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String Theory
Prior to the First Superstring Revolution
Early History S-Matrix Theory
Regge Trajectory
Bosonic String Theory Worldsheet
String
Bosonic String Theory
String Perturbation Theory
Tachyon Condensation
Supersymmetric Revolution Supersymmetry
RNS Formalism
GS Formalism
BPS
Superstring Revolutions
First Superstring Revolution GSO Projection
Type II String Theory
Type IIB String Theory
Type IIA String Theory
Type I String Theory
Type H String Theory
Type HO String Theory
Type HE String Theory
Second Superstring Revolution T-Duality
D-Brane
S-Duality
Horava-Witten String Theory
M-Theory
Holographic Principle
N=4 Super-Yang-Mills Theory
BFSS Matrix Theory
Matrix String Theory
(2,0) Theory
Twistor String Theory
F-Theory
String Field Theory
Pure Spinor Formalism
After the Revolutions
Phenomenology String Theory Landscape
Minimal Supersymmetric Standard Model
String Phenomenology

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Type IIA String Theory is a consistent 10-dimensional Superstring Theory. It has a Supersymmetry of $\mathcal N = 2$ (32 supercharges), or more specifically, $\mathcal N = 2A$, in string - theoretical terms, as opposed to Supergravity terms. It is GSO Truncated, so it does not have a Tachyon in its ground state. It is distinguished from the Type IIB String Theory by its opposite Ramond Sector GSO Projection on both left- and right- moving components of the state. This makes the theory chiral.[1]

Action PrincipleEdit this section

The Lagrangian Density (across the worldsheet) of the Type IIB String Theory is the RNS Lagrangian Density: [1]

$\mathcal{L_{RNS}} = {\frac{T }{2}}h^{ab} \left(\partial_a X^\mu \partial_b X^\nu - i\hbar c_0\bar{\psi_\mu }\not\partial\psi^\mu \right)g_{\mu \nu}$

Consequently, the action is:[1]

$S_{RNS} = \iint\left( \mathcal{L_{RNS}} = {\frac{T }{2}}h^{ab} \left(\partial_a X^\mu \partial_b X^\nu - i\hbar c_0\bar{\psi_\mu }\not\partial\psi^\mu \right)g_{\mu \nu} \sqrt{-\det h_{\alpha\beta}}\right)\mbox{d}^2\xi$

GSO ProjectionEdit this section

The GSO Projection in the Neveu-Schwarz Sector is obviously the Standard Neveu-Schwarz GSO Projection:[2]

$\mathsf{\mathcal{P}}_{\operatorname{NS}}^{-}\left| \psi \right\rangle =\left( 1-{{\left( -1 \right)}^{F}} \right)\left| \psi \right\rangle$

However, the GSO Projection in the Ramond Sector is more complicated. We may either use the following pair of projections:[2]

$\begin{matrix} \mathsf{\mathcal{P}}_{\operatorname{R}}^{-}\left| {{\psi }_{-}} \right\rangle =\left( 1-\left( \prod\limits_{k=1}^{10}{{{\gamma }_{k}}} \right){{\left( -1 \right)}^{F}} \right)\left| {{\psi }_{-}} \right\rangle \\ \mathsf{\mathcal{P}}_{\operatorname{R}}^{+}\left| {{\psi }_{+}} \right\rangle =\left( 1+\left( \prod\limits_{k=1}^{10}{{{\gamma }_{k}}} \right){{\left( -1 \right)}^{F}} \right)\left| {{\psi }_{-}} \right\rangle \\ \end{matrix}$

Or the following pair:[2]

$\begin{matrix} \mathsf{\mathcal{P}}_{\operatorname{R}}^{+}\left| {{\psi }_{-}} \right\rangle =\left( 1+\left( \prod\limits_{k=1}^{10}{{{\gamma }_{k}}} \right){{\left( -1 \right)}^{F}} \right)\left| {{\psi }_{-}} \right\rangle \\ \mathsf{\mathcal{P}}_{\operatorname{R}}^{-}\left| {{\psi }_{+}} \right\rangle =\left( 1-\left( \prod\limits_{k=1}^{10}{{{\gamma }_{k}}} \right){{\left( -1 \right)}^{F}} \right)\left| {{\psi }_{-}} \right\rangle \\ \end{matrix}$

I.e. both the left-moving components and the right-moving components must be GSO Truncated in the opposite way.[2] This is what makes the theory non-chiral.[3]

Cancellation of Conformal AnomalyEdit this section

Since the Type IIA String Theory is only a GSO Truncated version of the naive RNS String Theory, its critical dimension is 10 and its normal ordering constant is 0 in the Ramond Sector and $\frac12$ in order for cancellation of conformal anomaly to occur.

