Type IIB String Theory

String Theory
All Roads Lead to String Theory (Polchinski)
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 AdS CFT 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 IIB String Theory is a consistent 10-dimensional Superstring Theory. It has a Supersymmetry of $\mathcal N=2$ (32 supercharges), or more specifically, $\mathcal N = 2B$, 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 IIA String Theory by its same Ramond Sector GSO Projection on both left- and right- moving components of the state. This makes the theory non-chiral.

Action Principle

The Lagrangian Density (across the Worldsheet) of the Type IIB String Theory is the RNS Lagrangian Density:

$\mathcal{L_{RNS}} = {\frac{T }{2c_0}}h^{ab} \left(\partial_a X^\mu \partial_b X^\nu-2\psi_+^\mu\left(\frac{\partial \psi_-^\mu}{\partial \sigma}-\frac{\partial\psi_-^\mu}{\partial \tau}\right)\right)g_{\mu \nu}$

Consequently, the action is:

$S_{RNS} = \iint\left({\frac{T }{2c_0}}h^{ab} \left(\partial_a X^\mu \partial_b X^\nu-2\psi_+^\mu\left(\frac{\partial \psi_-^\mu}{\partial \sigma}-\frac{\partial\psi_-^\mu}{\partial \tau}\right)\right)g_{\mu \nu}\sqrt{-\det h_{\alpha\beta}}\right)\mbox{d}^2\xi$

GSO Projection

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

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

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

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

Or the following pair:

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

I.e. both the left-moving components and the right-moving components must be GSO truncated in the same way. This is what makes the theory non-chiral.

Cancellation of Conformal Anomaly

Since the Type IIB String Theory is only a GSO Truncated version of the naive RNS String Theory, its normal ordering constant is $a = 0$ in the Ramond Sector and $a = \frac12$ and its critical dimension is $D = 10, c = 15$ in order for cancellation of conformal anomaly to occur.

Gauge Group and Unsuitabilitiy as a Theory of Everything

The Type IIB String Theory is non-chiral, and consequently has no gauge group (unless the identity group is counted). 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 naive RNS String Theory, getting rid of Supersymmetry from the naive RNS String Theory 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). However, this would describe the other interactions, such as the strong force, the weak force and the Electromagnetic force incorrectly, as the experimentally verified Standard Model requires them to have gauge groups of $SU(3)$, $SU(2)$ and $U(1)$ respectively.

Therefore, the Type IIB 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 Type H String Theoryies. The Type HE string theory satisfies this requirement.

T-duality with the Type IIA string theory

Prior to the Second Superstring Revolution, the only known relationship between the Type IIB String Theory and the Type IIA String Theory 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.

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-duality with the "Sen string theory"

It is generally very difficult to find the strong-coupling limit of any theory, if it is in the peturbative regime, because this strong-coupling limit would obviously diverge. However, if one anaanalyses the massless fields of Type IIB String Theory, one sees that the fields $g_{\mu\nu}, F_{\mu\nu}, \Phi, A_\mu$, at strong coupling become unchanged, except with the F1 Strings transforming into the D1 Branes. Thus, the S-dual of the Type IIB String Theory is the Type IIB String Theory with the F1 Strings replaced with the D1 branes, which is occasionally called the Sen String Theory.

GS Formalism

In the GS Formalism, Type IIB 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:

${{\gamma }_{11}}{{\Theta }^{A}}={{\Theta }^{A}}$

Which makes it non-chiral.