchanged and reverses $p_R$. Equivalently, it reverses $X_R^9$ while keeping $X_L^9$, and reverses the corresponding right-moving worldsheet fermion to preserve worldsheet supersymmetry. Multiplication of the right Ramond ground-state spinor by $\Gamma^9$ reverses its ten-dimensional chirality. Left and right Ramond projections therefore change from opposite chirality to equal chirality, or conversely:
$$
\boxed{\text{IIA on }S_R^1\quad\longleftrightarrow\quad\text{IIB on }S_{\alpha'/R}^1.}
$$
The zero-mode mass contribution $(n/R)^2+(wR/\alpha')^2$ and the oscillator spectrum are unchanged. The <Buscher dilaton shift> gives $g_s'=g_s\sqrt{\alpha'}/R$, which preserves the nine-dimensional gravitational coupling proportional to $g_s^2/R$. The mixed metric and two-form components along the circle are exchanged; RR potentials with a circle leg map to one lower degree and those without it to one higher degree, in the corresponding gauge-covariant description.
For open strings, the derivative relations $\partial_\tau\widetilde X=\partial_\sigma X$ and $\partial_\sigma\widetilde X=\partial_\tau X$ exchange <Neumann boundary conditions> with <Dirichlet boundary conditions>. Consequently a wrapped D$p$-brane becomes a D$(p-1)$-brane, while a transverse one becomes a wrapped D$(p+1)$-brane. The even-dimensional IIA branes and odd-dimensional IIB branes therefore transform consistently with the closed-string chirality flip; transverse positions and compact gauge <Wilson lines> are exchanged.
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