RNS Formalism

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

This article or section 's content is very similar or exactly the same as that at Wikipedia. This is because the contributor of this article had initially contributed it to wikipedia.

The RNS Formalism, (naive form known as:RNS String Theory, also known as the RNS SuperString Theory), was an early attempt to introduce fermions through the means of Supersymmetry, into String Theory, which was then only Bosonic String Theory. "RNS" stands for "Ramond-Neveu-Schwarz". It was introduced as a theory with supersymmetry on the Worldsheet, but was later found to be equivalent to the GS String Theory, which has supersymmetry on the background spacetime. In the RNS Formalism, the fields describing the embedding of the Worldsheet in spacetime is actually a bosonic field, and the fermionic fields are spacetime vectors.

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Action principle

The RNS String Theory is given by the Lagrangian density:' " "

${{\mathsf{\mathcal{L}}}_{RNS}}=\frac{T}{2} h^{\alpha \beta} \left( \partial_\alpha X^\mu \partial_\beta X^\nu +i\hbar c_0 \bar{\psi_\mu} \not\partial \psi^\mu \right) g_{\mu\nu}$

The corresponding action is given by the RNS Action:

${{S}_{RNS}}=\iint {\left({\frac {T}{2}}h^{\alpha \beta }\left(\partial _{\alpha }X^{\mu }\partial _{\beta }X^{\nu }+i\hbar c_{0}{\bar {\psi _{\mu }}}\not \partial \psi ^{\mu }\right)g_{\mu \nu }\right)}{\sqrt {-\det {{h}_{\alpha \beta }}}}{\text{ }}{{\text{d}}^{2}}\xi$

Notice that this is only the Polyakov Action + the Dirac Action. The same action also continues to hold for some GSO truncated string theories, namely the Type IIB String Theory, the Type IIA Theory, and the Type I String Theory.

Sectors

For RNS Open Strings, there are 2 sectors. Namely, the

Ramond sector, with boundary condition:

$\psi _{-}\left(0\right)=\psi _{+}\left(0\right)$.

Neveu-Schwarz sector, with boundary condition:

$\psi _{-}\left(0\right)=-\psi _{+}\left(0\right)$.

For RNS closed strings, there are 4 sectors.

First of all, a periodic boundary condition in $A$ means that:

$A\left(\sigma +\ell _{s},\tau \right)=A\left(\sigma ,\tau \right)$.

Whereas an antiperiodic boundary condition in $B$ means that:

$B\left(\sigma +\ell _{s},\tau \right)=-B\left(\sigma ,\tau \right)$.

The Ramond Ramond sector is periodic on $\psi^\mu$.

The Neveu-Schwarz Neveu-Schwarz sector is antiperiodic in $\psi^\mu$.

The Ramond Neveu-Schwarz sector is periodic in $\psi _{-}^{\mu }$ and antiperiodic in $\psi _{+}^{\mu }$.

The Neveu-Schwarz Ramond sector is antiperiodic in $\psi _{-}^{\mu }$ and periodic in $\psi _{+}^{\mu }$.

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${\begin{aligned}&\left[{{\hat {L}}_{m}},{{\hat {L}}_{n}}\right]=-{\frac {i}{\hbar }}\left(\left(m-n\right){{\hat {L}}_{m+n}}+{\frac {D}{8}}{{n}^{3}}{{\delta }_{m+n,0}}\right)\\&\left[{{\hat {G}}_{r}},{{\hat {G}}_{s}}\right]=-{\frac {i}{\hbar }}\left(2{{\hat {L}}_{r+s}}+{\frac {D}{2}}\left(2r-1\right)\left(2r+1\right){{\delta }_{r+s,0}}\right)\\&\left[{{\hat {\tilde {L}}}_{m}},{{\hat {\tilde {L}}}_{n}}\right]=-{\frac {i}{\hbar }}\left(\left(m-n\right){{\hat {\tilde {L}}}_{m+n}}+{\frac {D}{8}}{{n}^{3}}{{\delta }_{m+n,0}}\right)\\&\left[{{\hat {\tilde {G}}}_{r}},{{\hat {\tilde {G}}}_{s}}\right]=-{\frac {i}{\hbar }}\left(2{{\hat {\tilde {L}}}_{r+s}}+{\frac {D}{2}}{{r}^{2}}{{\delta }_{r+s,0}}\right)\\\end{aligned}}$

