String Theory | ||
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All Roads Lead to String Theory (Polchinski) | ||
Prior to the First Superstring Revolution
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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
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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
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Phenomenology | String Theory Landscape Minimal Supersymmetric Standard Model String Phenomenology | |
- This page is about an object in String Theory. For other uses, see String (disambiguation).
A String is a 1-Dimensional weakly-coupled object in String Theory. In some old, canonical String Theoryies, it is the Fundamental "F1" String. In M-Theory, however, the M2-Brane and M5-Branes are fundamental.
Boundary Conditions on the StringEdit this section
In Bosonic String TheoryEdit this section
Closed Boundary ConditionsEdit this section
$ X\left(\sigma,\tau\right)=X\left(\sigma+\ell_s,\tau\right) $
Newmann Boundary ConditionsEdit this section
Newmann Boundary conditions are as follows.
$ {{\left. \left( \frac{\partial X}{\partial \sigma } \right) \right|}_{\sigma =0}}={{\left. \left( \frac{\partial X}{\partial \sigma } \right) \right|}_{\sigma ={{\ell }_{s}}}} $
Newmann Boundary Conditions basically state that the String becomes orthogonal to the walls at it's endpoints. When the walls are drawn vertically, the ends of the String appear to look horizontal.
The String may, however, transverse across these fake walls. It is thus, obviously, not fixed.
(Note: Under T-Duality, Newmann Boundary Conditions become Dirchilet Boundary Conditions, which are discussed in the section below.)
Dirchilet Boundary ConditionsEdit this section
Dirchilet Boundary Conditions are as follows.
$ X\left( 0,\tau \right)={{X}_{0}};X\left( {{\ell }_{s}},\tau \right)={{X}_{{{\ell }_{s}}}} $
Dirchilat Boundary Conditions basically state that the String is fixed onto walls ("D-Branes". D = Dirchilet).
The String may, however, make any angle with it's walls.
(Note: Under T-Duality, Dirchilet Boundary Conditions become Newmann Boundary Conditions, which are discussed in the section above.)
In Superstring Theory (the RNS Formalism)Edit this section
In the RNS Formalism, in addition to the above boundary conditions, there are also boundary conditions on the fermionic fields. The Boundary Conditions for the fermionic field actually divide the entire Formalism of RNS into two different sectors.
Ramond Boundary ConditionsEdit this section
The Ramond Boundary Condition is as follows:
$ \psi_+\left(\ell_s,\tau\right)=\psi_- \left(\ell_s,\tau\right) $
I.e. Periodic.
Neveu-Schwarz Boundary ConditionsEdit this section
The Neveu-Schwarz Boundary Condition is as follows:
$ \psi_+\left(\ell_s,\tau\right)=-\psi_- \left(\ell_s,\tau\right) $
I.e. Anti-Periodic.
Field Equations for the StringEdit this section
The Bosonic StringEdit this section
One may set the variation of the Polyakov Action to 0, obtaining the Field Equation. And then, one may apply the relevant boundary conditions to obtain the explicit form of the Equations of Motion.
The Field Equation is:
$ \frac{\partial^2X}{\partial\sigma^+\partial\sigma^-}=0 $
Closed StringEdit this section
The Mode Expansion (Explicit Equation of Motion) is:
Open String with Newmann Boundary ConditionsEdit this section
The Mode Expansion (Explicit Equation of Motion) is:
Open String with Dirchilet Boundary ConditionsEdit this section
The Mode Expansion (Explicit Equation of Motion) is:
The (RNS) SuperstringEdit this section
In addition to the previously discussed mode expansions and field equations (the ones for the Bosonic String, in the RNS Formalism, there are further more, field equations and mode expansions for the fermionic field $ \psi $. The Field Equation is:
$ \left[ \begin{matrix} {{\partial }_{+}} & {{\partial }_{-}} \\ \end{matrix} \right]\left[ \begin{align} & {{\psi }_{-}} \\ & {{\psi }_{+}} \\ \end{align} \right]=0 $
Ramond Boundary ConditionsEdit this section
The mode expansion for these can be shown to be:
$ {{\psi }^{\mu }}=\frac{1}{\sqrt{2}}\sum\limits_{n\in \mathbb{Z}}^{{}}{d_{n}^{\mu }\left[ \begin{align} & {{e}^{-in{{\sigma }^{-}}}} \\ & {{e}^{-in{{\sigma }^{+}}}} \\ \end{align} \right]} $
Neveu-Schwarz Boundary ConditionsEdit this section
The mode expansion for these can be shown to be:
$ {{\psi }^{\mu }}=\frac{1}{\sqrt{2}}\sum\limits_{n\in \mathbb{Z}+\frac12 }^{{}}{d_{n}^{\mu }\left[ \begin{align} & {{e}^{-in{{\sigma }^{-}}}} \\ & {{e}^{-in{{\sigma }^{+}}}} \\ \end{align} \right]} $