String

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 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 String

In Bosonic String Theory

Closed Boundary Conditions

Closed Boundary Conditions on a String.

$X\left(\sigma,\tau\right)=X\left(\sigma+\ell_s,\tau\right)$

Newmann Boundary Conditions

Newmann Boundary Conditions on a String.

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 Conditions

Dirchilet Boundary Conditions on a String.

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)

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 Conditions

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 Conditions

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 String

The Bosonic String

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 String

The Mode Expansion (Explicit Equation of Motion) is:

Open String with Newmann Boundary Conditions

The Mode Expansion (Explicit Equation of Motion) is:

Open String with Dirchilet Boundary Conditions

The Mode Expansion (Explicit Equation of Motion) is:

The (RNS) Superstring

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 Conditions

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 Conditions

The mode expansion for these can be shown to be:

${{\psi }^{\mu }}=\frac{1}{\sqrt{2}}\sum\limits_{n\in \mathbb{Z}+\frac12 }^{{}}{d_{n}^{\mu }\left[ \begin{align} & {{e}^{-in{{\sigma }^{-}}}} \\ & {{e}^{-in{{\sigma }^{+}}}} \\ \end{align} \right]}$