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Type IIB String Theory

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String Theory
Stringinteractions
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.[1]

Action PrincipleEdit this section

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:[1]

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 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 same way.[2] This is what makes the theory non-chiral.[3]

Cancellation of Conformal AnomalyEdit this section

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 EverythingEdit this section

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.[1]

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.[1]

T-duality with the Type IIA 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 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.[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-duality with the "Sen string theory"Edit this section

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.[5] 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 FormalismEdit this section

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.


ReferencesEdit this section


ReferencesEdit this section

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