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A $(p+1)$-form gauge potential $A_{p+1}$ couples electrically to a spatial $p$-brane through the Wess-Zumino brane couplingThe worldvolume dimension supplies exactly the degree required for integration. Under $A\mapsto A+d\Lambda$, the change is $e{\int }_{\partial \Sigma }{X}^{\ast }\Lambda$, so a closed worldvolume is gauge invariant; an open one needs suitable endpoint couplings. If $F_{p+2}=d{A}_{p+1}$ has an ordinary dual potential, $\ast F$ has degree $d-p-2$ and the dual potential degree $d-p-3$. Thus the electric and magnetic dimensions of a form-coupled brane areElectric charge is measured by ${\int }_{S^{d-p-2}}\ast F$, 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 $\left(\genfrac{}{}{0ex}{}{8}{k}\right)$ transverse polarizations. The RR fields distinguish the two theories:

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 $\left(\genfrac{}{}{0ex}{}{8}{4}\right)=70$ 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 $w$, withThe mapleaves $p_L$ 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:The zero-mode mass contribution $(n/R{)}^2+(wR/{\alpha }^{\prime }{)}^2$ and the oscillator spectrum are unchanged. The Buscher dilaton shift gives $g_s^{\prime }=g_s\sqrt{{\alpha }^{\prime }}/R$, 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 ${\partial }_{\tau }\tilde{X}={\partial }_{\sigma }X$ and ${\partial }_{\sigma }\tilde{X}={\partial }_{\tau }X$ 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.

The fields constitute an RNS string coupled to two-dimensional worldsheet supersymmetry. The zweibein and worldsheet gravitino impose constraints; they do not supply extra propagating string polarizations. Worldsheet diffeomorphisms, local frame rotations and Weyl transformations put the metric locally in conformal gauge, $h_{\mu \nu }=e^{2\omega }{\eta }_{\mu \nu }$. Local supersymmetry together with super-Weyl symmetry removes the worldsheet gravitino, giving superconformal gauge ${\chi }_{\mu }=0$. On a general closed worldsheet, moduli and spin structures remain and must still be summed or integrated over; gauge fixing is not a declaration that every surface is globally a flat cylinder.

Variation with respect to the metric and worldsheet gravitino before gauge fixing gives the stress-tensor and supercurrent constraints $T_{++}=T_{--}=0$ and $G_{+}=G_{-}=0$. The remaining matter consists of free $X^a$ and their worldsheet fermions. In covariant quantization, these become the N=1 super-Virasoro algebra physical-state conditions: positive Virasoro and supercurrent modes annihilate a physical state, $(L_0-a)|\mathrm{phys}⟩=0$, and the R sector also has the zero-mode condition $F_0|\mathrm{phys}⟩=0$. Null gauge states are quotiented out. Equivalently, physical states are BRST cohomology classes, so BRST-exact operators do not represent additional physical states.

The diffeomorphism $bc$ ghosts have central charge $-26$, while the bosonic superconformal ghosts $(\beta ,\gamma )$ have charge $11$. Matter contributes $D+D/2$. Therefore quantum gauge consistency requiresThis is the critical dimension of the RNS superstring. In light-cone gauge in string theory, choose nonzero $p^{+}$, make $X^{+}$ linear in worldsheet time and set ${\psi }^{+}=0$. The stress-tensor and supercurrent constraints solve for the longitudinal $X^{-}$ and ${\psi }^{-}$ oscillators. Only eight transverse bosons and eight transverse fermions remain, with positive norm. This explicitly eliminates the time-like and longitudinal unphysical states.

In a chiral sector the fermions have half-integral modes in the NS sector and integral modes in the R sector. The normal-ordering constant of a string is $a_{\mathrm{NS}}=1/2$ and $a_R=0$. The GSO projection retains odd fermion-excitation parity in the NS sector, removing its tachyonic vacuum. In the R sector it keeps one chirality of the zero-mode spinor, with oscillator parity included in the projection. The Ramond zero-mode Clifford algebra then leaves eight ground-state polarizations in light-cone gauge in string theory.