Gauge Group and Unsuitabilitiy as a Theory of EverythingEdit this section

The Type IIA String Theory is chiral, and consequently has a gauge group of $U(1)$. This is fine as a theory of quantum gravity, because a gauge group isn't required to describe Gravity (in fact, it can be checked, that since it is a GSO Truncated version of the naive RNS String Theory, getting rid of Supersymmetry from the RNS Formalism results in the Bosonic String Theory, and when its beta function is set to 0 to preserve Conformal invariance, one obtains the Einstein Field Equations, and thus General Relativity at the low-energy-limit) and as a theory of electromagnetism, which is supposed to have a gauge group of $U(1)$. However, this would describe the other interactions, such as the strong force and the weak force incorrectly, as the experimentally verified Standard Model requires them to have gauge groups of $SU(3)$ and $SU(2)$ respectively.[1]

Therefore, the Type IIA String Theory, even when compactified on a Calabi-Yau Manifold, cannot be a Theory of Everything. This initiated the main part of the First Superstring Revolution and the Heterotic String Theories. The Type HE String Theory satisfies this requirement.[1]

T-duality with the Type IIB string theoryEdit this section

Prior to the Second Superstring Revolution, the only known relationship between the Type IIB String Theory and the Type IIA String Theoryies was that both of them were GSO Truncated versions of the naive RNS String Theory, by applying the same and opposite GSO Projections to the left- and right- moving components of the state respectively. However, this relationship, was not a duality or an equivalence, and thus, it could not be used to derive one theory from the other.

During the start of the Second Superstring Revolution, it was realized that Type IIA String Theory is related to Type IIB String Theory by T-Duality.[4]

The reason is that it is clear that T-Duality negates the sign of the massless bosonic directional dimensional field $X^\mu$, much as how S-Duality negates the sign of the Dilaton field $\Phi$. Thus, by Worldsheet Supersymmetry, it should also negate the massless fermionic directional dimensional field $\mathbf{\psi}^\mu$ and therefore the Fermionic operator - valued modes of oscillation of the string $\hat d_{\mu}$ are Wick-rotated. So, therefore, the fermion number operator $F$ is also negated, and therefore, thus, the Klein Operator is also negated, and hence, thus, the GSO Projection $\mathcal P_{\operatorname{GSO}}$ transforms from the same on the left-moving component $|\psi_-\rangle$ and the right-moving component $|\psi_+\rangle$ to the opposite on the left- and right- components. This clearly transforms between the Type IIA String Theory and the Type IIB String Theory, unifying them into a single Type II String Theory.

S-DualityEdit this section

Here, we will use the uncommon word "Sen String Theory" to refer to the S-Dual (in this case, the strong-coupling limit) of the Type IIB. We know that the S-Dual of the Type IIB String Theory is the Type IIB String Theory with F1 strings replaced by D1 branes. [5] Then, the S-Dual of the Type IIA String Theory is the S-Duality of the T-Dual of Type IIB String Theory, which is the T-Duality of the "Sen String Theory", is the Type IIA String Theory with D1 Branes instead of F1 Strings. [5]

GS FormalismEdit this section

In the GS Formalism, Type IIA String Theory, being a Type II String Theory, has $\mathcal N=2$ Supersymmetry, which means that the spinor $\Theta^A$ has two components.

Furthermore, it is uniquely identified by the projection:

${{\gamma }_{11}}{{\Theta }^{A}}={{\left( -1 \right)}^{A+1}}{{\Theta }^{A}}$

Which makes it chiral.

ReferencesEdit this section

1. 1.0 1.1 1.2 1.3 1.4 Mohaupt, Thomas. Introduction to String theory.
2. 2.0 2.1 2.2 2.3
3. McMohan, David (2009). String theory DeMystified. Chicago: McGrawHill. ISBN 978-0071498708.
4. 5.0 5.1 Polchinski, Joseph (1998). String theory Vol. 2: Superstring theory and Beyond. Cambridge: Cambridge University Press. p. 181. ISBN 971-0-521-63303-6.