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RNS Formalism

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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.

([1][2][3][4])

Action principleEdit this section

The RNS String Theory is given by the Lagrangian density:' " " [1]

{{\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.

SectorsEdit this section

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.

([1][2][3][4][5]

\begin{align}
  & \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{align}

While in the Neveu-Schwarz sector,:[2][3][4]

\begin{align}
  & \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{align}

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:[2][3][4]

\begin{align}
  & {{{\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{align}

Imposing the Super-Virasoro constraintsEdit this section

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 \frac12 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, \frac32 of it, i.e., in this case, it is  15 .

Unsuitability as a Theory of EverythingEdit this section

Clearly, the mass spectrum, being given by:[4]

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,[4] 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.


ReferencesEdit this section


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