While in the Neveu-Schwarz sector,:

${\begin{aligned}&\left[{{\hat {\tilde {G}}}_{r}},{{\hat {\tilde {G}}}_{s}}\right]=2{{\hat {\tilde {L}}}_{r+s}}+D{{r}^{2}}{{\delta }_{r+s,0}}+2{{\hat {\tilde {\Gamma }}}_{rs}}\\&\left[{{\hat {L}}_{m}},{{\hat {L}}_{n}}\right]=\left(m-n\right){{\hat {L}}_{m+n}}+{\frac {D}{8}}{{n}^{3}}{{\delta }_{m+n,0}}\\&\left[{{\hat {G}}_{r}},{{\hat {G}}_{s}}\right]=2{{\hat {L}}_{r+s}}+{\frac {D}{2}}{{r}^{2}}{{\delta }_{r+s,0}}\\&\left[{{\hat {\tilde {L}}}_{m}},{{\hat {\tilde {L}}}_{n}}\right]=\left(m-n\right){{\hat {\tilde {L}}}_{m+n}}+{\frac {D}{8}}{{n}^{3}}{{\delta }_{m+n,0}}\\&\left[{{\hat {\tilde {G}}}_{r}},{{\hat {\tilde {G}}}_{s}}\right]=2{{\hat {\tilde {L}}}_{r+s}}+{\frac {D}{2}}{{r}^{2}}{{\delta }_{r+s,0}}\\\end{aligned}}$

This is clearly a central extension to the Super-Witt algebra. They are expressible in terms of the modes of the oscillations of the string as follows:

${\begin{aligned}&{{\hat {L}}_{n}}={\frac {1}{2}}\sum \limits _{m=-\infty }^{}{{{\hat {\alpha }}_{n-m}}\cdot {{\hat {\alpha }}_{m}}}+{\frac {1}{4}}\sum \limits _{r}^{}{\left(2r-n\right){{\hat {d}}_{n-m}}\cdot {{\hat {d}}_{m}}}\\&{{\hat {G}}_{r}}=\sum \limits _{m=-\infty }^{}{{{\hat {\alpha }}_{m}}\cdot {{\hat {d}}_{r-m}}}\\&{{\hat {\tilde {L}}}_{n}}={\frac {1}{2}}\sum \limits _{m=-\infty }^{}{{{\hat {\tilde {\alpha }}}_{n-m}}\cdot {{\hat {\tilde {\alpha }}}_{m}}}+{\frac {1}{4}}\sum \limits _{r}^{}{\left(2r-n\right){{\hat {\tilde {d}}}_{n-m}}\cdot {{\hat {\tilde {d}}}_{m}}}\\&{{\hat {\tilde {G}}}_{r}}=\sum \limits _{m=-\infty }^{}{{{\hat {\tilde {\alpha }}}_{m}}\cdot {{\hat {\tilde {d}}}_{r-m}}}\\\end{aligned}}$

Imposing the Super-Virasoro constraints

Imposing Super-Virasoro constraints to get rid of the Pauli-Villar ghost states, we see that the normal ordering constant must be $0$ in the Ramond sector and $\frac{1}{2}$ in the Neveu-Schwarz sector. Also, the critical dimension must be $10$ . Note, that unlike the Bosonic String Theory, the central charge is no longer equal to the critical dimension, but instead, $\frac{3}{2}$ of it, i.e., in this case, it is $15$ .

Unsuitability as a Theory of Everything

Clearly, the mass spectrum, being given by:

$m^{2}=N-a$

In the open string sector, has a tachyon at $N =0$ in the Neveu-Schwarz sector (since there, $a = \frac12$ ), . The same logic applies to the NS-NS, R-NS, NS-R, etc. sector of the closed strings, etc.

However, tachyons are unstable due to the Sen Conjecture, also known as Tachyon condensation. The reason being, that that would not allow stable ground states to exist. .

Thus, the naive RNS String Theory cannot be a Theory of Everything, which resulted in the need for the GSO Projection.

Note, however, that this only the naive RNS String Theory. The RNS Formalism, though, can still be used. I.e., the same formalism can be used, but with a GSO Projection, which results in different string theories.

References

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