Type II string theory

Type II superstring theory is a ten-dimensional closed oriented superstring theory with spacetime supersymmetry and no tachyon. Its massive closed-string levels have squared masses separated by $4/{\alpha }^{\prime }$.

A closed string sector with left-moving Ramond sector excitations and right-moving Neveu–Schwarz sector excitations. Its spacetime states are fermions. The closed-string level matching condition is $N_L=N_R-1/2$. After the supersymmetric GSO projection, the lowest states have $(N_L,N_R)=(0,1/2)$ and give the other massless gravitino and dilatino of type II string theory.

A closed string sector with left-moving Neveu–Schwarz sector excitations and right-moving Ramond sector excitations. Its spacetime states are fermions. The string intercepts differ, so the pair of oscillator vacua is not level matched: $N_L-1/2=N_R$. After the supersymmetric GSO projection, the lowest allowed states have $(N_L,N_R)=(1/2,0)$ and are a massless gravitino and dilatino.

The closed-string sector with Ramond sector states on both chiral sides. The left and right spinor tensor product describes differential-form gauge fields. Opposite chiralities in type IIA superstring theory give the massless one-form and three-form, with transverse dimensions $8$ and $56$. Same chiralities instead give the type IIB form-field content.

A differential-form gauge potential from the Ramond–Ramond sector. A D-brane carries electric or magnetic charge under the appropriate potential. Circle reduction of the M-theory metric gives the IIA one-form potential, while its three-form gives the IIA three-form potential.

The closed-string sector with NS sector excitations on both chiral sides. Its massless transverse tensor product is $8_v\otimes 8_v=35_v+28+1$, giving the graviton, Kalb–Ramond field and dilaton. These fields are common to the type II theories.

The antisymmetric two-form gauge potential couples electrically to a fundamental string. Its field strength is locally $H_3=d{B}_2$; its magnetic source is an NS5-brane.

A closed superstring theory whose left and right Ramond projections have the same ten-dimensional chirality. It has 32 real supercharges and is chiral. Its low-energy fields form type IIB supergravity. Circle T-duality exchanges this theory with type IIA superstring theory.

The NS-NS fields are a graviton, Kalb–Ramond field and dilaton, with $35+28+1$ physical polarizations. The RR fields are $C_0,C_2,C_4$, with $1+28+35$ polarizations; the five-form field strength of $C_4$ is self-dual. Two same-chirality Majorana-Weyl gravitini contribute $112$ polarizations, and two Majorana-Weyl dilatini of the opposite chirality contribute $16$. Potential components, dual potentials and physical polarizations are different counts.

The IIB five-brane doublet combines NS5-brane and Ramond–Ramond five-brane charges. Its toroidal M-description involves wrapped M5-branes or Kaluza-Klein monopoles, depending on whether the IIB circle lies along or transverse to the five-brane. Electric–magnetic charge bases fix the order and signs of the integer labels.

A (p,q) string carries $p$ units of fundamental string charge and $q$ units of D1-brane charge. Coprime integers label an elementary primitive charge vector; nonprimitive vectors describe multiple units. The displayed tension uses the string metric and vanishing axion. In the M-theory torus duality, it comes from an M2-brane wrapped on the corresponding primitive one-cycle.

At a junction of (p,q) strings, their oriented charge vectors sum to zero. A static junction also balances the tension vectors. Supersymmetry further correlates spatial directions with charge phases so that the arms share kappa symmetry projectors. Additional independent conditions reduce the common Killing spinor space; an aligned dyonic string alone does not reduce its fraction relative to an aligned fundamental string.

// Target: geometry-and-topology.bigb

The complex scalar combines the IIB axion and dilaton. Its imaginary part is $1/g_s$ for a constant background. S-duality acts by fractional linear transformations, corresponding to changes of torus basis in the M-theory torus duality.

A closed superstring theory whose left and right Ramond sector projections have opposite ten-dimensional Majorana-Weyl chiralities. It has 32 real spacetime supercharges and is nonchiral. Its massless fields form type IIA supergravity; the RR gauge potentials have odd degrees, including a one-form and a three-form.

Level matching combines two chiral massive sectors, each with 128 NS and 128 R states. The NS-NS and R-R combinations give $2(128{)}^2=32768$ bosons, and the mixed sectors give the same fermion count. Their total $2^{16}$ matches a massive uncharged long multiplet with 32 real supercharges and sixteen fermionic creation operators.

The bosons are the metric, two-form, dilaton, RR one-form and RR three-form, with physical dimensions $35+28+1+8+56=128$. Two opposite-chirality gravitinos and two dilatinos supply $2(56+8)=128$ fermionic degrees of freedom. They form the ten-dimensional type IIA supergravity multiplet. Counts refer to on-shell polarizations, not raw field components.

The four-point tree amplitude exposes poles at $M_n^2=4n/{\alpha }^{\prime }$ for $n=0,1,2,\dots$. The $n=0$ level is massless, and positive $n$ form an infinite tower of massive string excitations.

At generic fixed $t$, the factor $\Gamma (-{\alpha }^{\prime }s/4)$ has simple poles at the displayed nonnegative values of $s$. Its denominator $\Gamma (1+{\alpha }^{\prime }s/4)$ is finite there. The $t,u$ factors do not generically cancel those poles under $s+t+u=0$. Factorization identifies massless and massive intermediate states at those masses. The factor has no negative-mass-squared poles; special polarization or momentum choices can suppress individual residues and cannot establish the degeneracy of an entire level.

After the GSO projection, the first positive shifted level is one: $N_{\mathrm{NS}}=3/2$ and $N_R=1$. In a closed-string mass convention it corresponds to $M^2=4/{\alpha }^{\prime }$. The transverse oscillator basis has 128 NS states and 128 R states. They form the bosonic and fermionic pieces of a massive supersymmetry multiplet.

At the first massive chiral level, 128 NS bosons and 128 R fermions furnish a massive multiplet with 16 real supercharges and no central charge. Its rest-frame supersymmetry algebra has eight fermionic creation operators, producing $2^8=256$ states. The massive little-group representations are $44+84$ for the bosons and a 128-dimensional vector-spinor for the fermions.

At R level one, the families are ${\alpha }_{-1}^i|s⟩$ and $d_{-1}^i|s^{\prime }⟩$, each of dimension $8\cdot 8$. The zero-mode spinors have opposite chirality because the one fermionic oscillator reverses the oscillator contribution to the GSO projection. The resulting 128 states form a gamma-traceless Spin(9) vector-spinor, of dimension $9\cdot 16-16$.

At NS level $3/2$, the light-cone families are $b_{-3/2}^i|0⟩$, ${\alpha }_{-1}^i{b}_{-1/2}^j|0⟩$, and three distinct $b_{-1/2}$ oscillators. Their dimensions are $8+64+\left(\genfrac{}{}{0ex}{}{8}{3}\right)=128$. All have odd fermion-excitation parity. They assemble into the massive Spin group representations $44+84$, a symmetric traceless tensor and a three-form of Spin(9).