Worldsheet supersymmetry alone does not imply spacetime supersymmetry. The GSO projection makes the spinorial worldsheet currents mutually local with the retained vertex operators and pairs their NS and R states. More explicitly, let $S_A$ be an RNS spin field and ${\varphi }_{\mathrm{gh}}$ the bosonized superghost scalar. The spacetime supercharges are generated byand similarly in the other chiral sector. The spin field has conformal weight $10/16=5/8$, and its superghost factor has weight $3/8$, so the current has weight one. Its operator products generate the spacetime translation operator; schematically $\{Q_A,Q_B\}\propto (C{\Gamma }^a{)}_{AB}{P}_a$. This is the spacetime supercharge from an RNS spin field. For type IIA superstring theory, the left and right Ramond projections select opposite ten-dimensional Majorana-Weyl chiralities, giving 32 real supercharges and a nonchiral spacetime theory.

The massless physical states are best counted with the transverse little group $\mathrm{Spin}(8)$. Choose the left Ramond ground representation $8_s$ and the right one $8_c$. ThenThe NS-NS sector supplies the graviton, Kalb–Ramond field and dilaton, with $35$, $28$ and $1$ polarizations. The RR sector supplies a one-form potential and a three-form potential, with $\left(\genfrac{}{}{0ex}{}{8}{1}\right)=8$ and $\left(\genfrac{}{}{0ex}{}{8}{3}\right)=56$ polarizations. The mixed sectors supply two gravitinos of opposite chirality, each with $56$ polarizations, and two dilatinos, each with $8$. Covariantly the dilatino has chirality opposite to its corresponding supersymmetry parameter; the two chirality sets are both present. Thus the massless type IIA spectrum hasThese fields form the massless type IIA supergravity multiplet.

For the first massive level of a chiral RNS sector, restore ${\alpha }^{\prime }$ throughThe first positive value is one. The NS states therefore have level $N_{\mathrm{NS}}=3/2$. A complete transverse basis after GSO projection isAll contain odd fermion-excitation parity. Their respective dimensions are $8$, $8^2=64$ and $\left(\genfrac{}{}{0ex}{}{8}{3}\right)=56$, so the NS count is $128$. These are the first massive GSO-projected Neveu–Schwarz states. The middle family decomposes into $35_v+28+1$. Together the families assemble into the massive $\mathrm{Spin}(9)$ representations $44+84$: a symmetric traceless rank-two tensor and a three-form, since $44\downarrow \mathrm{Spin}(8)=35_v+8_v+1$ and $84\downarrow \mathrm{Spin}(8)=56_v+28$.

In the R sector, the first massive level is $N_R=1$. If the retained right-moving ground spinor is $8_c$, its two families areThe Ramond spinors have opposite zero-mode chiralities in these two families: inserting one fermionic oscillator reverses the oscillator contribution to the GSO condition, so the second family must use the opposite zero-mode chirality. Each family has $8\times 8=64$ states. HenceThese first massive GSO-projected Ramond states form the gamma-traceless vector-spinor representation of $\mathrm{Spin}(9)$, of dimension $9\cdot 16-16=128$. Its transverse branching is $8_s+56_s+8_c+56_c$. The right-moving sector therefore contains $128$ bosons and $128$ fermions, a massive chiral superstring supersymmetry multiplet.

For the closed string, closed-string level matching requires the left and right shifted levels to agree. Both chiral sectors at this first massive mass have $128$ NS states and $128$ R states. The bosons lie in the NS-NS sector and RR sector, while the mixed sectors are fermionic. ConsequentlyTheir total is $65536=2^{16}$. A massive ten-dimensional state with 32 real supercharges and no central charges has sixteen fermionic creation operators in its rest-frame supersymmetry algebra, giving a long multiplet of this size. Thus the chiral equality, the four closed sectors and level matching are consistent with the first massive type IIA long supermultiplet, rather than merely matching an isolated number of right-moving states.