Type II string theory

Type IIA string theory

At low energies, type IIA string theory is described by type IIA supergravity in ten dimensions which is a non-chiral theory (i.e. left–right symmetric) with (1,1) d=10 supersymmetry; the fact that the anomalies in this theory cancel is therefore trivial.

In the 1990s it was realized by Edward Witten (building on previous insights by Michael Duff, Paul Townsend, and others) that the limit of type IIA string theory in which the string coupling goes to infinity becomes a new 11-dimensional theory called M-theory. Consequently the low energy type IIA supergravity theory can also be derived from the unique maximal supergravity theory in 11 dimensions (low energy version of M-theory) via a dimensional reduction.

The content of the massless sector of the theory (which is relevant in the low energy limit) is given by $(8_v\oplus 8_s)\otimes (8_v\oplus 8_c)$ representation of SO(8) where ${\displaystyle 8_v}$ is the irreducible vector representation, ${\displaystyle 8_c}$ and ${\displaystyle 8_s}$ are the irreducible representations with odd and even eigenvalues of the fermionic parity operator often called co-spinor and spinor representations. These three representations enjoy a triality symmetry which is evident from its Dynkin diagram. The four sectors of the massless spectrum after GSO projection and decomposition into irreducible representations are

${\displaystyle \text{NS-NS}:8_v\otimes 8_v=1\oplus 28\oplus 35=\mathrm{\Phi }\oplus B_{\mu \nu }\oplus G_{\mu \nu }}$

${\displaystyle \text{NS-R}:8_v\otimes 8_c=8_s\oplus {56}_c={\lambda }^{+}\oplus {\psi }_m^{-}}$

${\displaystyle \text{R-NS}:8_c\otimes 8_s=8_s\oplus {56}_s={\lambda }^{-}\oplus {\psi }_m^{+}}$

${\displaystyle \text{R-R}:8_s\otimes 8_c=8_v\oplus {56}_t=C_n\oplus C_{nmp}}$

where ${\displaystyle \text{R}}$ and ${\displaystyle \text{NS}}$ stands for Ramond and Neveu–Schwarz sectors respectively. The numbers denote the dimension of the irreducible representation and equivalently the number of components of the corresponding fields. The various massless fields obtained are the graviton ${\displaystyle G_{\mu \nu }}$ with two superpartner gravitinos ${\displaystyle {\psi }_m^{\pm }}$ which gives rise to local spacetime supersymmetry, a scalar dilaton ${\displaystyle \mathrm{\Phi }}$ with two superpartner spinors—the dilatinos ${\displaystyle {\lambda }^{\pm }}$, a 2-form spin-2 gauge field ${\displaystyle B_{\mu \nu }}$ often called the Kalb–Ramond field, a 1-form ${\displaystyle C_n}$ and a 3-form ${\displaystyle C_{nmp}}$. Since the ${\displaystyle \text{p}}$-form gauge fields naturally couple to extended objects with ${\displaystyle \text{p+1}}$ dimensional world-volume, Type IIA string theory naturally incorporates various extended objects like D0, D2, D4 and D6 branes (using Hodge duality) among the D-branes (which are ${\displaystyle \text{R}}$${\displaystyle \text{R}}$ charged) and F1 string and NS5 brane among other objects.

The mathematical treatment of type IIA string theory belongs to symplectic topology and algebraic geometry, particularly Gromov–Witten invariants.